Industrial Organization Markets and Strategies 2nd Edition Belleflamme Solutions Manual

Industrial Organization Markets and Strategies 2nd Edition Belleflamme Solutions Manual

richard@qwconsultancy.com

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Industrial Organization: Markets and Strategies Paul Belle‡amme and Martin Peitz published by Cambridge University Press

Part I. Getting started Exercises Exercise 1 Free trade and competitive markets [included in 2nd edition of the book] Consider the market for shoes in country A. Demand is assumed to be 100 p where p is the …nal consumer price. Suppose that country A does not produce shoes and that there are two importers B and C. The export prices for shoes in both countries are pB = 59:99 and pC , respectively. Furthermore suppose that country A is a small country so that its demand does not in‡uence export prices. Suppose that, initially, country A levies a uniform import tari¤ of t = 10 on each pair of imported shoes.

1. Assume pC = 45. What is the e¤ect on demand and welfare in country A if country A signs a free trade agreement with country B? 2. Assume pC = 50. What is now the e¤ect on demand and welfare in country A if country A signs a free trade agreement with country B? Solutions to Exercise 1 In the absence of a free-trade agreement with country B, the consumers in country A always buy shoes from country C as pC + t < 59:99 + t whether pC = 45 or pC = 50. If there is a free-trade agreement with country B, then consumers in country A compare pC + t with pB = 59:99. As t = 10, we have that consumers buy shoes from country C if pC = 45 but from country B if pC = 50.

Comparing the two situations, we see that the free-trade agreement with country B has no e¤ect if pC = 45 (i.e., if country C is very inexpensive with respect to country B), as consumers in country A continue to buy shoes from country C. However, if pC = 50, the free-trade agreement makes consumers buy shoes from country B rather than from country C; this allows them to pay a lower price, which increases their surplus.

Exercise 2 Monopoly problem [included in 2nd edition of the book] Consider a monopolist with a linear demand curve: q = a bp, where a; b > 0. It produces at constant marginal cost c and has no …xed cost. Assume that 0 < c < a=b.

1. Find the monopoly price, quantity, and pro…ts.

2. Derive the inverse demand curve P (q). Draw P (q), the MR-curve, and the MC-curve in a diagram. Explain why we need the assumption c < a=b.

1

3. Does it matter that the monopolist sets price instead of quantity?

4. Calculate the deadweight loss of monopoly.

5. A change in b results in two opposing e¤ects on the deadweight loss. Calculate the e¤ect of a change in b on the deadweight loss.

6. Derive the price elasticity of demand with p?

for any price. How does

change

7. Show mathematically as well as graphically that the price elasticity of demand > 1 at the monopoly price.

Solutions to Exercise 2 1. The monopoly chooses p to maximize = (p c)(a bp). The …rst-order condition yields a bp bp+bc = 0, which is equivalent to pm = (a + bc) = (2b).

We compute then q m = a bpm = (a bc) =2 and pm c = (a bc) = (2b).

2 It follows that m = (a bc) = (4b).

2. The inverse demand curve P (q) is obtained by inverting q = a bp: bp = a q , p = a=b q=b. The intercept on the vertical axis (where price is measured) is a=b, and the intercept on the horizontal axis (where quantity is measured) is a. The MC curve is a horizontal line that cuts the vertical axis at c; if c were larger than a=b, inverse demand would be everywhere below the MC and no production would be pro…table. The MR curve is M R = a=b 2q=b, which has the same vertical intercept as the inverse demand curve but cuts the horizontal axis at a=2 instead of a.

3. No, because the monopoly controls the demand function, i.e., the relationship between p and q . To be sure, solve the monopoly problem in terms of quantity.

That is, let the monopoly choose q to maximize = (a=b q=b) q cq .

4. The …rst-best is achieved at marginal cost pricing: p = c; the corresponding quantity is q = a bc. Welfare is then equal to W = (1=2) (a=b c) (a bc) = 2 (a bc) = (2b). Under monopoly, the consumer surplus is equal to CS m = 2 (a bc) = (8b); adding the monopoly’s pro…t, we compute welfare under monopoly 2 as W m = 3 (a bc) = (8b). Hence, the deadweight loss of monopoly is W 2 W m = (a bc) = (8b).

5. The price elasticity of demand is de…ned as (p) =

q 0 (p)

p bp = : q (p) a bp

We compute 0

(p) =

ab (a

2 > 0;

bp)

meaning that the price elasticity of demand increases with p.

2

6. We have that (pm ) =

a + bc bpm = > 1. m a bp a bc

Exercise 3 Two-period monopoly problem [included in 2nd edition of the book] Consider a monopolist that produces for two periods. The demand curves in both periods are q t = 1 pt for t = 1; 2. The marginal costs are c in the …rst and and c q 1 in the second period. Here, is a small and positive number.

There is a discount factor of between the periods.

1. Explain brie‡y how the monopolist’s problem changes compared to a situation where the marginal cost is c in both periods.

2. Find the quantities q 1 and q 2 that the monopolist chooses in the two periods. Hint: Start by solving the monopolist’s problem in the second period and then continue to the …rst period.

3. Derive the restriction on concave.

that ensures that the pro…t function is strictly

Solutions to Exercise 3 1. If the marginal cost is c in both periods, then the two periods are unrelated (as the monopolist’s decisions in the …rst period do not a¤ect the demand nor the cost in the second period). In contrast, when the second-period marginal cost is c q 1 , the monopolist lowers his cost in the second period by producing more in the …rst period (this can result from some learning economies, for instance).

2. In the second period, the monopolist chooses q 2 to maximize 2 = 1 q 2 q 2 c q 1 q 2 . The optimum is easily found as q 2 = (1=2) 1 c + q 1 , result2

ing in a pro…t of

2

c + q 1 . In the …rst period, the monopolist

= (1=4) 1

chooses q 1 to maximize 1 + 2 = 1 The …rst-order condition yields

(1 + 2 (4 + ) ) (1

c)

q 1 q 1 cq 1 +( + 4) 1 2 1

(4 + )

2

2

c + q1 .

q 1 = 0;

and the second-order condition yields 2 1 (4 + ) 2 < 0, which is satisp …ed as long as (4 + ) 2 < 1 or < 4 + (this is the answer to question 3).

Then, the optimum is found as

q1 =

1 + 2 (4 + ) 1 c : 1 (4 + ) 2 2

It follows that

q2 = We check that for

q 1 > q 2 > (1

2 1

2+ (4 + )

= 0, q 1 = q 2 = (1

c) =2.

3

1 2

c 2

:

c) =2, while for 0 <

<

p

4+ ,

Exercise 4 Market structure for rating agencies1 In the recent …nancial crisis, rating agencies have become a focus of attention.

The market has traditionally been dominated by a few big agencies, currently Standard & Poor, Moody’s and Fitch. In 2006, the Securities and Exchange Commission (SEC) introduced measures to speed up the appoval process for rating agencies with the aim to increase competition. However, in response to the …nancial crisis, the Fed introduced lending programmes for which— this is what it said initially— it accepts only collateral that has been appraised by one of the big three. Discuss the likely consequences of such a decision on market structure.

1 see "the wages of sin", in: The Economist, April 25, 2009, page 76.

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Industrial Organization: Markets and Strategies Paul Belle‡amme and Martin Peitz published by Cambridge University Press

Part II. Market power Exercises & Solutions Exercise 1 Monopoly with quality choice Consider a monopolist who sells batteries. Each battery works for h hours and then needs to be replaced. Therefore, if a consumer buys q batteries, he gets H = qh hours of operation. Assume that the demand for batteries can be derived from the preferences of a representative consumer whose indirect utility function is v = u(H) pq, where p is the price of a battery. Suppose that u is strictly increasing and strictly concave. The cost of producing batteries is C(q) = qc(h), where c is strictly increasing and strictly convex.

1. Derive the inverse demand function for batteries and denote it by P (q).

2. Suppose that the monopolist chooses q and h to maximize his pro…t. Write down the …rst-order conditions for pro…t maximization assuming that the problem has an interior solution, and explain the meaning of these conditions.

3. Write down the total surplus in the market for batteries (i.e., the sum of consumer surplus and pro…ts) as a function of H and h. Derive the …rst-order conditions for the socially optimal q and h assuming that there is an interior solution. Explain in words the economic meaning of these conditions.

4. Compare the solution that the monopolists arrives at with the social optimum. Prove that the monopolist provides the socially optimal level of h. Give an intuition for this result.

Solutions to Exercise 1 1. The inverse demand for batteries is obtained by solving the following problem:

max u(qh) q

pq:

The …rst-order conditions for this problem can be written as

P (q) = hu0 (qh) which is the inverse demand function for batteries.

2. The monopolist’s maximization problem is given by

max qP (q) q;h

1

qc(h);

where P (q) is given in part 1. The …rst-order conditions for an interior solution are:

h(u0 (H) + Hu00 (H)) = c(h) and q(u0 (H) + Hu00 (H)) = qc0 (h) where h(u0 (H) + Hu00 (H)) is the marginal revenue from selling an extra battery.

First, let us interpret the expressions inside the brackets: u0(H) is the revenue from selling an extra hour of operation, while u00 (H) is the discount in the price per hour of operation that the monopolist must give in order to induce consumers to buy an extra unit. This discount has to be multiplied by H since the discount is given to all inframarginal units. Hence, u0 (H) + Hu00 (H) is the marginal revenue from selling an extra hour of operation. Since the monopolist sells hours of operation in packets of h units each (a battery provides h hours of operation), u0 (H) + Hu00 (H) has to be multiplied by h to give the marginal revenue from selling an extra battery. c(h) is the marginal cost of batteries. Therefore the …rst equation is the familiar monopoly pricing condition that says that at the optimum, the monopolist produces up to the point where his marginal revenue equals his marginal cost. Similarly, the second equation indicates that at the optimum, the marginal revenue from extending the life of each battery by one hour must equal to marginal cost (note that the marginal cost has to be multiplied by the number of batteries that the monopolist produces since the cost of each one of them increases by c0 (h)).

3. The total surplus in the market for batteries is given by u(H) qc(h). The …rst-order conditions for the social optimum (assuming that an interior solution to this problem exists) are

hu0 (H) = c(h) and qu0 (H) = qc0 (h) where hu0 (H) is the marginal utility of consumers from having an extra battery. It equals the consumers’ willingness-to-pay for an extra hour of operation multiplied by h which is the number of hours of operation they get when they buy an extra battery. The right-hand side of the …rst equation is the marginal cost of producing an extra battery; the …rst equation states that, at the social optimum, the marginal utility of consumers has to be equal to the marginal cost of production. The second equation says that, at the social optimum, the marginal utility of consumers from having batteries which provide one more hour each, has to be equal to the marginal cost of extending the life of each battery by one hour.

4. To compare the solutions in parts 2 and 3, we divide the …rst-order conditions for the monopoly problem by one another:

c(h) c(h) h = 0 , c0 (h) = : q qc (h) h This equation shows that the monopolist produces h by equating the marginal cost of h with the average cost of h. This implies in turn that h is produced at the minimum average cost.

By dividing the …rst-order conditions for social optimum by one another we get the same condition which implies that the monopolist chooses the socially optimal level

2

of h. Moreover, since the condition that determines h is independent of q , it follows that this result is true even though the monopolist provides too little quantity.

Exercise 2 Price competition [included in the 2nd edition of the book] Consider a duopoly in which homogeneous consumers of mass 1 have unit demand. Their valuation for good i = 1; 2 is v(fig) = vi with v1 > v2 . Marginal cost of production is assumed to be zero. Suppose that …rms compete in prices.

1. Suppose that consumers make a discrete choice between the two products.

Characterize the Nash equilibrium.

2. Suppose that consumers can now also decide to buy both products. If they do so they are assumed to have a valuation v(f1; 2g) = v12 with v1 + v2 > v12 > v1 . Firms still compete in prices (each …rm sets the price for its product— there is no additional price for the bundle) Characterize the Nash equilibrium.

3. Compare regimes from parts (1) and (2) with respect to consumer surplus.

Comment on your results.

Solutions to Exercise 2 1. Nash equilibrium given by p1 = v1 v2 , p2 = 0, 1 = v1 v2 and 2 = 0.

2. Nash equilibrium given by p1 = v12 v2 , p2 = v12 v1 , 1 = v12 v2 and v1 .

2 = v12 3. Consumer surplus in 1) is CSa = v2 , whereas it is given by CSb = v1 +v2 v12 in 2). As v12 > v1 by assumption, consumer welfare is strictly greater in 1) than in 2). In 2) the nature of competition changes because consumers have positive valuation to buy both products; this relaxes competition.

Exercise 3 Cournot competition Two …rms (…rm 1 and …rm 2) compete in a market for a homogenous good by setting quantities. The demand is given by Q(p) = 2 p. The …rms have constant marginal cost c = 1.

1. Draw the two …rms’ reaction function. Find the equilibrium quantities and calculate equilibrium pro…ts.

2. Suppose now that there are n …rms where n quantities and pro…ts.

2. Calculate equilibrium

Solutions to Exercise 3 This is the standard Cournot model, just in case students have forgotten about their intermediate micro class.

Exercise 4 Equilibrium uniqueness in the Cournot model

3

Consider an oligopoly with n …rms that produce homogeneous goods and compete à la Cournot. Inverse demand is given by P (Q) with P 0 (Q) < 0, and 0 00 each …rm P i has a cost function of Ci (qi ) with Ci (qi ) > 0 and Ci (qi ) 0. Denote q i = j6=i qj .

1. Compute the …rst- and second order condition of …rm i. Under which conditions is the pro…t function of …rm i, i , strictly concave? dqi . In which 2. Compute the slope of the best-reply function of …rm i, dq i interval is this slope?

A su¢ cient condition for uniqueness of a Cournot equilibrium is (see, e.g., Tirole (1999), page 226) @2 i + (n @qi2

1)

@2 i < 0; @qi @q i

3. Suppose that demand is concave and that marginal costs are constant.

For which number of n is the condition above satis…ed?

Pn 4. Suppose that P (Q) = a b i=1 qi and Ci (qi ) = cqi , for all i 2 f1; :::; ng.

Is there a unique equilibrium for any n?

Solutions to Exercise 4 see the 1999-book by Vives

Exercise 5 Industries with price or quantity competition Which model, the Cournot or the Bertrand model, would you think provides a better …rst approximation to each of the following industries/markets: the oil re…ning industry, farmer markets, cleaning services. Discuss!

Exercise 6 An investment game Consider a duopoly market with a continuum of homogeneous consumers of mass 1. Consumers derive utility vi 2 fv H ; v L g for product i depending on whether the product is of high or low quality. Firms play the following 2stage game: At stage 1, …rms simultaneously invest in quality: The more a …rm invests the higher is its probability i of obtaining a high-quality product. The associated investment cost is denoted by I( i ) and satis…es standard properties that ensure an interior solution: I( i ) is continuous for i 2 [0; 1), I 0 ( i ) > 0 and I 00 ( i ) > 0 for i 2 (0; 1), and lim #0 I 0 ( i ) = 0; lim "1 I 0 ( i ) = 1. Before the beginning of stage 2 qualities become publicly observable— i.e., all uncertainty is resolved. At stage 2, …rms simultaneously set prices.

1. For any given ( 1 ; 2 ), what are the expected equilibrium pro…ts? In case of multiple equilibria select the (from the view point of the …rms) Pareto-dominant equilibrium.

4

2. Are investments strategic complements or substitutes? Explain your …nding.

3. Provide the equilibrium condition at the investment stage.

4. How do equilibrium investments change as v H

vL

is increased?

Solutions to Exercise 6 1. Bertrand competition: If v1 = v2 , 1 = 2 = 0. If vi > vj , j = 0, i 6= j . The expected equilibrium pro…t is E i = i (1 2. At stage 1, each …rm solves max i

i (1

j)

j)

= I 0( i)

i

=

j)

.

and

I( i ). First-order condition

of pro…t maximization is

(1

Since I( i ) is strictly convex I 0 ( i ) is monotone and thus invertible.

0 1 [(1 i = (I )

j)

]

Since I 0 ( i ) is increasing the best response of …rm i is decreasing in investment decisions are strategic substitutes.

3. The equilibrium investment decisions (1

)

1 =

are characterized by

2

= I 0(

j and

):

There is a unique solution to this equation.

4. Rewriting the above equation as =

I 0( 1

)

we see that the numerator on the right-hand side is increasing in while the denominator on that side is decreasing in . Thus, an increase in implies a larger . Due to the nature of Bertrand competition only the quality di¤erence but not the absolute levels of qualities a¤ect investment incentives.

Exercise 7 Hotelling model Reconsider the simple Hotelling model in which consumers are uniformly distributed on the unit interval and …rms are located at the extremes of this interval. Now take consumers’participation constraint explicitly into account.

Derive the equilibrium depending on the parameter . [Be careful to distinguish between di¤erent regimes with respect to competition between …rms!]

Exercise 8 Price and quantity competition 5

Reconsider the duopoly model with linear individual demand and di¤erentiated products. Show that pro…ts under quantity competition are higher than under price competition if products are substitutes and that the reverse holds if products are complements.

Exercise 9 Asymmetric duopoly [included in the 2nd edition of the book] Consider two quantity-setting …rms that produce a homogenous good and choose their quantities simultaneously. The inverse demand function for the good is given by P = a q1 q2 , where q1 and q2 are the outputs of …rms 1 and 2 respectively. The cost functions of the two …rms are C1 (q1 ) = c1 q1 and C2 (q2 ) = c2 q2 , where c1 < a and c2 < (a + c1 )=2.

1. Compute the Nash equilibrium of the game. What are the market shares of the two …rms?

2. Given your answer to (1), compute the equilibrium pro…ts, consumer surplus, and social welfare.

3. Prove that if c2 decreases slightly, then social welfare increases if the market share of …rm 2 exceeds 1=6, but decreases if the market share of …rm 2 is less than 1=6. Give an economic interpretation of this …nding.

Solutions to Exercise 9 1. The Nash equilibrium of the game is obtained by solving the following system of equations:

@ 1 @q1 @ 1 @q2

=

(a

q1

q2 )

q1

c1 = 0;

=

(a

q1

q2 )

q2

c2 = 0:

The equilibrium is given by the pair

q1 =

a

2c1 + c2 a ; q2 = 3

2c2 + c1 : 3

Given q1 and q2 , the equilibrium market share are: 1

=

a2

=

q1 a 2c1 + c2 = ; q1 + q2 2a c1 c2 a 2c2 + c1 q2 = : q1 + q2 2a c1 c2

6

2. The equilibrium pro…ts are given by 1

= =

2

=

(a (a (a

q1 q2 )q1 c1 q1 = (q1 )2 2c1 + c2 )2 ; 9 2c2 + c1 )2 : 9

and the consumer surplus is given by

CS

=

Z q1 +q2

(a

q)dq

(a

q1

(2a (q1 + q2 )2 = 2

c1 18

c2 )2

q2 )(q1 + q2 )

0

=

:

Adding the three together, social welfare is given by

W

= CS + 1 + 2 (q1 + q2 )2 = + (q1 )2 + (q2 )2 : 2

3. Di¤erentiating W with respect to c2 yields @W @c2

= =

@(q1 + q2 ) @q1 @q2 + 2q1 + 2q2 @c2 @c2 @c2 1 q1 + q2 6(q2 )2 = 2(q1 + q2 )( 2 ): 3 6

(q1 + q2 )

Hence, a small reduction in c2 increases W 2 < 1=6.

if

2 > 1=6 and deceases W

is

Not that the result that a cost reduction may be socially undesirable was …rst demonstrated in Lahiri and Ono (1988), "Helping Minor Firms Reduces Welfare," Economic Journal.

Exercise 10 Selling independent products to budget-constrained consumers1 Consider two sellers 1 and 2 and a continuum of buyers. Seller i o¤ers product i at price pi and incurs zero marginal cost of production. Buyers are identical and derive utility ui from one unit of each product. Thus their utility is ui if they buy one unit of product i and zero units of product j, j 6= i; it is u1 + u2 if they buy one unit of each product. Additional units do not give any extra utility. Each buyer has a budget y, which she cannot exceed. Each seller is assumed to prefer not to sell a unit rather than setting a zero price. Suppose that u1 > u2 .

1 This exercise is inspired by Jeon, D.-S. and D. Menicucci (2006). Bundling Electronic Journals and Competition among Publishers. Journal of the European Economic Association 4: 1038-1083.

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1. Derive the demand function of each buyer.

2. Consider the game in which sellers simultaneously set price. Characterize the Nash equilibrium of the game.

3. Determine consumer surplus and total surplus that realize in equilibrium.

Is the equilibrium necessarily e¢ cient or are there ine¢ ciencies? Explain.

Solutions to Exercise 10 1. For p1 u1 , p2 For p1 u 1 , p2 if u1 p1 > u 2

u2 and p1 + p2 y , Q1 (p1 ; p2 ) = 1 and Q2 (p1 ; p2 ) = 1. u2 and p1 + p2 > y , Q1 (p1 ; p2 ) = 1 and Q2 (p1 ; p2 ) = 0 p2 and Q1 (p1 ; p2 ) = 0 and Q2 (p1 ; p2 ) = 1 if the reverse inequality holds. For pi maxfui ; yg and pj > uj , Qi (pi ; pj ) = 1 and Qj (pi ; pj ) = 0. For all other prices, Q1 (p1 ; p2 ) = 0 and Q2 (p1 ; p2 ) = 0.

2. For u1

u2 y , p1 = y and seller 2 does not sell. Thus, the buyer only buys product 1. Consider next the case u1 + u2 > y > u1 u2 . Note that when both products are bought under this inequality, we must have in equilibrium that p1 + p2 = y . Suppose that prices satisfy that u1 p1 = u2 p2 . Then, buyers are indi¤erent between product 1 and product 2 if they were to make a discrete choice. In this case, if …rm i slightly increases the price, the buyer can no longer a¤ord to buy both products. However, the buyer than buys product j and, therefore, …rm i does not have an incentive to deviate. Whenever u1 p1 6= u2 p2 and p1 + p2 = y , the …rm which o¤ers a larger net surplus has an incentive to increase its price. Hence, in equilibrium, u1 p1 = u2 p2 .

Solving u1 p1 = u2 p2 and p1 + p2 = y gives pi = (y + ui uj )=2. Clearly, for u1 + u2 y , each seller charges p1 = u1 and p2 = u2 .

3. Consumers make strictly positive surplus if u1 + u2 > y > u1

u2 . Here, …rms compete for the limited budget of the buyer. For a smaller as well as for a larger budget, consumer surplus is zero. Thus, consumer surplus in non-monotone in y . For u1 u2 y , there is a total surplus loss compared to the social optimum because in the social optimum each buyer obtains one unit of each product.

Exercise 11 Di¤ erentiated duopoly with uncertain demand 1. Consider a monopolist facing an uncertain inverse demand curve p=a

bq + :

When setting its price or quantity the monopolist does not know but knows that E[ ] = 0 and E[ 2 ] = 2 . The cost function of the monopolist is given by c2 q 2 C(q) = c1 q + ; 2 with a > c1 > 0 and c2 > 2b.

Show that the monopolist prefers to set a quantity if the marginal cost curve is increasing and a price if the marginal cost curve is decreasing.

Provide a short intuition for the result.

8

2. Now consider a di¤erentiated duopoly facing the uncertain inverse demand system p1 = a bq1 dq2 + and p2 = a

bq2

dq1 + ;

with 0 < d < b, E[ ] = 0 and E[ 2 ] = 2 . Again, the cost functions are 2 similar for both …rms and are given by C(q) = c1 q + c22q , with a > c1 > 0 2

2

and c2 > 2(b b d ) .

Both …rms play a one-shot game in which they choose the strategy variable and the value of this variable simultaneously.

Argue by the same line of reasoning as in (1) that (a) if c2 > 0 in the unique Nash equilibrium both …rms choose quantities (b) if c2 < 0 in the unique Nash equilibrium both …rms choose prices (c) if c2 = 0 there exist four Nash equilibria in pure strategies.

Solutions to Exercise 112 Price vs. quantity setting quantity setting:

E( )

= pq

C(q) = (a

@E( ) @q

= a

2bq

! E(q )

=

E(

=

q)

c1

bq)q

c1 q

c2 q 2 2

c2 q = 0

a c1 2b + c2 (a c1 )2 2(2b + c2 )

p =

ab + ac2 + bc1 + 2b + c2

price setting:

E( ) @E( ) @p ! E(p ) E(

p)

= pq =

C(q) = p

a b

c1

a b

p b

q =

(a c1 ) + 2b + c2 b

0 b(a c1 ) 2b + c2 (a c21 ) c2 2 2(2b + c2 ) 2b2

= a =

p b

2 This exercise is based on Klemperer&Meyer (1986).

9

c2 2

a b

p b

2

! compare pro…ts of the two settings (Note that E( 2 ) = p >

2

> 0):

c2 2 > 0 if c2 < 0 2b2

q ,

! …rm chooses price competition if MC are decreasing p <

c2 2 < 0 if c2 > 0 2b2

q ,

! …rm chooses quantity competition if MC are increasing Calculate best responses given the strategic choice of the other …rm (by symmetry we only have to look at …rm 1): Suppose that …rm 2 has decided to set quantity q2

! optimal quantity reaction by …rm 1: 1

=

(a

@ 1 @q1

=

0

E(q1 )

=

best response payo¤: E(

a

bq1

dq2 + )q1

c1 q1

q2 c2 1 2

c1 dq2 2b + c2

1) =

(a

c1 dq2 )2 2(2b + c2 )

! optimal price reaction by …rm 1: p1 (a

1

=

@ 1 @p1

=

E(p1 )

= a

p1

dq2 + )

c1

b

a

p1

dq2 + b

c2 2

a

p1

dq2 + b

0 dq2

b(a

best response payo¤: E(

dq2 c1 ) 2b + c2

1) =

(a

dq2 c1 )2 2(2b + c2 )

c2 2 2b2

! optimal reaction to quantity setting by the competitor is to also set quantities if c2 > 0, i.e. if MC are increasing! suppose …rm 2 has decided to set price p2

10

2

! optimal quantity reaction by …rm 1: plug in q2

a

=

! p1

= a

p2

dq1 +

b d)

(b

into p1 = a

(b2

d2 )

b

b

bq1

dq2 +

d (b d) q 1 + p2 + b b

rename:

a

(b

d) b

(b2

;

d2 ) b

1

=

(

@ 1 @q1

=

0

E(q1 )

=

E(

=

1)

d ; b

;

q1 + p2 + b)q1

b

c1 q1

(b

d) b

c2 q12 2

c1 + p2 2b + c2 (a c1 + p2 )2 2(2 + c2 )

! optimal price reaction by …rm 1: q1

=

@ 1 @q1

=

E(p1 )

=

E(

1)

=

0

p1 + p 2 + b

+ p2

( + p2 c1 ) 2 + c2

( + p2 c1 )2 2(2 + c2 )

2

c2b 2b2

! optimal reaction to price setting by …rm 2 is to set quantities if c2 > 0, i.e., if MC are increasing Summary: if c2 > 0, then it is a dominant strategy for both …rms to set quantities ! unique Cournot equilibrium if c2 < 0, it is a dominant strategy for both …rms to set prices ! unique Bertrand equilibrium if c2 = 0, …rms are indi¤erent between choosing prices and quantities ! there are four NE: {price, quantity}, {quantity, price}, {quantity, quantity}, {price, price}

11

Exercise 12 Cournot duopoly and cost information Consider a duopoly market for a homogeneous product in which …rms set quantity. Inverse demand is P (q) = 1 q with q = q1 + q2 . Firm 1 has marginal costs equal to 0.7. Firm 2 has marginal cost 0.65 with probability 1=2 and 1 with probability 1=2.

1. Suppose that the cost type is publicly observed by both …rms prior to the quantity setting. Characterize the equilibrium outcome of this game.

2. Suppose from now on that …rm 2 privately observes its cost type before setting its quantity. Determine the equilibrium of this game. What is the appropriate equilibrium concept? In particular, determine equilibrium quantities and pro…ts.

3. Would …rm 2 have an incentive to reveal its cost type to …rm 1 if it could do so at zero cost?

4. Would …rm 1 have an incentive to …nd out about …rm 2’s costs? Would it like to do so privately (assuming that …rm 2 does not know the cost of …rm 1 has when investigating) or publicly?

5. Are consumers better o¤ if …rm 2’s cost type remains private information?

Discuss.

Solutions to Exercise 12 1. …rms share information: high costs: …rm 1:

q1 q2 0:7)q1 1 = (1 @ 1 = 0:3 q2 2q1 = 0 @q1 0:3 q2 q1 = 2

…rm 2: 2 = (1

q2

q1

1)q2 = ( q2

q1)q2

H 2 =0

! will not produce ! !

0:3 = 0:15 q1 = 2 0:15 0:7)0:15 = (0:15)2 = 0:0225 1 = (1 12

low costs: …rm 2: 2

=

(0:35

q1

q2 )q2

@ 2 @q2

=

0:35

q1

2q2 = 0

q2

=

0:35 q1 2

…rm 1:

q1

=

q1

=

q2

=

1

=

2

= )

0:3 0:35 q1 + 2 4 4 1 4 1 = 0:0083 3 16 12 2 0:13 15 1 2 1 1 (0:3 ) = 0:0069 12 15 12 144 2 2 4 1 ) = 0:018 (0:35 12 15 15 225 the lower cost …rm makes higher pro…ts

2. Firm 2 privately observes its cost type: ! Solve for a Bayesian NE since its a static game with imperfect info …rm 2: 2

! max q

2

q2L q2H

=

(1

q2 c2 )q2 1 c2 q1 , q2 (c2 ) = 2 0:35 q1 = 2 = 0

13

q1

…rm 1: 1

=

1 (0:3 2

q1

1 q2L )q1 + (0:3 2

q1

q2H )q1 = (0:3

q1 )q1

1 1 ( q2L + q2L )q1 2 | {z 2 } E(q2 )

0:3

E(q2 ) q1 = 2 0:35 q1 E(q2 ) = 4 0:3 0:35 q1 17 q1 = = 2 8 140 4 L 0:114 q2 = 35 q2H = 0

0:121

17 4 NE strategies: fq1 = 140 ; q2L = 35 ; q2H = 0g

pro…ts:

E( 1 )

=

1 (0:3 2

L 2

=

(0:35

H 2

=

0

17 140 17 140

4 17 1 17 17 289 ) + (0:3 ) = 35 140 2 140 140 19600 4 4 4 16 ) = = 0:0131 35 35 35 1225

0:015

3. yes, because if he is of high cost, he makes zero pro…ts in both cases, but if he has lower costs, then his pro…t is higher if costs are revealed!

4.

If …rm 1 …nds out costs publicly:

E( 1 ) =

1 1 53 1 (0:15)2 + = 2 2 144 3600

0:01472

If costs are not revealed:

E( 1 ) =

289 19600

0:01474

If …rm 1 …nds out the costs privately: 4 It knows that …rm 2 thinks …rm 1 does not know and will therefore play q2L = 35 H and q2 = 0. Hence, …rm 1 will take this into account when maximizing pro…ts. high costs: 1 1

=

(0:3 q1 )q1 , q1 = 0:15 3 9 = ( )2 = 20 400 14

low costs:

4 13 )q1 , q1 = 35 140 13 2 169 = ( ) = 1 140 19600 1 9 1 169 61 E( 1 ) = + = 0:01556 2 400 2 19600 3920 1

=

(0:3

q1

) …rm 1 has an incentive to …nd out costs privately, whereas it does not want to …nd out publicly!

5. Are consumers better o¤ if …rm 2’s costs stay private information?

1. costs are publicly observed:

low cost: q

=

CS

=

high cost: q

=

2 13 1 + = 0:216 12 15 60 1 13 2 169 ( ) = 0:0235 2 60 7200

0:15 9 1 (0:15)2 = CS = 2 800 5 ! E(CS) = 0:01736 288

0:01125

2. costs are private info:

high cost: q

=

low cost: q

=

! E(CS)

=

17 289 ! CS = 0:00737 140 39200 17 4 1089 + ! CS = 0:02778 140 35 39200 689 0:01758 39200

(1)

3. …rm 1 observes costs without …rm 2 knowing it:

high cost: q

=

low cost: q

=

! E(CS)

=

9 800 4 13 29 841 + = ! CS = 35 140 140 39200 641 0:01635 39200

0:15 ! CS =

! Consumers are better o¤ if costs stay private info. They are worst o¤ when …rm 1 can …nd out the costs privately.

15

Exercise 13 Strategic capacity choice Consider a market in which …rms 1; :::; N set simultaneously capacities for a homogeneous product and afterwards a third party, which observes market demand and the capacity choice of each …rm sets the market clearing price.

Suppose that inverse demand is linear and of the form P (q) = a q, where PN p is the price, a a positive constant, and q aggregate output, q = i=1 qi ; qi is the quantity sold by …rm i. Suppose that …rm i has constant marginal costs of production ci (…rms have di¤erent marginal costs!). Suppose that a > maxfc1 ; :::; cN g. Suppose furthermore that the parameters of the model are such that each …rm produces positive output. Solve for the Nash equilibrium where capacity is the strategic choice of the …rms. Determine aggregate output and price level in equilibrium. Determine the output of each …rm i.

Exercise 14 Price setting in a market with limited capacity [included in the 2nd edition of the book] Suppose that two identical …rms in a homogeneous-product market compete in prices. The capacity of each …rm is 3. The …rms have constant marginal cost equal to 0 up to the capacity constraint. The demand in the market is given by Q(p) = 9 p. If the …rms set the same price, they split the demand equally. If the …rms set a di¤erent price, the demand of each one of the …rms is calculated according to the E¢ cient Rationing Rule. Show that p1 = p2 = 3 can be sustained as an equilibrium. Calculate the equilibrium pro…ts.

Solutions to Exercise 14 At the prices p1 = p2 = 3, both …rms produce at full capacity. We conclude that the …rms have no incentives to deviate to a lower price. This would result in the same amount of sales but at a lower price. The e¢ cient rationing rule implies that a …rm that deviates upwards faces a demand of D(p) = 6 p. Thus, the pro…ts are (6 p)p when deviating to some p > 3. It is easy to show that (6 p)p is decreasing in p for all p > 3. This, in turn, implies that deviating to a price above 3 is not pro…table. Hence, p1 = p2 = 3 is an equilibrium. The equilibrium pro…ts are: N E = 3(9 3)=2 = 9.

Exercise 15 Capacity-constrained imperfect competition Suppose two …rms in an industry face linear inverse demand curves Pi (qi ; qj ) = 7 qi qj , i = 1; 2, i 6= j. Firms compete in a two-stage game; …rst they set capacity and then they set price or output. At the …rst stage …rms set capacities, at this stage the marginal costs of capacity is 6. Suppose that …rms have zero marginal costs of production up until installed capacity and that production above capacity is not feasible. In case of rationing, rationing is assumed to be e¢ cient.

1. Suppose each …rm has a capacity of 7. Analyze competition at stage 2.

Determine the Nash equilibrium if both …rms set prices.

16

2. Consider the same situation as in (1) but suppose that …rms choose quantities, not prices at stage 2. Determine the Nash equilibrium.

3. Consider the same situation as in (1) but suppose that consumers do not observe price and incur a cost of 1=2 if, after visiting one …rm, they decide to visit the other …rm. [You can think of identical consumers with the demand function as given above]. Characterize the equilibrium if both …rms set prices. (What is the appropriate equilibrium concept here?).

Give an explanation (at most 2 sentences).

4. Suppose that …rms have given capacities q 1 and q 2 , respectively. If …rm 1 is the high-price …rm, what is its demand function? Determine the Nash 49=24, i = 1; 2). Show that equilibrium in prices (provided that q i equilibrium prices satisfy p1 = p2 = a q 1 q 2 .

5. Determine the subgame perfect equilibrium of the two-stage game in which …rms …rst set capacities and then prices. Give an explanation (at most 3 sentences).

6. Suppose that …rms collude at the stage at which they set capacity. What should they do?

7. Suppose that …rms are able to use a less costly technology (e.g., the marginal cost of capacity falls from 6 to 11=2). What are the competitive e¤ects of this reduction in capacity costs? What would happen if those costs fell to zero? Discuss your results.

Solutions to Exercise 15 1. Suppose that capacity is 7 ! NE in prices ! products are homogenous ! prices equal to MC B B ! pB i = 0, qi = 3:5 and i = 0 (i = 1; 2) 2. Suppose that capacity is 7 ! NE in quantities: i

=

(7

qi

qj )qi

@ i @qi

=

7

2qi

qj = 0

qi

=

7

q2 2

7 C ! q1C = q2C = 37 , pC 1 = p2 = 3 and

with i = 1; 2; i 6= j

C 1 =

17

49 C 2 = 9

3. Consumers do not observe prices before consumption The appropriate equilibrium concept is a Perfect Bayesian Equilibrium (PBE) since consumers’decision which …rm to choose depends on their beliefs. There are multiple PBE which can be supported by di¤erent belief systems. Focus will be on the PBE that yields highest pro…ts for the …rms.

Assume that consumers believe that both …rms set identical prices. ! Consumers are indi¤erent between visiting either …rm 1 or …rm 2 at …rst.

Therefore, they visit …rm i (i = 1; 2) with probability 1=2 and will never consider a second o¤er given that switching cost are larger than zero (they are equal to 1=2 in this exercise). In consequence, …rms will set their prices equal to the monopoly level. (We can therefore specify consumers’ beliefs even more precisely stating that consumers believe that …rms will both set monopoly prices.)

Do …rms have an incentive to deviate from setting monopoly prices given this belief system? No, since the probability of a consumer switching to check the o¤er of the second …rm is zero given the beliefs and consumers choose a …rm randomly in the …rst place.

Hence, consumers believing that …rms set monopoly prices and …rms actually choosing prices equal to monopoly level is a PBE.

4. Suppose that …rms have given capacities q1 and q2 : e¢ cient rationing rule implies the following demand for …rm 1 (residual demand for those not served by …rm 2):

b 1) = Q(p

( 0 Q(p1 )

q2

if q2 Q1 (p1 ) if q2 < Q(p1 )

maximum capacity chosen is such that its revenues minus costs are non-negative independent of its competitor: ! maximum revenue in period 2: M M

= p(7 =

p) !

@

M

@p

=7

2p = 0 , pM =

7 = 3:5 2

3:52

pro…t maximizing capacity must satisfy: tion is ful…lled by assumption

M

> 6qi , qi < 49 24 ! this condi-

pro…ts: 2 1

=

(p2 6)q2 b 1 ) 6q1 = p1 (Q=p1 ) = p1 Q(p

q2 )

6q1

Show that pN = p1 = p2 = 7 q1 q2 is the NE price, i.e., show that …rm i has no incentive to deviate by setting a price above or below pN ! [Note that pN

18

b 1 ) = (7 implies market clearing: Q(p

p1

q2 ) = (7

7 + q1 + q2

q2 ) = q1 and

Q(p2 ) = q2 ]

Suppose pi < p : ! at p …rm i sells all its capacity ! lowering pi would increase demand above capacity ! still sell qi , but at lower price ! i # Suppose pi > p : ! due to capacity limit of …rm j , …rm i obtains positive demand for pi > p ! in order to show that it is not optimal to raise prices, prove that the revenue maximizing price lies to the left of p (Note: costs of capacity are already incurred in period 1) ! if pi > p , then revenue is given as

b i) = pi Q(p

( pi (7 0

pi

qj )

if 7 else

pi > q j

7 q

! revenue maximizing price: pbi = 2 j ! this price must be smaller than p pbi < p ,

7

qj

2

<7

qi

qj , 7 > qj + 2qi

49 49 168 147 49 24 by assumption. Hence, 7 > 24 + 12 , 24 > 24 ! it is not optimal to set a price pi > p

! we know that qi

!p =7

q1

q2 is the NE price

5. SPNE of the two-stage game ! second stage: see 4) ! …rst stage: plug in NE prices in the pro…t function: 1

2

= =

(7 (7

q1 q2

q2 )q1 q1 )q2

) q1 = q2 =

6q1 , FOC: 1 6q2 , FOC: 1 1 and 3

1 =

2q1 2q2

2 = (7

q2 = 0 , q1 = q1 = 0 , q2 = 2 1 ) 3 3

6

1

q2 2

1

q1 2

1 1 = 3 9

! NE: …rms set quantities equal to qi = 31 and choose prices according to p = 7 q1 q2 ! chosen capacities are equal to the quantities that would result in a standard Cournot game with MC equal to 6

19

6. If …rms collude at the capacity setting stage, they maximize joint pro…ts:

M

@

M

1 = 1 2q = 0 , q = @q 2 M 1 1 q M = ) q1 M = q2 M = = and M 1 = 2 = 2 4 2 8 =

(7

q)q

6q !

7. Reduction in MC to c = 11 2 changes the results of the …rst stage: i = (7

qi

qj )qi

11 3 qi , qi = 2 4

qj 1 ) q1 = q2 = 2 2

! As costs of capacity fall, q1 and q2 increase. Furthermore, pro…ts increase to 1 = 2 = 14 As c ! 0, we arrive at capacities equal to Cournot quantities given zero marginal production costs (see question 1). In general, the …rst-stage game corresponds to solving a Cournot game with MC equal to the capacity costs and the unique second stage game NE corresponding to p1 = p2 = p = 7 q1 q2 .

Exercise 16 Competition, installed capacity, and demand uncertainty Suppose two …rms, …rm 1 and …rm 2, operate in a homogeneous good market.

The supply of …rm i, denoted by qi , is constrained by installed capacity ki , i.e., 0 qi ki for i = 1; 2. Firms have zero marginal costs of production for quantities weakly below their capacity. They cannot increase production beyond capacity.

There is a group of consumers of size e who all have unit demand with the same willingness-to-pay r. Demand e is uncertain. The size of demand is determined by the value a random variable takes. Suppose that this random variable is uniformly distributed on the unit interval (i.e., on the interval [0; 1]).

Due to regulatory intervention there is a price ceiling P r that …rms are allowed to charge (lower prices are admissible).

Consider the following two-stage game. At stage 1, before observing the demand realization e …rms make investment decisions simultaneously. The constant cost per unit of capacity is c. After the investment stage, information about capacities become public knowledge. Next, demand is realized and publicly observed. At stage 2, …rms compete in prices. Each …rm simultaneously and independently sets its price pi . Firms maximize expected pro…ts.

1. Determine the allocation at stage 2 for given capacity choice.

2. Determine the Nash equilibrium at stage 2 for given capacities. Here you have to distinguish between four di¤erent parameter regions. Note that 20

in two regions pure-strategy equilibria do no exist and you have to solve for mixed-strategy equilibria. Also note that since all consumer have the same willingness-to-pay you can be agnostic about the rationing rule that is applied. If prices are equal, demand is assumed to be equally split between …rms.

3. Analyze the 2-stage game and solve for subgame perfect Nash equilibria (HINTS: Do not treat …rms as symmetric. Consider capacity as a pure strategy). What is the expected pro…t at this stage? Draw the bestresponse functions of the two …rms in a diagram. Determine the aggregate equilibrium capacity K = k1 + k2 in this game. Implicitly characterize equilibrium capacities k1 and k2 . Can you say anything about equilibrium pro…ts (e.g., whether they are the same for the two …rms or which …rm enjoys higher pro…t)?

Exercise 17 Cournot equilibrium and competitive limit Suppose there are N …rms in a homogeneous good market which set their output sequentially (…rm i in period i). Suppose that …rms have identical constant marginal costs of P production c. The industry faces an inverse demand N P (q) = a q where q = i=1 qi is aggregate demand. Suppose that a > c.

1. What is the output level of …rm i in the subgame perfect equilibrium?

2. What is the aggregate output for N …rms?

3. Describe the equilibrium outcome when the number of …rms increases without bounds (N ! 1).

Solutions to Exercise 17 N …rms, M C = 0, homogenous good, output is set sequentially ) Search for SPNE (backward induction starting in period N with …rm N ) …rm N :

N

=

(p

c)qN = (a

q

c)qN = (a

N X1 j=1

!

@ N @qN

=

0 , qN =

1 (a 2

N X1 j=1

21

qj

c)

qj

qN

c) qN

…rm N

1:

N 1

=

(a

q

N X2

c)qN 1 = (a

qj

qN 1

qN |{z}

j=1

=

!

@ N 1 @qN 1

=

1 (a 2

qj

1 k = 2

(a

qN 1

e x p e c t e d : ) p l u g i n qN

c)qN 1

j=1

N X2

1 (a 2

0 , qN 1 =

...analogously for N Hence, qN

N X2

qj

c)

j=1

qN 2 = 12 (a

2:

PN k 1 j=1

PN 3 j=1

qj

c) or qi = 12 (a

qj

c)

qj

) Compare qi with qi 1 : qi

qi 1 , qi

= =

c) qN 1

1 (a 2 1 q 2 i 1

i 1 X j=1

1 (a 2

i 2 X

qi

c) Pi 1

j=1 qj

c).

c)

j=1

(I)

) …rms always produce less (half) than the …rm before them!

What does the …rst …rm i = 1 do?

qi=1 = 12 (a

c

0) = 12 (a

c)

(II)

) from (I) and (II) we …nd solutions for questions (1) to (3): 1. qi = 1i (a c) 2 PN PN 1 2. aggregate output: q = c) = (a i=1 qi = i=1 2i (a (a c) (1 12 N )

c)

PN

1 i=1 2i

=

3. For N ! 1: q ! (a c) Note that for q ! (a c): P ! c, i.e. Bertrand competition (But …rms are asymmetric in the sense of qi .)

Exercise 18 Competing in price-quantity pairs Consider a duopoly for a homogeneous product. Firms i = 1; 2 set pricequantity pairs (pi ; qi ) simultaneously. If at these pairs some consumers are rationed, rationing is assumed to be e¢ cient. Suppose …rms are constrained by capacities ki > 0 and inverse market demand is P (q) = q 1 . Does a Nash equilibrium in pure strategies exist? Is it unique? Characterize all Nash equilibria of this game 22

Exercise 19 Sequential price setting with di¤ erentiated products [included in the 2nd edition of the book] Consider a market with two horizontally di¤erentiated products and inverse demands given by Pi (qi ; qj ) = a bqi dqj . Set b = 2=3 and d = 1=3. The system of demands is then given by Qi (pi ; pj ) = a 2pi + pj . Suppose …rm 1 has cost c1 = 0 and …rm 2 has cost c2 = c (with 7c < 5a). The two …rms compete in prices. Compute the …rms’pro…ts: 1. at the Nash equilibrium of the simultaneous Bertrand game, 2. at the subgame perfect equilibrium of the sequential game (a) with …rm 1 being the leader, and (b) with …rm 2 being the leader.

3. Show that …rm 2 always has a second-mover advantage, whereas …rm 1 has a …rst-mover advantage if c is large enough.

4. Solve for the Nash equilibria of the endogenous timing game in which …rms simultaneously choose whether to play ‘early’ or to play ‘late’. If they both make the same choice (either ‘early’or ‘late’), the simultaneous Bertrand game follows; if they make di¤erent choices, a sequential game follows with the …rm having chosen ‘early’being the leader.

Solutions to Exercise 19 1. Nash equilibrium of the simultaneous Bertrand game. Firm 1 chooses p1 to maximize 1 = (a 2p1 + p2 )p1 . Solving the …rst-order condition for p1 , one …nds …rm 1’s reaction function as p1 = (a + p2 ) =4. Firm 2 chooses p2 to maximize 2 = (a 2p2 + p1 )(p2 c). Solving the …rst-order condition for p2 , one …nds …rm 2’s reaction function as p2 = (a + 2c + p1 ) =4. The solution to the system made of the two reaction functions is p1 = (5a + 2c) =15 and p2 = (5a + 8c) =15. We can then compute the equilibrium pro…ts as B 1 = 2 2 2 B 2 (5a + 2c) and = (5a 7c) .

2 225 225 2. Sequential game (a) If …rm 1 is the leader, it takes …rm 2’s price reaction into account when maximizing pro…ts; that is, it chooses p1 to maximize 1 = (a 2p1 + 1 4 (a + 2c + p1 ))p1 . The optimal price is found as p1 = (5a + 2c) =14.

Firm 2 then reacts by setting p2 = (19a + 30c) =56. Plugging the prices 2 F into the pro…t functions, one obtains L 1 = (5a + 2c) =112 and 2 = 2 (19a 26c) =1568.

23

(b) If …rm 2 is the leader, it takes …rm 1’s price reaction into account when maximizing pro…ts; that is, it chooses p2 to maximize 2 = (a 2p2 + 1 c). The optimal price is found as p1 = (5a + 7c) =14.

4 (a + p2 ))(p2 Firm 1 then reacts by setting p1 = (19a + 7c) =56. Plugging the prices into the pro…t functions, one obtains L 7c)2 =112 and F 2 = (5a 1 = 2 (19a + 7c) =1568.

F 2 L 3. Firm 2 always has a second-mover advantage: 2 = (19a p 26c) =1568 > 2 = p 2 14)a + (7 14 26)c > 0, which is (5a 7c) =112 ispequivalent to (19 5 p satis…ed as 19 5 14 = 0:292 > 0 and 7 14 26 = 0:192 > 0. Firm 1 has a 2 F 2 …rst-mover advantage if L 1 = (5a+2c) =112 > 1 = (19a+7c) =1568, which is equivalent to 7c2 + 14ac 11a2 > 0 or c > 0:604a. (Note: 0:604 < 5=7, i.e., there is a positive interval of c where the above condition is ful…lled.)

Exercise 20 Timing game [included in the 2nd edition of the book] Use the results of the previous exercise to solve for the Nash equilibria of the endogenous timing game in which …rms simultaneously choose whether to play ‘early’or to play ‘late’. If they both make the same choice (either ‘early’or ‘late’), the simultaneous Bertrand game follows; if they make di¤erent choices, a sequential game follows with the …rm having chosen ‘early’being the leader.

Discuss the economic intuition behind your result.

Solutions to Exercise 20 The following matrix represents the normal form of the game: Firm 1 / Firm 2 Early Late

Early

Late

B 1 ; F 1;

L 1; B 1 ;

B 2 L 2

F 2 B 2

1

1 L As far as …rm 2 is concerned, we already know from Exercise 4.1 that F 2 > 2 . It L 2 B 2 is also easy to see that 2 = 2(5a 7c) =224 is larger than 2 = 2(5a 7c) =225: We L B L 2 have thus F 2 > 2 > 2 . As for …rm 1, it is easy to see that 1 = 2(5a+2c) =224 > B B 2 F 1 = 2(5a + 2c) =225. We can also show that 1 > 1 : F 1

B 1

= =

2 2 1 2 1568 (19a + 7c) 225 (5a + 2c) 1 7c) (565a + 217c) ; 352 800 (5a

which is positive as we assume that 7c < 5a. Hence, there are two pure-strategy NE: (Early, Late) and (Late, Early). There also exist a mixed-strategy NE. To characterize it, let …rm 1 choose , the probability with which it plays (Early), such that …rm 2 is B F indi¤erent between (Early) and (Late); that is, ) L ) B 2 +(1 2 = 2 +(1 2 , F L B L B 2 2 ). We proceed in the same way which is equivalent to = ( 2 2 )=( 2 + 2 for …rm 2, which chooses , the probability with which it chooses (Early), to make …rm B F 1 indi¤erent between (Early) and (Late); that is, ) L ) B 1 +(1 1 = 1 +(1 1 ;

24

B F which implies that = L 1 1 =( 1 + level by their respective value, one …nds 2

7c) = 3175a14(5a 2 3760ac 878c2 and

L 1

2 B 1 ). Replacing the various pro…t 2

= 3175a214(5a+2c) 2590ac 1463c2 :

Exercise 21 Information sharing in Cournot duopoly [included in the 2nd edition of the book] Consider the Cournot duopoly with linear demand P (q) = 1 q with q = q1 + q2 and constant marginal cost. Firm one has marginal cost of zero. This is commonly known. The marginal cost of …rm 2 is privately known to …rm 2; …rm 1 only knows that they are prohibitively high or zero and that both events are equally likely (this is commonly known). High marginal costs are assumed to be prohibitively high such that …rm 2 does not produce. Consider the three-stage game in which, at stage 1, …rm 2 draws its marginal costs, then, at stage 2, it decides whether to share its information with its competitor and, at stage 3, in which both …rms compete in quantity.

1. Characterize the equilibrium of this game. Does …rm 2 have an incentive to share its private information?

2. Would …rm 1 be better o¤ under information sharing?

Solutions to Exercise 21 Suppose …rst that …rm 2 shares its information. In this case …rm 1 learns the cost type of …rm 2. If …rm 2’s costs are high, …rm 1 knows that …rm 2 will produce zero; thus, …rm 1 will produce the monopoly quantity q1m = 1=2; the monopoly pro…t is = 1=4. If the costs of …rm 2 are zero, we have a symmetric Cournot duopoly.

Firms will produce duopoly quantities. q1d = q2d = 1=3; and pro…ts are di = 1=9.

Now suppose that …rm 2 does not share its information. Then …rm 1 produces a quantity between q1d and q1m . Pro…ts of …rm 1 and the low-cost …rm 2 are 1

=

2

=

1 1 (1 q1 q2 )q1 + (1 2 2 (1 q1 q2 )q2

q1 )q1

First-order conditions of pro…t maximization at stage 2 can be written as:

q2

=

q1

=

1 2 1 2

1 q1 ; 2 1 q2 : 4

Solving this system we obtain q2 = 2=7 and q1 = 3=7. We note that q1m = 1=2 > 3=7 > 1=3 = q1d . Equilibrium at stage 2 under information sharing are

25

s 1

=

sL 2

=

1 2

3 7

1 1

3 7

2 7 2 7

3 1 + 7 2 2 4 = 7 49

1

3 7

3 9 = 7 49

Because of strategic substitutes …rm 1’s pro…t is larger than …rm 2’s pro…t.

We now turn to the decision of …rm 2 at stage 1. If the costs of …rm 2 are high, it obtains zero pro…t independent of whether it share information. If the costs of …rm 2 are low, then its pro…t will be higher if it shares information (1=9 rather than 4=49), because then …rm 1 will produce less. Thus, …rm 2 wants to share its information.

Note this is certainly true ex ante (i.e., if …rm 2 has to decide before knowing its cost). In the example, it is also true interim (after learning its cost), because the high cost type is indi¤erent whether to share information.

What about …rm 1’s pro…t? If …rm 2 shares information, …rm 1’s expected pro…t d is: (1=2) m 1 + (1=2) 1 = (1=2)(1=4) + (1=2)(1=9) = 13=72. If …rm 2 does not share information, its pro…t has been calculated to be 9=49. Since 9=49 > 13=72, …rm 1 would be better o¤ if …rm 2 did share information.

Exercise 22 Price competition and information sharing Consider the same setting except that …rms face a di¤erent demand function and that …rms set prices at stage 3. Let demand be Qi = 1 pi dpj with d > 0 so that products are substitutes and d < 1. Characterize the equilibrium of this game. Does …rm 2 have an incentive to share its private information?

Solutions to Exercise 22 Suppose that …rm 2 decides to share its information at stage 2. If its costs are high, …rm 1 knows that …rm 2 will produce zero. Thus, …rm 1 will set the monopoly price pm p1 )p1 = 1=2. If the costs are symmetric, we have a symmetric 1 = arg maxp1 (1 price competition model with linear demand. Each …rm i maximizes (1 pi dpj )pi subject to pi . The …rst-order condition of pro…t maximization can be written as pi = 1=2 (1=2)dpj . Using symmetry, we obtain that pdi = 1=(2 + d).

Now suppose that …rm 2 does not share information. Then …rm 1 sets an intermediate price that is between pd1 and pm 1 , as we will show next. Pro…ts are 1

=

2

=

1 1 (1 p1 )p1 + (1 2 2 (1 p2 dp1 )p2 :

p1

dp2 )p1 ;

First-order conditions of pro…t maximization can be written as

p1

=

p2

=

1 2 1 2

1 ap2 ; 4 1 ap1 : 2

With information sharing, the price of …rm 1 is

26

higher if the cost of …rm 2 are high (monopoly vs. intermediate price) lower if the cost of …rm 2 are high (duopoly vs. intermediate price) Let’s compare the pro…ts of …rm 2: Suppose that the costs of …rm 2 are high. Then it obtains zero pro…ts anyhow.

Suppose that the costs of …rm 2 are low. Then its pro…ts are lower if it has shared information, because in this situation …rm 1 will produce less. Thus, …rm 2 does not want to share information! Moral: it depends on Cournot vs. Betrand.

27

Industrial Organization: Markets and Strategies Paul Belle‡amme and Martin Peitz published by Cambridge University Press

Part III. Sources of Market Power Exercises & Solutions Exercise 1 Horizontal product di¤ erentiation1 [included in 2nd edition of the book] Hong Kong Island features steep, hilly terrain, as well as hot and humid weather. Travelling up and down the slopes therefore causes problems; this has led the city authorities to imagine rather unusual methods of transport. One famous example can be found in the Western District, where one of the busiest commercial area of Hong Kong can be found. This area stretches from Des Voeux Road in Central (which is at sea level) up to Conduit Road in the MidLevels (which is the mid section of the hill of Hong Kong Island). Because the street is so steep, sidewalks are made of stairs. To make travelling up the slope easier for pedestrians, the Mid-Levels escalators were opened to the public in October 1993. (See http://www.12hk.com/area/Central/MidLevelEscalators.shtml for some pictures of the escalators and the stairs of this area).

For the sake of this problem set, imagine the following story. Suppose that the street is one kilometre long (kilometre 0 is down at the crossroad with Des Voeux Road and kilometre one is up at the crossroad with Conduit Road).

Suppose that 100,000 inhabitants are uniformly distributed along the street.

Without loss of generality, we can approximate the consumer distribution by a continuum on [0; 1] with a mass set equal to 1 (i.e., we rede…ne all quantities by dividing them by 100,000).

There are only two shops selling sweet-and-sour soup in this area. For simplicity, we set their marginal cost of production to zero. As it happens, one shop (named ‘Won-Ton’and indexed by 1) is located at point 0, while the other shop (named ‘Too-Chow’ and indexed by 2) is located at point 1. Everyday, each inhabitant of the street may consume at most one bowl of sweet-and-sour soup, bought either from Won-Ton or from Too-Chow. The price per bowl of the two shops are respectively denoted by p1 and p2 . The net utility for a consumer located at x on the interval [0; 1] is given by 8 p1 if consumer buys at Won-Ton, < r 1 (x) r (1 x) p if consumer buys at Too-Chow 2 2 : 0 if consumer does not buy.

where it is assumed that r is large enough so that every consumer buys one bowl of soup.

1 This exercise draws from McCannon, B.C.. (2008). The Quality-Quantity Trade-o¤, Eastern Economic Journal 34, 95-100.

1

1. Before 1993 and the installation of the Mid-Levels escalators, walking up the street was much more painful than walking down. This is translated by the following assumptions: 1 (x) = tx and 2 (1 x) = (t + ) (1 x), with t; > 0. (a) Derive the identity of the consumer who is indi¤erent between the two shops. (b) Compute the equilibrium prices and pro…ts of the two shops. (c) Show that Two-Chow’s pro…ts increase if walking up the street becomes more costly for consumers, that is if increases (e.g., because the temperature has risen). Explain the intuition behind this result.

2. After 1993, the Mid-Levels escalators made going up and down equally painful for consumers. However, consumers had to pay a …xed fee f (independent of distance) to use the escalators. This is translated by the following assumptions: 1 (x) = tx and 2 (1 x) = t (1 x) + f , with f > 0. (a) Derive the identity of the consumer who is indi¤erent between the two shops. (b) Compute the equilibrium prices and pro…ts of the two shops. (c) Express the condition (in terms of f and t) under which the previous answers are valid (i.e. the condition for Too-Chow to set a price above its zero marginal cost). (d) Show that Two-Chow’s pro…ts increase if taking the escalator becomes less expensive, that is if f decreases. Explain the intuition behind this result and contrast with your answer at (1c).

3. Comparing your answers for (1) and (2), establish and explain intuitively the following results. (a) Too-Chow su¤ers from the installation of the escalators (even when its access is free, i.e., for f = 0). (b) Won-Ton bene…ts from the installation of the escalators, unless the extra transportation cost of climbing the stairs (i.e., ) is too large.

(To show this, set t = 2, f = 3 and compare Won-Ton’s pro…ts for = 2 and = 4).

Solutions to Exercise 1 1. Before 1993. (a) The indi¤erent consumer is identi…ed by xo such that r txo p1 = r (t + ) (1 xo ) p2 ; that is, xo = (t + p1 + p2 ) = (2t + ).

2

(b) Firm 1 maximizes

1 = p1 xo . Solving the …rst-order condition for p1 , we …nd …rm 1’s reaction function as p1 = (t + + p2 ) =2. Firm 2 maximizes xo ). Solving the …rst-order condition for p2 , we …nd …rm 2’s 2 = p2 (1 reaction function as p2 = (t + p1 ) =2. The solutionof the system made of the two reaction functions is p1 = t+ 32 and p2 = t+ 13 . The equilibrium pro…ts are

1 (3t + 2 <93 = 1 9

2

)

2t +

and

1 (3t + <93 = 2

9 2t +

2

)

:

2

(c) We compute @ 2 =@ = (3t + ) (t + ) =(9 (2t + ) ) > 0. An increase in raises the horizontal di¤erentation between the two shops, therby realxing price competitionand increasing pro…ts. (Note: this is true up to the limit where consumers stop buying from Two-Chow.)

2. After 1993. (a) The indi¤erent consumer is identi…ed by xs such that r txs r t (1 xs ) f p2 ; that is, xs = (f + t p1 + p2 ) = (2t).

p1 =

(b) Firm 1 maximizes

1 = p1 xs . Solving the …rst-order condition for p1 , we …nd …rm 1’s reaction function as p1 = (t + f + p2 ) =2. Firm 2 maximizes xs ). Solving the …rst-order condition for p2 , we …nd …rm 2 = p2 (1 2’s reaction function as p2 = (t f + p1 ) =2. The solutionof the system made of the two reaction functions is p1 = t + 31 f and p2 = t 13 f . The equilibrium pro…ts are 2

>93 = 1

1 (3t + f ) and 18 t

2

>93 = 2

1 (3t f ) : 18 t

(c) We need to assume that 3t > f . (d) We compute @ 2 =@f = (f 3t) =(9t), which is negative given the assumption 3t > f . Here, an increase in f reduces the attractiveness of Two-Chow; it is an increase in vertical product di¤erentiation (the “quality” of …rm 2 is deteriorated), which is detrimental to …rm 2.

3. Comparison (a) We have <93 2

1 (3t + >93 = 2 f =0

9 2t +

2

)

2

1 (3t) 3t + 2 = > 0: 18 t 18 2t +

Two-Chow prefers the pre-1993 situation to the post-1993 situation because the installation of the escalator (even if it is free for the consumers) reduces product di¤erentiation and thus intensi…es price competition.

3

(b) We have <93 1

4( >93 = 1 t=2;f =3 9

2

+ 3) +4

9 1 15 + 16 2 = 4 36 +4

180

:

= 2 and = 4, the latter expression is equal to 43=108 and 17=36 respectively. So, with = 2, Won-Ton prefers the post-1993 situation and the reverse applies with = 4.

With

Exercise 2 Competition on the Salop circle Consider a market in which …rms 1; :::; N are equidistantly distributed on a circle with circumference 1. Firms have constant marginal costs of production c, which are the same for all …rms. Consumers are uniformly distributed on the circle (and have mass 1). A consumer x incurs a transportation jx li j when buying from …rm i. Here the distance between consumer and …rm is the arc distance on the circle (that is consumers move on the circle). Suppose that all consumers are active in the market.

1. Determine the demand function of …rm i as a function of all prices. [Be careful!]

2. Determine equilibrium prices in the game in which all …rms set prices simultaneously.

3. How do transport costs a¤ect pro…ts?

4. Argue informally whether or not you think that an equilibrium exists for all location con…gurations.

Solutions to Exercise 2 This model is often referred to as the circular city. As in the standard Hotelling model, we can determine demand for each individual …rm by …nding the indi¤erent consumer (assuming full-market coverage). Be …rm i located at zi , then there exist two indi¤erent consumers, one to the left, call him z and one to the right, call him z .

Hence, demand then can be derived by solving for z and z which are indi¤erent between buying from …rm i or its closest neighbor. (Note that the underlying assumption is that none of the …rms is setting its price so signi…cantly below its competitors that it could catch full demand on the circle. Instead, we assume that all …rms are identical, as they have the same marginal cost c and are equidistantly distributed on the circle (as can be seen in second …gure below.)

4

z :pi + (z

zi ) = pi+1 + (zi+1

z)

suppose without loss of generality that zi = 0 ! zi+1 = !z=

1 (pi+1 2

z :pi + (z

pi +

N

zi ) = pi 1 + (z

1 N

) zi 1 ) , z =

1 (pi 1 2

pi +

N

1 2 (pi+1 + pi 1 2pi + ) 2 N 1 ! by symmetry: pi+1 = pi 1 = p ! qi (pi ; p) = (p pi + ) N ! demand for …rm i: qi = z + z =

on 2. equilibrium prices

max pi

FOC:

i = (pi

1

(p

1 c) (p

c)qi (pi ; p) = (pi pi +

N

best response: pi =

)

1

(pi

c) = 0

(p + N + c) 2

5

pi +

N

)

)

due to symmetry (equidistant locations, identical costs) pi = p:

!p=

(p + N + c) ,p =c+ 2 N

Hence, prices are as in the standard Hotelling model. (Note that p ! c as N ! 1.) on 3.

! 0 or

= N 2 ) pro…ts increase as transport costs rise

on 4. Does an equilibrium exist for all locations?

No! Proof by contradiction with an example where locations are still …xed (and symmetric), but may be no longer equidistant.

Assume full-market coverage.

1. suppose that N = 2: ! p1 = p2 still seems to be a reasonable NE candidate since the two …rms still

share total demand equally.

2. suppose that N = 3:

Can p1 = p2 = p3 be a NE? …rm 2 can only gain a small share of total demand, maximally all consumers between F1 and F3 …rm 1 and 3 share consumers on the rest of the circle ! they have much higher demand!

6

! …rm 1 and 3 have a higher incentive to raise prices than …rm 2: i =(pi

@ i = @pi

c)Q(pi ; p i ) Q(pi ; p i ) | {z }

+

larger for F1 and F3

@Qi (pi c) @pi | {z }

identical for all …rms

compared to F2

because the marginal decrease in demand is the same

! there exists no equilibrium with symmetric prices!

Can there be an asymmetric equilibrium? Note that if p2 is verly low s.t. p2 p1 1 z2 e, then …rm 2 can steal all of …rm 1’s demand, i.e. there is discontinuity in demand. Therefore, …rm 1 cannot raise p1 above p1 = p2 + jz1 z2 j. On the other hand, the lowest bound for p2 is c. ! While …rm 1 (3) has an incentive to raise its price to some optimal level p1 (p3 ), …rm 2 always wants to slightly undercut …rm 1’s (3’s) price. If p2 = c and p1 = p2 + jz1 z2 j, then at least …rm 1 wants to raise its price to p1 (if p1 p2 + jz1 z2 j) to increase pro…ts. In consequence, a pure-strategy equilibrium might not exist.

jz

Exercise 3 Quality-augmented Hotelling model [included in 2nd edition of the book] Consider the Hotelling model in which consumers are uniformly distributed on the [0; 1]-interval and …rms A and B are located at the extreme points.

Firms produce a product of quality si . Consumer x 2 [0; 1] obtains utility uA = (r tx)sA pA if she buys one unit of product A and uB = (r t(1 x))sB pB if she buys one unit of product B. Each consumer buys either one unit of product A or one unit of product B. 1. Describe the property of the utility function with respect to quality in two or three sentences.

2. Determine the demand for products A and B at given prices and given qualities.

3. Suppose that qualities sA and sB are given and that marginal costs of production are zero. Determine the Nash equilibrium in prices under the assumption that qualities are not too asymmetric implying that both …rms have a strictly positive market share in equilibrium.

4. Suppose that qualities are symmetric and that the cost of quality C(si ) is increasing and strictly convex in si . How does the equilibrium pro…t depend on quality?

5. Compare this …nding to the standard quality-augmented Hotelling-model in which consumer x obtains utility uA = r + sA tx pA if she buys product A and uB = r + sB t(1 x) pB if she buys product B. 7

Exercise 4 Market integration Somewhere far away there exist two villages Applecastle (A) and Orangevillage (B). Each village has its grocery store which sells a particular brand. Suppose that initially connection are bad so that all inhabitants of A do their shopping in A and all inhabitants of B do their shopping in B. Some villagers propose a better connection between A and B. 1. Do the grocery owners support this connection? Is it possible that both, one or none of the owners support the project? Explain.

2. What is the likely position the two city councils will take?

3. How may the opinion of the city council be di¤erent if there is a local sales tax?

Exercise 5 Spatial competition Consider a horizontally di¤erentiated product market in which two …rms are located at any points l1 and l2 on the real line, respectively, with the notation l1 l2 . Firms produce at marginal costs c. There is a continuum of consumers of mass 1 who are uniformly distributed on the unit interval. They have unit demand and have an outside utility of 1. A consumer located at x 2 [0; 1] obtains indirect utility v = maxfv1 ; v2 g with v1 = r (x l1 )2 p1 if she buys 2 one unit from …rm 1 and v2 = r (l2 x) p2 if she buys from …rm 2. Firms have marginal costs equal to c.

1. Suppose that prices are regulated at pi = 2c. In the game in which …rms simultaneously decide where to locate their product, characterize the Nash equilibrium. Determine the demand function for each …rm for each admissible price pair (p1 ; p2 ) given locations l1 and l2 .

2. Suppose that the two …rms simultaneously set prices. Determine the market equilibrium for all possible combinations of (l1 ; l2 ).

3. Suppose that the social planner chooses …rst-best optimal prices. Which price pairs would be socially optimal the pair of locations l1 = 0 and l2 = 1=2?

4. Compare you results obtained in (3) and (4) for locations l1 = 0 and l2 = 1=2. Is the equilibrium socially e¢ cient? Depending on your answer elaborate on the sources of the ine¢ ciency or give the reason for e¢ ciency.

5. Consider the two-stage game in which …rm …rst set locations on the real line simultaneously and then set prices simultaneously. Characterize the set of pure-strategy subgame-perfect equilibria. Determine equilibrium pro…ts. Note: Calculations are tedious; explain the steps in your calculations. You may then work with functions l1 = (l2 2)=3 and l2 = (l1 +4)=3.

8

6. Determine the socially optimal solution of a welfare-maximizing social planner who can choose …rm locations in a market in which price decision are decentralized (second-best optimum). Compare your results obtained in (1) and (6) to second-best socially optimal locations and explain any di¤erences you obtained.

Solutions to Exercise 5 1. At symmetric exogenous prices, …rms locate at (1=2; 1=2) in the unique Nash equilirbrium of the game.

2. The indi¤erent consumer is located at x b=

Hence, demand is

l1 + l2 p2 p1 + : 2 (l2 l1 ) 2

q1 (p1 ; p2 )

=

q1 (p1 ; p2 )

=

3. Firms maximize pro…ts (pi

p2 p 1 l1 + l2 + ; 2 (l2 l1 ) 2 p2 p 1 l 1 + l2 : 1 2 (l2 l1 ) 2

c)qi (pi ; pj ) with respect to pi . First-order condi-

tions are

p1 c p 2 p1 l1 + l 2 + + 2 (l2 l1 ) 2 (l2 l1 ) 2 p 2 p1 l1 + l 2 p2 c +1 2 (l2 l1 ) 2 (l2 l1 ) 2

=

0

=

0

Rewriting the …rst-order conditions we obtain

p1

=

p2

=

p2 + c + (l2 2 2 p1 + c + (l2 2 2

l1 )(l1 + l2 ); l1 )(2

(l1 + l2 )):

Solving this system of two equations we obtain

p1 (l1 ; l2 )

= c+

p2 (l1 ; l2 )

= c+

3 3

(l2

l1 )(2 + l1 + l2 );

(l2

l1 )(4

(l1 + l2 )):

4. Given locations l1 = 0 and l2 = 1=2, any prices p1 = p2 are welfare-maximizing since there is full participation and equal prices minimize total transport costs.

9

5. Given locations l1 = 0 and l2 = 1=2, equilibrium prices are p1 (0; 1=2) = c + 5 =12 and p2 (0; 1=2) = c + 7 =12. Clearly, since …rm 2 has a larger set of inframarginal consumers it prices less aggressively and has a market share which is less than the social optimum.

6. At the …rst stage …rm i maximizes (pi (l1 ; l2 ) c)b x(p1 (l1 ; l2 ); p2 (l1 ; l2 )) with respect to li . First-order conditions can be written as 18 18

( 3l12

2l1 l2 + l22

( l12 + 2l1 l2

3l22

8l1

4)

=

0;

16l2

16)

=

0:

The solution of each equation which satis…es the second-order conditions is

l1

=

l2

=

1 (l2 2); 3 1 (l1 + 4): 3

Hence, in subgame-perfect equilibrium, l1 =

1=4 and l2 = 5=4.

7. The social planner would choose l1 = 1=4 and l2 = 3=4, as these locations minimize total transport costs. In this symmetric setting with full participation, decentralizeing prices does not a¤ect total surplus. For …xed prices, there is too little di¤erentiation, as …rms try to increase there market share by moving closer to the center. By contrast, under endogenous pricing, …rms foresee that they sacri…ce margins if they move closer to the center. Firms balance the incentive to be closer to the marginal consumer with the incentive to avoid intense price competition by choosing equilibrium locations with too much di¤erentiation from a welfare perspective.

Exercise 6 A model of vertical product di¤ erentiation Suppose there are 2 …rms in a vertically di¤erentiated market. Consumers buy either one unit of any of the two goods or they do not purchase in the market. If they do not purchase in the market their indirect utility is 0. If they purchase good i their indirect utility is si pi , where is the preference parameter of a consumer, si is the quality of good i and pi is the price of good i. Assume that there exist consumers of mass 1 whose preference parameter is uniformly distributed on [0; 1].

1. Determine the demand function of each …rm depending on prices and qualities. [Hint: this is not the same model as the one presented in the book.]

2. Suppose that qualities are given, s1 < s2 , and that …rms face constant marginal costs of production c, which are independent of quality. Determine equilibrium prices and pro…ts in the game in which …rms simultaneously set prices.

10

3. Suppose …rms set qualities from an interval [0; s]. Which qualities will result in subgame perfect equilibrium in the game in which …rms set qualities simultaneously at stage 1 and set prices simultaneously at stage 2?

Solutions to Exercise 6 1. derive demand functions by solving for indi¤erent consumers (see …gure):

1. (a) Suppose s1 < s2 : : s1 p1 = s2 p2 , = ps22 sp11 : si p1 = 0 , = ps11 case 1: full market coverage

p1 0 D2 (s; p) = 1 ps22 ps11 D1 (s; p) = ps22 ps11 case 2:

= p1 > 0 D2 (s; p) = 1 ps22 ps11 p1 D1 (s; p) = ps22 ps11 s1 (b) Suppose that s1 = s2 = s: products are homogeneous. ! …rms share the market equally if p1 = p2 and gain whole market if p1 < p2 8 > if pi > pj <0 p 1 Di (s; q) = 2 2s if pi = pj = p > : 1 psi if pi < pj (c) Suppose that s2 > s1 : ! symmetric to case (a)

2. Suppose that s1 < s2 , MC equal to c and independent of s

11

! prices and pro…ts for the price setting game ) Demand as in (a) case 1: full market coverage: 1 = (p1

c)D1 (s; p) = (p1

) cannot be an equilibrium

c) ps22 ps11 p1 < 0 , not optimal for …rm 1

case 2:

> 0 , p1 > 0 c)( ps22 ps11 1 = (p1 s1 p2 p1 = 2s2 + 2c

p1 p2 p1 @ 1 s1 ) ) @p1 = s2 s1

c)(1 ps22 ps11 ) ) @@p22 = 1 2 = (p2 s2 s1 1 p2 = 2 + c+p 2

p1 s1

p2 p1 s2 s1

p1 c s 2 s1

(p1 c) ! =0 s1

p2 c ! s2 s1 = 0

! Solve reaction functions for p1 and p2 : p2 =

s2 (2(s2 s1 ) + 3c) s1 (s2 and p1 = 4s2 s1

s1 ) + c(2s2 + s1 ) 4s2 s1

! this yields: s2 (s1 2c) s1 (4s2 s1 ) s2 (s2 s1 )(s1 c)2 1 = s1 (4s2 s1 )2

and

q1 =

2s2 c 4s2 s1 (2s2 c)2 (s2 s1 ) 2 = (4s2 s1 )2

q2 =

and

3. si 2 [0; s] ! SPNE! Maximize pro…ts with respect to si s1 s2 (4s2 s1 )2 [ (s1 2c)2 +(s2 s1 )2(s1 2c)] (s2 s1 )(s1 2c)2 s2 [(4s2 s1 )2 2s1 (4s2 s1 )] @ 1 @s1 = (4s2 s1 )4 s21 (4s2 s1 )2 [(2s2 c)2 +4(s2 s1 )2(2s2 c)] (s2 s1 )(2s2 c)2 8(4s2 s1 ) @ 2 @s2 = (4s2 s1 )4

Remarks

8 s1 > <for c 2 for s21 < c 2s2 > : for c > 2s2

both …rms will produce ! relevant case! only …rm 2 will produce no …rm will produce

!

! @@s11 = 0 has multiple solutions:

12

s1 = 2c )

1 = 0 , i.e. cannnot be optimal

s2 (2s2 3c)

s1 =

p

(2s2 c)(23c+2s2 if s2 6= 74 c and c 4c 7s2

s1 2

if c > s21 …rm 2 will not produce !

! @@s22 = 0 has multiple solutions: s2 = 2c ) s2 =

2 = 0 cannot be optimal

2c+3s1

+

p

(s1 2c)(2c+23s1 ) ! square root only positive if s1 < 2c 8

+ 1. if …rm 1 does not produce, i.e. if s1 < 2c, …rm 2 sets 8 p < 2c+3s1 + (s1 2c)(2c+23s1 ) if c > 4 s 7 1 p 8 s2 = (s1 2c)(2c+23s1 ) : 2c+3s1 if c < 74 s1 8

2. if …rm 1 produces, i.e. if c < s21 , then …rm 2 will set s2 = s ) this is the relevant case!

Exercise 7 Vertical product di¤ erentiation and cost of quality A consumer with income m who consumes a product of quality si and pays pi obtains the utility si m=6 pi . If instead the consumer decides not buy the good, the resulting utility is zero. Consumer income m is uniformly distributed on the interval [2; 8] with the density 1=6. The total mass of consumers is equal to 1. There are two …rms in the market. Firms 1 and 2 o¤er the qualities s1 and s2 , respectively. We assume that s1 s2 and s1 ; s2 2 [1; 2]. Suppose that …rm i has constant marginal cost equal to c si . It is, thus, more expensive to produce higher quality.

1. Derive the demand of …rms 1 and 2, and calculate the reaction functions of the two …rms.

2. Calculate the Nash Equilibrium in prices and …nd the equilibrium pro…ts as a function of s1 and s2 . What are the equilibrium quality choices of the two …rms?

3. How does an increase in c a¤ect the pro…ts of the two …rms? Provide the economic intuition behind this result. Show that the high quality …rm, …rm 2, continues to earn higher pro…ts than …rm 1 as long as c < 5=6.

13

Solutions to Exercise 7 Assumptions:

vi = si6m

pi (indirect utility) and v0 = 0 if no good is consumed uniform, f (m) = 8 1 2 = 16

if consuming good i income m 2 [2; 8] , m 2 …rms: quality s1 < s2 ; si 2 [1; 2]

M Ci = c si …rm 2 has only one margin …rm 1 has two margins ! two cases depending on whether m b binds or not!

1. Find indi¤erent consumer m ^:

s1 m ^ 6

p1 =

assume m ^ 2 [2; 8]

,m ^ =

s2 m ^ 6

6(p1 s1

p2

p2 ) s2

Full market coverage (condition for all consumers to buy: even for m ^ =2

0): 2s1 6

p1

0

,

s1 3

p1

! Find demand for given s1 ; s2 : Case 1: if p1

s1 3

:

Q1 (p1 ; p2 ) =

^ Zm

1 m ^ m ^ dm = ( )m = 6 6 2 6

2

Q2 (p1 ; p2 ) =

Z 8

1 4 dm = 6 3 m ^

p1 s1

p2 s2

z }| { 1 p 1 p2 = 3 s s | 1 {z 2}

1 3

Case 2: what if p1 > s31 ? Marginal consumer m e such that v1 (p1 ) = 0

s1 m e 6

thus,Q1 (p1 ; p2 )

,

Z m ^

1 m ^ dm = 6 6

= s p1 > 31

m e =

p1 = 0

m e

14

6p1 s1

m e p 1 p2 = 6 s1 s2

p1 s1

v1

…nd best responses for both cases: i = (pi

max pi

p1 (p2 ) =

( 1 6

(s1

csi ) Qi (pi ; pj )

s2 ) + s p2 s1 +cs1 2 2

p2 +cs1 2

s1 3

o/w

2 (s1 3

p2 (p1 ) =

if p1

s2 ) +

p1 + cs2 2

Firm 1 exhibits a kinked best-response function. Hence, there may exist two equilibria for varying c. (See case 1 and 2) 2. SPNE s

1 Case 1: if p1 3 : second stage: ! solve p1 (p2 (p1 )) for p1

1 c(s2 + 2s1 ) 3 1 p2 = c(s1 + 2s2 ) 3

2 (s1 9 7 (s1 9

p1 =

s2 ) s2 )

1 2 c+ 3 9 1 7 Q2 = c+ 3 9 Q1 =

1 2 c + )2 (s2 3 9 1 7 2 ) (s2 2 (s1 ; s2 ) = ( c 3 9

1 (s1 ; s2 ) = (

s1 ) s1 )

…rst stage: s1 ; s2 = ? @ 2 @ 1 @s1 < 0 and @s2 > 0

! …rms will choose s1 , s2 to maximally di¤erentiate: s2 = 2 and s1 = p1 3 (Hence, s2 is equal to the maximum quality level and s1 corresponds to the quality level that guarantees that all consumers buy.)

plug p1 into s1

2 2 s1 9 9 6c + 4 ! s2 = 2 and s1 = 5 6c :

s1 =

2 2 c + s1 3 3

3c + 2 1 2 ; 1) = ( c + )2 (1 5 6c 3 9 3c + 2 1 7 2 ; 1) = ( c + ) (1 2(c) = 2 ( 5 6c 3 9 1(c) =

1(

15

3c + 2 ) 5 6c 3c + 2 ) 5 6c

s

Case 2: if p1 > 31 : second stage: use 2nd part of …rm 1’s best response function

p1 (p2 (p1 ))

s1 (4s1 (4 + 9c)s2 ) 3(s1 4s2 ) s2 ((8 3c)s1 ) (8 + 6c)s2 ) p2 = 3(s1 4s2 ) s2 (3c 4) Q1 = 3(s1 4s2 ) 2s2 (3c 4) Q2 = 3(s1 4s2 )

!

p1 =

3c)2 s1 s2 (s2 s1 ) 9(s1 4s2 )2 4(4 3c)2 s22 (s2 s1 ) e2 (s1 ; s2 ) = 9(s1 4s2 )2 e1 (s1 ; s2 ) =

(4

…rst stage: s1 ; s2 = ? di¤erentiation of pro…t functions w.r.t. si yields best responses for si

s1 (s2 ) = 74 s2 s2 (s1 ) no interior solution ! use Kuhn-Tucker ! s2 = 2 ! in equilibrium s2 = 2 and s1 = 87 !e1(c) = 3.

@

1

@c @

2

@c !

(4

3c)2 7(4 3c)2 and e2(c) = 216 216 (4

3c) 4 < 0 if c < 36 3 7(4 3c 4 < 0 if c < 36 3

= =

2 falls more strongly than

1

di¤erentiation becomes more costly for …rm 2 (s2 = 2) as c increases ! p2 has to increase and …rm 2 will loose consumers to …rm 1 ! pro…ts fall! as c increases, …rm 1 faces higher demand because consumers switch from 2 to 1; p1 will also increase, since …rm 1 adjusts its quality upwards (moving closer to …rm 2) ! in total, its pro…ts rise!

16

Extension: Will …rm 1 …nd it optimal to cover the whole market or does it choose to not serve all customers?

! as costs increase, the price per quality has to go up ! marginal consumers at the left end of the utility distribution have the lowest reservation price ! at some cost level, it is not useful to serve them anymore ! at a low cost level, …rm 1 …nds it optimal to o¤er a low quality in order to di¤erentiate from …rm 2 ! …rm 1 will capture whole market show that case 1:

1 >

2

1 2 1 7 2 ( c + )2 < ( c ) 3 9 3 9 2 2 c 4 4 c 14 49 + c+ < c+ 9 27 81 9 27 81 45 6 c < 9 81 5 c < 6 case 2:

2 >

1 always ful…lled!

Exercise 8 The quality-quantity trade-o¤ under vertical di¤ erentiation2 [included in 2nd edition of the book] Consider the vertical di¤erentiation model presented in Section 5.3. Suppose that the quality of the product can be described by some number si 2 [s; s] R+ . Consumers are identi…ed by 2 ; R+ , which measures their preference for quality. Consumers are distributed uniformly on ; and are of mass 2 This exercise draws from McCannon, B.C.. (2008). The Quality-Quantity Trade-o¤, Eastern Economic Journal 34, 95-100.

17

M=

. A consumer of type

receives a utility of

vi (p; y; ) = r

pi + si

when consuming a unit of good i (where r is supposed to be su¢ ciently large, so that all consumers buy in the market).

Two …rms compete in the market. We look for the subgame-perfect equilibria of the following two-stage game: …rms …rst choose the quality of their product and then compete in prices. Contrary to what was assumed in Section 5.3, we assume now that the marginal cost of production depends on quality. We denote

by C (qi ; si ) the cost of …rm i producing qi units at a quality si and we assume C (qi ; si ) = aqi si .

With a > 0, this formulation introduces a trade-o¤ between quality and quantity as the marginal cost of production, asi , increases with quality. That is, if the …rm increases one dimension (quality or quantity), the cost of providing the other dimension increases and the amount of this other dimension is thus reduced. An example of such an inverse relationship between quality and quantity can be found in the way an instructor teaches a course: as the number of students enrolled (i.e., quantity) increases, the cost for the instructor of providing a high-quality teaching increases (given the time available, the instructor’s ability to meet students outside of class, or to provide students with feedback on their assignments, inevitably decreases with the number of students enrolled).

To guarantee interior solutions in the pricing game, we assume > max 2

a;

+a 2

:

(A1)

1. Consider the second stage of the game where …rms set prices simultaneously, taking the qualities as given. Firm 1 produces quality s1 and …rm 2 produces quality s2 , with the convention that s1 < s2 . Derive the Nash equilibrium in prices and express the equilibrium quantities and pro…ts of the two …rms at stage 2.

2. Consider now the …rst stage of the game where …rms simultaneously choose the quality of their product. (a) Show that (s1 ; s2 ) = (s; s) or (s; s) are the equilibrium quality choices of the game. (b) What is the e¤ect of a stronger quality–quantity trade-o¤ (i.e., of a larger value of parameter a)? Discuss.

Solutions to Exercise 8

18

p1 + bs1 = r

1. The indi¤erent consumer satis…es r gives

p2 + bs2 . Solving for b

b = p2 p1 for b 2 ; : s2 s1 b Consumers of type > buy the high-quality product s2 whereas consumers b. of type < b buy the low-quality product. That is, q1 = b and q2 = For prices such that (s2 s1 ) p2 p 1 (s2 s1 ), the …rms’ pro…t functions are

1 (p1 ; p2 ; s1; s2 ) = (p1

as1 )

2 (p1 ; p2 ; s1; s2 ) = (p2

as2 )

p2 p1 s2 s1 p2 p1 s2 s1

; :

The system of …rst-order conditions of pro…t maximization (with respect to prices) is:

(

@ @p1 @ @p2

1 1 (p1 ; p2 ; s1; s2 ) = s2 s1 (p2 1 2 (p1 ; p2 ; s1; s2 ) = s2 s1 p1

(s2 2p1 + as1 2p2 + as2 + (s2

s1 )) = 0; s1 ) = 0:

The solution is

p1

=

p2

=

1 3 1 3

2 2

(s2

s1 ) + 31 a (2s1 + s2 )

(s2

s1 ) + 31 a (s1 + 2s2 )

We identify the indi¤erent consumer as

b = p2 s2

p1 = 31 s1

+ +a

and we check that it lies between and under assumption (A1). It follows that the equilibrium quantities are equal to

q1 = 13

2 + a and q2 = 31 2

a :

We can then compute equilibrium pro…ts at stage 2: 1 (s1; s2 ) 2 (s1; s2 )

= =

1 9 (s2 1 9 (s2

s1 ) s1 ) 2

2 +a a

2 2

; :

2. Quality choice (a) The answer follows from the fact that both crease with the quality di¤erence (s2 s1 ).

1 (s1; s2 ) and

2 (s1; s2 ) in-

(b) As a increases, the high-quality …rm sells less and the low-quality …rm sells more. This is because an increase in a disproportionately a¤ects the marginal cost of production of the high-quality producer. A stronger trade-o¤ also pushes up the price of both goods as the marginal cost of production is increased for both …rms.

19

Exercise 9 Examples of product di¤ erentiation Give …ve examples of product markets in which product di¤erentiation is likely to be a determining factor for competition in the market place. Give …ve examples in which the imperfections in competition are likely to be the result of factors di¤erent from product di¤erentiation.

Exercise 10 Price transparency and imperfect competition [included in 2nd edition of the book] Consider a Hotelling model in which two …rms are located at the opposite ends of the unit interval and serve a unit mass of consumers, who are uniformly distributed on this interval. Each consumer has unit demand and her utility if she buys from …rm 1, located at 0, is r x p1 , where x is the consumer’s location on the line, her utility if she buys from …rm 2, located at 1, is r (1 x) p2 . Her utility if she does not buy at all is 0. For simplicity, both …rms are assumed to have zero costs.

The two …rms compete by simultaneously setting their prices. Consumers >0 fall into two categories: at every point on the line, a fraction with of consumers observe p1 and p2 and then decide whether to buy from …rm 1, …rm 2, or not to buy at all (these consumers behave as in a standard Hotelling model). A fraction 1 of consumers at every point x, do not observe p1 and p2 (i.e., they are “uninformed” about prices). Instead, each uninformed consumer forms an expectation about p1 and p2 , and uses these expectations to choose whether to visit …rm 1, …rm 2, or none of the …rms.

Visiting one …rm is possible at zero costs, visiting both …rms is infeasible or prohibitively costy. If an uniformed consumer chooses to visit one of the two …rms, she learns its actual price, and then either buys from that …rm or does not buy at all. In equilibrium, the beliefs of uninformed consumers are correct.

For simplicity, assume that r is su¢ ciently high to ensure that the market is fully covered for all values of .

1. Solve for the equilibrium when …rms 1 and 2 choose p1 and p2 , respectively.

2. Let us interpret as “market transparency”: An increase in makes the market “more transparent”. What happens to prices and what happens to consumer surplus when the market becomes more transparent? What is the intuition for your answer?

3. Suppose that a policy maker maximizes total surplus as the sum of consumer surplus and pro…ts. Should the policy maker enforce high transparency or not? Explain the intuition for your answer.

4. Now suppose that consumers always observe …rm 2’s price, p2 , but, as before, only a fraction of consumers observe p1 while the others are uniformed and base their decision on their expectations regarding p1 , which are correct in equilibrium. Solve again for the Nash equilibrium. How does a¤ect the pro…t of …rm? Does it pay …rm 1 to have non-transparent prices? Provide an intuition for your result.

20

Solutions to Exercise 13 1. To solve for the Nash equilibrium, let us …rst determine the consumer who is indi¤erent between the two …rms. Given prices p1 and p2 , the location of the indi¤erent consumer satis…es

x + p1 = (1

x) + p2 :

Thus,

x=

1 2

p1

p2 2

:

If a consumer is informed observes the prices, then p1 and p2 are the actual prices. If a consumer is uninformed then p1 and p2 are the expected prices (which, in equilibrium, are equal to the actual prices but di¤er from the actual price if a …rm deviates from the equilibrium). The pro…ts of …rm i, i = 1; 2; is:

p1

1 2

i = pi

p2 2

+ (1

)

pe1

1 2

pe2 2

;

where j 2 f1; 2g and j 6= i and pei is the expected price of …rm i. The …rst-order condition for …rm 1’s problem is:

@ 1 = @p1

1 2

p1

p2 2

+ (1

)

1 2

pe1

pe2 2

p1 = 0: 2

In equilibrium, pe1 = p1 = p1 and pe2 = p2 = p2 . Moreover, by symmetry, p1 = p2 = p . Hence, the equilibrium price satis…es

1 p + = 0; 2 2 and, therefore, p =

.

2. It is easy to see that an increase in transparency, , leads to lower prices. The reason is simple: When some consumers are uninformed, the …rm does not lose them when it increases its actual price. This can be seen by looking at the second term in the …rst-order condition which represents the cost to a …rm from raising its price: the higher , the bigger the cost so the more reluctant the …rm is to raise its price. When is low the cost of raising the price is low, so the …rm raises it more. In equilibrium, both …rms have the same market shares so consumers on the interval [0; 1=2] buy from …rm 1 and those on [1=2; 1] buy from …rm 2, independent of . Hence, less transparency leads to lower consumer surplus.

3. From a welfare perspective, the market is covered so there is no deadweight loss.

The prices are then a wash (the …rms gain and consumers lose but by the same amount), and total surplus is not a¤ected by transparency. A policy maker who is only interested in total surplus should be indi¤erent to the degree of market transparency.

21

4. When only the price of …rm 1 may not be fully transparent, the pro…ts are given by: 1

= p1

2

= p2

1 p 1 p2 2 2 1 p 1 p2 + 2 2

+ (1

)

+ (1

)

1 pe1 p2 2 2 1 pe1 p2 + 2 2

; :

The …rst-order conditions for pro…t maximization are:

@ 1 @p1 @ 2 @p2

= =

1 p 1 p2 2 2 1 p 1 p2 + 2 2

+ (1

)

+ (1

)

1 pe1 p2 2 2 e 1 p1 p2 + 2 2

p1 = 0; 2 p2 = 0: 2

In equilibrium, pe1 = p1 = p1 and pe2 = p2 = p2 . Hence, …rst-order conditions can be rewritten as

1 2

p1 1 + 2

p2

2 p1

p1 2 p2 2

p2 2

=

0;

=

0:

Solving this systen gives equilibrium prices

p1

=

p2

=

3 ; 1+2 (2 + ) : 1+2

Notice that …rm 1, which lacks transparency, charges a higher price than …rm 2 ( < 1 and, thus, 3 > 2 + ). Hence, …rm 1 has a market share of less than 1/2. Substituting the prices in …rm 1’s pro…t, yields 1 =

2 ; 2(1 + 2 )2

which is the pro…t maximum. Di¤erentiating with respect to

gives

9(1 2 ) @ 1 = : @ 2(1 + 2 )3 Hence, …rm 1 bene…ts from having some non-transparency and the optimal degree of non transparency is = 1=2. Being non-transparent allows …rm 1 to be less aggressive and maintain high prices. Firm 2 responds with high prices as well (strategic complements) and, hence, …rm 1 bene…ts from facing a less aggressive rival. This is bene…cial for …rm 1 if < 1=2. When > 1=2, …rm 1 loses more from being too soft than it gains by making …rm 2 soft.

22

Exercise 11 Advertising intensity Consider the elasticities reported in the table below. The easiest way to think about the advertising elasticities is the following: Total demand consists of demand today and tomorrow. The short-run elasticity is the e¤ect that advertising today has on demand today whereas the long-run elasticity is the e¤ect that advertising today has on demand tomorrow. In which industries do you expect advertising intensity to be high? Distinguish between short run and long run.

Short-run Long-run Income Price advertising advertising elasticity elasticity elasticity elasticity Bakery products 0.7 0.3 0.2 0.3 Books 2.2 0.8 0.3 0.4 Drugs 0.7 1.1 0.7 1.0 Tobacco products 0.0 1.8 0.4 0.6 Exercise 12 Wasteful advertising Suppose that advertising expenditures are wasteful in the sense that they only redirect existing demand and do not increase consumer utility. Can such advertising be total surplus increasing? Explain.

Exercise 13 Surplus-increasing advertising in the Hotelling model Consider a horizontally di¤erentiated product market in which …rms are located at the extreme points of the unit interval. Firms produce at marginal costs equal to zero. A continuum of consumers of mass 1 are uniformly distributed on the unit interval. They have unit demand and have an outside utility of 1. A consumer located at x 2 [0; 1] obtains indirect utility v1 = r1 tx p1 if she buys one unit from …rm 1 and v2 = r2 t(1 x) p2 if she buys from …rm 2. Firms have marginal costs equal to zero.

1. Suppose that …rms have set prices at p1 and p2 respectively. Determine the demand function for each …rm for each admissible price pair (p1 ; p2 ).

2. Suppose that the social planner chooses …rst-best optimal prices. Which price pairs would be socially optimal.

3. Suppose that the two …rms simultaneously prices. Determine the market equilibrium for all possible combinations of (r1 ; r2 ).

4. From now on consider the special case that t = 1. Suppose that each …rm i can use advertising to increase the willingness to pay from ri = 1 to ri = 2. Consider the two-stage game in which …rms choose advertising at the …rst stage and price at the second stage. Characterize the subgameperfect Nash equilibrium of the game depending on the advertising cost A. Consider the cases A = 2=9, A = 3=9, and A = 4=9. What is the welfare ranking?

23

5. What are the equilibria for A = 5=18 and A = 7=18?

6. What are the welfare consequences of a reduction in advertising the advertising cost from A = 5=18 + " to A = 5=18 " for the limit where " ! 0 (determine whether total surplus increases or decreases and by how much)? Comment on your result in one sentence.

7. What are the welfare consequences of a reduction in advertising the advertising cost from A = 7=18 + " to A = 7=18 " for the limit where " ! 0 (determine whether total surplus increases or decreases and by how much)? Comment on your result in one sentence.

Solutions to Exercise 13 1. For prices such that demand is strictly positive for each …rm, demand of …rm 1 is

Q1 (p1 ; p2 ) =

1 (r1 + 2

r2 )

(p1 2t

p2 )

:

2. Maximize social surplus x br1 tb x2 =2 + (1 x b)r2 t(1 x b)2 =2 by choosing x b. Clearly, price equal marginal costs implement the …rst best. Also, any prices p1 = p2 implement the …rst best. The optimal (interior) allocation is characterized by

3. Firm 1 solves maxp1 p1

x bW =

1 (r1 r2 ) + : 2 2t

(r1 r2 ) (p1 p2 ) 1 2 + 2t

, …rm 2 solves maxp2 p2

1 2

(r1 r2 ) (p1 p2 ) 2t

First-order conditions can be written as best-response functions pi = t=2+(ri rj + pj )=2. In equilibrium, …rms set prices

pi = t + (ri

rj )=3:

Thus, from a social point of view the high-quality …rm i with ri > rj sets a too high price, whereas the low-quality …rm j sets a too low price. The equilibrium allocation is characterized by

x b =

1 (r1 r2 ) + 2 6t

such that too few consumers buy from the high-quality …rm. Equilibrium pro…ts are i =

1 2t

t+

ri

rj 3

2

:

4. Pro…ts in asymmetric advertising case: advertising …rm makes (1=2)(4=3)2 A = 8=9 A. no ad ad

no ad

ad

1=2; 1=2 8=9 A; 2=9

2=9; 8=9 A 1=2 A; 1=2 A 24

.

case A = 2=9: no ad ad

no ad

ad

1=2; 1=2 2=3; 2=9

2=9; 2=3 5=18; 5=18

equilibrium in which both …rms advertise. case A = 3=9: no ad ad

no ad

ad

1=2; 1=2 5=9; 2=9

2=9; 5=9 1=6; 1=6

two asymmetric equilibria in which one of the …rm advertises case A = 4=9: no ad ad

no ad

ad

1=2; 1=2 5=9; 2=9

2=9; 5=9 1=18; 1=18

equilibrium in which none of the …rm advertises. welfare with A = 2=9: T S = 2

1=4

4=9 = 47=36

welfare with A = 3=9: T S = 5=3

5=18

3=9 = 19=18

welfare with A = 4=9: T S = 1

1=4 = 3=4

5. case A = 5=18: no ad ad

no ad

ad

1=2; 1=2 11=18; 2=9

2=9; 11=18 2=9; 2=9

case A = 7=18: no ad ad

no ad

ad

1=2; 1=2 1=2; 2=9

2=9; 1=2 1=9; 1=9

6. welfare if one …rm advertises 5=3

5=18

5=18 = 20=18

welfare if both …rms advertise: 2

1=4

10=18 = 43=36

Discontinuous increase in total surplus at a marginal decrease of advertising costs at 5=18. Here increase of total consumer valuations for one third of the population + reduction in transport cost more than o¤set the advertising cost (the …rst of the two positive e¤ects alone already dominates).

7. welfare if none advertises 3=4 welfare if one …rm advertises: 5=3

5=18

7=18 = 1

"quality" e¤ect dominates transport cost and cost of advertising. Marginal decrease in advertising cost at 7=18 leads to a discontinuous upward jump of total surplus.

25

Exercise 14 Negative advertising and information disclosure [included in 2nd edition of the book] Consider the linear Hotelling duopoly in which each …rm produces a product with a …rm-speci…c undesirable ingredient at zero marginal costs. Suppose that, absent advertising, consumers are not aware of this ingredient. In this case a consumer of type x derives utility r tx p1 if she purchases product 1 and utility r t(1 x) p2 if she purchases product 2. If a consumer learns that product i has the undesirable ingredient utility is decreased by d. Suppose that parameter values are such that in the equilibria to be characterized below the market is fully covered. Firms set prices simultaneously.

1. Derive the equilibrium if …rms cannot inform consumers that their product contains the undesirable ingredient.

2. Suppose that, at an initial stage, …rm i simultaneously decide whether to inform consumers that its product contains an undesirable ingredient (suppose that such informative advertising is possibly costless). Characterize the equilibrium of the two-stage game.

3. Suppose now that, at an initial stage, …rms can simultaneously launch costly attack ads in which they reveal that their competitor’s product contains an undesirable ingredient. Characterize the equilibrium of the two-stage game depending on the advertising cost A. Are consumers better o¤ in this equilibrium compared to the solutions in (1) and (2). Explain your result.

4. Should attack ads be allowed in this setting?

Solutions to Exercise 14 1. …rms cannot inform consumers

! indi¤erent consumer: r

tx

p1

Firms 1

2

p1

= r

(1

x)t

p2 () x =

1 p2 p 1 + 2 2t

: t p2 1 p2 p 1 = p1 x = p1 ( + ) =) FOC: p1 = + 2 2t 2 2 1 p1 p2 t p1 = p2 (1 x) = p2 ( + ) =) FOC: p2 = + 2 2t 2 2 t = p2 = t and 1 = 2 = 2

2. two-stage game:

26

=) solve for SPNE by backward induction! stage 2: if consumers are uninformed, NE as in 1 if both …rms informed their consumers, indi¤erent consumer’s choice stays the same and if advertisement is costless, pro…ts stay the same as well if one …rm informs and the other does not? suppose …rm 1 informed:

r

d

tx

p1 = r

1 = p1 x = p1 ( 2 = p2 (1

t

(1

d

+

x)t

p2

p1

p2 2 2 t + d p1 =) FOC: p2 = + 2 2 d d ! p1 = t and p2 = t + 3 3 1 < if d > 0 2

)

2t 2t t + d p1 p2 x) = p2 ( + ) 2t 2t

1 x = (t 2t | {z

x>0,t> d 3

x = t2td + p22tp1

p2 ()

d ) 3}

=)

FOC: p1 =

!

1 =

1 (t 2t

t

d

+

d 2 ) and 3

2 =

1 d (t + )2 2t 3

=) if …rm 2 informs the consumers, then the results are reversed! stage 1: …rms decide simultaneously whether to inform or not Firm 1 / Firm 2 inform

: inform

inform t t 2;2 1 d 2 1 2t (t+ 3 ) ; 2t (t

d 2 3)

1 2t (t

: inform d 2 1 d 2 3 ) ; 2t (t+ 3 ) t t 2;2

Note: 1 (t 1. 2t < 2t

d 2 d 3 ) () t < 6

non-negative demand

1 2. 2t < 2t (t + d3 )2 , t >

! not possible since t > d3 in order to have

d 6 , if t > 0 this is always ful…lled!

! SPNE is {: inform, : inform} 3. stage 2: if no …rm launches a costly ad, see 1.

27

if both …rms launch attack ad (pro…t is reduced by d for both …rms):

! p1 = p2 = t and

1=

2=

t A 2

if one …rm (say …rm 1) chooses to launch an attack ad, …rm 2’s ingredient is revealed:

!

d d and p1 = t + 3 3 1 d 2 (t + ) A and 1 = 2t 3

p2 = t

2 =

1 (t 2t

d 2 ) 3

stage 1:

Firm 1 / Firm 2 attack ad : attack ad

attack ad 1 2t (t

t A; 2t A 2 d 2 1 d 2 3 ) ; 2t (t + 3 )

A

if A

d 3

d2 18t ! NE: {attack ad, attack ad}

if d3

d2 18t

A > d3

if A > d3

: attack ad 1 A; 2t (t t t 2; 2

1 d 2 2t (t + 3 )

d 2 3)

d2 18t ! 2 NE: {attack, : attack} and {: attack, attack}

d2 18t ! NE: {: attack ad, : attack ad}

=) if the ingredient has a high disutility, …rms want to use attack ads!

What about consumers?

1. symmetric equilibria ! consumers split at 21 and pay price of t ! e¢ cient 2. asymmetric equilibria are not optimal because indi¤erent consumer is not at 12 ! if we want consumers to be informed per se, we should encourage attack ads if consumers su¤er from d in any case, costs of attack ads should be avoided

Exercise 15 Informative advertising

28

It is not di¢ cult to navigate in Lonely-Line City: a single street runs from kilometer 0 to kilometer 1 along which 100 inhabitants are equidistantly distributed. [Approximate the consumer distribution by a continuum on [0; 1] with a mass of 100.] To keep the place residential the local government has decided that no shops are allowed within the city limits. As it happens, there exists one shop at each boundary of the city [one at point 0 and one at point 1].

Each morning each inhabitant drinks one liter of fresh milk. Assume that transporting one liter of milk costs t cents per kilometer (this is the disutility incurred by an inhabitant if he walks or the cost for the shop for delivery), each shop pays a wholesale price of c cents per liter.

1. Suppose that each shop i sells one liter at price pi at the shop and that all inhabitants get up each morning and walk to one of the two shops to get the milk. What is the price set by each of the two shops, what are the shops’ pro…ts? [Characterize the Nash equilibrium of the corresponding game!]

2. Suppose that instead of consumers walking to one of the shops, both shops have a delivery service and that shops set a price that depends on the address of the inhabitant who buys. What are the prices charged by the shops, what are the shops’ pro…ts? [Characterize the Nash equilibrium of the corresponding game in which shops simultaneously set prices pi !

Illustrate your analysis by a …gure!] Compare your …ndings to those in (1.).

3. Return to the situation in (1.) but suppose that shops sometimes do not have fresh milk available and that inhabitants only make the walk if they know that they get the milk for sure. Therefore, each shop can buy the right to use the city’s public speakers to advertise the availability of the milk. There is time for two ads. The inhabitants of Lonely-Line City, however, do not always pay attention to the ads. Each inhabitant listens to ad 1 with a 50% chance and to ad 2 also with a 50 % chance. Assume furthermore that for each inhabitant the probability to listen to ad 2 is independent of whether he or she has listened to ad 1. Consequently, there is a 25% chance that an inhabitant listens to ad 1 only, a 25% chance that an inhabitant listens to ad 2 only, a 25% chance that an inhabitant listens to ads 1 and 2, and a 25% chance that an inhabitant listens to none of the ads.

4. Consider a day at which both shops have milk available. Shops have the following two options: (a) they jointly announce the availability of milk in each ad, i.e., both ads contain information on both shops, (b) ad 1 contains information on shop 1, ad 2 contains information on shop 2.

Determine the equilibrium in each of the two cases (the advertising costs are assumed to be the same in each case). Which option do shops prefer?

[Characterize the equilibrium for options 1 and 2. In each case, you can assume that parameter constellations are such that …rst-order conditions 29

of pro…ts maximization characterize the equilibrium.] Provide an intuition for your result.

Exercise 16 Comparative advertising [included in 2nd edition of the book] Consider a Hotelling duopoly in which …rms are located at the extreme points of the unit interval and consumers of mass 1 are uniformly distributed on the unit interval. The price of the two products is given and equal to 1, p1 = p2 = 1 (e.g. because the price is …xed upstream); production costs are zero. The quality of product i is denoted by si 2 [2; 3]. The quality of each product is drawn independently from the uniform distribution on this interval [2; 3]. Both …rms observe the two qualities of the product, consumers do not observe the qualities. A consumer located at x 2 [0; 1] derives utility Es1 x p1 from product 1 and Es2 (1 x) p2 from product 2, where Esi is the expected quality of product i given the information available to consumers.

1. Suppose that …rms can simultaneously disclose their own quality si at zero cost. Consumers then decide which product to buy. Characterize the equilibrium of this game. Prove that it is the unique equilibrium.

2. Suppose now that disclosure is costly, i.e. a …rm has to spend a given advertising cost a 2 (0; 1=4] to disclose its own quality. Suppose that …rm i conditions its action on si only. Characterize the equilibrium of the game in which …rms …rst decide whether to advertise their own quality truthfully or not to disclose any information and then consumers make their choices. Note that …rms know the cost a and make their disclosure decisions simultaneously.

3. Suppose that instead of advertising their own quality, …rms can costly advertise only the quality di¤erence s1 s2 , i.e. …rms can only engage in comparative advertising. Characterize the equilibrium in which both …rms simultaneously decide whether to disclose the quality di¤erence at cost a 2 (0; 1=4].

4. Discuss verbally the welfare properties of the equilibria determined in (2) and (3).

5. Consider now a model in which …rms can choose not to advertise, to use non-comparative advertising or to use comparative advertising. In the last two cases the same advertising cost a applies. Provide verbally an intuition about the properties of the equilibrium of this game. To simplify the argument, consider an alternative setting in which there are only two discrete types si 2 f2; 3g.

Solutions to Exercise 163 3 This exercise is inspired by the paper by W. Emons and C. Fluet (2008), Non-comparative versus Comparative Advertising as a Quality Signal, mimeo.

30

1. Independent of sj , …rm i always discloses if si 2 (2; 3]. To see this, note that pro…t of …rm i is i = qi 1. Demand qi is determined as follows: The indi¤erent consumers satis…es Es1 x b 1 = Es2 (1 x b) 1. Hence, x b = 1=2 + (Es1 Es2 )=2 and q1 = x b, q2 = 1 x b. Thus i is increasing in Es1 . Consequently, it is a dominant strategy for any type si = 3 ", " su¢ ciently small, to disclose its quality. Suppose that all si 2 [e s; 3] for any se > 2 have disclosed. Then there exists an " > 0 such that all types with si 2 (e s "; se) prefer to disclose.

2. There exists a sb 2 [2; 3] such …rm i discloses for si sb and does not disclose for si sb. With disclosure the pro…t of the …rm of type sb is 1=2 + (b s Esj )=2 a.

If the …rm does not disclose, the expected quality of that …rm is 2 + (b s 2)=2.

Hence, this …rm’s pro…t is 1=2 + [2 + (b s 2)=2 Esj ]=2. The indi¤erent type satis…es s b 2a = 2 + (b s 2)=2, which is equivalent to sb=2 1 = 2a. Thus, sb = 4a + 2.

3. Clearly, only the …rm with the higher quality may have an incentive to advertise.

There exists a critical quality di¤erence db above which …rm i prefers to disclose the quality di¤erence for si > sj . By symmetry, there is a quality di¤erence db below which …rm j prefers to disclose the quality di¤erence. In an intermediate range none of the …rms advertises. Thus, if none of the …rms has advertised consumers expect that the quality di¤erence between the two …rms is on average zero. Hence, without disclosure the …rms’pro…ts are 1=2. If …rm i discloses its pro…t is 1=2 + (si sj )=2 a. Thus, the indi¤erent type db satis…es that

1=2 + (si

sj )=2

a = 1=2, which is equivalent to db = 2a.

4. From a social point of view, disclosure is socially bene…cial if qualities are suf…ciently asymmetric (not that the is full participation). When both …rms advertise (which may happen under non-comparative advertising) this advertising tends to be socially wasteful as the two qualities are rather similar. By contrast, under comparative advertising no such socially wasteful advertising occurs.

5. If qualities are strongly asymmetric the high-quality …rm has an incentive to reveal this quality di¤erence. This means that non-comparative can only be an equilibrium strategy for …rms that are not very asymmetric. However, consumers know that whenever they do not observe comparative advertising, quality di¤erences cannot be very pronounced. This suggests that non-comparative advertising may not be an equilibrium strategy for any type. This is clearly seen in case of binary types si 2 f2; 3g.

Exercise 17 Price dispersion A study by Brynjolfsson and Smith on retail price for books and CDs …nds that price dispersion (weighted by market shares) is lower for internet retailers than for conventional retailers. Discuss.

Exercise 18 Spatial price dispersion

31

Two …rms (1 and 2) produce a homogeneous good at zero marginal cost.

They face two types of consumers: a mass N of consumers are informed about the prices, p1 and p2 of the two …rms and therefore buy from the cheapest …rm; a mass M of consumers are uninformed in the sense that they only know the price of one …rm and therefore have a demand only for this …rm. We assume that M = M1 + M2 , where Mi is the mass of consumers who can only observe pi and where M2 > M1 . All consumers have an inelastic demand: they buy at most one unit of the good, as long as the price is not larger than their reservation price R > 0.

1. Suppose that …rm 1 sets its price before …rm 2. Characterize the subgameperfect equilibrium of this two-stage game.

2. Repeat the previous question by supposing instead that it is …rm 2 that sets its price …rst.

3. Show that …rm 1 has a second-mover advantage, while …rm 2 is indi¤erent between playing …rst or second.

Solutions to Exercise 18 1. We start by solving the second-stage for any price pi 2 [0; R] set by …rm 1. Firm 2 has two options. Either, it focuses on its ‘captive’consumers and sets p2 = R to achieve a pro…t of 2 = M2 R, or it undercuts …rm 1 by setting p2 just below p1 , thereby obtaining a pro…t almost equal to 2 = (M2 + N ) p1 . The …rst option dominates as long as p1 < (M2 R) = (M2 + N ) p1 . We can now look at period 1. Anticipating …rm 2’s conduct, …rm 1 faces two option. Either it sets the largest possible price so that …rm 2 does not …nd pro…table to undercut and attracts the informed consumers: p1 = p1 and 1 = (M1 + N ) p1 , or it accepts …rm 2’s undercutting and focuses on its ‘captive’ consumers: p1 = R and 1 = M1 R. The …rst option is more pro…table if

(M1 + N ) p1 = R > M1 R , M2 > M1 ; which is true according to our initial assumption. Hence, the subgame-perfect equilibrium is such that p1 = M2 R= (M2 + N ) and p2 = R, yielding pro…ts F equal to L 1 = M2 (M1 + N ) R= (M2 + N ) and 2 = M2 R (with L and F standing, respectively for leader and follower).

2. The analysis of the second stage remains the same. As for stage 1, because M1 < M2 , …rm 2 prefers to set p2 = R and have L 2 = M2 R. This allows …rm 1 to undercut by setting a price just below R and achieve a pro…t almost equal to F 1 = (M1 + N ) R. 3. We see that

L 2 =

F 2 , whereas

F 1 = (M1 + N ) R >

32

L M2 1 = M2 +N (M1 + N ) R:

Exercise 19 Another model of sales [included in 2nd edition of the book] Consider a market for a homogenous product with n identical price-setting stores, where n is determined by free entry. Each store has a cost function p C(q) = q, where q is the number of customers the store serves. There are M + 15 consumers in the market, each of whom wishes to buy up to one unit and is willing to pay for it up to r = 1. The number of 15 consumers know the prices charged by all the stores in the market (i.e., have zero search costs), while M consumers do not know the prices at all (i.e., have prohibitively high search costs). Of the latter M=n visit store i and none of the other stores, i = 1; :::; n.

1. Prove that there cannot exist a symmetric pure-strategy equilibrium in this market.

2. Suppose that all the stores in the market use the same mixed strategy.

What is the support of the mixed strategy as a function of n?

3. Write the pro…t of a store when it charges p = 1 (hint: what is the probability that when a store charges p = 1 it will have the lowest price in the market?). Prove that this pro…t is zero. Use the zero pro…t condition to compute the equilibrium number of …rms, n . Given n , write the support of the equilibrium mixed strategy of prices.

4. Compute the pro…ts of a store when it happens to be charging the lowest price in the market and when it does not. Using these expressions, compute the equilibrium distribution of prices at each store, F (p).

5. What happens to the distribution of prices when the number of uninformed consumers, M , increases? What does this result mean for the uninformed consumers? Give an intuition for this.

6. What happens to the distribution of price paid by informed consumers when the number of uninformed consumers, M , increases? Provide an intuition for this result.

Solutions to Exercise 19 1. First, if n = 1 then the store will maximize pro…ts by setting p = 1 and will earn pro…t (M + 15) (M + 15)1=2 > 0. This will induce entry. Hence, there cannot exist a pure-strategy equilibrium with n = 1. Now suppose that there exists a pure strategy equilibrium with n 2 stores. Each store serves q = (M + 15)=n customers and its average cost is (1=q )1=2 . Since there is free entry, each store earns zero pro…ts. Hence, p = (1=q )1=2 . But if a store charges p , it gains all 15 informed customers, while its revenue per customer falls only by . This deviation upsets the putative pure-strategy equilibrium.

33

2. The largest number of customers one store may serve is qh = 15 + M=n. Hence the lowest average cost a store may have is (1=qh)1=2 . Thus no store will ever charge a price below (1=qh )1=2 . Similarly, no store will ever charge more than 1, otherwise it will have no customers. Propositions 6-8 in Varian establish that the support of the distribution of prices, F (p), is [(1=qh )1=2 ; 1] (the support depends on n since qh = 15 + M=n). See also the book.

3. When a store charges p = 1, it almost surely does not charge the lowest price in the market since stores use mixed strategies with no mass points so the probability that all stores charge p = 1 is zero. Hence the store serves only M=n uninformed customers (recall that uninformed consumers split evenly between all n stores) so its expected pro…t (the expectation is with respect to the mixed strategies of other stores) is

M (1) = n

r

M : n

Since there is free entry, each store makes zero expected pro…ts overall (here the expectation is w.r.t the mixed strategies of other stores and w.r.t. to the mixed strategy of the store itself). But in order to use a mixed strategy, each store must be indi¤erent between all prices in the support of F (p); hence, the expected pro…t at every price in the support must be zero. In particularly, (1) = 0. Solving for n yields, n = M . That is, there are as many stores as the number of uninformed consumers. Given n , qh = 16, so the support of the equilibrium distribution of prices, F (p), is [1=4; 1].

4. When a store charges the lowest price in the market, it attracts 16 customers (all 15 informedpconsumers plus 1 uninformed consumer). The store’s pro…t is 16 = 16p 4:When a store does not charge the lowest price s (p) = 16p in the market, it attracts only M=n = 1 uninformed consumers. The store’s pro…t is p f (p) = p

1=p

1:

We can calculate the equilibrium price distribution: p 1 3 15p

F (p) = 1

1 M 1

:

5. As M increases, F (p) decreases, so each store is more likely to charge high prices. This means that the more uninformed consumers in the market, the less aggressive is the pricing strategy of stores. This result is intuitive because when there are more uninformed consumers, stores will be more tempted to take advantage of them by charging high prices. The cost of charging high prices is that a store is more likely to lose informed consumers who go to the store that charges the lowest price. But when there are more uninformed consumers around, the relative importance of serving informed consumers is declining.

34

6. We label …rm such that prices are in an increasing order, p1 < p2 < ::: < pn .

Informed consumers pay the lowest price on the market, i.e. p1 . The distribution of p1 is given by 1 (1 F (p1 ))n : Since F (p) = F (p) and n = M , the equilibrium probability of p1 is given by

1

p 3

1 15p

M M 1

:

As M increases, F1 (p) decreases, so the lowest price is more likely to be higher than before. The reason for this result is as follows: As M increases, stores charge higher prices than before but there are more of them in the market. The …rst e¤ect tends to lower F1 (p) while the second tends to increase F1 (p). In other words, since there are more stores, it is more likely that one of them will charge a low price but even this store charges a higher price than before. In the current example, it turns out that the …rst e¤ect dominates. (Note that this is in contrast to the example in Varian’s article where the second e¤ect is stronger.)

Exercise 20 Yet another model of sales Suppose that two …rms with constant marginal costs compete in prices in a homogeneous product market. All consumers have unit demand and a willingness-to-pay r. A share of consumers is informed about the prices in the market. The share (1 )=2 goes to …rm i = 1; 2 and decides whether to buy (these consumers do not know that a product from …rm j 6= i exists). Firms set prices and then consumers make their consumption decisions.

1. Show that there does not exist a symmetric Nash equilibrium in pure strategies.

2. Characterize equilibrium prices in the unique mixed-strategy Nash equilibrium.

3. How do prices change if

is increased?

4. How do equilibrium pro…ts change as

increases? (Calculate @

=@ .)

Solutions to Exercise 20 This is a variation of the Varian-model, see, e.g., Schultz, European Economic Review (2005).

Exercise 21 Bargains and ripo¤ s Consider a market for a homogenous product with n identical stores, where n is determined by free entry. Each store has a cost function C(q) = 4 + q, for q 4 and c(q) = 1 for q > 4 (in other words, each store can sell up to 4 units 35

and its cost of selling the …rst q units is 4 + q). There are L consumers in the market, each of whom wishes to buy up to 1 unit and is willing to pay for it up to r = 5. Suppose that a fraction of all the consumers is fully informed about the prices that the di¤erent stores charge. The remaining (1 )L consumers are uninformed and have to pay a cost z in order to learn the prices that di¤erent stores charge. If an uninformed consumer does not pay z, she knows only the distribution of prices but not the actual prices charged by each store. Such a consumer then picks a store at random. However, once an uninformed consumer pays z, she becomes completely informed and knows all prices charged by all stores.

1. Compute the marginal and average costs of stores and illustrate it in a …gure.

2. Suppose that z = 0. Solve for the long-run competitive equilibrium in the market.

3. Now suppose that z > 0. Prove that there can be at most 2 prices in a Nash equilibrium.

4. Assume that there are two prices being charged in equilibrium. What is the low price, pl ? Given your answer, compute the high price, ph (hint: assume that a fraction of all stores charge pl and a fraction 1 charge ph and use the condition that ensures that uninformed consumers do not …nd it worthwhile to search).

5. Compute the demand faced by low and high price stores (note that uninformed consumers pick stores at random so each store gets an equal share of the (1 )L uninformed consumers; informed customers are indi¤erent among all stores that charge low prices, so each one of these stores gets an equal share of the L informed consumers).

6. Use your answers in (4) and (5) to express the zero pro…t conditions for high and low price stores (recall that there is a free entry so in equilibrium, each store must earn a zero pro…t).

7. Solve the conditions you wrote in (6) for 8. How do the equilibrium values of intuition for your result.

and n.

and n vary with z? Explain the

9. Compute the average price on the market and the standard deviation of prices. Using these calculations, let P D = SD=AP be a measure of price dispersion, where SD is the standard deviation of prices, and AP is the average price. How is P D a¤ected by s? How is P D a¤ected by ? Are these results intuitive? Explain.

36

Solutions to Exercise 214 1.

( 1 if q 4 M C = C (q) = 0 if q > 4 ( 4+q if q 4 q AC(q) = 1 if q > 4 0

2. z = 0 ) all consumers can become informed at zero cost ! …rms can only set one price ! price must be such that no more entry occurs, i.e. all …rms must earn zero pro…ts ! monopolistic competition = pq

C(q) = 0 , p = AC(q)

! demand for each store is equal to L n ! all stores sell q = 4 units at p = 2 and there will be n = L4 stores in the market (abstracting from integer problems)

3. ! equilibrium description: A price vector p = fp1 ; :::; pn g, a number n of …rms in the market and a percentage

of consumers that gather information that full…ll

4 The exercise is based on the "Bargains and Ripo¤s" model of Salop and Stiglitz (RES, 1977, Vol. 44, 493-510).

37

(a) pro…t maximization: each …rm chooses a price to maximize pro…ts given prices of the other …rms and the search strategy of consumers

(b) zero pro…ts: the maximized value of pro…ts for every …rm equals zero ! an equilibrium is characterized by n …rms charging identical or di¤erent prices and each producing and selling enough output to place them on the downward sloping part of their common AC-Kurve and with enough …rms to serve every customer

(c) search equilibrium: in equlibrium, consumers gather information optimally

=

(

for z p 1 for z < p

pmin pmin

) Proof by contradiction that there cannot be more than 2 prices: Suppose there are 3 prices pl < pm < ph (see …gure below)

Note that all those prices lie between [2; 5] ! prices must be equal to AC to guarantee that = 0 ! all L informed consumers buy from pl …rms ! if the (1 )L uninformed consumers do not inform themselves (if they would, we can only have one price), then each …rm obtains a n1 share of those (1 )L consumers

38

) the sales of pm and ph …rms will be identical ) ph must therefore earn a positive pro…t as ph > pl , a contradiction to

the zero pro…t condition!

4. two prices charged in equilibrium: pl ; ph ) uninformed consumers stay uninformed and choose randomly )(ph pl ) ) 5 p > 5 pl z , z p pl , z (1 ! the lower price will be equal to the one in question 2.: pl = 2 and ph = 4 (Note that setting p > 2 will make it pro…table to deviate and capture all of L consumers.) ) the higher price must be low enough to avoid that uninformed consumers search: ! assume a fraction (1

) of stores charges the high price: z (1 )ph + pl pl , ph pl + 1 z ) high price is given by ph = min[r; pl + 1 z ]

(Note: we assume that a consumer that is indi¤erent will choose not to search)

5. ph -…rms: qh = (1 )L n pl -…rms (there are of them): L ql = + n |{z} share of inform ed consum ers

6.

L (1 ) | {z n}

=L n (1

+ )

share of uninform ed consum ers

C(ql ) , 2 = 4+q l = 0 , pl = ql q + ) 4+ L n (1 2 = L (1 + ) n h) = 0 , ph = C(q , 2 + 1z h qh

=

4+(1 )L n (1 )L n

7. solve the two equations stated in 6. for , = +z(1 ) and n = L4 (1 + z(1

and n:

))

(1 ) 8. @@z = ( +z(1 ))2 < 0 ! as it becomes harder to inform, i.e. as z ", the fraction of …rms that demand

a low price falls. If information cost lare low, there will be only a small share of …rms that can charge high prices in equilibrium. @n L @z = 4 (1

)>0 ! as costs of information increase, there willl be more …rms that charge a higher price ! since they have to make zero pro…ts, they sell less. In consequence, there have to be more …rms in the market to serve all customers L.

39

9.

AP

=

AP

= p=2+z p = (pl p)2 + (1

SD PD @P D @z @P D @

= =

=

pl + (1

SD = AP

(2

r

s

z)

q

2 (1

)ph =

(1 z (1

2+

+ z(1

)

)(ph

p)2 =

z )(2 + z)2 )

2z(2 + z)2

r

z(1 ) + z(1 )

2+

z( + z(1 )) z(1 )

z (1

)

> 0 if z < 2

z (1

)

)(2 + z)

>0

! as the share of informed consumers increases, the price dispersion becomes larger. This is because the number of stores setting low prices (pl = 2) increases, i.e. @@ > 0. In consequence, the bene…t from searching for the informed consumers, (1 )(ph pl ), which must be smaller than their search costs, falls and there is scope for high price …rms to raise ph . In sum, the average price stays constant but ph increases, leading to an increase in PD. ! when z increases, this has an e¤ect on the incentive of information gathering:(1 )(ph

pl )

z

(a) the cost increase introduces a slack in the incentive ! …rms raise ph ) p "

(b) as ph increases, qh has to fall because of

falls and the h = 0, therefore share of high price shops increases ! the standard deviation grows ! the positive increase in ph becomes smaller as (1 ) grows ! at some point (z = 2), the SD grows less than the AP and the measure of price dispersion falls as a function of z

Exercise 22 Switching costs and competition Switching costs relax competition and their presence are therefore pro…tenhancing. Is this statement necessarily correct? Explain.

Exercise 23 Switching costs and pricing strategy Suppose that two software companies launch a new software each. One of them is called COOL, the other GREAT. There is a unit mass of consumers.

All of them consider the two software o¤ers as identical. Both softwares are produced at zero marginal costs and consumers are willing to pay r for the software.

40

1. Suppose that …rms set prices and compete only in one period. Characterize equilibrium prices, allocation and pro…t. (If a group of consumers is indi¤erent suppose that half of them buys software COOL and the other half GREAT.) 2. Suppose that each …rm sold to half of the consumers their software and that they launch new products COOL2 and GREAT2. Consumers are willing to pay r for the new products. Suppose, however, that consumers who buy the product from a di¤erent …rm than in the …rst period incur a disutility . Suppose …rms set prices. Characterize equilibrium prices, allocation and pro…t. Does an equilibrium (in pure strategies) always exist? Discuss.

3. Consider now the market environment in which the …rm that produces COOL in period 1 and COOL 2 in period 2 is aware of the fact that it will launch COOL2 in period 2. Does this a¤ect its incentive in period 1? In particular, does it have an incentive to deviate from the price calculated in (1)? (NOTE: Suppose that there is no discounting.)

4. Suppose now that consumers who bought COOL will not consider buying GREAT2 in period 2 and that consumers who bought GREAT will not consider buying COOL2 in period 2. Characterize the subgame perfect equilibrium in the two-period model. (Again, suppose that there is no discounting.)

5. Provide some real-world examples that have some similar features as the theoretical market described in part (4). Explain in up to …ve sentences the general economic principles at work in markets such as the one describend in (4).

6. Consider the market environment as described in (4). Suppose the courts rule that prices cannot be set below marginal costs. Analyze the e¤ects of such a policy on consumer surplus, pro…ts, and welfare.

41

Industrial Organization: Markets and Strategies Paul Belle‡amme and Martin Peitz published by Cambridge University Press

Part IV. Pricing strategies and market segmentation Exercises Exercise 1 Geographical pricing [included in 2nd edition of the book] "Purple Dream"has the monopoly on the production of purple light-emitting diodes (LEDs). It faces geographically separated markets, market 1 and 2. The demands are qA = 1 pA and qB = 1=2 pB , respectively. The transport and production costs are set to zero.

1. Assume that the …rm chooses to set a uniform price across the two markets.

What is the pro…t maximizing uniform price? What are the quantities sold on the two markets at this price?

2. Assume that the …rm uses third-degree price discrimination. What are the pro…t maximizing prices and quantities on the two markets?

3. Calculate consumer surplus and pro…t under a uniform price and under third-degree price discrimination. Compare the two situations and comment on the result.

4. Does the result from question 3 hold generally? How would the results change if qB = 1=3 pB ?

Solutions to Exercise 1 1. ‘Purple Dream’ has the monopoly on the production of purple light-emitting diodes (LEDs). It faces geographically separated markets, noted A and B . The demands on these two markets are respectively given by qA = 1 pA and qB = 1=2 pB . The transport and production costs are set to zero.

2. The …rm chooses a uniform price p to maximize = p (qA (p) + qB (p)) = p (3=2 2p). The pro…t-maximizing price is easily found as pu = 3=8. The quantities sold are qA (3=8) = 5=8 and qB (3=8) = 1=8.

3. Here, the …rm chooses pA to maximize A = pA (1 pA ) and pB to maximize pB ). The optimal prices are found from the …rst-order condiB = pB (1=2 tions, respectively as pA = 1=2 and pB = 1=4; the corresponding quantities are qA (1=2) = 1=2 and qB (1=4) = 1=4.

4. Pro…ts under uniform pricing and under third-degree price discrimination are respectively equal to uni

=

3rd

=

3 5 1 9 8 8 + 8 = 32 ; 11 11 10 2 2 + 4 4 = 32 ;

1

which clearly shows that the …rm is better o¤ under price discrimination. As for 2 consumer surplus, it is computed as CSA = (1=2) (1 p) on market A and 2 as CSB = (1=2) (1=2 p) for a given price p. We check thus that, on both markets, consumer surplus is a decreasing function of price. As pA > pu > pB , we see that consumers on market A (resp. B ) are worse (resp. better) o¤ under price discrimination than under uniform pricing. As for total consumer surplus, we see that uniform pricing is globally preferred:

CS uni = 21 1

3 2 1 8 +2

1 2

3rd 3 2 = 13 = 12 8 64 > CS

1

1 2 1 2 +2

1 2

1 2 = 10 4 64

5. If qB = 1=3

pB , then pu = 1=3, while pB = 1=6. Here, the optimum for the monopolist is to sell only on market A. We still have that consumers on market A prefer uniform pricing whereas consumers on market B prefer price discrimination. Although consumers on market B make no surplus under uniform pricing, it is still the case that uniform pricing is preferred globally: CS uni = 12 1

1 2 1 3 +2

1 3

8 1 2 = 36 > CS 3rd = 12 3

1

1 2 1 2 +2

1 3

5 1 2 = 36 6

Exercise 2 Price discrimination and pharmaceuticals Do you think that price discrimination between rich and poor countries is a feasible solution for giving poor developing countries better access to patented pharmaceuticals? In the answer you should draw upon the theory covered in the book.

Exercise 3 Spatial price discrimination A monopoly faces a continuum of consumers who are distributed uniformly on the unit interval (i.e., the "numberöf consumers in each given interval is the same). The total mass of consumers is 1. Each consumer is interested in buying at most one unit. Consumers di¤er in the way they perceive the monopoly’s output. Assuming for simplicity that the monopoly is located at point 0, the utility of a consumer who is located at some point x between 0 and 1 if he buys from the monopoly is r p tx2 , where p is the monopoly’s price, r > 0, and t > 0 is the transportation cost per-unit of distance. If a consumer does not buy, his utility is 0. The monopoly’s per-unit cost of production is c.

1. Given p, …nd the location (i.e., äddress") of the consumer who is just indi¤erent between buying and not buying. Show this consumer on a graph.

2. How high can p be such that the market will be still covered (i.e., every consumer will buy)?

3. Write the monopoly pro…t as a function of p (hint: distinguish the case where the market is covered from the case where the market is uncovered).

2

4. Show that if t < (r c)=3, then at the pro…t maximum, the monopoly will choose a price that ensures that the market is covered. (Hint: this part might be a little di¢ cult: essentially you need to show that if t < (r c)=3, then raising p slightly above the point at which the market is just covered leads to a lower pro…t).

5. Suppose that the monopoly incurs a …xed cost F whenever it opens a plant. Compute the monopoly’s maximal pro…t given the assumption that the market is fully covered by the monopolist and that the monopolist has a single plant at point 0.

6. Now suppose that the monopoly opens a second plant at point 1. The utility for the consumer who buys at this plant is r p2 t(1 x)2 , where p2 is the price of the product at the second plant. Compute the monopoly’s maximal pro…t in this case under the assumption that the market is covered (note that now the monopoly bears a …xed cost of 2F since it operates two plants).

7. Based on your answers in (5) and (6), compute the range of F for which the monopoly will operate two plants, one plant, or no plants (i.e., will exit the market altogether).

8. Explain the intuition for your result in (7). In particular, explain why the monopoly might bene…t from opening a second plant at point 1.

Solutions to Exercise 3 1. Given p, the “address” x ^ of the consumer who is just indi¤erent between buying and not buying is given by r p t^ x2 = 0. Hence, r r p : x ^= t 2. For the market to be covered it must be the case that x ^

1. This ensures that every consumer gets a positive utility from buying. From the above equation it is easy to see that this happens when p r t.

3. The monopoly pro…t as a function of p is given by (p) =

x ^(p c) if x ^ 1; p c if x ^ > 1:

The …rst line describes the monopoly pro…t if p is high such that the market is not covered. The second line describes the monopoly pro…t if p is low such that the market is covered.

4. First, as long as x ^ > 1, the monopolist can raise its price without losing any customers. Hence, among all prices for which x ^ 1, the best price from the monopoly’s perspective is the one at which x ^ = 1. From (2) we already know 3

that this price is p = r t. We now want to derive a condition under which the monopolist will not wish to raise p above that level (if it does the market will not be covered). To establish the condition that ensures that it is never optimal to raise p above r t, note …rst that (p) is continuous in p including at p = r t. Second, note that (p) is concave in p. Hence, the monopolist surely does not raise p above r t if the

d (p) < 0: p#r t dp lim

This condition is equivalent to t < (r c)=3. Hence, the monopolist will set a price equal to r t if and only if t < (r c)=3. This implies that the market is fully covered.

5. Assuming that the market is fully covered, the monopoly price is determined by solving the equation x ^ = 1 to obtain p = r t. The monopoly pro…t then is 1

=r

t

c

F:

6. If the monopolist opens a second plant at point 1, then its pro…t-maximizing prices must be such that x ^ = 1=2: The plant at 0 covers the left hand side of the market while the plant at 1 covers the right hand side of the market.

Pro…t-maximizing prices are p = r t=4. Hence, the monopoly pro…t in this case is 2

7. Comparing

1

and

2

=r

t=4

c

2F:

reveals that if F < 3t=4, the monopolist will operate two

plants.

If 3t=4 < F < r c t, the monopolist will operate one plant, and if F > r c t, the monopolist will exit the market.

If F < 3t=4, the monopolist opens two plants because by doing so it can better segment the market and raise prices. The better segmentation of the market is re‡ected by the fact that the monopolist can o¤er consumers products that better suit their tastes and involve a smaller loss of utility from having to compromise on what they like (i.e., incur “transportation costs”). This price increase more than compensates the monopolist for having to incur the …xed cost F associated with the new plant. If F > 3t=4 the price increase associated with opening a second plant does not compensate the monopolist for the cost of opening a second plant. In this range, it is optimal to open only a single plant.

Exercise 4 Uniform vs. local pricing for a chain-store1 [included in 2nd edition of the book] 1 This exercise draws from Dobson, P.W. and Waterson, M. (2005). Chain-Store pricing Across Local Markets, Journal of Economics and Management Strategy 14, 93-120.

4

Consider a country that can be divided into two distinct markets of di¤erent sizes: market 1 is small (in the sense that it can only accommodate one …rm), while market 2 is larger (in the sense that it can accommodate two …rms). Two …rms are active in the country: …rm A is a national company that is active on both markets; …rm B is a local company and is only active on market 2. Demand conditions on the two markets are as follows. On market 1 (where only …rm A is active), inverse demand is given by qA1 = a

pA1 :

On market 2 (where both …rms are active and their products are seen as imperfect substitutes by the consumers), the system of inverse demands is qA2 = 32 (1620 qB2 = 23 (1620

2pA2 + pB2 ) 2pB2 + pA2 ) ;

where qKi (resp. pKi ) is the quantity demanded to (resp. the price set by) …rm K in market i (K = A; B and i = 1; 2). To translate the fact that market 2 is larger than market 1, it is assumed that a < 1620. Both …rms produce at a constant marginal cost, which is assumed to be equal to zero for simplicity.

1. Local pricing. Suppose that the national …rm (…rm A) chooses to adapt its prices to the local market conditions. Firm A has thus two choice variables: pA1 and pA2 . As for the local …rm (…rm B), it has, by de…nition only one choice variable: pB2 . Find the equilibrium prices of the two …rms and then compute their equilibrium pro…ts.

2. National pricing. Suppose now that …rm A commits to set the same price in the two local markets. Denote this price by pA . As for …rm B, nothing changes: it still sets its single price pB2 . Find the equilibrium prices of the two …rms and then compute their equilibrium pro…ts.

3. Compare your answers to questions 1 and 2 by taking three speci…c values for the parameter a, namely a = 540, a = 1188, and a = 1260. In which scenario(s) does …rm A prefer national pricing over local pricing? Explain the intuition behind your results. In which scenario(s) does …rm B prefer that …rm A sets the same price in the two markets (national pricing)?

Explain the intuition behind your results.

Solutions to Exercise 4 1. Local pricing. On market 1, …rm A’s pro…t is given by pA1 (a pA1 ); the optimal price is thus pA1 = a=2. On market 2, …rm A’s pro…t is given by pA2 32 (1620 2pA2 + pB2 ); solving the …rst-order condtion, we derive …rm A’s reaction function: pA2 = 14 (1620 + pB2 ). We proceed in a similar way for …rm B on market 2: …rm B ’s reaction function is pB2 = 41 (1620 + pA2 ). The Nash

5

equilibrium on market 2 is symmetric: pA2 = pB2 = p2 = 540. Equilibrium pro…ts are then computed as loc A loc B

2 a2 2 540) = a4 + 388 800; 4 + 3 540 (1620 2 540) = 388 800: 3 540 (1620

= =

2. National pricing. Firm A chooses pA to maximize its joint pro…t on the two markets; that is, …rm A’s problem is max pA (a pA

FOC:

(a

pA ) + pA 23 (1620

2pA ) + 23 (1620

2pA + pB2 ) ;

4pA + pB2 ) = 0

1 (3240 + 3a + 2pB2 ). As which yields the following reaction function: pA = 14 for …rm B , it still reacts in the same to …rm A’s price: pB2 = 41 (1620 + pA ).

Solving the system of two reaction functions for the two prices yields pA = 2 1 9 a + 300 and pB2 = 18 a + 480. On e can then compute the equilibrium pro…ts under national pricing: 2 nat 28 A = 243 (a + 1350)

and

2 nat 1 B = 243 (a + 8640) .

3. Comparisons a = 540 a = 1188 a = 1260

ploc A1 270 594 630

ploc 2 540 540 540

pnat A 420 564 580

pnat B2 510 546 550

loc A

nat A

loc B

nat B

461 700 741 636 785 700

411 600 742 224 784933

388 800 388 800 388 800

346 800 397 488 403333

Firm A prefers to commit to a uniform national price rather than setting two di¤erent local prices for intermediate values of a (here, a = 1188), i.e. when market 1 is neither too small nor too large with respect to market 2. Why? Note …rst that committing to a national price necessarily reduces …rm A’s pro…ts on market 1 as the uniform price di¤ers from the monopoly price on this market.

It follows that national pricing can only be bene…cial if the reduction in pro…ts on market 1 is compensated by an increase in pro…ts on market 2. How can national pricing raise …rm A’s pro…ts on market 2? If the uniform price is larger than the price …rm A would set if it priced locally, then it is pro…table for …rm A to commit to this larger price as it relaxes price competition. Indeed, because prices are strategic complements, committing to a higher price leads …rm B to raise its price too, which increases …rm A’s pro…ts. This is true for a = 1188 or a = 1260, but not for a = 540. Yet, as market 1 grows in size, the losses on market 1 become relatively more important than the gains on market 2. This explains why national pricing is the preferred strategy for a = 1188 but not for a = 1260. Firm B prefers that …rm A prices nationally when market 1 (i.e., the market on which it does not operate) is large enough (i.e., for a = 1188 or a = 1260). As we have seen above, for these values of a, national pricing drives …rm A to raise its price on market 2 (with respect to local pricing), which softens price competition and increases the pro…ts of both …rms on market 2 (which is the only market that matters for …rm B ).

6

Exercise 5 Spatial market segmentation [part 2 and 3 of the exercise are included in the 2nd edition of the book] Consider a horizontally di¤erentiated product market in which two …rms are located at points l1 = 0 and l2 = 1 on the line. Firms produce at marginal costs c. There is a continuum of consumers of mass 1 who are uniformly distributed on the unit interval. They have unit demand and have an outside utility of 1.

A consumer located at x 2 [0; 1] obtains indirect utility v1 = r (x)2 p1 if 2 she buys one unit from …rm 1 and v2 = r (1 x) p2 if she buys from …rm 2. Firms have marginal costs equal to c.

1. Suppose that …rms simultaneously set a uniform price for all consumers.

Characterize the equilibrium of the game. Determine equilibrium pro…ts.

2. Suppose now that …rms can price-discriminate between consumers located on [0; 1=2] (segment A) and [1=2; 1] (segment B). Determine the pro…t function of each …rm. Characterize the pure-strategy Nash equilibrium of the game in which …rms simultaneously set prices. Note: You are allowed to restrict attention to the part of the demand function, which is relevant for the equilibrium analysis.

3. Compare your result in (2) to the game in which …rms cannot discriminate.

In which environment obtain …rms larger pro…ts? Explain your …ndings.

4. Suppose now that …rm 1 can discriminate between the two consumer segments and that …rm 2 cannot. Characterize the Nash equilibrium of the price game in which …rm 1 sets a possibly di¤erent price for each consumer segment, while …rm 2 sets the same price to all consumers.

5. Compare the …rms’pro…ts in situation (4) to those in situation (2).

6. Consider the possibility for …rms to “invest” (non-negative number) into the possibility to price discriminate between consumers in segments A and B at investment cost I. Characterize the equilibrium of the two-stage game in which …rms simultaneously decide whether to invest in stage 1 and simultaneously set prices in stage 2 depending on the level of the investment cost I. Comment on your result.

Solutions to Exercise 5 1. The indi¤erent consumer is located at

Hence, demand is

x b=

p1

1 + : 2

p2

p1

p2 2

q1 (p1 ; p2 )

=

q1 (p1 ; p2 )

= 7

2 1

1 + ; 2 p 2 p1 1 : 2 2

Firms maximize pro…ts (pi order conditions we obtain

c)qi (pi ; pj ) with respect to pi . Rewriting the …rstp1

=

p2

=

p2 + c + ; 2 2 p1 + c + : 2 2

Solving this system of two equations we obtain

p1 p2

= c+ ; = c+ :

Equilibrium pro…ts are =2.

2. In each segment, there is an indi¤erent consumer, pA 2

pA 1

1 pB pB 1 1 and x bB = 2 + : 2 2 2 2 A B B Pro…t functions are 1 = (pA c)b x + (p c)(b x 1=2) and 2 = (pA 1 1 2 A B B B c)(1=2 x b ) + (p2 c)(1 x b ). Maximizing i with respect to pA i and pi for each …rm i gives rise to four …rst-order conditions. We note that segments A and B are independent. First-order conditions in segment A can be rewritten x bA =

+

as

1 1 A A (c + pA 2 + ) and p2 = (c + p1 ): 2 2 2 1 A Hence, equilibrium prices are pA 1 = c+ 3 and p2 = c+ 3 . Correspondingly, 1 2 B B in segment B we obtain p1 = c + 3 and p2 = c + 3 . pA 1 =

3. When …rms can discriminate between the two segments they set lower prices in equilibrium, as competition has become more intense (as a price cut leads to a relatively lower reduction in revenues since there are relatively fewer inframarginal consumers). Since the overall market shares are the same, under price discrimination pro…ts are unambiguously lower than without price discrimination.

4. There continues to be an indi¤erent consumer in each segment, pA 1

1 1 p2 p B 1 and x bB (pB + : 1 ; p2 ) = 2 2 2 2 A A A B B B Pro…t functions are 1 = (p1 c)b x (p1 ; p2 )+(p1 c)(b x (p1 ; p2 ) 1=2) and B B ; p ))+(p c)(1 x b (p ; p bA (pB 2 2 2 )). Firm 1 maximizes 2 = (p2 c)(1=2 x 1 1 A B 1 with respect to p1 and p1 , while …rm 2 maximizes 2 with respect to p2 . x bA (pA 1 ; p2 ) =

p2

+

We can write …rst-order conditions as

pA c pA p2 1 1 2 2 pB c p2 pB 1 1 + 2 2 p2 c pA p2 p2 pB 1 + 1 2 2 2 1 2

1 2

8

=

0;

=

0;

=

0:

B Solving these three equations we obtain pA 1 = c + 3 =4, p1 = c + =4, and p2 = c + =2.

5. If both …rms discriminate, each …rm obtains equilibrium pro…t 5 =18. If …rm 2 does not discriminate, …rm 1 obtains equilibrium pro…t 5 =16, while …rm 2 obtains a pro…t of =4. Hence, …rm1 is better o¤, if …rm 2 cannot discriminate, while …rm 2 is worse o¤.

6. Equilibrium pro…ts without discrimination are =2. We thus have I no I

I 5 =18 I , 5 =18 =4, 5 =16 I

no I 5 =16 I , =4 =2, =2

I

Hence, for any I 0, there exists a subgame-perfect equilibrium in which none of the …rms invests in the ability to price discriminate. If 5 =18 I > =4 there is also an equilibrium in which both …rms invest in the ability to price disriminate. Thus, the condition for this alternative equilibrium to exist is I < =36. Firms may fail to coordinate and both invest in the ability to spatially discriminate which leads to lower prices and lower pro…ts.

Exercise 6 Markets with damaged goods Some …rms incur costs to o¤er a lower quality: i) Intel dismantled the mathematical coprocessor in some versions of the 486 CPU, ii) IBM has developed software to make some of their printers slower, and iii) Sony deliberately limited the capacity of some MiniDiscs to 74 instead of 80 minutes. Why do you think that the …rms do this? What do you think that the welfare consequences are?

Exercise 7 Damaged-good strategy [included in 2nd edition of the book] A …rm sells a product in a market where there are two types of consumers, high and low-valuation consumers. There are equally many of the two types of consumers, and the total number of consumers is normalized to 1. The product has value 3 to the high-valuation consumers and value 1 to the low-valuation consumers. All consumers have unit demand, i.e., they buy either one unit or do not participate. The product is produced at constant marginal cost equal to 0.

1. Find the pro…t maximizing price and calculate the …rm’s pro…t. The …rm considers introducing a damaged version of the product. The damaged version is produced at constant marginal cost equal to 1/10. It results in a utility of 5/10 to the low-valuation consumers and of 6/10 to the high valuation consumers.

2. Find the optimal price of the normal and of the damaged version of the product. Should the …rm introduce the damaged version? What are the welfare consequences of the introduction of the damaged version?

9

Solutions to Exercise 7 1. It is immediate that p = 3 and = 1).

= 3=2 (the alternative being p = 1, yielding

2. The damaged version is sold to the low valuation consumers and the normal version to the high valuation consumers. Hence, p damaged = 5=10. To satisfy the incentive constraint of the high type, p normal must be chosen such that 6=10 p damaged = 1=10 = 3 p normal or, equivalently, p normal = 29=10 which is less than 3. This results in pro…ts: (1=2)(5=10 1=10) + (1=2)(29=10) = 33=20 > 3=2.

3. Hence, the …rm makes a higher pro…t by introducing the damaged version in spite of the higher cost of production for this version. The high valuation consumers are also better o¤ when the damaged version is introduced, as they face a lower price. Finally, the low valuation consumers obtain zero utility in both cases. The introduction of the damaged version results thus in a Pareto improvement.

Exercise 8 Non-linear pricing [included in 2nd edition of the book] A monopolist produces a good with constant marginal cost equal to c, c < 1. Assume for now that all consumers have the demand Q(p) = 1 p. The population is of size 1.

1. Suppose that the monopolist cannot discriminate in any way among the consumers and has to charge a uniform price, pU . Calculate both the price that maximizes pro…ts and the pro…ts that correspond to this price.

2. Suppose now that the monopolist can charge a two-part tari¤ (m; p) where m is the …xed fee and p is the price per unit. Expenditure then is m + pq.

Calculate the two-part tari¤ that maximizes pro…ts and the pro…ts that correspond to this tari¤. Compare pU and p and comment brie‡y. Compare the situation with a uniform price and a two-part tari¤ in terms of welfare (a verbal argument is su¢ cient).

3. Assume now instead that there are two types of consumers. The consumers of type 1 have the demand Q1 (p) = 1 p, and the consumers of type 2 have the demand Q2 (p) = 1 p=2. The population is of size 1 and there are equally many consumers of the two types. Finally, it is assumed in this question that c = 1=2. Calculate the two-part tari¤ that maximizes the pro…ts of the monopolist. Compare the two-part tari¤s found in questions (2) and (3) for c = 1=2 and comment brie‡y.

Solutions to Exercise 7 1. The monopoly chooses p to maximize = (p c) (1 p). The pro…t-maximizing price is easily found as pU = (1=2) (1 + c). The corresponding pro…t is U = 2 (1=4) (1 c) .

10

2. Facing a tari¤ (m; p), the participation constraint of a consumer is CS (p) m 0, where CS (p) is the consumer surplsu at price p. With demand Q(p) = 1 p, 2 we have CS (p) = (1=2) (1 p) . As the monopolist’s best intersest is to set m = CS (p), its problem is thus to set p so as to maximize = CS (p) + (p c) (1 p) = (1=2) (1 p) (1 + p 2c). The …rst-order condition yields 2 p c = 0. Hence, the optimal two-part tari¤ is (m; p) = ((1=2) (1 c) ; c).

The optimal two-part tari¤ consists in selling the good at marginal cost, so as to generate the largest consumer surplus, and in capturing this surplus fully 2 through the …xed fee. The corresponding pro…t is T P = m = (1=2) (1 c) .

Welfare is maximized under the two-part tari¤ as it involves marginal cost pricing (whereas uniform pricing results in a deadweight loss).

3. Consumer surplus for agents of type 2 at any price p 2 is computed as 2 2 CS2 (p) = (1=4) (2 p) . We recall from (2) that CS1 (p) = (1=2) (1 p) for any p 1. For any price where the two types of consumers buy (i.e., p 1), we check that CS2 (p) CS1 (p). One option for the monopolist is to make sure that consumers of both types buy; for this, m = CS1 (p) and p 1.

We have then the same problem as in (2), with c = 1=2: (m; p) = (1=8; 1=2) and 1 = 1=8. The alternative option is to sell only to type-2 consumers with m = CS2 (p) and p 2. The monopolist’s problem is then to choose p to 2 maximize 2 = (1=2) ((1=4) (2 p) + (p 1=2) (1 p=2). The …rst-order condition yields p = 1=2. As p = 1=2 (i.e., marginal cost pricing violates the constraint), the monopolist chooses p = 1, so that m = 1=4. Theresulting pro…t is 2 = (1=2) (1=2) = 1=4. We see that 2 > 1 : the monopolist prefers to sell to type-2 consumers only by setting a price above marginal cost.

Exercise 9 Multi-stop shopping2 [included in 2nd edition of the book] Suppose that a supermarket o¤ers a product selection consisting of of products A and B. Consumers are willing to pay 10 Euro for one unit of product A and 10 Euro for product B. Consumers have heterogeneous shopping cost z.

This shopping cost is uniformly distributed over the interval [0; 10]. Consumers are of mass 10. The …rm has marginal cost of 6 for product A and 0 for product B. 1. Calculate the pro…t-maximizing prices pA and pB . How much pro…t can the supermarket make.

2. Suppose that a discounter has entered the market who sells product A at its (lower) marginal costs of 4 Euro. Now consumers can opt for onestop shopping at the supermarket, two-stop shopping at the supermarket and the discounter, or not to shop at all. The shopping cost z applies to each stop. Determine the pro…t maximizing prices of the supermarket.

Determine the supermarket’s pro…t.

3. Compare your results in (2) to those in (1). Interpret your …ndings.

2 This exercise is inspired by Patrick Rey and Zhijun Chen (2012), "Loss Leading as an Exploitative Practice", American Economic Review 102, 3462-3482.

11

Solutions to Exercise 9 1. Pro…t-maximizing prices satisfy pA

10 and pB 10. Then consumers buy either both products or do not buy at all. Denote the consumer who is indi¤erent between not buying at all and buying both products by z2 : z2 = 20 pA pB .

Thus demand is Q(pA ; pB ) = 20 pA pB . Denote the total price as p = pA + pB . The maximization problem is maxp (p 6)(20 p). Thus pm = 13 and m = 49. Any prices pA ; pB with pA 10, pB 10, and pA + pB = 13 maximize monopoly pro…ts.

2. Consumers with low shopping cost buy product A from the discounter and product B from the supermarket. Denote the consumer who is indi¤erent between this two-stop shopping and one-stop shopping at the supermarket by z1 . This consumer satis…es (10 4) + (10 pB ) 2z1 = 20 pA pB z1 which is equivalent to z1 = pA 4. Hence, consumers between 0 and z1 buy product A at 4 Euro from the discounter and product B at price pB . Consumers between z1 and z2 buy both products at the supermarket and consumers with shopping costs above z2 do not buy.

The pro…t-maximization problem is maxpA ;pB pB (pA 4)+(pA +pB 6)(24 2pA pB ). Rewriting the …rst-order conditions gives pA + pB = 13 and 2pA + pB = 18. Thus, pro…t-maximizing prices are pA = 5 and pB = 8. These prices are lower than the willingness-to-pay of consumers and involve product A being sold below costs. Given these prices, z1 is located at 1 and z2 at 7. Pro…ts under competition from the discounter are c = 1 8 + (7 1)(13 6) = 50.

3. The supermarket makes higher pro…t under competition from a more e¢ cient discounter than under monopoly. Product A can be seen as a loss leader since it is sold below marginal costs. Under competition the supermarket is better able to price-discriminate between di¤erent consumer types. Setting the price for product A below marginal costs allows the supermarket to raise the price of product B , which it sells to all consumers including those with low shopping costs. By contrast, under monopoly, the supermarket would sell product B at a loss to all consumers if it mimicked the price strategy under competition (which is one of the many solutions to the monopoly problem).

Exercise 10 Price discrimination among sequentially arriving consumers and …xed supply [included in 2nd edition of the book] Suppose that a monopoly retailer has exactly 2 units of a perishable product available. It cannot increase its stock in the relevant period. Customers have unit demand and either a high or a low willingness to pay: 1 consumer is willing to pay r = 10 and 2 consumers are willing to pay r = 6. Customers arrive in random order at the shop. The retailer has to set a price for each of the 2 units.

The retailer’s opportunity cost of selling is zero.

1. What is the optimal pricing of the 2 units of the product if the retailer has to set the same price for all units?

12

2. What is the optimal pricing of the 2 units of the product if the retailer is allowed to set di¤erent prices for these units? What are the monopolist’s expected pro…ts?

3. Provide a pro…t comparison. Discuss your result. Is welfare larger or smaller in (2) than in (1)?

Solutions to Exercise 10 1. The retailer’s pro…t-maximizing price is either p = 10 or p = 6. If it sets p = 10 it sells to 1 consumer and makes pro…t 10. If it set p = 6 it sells both units and makes pro…t 12.

2. The retailer’s optimal pricing is to sell 1 unit at p = 10 and the other unit at p = 6. With probability 1=3 the high valuation consumer arrives …rst, with probability 1=3 second and with probability 1=3 last. If she arrives …rst, she buys the product at p = 6 and the second unit is not sold. With probability 2=3 the unit sold at p = 6 has already been bought by another consumer and she buys the product at p = 10. Hence, expected pro…ts are (1=3)6+ (2=3)16 = 38=3 > 12.

3. It is optimal for the retailer to o¤er the two units at di¤erent prices. The consumer who arrives …rst buys the cheaper unit. Thus, the retailer has the risk no to be able to sell one of the units. Otherwise, it is able to extract all surplus from two consumers. The expected total surplus is the sum of expected pro…t 38=3 and expected consumer surplus. The latter is the probability that the high type consumer gets the product at the low price times her net surplus, (1=3)(10 6) = 4=3. Hence, total surplus is 42=3.

Total surplus in case of the unit price p = 6 is pro…ts of 12 plus expected consumer surplus. The latter is the probability that the high type buys the product times her net surplus, (2=3)(10 6) = 8=3. Hence, total surplus is 44=3.

In terms of total surplus there are two countervailing e¤ects: with discriminatory pricing the high type always buys the product whereas under uniform pricing, this occurs only with probability 2=3. This generates an increase of total surplus of (1=3)(10 6) = 4=3. The downside of discriminatory pricing is that with probability 1=3, one of the two units remains unsold, whereas they generate a surplus of 6 under uniform pricing. This reduces total surplus by (1=3)6 = 2.

The negative e¤ect outweighs the positive e¤ect and, thus, total surplus is lower under discriminatory pricing. (Clearly, if parameters where such that the optimal uniform price would lead to participation of only the high type, the total surplus ranking is reversed.)

Exercise 11 Dynamic pricing and consumer storage [included in 2nd edition of the book]

13

Consider a monopolist providing a product over several periods. The monopolist has constant marginal costs of production of c in each period. Consumers consider to consume one unit of a good in each period. The consumer population is of mass 1. Half of all consumers have a high valuation rH for the product, which is the same in each period. The other consumers have a low valuation rL with rH > rL > c. While high-type consumers have to buy the product in the period in which they consume it, low-type consumers are assumed to be able to store one unit for one period at no cost. The common discount factor is .

We assume that even a product which is consumed one period later generates a value larger than production costs, rL > c.

1. Suppose that there is an in…nite time horizon and that the monopolist commits to a price path fpt gt=1;2;::: before the market opens. Determine the pro…t-maximizing price path under the constraint that the price has to be the same in all periods. Determine the monopolist’s pro…ts.

2. Consider the same setting as in (1) without the restriction that the prices have to be constant over time. What is the pro…t-maximizing price path?

Under which conditions does the monopolist prefer non-constant prices over constant prices? Determine the maximal pro…ts of the monopolist?

Explain your result.

3. Consider now a two-period setting. Suppose that the monopolist cannot commit to a price path. In particular, the monopolist sets p1 and, after selling in the …rst period, he sets p2 at the beginning of the second period.

Are there parameter constellations such that the monopolist sets di¤erent prices in the two periods? If your answer is negative provide a proof why non-constant prices cannot be a subgame perfect Nash equilibrium. Otherwise, characterize the set of parameters under which a subgame perfect Nash equilibrium with non-constant prices is supported.

Solutions to Exercise 11 1. The pro…t-maximizing price is either p = rH or p = rL . If the monopolist commits to set price p = rH in every period, he will make per-period pro…t of (rH c)=2. If he sets price p = rL in every period, he will make per-period pro…t of rL c. Hence, if (rH c)=2 > rL c or, equivalently, (rH + c)=2 > rL the monopolist sets the high price. Otherwise he sets the low price. The present value (1+ )(rH c) (r c) H c) = (1+ 1)(rL2 c) or (r . of discounted pro…ts is either 1L 2(1 ) = 2(1 2 )

2. If the monopolist can commit to a non-monotone price path, he can commit to set p = rH in even periods and p = rL in uneven periods. At a price of rL , low-type consumers are willing to store one unit to consume it the next period.

If all high-type consumers buy every period and low-type consumers buy two units in every odd period, discounted pro…ts are A (3=2)( rL c)+ =2(rH

c) +

2

(3=2)( rL

c) +

3

=2(rH

rL c) + c) + ::: = (3=2)( 1 2

14

=2(rH c) . This 1 2

price path generates higher pro…ts than the constant low price with p = rL in every period if

(3=2)( rL 2 1

c)

=2(rH c) (1 + )(rL > 2 2 1 1

+

c)

which is equivalent to

=2(rH

c) > (1

=2)(rL

c);

(*)

which is satis…ed for su¢ ciently large. The proposed price path generates higher pro…ts than the constant high price with p = rH in every period if

(3=2)( rL 2 1

c)

+

(rH 2(1

c) (1 + )(rH c) > 2 ) 2(1 )

2

which is equivalent to

(3=2)( rL

c) > (1=2)(rH

c):

(**)

Hence, if (rH c)=2 > rL c and rL c > (rH c)=3 the monopolist set the proposed cyclical price path rather than a constant price path. These inequalities are satis…ed for su¢ ciently large and rH in some intermediate range.

If the monopolist started with p = rH in the …rst period, he would obtain (rH c)=2 + A. This is larger than A if 1=2(rH c) + (3 =2)( rL c) > (3=2)( rL c)+ =2(rH c), which is equivalent to 1=2(rH c) > (3=2)( rL c) and thus dominated whenever A is larger than the pro…t with a constant price path p = rH .

3. Consider now the two-period problem where the monopolist cannot commit to the second-period price. We want to establish conditions for an equilibrium with p1 = rL and p2 = rH to exist. Low-type consumers buy two units in period 1 and zero units in period 2, while high-type consumers buy one unit in each period. Pro…ts are (3=2)( rL c) + ( =2)(rH c). Clearly, in such an equilibrium, the monopolist does not have an incentive to deviate in period 2 as no low-type consumers are around. In the …rst period, the …rm may set a higher price. For any price p 2 ( rL ; rL ] the …rm looses demand for a second unit by low-type consumers.

If (rH c)=2 > rL c, then loosing these sales in period 1 does not have any implications for period 2. In this case, the highest pro…t such that lowtype consumers buy in period 1 is (rL c) + ( =2)(rH c). However, under (rH c)=2 > rL c the monopolist makes a higher pro…t by setting p1 = rH .

This generates a present value of [(1 + )=2](rH c). The monopolist rather sets p1 = rL if (3=2)( rL c) + ( =2)(rH c) > [(1 + )=2](rH c) which is equivalent to (3=2)( rL c) > (1=2)(rH c). This is the same condition established as (*).

15

If (rH c)=2 < rL c, the monopolist will sell in period 2 at price p2 = rL if all consumers are still around. Under this inequality, the monopolist sets p1 = rL rather than p1 = rH . Hence, the monopolist may set p1 = rL and obtain a present value of discounted pro…ts of (1 + )(rL c). Alternatively, he could set p1 = rL . There is an equilibrium in which all low-type consumers buy in period 1, which implies that p2 = rH . The monopolist sets the lower price p1 = rL if (3=2)( rL c) + ( =2)(rH c) > (1 + )(rL c) which is equivalent to =2(rH c) > (1 =2)(rL c) and the same condition as (**). Hence, even though the monopolist cannot commit to period-2 prices there are equilibria which replicate the prices under commitment.

Exercise 12 Behavior-based price discrimination [included in 2nd edition of the book] Consider a market with network e¤ects (i.e., a consumer’s utility depends on the number of users of a product) in which each consumer has a willingness to pay equal to xi where xi is the number of consumers buying product i = 1; 2. The products are functionally identical and thus consumers are indi¤erent between any products in the market, given equal numbers of units sold. Suppose that the incumbent …rm has served mass 2=3 of consumers in the previous period.

These old consumers already have experienced product 1 and are not willing to consider product 2. There is mass 1=3 of new consumers, who have not previously experienced product 1. All costs are assumed to be equal to zero. In (1) to (3) …rms …rst set prices and after observing prices, consumers make their purchasing decisions.

1. Suppose that the incumbent …rm cannot distinguish between new and old consumers and that …rm 2 sets its price before …rm 1. What is a subgameperfect Nash equilibrium that gives the highest pro…t for the incumbent …rm among all equilibria? Characterize this equilibrium.

2. Under the same circumstances as in (1), what is a subgame-perfect Nash equilibrium that gives the highest pro…t for the entrant …rm among all equilibria? Characterize this equilibrium.

3. Suppose now that the incumbent …rm can distinguish between new and old consumers and that …rm 2 sets its price before …rm 1. What are the highest pro…ts that the entrant can make in any subgame-perfect Nash equilibrium? Provide a formal justi…cation of your answer.

4. Discuss the economics behind your results in (1) to (3).

16

Solutions to Exercise 12 1. It is easily found that p1 = 1, p2 = 0, x1 = 1, x2 = 2. Firm 1 makes pro…t 1 = 1, whereas …rm 2 makes zero pro…t.

2. Here, p1 = 2=3, p2 = 1=9, x1 = 2=3, x2 = 1=3. Suppose there is an equilibrium with x1 = 2=3; x2 = 1=3. In such an equilibrium …rm 1 optimally sets p1 = 2=3 and thus makes pro…t 4=9. A new consumer derives net utility 1=3 p2 if all other new consumers buy product 2. A consumer is indi¤erent between this o¤er and an alternative o¤er by …rm 1 if 2=3 p1 = 1=3 p2 . If …rm 1 deviated to a price slightly below this critical price, all consumers would buy from …rm 1 and its pro…t is (almost) 1 p1 . Hence, p2 must be set such that the potential deviation pro…t is weakly less than 4=9. Hence, 2=3 4=9 = 1=3 p2 or, equivalently, p2 = 1=9. Thus, …rm 2 makes pro…t 1=3 1=9 = 1=27. Firm 1’s equilibrium pro…t is 1 = 4=9.

3. Firm 1 can set two di¤erent prices. Hence, it can always undercut the price o¤ered to new consumers that is set by …rm 2 without losing pro…ts earned from old co nsumers. Hence, …rm 2 cannot make any pro…t in equilibrium.

4. Behavior-based price discrimination allows …rm 1 to respond more aggressively to the pricing of …rm 2 since it does not su¤er from lower prices for old and thus inframarginal consumers.

Exercise 13 Software bundling [included in 2nd edition of the book] A software company sells two applications, noted A and B, that are totally unrelated to one another. The marginal cost of production for each application is constant and is equal to 10. The company faces four categories of potential buyers, which are characterized by a pair of reservation prices as depicted in the following table; it is assumed that each category counts the same mass of consumers, which is set to 1.

Category 1 Category 2 Category 3 Category 4

Application A 100 80 60 30

Application B 30 80 60 100

1. What price should the company set for each application if it decides to sell them separately? What pro…ts will the company achieve in this case and which categories of consumers will buy which application?

2. Suppose now that the company pursues a mixed bundling strategy. Which price should it set for the bundle and for the separate applications? What pro…ts will the company achieve in this case and which categories of consumers will choose which option? Is mixed bundling more pro…table than spearate selling? Discuss.

17

3. How would your answers to (1) and (2) change if the marginal cost of production increased from 10 to 40?

Solutions to Exercise 13 1. The two applications have similar demand schedules. It is easily found that for each application the pro…t-maximizing prices are pA = pB = 60; at these prices, category 1 buys application A, categories 2 and 3 buy both applications, and category 4 buys application B . Total pro…t is computed as S = 2 3 (60 10) = 300.

2. The reservation price for the bundle is simply the sum of the reservation prices for the two applications. Hence, the reservation price for the bundle is equal to 160 for category 1, to 130 for categories 1 and 4, and to 120 for category 3.

All categories have a reservation price for each application that is above their marginal cost of production. As 4 (120 20) = 400 > 3 (130 20) = 330, the company sets pb = 120 and sells the bundle to all four categories. This allows the company to improve its pro…t compared to separate selling. Mixed bundling does not add any pro…t in this case.

3. Under separate selling, the optimal prices are pA = pB = 80; at these prices, category 1 buys application A, category 2 buys both applications, category 4 buys application B , and category 3 buys nothing. The corresponding pro…t is 40) = 80. Consider now bundling. With a marginal cost of 40, it S = 2 (80 appears that categories 1 and 4 value one application below its cost. Hence, the company has no interest in selling the bundle to them and prefers to apply mixed bundling. As far as the bundle is concerned, the company must set pb > 130 to discourage categories 1 and 4 to buy the bundle. At such prices, only category 2 is willing to buy the bundle; hence, the pro…t-maximizing bundle price is pb = 160.

The separate applications are then intended to categories 1, 3 and 4. It is easily found that the optimal prices are pA = pB = 100; at these prices, category 1 buys application A, category 4 buys application B , and category 3 buys nothing.

Total pro…t is equal to M = 1 (160 80) + 2 (100 40) = 200 > S .

Mixed bundling improves pro…t with respect to separate selling.

Exercise 14 Monopoly bundling [included in 2nd edition of the book] Suppose that a monopolist produces two products, product 1 and product 2. There is a mass 1 of consumers. A share of consumers are heterogeneous among each other and are described by their type . This type is distributed uniformly on the unit interval. The willingness-to-pay for product 1 is assumed to be r1 = and r2 = 1 . A share (1 )=2 of consumers has willingness to pay r1 = 2=3 and r2 = 0. The remaining share (1 )=2 of consumers has willingness to pay r1 = 0 and r2 = 2=3. The …rm can sell products 1 and 2 independently at prices p1 and p2 , respectively. Alternatively, it may only sell a bundle at price p. This is a situation referred to as pure bundling. A third possibility is that the …rm sells the bundle and the independent products, a situation referred to as mixed bundling.

18

1. Suppose that = 1. Determine whether independent selling, pure or mixed bundling are pro…t maximizing. Calculate associated prices and pro…ts.

2. Suppose that > 0 and characterize the solution under independent selling for all > 0.

3. Suppose that = 4=5. Characterize the pro…t-maximizing solution under independent selling, pure bundling, and mixed bundling. Show which of the selling strategies is pro…t-maximizing. Discuss your result.

4. Repeat the previous question with

= 2=3.

Solutions to Exercise 14 1. Under independent selling, demand for product i (i = 1; 2) is determined by Qi = 1 pi ; the optimal price are easily found as p1 = p2 = 1=2. Thus, pro…t under independent selling is ind = 21 21 + 12 21 = 12 . Under pure bundling, demand is determined by + (1 ) p. It follows that the optimal price is p = 1 and as all consumers buy the bundle, the corresponding pro…t is pb = 1.

Consider …nally mixed bundling. Given that the only sensible bundle price is p = 1, mixed bundling cannot improve relative to pure bundling. No consumer would buy the individual good at p = 1.

2. Consider product 1. If the …rm sells to the …rst and the second group (i.e. as long as p1 2=3), demand from the …rst group of consumers is (1 p1 ) and from the second group (1 )=2. Thus, the …rm chooses p1 to maximize = p ( (1 p ) + (1 )=2) .

Solving the …rst-order condition, one …nds 1 1 1 p1 = (1 + ) = (4 ). For 3=5, this price would be larger than 2=3, which would result in exclusion of the group 2 consumers. While serving only group 1 though, the monopolist would like to set the price equal to 1=2, which again would mean to include group 2 consumers. Thus, the pro…t maximizing price is pI1 = minf2=3; (1 + ) = (4 )g. The same holds for product 2.

1 9 3. Suppose = 4=5. Under independent selling, p1 = p2 = 41 + 4 4=5 = 16 0:563. Then, pro…t for each product becomes i ind

= =

9 9 ( 12 + 21 pi )pi = 32 + 40 162 0:506: 1 + 2 = 320

81 81 320 = 320

Under pure bundling, the …rm has two choices. The …rst option is to sell to all consumers; then the optimal bundle price is p = 2=3, which yields a pro…t of pb1 = 2=3. The second option consists in selling only to the …rst group of consumers at p = 1. Pro…t then is pb2 = p = 4=5. As pb2 > pb1 , the second option is preferred. Under mixed bundling, the only candidate for truly mixed bundling is to set pIi = 2=3 and p = 1. Consumers of group 1 face the choice between: (a) Buy the bundle at p, yielding a net utility of +1 p = 0; (b) Buy product 1 only, yielding p1 = 2=3; thus all consumers with 2=3 buy only product 1; (c) Buy product 2 only, yielding p2 = 2=3;

19

thus all consumers with 1=3 buy only product 2. In this case all consumer in the …rst group with 2 ( 31 ; 23 ) prefer the bundle rather than the independent product. Thus pro…ts are given by mb =

12 33 +

12 33 +

1 1 3p + 2 2

Evaluated at = 45 this yields a pro…t of bundling that o¤ers the highest pro…t.

mb

2 2 3 = 3 + 9:

0:756. Hence, it is pure

4. With

= 2=3, you should obtain the following pro…ts respectively for independent selling, pure bundling and mixed bundling: ind = 25=48 0:521, 0:667, and M i = 20=27 0:741. Therefore, it is now pro…tpb = 2=3 maximizing to discriminate between the di¤erent groups and to use mixed bundling.

Exercise 15 Competitive bundling [included in 2nd edition of the book] Suppose that, as in Section 11.3.2, each of two …rms 1 and 2 provides two components A and B. The o¤erings of both …rms are horizontally di¤erentiated.

A consumer of type ( A ; B ) 2 [0; 1]2 derives a net surplus r (1 A B) p1A p2B if she buys component A from …rm 1 at price p1A and component B from …rm 2 at price p2B . Correspondingly, for other systems of A and B. The gross surplus is assumed to be zero if the consumer does not buy a system. Consider only values of r such that the market is fully covered in equilibrium. Consumers of mass 1 are uniformly distributed on the unit square.

1. Suppose that …rm 1 incurs a constant marginal cost of zero for each component and that …rm 2 incurs a marginal cost of c (with c < 3). Thus, …rm 1 is more e¢ cient. Determine the pro…t function of each …rm when each …rms sells each component separately. Determine equilibrium prices and equilibrium pro…ts in the setting in which …rms simultaenously set prices for both components.

2. Consider the same setting as before when both …rms o¤er only a system (pure bundling). Determine equilibrium prices and equilibrium pro…ts in the setting in which …rms simultaneously set prices for their system.

3. Compare your results in (1) and (2). When is it more pro…table for …rm 1 to sell components in a bundle? When is it more pro…table for …rm 2 to sell components in a bundle?

4. Suppose now that …rm 1 is more e¢ cient producing component A and …rm 2 is more e¢ cient producing component B. In particular, suppose that, for component A, …rm 1 incurs a constant marginal cost of zero and that …rm 2 incurs a marginal cost of c, while the reverse holds for component B. Determine equilibrium prices and equilibrium pro…ts under independent selling and pure bundling. Discuss the pro…tability of pure bundling in this setting.

20

Solutions to Exercise 15 1. Under independent selling, …rm i faces Hotelling demand 1=2 for component z . The …rms pro…ts are 1 2

(piz

pjz )=2

= p1A (1 p1A + p2A )=2 + p1B (1 p1B + p2B )=2; = (p2A c)(1 p2A + p1A )=2 + (p2B c)(1 p2B + p1B )=2:

Solving the system of four …rst-order conditions gives equilibrium prices p1A = p1B = 1+c=3 and p2A = p2B = 1+2c=3. In equilibrium, the marginal consumer is located at (1=2) (1 + c=3). Equilibrium pro…ts are 1ind = (1 + c=3)2 and 2 c=3)2 . ind = (1

2. Denote the price of a bundle of …rm i by pi . A consumer of type ( A ; B ) is indi¤erent between the two bundles if A

B

p1 =

(1

A)

(1

p2 :

B)

Hence, the demand of …rm 1 is

1

1

2

D (p ; p ) =

8 > < 1 > :

1

1 2

2

1 + p 2p

1 2

2

1

1 + p 2p

2

2

if p1 < p2 ; if p1

p2 :

Correspondingly, for …rm 2. Firm 1 solves max p1 D 1 (p1 ; p2 ) and …rm 2 solves max(p2 2c)D2 (p1 ; p2 ). In equilibrium, the indi¤erent consumer with A = B p is located at ^(c) = 7 + c c2 2c + 9. Equilibrium prices are p p1 = 54 c 54 + 34 c2 2c + 9; p p2 = 74 c + 14 + 14 c2 2c + 9: Equilibrium pro…ts are 1 bun 2 bun

= =

1 32 1 128

c2 1

2c + 19 p c + c2

p c2

2c + 9

2c + 9

1 32 (c

1)(c2

2c

41);

3

:

3. While calculations are somewhat tedious, we con…rm the result that separate selling dominates pure bundling under symmetry (c = 0) for small cost asymmetries. It can be veri…ed that the inequality 1ind > 1bun holds if and only if c < 1:415. The inequality 2ind > 2bun holds if and only if c < 2:376. Hence, for large cost asymmetries the result is reversed and both …rms prefer bundling.

For an intermediate range of cost di¤erences, the more e¢ cient …rm prefers pure bundling, while the less e¢ cient …rm prefers independent selling. By bundling its products, …rm 1 may force that consumers choose between the system provided by …rm 1 and a system consisting of the components of …rm 2. Then, the bundling decision of …rm 1 increases its own pro…ts, but decreases the competitors

21

pro…ts. (As a preview to Chapter 16, in the present model, bundling may be a strategy to blockade entry. For this to be the case, entry costs must be such that …rm 2 cannot recover its entry costs under bundling while it would do so under independent selling.)

To better understand this result, we observe that two forces are at play which determine equilibrium pro…ts. Note that under bundling, we are interested in the distribution of the average consumer = ( A + B )=2. Under the uniform distribution over the unit square, the corresponding density is linearly increasing in on [0; 1=2] and linearly decreasing in on [1=2; 1]. We note that for any given prices with p1 = p1A + p1B and p2 = p2A + p2B and p2 > p1 we must have that the …rm 1’s demand is larger under bundling than under independent selling (this not only holds for the uniform distribution but holds more generally).

The opposite holds for …rm 2. Hence, due to this demand e¤ect, …rm 1 tends to bene…t from pure bundling, while …rm 2 tends to su¤er. However, moving from independent selling to bundling changes the pricing incentives, which are determined by the price elasticity e¤ect. The way this e¤ect plays out depends on the asymmetry of the market. If the marginal cost di¤erence is small, the indi¤erent consumers with average will be close to 1=2 in equilibrium. Then, demand under bundling is more sensitive to price changes. This tends to increase competition between …rms when moving to bundling. Hence, …rms are unlikely to bene…t from bundling their components. By contrast, if the marginal cost di¤erence is large (but smaller than 3) the indi¤erent consumers with average will be close to 1 in equilibrium. Then, demand under bundling is less sensitive to price changes than under separate selling. As this tends to lead to less intense competition under bundling compared to separate selling, both …rms are likely to bene…t from bundling their components.

4. In this setting, …rms are symmetric under bundling. Equilibrium prices are 1 + c and equilibrium pro…ts are 1=2. Under separate selling, the two component markets can be analyzed separately. Using the results from (1), one obtains that equilibrium pro…ts are 1 + c2 =9. Here, marginal costs a¤ect pro…ts under separate selling as a higher marginal cost increases the market power of the more e¢ cient …rm and leads to higher industry pro…ts. By contrast, under bundling both …rms are a¤ected in the same way by an increase in c. Due to the price elasticity e¤ect explained in (3), equilibrium pro…ts are lower under bundling than under separate selling for c = 0. An increase in c does not a¤ect pro…ts under bundling while it increases pro…ts under separate selling. Thus, bundling is always dominated by separate selling.

22

Industrial Organization: Markets and Strategies Paul Belle‡amme and Martin Peitz published by Cambridge University Press

Part V. Product quality and information Exercises Exercise 1 Lemons problem Consider the following version of the lemons problem. There is a continuum of buyers and sellers in the market; the total mass of each group is 1. Each seller has one car to sell and each buyer wishes to buy at most one car, but only sellers know the quality of their cars before trading. It is common knowledge however that the quality of cars, denoted s, is drawn from a uniform distribution on the interval [0; 1] (hence, the probability that a car’s quality is below some number x is equal to x if 0 x 1 and is equal to 1 if x 1). It is also common knowledge that a fraction the sellers are of type 1 and have a payo¤ U1 = p s=8 if they sell their cars and 0 otherwise, and a fraction 1 of the sellers are of type 2 and their payo¤ is U2 = p s=4 if they sell their cars and 0 otherwise, where p is the price of the car (note that the two types of sellers di¤er only with respect to their payo¤s but not with respect to the quality of cars they have to sell). There is a continuum of buyer types: the payo¤ of a type- buyer if he buys a car whose quality is s is U ( ) = s p, where is distributed uniformly on the unit interval. If a buyer does not buy a car his payo¤ is 0. The buyers cannot observe the quality of cars before they buy nor can they observe the type of seller they face.

1. Compute the supply of cars by type 1 sellers, type 2 sellers, and the aggregate supply of cars (i.e., compute the fraction of cars that will be supplied at a given price by each type of sellers and then add the two to obtain the aggregate supply). Show your answer in a …gure.

2. Let sb(p) denote the average quality of cars supplied on the market as a function of p. Using your answer to (1), compute sb(p). How does sb(p) vary with p and with ? Explain the intuition for this.

3. Assume that buyers correctly anticipate sb(p) and compute the demand for cars (i.e., the fraction of buyers that will wish to buy a car at a given price) and show your answer in the …gure you drew in Part (1). Explain the shape of the demand function.

4. Assume that the market is perfectly competitive and solve for the equilibrium price, p .

1

Solutions to Exercise 1 1. Type 1 sellers will o¤er their cars provided that p s=8, or, equivalently, provided that s 8p. Since s is distributed uniformly over the interval [0; 1], the probability that a type 1 seller will o¤er his car is 1 if 8p 1 (recall that s 1), or 8p if 8p 1. Given that type 1 sellers account for a fraction of the population, their supply is

S1 (p) =

minf8p; 1g:

Type 2 sellers will o¤er their cars provided that p s=4, or, equivalently, provided that s 4p. Hence, the probability that a type 2 seller will o¤er his car is 1 if 4p 1, or 4p if 4p 1. Given that type 2 consumers account for a fraction 1 of the population, their supply is

S2 (p) = (1

) minf4p; 1g:

The aggregate supply functions is therefore:

S(p) = minf 8p; g + minf(1

)4p; 1

g:

2. Given p, type 1 seller o¤er their cars provided that s 8p. Hence, the average quality of cars they o¤er is 4p. Type 2 sellers o¤er their cars provided that s 4p. Hence, the average quality of their cars is 2p. When a consumer goes to the market he needs to assess the likelihood that a seller he meets is either type 1 or type 2. At a price p, the frequency of type 1 sellers is S1 (p)=S(p) and the frequency of type 2 sellers is S2 (p)=S(p). To simplify, let us assume that 8p 1 so that we do not need to worry about the constraints that 8p and 4p exceed 1. Then, when the price is p the average quality of a car is given by:

s^(p) =

S2 (p) 2(1 + 3 ) S1 (p) 4p + 2p = p: S(p) S(p) 1+

Obviously, s^(p) increases with p (higher p means more sellers are willing to sell) and decreases with (the average quality of cars o¤ered by type 2 consumers is lower so when the probability that a seller is of type 1 increases, the average quality increases).

3. A buyer with valuation will buy a car provided that s^(p) p, or Since is distributed uniformly on [0; 1], the likelihood that 1 p=^ s(p). Hence, the demand for cars is D(p) = 1

p=^ s(p). p=^ s(p) is

1+ 1+5 = : 2(1 + 3 ) 2+6

The demand function does not depend on p because the negative impact of p on total demand for given quality and the positive impact of p on the average quality of cars cancel out.

2

4. In a perfectly competitive market, the equilibrium price, p , is given by the solution to S(p) = D(p). Solving the equation yields: p =

1+5 : 8 + 32 + 24 2

We need to check though that p 1=3.

5. Given that increase of

is such that 8p

1; this requires that

1=3, p is a decreasing function of , implying that after an (i.e., there are more type 1 sellers) the equilibrium price is lower.

Exercise 2 Quality and information A …rm sells a product which may be of high or low quality, sH or sL , respectively. High quality is to occur with probability and low quality with probability 1 . There is a unit mass of consumers with unit demand and the same willingness-to-pay for the product of a particular quality. Consumers like high quality more than low quality— i.e., consumer valuations (= willingnessto-pay) satisfy rH > rL . The consumer valuation for high quality is larger than costs c, which is independent of quality. The marginal cost is assumed to satisfy that rH + (1 )rL > c.

There are two groups of consumers. A share is informed about product quality and a share 1 is uninformed; there is no communication between informed and uninformed consumers. We consider the following situation: First, Nature determines product quality. The quality is observed by the …rm and a share of consumers. Second, the …rm sets the price of the product. Third, uninformed consumers update beliefs and all consumers make their purchasing decision.

1. Suppose that = 1. Characterize the equilibrium (price, allocation, pro…t). Distinguish between case c < rL and c > rL .

2. Suppose that = 0. Characterize the equilibrium (price, allocation, pro…t). Distinguish between case c < rL and c > rL .

3. Suppose that < 1 and c < rL . Do there exist parameter constellations under which the same allocation as under (1) can be supported? If your answer is negative give a proof that shows that the outcome in (1) cannot be replicated for < 1. Otherwise, give the exact parameter range for for which the outcome in (1) can be replicated.

4. Suppose that > 0 and c > rL . Do there exist parameter constellations under which the same allocation as under (2) can be supported? If your answer is negative give a proof that shows that the outcome in (2) cannot be replicated for > 0. Otherwise, give the exact parameter range for for which the outcome in (2) can be replicated.

3

5. Comment on what is happening in this market.

6. What kind of government interventions would matter in such a maket with < 1? Discuss the welfare consequences of such government interventions (try to refer to (2)-(4)).

Solutions to Exercise 2 1. rL > c: high quality: p = rH , low quality: p = rL rL < c: high quality: p = rH , low quality: DO NOT SELL 2. p = rH +(1 3.

H = rH 0 H H

)rL

c

L = r L c; 0 )(rH c): L = (1 0 L L is equivalent to

4.

H= 0 H= H

rH +(1

rH r L rH c

)rL c

(rH c) 0 H is equivalent to rH +(1 )rL c rH c

L = (1 0 L = rL

)[ rH +(1

)rL c]

c < 0 by assumption. The low quality …rm has no incentive to

deviate.

Hence, we need ( rH + (1 )rL to average quality level for both …rms.

c)=(rH

c) to sustain prices equal

5. Price signalling is viable if a share of consumers is fully informed. Informed consumers generate an information spillover: uninformed consumers obtain information through the …rm’s price because they know that some other consumers have better information and act according to this information.

6. Suppose that c < rL . Welfare under full information as under full separation (see question 2):

= (rh c) + (1 )(rL c) CS = 0 W = (rh c) + (1 )(rL c) ) welfare under pooling (see question 3): = (1 )(1 )[ rH + (1 )rL c] + [ rH + (1 CS = (1 ) (rH rL ) W = (rh c) + (1 )(rL c) (1 )(rL c)

)rL

c]

Welfare is smaller under pooling than under separation. Government interventions should increase the fraction of informed consumers such that separating equilibrium becomes feasible.

4

Exercise 3 Prices as a signal of quality [included in 2nd edition of the book] A …rm has either a high quality or a low quality product (the …rm cannot choose the quality of its product). The …rm faces a continuum of consumers with a total mass one. Each consumer wishes to buy at most one unit. There are 2 types of consumers: 2=5 of the consumers are of type 1 and their utility is 10 p if they buy a high quality product and 5 p if they buy a low quality product; 3=5 of the consumers are of type 2 and their utility is 6 p if they buy a high quality product and 3 p if they buy a low quality product. Both types of consumers obtain a utility of 0 if they do not buy. The per unit cost of production is 2 if the …rm has a high quality product and 0 if it has a low quality product.

1. Suppose that consumers can tell the quality of the product before they buy. Determine the prices that each type of …rm would charge. (Hint: note that if quality is high, the …rm can sell only to type 1 consumers if p > 5 but to all consumers if p 5; likewise, if quality is low, the …rm can sell only to type 1 consumers if p > 3 but to all consumers if p 3.)

2. Suppose now that consumers cannot tell the quality of the product before they buy and can only infer it from the price that the …rm charges. Show that there exists a separating equilibrium in which high- and low-quality …rms behave di¤erently and therefore consumers can infer the quality of the product from the price that the …rm charges. In this equilibrium, a low-quality …rm behaves as in (1). What is the price that the high-quality …rm needs to charge in order to separate itself from the low-quality …rm?

Show that charging this price is pro…table for the high quality …rm. Does the high-quality …rm signal its quality by charging a high price or a low price? Provide an intuition.

Solutions to Exercise 3 1. If consumers can tell the product’s quality, then the …rm will charge either 10 or 6 if quality is high and 5 or 3 if quality is low. To see which price the …rm will charge, suppose quality is high. If the …rm charges 10, it will serve only type 1 customers and its pro…t will be h (10) = (2=5) (10 2) = 16=5. If the …rm would charge 6, then it will serve all consumers and its pro…t will be 2 = 4. Since 16=5 < 4, the …rm is better o¤ charging 6 and h (6) = (6 serving all consumers. Likewise, if quality is low and the …rm charges a price of 5, it will serve only type 1 customers and its pro…t will be l (5) = (2=5)5 = 2.

If the …rm charges 3, it will serve all consumers and its pro…t will be l (5) = 3.

Once again, charging a low price that would appeal to all consumers is better than charging a high price and serving only type 1 consumers.

2. If the low quality …rm behaves as in (1), then it charges a price of 3 and serves all consumers. Its pro…t therefore is 3. If the high-quality …rm charged 6 as in (1), then the pro…t of the low quality …rm from mimicking would be 6 since it has

5

no cost and a price of 6 would allow it to serve all consumers (who will believe, incorrectly, that the …rm is a high quality …rm). Hence, to separate itself, the high quality …rm must charge a price above 6. At this price, however, only type 1 customers buy so the sales are merely 2/5. The price of the high quality …rm, p, must be such that the low quality …rm would earn less than 3 (otherwise it would mimic). Hence, (2=5)p < 3, implying that p < 7:5. Suppose then that the high quality …rm charges 7.5. Consumers should anticipate that the …rm’s product is of a high quality. The remaining question is whether it pays the high quality …rm to separate itself. If it charges 6, consumers will think its quality is low and would not buy. If it charges less than 5 but above 3, consumers will believe its quality is low but only type 1 consumers would agree to buy. Hence, the …rm’s sales would be 2/5. But then, the high quality …rm might as well charge 7.5 and signal its high quality. If the high quality …rm charges 3, it will sell to all consumers and its pro…t will be 3 2 = 1. In the proposed equilibrium, the high quality …rm earns a pro…t of (2=5)(7:5 2) = 11=5. Since 11/5 > 1, it pays the high quality …rm to charges 7.5. Summing up, the high quality …rm signals its type by raising its price above the full-information price of 10, so we have the opposite of an introductory o¤er. The reason why this o¤er deters the low quality …rm from mimicking is due to the fact that the high quality …rm serves only 2/5 of the market, while the low quality …rm serves all consumers.

The signal then is the restricted sales, which are due to the higher than the full-information price. Once consumers learn the product’s quality, prices will come down so over time we will see declining prices.

Exercise 4 Advertising as a signal of quality [included in 2nd edition of the book] A …rm produces a single product whose quality is either high or low (the …rm knows the quality but cannot choose it) and sells it to consumers in each of two periods. The marginal cost of production is 4 if quality is high and 3 if quality is low. In each period there are N consumers, each of whom is interested in buying at most one unit in each period and is willing to pay 10 if quality is high and 5 if quality is low. However, consumers cannot tell the product’s quality before they consume it in period 1. Suppose that the …rm can advertise its product on TV in period 1. Although advertising itself does not convey direct information about the product’s quality, it can serve as a signal - consumers might be able to infer the product’s quality from the fact that the …rm was willing to spend money on advertising. Suppose that the cost of a TV ad is A if quality is low and A if quality is high, where < 1 (e.g., it is cheaper to design an ad for a high quality product). The intertemporal discount factor is .

1. What is the minimum amount of TV ads that the …rm needs to sponsor in order to signal that its product’s quality is high?

2. How does your answer depend on the discount factor ? How does it depend on ? How does it depend on N ? Explain your answer in detail.

6

Solutions to Exercise 4 1. Suppose that consumers believe that if the …rm sponsors x TV ads, then its quality is high. If the low quality …rm does not advertise, consumers will agree to pay at most 5 for its product. Hence, the pro…t of the low quality …rm will be

N (5

3)(1 + ) = 2N (1 + ):

If the low quality …rm advertises, consumers will pay 10 in the …rst period and 5 in the second period. The …rm’s pro…t then will be

N (10

3)

Ax + N (5

3) = 7N

Ax + 2N:

Comparing the two pro…t levels, it does not pay the low quality …rm to advertise if and only if 7N Ax + 2N < 2N (1 + ), which is equivalent to x > 5N=A. We need to check that the high-quality …rm is willing to sponsor 5N=A ads. If it sponsors x TV ads (thereby signaling its high quality) its pro…t is:

N (10

4)

Ax + N (10

4) = 6N (1 + )

Ax:

If the …rm does not advertise in period 1, consumers will believe in period 1 that its quality is low and would agree to pay only 5. Since consumers learn the …rm’s quality in period 2, the pro…t of …rm 1 in this case is:

N (5

4) + N (10

4) = N + 6N:

Comparing the two pro…t levels, it pays the high quality …rm to advertise if and only if 6N (1 + ) Ax > N + 6N , which is equivalent to

x<

5N : A

Since < 1, sponsoring 5N=A ads will enable the high-quality …rm to separate itself. As shown above, this is the minimal number of ads needed for separation.

2. The answer does not depend on

since the second period pro…ts are completely independent of advertising. Intuitively this is because the …rm’s quality becomes common knowledge in period 2 regardless of what happened in period 1.

The answer also does not depend on since any < 1 ensures that the high quality …rm will …nd it pro…table to sponsor 5N=A ads given that its advertising cost is smaller.

However, the answer does depend on N because a higher N increases the value of advertising given that the bene…t from advertising is in the form of higher per unit pro…ts in period 1. The higher N the higher the total pro…t from advertising. Thus, the high-quality …rm needs to spend a larger amount in period 1 in order to separate itself from the low-quality …rm.

7

Exercise 5 Advertising and information Consider a monopolist who has one unit of a product. The outside option of not selling the product is c. This product has high quality sH with probability and low quality sL with probability 1 . There are two consumers, each of whom has valuation vH for high quality and vL for low quality. We assume that vH > c > vL . Furthermore, we assume that vH + (1 )vL > c. Consumers bid for the object using a second-price auction. Suppose that the …rm learns its quality, but that consumers are initially uncertain about product quality.

1. Determine the equilibrium price of the game in which the …rm …rst decided whether to post the item and consumers then make their bids.

2. Consider a two-period extension in which quality is constant over time.

Suppose that consumers learn the quality after period 1— i.e., quality becomes public information— and that they bid for a second unit of the product (both consumers are also identical in period 2) in period 2. Characterize the equilibrium of the two-period game in which in period 1a Nature draws quality, in 1b the …rm becomes privately informed about quality and decides whether to o¤er its product, and in 1c consumers bid for the period-1 product and in period 2a consumers learn the true quality, in 2b the …rm decides whether to o¤er the product, and in 2c consumers bid for the period-2 product. Compare your …ndings to (1).

3. Consider a modi…ed model in which consumers are ex ante uninformed also about the existence of the product. Suppose that the …rm can make the advertising expenditure A( ) at the beginning of period 1, where is the probability that both consumers are informed about the existence of the product. A( ) is assumed to be continuously di¤erentiable and strictly convex with the appropriate limit properties such that you can restrict attention to interior solutions. Note that either both or none of the consumers become informed. Consider the game from (2) with the addition that when the …rm decides whether to o¤er the product it also chooses and consumers observe A( ). A consumer can only buy in period 2 if she has become informed about the existence of the product in period 1. Characterize the separating equilibrium that gives the highest pro…t to the high-quality …rm.

4. Compare your result in (3) to a situation in which consumers observe the quality ex ante (but which is otherwise the same as in part 3).

Solutions to Exercise 5 1. Solved by backward induction. Consumers will bid according to their valuation in the second stage (because of the second-price auction). Since they do not observe the true quality, they both bid their average valuation vH + (1 )vL .

Since this bid is larger than c by assumption, the monopolist will post the object in period 1.

8

2. Solved by backward induction. Consumers will bid according to their valuation in the second stage. Hence, in period 2, they bid vH if the product is of high quality and vL if the product is of low quality. In consequence, the monopolist posts its product in period 2 only if it is of high quality (because c > vL ). In period 1, both consumers and monopolist behave as in a). Hence, the monopolist does not bene…t from the information revelation in period 2 while being in period 1.

3. In a separating equilibrium where only the high quality type advertises, consumers believe that the product is of high quality if they see an advertisement (i.e. a positive amount of advertising expenditures). Then, the low quality type should not have an incentive to mimic the high quality type by choosing the same amount of advertising. Hence, the following incentive constraint must A( ) hold to have a separating equilibrium: (vH c) A( ) 0, (v c) . h

(Note that the low quality type will gain pro…t (vH c) from all consumers that know about the existence of the product in period 1 if it mimics the high quality type. As consumers become informed in the second period, however, the low quality type will always choose the outside option c in the second period.)

4. Under full information about product quality, the high quality type will choose the pro…t-maximizing level of advertising expenditures:

= arg max [2 (vH

c)

A( )] :

This might be smaller than the which is necessary to ful…ll the incentive constraint in c). In this case, the high quality type has to overinvest into advertising in order to separate from the low quality type when quality is not observed.

Exercise 6 Prices and advertising as signals of quality A monopoly operates for two periods and produces a homogenous good whose quality is either high or low (the monopoly cannot choose the quality of the good). In the …rst period, the quality of the good is unobserved by consumers and their demand is q1 = s1 p1 , where s1 is the perceived quality of the good and p1 is the price in period 1. In the second period, the quality of the good becomes common knowledge and the demand for the good is q2 = 4 p2 if the quality is high and q2 = 2 p2 if the quality is low, where p2 is the price in the second period. The per unit cost of production is 1 in the …rst period, and 1 q1 in the second period, where is a positive constant that re‡ects a learning-by-doing e¤ect: the more the …rm produces in period 1, the lower is its per unit cost in period 2. Assume that = 1=4 if the monopoly produces a high quality product and = 1=2 if the monopoly produces a low quality product.

For simplicity, assume that there is no discounting.

9

1. Solve the monopoly’s problem in period 2 and compute the monopoly’s pro…t at the optimum, taking q1 as given (recall that q1 determines the per-unit cost of production in period 2).

2. Write out the sum of the monopoly’s pro…ts in periods 1 and 2 as a function of p1 , given the monopoly’s type, assuming that consumers believe that (i) s1 = 4, and (ii) s1 = 2.

3. Now suppose that in period 1 the monopoly chooses a price, p1 , and a level of uninformative advertising, A. Solve for the strategy of a low type monopoly in a separating equilibrium.

4. Let A(p1 ) de…ne, for each period 1 price p1 , the minimal amount of advertising required by a high quality monopoly in order to deter a low quality monopoly from mimicking it. Given your answers to parts (2) and (3), compute A(p1 ) and show it in a …gure. Moreover, compute the prices at which A(p1 ) crosses the horizontal axis. Explain the meaning of these crossing points.

5. Solve for the price that a high quality monopoly will charge in a Pareto undominated separating equilibrium (one where a high quality monopoly advertises just enough to induce separation, or more precisely, one where consumers believe that the monopoly must be of a high quality if they observe a pair (p1 ; A) which is a weakly dominated strategy for a low quality monopoly) and compute the amount of advertising that it will choose.

6. Compare your answer in part (5) to the optimal strategy of a high quality monopoly in the full information case (the case where the quality is common knowledge even in period 1). Does the monopoly underprice or overprice in equilibrium, relative to the full information case? Explain why the price distortion could serve as a signal for quality in this particular case.

Solutions to Exercise 6 1. In period 2, the quality of the good is common knowledge. Hence, the maximization problem of the monopoly when the quality of the good is s 2 f2; 4g is given by

max(s p

p)(p

(1

where = 1=4 if the quality is high and this problem yields

p2 (s) =

q1 ));

= 1=2 if the quality is low. Solving

s+1 2

q1

(1

q1 ))2

:

Hence, the second period pro…t is 2 (s) =

(V

4 10

:

2. Suppose that the quality is high. Then, the per-unit cost of production in period 2 is 1 q1 =4, so the monopoly’s pro…t as a function of p1 , the belief s1 = 4, and the true quality s = 4 is (p1 ; 4; 4)

=

(4

p1 )(p1

=

9p1 (32 7p1 ) : 64

4

1) +

4 p1 2 4 )

(1 4

If out of equilibrium beliefs are s1 = 2, the monopoly’s pro…ts as a function of p1 would be

(p1 ; 2; 4)

=

(2

p1 )(p1

1) +

=

68 + 164p1 64

63p21

4

2 p1 2 4 )

(1 4

:

Likewise, if quality is low, the per-unit cost of production in period 2 is 1 q1 =2, the monopoly pro…t as a function of p1 , the belief s1 = 4, and the true quality s = 2 would be

(p1 ; 4; 2)

=

(4

=

p1 )(p1

1) +

28 + 68p1 16

15p21

4

4 p1 2 2 )

(1 4

:

and if s1 = 2, the monopoly pro…t as a function of p1 is

(p1 ; 2; 2)

= =

(2

p1 )(p1

1) +

16 + 40p1 16

15p21

4

2 p1 2 2 )

(1 4

:

3. In a separating equilibrium, the identity of the low quality monopoly is revealed so it will obviously choose zero advertising AL = 0 (no need to advertise if consumers already know that the quality is low) and will choose p1 to maximize (p1 ; 2; 2). Using the equation for (p1 ; 2; 2), it follows that the equilibrium price of a low type monopoly is pL = 4=3 and its equilibrium pro…t is L = (4=3; 2; 2) = 2=3.

4. The A(p1) curve is de…ned implicitly by the solution to the equation (p1 ; 4; 2) A = L . Using the equation for (p1 ; 4; 2) we get: A(p1 ) =

116 + 204p1 48

45p21

:

Using this expression, it is straightforward to compute that the A(p1) curve crosses the horizontal axis at 2/3 and 58/15. If p1 < 2=3 or p1 > 58=15, then

11

the low quality monopoly is better-o¤ choosing AL = 0 and pL = 4=3 than choosing p1 and there is no need for a high-quality monopolist to also advertise in order to induce separation.

5. The strategy of a high quality monopoly in a separating equilibrium is given by: max (p1 ; 4; 4) p1

s:t: A

A

A(p1 )

Substituting from the constraint into the objective function and solving, we get that pH = 8=3. The equilibrium amount of advertising is AH = A(8=3) = 9=4.

6. In the full information case the monopoly does not need to advertise so AH = 0.

The pro…t-maximizing price is pH = arg maxp1 (p1 ; 4; 4) = 16=7. This price is less than the price in a separating equilibrium so the monopoly separates himself by overpricing and by advertising. Intuitively, the monopoly wishes to produce as much as possible in the …rst period to lower the cost in the second period. By increasing the price above pH , the monopoly restricts its output in the …rst period and, thus, raises his cost in the second period. This sacri…ce by the high-quality monopoly is not worthwhile for the low-quality monopoly because it enjoys a stronger learning-by-doing e¤ect (twice as strong) and, hence, it is more reluctant to cut its output in period 1. By cutting its output in the …rst period (and burning money by advertising), the monopoly signals to consumers that it cannot possibly be of low quality.

Exercise 7 Pricing and quality information Suppose there are two groups of consumers, group 1 of size and group 2 of size 2 . Consumers have unit demand. Consumers in group 1 have willingness-to-pay for a high-quality good equal to 3 and for a low-quality good equal to 2. Consumers in group 2 have willingness-to-pay equal to 2 independent of the quality of the good.

A monopolist sells the product to consumers. With probability 1=2 it produces high quality and with the remaining probability 1=2 it produces low quality. High quality is produced at unit cost c = 3=2 and low quality is produced at 0 unit cost.

1. Determine pro…t-maximizing prices for each type of the …rm under the assumption that consumer observe the quality before purchase.

2. Suppose now that consumers only observe price but not quality prior to purchase. Consider the case = 1. Does a separating equilibrium exist?

If your answer is a¢ rmative characterize the separating equilibrium which is best for the high-quality …rm. Do not forget to specify consumer beliefs.

12

3. What happens if = 3=2? Does a separating equilibrium exist? If your answer is a¢ rmative characterize the equilibrium which is best for the high-quality …rm. Again do not forget to specify consumer beliefs.

Exercise 8 Content Advertising1 [included in 2nd edition of the book] Consider a monopolist who sells a product that contains two attributes A and B. Each of these attributes can be of high or of low quality. Low quality gives utility ui = 0 and high quality utility ui = 1=2, i = A; B. The willingness to pay for the product is the sum of the attributes’ utilities plus some small, positive …xed value, u0 > 0— i.e., u0 + uA + uB . Nature draws high and low quality with probability 1=2 each, independently across attributes. The cost of production is equal to zero. The realization of the two qualities is private information of the …rm. The …rm can advertise that the product exists at zero costs. The …rm chooses its marketing strategy, consisting of its advertising strategy and its price.

1. Suppose that the monopolist can reveal the quality of both attributes at zero cost. What is the equilibrium outcome?

2. Suppose that the monopolist can at most truthfully advertise the quality of one of the two attributes. What is the equilibrium outcome of such a game?

3. Consider the situation in (2), but suppose that, after the advertising decision, consumers can search at cost z to receive noisy information about the quality of one of the attributes. Clearly, consumers may only search for the quality of an attribute whose quality has not been discloses. Speci…cally, suppose that consumers learn with probability 3=4 that a low-quality attribute is indeed low quality. With the remaining probility 1=4 they obtain the same signal that they receive if the attribute is of high quality.

Show that there exists a semi-separating equilibrium in the which a product with two high-quality attributes pools with a product with two lowquality attributes by not revealing any attribute information. By contrast, products (H; L) and (L; H) inform consumers through advertising about the high quality of one of the attributes.

Characterize the equilibrium search behavior of the consumers.

Characterize the maximal equilibrium prices and pro…ts that can be supported along the equilibrium path.

Show that none of the …rms has an incentive to deviate.

1 Parts (3) and

(4) of this exercise are inspired by D. Mayzlin and and J. Shin (2011) Uninformative Advertising as an Invitation to Search, Marketing Science 30: 666-685 .

13

Give a condition on the level of the search cost such that the equilibrium exists.

4. Discuss your result obtained in (3) with respect to the informativeness of ads depending on product quality.

Solutions to Exercise 8 1. The …rm advertises the existence of the product. By the unravelling argument, it also reveals any attribute with high quality and consumers correctly infer that any attribute that is not advertised must be of low quality.

Prices are p(L; L) = u0 , p(L; H) = p(H; L) = u0 + 1=2, p(H; H) = u0 + 1.

2. The …rm with two high-quality attributes pools with the …rm with only one highquality attribute by informing consumers that one of their attributes is of high quality. The expected utility of such a product is [(1=2)(1=2)+(1=4)1]=(3=4)+ u0 = (1=2)=(3=4) + u0 = 2=3 + u0 . Thus, p(L; L) = 2=3, p(H; L) = p(L; H) = p(H; H) = 2=3.

3. 2 All types advertise; the (H; L) and (L; H)-types choose informative advertising. If consumers see informative advertising they believe that, with probability one, the other attribute is of low quality and do not search. If consumers see advertising that is uninformative in the sense that is does not contain any information on content, they search. Suppose that the product sold by the (L; L) and (H; H) …rm is sold at price p. Consumers who observe that the attribute they investigated is of low quality, do not buy the product. Otherwise, they buy if the price is not too high. The expected net utility prior to search is (1=2)(1=4) (u0 p) + (1=2)1 (1 + u0 p) z . If the …rm extracts the full expected surplus this expression is equal to zero. Solving for p, we obtain that p = u0 + (4=5) (8=5)z .

Hence,

(L; L)

= (1=4)p = (1=4)[u0 + (4=5) = u0 =4 + 1=5(1 2z)

(8=5)z]

which must be greater than u0 to constitute an equilibrium— i.e., u0 must be su¢ ciently small. Otherwise, the (L; L)-…rm has an incentive to separate itself by setting the price equal to u0 .

(H; H) = p = u0 + (4=5) (8=5)z, (H; L) = (L; H) = u0 + 1=2.

2 Parts (c) and (d) of the exercise are inspired by D. Mayzlin and and J. Shin, 2009, Uninformative Advertising as an Invitation to Search, mimeo.

14

The latter follows from the fact that p(H; L) = p(L; H) = u0 +1=2 extracts the full surplus of a product with one high-quality attribute. Clearly, for z 1=2 the (H; H)-…rm does not have an incentive to mimic the (H; L) and (L; H)…rms. It remains to be shown that the (H; L) and (L; H)-…rms do not have an incentive to pool with the other …rms. If they do so with probability 5=8 they can sell the product at price p because consumers will obtain a positive signal with this probability. With the remaining probability consumers believe that both attributes are of low quality.

Thus, the deviation pro…t is (5=8)p = (5=8)[u0 + (4=5) (8=5)z] = (5=8)u0 + z which is strictly less than the pro…t along the equilibrium path. The search cost must satisfy: z < 1=2 (15=8)u0 (to satisfy the IC-constraint of the (L; L)-type).

1=2

4. The …rm with two high-quality attributes prefers not to reveal high quality of one of the attributes in order to invite consumers to search. The noisy information on one of the attributes obtained through search is more valuable because consumers have negatively correlated beliefs in case of informative advertising, but positively correlated beliefs in case of non-informative advertising combined with search.

Exercise 9 Quality of business school teaching A business school o¤ers an MBA program with two areas of specialization: …nance and marketing. For simplicity, suppose that the school is facing only two potential students: one who is interested only in …nance and his utility from studying in the school is 2sF p and the other who is interested only in marketing and his utility is sM p, where sF is the quality of the …nance courses, sM is the quality of the marketing courses and p is the tuition. If either one of the two students decides not to enroll his utility is 0. (We can easily extend the problem and consider many students of each kind but this will not change any of the results). Suppose that the business school can choose the quality of the …nance and the marketing courses, but the cost is increasing with the quality of the courses: The cost of …nance courses for the school is (sF )2 =2 and the cost of the marketing courses is (sM )2 =2. For simplicity, assume that the tuition, p, is determined by the government (the school cannot choose it) and is equal to 2. The objective of the school is to choose the quality of the …nance and the quality of the marketing courses in order to maximize its income from tuition minus the cost of providing courses.

1. Suppose the students can tell the quality of the courses before they enroll.

What is the quality above which each student will decide to enroll.

2. Given your answer to (1), compute the quality of the courses that the school will o¤er and the school’s pro…t. Explain what will happen if p > 2.

15

3. Which courses will have higher quality: …nance or marketing? Which student cares more about quality? Does the school provide e¢ cient level of quality or not?

4. Now suppose that the students cannot observe the quality of the courses before they enroll. Compute once again the quality of the courses that the school will o¤er and the school’s pro…t (note that the school chooses the quality although the students cannot observe it before they enroll).

5. Now suppose that the students can only observe the average quality of the courses provided by the school. That it, students only observe s (sF + sM )=2 but they cannot observe sF alone and sM alone. Suppose that the students observe an average quality s and both enroll. What will be the qualities sF and sM that the school would like to choose? (Hint: think about the combination of sF and sM that produces a given s at a minimum cost) 6. Now suppose that the students anticipate the quality choices that the school makes. What is the value of s for which both students will enroll?

7. Are the students better-o¤ when they observe both sF and sM or are they better o¤ when they only observe s? What about the school: is the school better o¤ when the students observe both sF and sM or when they only observe s? Explain the intuition for your answer in detail.

Solutions to Exercise 9 This problem is based on the paper “The Economics of Quality Indexes” by Glazer and McGuire.

1. If the students can tell the quality of the courses before they enroll, each will enroll if by doing so she gets a positive utility. Since the utility of the student who is interested in …nance is 2sF 2, it is clear that he will enroll only if sF 1. Likewise, the student who is interested in marketing will enroll only if sM 2.

2. Since quality is costly, the school will o¤er the minimal quality that will induce each student to enroll. That is, the school will o¤er the qualities sF = 1 and sM = 2 and its pro…t will be 22 3 = : 2 2 If p > 2, the school will be better o¤ setting sM = 0 in which case the school 12 2

=2 2

will essentially o¤er only the “cheap” …nance courses that do not require a lot of quality (the marketing student will not enroll). The school’s pro…t in that case is 2 1=2 = 3=2. By o¤ering marketing courses with positive quality, the school can only lose money: if sM < p, the marketing student will not enroll anyway and the school will have higher costs but the same income as before.

To attract the marketing student, the school needs to set sM = p, but then the cost of the marketing courses will be p2 =2. Since p > 2, the cost is more than 2 which is the tuition paid by the marketing student.

16

3. From (2) it is clear that the quality of the marketing courses will be twice as much as the quality of the …nance courses despite the fact that the …nance student cares about quality twice as much as the marketing student. Hence, the provision of quality is clearly ine¢ cient since the students who cares less about quality gets more of it than the one who actually cares more about it.

4. The school will now set sF = sM = 0. To see why, suppose that the students believe that the quality is above 1 for …nance and 2 for marketing and they are willing to enroll. The school then gets an income of 4 for sure and can maximize pro…ts by setting sF = sM = 0. If the students do not enroll then it is surely optimal to set sF = sM = 0 in order to save costs.

5. Holding the average quality …xed, the school minimizes its cost by setting sF = sM . Hence, if the average quality is s, the school minimizes its costs by choosing sF = sM = s.

6. If the students anticipate that sF = sM = s, it must be the case that s

2

to ensure that the marketing student wishes to enroll. Given this quality, the …nance student will surely enroll.

7. When the students only observe s, the school will have to set sF = sM = 2 in order to induce both students to enroll. The school’s pro…t will then be

=2 2

22 2

22 = 0: 2

Surely, the school is now worse o¤, the …nance student is better o¤ as he gets twice as much quality as under full information, while the marketing student is indi¤erent. Intuitively, when students know only the average quality of the courses, the school’s incentive to o¤er courses of the exact same quality (to minimize costs) implies that the school would have to set both courses at a high level to convince the marketing student that the quality of the marketing courses are indeed high. Any lower average quality will signal to the marketing student that the quality of the marketing courses is not su¢ ciently high and he will simply not enroll. The …nance student then enjoys a higher quality as a results and, thus, he is better o¤ (interestingly, it is because the school wants to attract the marketing students that the …nance students gets higher quality).

Remark: Another option that the school has is to choose s = 1. Now the students will correctly anticipate that the school chose sF = sM = 1. In that case, only the …nance student will enroll and the school’s pro…t will be:

=1 2

12 2

12 = 1: 2

Since this is more than 0, the school will actually choose this option rather than set sF = sM = 2 and induce both students to enroll. Therefore, we are in a strange situation in which the school invests in marketing even though there is no demand for it (and the school anticipates that there will be no demand

17

for marketing). Yet, the school invests in marketing nonetheless, in order to convince the …nance student that the quality of …nance is at least 1. Relative to full observation, the school earns less money, 1 vs. 3/2, and the two students are indi¤erent: the …nance student gets a quality of 1 and pays a tuition of 2 either way so his utility in both cases is 0, while the marketing student gets a quality of 2 and pays 2 (and hence gets 0 utility) under full information, and in the case where only the average quality can be observed, he does not enroll and his utility is 0.

Exercise 10 Warranties Manufacturers issue warranties for their products, often beyond the mandatory minimum length. However, the warranty is restricted to require consumers to take proper care of the product. Discuss this restriction. Should consumer protection apply to any malfunctioning of the product within a given time period?

Exercise 11 Money-back guarantees Consider a monopolist who sells a single unit of a product that may be of high or low quality. Both events occur with probability 1=2. There are at least two identical consumers. Consumers are willing to pay 1 for a functioning product and 0 for a product that breaks down. A high quality product breaks down with probability 1=4 whereas a low-quality product breaks down with probability 3=4. Consider the game in which the …rm …rst observes its quality (but not its actual performance, which is only observed after consumption), then decides whether to enter the market or to choose the outside option. Afterwards consumers simultaneously bid for the product, e.g., in a second-price auction.

Assume that consumers bid their true willingness-to-pay. (The product goes to the highest bidder at the price of the second highest bid. With equal bids the product is assigned with equal probability to each bidder.) The outside option gives pro…t 0.

1. Suppose that the monopolist has a cost of 0 to produce and sell the product independent of its quality. Determine the equilibria in the game in which consumers simultaneously bid for the product and in which consumers bid their true willingness-to-pay.

2. Suppose now that the …rm incurs a cost of c > 0 for high quality (and 0 for low quality). This cost is incurred if the …rm decides to produce the product and o¤er it in the market. If it decides not to o¤er the product, it makes zero pro…t. Determine equilibria for all values of c 2 (0; 1).

(Consider only equilibria in which high and low-quality …rms choose pure strategies.)

3. Return to the case in which the …rms’costs are zero. Suppose now that, at the stage between entering and consumer bidding, the monopolist can 18

commit to a money-back guarantee. This money-back guarantee has the feature that for each faulty unit that is returned the …rm pays back the full price. In addition it incurs a transaction cost t 2 (0; 3) per unit returned.

Would the …rm o¤er money-back guarantees in equilibrium? Characterize the equilibrium with money-back guarantee (depending on t).

4. The monopolist is considering to advertise its product by spending A after entering (instead of o¤ering a money-back guarantee). Discuss the e¤ect of such a strategy. Is advertising possibly pro…table. Analyze the equilibria in this market. Discuss informally which equilibrium should be selected and what are the features of this equilibrium.

Exercise 12 Warranties as signal of quality [included in 2nd edition of the book] Consumers wish to buy a product and get a utility 10 if the product is of high quality and is working, and a utility of 4 if the product is of low quality and is working. If a product does not work, then the utility from having it is 0 irrespective of its quality. Ex ante, only the …rm knows the quality of its product (but cannot choose it). Consumers expect that the product is of high/low quality with equal probabilities. The likelihood that a high quality will not work is 1/5 and the likelihood that a low quality product will not work is 4/5. The cost for the …rm of replacing a product that does not work is c (this cost is independent of the product’s quality) 1. Suppose the …rm does not o¤er a warranty. What is the price that consumers will pay for the product?

2. Suppose that the …rm does o¤er a warranty. What are the conditions on c such that o¤ering a warranty can serve as a signal of high quality?

3. What happens if the conditions you found in (2) are violated?

4. Compare the pro…ts of …rms in (2) and in (3). Comment on your …ndings.

Solutions to Exercise 12 1. Absent a warranty, consumers agree to pay (10 4=5+4 1=5)=2 = 44=10 = 4:4.

2. Suppose the …rm o¤ers a warranty and consumers believe that its quality is high. Hence, they agree to pay 10 if there is a warranty. If there is no warranty, consumers believe that the product is of low quality (because if quality was high the …rm would have o¤ered a warranty) and, hence, they agree to pay 4 x 1/5 = 4/5. A high quality …rm earns under these assumptions 10 c=5 if it o¤ers a warranty and 4=5 otherwise. A low-quality …rm earns 10 4c=5 if it o¤ers a

19

warranty and 4=5 otherwise. The consumers will correctly believe that only a high quality …rm o¤ers a warranty is correct if:

10

c=5 > 4=5, and 10

4c=5 < 4=5;

or, equivalently, 46 > c, and 46=4 < c. Thus, we must have 11:5 < c < 46 to sustain warranties as a signal.

3. Suppose that c < 11:5. Then, even a low quality …rm …nds it pro…table to o¤er a warranty, so o¤ering one is not enough to convince consumers that quality is high. In that case, consumers will o¤er 4:4 for the product. It is reasonable to assume that in this case, …rms will not o¤er warranties since o¤ering one does not enable them to charge higher prices. Note however that if c is particularly low, then …rms o¤ering warranties can be supported as an equilibrium if not o¤ering a warranty is interpreted by consumers as a signal for low quality. In that case, …rms o¤er warranties even when their products are of low quality. Of course, this can be an equilibrium only of both types of …rms have an incentive to o¤er warranties. A low quality …rm will o¤er a warranty in this case provided that 4:4 4c=5 > 4=5, since if it does not o¤er a warranty, it is believed to be of low quality and its pro…t then is 4=5.

If c > 46, then even a high quality …rm does not …nd it pro…table to o¤er a warranty, so no …rm will o¤er a warranty and again consumers will agree to pay 4.4 for the product.

4. In (2), the high-quality …rm earns 10

c=5 while in (3) it earns 4.4 (assuming that in equilibrium no warranties are o¤ered). If c > 5:6 5 = 28, the …rm is actually better o¤ in (3). It o¤ers warranty nonetheless because consumers expect it to do so and if it does not, they agree to pay merely 4=5, not 4:4.

The low-quality …rm earns 4/5 in (2) and 4.4 in (3). It de…nitely prefers c to be above 46 because then consumers will not be able to tell it apart from a high quality …rm. Then, the low-quality …rm will be able to charge a higher price.

Exercise 13 Quality inspection [included in 2nd edition of the book] A monopolist sells a product whose quality is unknown to consumers before they buy. It is common knowledge however that the product’s quality, s, is drawn from a uniform distribution on the unit interval. There is continuum of consumers with a total mass of 1. Each consumer is interested in buying at most 1 unit and gets a utility of s p if he buys and 0 otherwise. The monopoly can send its product to inspection before consumers buy. The cost of inspection is c. The inspection perfectly reveals s to consumers with probability . With probability 1 , it reveals nothing in which case consumers cannot tell whether the monopolist has sent its product to inspection or did not.

1. Write out the monopoly’s pro…t if it sends its product to inspection and if it does not.

20

2. Prove that the monopoly will send its product to inspection if and only if s is above some threshold sb.

3. Compute the expected quality of the monopoly’s product if consumer do not see the inspection results (hint: you should compute a weighted average of two conditional expected values: one for cases in which s < sb and one for s > sb; the weights depend on the probability that s < sb and the probability that s > sb and that the inspection reveals nothing).

4. How does the expected quality that you computed in (3) vary with Explain the intuition.

?

5. Given your answer to (3), compute the threshold sb.

6. How does s vary with c and with ? Explain the intuition.

Solutions to Exercise 13 1. The monopolist will set a price equal to the maximal willingness of consumers to pay. Hence, if it sends its product for inspection, its pro…t is I

(^ s; s) = s + (1

)^ s

c;

where s^ is the expected quality of the product if the inspection reveals nothing.

If the monopolist does not send its product for inspection, its pro…t is NI

(^ s; s) = s^:

(^ s; s) is independent of s while I (^ s; s) increases in s, it is clear that there exists a threshold s such that the monopoly will send its product to inspection if and only if its quality s is higher than s.

2. Since

NI

3. Since s is distributed uniformly on the unit interval, the probability that s < s is s and the probability that s > s and the inspection reveals nothing is (1 )(1 s). Hence, Rs sdF (s) s 0 s^ = s s + (1 )(1 s) R1 sdF (s) (1 )(1 s) s + s + (1 )(1 s) 1 s 1 (1 s2 ) = : 2(1 (1 s)) 4. Di¤erentiating s^ with respect to

reveals that it falls with . Intuitively, when increases there is a smaller likelihood that when no inspection results are observed this might be due to the bad luck, i.e., that the monopoly has a high-quality product, sent it to inspection, but the inspection did not reveal any results. This argument is of course incomplete because we hold s constant despite the fact that it is actually a function of .

21

5. The threshold s is de…ned implicitly by the equation N I (^ s; s) = I (^ s; s).

Substituting for s^ into this equation, using the pro…t function we wrote in (1), and simplifying, yields

s=

+c

I+

p

1 + c2

:

6. Clearly s increases with c. The reason is that when by revealing the product’s quality the monopolist can sell the product at a premium (above the average price), but unless s is high, this premium is not big enough to cover the cost of inspection. As inspection is more costly, the …rm prefers not to send its product for inspection for a wider range of values of s since the gain is outweighed by the cost. Di¤erentiation also reveals that s decreases with : the higher is (the more "accurate" the inspection is), the more the monopolist is inclined to send the product to inspection. The intuition here is that the bene…t from inspection is that the product is sold at a premium. But the monopolist gets this bene…t only if the inspection succeeds which occurs with probability . Hence, as increases the monopolist bene…ts more from inspection and has an incentive to send its product to inspection for a wider range of values of s.

Exercise 14 Quality information and testing Consider a monopolist selling computer software. Software is of high or low quality and is chosen by Nature: quality is high with probability and low with probability 1 . Consumers can buy the software in period 1 and consume it in periods 1 and 2, where the discount factor is < 1. Software is produced at …xed cost Fi and marginal costs ci , i = H; L. The software producer learns quality before period 1 whereas consumers only learn the quality after purchase.

Each consumer has the same willingness-to-pay rH for high quality and 0 for low quality. There is mass M of consumers.

1. Suppose rH cH and (1 + )(rH cH )M FH < 0 but (1 + )( rH cH )M FH < 0. What is the equilibrium in this market? Interpret your result. Compare your result to the solution under full information.

2. Suppose now that consumers can copy the software at an opportunity cost c c pc and value the copy at ric with rH > rH > rL = 0, i.e., copies of highquality software are an imperfect substitute for the original. Suppose that c c (1 + ) rH > pc and rH rH > maxfcL ; cH g for a share of consumers who are considering to copy the product (“pirates“). The remaining share 1 will never copy the software. Copiers learn the product quality after period 1 and have the option to buy the product in period 2 given the price set in period 2. Is it possible that the …rm is better o¤ under the presence of copying compared to the situation where = 0? Prove your answer formally. What is the intuition?

22

3. Discuss what would happen if the …rm could choose the degree of copyright enforcement (without any cost). What is the optimal level of copyright protection? Provide some intuition and discussion without necessarily doing the calculations.

Exercise 15 Information disclosure Consider a market in which …rms have private information about their quality s 2 [0; 1]. Quality is drawn from the uniform distribution on the 0-1 interval; this is common knowledge among …rm and consumers. After observing its type the …rm decides whether to reveal its quality to consumers (it has the choice whether or not to reveal its quality but not to mislead; a justi…cation is that the …rms sends a sample to a certifying authority for testing but that publishing the test result is within its own control) and sets a price p. Consumers derive utility u = se p, where se denotes the expected quality.

1. Suppose the government mandates information disclosure. Characterize the solution to the …rm’s pricing problem.

2. Suppose now that the …rm can reveal information at zero cost. Characterize the solution to the …rm’s pricing and information disclosure problem.

3. Suppose now that there is a dislosure costs k > 0. Characterize the solution to the …rm’s pricing and information disclosure problem as a function of k.

4. Above which level of k will the …rm never disclose private information?

Exercise 16 Mandatory disclosure rules Consider as above a market in which …rms have private information about their quality s 2 [0; 1]. Quality is drawn from the uniform distribution on the 0-1 interval; this is common knowledge among …rm and consumers. After observing its type the …rm decides whether to reveal its quality to consumers (it has the choice whether or not to reveal its quality but not to mislead; a justi…cation is that the …rms sends a sample to a certifying authority for testing but that publishing the test result is within its own control) and sets a price p.

Consumers derive utility u = se p, where se denotes the expected quality. With probability (which is independent of quality) the …rm can costlessly disclose information, with the remaining probability it cannot disclose information on product quality.

1. Characterize the solution to the …rm’s pricing and information disclosure problem.

23

2. Discuss your result in light of the resuts obtained in the previous exercise.

3. Does mandatory disclosure increase welfare? Discuss.

Exercise 17 Product quality and repeated interaction Consider an industry in which …rms produce an experience good and sell it to a continuum of consumers with a total mass of one (i.e., we normalize the number of consumers to 1). There are in…nitely many periods. In every period, each consumer is willing to buy at most one unit of the good provided that its quality is high. Consumers know that …rms can choose every period whether to produce a high quality good at a cost of 10 per unit or a low quality good at a cost of 4 per unit. Suppose that consumers adopt a boycott strategy and never buy from a …rm that sold a low quality good in the past. Let be the intertemporal discount factor.

1. Compute the lowest price that guarantees the production of high quality in every period.

2. Now suppose that each period, there is a probability 1 that a superior good will be invented and that the current period will be the last. Repeat your answer to part (1) and explain how the lowest price that guarantees the production of high quality in every period depends on . Make sure you explain in detail the economic intuition for your answer. (Hint: in the presence of , the expected discounted value of 1 dollar tomorrow from today’s point of view is ).

3. Now suppose that each period, the number of consumer grows at a rate of g so that there are 1 + g consumers in the period 2, (1 + g)2 consumers in period 3, (1 + g)3 consumers in period 4 and so on (note that in period t there (1 + g)t 1 consumers). Repeat your answer to part (1) and explain how the lowest price that guarantees the production of high quality in every period depends on g. Once again, make sure you explain in detail the economic intuition for your answer. (Hint: in the presence of g, the expected discounted value of 1 monetary unit tomorrow from today’s point of view is (1 + g)).

4. Suppose that = 1 and g = 0 (the case we considered in part (1)) and assume that there are n …rms in the industry and that these …rms are competing by setting prices. Consumers buy from the lowest price …rm (provided that buying at this low price is better than not buying at all!); if several …rms charge the same low price, then consumers pick one of them at random and buy from it. What will be the price that …rms will set in equilibrium (i.e., when no …rm can improve its pro…t by changing its price)? Compute the per-period pro…t of each …rm and the discounted in…nite sum of its per-period pro…ts (i.e., the "net present value" of the …rm).

24

5. Given your answer in (3), how many …rms will enter the industry in the …rst place if entry requires a one-time investment of 18? (Hint: if a …rm stays out of the market its pro…t is 0; entry makes sense only if a …rm can earn a positive pro…t).

6. How does the number of …rms that you computed in (4) vary with ? That is, are there more or less …rms in the industry when is higher? Explain the intuition for this.

7. Consider two geographical markets, A and B, that behave according to the model described in this question. Assume that the two geographical markets are the same in every respect, save for which is higher in market A than in market B. Which market will have more …rms? Which market will have higher prices? If you compare the two markets by observing the number of …rms and the prices in each market what will you conclude about the correlation between the number of …rms and prices: Is there a positive or a negative correlation (i.e., are more …rms associated with higher or lower prices)? What is the intuition for the correlation you …nd?

8. On the basis of your previous answers, would you expect more or less …rms in a fast growing market relative to stagnant markets? Explain your answer in detail and explain the intuition.

Solutions to Exercise 17 1. If a …rm produces high quality in every period then it can sell at a price of p forever. The present value of the …rm’s pro…ts is

(p

10) + (p

10) +

2

(p

10) + : : : =

p 10 : 1

On the other hand, if the …rm sells a low-quality product once, then it cannot sell in latter periods since consumers will boycott it, so its pro…t is p 4.

The lowest price that guarantees the provision of high quality is such that

p 10 =p 1

4:

Hence, this price is

6 p =4+ : 2. Now, the present value of the …rm’s pro…ts if it sells a high-quality product in every period becomes

(p

10) +

(p

10) +

2 2

(p

10) + : : : =

Since the one-time pro…t from selling low quality is still p that guarantees the provision of high quality is such that

p 1

10

25

=p

4:

p 1

10

:

4, the lowest price

Hence, this price is

p =4+

6

:

It is easy to see that the minimal price needed to ensure the provision of high quality decreases with . This is not surprising since when increases, there is a higher chance that the market will continue one more period and, thus, …rms care more about the future. Consequently, they are less willing to adopt a ‡yby-night strategy (selling low quality once before exiting the market forever) and a lower price is su¢ cient to induce them to produce high quality than adopt a ‡y-by-night strategy. Another way to look at it is to note that when increases, …rms have a higher chance to reap the bene…ts from a good reputation. Hence, there is less need for a high price to ensure that …rms wish to produce high quality.

3. Given a growth rate g in the number of consumers, the present value of the …rm’s pro…ts if it sells a high quality product in every period becomes

(p

10) + (1 + g)(p

2

10) +

(1 + g)2 (p

10) + : : : =

p 1

10 : (1 + g)

Since the one time pro…t from selling a low quality is still p 4, the lowest price that guarantees the provision of high quality is now such that

p 1

10 =p (1 + g)

p

=4+

4:

Hence, this price is

6 : (1 + g)

It is easy to see that an increase in g lowers the minimal price needed to ensure the provision of high quality. The reason for this is that as g increases, staying in the market by selling a high quality today becomes more important for …rms (a higher g means that there are more consumers around and, hence, higher pro…ts to be earned). Consequently, …rms are less willing to adopt a ‡y-bynight strategy, so a lower price is su¢ cient to induce them to produce high quality than adopt a ‡y-bynight strategy.

4. Competition among …rms will induce them to cut prices in order to get a higher market share. Yet, …rms cannot lower their prices below p otherwise consumer will expect them to provide low quality. Hence, the equilibrium price must be p . Since all …rms will set the same price, each will get a share 1=n of total sales. That is, the pro…t of each …rm in every period will be

=

1 (p n

10) =

6(1

) n

:

The discounted sum of pro…ts is:

+

+

2

+ ::: =

26

1

=

6 : n

5. Entry will take place up to the point where the discounted sum of pro…ts equals 18. Hence, 6=( n) = 18, which give the number of …rms with entry, n = 1=(3 ).

6. As we can see, n decreases with : the higher is, the fewer …rms will operate in the industry. To see why, note that p falls when increases: that is, the higher is , the lower is the price that …rms can charge. The reason for this is that when is high, the quality premium is small, since even a relatively small premium is su¢ cient to induce …rms to choose high quality— after all with a high , the "future is important" so even with a relatively low price …rms do not wish to risk their future sales for a one time gain due to quality cuts. Hence, when is high pro…ts are low and, hence, the market supports only few …rms.

7. From part (1) we know that p decreases with and from part (6) we know that n decreases with . Hence, the comparison between markets A and B will reveal positive correlation between prices and the number of …rms: Higher prices will be associated with more …rms operating in the market. The intuition is that higher prices allow more …rms to enter the market so that the causality is not from the number of …rms to prices but in the opposite direction: Higher prices induce more …rms to enter.

8. The higher

is, the fewer …rms will operate in the industry. In part (3) we saw that when the market is growing, the e¤ective discount becomes (1 + g).

Hence, the faster the market is growing, the higher the e¤ective discount factor and hence the fewer …rms will operate in the market. The reason for this is that the faster the growth rate, the lower the prices and, hence, the fewer will be the number of …rms that will enter.

Exercise 18 Moral hazard and reputation A …rm operates for two periods and sells in each period an experience good.

There is a continuum of consumers with a total mass of one. Each consumer wishes to buy at most one unit. The utility of a consumer who buys the good at a price p, is V p if the quality of the good is high and p if the quality is low, where V > 0. If a consumer does not buy his utility is 0. The …rm can choose in each period whether to produce high or low quality. It costs c > 0 to produce a high quality good and 0 to produces a low quality good. The intertemporal discount factor is .

1. Solve for the quality and pricing decisions in the second period.

2. Using your answer in (1), solve for the quality and pricing decisions in the …rst period.

3. Now suppose that with probability , the …rm believes that it is morally wrong to produce low quality and it therefore produces only the high quality good no matter what. With probability 1 the …rm is as before and can choose the quality of its good in every period. Only the …rm knows 27

its type. Consumers only know that with probability the …rm will only produce high quality and they also know in period 2 which quality the …rm sold in period 1. Restate your answer to (1). (Hint: in order to solve for the optimal price, you should distinguish 3 possible cases depending on the type of quality consumers have bought in period 1 and depending on whether consumers believe that a …rm that can choose the quality will provide high or low quality in period 1).

4. Given your answer to (3), …nd a condition on that ensures that a …rm which can choose the quality of its good will provide a high quality product in period 1. (Hint: think about the options that the …rm has: it can either sell a low quality in period 1 in which case its identity is known in period 2 or sell a high quality in period 1 in which case consumers cannot tell its type in period 2). Discuss 3 factors that make this condition more likely to be satis…ed and explain why.

5. Solve for the equilibrium quality and pricing decisions in period 1 assuming that the condition you found in (4) holds.

6. Explain what you think happens when the condition you found in (4) fails (you are not required to solve for this case since the solution is quite involved; you are expected to think about what might happen even if you cannot fully characterize the outcome).

Solutions to Exercise 18 1. p = V for high quality and p = 0 for low quality.

2. The …rm will provide low quality in both periods and will charge them 0 since consumers will not agree to pay positive prices.

3. Since the game ends after period 2, a type 2 …rm will surely produce a low quality in period 2 (there is no future after period 2 so there is nothing to gain by being “honest” and producing a high quality). By contrast, a type 1 …rm always produces a high quality. As for the price in period 2, there are three possible cases: Case 1: If consumers bought a low quality in period 1, then they know that the …rm’s type must be type 2 and, hence, they will correctly anticipate a low quality in period 2 as well. Therefore, they will not agree to pay a positive price for the product.

Case 2: If consumers bought a high quality in period 1 and it is known that a type 2 …rm can bene…t from o¤ering a high quality in period 1 (i.e., it pays a type 2 …rm to build a good reputation in period 1), then consumers cannot tell whether the …rm’s type is high or low. In that case, consumers expect a high quality with probability (if the …rm’s type turns out to be 1) and a low quality with probability 1 (if the …rm’s type turns out to be 2), and will therefore agree to pay no more than V which is what both types will charge.

28

Here, a type 2 …rm that provides high quality in period 1 invests in reputation and builds it in order to “milk” it in period 2.

Case 3: This case is similar to case 2 except that now it is known that it does not pay a type 2 …rm to build a good reputation in period 1. That is, now it is known that a type 2 …rm adopts a “‡y-by-night” strategy whereby it sells low quality in period 1 and makes a one-time gain (the gain arises from the fact that consumers cannot tell the …rm’s type in period 1 and will pay V ; the …rm’s pro…t then is V since the cost of providing low quality is 0). This time, after buying high quality in period 1, consumers know that the …rm’s type must be 1 (only type 1 …rm will provide a high quality in period 1) and, therefore, they will agree to pay a price of V in period 2 as they except high quality with probability 1.

4. In period 1, a type 2 …rm can either sell a high quality at a price of V in which case it can sell low quality in period 2 at a price of V (i.e., the …rm’s strategy is to build reputation in order to “milk” it later), or can sell a low quality at a price of V but then not sell at all in period 2 (i.e., adopts a “‡y-by-night” strategy). Given , the condition that ensures that the type 2 …rm will provide a high quality product in period 1 is given by (V c) + V V , which is equivalent to

c : V

As one can see, this condition is more likely to be satis…ed when (i) c is low (the “temptation to cheat”is low), (ii) is high (the future is “important”), and (iii) V is high (the “reward” is large).

5. If

c=( V ), then both types will provide a high quality, will charge V , and consumers will buy the product.

6. Things are more complex when

< c=( V ). In that case it is possible to …nd a condition that will ensure that a type 2 …rm will adopt a ‡y-by-night strategy by producing a low quality in period 1. To …nd this condition, note that if consumers expect that a type 2 …rm will sell a low quality in period 1, then in period 1 they will not agree to pay more than V for the product. Since after buying low quality in period 1 consumers know that the …rm’s type is 2 and, hence, expect it to sell them a low quality in period 2 as well, the pro…t from adopting a ‡y-by-night strategy is V .

A ‡y-by-night strategy is optimal, however, only if a type 2 …rm cannot gain from building a good reputation in period 1 in order to “milk” it in period 2. If a type 2 …rm deviates from the ‡y-by-night strategy and provides a high quality in period 1 after all, then since consumers expects it to provide low quality, they believe after buying high quality in period 1 that the …rm must be of type 1 and, hence, they will anticipate (incorrectly!) a high quality in period 2 as well. A type 2 …rm’s pro…t from such a deviation is therefore ( V c) + V . That is, the …rm will make a pro…t V c in period 1 and will sell low quality in period 2 at a price of V thus exploiting the fact that consumers (wrongly) believe that

29

its type is 1. We now want to make sure that such a deviation is not pro…table (if it were then the …rm would not have adopted the ‡y-by-night strategy).

The condition that ensures that this is the case is given by or, equivalently,

V >( V

c)+ V

c > 1: V

This condition says that a ‡y-by-night strategy is optimal only when c is relatively large and and V relatively small.

< 1, it is impossible that both c= V > 1 and c=( V ). If < 1, then a type 2 …rm is indi¤erent between high and low quality

Note that since

c=( V ) < in period 1.

Exercise 19 Milking reputation Give an example in which a …rm “milks its reputation”over time. Ideally you …nd a real world case which has been documented in the business press. Note that you have to argue that milking its reputation was a deliberate decision by the …rm in question. Document your exposition by providing the sources that you have used.

Exercise 20 Umbrella branding3 [included in 2nd edition of the book] Suppose that a single …rm sells two products of potentially di¤erent qualities.

Qualities are described by numbers H and L , measuring the willingness to pay of all consumers. By de…nition, consumers are willing to pay more for high than for low quality, H > L . Qualities, viewed as random variables, are assumed to be independent across products: Each product is of high quality with probability p. The realized product quality is observed by the …rm but not always by consumers: The product is tested with probability by a third party, in which case quality realization is truthfully communicated to consumers. Before these tests are performed, the …rm has to take several decisions. It decides which products to o¤er in the market. It may not want to o¤er a low-quality product on the market because the revenues from selling low quality may not recover the sunk cost f . Suppose that selling low quality is socially undesirable, i.e., L f < 0, but privately pro…table if low quality is wrongly percieved to be of low quality if not tested, i.e., L + (1 ) H f > 0.

The …rm can sell its products under an umbrella brand at a cost k. The timing of the game is as follows: Stage 1: Nature chooses the quality of both products as independent draws from a pool in which high quality occurs with probability p. The product qualities are observed by the …rm but not by consumers.

3 This exercise is largely inspired by Hakenes and Peitz (2009), Umbrella Branding and Exgternal Certi…cation, European Economic Review 53, 186–196.

30

Stage 2: The …rm decides which products to o¤er on the market and pays the associated …xed cost per product f . It also decides whether to use umbrella branding at a cost k.

Stage 3: Consumers observe whether products are sold under an umbrella brand. They also observe the true quality of a product with probability (where the underlying random variables are independent across products). Consumers update their beliefs and bid for the two products.

Consider perfect Bayesian equilibria (PBE) of this game and suppose that consumers bid their expected surplus. Provide conditions under which a …rm with two high-quality products uses umbrella branding to signal its quality to consumers.

Solutions to Exercise 20 Suppose that umbrella branding may signal product quality and is used by a …rm with two high-quality products but not by a …rm of a di¤erent type. Then a …rm with only one high-quality product cannot distinguish itself from a …rm with two low-quality products in the event that product quality is not detected. We will now provide conditions under which entry with this kind of (partial) pooling is pro…table and only a …rm of type ( H ; H ) sells its products under an umbrella brand.

In the candidate equilibrium, a …rm with two high-quality products uses umbrella branding and all other types enter but do not use umbrella branding and sell each product under a separate brand (alternatively we could provide conditions under which any …rm of a type other than ( H ; H ) does not enter). A …rm belongs to this latter group with probability 2 p (1 p) + (1 p)2 = 1 p2 . Since the unconditional probability that a high-quality product is not sold under an umbrella brand is p (1 p), we have that p (1 p)=(1 p2 ) = p=(1 + p) is the probability that a product is of high quality if it is sold under a separate brand. Correspondingly, 1=(1 + p) is the probability that a product is of low quality if it is sold under a separate brand. Along the equilibrium path we thus have the following conditional beliefs: b( i ju = 0; Ii = f H ; L g; Ij ) = 1=(1 + p) for all Ij . In addition, two products that are sold under an umbrella brand are believed to be of high quality unless there is contradictory evidence, i. e., b( i ju = 1; Ii = f H ; L g; Ij = f H ; L g) = b( i ju = 1; Ii = f H ; L g; Ij = f H g) = 1. To complete the belief system, we specify out-of-equilibrium beliefs that support this equilibrium. Suppose that when consumers observe low quality for a product within an umbrella brand, they are pessimistic and believe that the quality of the other product is also low, b( i ju = 1; Ii = f H ; L g; Ij = f L g) = 0.

We now provide a condition for a …rm with one high- and one low-quality good to enter. If a …rm enters with two unbranded products and the qualities of both products H L p 1 + 1+p provided that are not detected, its expected pro…ts per product are 1+p types ( H ; L ), ( L ; H ), and ( L ; L ) pool. Therefore, a …rm of type ( H ; L ) or H L p 1 ( L ; H ) makes expected pro…ts of ( H f ) + 2 (1 ) 1+p + 1+p f +

( L

f ). A …rm of type ( L ; L ) obtains lower pro…ts as both of its products can H L p 1 + 1+p f . Participation of be of low quality, 2 ( L f ) + 2 (1 ) 1+p

31

a …rm of type ( L ;

L

) is pro…table if and only if

(1

)

p ( H 1+p

1+ p (f 1+p

f)

L

):

(1)

This condition is satis…ed if f is su¢ ciently close to L . Hence, if (1) holds, a …rm with two low-quality products enters, provided it does not use an umbrella. This implies already that …rms of type ( H ; L ) or ( L ; H ) also enter. In addition, we have to show that in equilibrium, …rms of type ( H ; L ), ( L ; H ), or ( L ; L ) cannot make higher expected pro…ts by selling their products under an umbrella brand. Consider …rst a …rm of type ( H ; L ) or ( L ; H ). With probability 1 , the low quality is not discovered by consumers, in which case consumers believe that both products are of high quality. Pro…ts in this case are 2 ( H f ). With probability 2 , the quality of both products is observed. Pro…ts in this case are H + L 2 f . With the remaining probability (1 ), consumers only obtain the information that one of the products under the umbrella is of low quality. In this case we have imposed out-of-equilibrium beliefs that the other product is also of low quality. Pro…ts are then 2 ( L f ).

Expected deviation pro…ts are therefore (1 ) 2 ( H f) + 2 ( H + L 2 f) + (1 ) 2 ( L f ) k . These deviation pro…ts have to be lower than pro…ts along the proposed equilibrium path. Hence, a deviation is not pro…table if and only if

1 (2 1+p

p )( H

L

is satis…ed. Second, consider a …rm of type ( L ;

(1

2

) 2(

H

f ) + (1

(1

2

) )2(

L

)

k

0:

(2)

L

). Expected deviation pro…ts are k . These deviation pro…ts have to

f)

be lower than pro…ts along the proposed equilibrium path. Hence, a deviation is not pro…table if and only if

1 (1 1+p

p )( H

L

k 2

)

0

This inequality is implied by (2). Finally, a …rm of type ( H ; H ) must have an incentive to actually use the umbrella. The non-deviation constraint of the …rm of type ( H ; H ) is

2( H

f)

k

2 ( H

f ) + 2 (1

p 1+p

)

H

+

1 1+p

L

f

This condition can be rewritten as

1 ( H 1+p

L

)

k 2

0.

(3)

We thus have shown that under the conditions (1), (2), and (3) there exist separating equilibria in which a …rm with two high-quality products uses umbrella branding as a signal of product quality, and all other …rm types sell their products under separate brands.

32

Exercise 21 Discussion of umbrella branding Give an example of a …rm using umbrella branding using reputation concerns. Provide a detailed account of the relevant industry/market and discuss whether the possibility of umbrella branding constitutes an important incumbency advantage in this example.

33

Industrial Organization: Markets and Strategies Paul Belle‡amme and Martin Peitz published by Cambridge University Press

Part VI. Theory of competition policy Exercises & Solutions Exercise 1 Industries with cartels Brie‡y describe and analyze a case of your choice concerning a price- or quantity-…xing cartel (please not OPEC). The following questions may be useful to bear in mind: What are the relevant characteristics of the industry? What was the scope of the cartel? How was the cartel enforced? What were the e¤ects of the cartels? How did the competition authority or court argue and what was the decision, if any?

Exercise 2 Collusion and pricing Two (advertising-free) newspapers compete in prices for an in…nite number of days. The monopoly pro…ts (per day) in the newspaper market are M and the discount rate (per day) is . If the newspapers compete in prices, they both earn zero pro…ts in the static Nash equilibrium. Finally, if the …rms set the same price, they split the market equally and earn the same pro…ts.

1. The newspapers would like to collude on the monopoly price. Write down the strategies that the newspapers could follow to achieve this outcome.

Find the discount rates for which they are able to sustain the monopoly price using these strategies.

2. On Sundays, the newspapers sell a weekly magazine (that can be bought without buying the newspaper). The monopoly (competitive) pro…ts when selling the magazine are also M (zero).

3. For which discount rates can the monopoly price be sustained only in the market for magazines? (Write down the equation that characterizes the solution.) Compare the solution found in question 1 and 2 and comment brie‡y.

4. For which discount rates can the monopoly price be sustained both in the market for newspapers and in the market for magazines? (Write down the equation that characterizes the solution.)

Exercise 3 Collusion and pricing II [included in 2nd edition of the book]

1

Consider a homogeneous-product duopoly. The two …rms in the market are assumed to have constant marginal costs of production equal to c. The two …rms compete possibly over an in…nite time horizon. In each period they simultaneously set price pi , i = 1; 2. After each period the market is closed down with probability 1 .

Market demand Q(p) is decreasing, where p = minfp1 ; p2 g. Suppose, furthermore, that the monopoly problem is well de…ned, i.e. there is a solution pM = arg maxp pQ(p). If …rms set the same price, they share total demand with weight for …rm 1 and 1 for …rm 2. Suppose that 2 [1=2; 1). Suppose that …rms use trigger strategies and Nash punishment.

1. Suppose that

= 0. Derive the equilibrium of the game.

2. Suppose that > 0 and = 1=2. Derive the condition according to which …rm 1 and …rm 2 do not …nd it pro…table to deviate from the collusive price pM .

3. Suppose that > 0 and > 1=2. Derive the condition according according to which pM is played along the equilibrium path. Show that the condition is the more stringent the higher .

4. Show that previous results in (3) also hold for any collusive price pC 2 (c; pM ).

5. Suppose that > 0 and > 1=2 and that …rm 1 can only adjust its price every periods. Derive the condition according to which pM is played along the equilibrium path. How does the time span in‡uence the condition?

Solution to Exercise3 1. Standard Bertrand model, p1 = p2 = c. Equilibrium pro…ts are equal to zero.

2. Pro…t with collusive price pM is (1=2) M =(1 The deviation pro…t is

(1=2) M =(1

)

), where

M

pM Q(pM ).

M

M

. Thus, …rms do not have an incentive to deviate if , which is equivalent to 1=2.

3. Consider …rm 2. The deviation pro…t is M . Pro…t with collusive price pM is (1 ) M =(1 ). Thus, …rm 2 does not have an incentive to deviate if M M (1 ) =(1 ) , which is equivalent to . This shows that the condition is more stringent the higher 1=2.

4. As follows from the inequality (1

)e=(1

same inequality holds.

)

e for any e 2 (0;

M

) the

1 M 5. Consider …rm 2. The deviation pro…t is M + M + ::: + . Pro…t with M M collusive price p is (1 ) =(1 ). Thus, …rm 2 does not have an incentive M to deviate if (1 ) M =(1 ) (1 )=(1 ), which is equivalent to . This shows that the condition is more stringent the higher 1=2.

2

Exercise 4 Parallel pricing and evidence of collusion A competition policy authority has noticed that the …rms in the Lysine industry consistently charge very similar prices, and the suspicion is that they are colluding. Do you think that parallel pricing is proof of collusion? If not, what kind of evidence would you look for?

Solution to Exercise 4 The problem of using parallel pricing as evidence of collusion is that …rms can set similar prices both when they are colluding and when they are competing …ercely. For example: If …rms compete in prices and have symmetric costs, the equilibrium is that all …rms charge the same price (= the marginal cost).

Another piece of evidence that would point in the direction of collusion is a very high mark-up in the industry. This requires, however, knowledge of the …rms’ cost structure. Often, direct evidence of collusion (e-mails, letters, etc.) is therefore needed to prove that collusion has taken place.

Exercise 5 Collusion and quantity competition [included in 2nd edition of the book] Consider the following market: Two …rms compete in quantities, i.e., they are Cournot competitors. The …rms produce at constant marginal costs equal to 20. The inverse demand curve in the market is given by P (q) = 260 q.

1. Find the equilibrium quantities under Cournot competition as well as the quantity that a monopolist would produce. Calculate the equilibrium pro…ts in Cournot duopoly and the monopoly pro…ts. Suppose that the …rms compete in this market for an in…nite number of periods. The discount factor (per period) is , 2 (0; 1).

2. The …rms would like to collude in order to restrict the total quantity produced to the monopoly quantity. Write down grim trigger strategies that the …rms could use to achieve this outcome.

3. For which values of is collusion sustainable using the strategies of question (b)? [Hint: Think carefully about what the optimal deviation is.]

Solution to Exercise 5 1. Here, q1 = q2 = 80 and q m = 120. Furthermore, m = 14400:

1

=

2

= 6400 and

2. Start by setting qi = q m =2 = 60 in the …rst period. In the later periods, set qi = 60 if q1 = q2 = 60 in all previous periods. Otherwise, set qi = qi = 80: 3. We have that Ri (qj ) = 120 qj =2. This implies that the optimal deviation to qj = 60 is qi = 90. Hence, Deviation = (260 90 60 20)90 = 8100: We conclude that collusion can be sustained if and only if

14400 2(1 )

8100 +

3

6400 , (1 )

9 .

17

Exercise 6 The European air cargo cartel [included in 2nd edition of the book] Read the following press release of the European Commission (Brussels, March 28, 2012): http://europa.eu/rapid/press-release_IP-12-314_en.htm. For the sake of the exercise, we model the European air cargo market during the 72 months of the cartel existence as follows. First, only the 14 …rms associated in the cartel were active on the market; second, these …rms were symmetric; they all had the same constant marginal cost, c, for supplying airfreight services; third, these …rms competed à la Cournot (i.e., by choosing the quantity of airfreight services); …nally, the inverse demand for airfreight services (per month) was given by p = a 2q, where q denotes the total quantity of airfreight services supplied by the 14 …rms.

As written in the text, the Commission …ned the cartel members a total of e169 million. Given that the cartel lasted for 72 months, this is roughly equivalent to a …ne of e2.35 million per month. This …ne is meant to compensate the European consumers for the surplus reduction that they su¤ered because of the existence of the cartel.

1. Show that (a c) had to be equal to e3.89 million (per month) to justify the …ne of e2.35 million per month imposed by the Commission.

2. Set (a c) to e3.89 million. Assuming that the cartel pro…ts were equally shared among the 14 cartel members, show that any of these …rms would have been better o¤ by unilaterally leaving the cartel (which would have then counted only 13 members) and by acting independently. What does this tell you about the stability of cartels? Discuss.

3. Continue to set (a c) to e3.89 million. Take now a tacit collusion perspective. Suppose that the 14 …rms were following a grim trigger strategy and were expecting to continue to compete inde…nitely on that market.

Compute the minimum discount factor that allowed the 14 …rms to sustain full collusion (i.e., to behave as a cartel).

Solution to Exercise 6 1. We recall from the analysis of the symmetric Cournot oligopoly of Chapter 3 (with demand given by P (q) = a bq and linear marginal cost c) that the total quantity at the Nash equilibrium is q (n) = n (a c) = (b (n + 1)). As for consumer surplus, it is equal to CS (n) = (b=2) (q (n)). The reduction in consumer surplus is given by CS (14) CS (1), where CS (14) is the surplus that consumer ould obtain if the 14 …rms were competing à la Cournot and CS (1) is the surplus that they actually obtain when the …rms collude (and act thus as a monopoly). Setting b = 2, we …nd CS (14)

CS (1) =

49 225

1 16

(a

2

c) = 2:35 , a

c = 3:89:

2. We also recall from Chapter 3 that the equilibrium pro…t is equal to 2 2 (n) = (a c) =(b (n + 1) ). Each cartel member receives a share 1=14 4

2

of the monopoly pro…t, i.e., (1) =14 = (a c) =(56b). If a …rm leaves the cartel, then competition takes place between two decision-makers (the cartel made of the remaining 13 …rms and the deviating …rm); so the 2 deviating …rm earns (2) = (a c) =(9b). As 9b < 56b, deviation is pro…table, meaning that the cartel is unstable. Note that we do not need a speci…c value for b and (a c) to make that argument.

3. Again, the answer does not depend on the speci…c value of b and (a c) (you should nevertheless check this!). We can refer to Section 14.2.1: the minimal discount factor in the symmetric linear Cournot oligopoly is 2 Cour Cour 2 min (n) = (n + 1) = n + 6n + 1 . Setting n = 14, one …nds min (14) = 225=281 ' 0:8.

Exercise 7 Cournot mergers: pro…tability and welfare properties Consider a homogeneous good duopoly with linear demand P (q) = 12 where q is the total industry output, and constant marginal costs c = 3.

q,

1. Suppose that …rms simultaneously set quantities. Determine the equilibrium (price, quantities, pro…t, welfare).

2. The …rms consider to merge although their production costs are not a¤ected. Determine the solution to this problem. Is such a merger pro…table?

What are the welfare e¤ects of such a merger?

3. Suppose that the merger is e¢ ciency enhancing, leading to marginal costs cm < c. What are the welfare e¤ects of such a merger. Do you possibly have to qualify your answer in (2)?

4. Consider the possibility of …rm entry after the merger (the entrant produces at marginal costs c = 3 and has entry cost e). Suppose …rst that the merger is not e¢ ciency-enhancing. Analyze such a market and comment on your result (depending on the entry cost e). Suppose next that the merger is e¢ ciency-enhancing, i.e. cm < 3. Depending on cm and e, when is a merger pro…table? [Hint: Calculate pro…ts for cm = 1=2.]

5. Many countries scrutinize merger and sometimes block them (or impose remedies)? Discuss which factors should make the courts or the competition authority more inclined to block a merger.

Exercise 8 Cournot mergers and synergies. [included in 2nd edition of the book] Consider a homogeneous-product Cournot oligopoly with 4 …rms. Suppose that the inverse demand function is P (q) = 64 q.

5

1. Suppose that …rms incur a constant marginal cost c = 4. Characterize the Nash equilibrium of the game in which all …rms simultaneously choose quantity.

2. Suppose that …rms 1 and 2 consider to merge and that there are synergies leading to marginal costs cm < c. Characterize the Nash equilibrium. At which level cm (you may want to give an approximate number) are the two …rms indi¤erent whether to merge?

3. Is such a merger that just makes the two …rms indi¤erent between merging and non-merging consumer-welfare increasing?

4. At which level cm would the merger be consumer-welfare neutral?

5. Suppose that instead …rms 1, 2, and 3 consider to merge. The new marginal cost of the merged …rms is cn < c. At which level cn are the three …rms indi¤erent whether to merge?

6. Compare your …ndings in (5) and (2). What can you say about incentives to merge in this case?

Solution to Exercise 8 1. Since all …rms have the same all choose h costs, they i the same quantity in equiP librium: q = arg maxqi (60 qi j6=i qj )qi ) q = 12, p = 16 and = 144.

2. The merged …rm sets qm = arg maxqm [(64 qm q3 q4 cm )qm ] while the non-merged …rms set q3 = arg maxq3 [(60 qm q3 q4 )q3 ] and q4 = arg maxq4 [(60 qm q3 q4 )q4 ]. By symmetry, q3 = q4 and optimal quantities result as functions of cm : qm = 18 43 cm and q3 = q4 = 14 + 14 cm .

2 2 It follows that m = 18 43 cm and 3 = 4 = 14 + 41 cm . Firms 1 and 2 are indi¤erent to merge if

m =2

, 18

2 3 = 288 , cm 4 cm

1:373.

3. Consumer welfare before the merger was CSc = 1152. After the merger, con2 (46 14 cm ) . Plugging in cm = 1:373 yields sumer surplus is equal to CSm = 2 CSm = 1042:27 which is smaller than CSc . Hence, a merger that makes the merging …rms just indi¤erent is detrimental to consumers.

4. The merger will never be consumer welfare neutral because CSc = CSm , cm = 8.

5. Proceed as in (2) but with only two players left (the merged entity and one remaining competitor). The resulting optimal quanities are qn = 68 32cn for n the merged …rm and q4 = 56+c for the non-merged …rm. The …rms will be 3 indi¤erent to merge when m = 3 , cn 2:82.

6

6. A merger of 3 …rms is already pro…table for a smaller amount of synergies than a merger of 2 …rms. This is due to the fact that merger-speci…c synergies balance two opposing e¤ects. First, the merger leads to an internalization of competition between the merging …rms. Second, it has an e¤ect on the remaining outsiders’ quantity choice that leads to lower pro…ts of the merged entity. When 3 instead of only 2 …rms merge, the outsiders’e¤ect is smaller and therefore smaller synergies are needed to balance both e¤ects and make the merging …rms indi¤erent.

Exercise 9 Cournot mergers and demand Consider the following Cournot merger game. The inverse demand function is of the form P (q) = a q where c.

> 0 and there are n …rms with constant marginal costs of production

1. Discuss the shape of the inverse demand function depending on .

2. Determine the Cournot equilibrium pro…ts in this set-up.

3. Determine all n for which a single merger, i.e. going from n to n is pro…table.

1 …rms,

Exercise 10 Cournot mergers and asset complementarity [included in 2nd edition of the book] Consider an n …rm symmetric homogeneous product Cournot oligopoly.

Each …rm i has a physical asset Ki = K normalized to 1. Its cost function 1 . The demand side is Ci (qi ; Ki = 1) = qi2 + 32 P of the market is given by an inverse demand curve P (q) = 1 q where q = i qi .

1. Determine equilibrium price, quantity, and pro…t of the n …rm Cournot model speci…ed above. What is the upper bound on n for all …rms in the industry to make non-negative pro…ts?

2. Consider now a merger between the two …rms n and n 1. Suppose …rst that physical assets cannot be sold and that the assets of one of the merging partners are liquidated at zero cost, i.e. the cost function of the mer1 . Determine equilibrium price, quantity, and ged …rm is Ci (qi ; 1) = qi2 + 32 pro…t of the …rm Cournot model after the merger. For which n are mergers pro…table? For which n are mergers consumer surplus increasing?

7

3. Consider again a two-…rm merger, but suppose now that the merged …rm’s assets can be combined giving rise to the cost function Ci (qi ; Ki ) = K1i qi2 , 1 i.e. the merged …rm n 1 has cost function Cn 1 (qi ; 2) = 21 qi2 + 16 . Determine equilibrium price, quantity, and pro…t of the …rm Cournot model after the merger. For which n are mergers pro…table? For which n are mergers consumer surplus increasing? [Hint: you may want to solve parts 3 and 4 together] 4. Consider once again a two-…rm merger, but suppose now that the merged …rm’s assets are complementary. More speci…cally, suppose that the 1 . Determine merged …rm n 1 has cost function Cn 1 (qi ; 2) = 41 qi2 + 16 equilibrium price, quantity, and pro…t of the …rm Cournot model after the merger. For which n are mergers pro…table? For which n are mergers consumer surplus increasing? What happens when the complementary is even stronger?

5. Based on your …ndings in parts 2-4 what are the policy conclusions for an antitrust authority?

Solution to Exercise 10 1 2 3 , q (n) = n+3 , and (n) = (n+3) 1. The answers are p (n) = n+3 2 2 2 (n) 0 if and only if (n+3)2 or equivalently n 5 .

64

1 32 . Hence,

2. Firms are still symmetric after the merger. There are now n 1 …rms left in the 3 1 market and one compute p (n 1) = n+2 , q (n 1) = n+2 , and (n 1) = 1 2 1 2 (n 1) > 2 (n), i.e. (n+2) 2 (n+2)2 32 . A merger is pro…table if 32 > 2 1 2( (n+3)2 32 ). It is easily checked that a merger in an industry with 3 or more …rms is pro…table, whereas a merger from duopoly to monopoly is not pro…table. Thus a merger to monopoly is non-pro…table whereas mergers in less concentrated industries are. This non-standard result holds because of the high cost incurred by a single …rm. Since p is decreasing in n, all mergers are consumer surplus decreasing. Any pro…table merger should be prohibited.

3. The merged …rm n 1 is di¤erent from all other …rms i = 1; :::; n 2. Looking for an equilibrium in which all non-merged …rms set the same quantity (equilibrium values are denoted by superscript ), we can rewrite the system of …rst-order conditions of pro…t maximization (with marginal costs of the merged …rm being equal to qn 1 ) as a two equation system:

1 1

2qn 1 (n 2)qi = qn 1 ; qn 1 (n 1)qi = 2qi :

3 = 1, we obtain post-merger equilibrium values : qn 1 = 2n+5 , qi = 6 27 1 (i = 1; :::n 2), p = 2n+5 and n 1 = 2(2n+5)2 .

A merger 16 between …rm n and n 1 is pro…table if n 1 > 2 (n), which is equivalent 27 4 to 2(2n+5) 2 > (n+3)2 . Thus a 2-…rm merger in an n-…rm industry is pro…table

For

2 2n+5

8

if 5n2 + 2n + 43 > 0 which is satis…ed for n up to n = 3. In other words, mergers to monopoly and to duopoly are pro…table in this example. The merger to duopoly increases the price from 1=2 to 6=11 and is therefore consumer surplus decreasing. Any pro…table merger should be prohibited.

4. First-order conditions from part c with values after the merger by superscript 1 n+3

= 1=2 apply. We denote equilibrium 2 . We compute: qn 1 = n+3 , qi =

3 2), p = n+3 and between …rm n and n 1 is pro…table if

(i = 1; :::n

6 1 = (n+3) 2 16 . A merger (n), which is always n 1 > 2 n 1

satis…ed. Here, because of strong complementarities between assets, a merger by two …rms is always pro…table. The level of complementarity here is a knifeedge case in which consumer surplus is not a¤ected, as the price is the same as prior to the merger. Hence, the competition authority can approve any proposed merger on the grounds that the merger does not reduce consumer surplus. If the complementarity is even stronger the merger is strictly consumer surplus increasing.

5. If complementarities between merging …rms are su¢ ciently strong such that the post-merger price is lower than the pre-merger price, the competition authority should approve any proposed merger based on the consumer surplus criterion.

Depending on the cost function of the merged …rm a possible merger may not be pro…table and, therefore, will never be proposed, be pro…table but consumer surplus decreasing, or be pro…table and consumer surplus increasing. To distinguish between the last two cases, the competition authority must be able to predict the post-merger price. To be able to do so, the merged …rm must present evidence on the complementarity between the two assets.

Exercise 11 Mergers and free entry [included in 2nd edition of the book] Consider a homogeneous product market with in…nitely many quantitysetting …rms. The inverse demand function is given by P (q) = a q where q is industry quantity. The cost function of each …rm is Cs (qi ) = cs qi + Fs for qi > 0 and Cs (0) = 0. Suppose that parameters are such that, in any equilibrium, more than one …rm is active (Note: In the analysis below it is NOT required to derive the parameter restriction which guarantees that this property is satis…ed.) For simplicity, the analysis below should be carried out under the assumption that the number of …rms is a real number.

1. Characterize the set of pure-strategy free-entry equilibria of the game in which all …rms simultaneously quantities. Determine equilibrium quantity of each active …rm, equilibrium price, industry quantity, number of …rms, and consumer welfare in equilibrium.

2. Consider a single merger between two …rms. Suppose that the merged …rm has cost function Cm (qi ) = cm qi + Fm for qi > 0 and Cm (0) = 0 and that cm = cs , while Fm 2Fs . Derive the exact condition when a merger is pro…table when there is free entry before and after the merger.

9

3. In the setting of (2) is a pro…table merger welfare-increasing? Or is a pro…table merger welfare-decreasing? Explain your …ndings.

4. Consider now a single merger after which the merged …rm has costs with cm cs and Fm = Fs . Derive the exact condition when a merger is pro…table when there is free entry before and after the merger.

5. In the setting of (4) is a pro…able merger welfare-increasing? Or is a pro…table merger welfare-decreasing? Explain your …ndings.

Solution to Exercise 11 1. Pro…t of …rm i is i = (a qi q i cs )qi Fs . FOCs can be written as qi = (a q i cs )=2. In symmetric equilibrium with n …rms, we must have qi = qi and q i = (n 1)qi . Using the FOC of …rm i, we obtain qi = (a cs )=(n + 1).

In free-entry equilibrium, i =

a cs n+1

2

Fs = 0:

Solving for n we obtain that, in free-entry equilibrium,

a cs n= p 1: Fs p p Fs , p = cs + Fs , CS = (a We that q = nqi = a cs p have Fs )2 =2.

cs

2. The best response function of the merged …rm is the same as non-merged …rms.

Hence, we obtain the same aggregate output and the same total number of active …rms (one additional …rm enters as a result of the merger). The only di¤erence is that the merged …rm has di¤erent …xed costs. Clearly, the merger is pro…table if and only if Fm < Fs as in this case pro…ts increase from zero to Fs Fm :

3. A pro…table merger is always welfare-increasing. Consumer surplus is not a¤ected by the merger, the pro…ts of outsiders are zero before and after the merger.

Thus the only change in welfare is due to the change of the …rms involved in the merger.

4. Denote the newly merged …rm as …rm 1. Then q1 = (a q 1 cm )=2 and qi = (a q i cs )=2 P for i 6= 1. For the non-merged …rms, in free-entry equilibrium, (P (q1 + i6=1 qi ))qi = Fs . Since the best response of …rm i and the inverse demand function P are the same before and after the merger, in

free-entry equilibrium after the merger the output of active outsider …rms is the same as before the merger. In addition total industry output q remains the same, which implies that consumer surplus is unchanged. Using the FOC of …rm 1, we know that q1m = (a (q q1m ) cm )=2 after the merger. Hence, m q1 = a q cm which is larger than the quantity of a single …rm prior to the merger, q1 , since marginal costs have been decreased due to the merger. Since

10

the merger does not a¤ect aggregate output, we have q1m merger is pro…t-increasing if and only if cs > cm .

q1 = cs

cm . The

5. Since consumer surplus and outsider pro…ts are not a¤ected by the merger, the merger is welfare-increasing if and only if it is pro…table. The …ndings illustrate that barriers to entry are at the heart of merger analysis, as private and social incentives are aligned under free entry.

Exercise 12 Burning ships Hernan Cortéz, the Spanish conqueror (”conquistador”), is said to have burned his ships upon the arrival to Mexico. Why would he do such a thing?

Exercise 13 Quantity commitment [included in 2nd edition of the book] Up to two …rms are in a market in which quantities are the strategic variable.

There are two periods; in the …rst period …rm 1 is a protected monopolist.

In each of the two periods t = 1; 2 the inverse demand function P t is given

by P t (xt ) = 20 xt . In each period the cost function of …rm i is given by Cit (xti ) = 9 + 4xti . Pro…ts of a …rm are the sum of its pro…ts in each period (no discounting). Firms maximize pro…ts by setting quantities.

1. Determine the monopoly solution.

2. Because of technological restrictions …rm 1 has to choose the same quantity in each period (x11 = x21 ). Observing x11 , …rm 2 is considering to enter in period 2. Determine the pro…t maximizing x22 given x11 .

3. Assume that …rm 2 will enter in period 2. What quantity will …rm 1 produce? Determine equilibrium prices, quantities, and pro…ts.

4. Firm 2 only enters in period 2 if it can make positive pro…ts. Determine the subgame perfect equilibrium of the two-period model.

Solution to Exercise 13 1. In the monopoly situation, we can analyze a single period (both periods are identical). The …rm chooses q1 to maximize t1 = (20 q1 )q1 9 4q1 . The …rst-order condition is 16 2q1 = 0. Thus, q1 = 8, P = 12 and t1 = 55. In consequence, total pro…ts are 1 = 2 t1 = 110.

2. Firm 2’s pro…t given q1 is

t=2 = (20 2

q1

q2 )q2

best response from the …rst-order condition: q2 = 8 best reposnse, …rm 2 achieves the following pro…t: t=2 2

= =

(20 q1 (8 1 10) (q1 4 (q1

1 2 q1 ))(8

1 2 q1 )

9

4q2 . We …nd …rm 2’s 1 q 1 2 . When playing this 9

4(8

1 2 q1 )

22) :

As q2 < 0 for q1 > 22, we see that the condition for …rm 2 to make a positive pro…t (and thus to enter) is q1 10.

11

3. Assuming that …rm 2 enters, …rm 1 has to consider both pro…ts on the …rst and second period in its decision (as q1 can not be changed between the periods per assumption): 1 =

t=1 + 1

t=2 = (20 1

q1 )q1

9

4q1 + (20

q1

q2 )q1

9

4q1 :

1 Now, we can set in q2 = 8 2 q1 as …rm 1 knows how …rm 2 will react on a given q1 ; doing so, we have 1 = 32 16q1 q12 12 . The …rst-order condition yields 16 2q1 = 0 , q1 = 8. As a result, q2 = 4, P 1 = 12, P 2 = 8, 1 = 78 and 2 = 7.

4. To …nd the subgame-perfect equilibrium, we need to be determined whether the incumbent …rm 1 would prefer to accommodate entry of …rm 2 in period 2 or deter entry. Given accommodated entry, we know from (3) that 1 = 78.

To deter entry, we know from (2) that …rm 1 then needs to set q1 = 10 (or higher). We thus determine the pro…t for …rm 1 under this strategy: 1 (q1 = 10) = 2 [(20 q1 ) q1 9 4q1 ] = 102.Thus, pro…ts for …rm 1 under entry deterrence are higher than under accommodated entry.

Exercise 14 Strategic quantity choice Consider a market with two …rms, A and B. The …rms produce homogenous goods, compete in quantities, and face a constant marginal cost equal to 1/4.

The timing is the following: First, …rm A chooses its quantity qA . Then, after observing qA , …rm B chooses its quantity qB . The price in the market is given by the inverse demand function P (q) = 1 q, where q = qA + qB .

1. Find the subgame perfect Nash equilibrium.

2. Assume from now that there is an entry cost of e. Firm A is already established in the market, and …rm B is considering whether to establish itself in the market or not. The timing is the following: First, …rm A chooses its quantity qA . Then, after observing qA , …rm B decides whether to enter the market and, in case of entry, how much to produce.

3. Write down …rm B’s pro…t function.

4. Assume for now that e = 1=10. Illustrate …rm A’s pro…t as a function of qA when the reaction of …rm B is taken into account.

5. Find the subgame perfect Nash equilibrium for all values of e > 0.

Exercise 15 Taxonomy of entry-related strategies I

12

Consider a market with di¤erentiated products. In the …rst stage …rm 1 is the incumbent …rm and can invest an amount K1 0 in reducing its marginal costs, c(K1 ) = c K1 =10. In stage two …rm 2 can decides about entering the market with constant marginal costs of c and entry costs of e. In stage three if entry takes place …rms engage in price competition and face symmetric demand functions given by Di (pi ; pj ) = A api + bpj (A > a > b > 0). If no entry takes place, …rm 1 acts as a monopolist with demand, D1 (p1 ) = A ap1 .

1. Calculate the best response functions for both …rms. Draw a graph.

Argue graphically from now on: 2. Does an increase in K1 increase or decrease the pro…t of the entering …rm?

Does an increase in K1 make the incumbent tough or soft?

3. Does a marginal investment K1 increase or decrease the pro…t of the incumbent? (Assume that e is su¢ ciently low such that entry takes place for K1 close to zero. Moreover, assume A 10) 4. Use your answer of (2): Is entry deterrence via cost reduction possible in this setting? If your answer is YES, which numbers would you have to compare to decide whether entry deterrence is optimal? If your answer is NO, what do we have change in this model to induce entry deterrence?

5. Use your answer of (3): If entry accommodation is optimal how much should …rm 1 invest in cost reduction?

6. How would you answer to (4) change if we consider a Cournot game instead?

7. How would you answer to (5) change if we consider a Cournot game instead?

Solution to Exercise 15 demand Di (Pi ; Pj ) = A

api + bpj ;

A>a>b>0

incumbent can invest in cost reduction K1

0, M C = c(K1 ) = c

K1 10

entrant faces M C = c and entry costs = e

1. Best responses: max FOC:

a(pi

ci ) + A

i (pi ; pj )

api + bpj 2api

=

pi (pj )

=

and ci

=

13

(pi

ci )(A

api + bpj )

!

= 0 = A + aci + bpj A+aci +bpj 2a

…rm i’s best response

c(K1 ) = c c;

K1 10 ;

if i = 1 if i = 2

P2 P1* (P2)

P2* (P1)

P1 Abbildung 1: Strategic complements (prices)

Best-response functions and iso-pro…t lines (upper contour sets opened towards the north east and tangent on vertical resp. horizontal line by FOC).

2. Graphical argument: p1 (p2 ) = K1 "

!

A + a(c 2a

K1 " 10 )

#+

b pj 2a

p1 -intercept of …rm i’s best response function moves to the left K1 "!

2 #

Alternatively:

z }| { z}|{ z}|{ z }| { @p1 @c d 2 @ 2 @ + 2 <0 jK1 =0 = dK1 @K @p @c @K1 | {z 1} | 1 {z } =0

s t r a t e g i c e ¤ e c t <0

Hence, ivestment in cost reduction makes …rm 1 tough.

14

3.

d 1 jK =0 = dK1 1

z

}| @ 1 @K | {z 1}

{

d ire c t e ¤ e c t ! 0

+

@ 1 @p1 |{z}

by E nve lo p e T h . = 0

}| { z z}|{ z}|{ z}|{ z}|{ @p1 @c @ 1 @p2 @p1 c <0 + @c @K1 @p2 @p1 @c K1 {z } | ; strategic e¤ect

4. Entry deterrence (D): Is possible here: Since

d 2 < 0 (see b)); 9 K1D > 0; s.t. dK1

Entry deterrence is optimal, if

M onopolist (K1D ) 1

K1D

D 2 (K1 ) = e D 1 (K1 )

5. Entry accommodation (A); see (3): Since the strategic e¤ect is negative, …rm 1’s pro…t decreases in the level of cost d reduction, dK11 < 0. Thus, K1A = 0 is optimal if entry occurs.

6. Entry deterrence under Cournot: Entry deterrence is possible because the deterrence level of cost reduction is also positive, i.e. K1D > 0. Entry deterrence D Cournot is optimal under Cournot if M K1D 1 (K1 ) 1 7. Entry accommodation under Cournot: K1A > 0 is optimal here, because the strategic e¤ect is positive under Cournot (and we implicitly assume it to be larger than the direct e¤ect)

z }| { z }| { z}|{ z}|{ z}|{ z }| { d 1 @ 1 @ @q2 @q1 @c jK1 =0 = + 1 >0 dK1 @K @q @q1 @c @K1 | {z 1} | 2 {z } d ire c t e ¤ e c t ! 0 ; strategic e¤ect Best-response functions and iso-pro…t lines (upper contour sets opened towards the south resp. west and tangent on vertical resp. horizontal line by FOC)

Exercise 16 Taxonomy of entry-related strategies II

15

q2

c1 q1* (q2) q1* (q2)

q2*(q1)

_

π1

q1

qM

Abbildung 2: Strategic substitutes (quantities)

Consider the market from the previous exercise again.

1. Use the best-response functions from the previous exercise to calculate equilibrium prices for c1 6= c2 in the Nash equilibrium in which …rms set prises.

2. Use your result from (1) to show that pi = (A + ac)=(2a if c1 = c2 = c.

b), for i = 1; 2,

Now, use again that c1 (K1 ) = c (K1 =10) and c2 = c.

Set c = 4, a = 2, b = 1, A = 10, and e = 7:95.

3. Show that the critical level of K1D to deter entry is below 0:5.

4. Suppose that investment in cost reduction is restricted to half units, i.e.

K1 2 f0; 0:5; 1; 1:5; : : :g. Will …rm 1 deter entry in a subgame perfect Nash equilibrium? State …rm 1’s optimal business strategy.

5. Reconsider your answer to (4) if …rm 1 as a monopolist faces a demand of D1 (p1 ) = A

ap1 + bp1 = A

(a

16

b)p1 .

Solution to Exercise 16 1. obvious 2. c1 = c2 ) p1 = p2 : pi =

2a(A + ac) + b(A + ac) (2a + b)(A + ac) A + ac = = (2a b)(2a + b) (2a b)(2a + b) 2a b

3. Entry deterrence if =

2

Q:E:D:

0:

2

(p2 c)(A ap2 + bp1 ) e 0 ! plug in numbers: 2 1 K1 )(10 12 + 75 K1 + 6 (2 75 1 2 (2 75 K1 )(4 75 )K1 0 2 8 2 0:05 752 K1 75 K1 8K1 75 2 752

K12 K1=2

K12 = 150

! K1 [K2

= 0:4695 = 299:5305]

4 75 K1 )

7:95

0

+ +0:05 =0 2 752

300K p 1 + 140:625 = 0 1502 + 140:625

4. Compare pro…ts under entry determination and entry accommodation: Accommodation case: From the previous exercise we know K1A = 0 1

! numbers:

p A 1

=

(p

c)(A

ap + bp)

10+8 = A+ac 2a b = 4 1 = 2 (10 12 + 6)

= 6 = 8

Deterrence case: Use K1D = 0:5 from now on Demand is now given by D1 (p1 ) = A ap1 . Note that this assumption is not very plausible because it implicitly implies that p2 = 0 (the market is too small with only one product, because consumer behave as if there exists a close substitute for free). Thus, the bene…ts of entry deterrence are underestimated.

1 @ 1 @p

p D 1

Since

A 1 >

= =

(p

3:95)(10

2p)

0:5 !

4p + 17:9

= 4:475 = (0:525)(10

=

8:95)

0:5

= 0:05125

D 1 in SPNE …rm 1 will accommodate entry.

17

0

5.

D 1

=

@ @p

=

p

= =

D 1

D 1

(p

3:95)(10

2p + p)

0:5 !

2p + 13:95 6:975 (3:025)(10

6:975)

0:5

=

0

=

8:650625

>

A 1

Hence, …rm 1 deters entry in SPNE.

Exercise 17 Sequential quantity choice and entry Consider a market for a homogenous good with one incumbent …rm (…rm 1) and one potential entrant (…rm 2). The interaction between the two …rms evolves in two stages. In stage 1, …rm 1 chooses its quantity q1 . In stage 2, after observing q1 , …rm 2 decides whether or not to enter the market. If it enters, it incurs an entry cost e and chooses its own quantity, q2 . If …rm 2 does not enter then q2 = 0 and …rm 2 does not pay the entry cost e (…rm 1 then is a monopoly). Assume that the inverse demand for the good is P = a (q1 + q2 ), and that the cost of production of each …rm i is C(qi ) = qi =2.

1. Compute the range of e for which entry is blockaded. That is, compute …rm 1’s output when it operates as a monopolist, then given this quantity, compute the highest pro…t that …rm 2 can earn if it decides to enter, and …nally, compute the range of e for which entry is blockaded.

2. Now, suppose that e is su¢ ciently low to ensure that entry is not blockaded. Compute the quantities and pro…ts of each …rm when entry is accommodated. That is, compute the outputs that will be selected in a Stackelberg equilibrium and the resulting pro…ts. (Instruction: …rst, compute …rm 2’s best response function, br2 (q1 ). Second, substitute for br2 (q1 ) into …rm 1’s pro…t function and compute …rm 1’s pro…t-maximizing quantity q1 . Third, …nd …rm 2’s best response against q1 , using …rm 2’s best response function. Finally, given the pair of quantities you found, compute the equilibrium pro…ts).

3. Compute the lowest q1 for which entry is deterred. Compute …rm 1’s pro…ts at this output level.

4. Given your answer in (3), show …rm 2’s best response function graphically in the quantities space (recall that …rm 2 may wish to stay out of the market when q1 is relatively high). Show on the same graph the Stackelberg equilibrium you found in Section (3) and the lowest q1 for which entry is deterred.

18

5. Given your answers in (2) and (3), …nd the range of e for which entry is accommodated, and the range of e for which it is deterred. Explain in no more than 3 sentences the intuition for the result (i.e., why is it natural to expect that entry is accommodated/deterred when e is relatively low/high).

Solution to Exercise 17 1. If …rm 1 is a monopolist it produces q1 = a=3. Standard calculations reveal that …rm 2’s best response function is br2 (q1 ) = (a q1 )=3. Substituting this expression in …rm 2’s pro…t gives that …rm 2’s pro…t, when it plays a best response against …rm 1, is (a q1 )2 =6 e. For q1 = a=3, …rm 2’s pro…t is 2 2a =27 F . This implies that entry is blockaded for all e > 2a2 =27.

2. If entry is accommodated, then …rm 1 chooses q1 subject to q2 = br2 (q1 ), which result in q1 = (2=7)a. Firm 2 then chooses q2 = (5=21)a. For these quantities, 2 …rm 1’s pro…t is A 1 = 2a =21.

3. and 4. As we saw in (1), …rm 2’s pro…t, provided that it plays a best-response against …rm 1, is (a q1 )2 =6 e. Entry is deterred if this pro…t is less than or equal to 0. Solving, the equation (a q1 )2 =6 e = 0, implies p that to deter entry 6e. Note that, since we …rm 1 must produce the deterring quantity q1D = a assume that entry is not blockaded, e 2a2 =27, we have r p 1 12 2 D q1 = a a = a; 6e a 27 3 where a=3 is the monopoly output of …rm 1. Hence, to deter entry, …rm 1 must produce strictly above its monopoly output and more so as e continues to fall from 2a2 =27, which is the critical level of e that separates deterred from blockaded entry. (For e 2a2 =27, entry is blockaded. Hence, …rm 1 can behave D as a monopolist p and 2simply ignore …rm 2.) Firm 1’s pro…t when it produces q1 is D = 2a 6e a =2 9e .

1

5. To solve this problem we need to compare the pro…t under accommodation with the pro…t under deterrence. Firm 1’s pro…t under accommodation is

A 1

= 2a2 =21, which is clearly independent of e. Firm 1’s pro…t under deterrence D 1 is increasing in e because the higher e is, the lower is the entrant’s pro…t, and, hence, the lower …rm 1’s quantity that is necessary to deter entry.

Hence, it is more pro…table to accommodate when e is relatively small and D to deter if e is relatively large. Solving thepequation A 1 we obtain 1 = 2 that entry is accommodated if e < (31 4 21)a =378 and it is deterred if p (31 4 21)a2 =378 < e < 2a2 =27.

Exercise 18 Capacity choice and entry [included in 2nd edition of the book]

19

Consider an industry for a homogenous product with a single …rm (…rm 1) that can produce at zero cost. The demand function in the industry is given by Q = a p. Now suppose that a second …rm (…rm 2) considers entry into the industry. Firm 2 can also produce at zero cost. If …rm 2 enters, …rms 1 and 2 compete by setting prices. Consumers buy from the …rm that sets the lowest price. If both …rms charge the same prices, consumers buy from …rm 1.

1. Solve for the Nash equilibrium if …rm 2 chooses to enter the industry.

Would …rm 2 wish to enter if entry required some initial investment?

2. Now suppose that before it enters, …rm 2 can choose a capacity, x2 , and a price p2 (the capacity x2 means that …rm 2 can produce no more than q2 = x2 units). Given q2 and p2 , …rm 1 chooses its price and then consumers decide who to buy from. Compute the Nash equilibrium in the product market if …rm 1 chooses to …ght …rm 2. What is …rm 1’s pro…t in this case? Show …rm 1’s pro…t in a graph that has quantity on the horizontal axis and price on the vertical axis. Would …rm 2 choose to produce in that case?

3. Now suppose that …rm 1 decides to accommodate the entry of …rm 2.

Compute the residual demand that …rm 1 faces after …rm 2 sells q2 units, and then write the maximization problem of …rm 1 and solve it for p1 .

What is …rm 1’s pro…t if it decides to accommodate …rm 2’s entry? Draw …rm 1’s pro…t in a graph that has quantity on the horizontal axis and price on the vertical axis. Would …rm 2 wish to enter in this case?

4. Given your answers to (2) and (3), compute for each p2 the largest capacity that …rm 2 can choose without inducing …rm 1 to …ght it. (Hint: to answer the question you need to solve a quadratic equation. The solution is given by the small root).

5. Show that the capacity you computed in (4) is decreasing with p2 . Explain the intuition for your answer. Given your answer, explain how …rm 2 will choose its price. Computing p2 is too complicated; you are just asked to explain in words how …rm 2 chooses p2 .)

Solution to Exercise 18 1. In a Nash equilibrium, both …rms will charge prices equal to 0. This is the only pair of prices for which no …rm can bene…t from deviation. If prices are negative, …rms lose money and are better o¤ charging 0 (in which case they do not lose money). If prices are positive it pays to cut the price by a cent below the price of the rival and, thereby, capture the entire market. Hence, if entry requires even a small initial investment, …rm 2 will choose to stay out.

2. If …rm 1 …ghts it charges p2 and captures the entire market because when both …rms set equal prices, all consumers prefer to buy from …rm 1. Firm 1’s pro…t then is F p2 )p2 . Firm 2 obtains 0 pro…ts and, hence, would prefer to 1 = (a stay out.

20

3. If …rm 1 accommodates …rm 2, its residual demand is Q1 = a q2 p1 . The problem of …rm 1 is to maximize p1 Q1 . The price that maximizes …rm 1’s pro…t is p1 = (a q2 )=2, so …rm 1’s pro…t is A q2 )2 =4. In this case, …rm 2 1 = (a earns p2 q2 > 0 and hence would choose to enter.

A 4. We need to compare F 1 and 1 . This comparison reveals that for each p2 , the largest x2 that p …rm 2 can choose without inducing …rm 1 to …ght it is x2 (p2 ) = a 2 (a p2 )p2 .

5. x2 (p2 ) is decreasing in p2 if p2 > a=2 and increasing otherwise. To determine whether p2 is above or below a=2, note that the entrant’s pro…t is p2 x2 (p2 ).

If p2 < a=2, then since x2 (p2 ) is increasing in p2 , the entrant would want to raise p2 as much as possible since both his capacity and his pro…t per unit will increase. This happens until a=2. Thus, in equilibrium, it must be the case that p2 > a=2 so that x2 (p2 ) is decreasing in p2 .

Exercise 19 Investment and incumbency Consider a di¤erentiated product market. At the …rst stage …rm 1 is the incumbent …rm and can invest an amount I1 0 in reducing its marginal costs, c(I1 ) = c I1 =10. At stage two …rm 2 decides whether to enter the market with constant marginal costs of c and entry costs of e, which is sunk at this stage. At stage three, if entry has taken place, …rms engage in quantity competition and face inverse demand functions given by Pi (qi ; qj ) = a bqi dqj (a > b > d > 0). If no entry takes place, …rm 1 acts as a monopolist with inverse demand, P m (q1 ) = a bq1 .

1. Calculate the best response functions for both …rms. Draw a graph.

2. Does an increase in I1 increase or decrease the pro…t of the entering …rm?

Does an increase in I1 make the incumbent tough or soft?

3. Does a marginal investment I1 > 0 increase or decrease the pro…t of the incumbent? (Assume that e is su¢ ciently low such that entry takes place for I1 close to zero.)

4. If entry accommodation is optimal how much should …rm 1 invest in cost reduction? (Use your answer of (3).)

5. Is entry deterrence via cost reduction possible and pro…table in this setting? Discuss your results in the light of the taxonomy developed in the book.

Exercise 20 Competition and entry [partly included in 2nd edition of the book]

21

Consider a homogeneous good duopoly with linear demand P (q) = 1 q, where q is the total industry output. Suppose that …rms are quantity setters and …rms incur constant marginal costs of production ci .

1. Suppose that …rms have constant marginal costs of production c. Determine the Nash equilibrium in quantities (report prices, quantities, pro…t, welfare) 2. Reconsider your answer in (1) because of the following: A tabloid runs a series on consumers paying “excessive” prices. The government considers introducing a non-negative special sales tax t 0 per unit on this product (and plans to use the revenues for some project from which nobody bene…ts). Determine the welfare-maximizing tax rate (the government is assumed to be able to commit to the tax; welfare is total surplus which includes tax revenues). Discuss your result. What would be your conclusion if the government was considering subsidizing the …rm?

3. Return to the case without taxes. Consider now the duopoly with c1 = 0 and c2 = c 2 [0; 1]. Determine the equilibrium (price, quantities, pro…t, welfare).

4. Consider now an extended model in which only …rm 1 is necessarily present. At stage 1, …rm 1 can make an investment I after which …rm 2’s marginal costs is c2 = 1=2 instead of c2 = 0. Afterwards, …rm 2 observes the investment decision of …rm 1 and, at stage 2, decides whether to enter at a negligible entry cost e > 0. At stage 3, active …rms set quantities simultaneously. Determine the subgame perfect equilibrium of this game.

Discuss your result in the light of what you have learnt reading about entry-related strategies (max 3 sentences).

5. Consider now a di¤erent entry model. Both …rms have zero marginal costs of production but consumers have become accustomed to product 1 (even if they did not consume it themselves). Therefore, consumers are willing to pay 1=2 money units less for product 2 than for product 1. The inverse demand function P (q) = 1 q gives demand for product 1. At stage 1, the potential entrant, …rm 2, considers to enter the market at an entry cost e > 0. (There is no investment stage in this game.) At stage 2, …rms set quantities simultaneously. Report the pro…t function for each …rm. Determine the equilibrium in case …rm 2 has entered (report prices, quantities, pro…t, welfare). Determine the subgame perfect equilibrium and comment on your result. You may want to reuse some of the results derived above.

Solution to Exercise 20 1. maxqi (1 foc: 1

2qi

q

c)qi qj

c = 0.

22

symmetric equilibrium: 1

3qi

qi = (1

c)=3

c)=3, q = 2(1

c = 0.

p = P (q ) = 31 + 23 c 2 = 13 c

R1 2 CS = p (t) qdp = [1 2p ] = 92 (1 + CS = 94 (1

T S(t) = 2 2. maxqi (1

q

t

c)qi

q (t) = 2(1

c

t)=3

2

c)

c)2

p (t) = P (q ) = 31 + 32 (c + t) p (t) c t = 1 3c t 2 (t) = 91 (1 c t) CS(t)

Z 1

=

qdp =

2

[1

p (t)] 2

p (t) 2 3 (1

=

2 (1 9

=

T S(t)

=

c 2

t)

c

t)2

2

2 (t) + CS(t) + tq (t) 2 2 (1 c t)2 + (1 c 9 9

=

dT S(t) dt

8 (1 9

=

2 t)2 + t (1 3

2 t) + (1 3

c

c

t)

c

t)

2 t 3

< 0 The optimal tax is zero. The government should not impose a tax as a response to market power.

3. Suppose that …rm 2 is active: foc …rm 1: 1

2q1

foc …rm 2: 1

c

q2 = 0. Hence, q1 = 1 2q2 : q1

2q2 = 0. Hence, q2 = 1 c2 q1 :

Substitute from above:

1

c

2q2

1

q2 2 q2

23

=

0

=

1 3

2 c 3

Substitute back: q1 = 13 + 31 c.

p

1 1 1 [ + c+ 3 3 3

= P (q ) = 1 = 1

=

2

=

2 c] 3

1 1 + c 3 3 1 (1 + c)2 9 1 (1 2c)2 9

Equilibrium demand q2 positive as long as c 1=2. For higher marginal costs q2 = 0 and …rm 1 maxq1 (1 q1 c)q1 under the constraint that the best response of …rm 2 is zero, i.e. 1 c2 q1 = 0. Solving for q1 in the constraint gives q1 = 1 c. P (q1 ) = c in this case. Firm 1’s pro…t is c(1 c). The unconstrained problem gives q1 = 1=2. Hence, the constraint is not binding for c > 1=2 and the …rm sets the unconstrained monopoly price (i.e. …rm 1 does not want to set a quantity lower than 1=2 in monopoly).

4. Solve by backward induction. The investment will lead to q2 = 0. Thus …rm 1 makes (monopoly) pro…t 1=4. If …rm 1 does not invest it will make pro…t 1=9 (take solution from (1): 1 3 c

2

). Hence the investment is pro…table as long as

1 4

I

1 9

or I 5=36. Note that the investment here raises the rival’s costs. However, the strategy is here NOT used as a deterrence strategy because …rm 2 would not be active in any case (even if it does not have to decide whether to enter).

5. Given q1 and q2 , prices are p1 = P (q1 + q2 ) = 1 q2 ) 1=2 = 1=2 q1 q2 .

FOC …rm 1: 1

2q1

q1

q2 and p2 = P (q1 +

q2 = 0.

FOC …rm 2: 1=2 q1 2q2 = 0.

Use analysis from part (3) with c = 1=2. Hence, using the formulas from above, …rm 2 does not have an incentive to enter the market since it would make zero sales. The presence of switching costs leads to zero quantity by …rm 2. This would even hold if entry was not costly. Here, switching costs do not lead to entry for any level of entry costs. For non-negligible entry costs, entry can even be avoided for lower levels of the switching cost.

The equilibrium values are q2 = 0, q1 = 1=2, p = 1=2, 1 = 1=4, 2 = 0, CS = 1=8 and T S = 3=8.

Exercise 21 Upstream merger

24

Consider an industry with two symmetric upstream …rms A and B and one downstream …rm D. The downstream …rm may sell the product of none, one, or both upstream …rms. Total industry pro…ts are a function of the number of upstream …rms selling through D, denoted by V (n), n 2 f0; 1; 2g, which is assumed to be increasing in n. The outside option for each …rm of not selling is zero. Thus, V (0) = 0.

Rents are the outcome of Nash bargaining between any pair of one upstream …rm and the downstream …rm (equal sharing of the surplus within the pair above the pro…ts that would occur if this pair did not agree).

1. What will be the pro…ts of …rms A, B and D? 2. Suppose that the two upstream …rms merge (and become a two-product …rm). Thus Nash bargaining takes place between the merged upstream …rm and the downstream …rm. What will be the pro…ts of the merged upstream …rm AB and the downstream …rm D? 3. Provide the exact condition for the merger to be pro…table.

4. Explain your …ndings.

Solution to Exercise 21 1. Suppose that there is an agreement between B and D. If A and D do not reach an agreement, neither A nor D obtain any pro…t from product 1. The additional industry pro…t generated by the agreement between B and D is V (2) V (1).

This gain is shared equally and thus B = [V (2) V (1)]=2. Symmetrically, V (1)]=2. Hence, D = V (2) A = [V (2) A B = V (1).

2. The industry pro…t is shared equally between both …rms, D = V (2)=2.

3. The merger is pro…table if AB > V (2) V (1) or V (1) > V (2)=2.

A +

AB = V (2)=2 and

B , which is equivalent to V (2)=2 >

4. Prior to the merger, any pair with one upstream and the downstream …rm bargain under the presumption that the other upstream …rm sells through the downstream …rm. This leads to lower upstream …rm pro…ts than after the merger if V is concave in n.

Exercise 22 Non-linear pricing in the supply chain A monopolist produces a good with constant marginal cost equal to c, c < 1. Assume for now that all consumers have the demand Q(p) = 1 p. The population is of size 1.

25

1. Suppose that the monopolist cannot discriminate in any way among the consumers and has to charge a uniform price, pU . Calculate both the price that maximizes pro…ts and the pro…ts that correspond to this price.

2. Suppose now that the monopolist can charge a two-part tari¤ (m; p) where m is the …xed fee and p is the price per unit. Expenditure then is m + pq.

Calculate the two-part tari¤ that maximizes pro…ts and the pro…ts that correspond to this tari¤. Compare pU and p and comment brie‡y.

3. Compare the situation with a uniform price and a two-part tari¤ in terms of welfare (a verbal argument is su¢ cient).

4. Assume now instead that there are two types of consumers. The consumers of type 1 have the demand Q1 (p) = 1 p, and the consumers of type 2 have the demand Q2 (p) = 1 p=2. The population is of size 1 and there are equally many consumers of the two types. Finally, it is assumed in this question that c = 1=2. Calculate the two-part tari¤ that maximizes the pro…ts of the monopolist. Compare the two-part tari¤s found in questions (2) and (3) for c = 1=2 and comment brie‡y.

Exercise 23 RPM RPM was common in a number of industries. In particular, it could be observed in the clothing, consumer electronics, and food industry. What is the probable motivation for …rms to use RPM in these industries? Discuss the likely welfare consequences in these industries.

Exercise 24 Vertical contracting [included in 2nd edition of the book] A buyer wants to buy one unit of a good from an incumbent seller. The buyer’s valuation of the good is 1, while the seller’s cost of producing it is 1/2. Before the parties trade, a rival seller enters the market and his cost, c, is distributed on the unit interval according to a distribution function with density g(c). The two sellers then simultaneously make price o¤ers and the buyer trades with the seller who o¤ers the lowest price. If the two sellers o¤er the same price, the buyer buys from the seller whose cost is lower.

1. Determine the price that the buyer pays in equilibrium, p , as a function of c. Given p , write the payo¤s of the expected payo¤s of the buyer and the two sellers.

2. Suppose that the distribution of c is uniform. Show p and the expected payo¤s of the parties graphically (put c on the horizontal axis and the equilibrium price function on the vertical axis and show the payo¤s by pointing out the appropriate areas in the graph).

26

3. Now suppose that the incumbent seller o¤ers the buyer a contract before the entrant shows up. The contract requires the buyer to pay the incumbent seller the amount m regardless of whether he buys from him or from the entrant, and gives the buyer an option to buy from the incumbent at a price of p (this is equivalent to giving the buyer an option to buy at a price m + p and requiring him to pay liquidated damages of m if he switches to the entrant). If the buyer rejects the contract things are as in part (1).

Given p and c, what is the price that the buyer will end up paying for the good? Using your answer, write the expected payo¤s of the buyer and the two sellers as a function of p and m.

4. Explain why the incumbent seller will choose p by maximizing the sum of his expected payo¤, I , and the buyer’s expected payo¤s, u.

5. Write the …rst-order condition for p and show that the pro…t-maximizing price of the incumbent seller, p , is such that p < 1=2. Also show that if g(0) > 0 then p > 0.

6. Explain why the contract is socially ine¢ cient. Is the outcome in part (1) socially e¢ cient? Explain the intuition for your answer.

7. Compute p assuming that the distribution of c is uniform, and show the expected payo¤s of the parties and the social loss graphically (again, put c on the horizontal axis and the equilibrium price function on the vertical axis).

8. Compute p under the assumption that G(c) = c , where does p vary with ? Give an intuition for this result.

> 0. How

Solution to Exercise 24 1. In equilibrium, the price is p = 1=2 if c 1=2 (hence, the buyer buys from the entrant), and p = c, otherwise (hence, the buyer buys from the incumbent).

Given this price, the buyer’s expected bene…t is

u=1

Z 12 0

1 g(c)dc 2

Z 1

cg(c)dc

1 2

The expected pro…t of the incumbent seller and the entrant are:

I

E

=

Z 1

(c

1 2

=

Z 21 0

27

(

1 2

1 )g(c)dc; 2 c)g(c)dc:

Using integration by parts, net bene…t and pro…ts are:

u

=

1 1 G( ) 2 2

1

=

"

Z 1

G(c)dc; 1 )G(c) 2 1

1 2

I

=

(c

E

=

(c

Z 1

1 cG(c)j1=2

G(c)dc

1 2

#

Z 1

Z 1 1 G(c)dc = G(c)dc; 1 1 2 2 2 2 1 Z 21 Z 21 2 1 )G(c) + G(c)dc = G(c)dc: 2 0 0 0 1

2. Draw …gure.

3. Given p and c, the buyer will buy from the entrant if c p and from the incumbent seller otherwise and will pay p. The buyer’s expected payo¤s is therefore u=1

p

m;

and the expected payo¤s of the incumbent seller and the entrant are

I

=

Z 1

p

p

E

=

Z p

(p

1 2

g(c)dc + m =

p

1 2

[1

G(p)] + m;

c)g(c)dc:

0

4. First, note that the contract must be designed so as to ensure the buyer the same expected payo¤ as in Part 1, otherwise the buyer will reject the contract.

Second, since the incumbent seller can choose m to extract the buyer’s expected payo¤, the expected payo¤ of the incumbent seller is equal to the sum of his expected payo¤, I , and the buyer’s expected payo¤ u minus the expected payo¤ of the buyer absent a contract. Since the latter is a constant, p will be chosen to maximize I + u.

5. Di¤erentiating

I + u, the …rst-order condition for p

p

1 2

g(p)

is given by

G(p) = 0:

At p = 1=2, the left-hand side of the …rst-order condition is negative. Thus, < 1=2. On the other hand, if g(0) > 0 then, at p = 0, the left-hand side of the …rst-order condition is positive. Thus, p > 0. The …rst expression on the left-hand side of the …rst-order condition is the marginal reduction in the likelihood the buyer buys from the incumbent seller. This expression is positive since p < 1=2: In other words, the less often the buyer exercises the option to buy the lower are the losses to the incumbent seller from having to sell at a

p

28

price below his costs. The second expression on the left-hand side of the …rstorder condition represents the marginal decreases in the sum of the expected payo¤s of the buyer and the seller. This expression is negative because when p increases by a dollar, the buyer pays one more dollar with probability 1, while the incumbent seller gets an additional dollar only when the buyer buys from him. The latter occurs with probability (1 G(p)). Thus, the total change is G(p).

6. Since the buyer always buys the good, e¢ ciency is achieved if and only if the lowest cost seller produces the good. In Part 1 competition ensured that this is indeed the case. Under a contract, the incumbent produces whenever c p .

Since p < 1=2, production is ine¢ cient whenever p < c < 1=2 since then the incumbent produces even though the entrant is more e¢ cient.

7. When c is distributed uniformly on the unit interval, the …rst-order condition for p becomes 1 p = 0: p 2 Hence, p

= 1=4. The social loss is given by L=

Z 21

1 2

p

c g(c)dc:

8. When G(c) = c , the …rst-order condition for p 1 2

p

p

1

becomes

p = 0:

Solving this equation we obtain

p

=

1 2

+1

:

This equation shows that p is increasing with . Intuitively, an increase in gives rise to two e¤ects: First, the higher is , the more likely is the entrant to have high costs. Hence, the marginal cost of raising p which is G(p) (see Part 5) becomes smaller. Second, the marginal bene…t of raising p, given by (1=2 p)g(p), changes as well and this e¤ect may work in the opposite direction.

In our case though, the …rst positive e¤ect dominates. Hence, the less e¢ cient the entrant is in expectation, the higher will be the price that the buyer is required to pay.

Interestingly, this is not always true. In particularly, if G( )=g( ) is an increasing function and there is a shift parameter that shifts both G( ) and G( )=g( ) upwards then p will decrease although the entrant is less likely to be e¢ cient.

Exercise 25 Franchising

29

A monopolistic manufacturer produces a good that is sold to three retailers.

The manufacturer has constant marginal cost equal to c < 21 . The retailers are monopolists in three di¤erent cities. The marginal cost of the retailers is equal to the wholesale price of the good. Each of the retailers faces a demand function Qb = 1 b p where p is the retail and b is a variable characterizing the local demand function. Each retailer knows the value of b in its city. Furthermore, it is common knowledge that b = 1 in one city and b = z in two cities, z 2 (1; 2].

The manufacturer o¤ers each of the retailers a two-part tari¤ consisting of a wholesale price w and a franchise fee f . Let us assume that the retailers accept any contract that results in nonnegative pro…ts. The retailers choose the retail price p in their market (i.e., there is no resale price maintenance).

1. Let w < 12 and take f as given. Find the price that a retailer sets as a function of b. Who earns the highest pro…t, retailers with b = 1 or b = z?

2. Assume in the rest of the exercise that z = 2. Suppose …rst that the manufacturer knows the value of b in all three cities and that z = 2. What contract will the manufacturer o¤er to the retailers? Is the franchise fee the same for all retailers? Is it possible for the retailer to extract all pro…ts in the vertical chain? Assume from now on that the manufacturer does not know the retailers’ individual b. However, the manufacturer knows that two of the retailers have b = z = 2 and that one retailer has b = 1.

Assume also that the manufacturer wishes to serve all retailers.

3. Find the optimal franchise fee as a function of the wholesale price w.

4. Find the optimal wholesale price as a function of c. Is the wholesale price greater or less than c?

5. Can the manufacturer extract all pro…ts by setting w and f optimally?

6. Assume now instead that c = 1=4. Show that it is optimal for the manufacturer only to sell to the retailer with b = 1. What happens if c ! 0?

Exercise 26 Exclusive dealing [included in 2nd edition of the book] Suppose that two …rms produce at constant marginal costs c. There are two periods and m buyers. Each buyer has an inverse demand curve: P (qI + qE ) = 1 (qI +qE ) where qI is the quantity sold by the incumbent and qE is the quantity sold by the entrant. In the …rst period, there is only the incumbent in the market. Thus, the incumbent produces the monopoly quantity. The incumbent has marginal cost equal to cI . In the second period, an entrant with constant marginal cost equal to zero enters into the market. The entry is foreseen by the buyers. After entry, the two …rms compete à la Cournot. In the …rst period, the incumbent o¤ers the buyers a fee for an exclusive dealing agreement (take-itor-leave-it). If a buyer accepts the o¤er, she cannot buy from the entrant in the second period.

30

1. Suppose that the incumbent and entrant are equally e¢ cient, i.e. cI = 0. What is the maximal fee for an exclusive dealing agreement that the incumbent is willing to o¤er to a buyer? What is the minimum fee that a buyer is willing to accept for signing an exclusive dealing agreement? Will there be exclusive dealing in equilibrium?

2. Consider a general marginal cost of the incumbent cI . For which values of cI will exclusive dealing arise in equilibrium?

Solution to Exercise 26 The value an exclusive dealing is worth VI = 1=4(1 cI )2 1=9(1 2cI )2 to the incumbent. A buyer accepts if she is o¤ered a payment of, at least, VB = 1=18(2 cI )2 1=8(1 2cI )2 to sign an exclusive dealing agreement . Setting cI = 0 provides the solution to (1). Exclusive dealing occurs in equilibrium.

Solution to part (2): Exclusive dealing arises in equilibrium if and only if VI VB 0, which is equivalent to cI 1=3.

Explanation: Unlike the case of Bertrand competition considered in Rasmusen et al., exclusive dealing can arise in equilibrium under Cournot competition. The reason is that exclusive dealing allows the incumbent and the buyer to obtain some of the rents that the entrant would earn under no exclusive dealing. Notice: This is not possible under Bertrand competition as the entrant earns zero pro…ts. Exclusive dealing only arises if the incumbent is not too ine¢ cient compared to the entrant. Otherwise, the ine¢ cient incumbent cannot o¤er a high enough fee to make it worthwhile for the buyer to exclude the e¢ cient entrant.

Exercise 27 Long-term contracts, upgrades, and exclusion Consider a market for a base good that is sold over two periods. In period 2, also an upgrade may become available. For simplicity, set the mass of consumers equal to 1 and marginal costs equal to zero. Firm 1 can be active in periods 1 and 2, …rm 2 can only be active in period 2. At the beginning of period 2, …rms decide simultaneously whether to upgrade. Suppose that …rms maximize the sum of pro…ts in periods 1 and 2.

The willingness-to-pay without upgrades is V per consumer in each period.

An upgrade by …rm 1 leads to a surplus of r + 1 , while an upgrade by …rm 2 would lead to a surplus of r + 2 . Firm 2 is assumed to be more e¢ cient, 2 > 1 . The upgrading cost is C. Suppose furthermore that 1 > C. This assumption means that upgrading is socially superior to not upgrading even if it is done by the less e¢ cient …rm.

1. Characterize the equilibrium if …rms can only o¤er short-term contracts, i.e., …rm 1, when selling to consumers in period 1, cannot make them sign a contract that binds consumers to buy from it in period 2.

2. Characterize the equilibrium if …rm 1 can o¤er a long-term contract that does not allow consumers or prevents them from buying from …rm 2. Discuss your result.

31

Solutions to Exercise 27 1. Bertrand competition in period 2 implies that consumers make a net surplus of r in period 2. In the case of short-term contracts …rm 2 upgrades in period 2 and makes a pro…t of 2 C (because, in equilibrium, …rm 1 does not upgrade).

Firm 1 makes a pro…t of r in period 1.

2. Alternatively, …rm 1 could o¤er a long-term contract requiring the consumer not to buy from the rival in period 2. Consumers are willing to take this option if their surplus is weakly larger than r . Hence, …rm 1 can charge r in period 1. In addition, in period 2 it can release an upgrade and charge 1 for it. Since …rm 1’s pro…t are r + 1 C > r , it is in the interest of …rm 1 to o¤er such a contract.

If consumers have signed with …rm 1 in period 1, …rm 2 has no incentive to introduce an upgrade at the beginning of period 2. Long-term contracts result because of the market power of …rm 2 in the market with upgrades. Through long-term contracting, the more e¢ cient …rm 2 is excluded from the market.

Exercise 28 Vertical integration. [included in 2nd edition of the book] 1. Suppose that two downstream retailers sell a homogeneous product in a downstream market. They have to pay w < 1 for each unit of the product that they sell on to …nal consumers and do not incur any further variable costs. The inverse downsteam market demand P (q) = 1 q , where q = q1 + q2 . Determine the Nash equilibrium when both …rms set quantities simultaneously.

2. Suppose that there are many such downstream markets (to be precise, a continuum of mass 1) and that prior to the quantity setting in those downstream markets two upstream …rms simultaneously set quantities xi .

For each unit they incur marginal costs c. Determine the subgame perfect equilibrium of the two-stage game.

3. What would change if there is only one instead of a continuum of downstream markets? Is there any conceptual di¤erence between those two settings? Explain in at most three sentences.

4. Suppose that upstream …rm 1 merges with downstream retailer 1 in each of the many downstream markets. Suppose furthermore that …rm 1 commits neither to sell to any downstream retailer 2 nor to buy any units from upstream …rm 2. The timing of the game is that, at stage 1, upstream …rms simultaneously set xi and that, at stage 2, retailers set qi . Determine the subgame-perfect equilibrium of this two-stage game.

5. Compare your results in (2) and (4). Does the vertical merger increase the input price for non-integrated downstream …rms? Does the vertical merger make consumers better o¤? Explain in one or two sentences.

32

Solutions to Exercise 28 1. The pro…ts of each downstream …rm is

D i = (1

qi

qj

w)qi . Solving the w)=3. Hence,

…rst-order conditions imposing symmetry gives q1 = q2 = (1 q = (1 w)2=3.

2. Since q = x, the inverse demand in the upstream market is w = 1 3x=2.

Upstream pro…ts are U 3(xi + xj )=2 c)xi . Solving for the FOCs i = (1 gives x1 = x2 = 2=9(1 c). Thus x = 4=9(1 c), w = 1=3 + (2=3)c and p = 5=9 + (4=9)c.

3. With a continuum of downstream markets the demand by each downstream …rm for inputs leaves the market price unchanged. With a single downstream market, if a …rm increases its quantity this will a¤ect the input price.

4. At stage 2, downstream …rm 2 obtains the input at wholesale price w2 , whereas downstream …rm 1 obtains the input through internal production and, thus, at a price of 0. Thus, …rm 1’s pro…t function is D 1 = (1 q1 q2 c)q1 , while …rm 2’s pro…t function is D 2 = (1 q1 q2 w2 )q2 . Solving the system of …rst-order conditions gives q1 = (1+w2 2c)=3 and q2 = (1 2w2 +c)=3. Since q2 = x2 , the upstream …rm faces an inverse demand of w2 = (1 3x2 + c)=2. Its pro…t is (w2 c)x2 . Maximizing pro…ts with respect to x2 yields x2 = 1=6(1 c) and w2 = 1=4(1 + 3c). Hence, x1 = q1 = (5=12)(1 c). Since q = q1 + q2 = (1=6 + 5=12)(1 c) = (7=12)(1 c), the retail price is 5=12 + (7=12)c.

5. The wholesale price paid by the non-integrated …rm is less than without the vertical merger. This may sound surprising as upstream …rm 2 exerts monopoly power. However, the upstream …rm faces a more elastic demand and, therefore, charges a lower input price. In equilibrium, consumers pay a lower retail price under vertical integration than absent vertical integration. Thus, consumers are better o¤.

Exercise 29 Exclusive dealing and vertical integration.

Consider a vertical duopoly with exclusive dealing contracts in place, i.e., upstream …rm i only sells to downstream …rm { , i = 1; 2. Suppose that, at stage 1, upstream …rms and then, at stage 2, downstream …rms set prices. Downstream demand is of the form Qi (pi ; pj ) = 1 bpi + dpj . Upstream …rms have zero marginal costs of production and set their wholesale price. Consider subgame perfect equilibria.

1. Characterize equilibrium upstream and downstream prices.

2. Suppose that b = d = 1. Characterize the equilibrium if …rms indexed by 1 have vertically integrated (so that the integrated …rm’s transfer price is 0).

3. Are there incentives for vertical integration (for b = d = 1)? Discuss your results.

33

Industrial Organization: Markets and Strategies Paul Belle‡amme and Martin Peitz published by Cambridge University Press

Part VII. R&D and intellectual property Exercises & Solutions Exercise 1 Incentives to innovate and competition Reconsider the incentives to innovate in monopoly versus under competition in the same setup as the one in the book.

1. Take the linear demand P (q) = a bq and establish the result that a competitive …rm places a larger value on a minor process innovation than a monopoly does analytically.

2. Check that the previous conclusion still holds for the case of a major innovation.

Exercise 2 Incentives to R&D and market structure [included in 2nd edition of the book] At a hotel in Munich in May 2007, Webasto, a German auto parts maker, decided to license the rights to one of its best-selling products - a roof-top solar panel for cars and trucks - to the highest bidder at a public auction. (See International Herald Tribune, May 13, 2007). Firms coming from di¤erent industries attended this public auction. From which type of industry do you think the highest bidder for Webasto’s products came from? From a concentrated (monopoly-like) industry? From a much more competitive industry with numerous small players? Or from some intermediate (oligopoly-like) industry?

The following exercise will help you answering these questions.

Assume that the demand for trucks is p = 100 q (where q is the quantity and p is the price), and that Webasto’s roof-top solar panel allows truck manufacturers to reduce the constant marginal cost of production from 70 to 60.

1. Con…rm that this is a nondrastic (or minor) innovation and that marginal cost would have to be reduced to less than 40 for the innovation to be drastic (or major).

2. Suppose that the industry is a monopoly (not threatened by entry). How much is this …rm willing to pay (per period of time) to acquire the innovation?

1

3. Suppose that the industry is a Bertrand oligopoly. That is, there are n …rms (with n 2) that compete in price. Before the innovation, all …rms have the same marginal cost of 70. After the innovation, one of them has a lower cost of 60. Compute how much the latter …rm is willing to pay for the innovation.

4. Now assume that the market is served by Cournot duopolists who have identical marginal costs of 70 before the innovation. (a) Con…rm that the pre-innovation price is 80 and that at this price each …rm has pro…ts per period of 100. (b) Suppose that one of these …rms is granted use of the innovation.

Con…rm that the price falls to 76.67, and compute the per period pro…ts of the two …rms. (c) How much is any of these duopolists willing to pay to acquire the innovation?

5. Suppose that the industry is a monopoly threatened by entry. More precisely, with the existing technology, production at a marginal cost of 70 does not make entry pro…table. However, by lowering the marginal cost to 60, the new technology makes entry pro…table. So, by acquiring the innovation, the incumbent …rms precludes entry: it remains a monopolist and produces now at a marginal cost of 60. On the other hand, if the monopolist does not acquire the innovation, another …rm does, which allows it to enter the market. The market structure becomes thus an asymmetric Cournot duopoly in which the incumbent …rm has a marginal cost of 70, while the entrant has a marginal cost of 60. (a) How much is the incumbent …rm willing to pay for the innovation? (b) How much is the entrant willing to pay for the innovation? (c) If the innovation goes to the highest bidder, what is the in‡uence of innovation on market structure? Discuss.

6. Finally, by collecting your answers to questions 1 to 5, rank the various market structures according to the incentives to innovate that they convey to …rms. Comment your ranking.

Solutions to Exercise 2 1. Let c denote the marginal cost. A monopolist chooses its quantity to maximize pro…t = (100 q) q cq . The optimal quantity is easily found as q m (c) = (100 c) =2 and the corresponding price is pm (c) = 100 q m = (100 + c) =2.

Using this expression, we compute the monopoly price corresponding to the post-innovation cost of 60: pm (60) = 80. As this price is larger than the preinnovation marginal cost (i.e., 70), we check that the innovation is nondrastic.

For the innovation to be drastic, one would need pm (c) = (100 + c) =2 < 70, which is equivalent to c < 40.

2

2. Using the analysis of the previous question, we can compute the monopoly optimal pro…t for any given marginal cost: m (c) = q m (c) [pm (c) c] = 2 1 c) . The incentive to innovate for the monopoly is measured by the in4 (100 m crease in its pro…t dut ot the innovation. That is: P I m = m (60) (70) = 400 225 = 175.

3. As the innovation is nondrastic, we know that the innovator’s optimal price strategy is to set a price just below the marginal cost of the rival …rms, i.e., p = 70 ", where " > 0 can be arbitrarily small. The innovator captures the whole demand and sells a quantity equal to q = 100 (70 ") = 30 + ". Given that the innovator’s cost is equal to 60, its pro…t is computed as b = (p 60) q ' 300. As the pre-innovation pro…t is zero, the incentive to innovate for a Bertand oligopolist is simply: P I b = b = 300:

4. Cournot duopoly (a) Before the innovation, …rm 1 chooses its quantity q1 to maximize 1 = (100 q1 q2 ) 70q1 . The …rst-order condition yields: 30 2q1 q2 = 0.

We derive …rm 1’s reaction function from the latter expression: q1 (q2 ) = 1 q1 ). As …rm 2 faces the exact same conditions, it has a similar re2 (30 action function: q2 (q1 ) = 12 (30 q2 ). We …nd the Cournot-Nash equilibrium by solving for the system of the two reaction functions: q1 = q2 = 10.

The equilibrium price is then equal to p = 100 20 = 80, meaning that each …rm makes a margin of p 70 = 10. Hence, equilibrium pro…ts for both …rms are equal to cpre = 100. (b) Let the innovator be …rm 1. Its pro…t is now written as 1 = (100 q1 q2 ) 60q1 . From the …rst-order condition for pro…t maximization, we derive …rm 1’s reaction function: q1 (q2 ) = 21 (40 q2 ). As for …rm 2, the marginal cost has not changed and thus, we have the same reaction function as in the previous question: q2 (q1 ) = 12 (30 q1 ). Solving for the system of equa20 tions given by the two reaction functions, we …nd q1 = 50 3 and q2 = 3 .

50 20 It follows that the equilibrium price is equal to p = 100 3 3 = 230 = 76:67 .

The pro…ts of the two …rms are then computed as 3 1

=

2

=

230 3 230 3

60 70

50 2500 3 = 9 = 277:78 20 400 3 = 9 = 44:44:

c post

(c) We compute the incentive to innovate as the pro…t increase for the innoc vator: P I c = cpost 100 = 177:78. pre = 277:78 5. Monopoly threatened by entry (a) In case the incumbent gets the innovation, it remains a monopoly with a marginal cost of 60. We know from (2) that its pro…t is then equal to m (60) = 400. In case the incumbent does not get the innovation, it will be a Cournot duopolist with a cost of 70 facing a rival …rm with a cost of 60. This is the equilibrium pro…t of …rm 2 as computed in (4.b):

3

2 = 44; 44. Hence, the incumbent’s incentive to innovate is equal to P I inc = 400 44:44 = 355:56.

(b) The entrant makes zero pro…t if it does not get the innovation and if it c does, it makes the duopoly pro…t identi…ed as 1 pre = 277:78 as computed in (4.b). Hence, the entrant’s incentive to innovate is equal to P I ent = 277:78.

(c) It is the incumbent that will end up with the innovation, which will allow it to maintain its monopoly position through time.

6. In this example, we observe P I inc > P I b > P I ent > P I c > P I m . The fact that P I b > P I m is known as the replacement e¤ect. The fact that P I inc > P I ent is known as the e¢ ciency e¤ect. The fact that P I inc > P I m shows that the threat of entry increases the incentive to innovate for a monopoly. The fact that P I c > P I m is not a general result.

Exercise 3 Incentives to invest in product and process innovations1 [included in 2nd edition of the book] Consider the following duopoly. Each …rm i (i = 1; 2) incurs a constant marginal cost equal to ci and produces a di¤erentiated product, qi , sold at price pi . The demand system is obtained from the optimization problem of a representative consumer. We assume a quadratic utility function which generates the linear inverse demand schedule pi = a qi qj in the region of quantities where prices are positive. The parameter 2 [0; 1] is an inverse measure of the degree of product di¤erentiation: the lower the more products are di¤erentiated (if = 1, products are perfect substitutes; if = 0, products are perfectly di¤erentiated). Firms compete à la Cournot on the product market.

Initially, both …rms produce at cost ci = c. A new process innovation allows …rms to reduce the constant marginal cost of production from c to c0 = c x (with 0 < x < c). We assume that the innovation is nondrastic. That is the cost reduction does not allow the innovator to behave like a monopolist. A su¢ cient condition is that the monopoly price corresponding to c0 is larger than the initial cost c; that is, (a + c x)=2 > c. Equivalently, assuming without loss of generality that the di¤erence a c is equal to unity, we assume: x < a c = 1.

1. Compute how much a duopolist is willing to pay for acquiring the innovation and being its single user (i.e., compute the di¤erence between the pro…t a …rm makes when it is the sole user of the innovation and the pro…t it makes when no …rm uses the innovation).

2. Suppose now that a product innovation allows …rms to increase product di¤erentiation (i.e., to reduce the parameter ). Show that the adoption 1 This exercise draws from Belle‡amme, P. and Vergari, C., "Incentives to Innovate in Oligopolies" (Manchester School, 2009).

4

of this product innovation raises the incentives to adopt the process innovation (computed at the previous question) if and only if the initial degree of product substitution, , is lower than 2=3.

Solutions to Exercise 3 1. We …rst derive the Cournot equilibrium for given c1 and c2 . Firm i’s problem is maxqi (a ci qi qj ) qi . From the FOC, we …nd …rm i’s reacqj ). Similarly, we have for …rm j : tion function as qi (qj ) = 12 (a ci qi ). Solving the system of the two reaction functions, qj (qi ) = 12 (a cj we have

qi =

(2 (2

)a 2ci + cj )(2+ ) ;

To …nd the pre-innovation pro…t, Using a c = 1, we …nd

2 i = (qi ) :

(1)

pre , we set ci = cj = c in expression (1).

1 pre = (2+ )2 :

To …nd the post-innovation pro…t, post , we set ci = c expression (1). Using a c = 1, we …nd 2 (2

post =

+2x )(2+ )

x and cj = c in

2

:

Incentives to innovate are computed as post

pre

2

=

2 (2

+2x )(2+ )

=

2 (2

+2x 1 )(2+ ) + (2+ )

=

2(2 +x) (2 )(2+ ) (2

1 (2+ )2 2 (2

+2x )(2+ )

1 (2+ )

4x(2 +x) 2x : )(2+ ) = (2 )2 (2+ )2

2. To measure the sensitivity of the incentive to innovate in the process innovation with respect to a product innovation, we need to di¤erentiate ( post pre ) with respect to :

d d

4x(2 +x) (2 )2 (2+ )2

(3 = 4x (2

2)(2 ) >0, )3 (2+ )3

> 23 :

Suppose that initially, > 23 . Then, a product innovation that has the e¤ect of decreasing will induce a decrease in the incentive to innovate in the process innovation. On the other hand, if the inital value of is below 23 , then a decrease in leads to an increase in ( post pre ).

Exercise 4 Cumulative innovations [included in 2nd edition of the book]

5

Consider the following market structure. There are two …rms, noted 1 and 2. In the …rst period, …rm 1 exogenously makes a discovery. If it incurs costs c1 , it can turn this discovery into a new product. A mass 1 of consumers has a valuation for this product of r1 . In the second period (no discounting), …rm 2 exogenously makes a discovery if (and only if) …rm 1 has developed the product in period 1. Incurring costs c2 , …rm 2 can turn this discovery into a competing product for which consumers have a valuation of r2 . Assume that consumers have unit demand and buy either one unit of product 1, one unit of product 2 or nothing after the second period. Throughout the exercise, assume: A1 : r2 > c2 + r1 and A2 : r1 > c1 > 0.

1. Brie‡y interpret A1 and A2. Which investment decisions are made in the subgame-perfect Nash equilibrium of the game without an allocation of intellectual property rights, i.e., with standard competition in period 2?

Explain why this result may be ine¢ cient from a welfare standpoint.

2. For the same parameter constellation, show that a license fee payable from …rm 2 to …rm 1 for every unit of the good sold can induce a welfare optimal allocation. (Assume the following timing of the game: …rst the fee is set; then …rm 1 makes the investment decision; …nally …rm 2 makes the investment decision.) In which range must lie to be e¤ective? What happens if it is too high/too low?

3. Now let us further expand the game in the second period. (For this last part of the problem, assume for simplicity that r1 = 0.) Assume that courts only enforce …rm 1’s license claims against …rm 2 with probability p. Firm 2 can choose two types of monetary investment: ca2 increases the value of the product for consumers, such that @r2 =@ca2 > 0, with r2 (0) > r1 and @ 2 r2 =@ca2 2 < 0. Parameter cb2 does not a¤ect the value of the product to consumers, but it reduces the probability that …rm 2 would be required to pay the license fee by a court, i.e. @p=@cb2 < 0, with limcb2 !1 p > 0 and 2

@ 2 p=@cb2 > 0. Find a subgame-perfect Nash equilibrium of the following game. In the …rst stage, …rm 1 chooses a …xed . In the second stage, …rm 2 chooses both its investment levels. In the third stage, consumers make their purchase decisions and courts enforce the license fee with probability p(cb2 ). What changes if …rm 1 can set as a share of r2 instead of a …xed fee?

Solutions to Exercise 4 1. The two assumptions state that (1) the development of product 2 is welfare increasing given that product 1 has been designed and (2) the development of product 1 as a stand-alone product (without the development of product 2) is welfare increasing as well.

Solve via backward induction: If 1 has not developed its product, 2 cannot develop. If one has developed its product, the following sub-game ensues: If 2

6

develops its product at cost c2 , the two …rms engage in Bertrand competition afterwards. All consumers choose product 2 at price r2 r1 (development costs are sunk at this stage), therefore …rm 2 makes a pro…t by A1. Firm 1 makes a loss of c1 . Anticipating this, it does not develop its product in stage 1 of the game. As both developments increase total surplus, this is ine¢ cient.

2. The license fee has to achieve the following: It must give …rm 1 an incentive to invest in the …rst period and …rm 2 to invest in the second, while …rm 2 sells the product to consumers in the end. Looking at the participation constraints, the following therefore must hold: (P2 ) : r2 c2 0$ r2 c2 and (P1 ) : c1 > 0 $ > c1 . Further, we must ensure that it is …rm 2 selling the product and not …rm 1, so either of the following must hold: (IC1 :) > r1 , in which case 1 prefers licensing to selling its own product or ("IC1 ") : r2 r1 > c2 , in which case …rm 2 can sell the product in the market at a pro…t despite the license fee.

If P2 is violated, then …rm 2 cannot make a pro…t and does not invest. If P1 is violated, …rm 1 cannot make a pro…t from licensing and will only invest if the parameters do not allow …rm 2 to enter the market later on. If both IC1 and "IC1 " are violated, then …rm 1 will sell its product to consumers in the last stage and …rm 2 does not invest.

3. With its investment decisions, …rm 2 maximizes 2 = r2 (ca2 ) p(cb2 ) ca2 cb2 . @p @r2 This gives the …rst order conditions @ca = 1 and j @cb j = 1. Therefore the 2

2

investment incentive regarding quality is not distorted through the …xed fee, while a higher fee increases the incentive for …rm 2 to try to invent around the prior patent. Note that if p(cb2 ) < c1 , then again the incentives for …rm 1 to invest break down and there will be no innovation.

For a percentage fee, …rm 2 maximizes 2 = (1 p(cb2 ) )r2 (ca2 ) ca2 cb2 . It is straightforward to show from the …rst-order conditions that here the quality investment incentive is distorted, as well.

Exercise 5 Cumulative innovations and the Human Genome Project In the Human Genome News Archive Edition of November 2000, one can read the following: “The deluge of data and related technologies generated by the Human Genome Project (HGP) and other genomic research presents a broad array of commercial opportunities. Seemingly limitless applications cross boundaries from medicine and food to energy and environmental resources, and predictions are that life sciences may become the largest sector in the U.S. economy.

Established companies are scrambling to retool, and many new ventures are seeking a role in the information revolution with DNA at

7

its core. IBM, Compaq, DuPont, and major pharmaceutical companies are among those interested in the potential for targeting and applying genome data.

In the genomics corner alone, dozens of small companies have sprung up to sell information, technologies, and services to facilitate basic research into genes and their functions. These new entrepreneurs also o¤er an abundance of genomic services and applications, including additional databases with DNA sequences from humans, animals, plants, and microbes.”

1. Innovations in the genomic …eld are in essence cumulative. De…ne two di¤erent types of cumulative innovations.

2. For each type of cumulative innovations, describe the generic problem that is likely to arise and discuss how the patent system could be complemented in order to solve these problems.

3. What is your opinion about the fact that “dozens of small companies have sprung up to sell information, technologies, and services to facilitate basic research into genes and their functions”? Is this likely to foster, or rather to impede, innovation in the genomic …eld?

Solutions to Exercise 5 1. Sequential vs. complemetary innovations; see Subsection 19.3.2.

2. The main problem with sequential innovations is the potential hold up problem; ex ante licensing may be used to solve this problem. As for complementary innovations, the main problem is called the ‘tragedy of the anticommons’; patent pools and other collaborative mechanisms may alleviate this problem (see Subsection 19.3.2 for more).

3. Small …rms may foster innovation by increasing the tradability of technologies and making sure that they are used by those who value them the most. On the other hand, the more right-holders there are on complementary innovations, the more accute the problems linked to the tragedy of the anticommons.

Exercise 6 Secrecy versus patenting2 Consider an innovative environment where independent or nearly simultaneous discoveries are possible. More speci…cally, we assume that two …rms are engaged in R&D that results either in an innovation (with probability ) or failure (with probability 1 ). It is assumed that the probability of success ( ) is independent across …rms. Firms can protect their innovation either by secrecy or by …ling for a patent.

2 This exercise is based on Kultti, K., Takalo, T., and Toikka, J. (2007). Secrecy versus Patenting. Rand Journal of Economics 38: 22–42.

8

If the innovation is protected by secrecy, it can leak out with probability 1 s . When this happens, the innovation is publicly available and production is at the competitive level, driving the innovator’s pro…ts down to zero.

Patent protection is measured by the probability that a patent holder can exclude competitors from using the innovation, which is denoted p .

Hence, with probality 1 p , the innovation becomes public, resulting again in zero pro…ts for the innovator.

If only one …rm succeeds in R&D and the innovation does not become public, the …rm earns monopoly pro…t m . If both …rms succeed and their innovation does not become public, each …rm earns duopoly pro…t d < m . In the case where both …rms are successful and …le for the patent, each …rm obtains it with probability 1=2.

The two …rms have to decide whether to …le for a patent (strategy noted P ) or resort to secrecy (strategy noted S). This decision has to be made before learning whether the competitor has succeeded or not.

1. Using the above information, compute the …rms’expected pro…ts for the four combinations of strategies. Denote (a1 ; a2 ) the expected pro…t for a …rm when it chooses strategy a1 and its opponent chooses strategy a2 , with a1 and a2 2 fP; Sg. You are thus asked to compute (P; P ), (P; S), (S; P ), and (S; S).

2. Suppose that p = s . That is, the innovation has the same probability of becoming public whether it is protected by secrecy or by a patent (in other words, patent and secrecy o¤er the same level of protection). (a) Show that patenting is a dominant strategy. That is, show that both (P; P ) (S; P ) and (P; S) (S; S) are true. (b) Show also that successful …rms prefer the situation where they both …le for a patent over the situation where they both keep the innovation secret. That is, show that (P; P ) (S; S) is true.

3. Suppose now that p 6= d = 4, and = 1=2.

s.

(a) Compute the values of der these assumptions.

To ease the computations, set (P; P ),

(P; S),

(S; P ), and

m

= 16,

(S; S) un-

(b) Characterize the Nash equilibrium (in pure strategies) of the game for all p , s 2 [0; 1]. Represent graphically the characterization of the equilibrium in the plane ( p ; s ). (c) Show that both …rms may choose to protect the innovation via a patent even though patents o¤er a weaker protection than secrecy.

Explain the economic intuition behind this result.

9

Solutions to Exercise 6 1. We …nd (S; S) = (1 ) s m + 2 s d , (S; P ) = (1 m m (P; S) = (1 ) p + 2 p m= , and (P; P ) = p 21 1 m m = (2 ) . p 2 p 2 2. Let

p =

s =

) (1

s

m

,

)

p

, then

(a) (P; P )

(S; P )

= =

(P; S)

(S; S)

1 2 1 2

)

2 m

> 0;

m

= =

m

(2

2

(1 m

d

(1

)

m

) m > 0;

2

d

m

2

(b) (P; P )

(S; S)

= =

1 2 1 2

(2 2

) m

m

(1

2 d

0;

(with equality if …rms can perfectly collude, i.e., if

3. Let (a)

m

= 16,

d

= 4, and

(S; S) = 5 s ,

)

d

=

m

d

=2).

= 1=2. Then,

(S; P ) = 4 s ,

(P; S) = 8 p ,

(P; P ) = 6 p .

(b) First, (S; S) is a Nash equilibrium i¤ (S; S) (P; S) or p = s 5=8 = 0:625. Second, (P; P ) is a Nash equilibrium i¤ (P; P ) (S; P ) or p = s 4=6 ' 0:666. Finally, (S; P ) and (P; S) are Nash equilibria i¤ 0:625 0:666. p= s (c) As (P; P ) is an equilibrium for

0:625 s , it obtains for values of p p that are strictly lower than s . The intuition is explained by Kultti et al.

(2007, p. 23): “under a patent system, it only pays to keep the innovation secret when the probability that a competitor comes up with the same innovation and patents it is su¢ ciently small. If the probability is large, it pays to apply for the patent even if it confers only weak protection because, otherwise, someone else gets it, and the innovator risks infringement if she tries to capitalize her innovation. In other words, when innovators contemplate patenting, the typical choice is not between patenting or keeping the innovation secret but between patenting or letting the competitors patent.”

Exercise 7 Strategic patenting [included in 2nd edition of the book]

10

m

+

Consider a market where demand is given by P (q) = a q. An incumbent …rm has a proprietary technology with a constant marginal cost of cI (with cI < a). One other …rm could enter the market as a Cournot duopolist, but the technology available to this …rm does not allow it to make any positive pro…t if it enters. Precisely, the marginal cost corresponding to the entrant’s technology, cE , is such that cE = (a + cI ) =2 = c~.

1. Check that this condition implies the nonpositivity of the entrant’s quantity at the Cournot-Nash equilibrium.

Suppose now that alternative technologies become available with a constant marginal cost c comprised between cI and c~.

2. Show that, although the incumbent has no incentive to switch to any of these technologies, it has a higher incentive to acquire a patent on them than the entrant has.

3. What does the previous result tell you about …rms’ motivations to …le patents? Discuss.

Solutions to Exercise 7 1. Suppose that the incumbent and the entrant compete as Cournot duopolists.

The incumbent chooses qI to maximize (a qI qE cI ) qI . We derive the incumbent’s reaction function from the …rst-order condition for pro…t maximization, i.e., qI (qE ) = (1=2) (a cI qE ). Proceeding in a similar way, we …nd the entrant’s reaction function as qE (qI ) = (1=2) (a cE qI ). Solving the system of equations made of the two reaction functions allows us to compute the equilibrium quantities as (with the superscript d for duopoly): d = (1=3) (a 2cE + cI ). Replacing cE qId = (1=3) (a 2cI + cE ) and qE d by its value (a + cI ) =2, we …nd qE = (1=3) (a (a + cI ) + cI ) = 0, which completes the proof.

2. If the incumbent acquires the patent on the new technology, it remains a monopolist (since the entrant has no pro…table way to enter). Naturally, the incumbent prefers to keep on using its existing technology as the associated cost of production is lower (cI < c). The incumbent’s pro…t is then computed as follows.

From the reaction function computed above, we see that the incumbent’s optimal quantity when the entrant stays out (i.e., when qE = 0) is the monopoly quantity qIm = (1=2) (a cI ). The market price is then pm = (1=2) (a + cI ) 2 and the incumbent’s pro…t is m cI ) , while the entrant pro…t is I = (1=4) (a zero. On the other hand, if the entrant acquires the patent, it enters the market and the two …rms act as Cournot duopolists. The incumbent has cost cI while the entrant has cost c. Using our previous analysis, we compute the equilibrium d quantites as qId = (1=3) (a 2cI + c) and qE = (1=3) (a 2c + cI ). Using the …rst-order conditions for pro…t-maximization, it is easily seen that for each …rm, the equilibrium pro…t is equal to the square of the equilibrium quantity, so

11

2

2

that dI = (1=9) (a 2cI + c) and dE = (1=9) (a 2c + cI ) . We can now compute the incentive to acquire the patent for each …rm. For the incumbent, d the value of the patent is given by the di¤erence m I I , whilst it is given by d d d 0 for the entrant.

To prove the claim, we need to show that m E I I > E for all c comprised between cI and c~ = (a + cI ) =2. We compute m I

d I

d 1 E = 36 (a + 10c

11cI ) (a

2c + cI ) :

The …rst bracket can be rewritten as a cI + 10 (c cI ), which is positive as we assumed that cI < a and c > cI . As for the second bracket, we see that it is positive as long as c < (a + cI ) =2, which is what we assumed. Hence, we have d d m d d proved that m I I E > 0 or I I > E.

3. The previous result can be seen as the evidence of strategic patenting. It may help to explain why some …rms …le for many more patents than they actually intend to use. Such patents are sometimes called “blocking patents”.

Exercise 8 The patent thicket3 Suppose that N …rms, i = 1; : : : ; N , each own a patent that is essential to the production of a given product. For simplicity, we assume that there is a competitive industry that produces this product, buying and assembling the necessary components from each of these N …rms. Firm i sets a license fee for the use of its patent by the competitive assembly industry. The cost to …rm i per unit (for licensing its patent to assemblers) is denoted by ci . For simplicity, assume that ci = 0 8i. The license fee charged by …rm i is denoted by pi . The price of the …nal product is denoted by p. We assume that the assembly …rms incur no other assembly cost in addition to paying PN royalties. Competition in the assembly industry therefore ensures that p = i=1 pi . Demand for the …nal product is given by q = a p.

1. Suppose that the N patent holders set their license fee independently and noncooperatively. Derive the Nash equilibrium in prices, compute (i) the total license fee for the N patents (denote it by ps ) and (ii) the pro…t of each patent holder at equilibrium (denote it by s ).

2. Suppose now that all …rms form a patent pool and choose a common license fee p to maximize their joint pro…t. Assembly …rms that pay p have access to the whole pool of patents. Derive the optimal p. Supposing that the pool’s pro…t is equally distributed among the N …rms, compute each individual …rm’s pro…t (denote it by p ).

3. Comparing your answers to the previous two questions, show that the formation of the pool (i) reduces the total license fee (p < ps ) and (ii) increases each patent holder’s pro…t ( p > s ). Discuss.

3 This exercise draws from the technical appendix of Shapiro, C. (2001). Navigating the Patent Thicket: Cross Licenses, Patent Pools, and Standard Setting. NBER/Innovation Policy and the Economy 1: 119-150.

12

4. Consider that k …rms, with 1 < k < N , form a patent pool and coordinate their pricing decision, while the other N k …rms continue to set their fee independently. (a) Derive the equilibrium pro…t of each …rm, independent and pool member, at equilibrium. (b) Suppose that N 3 and show that if the pool has to be formed through simultaneous individual decisions, no pool is formed, i.e., it is a Nash equilibrium for all …rms to set their fee independently.

Discuss.

Solutions to Exercise 8 1. Firm i’s maximization program is maxpi pi (a pi P i ), where P i denotes the sum of the license fees of all …rms but i. The …rst-order condition yields the reaction function pi (P i ) = 21 (a P i ). As …rms are symmetric, they all set the same fee, p , at equilibrium. Substituting into the previous equation, we get a p = 21 (a (N 1) p ) , p = ; N +1 which implies that the total fee paid for the N patents is

Na : N +1

ps =

Each …rm makes the following pro…t at equilibrium: s

=

a N +1

Na N +1

a

=

2. The pool’s maximization program is maxp p (a

a2

2:

(N + 1)

p). From the …rst-order con-

dition, it is easily found that the optimal price is

p=

a : 2

It follows that the pool’s pro…t is equal to a2 =4, which implies that each …rm makes a pro…t of p

=

a2 : 4N

3. Comparing total license fee, we observe that ps

p=

Na N +1

a (N 1) a = > 0: 2 2 (N + 1)

The total license fee is larger when …rms set their fees independently. The di¤erence with the pool’s fee increases with N . Comparing pro…ts, we observe that p

s

=

a2 4N

a2

2

(N 1) a2 2 = 2 > 0; (N + 1) 4N (N + 1) 13

which establishes that …rms earn higher pro…ts when they form a pool. The intuition is simple (see the description of the tragedy of the anti-commons in Chapter 19): each individual …rm ignores the positive e¤ect that a decrease in its own price has on the demand for the other …rms’ patents; in contrast, the pool internalizes the complementarity between the patents and, hence, has a further incentive to decrease prices; also, the internalization of the externality allows …rms to reach a higher pro…t.

4. Pool formation (a) There are now N

k +1 independent price setters (the N k independent …rms and the pool). Repeating the analysis of question 1, we easily …nd that each price setter chooses p = a= (N k + 2) and obtains a pro…t equal to the square of this price. Hence, an independent …rm and a pool member achieves, respectively, the following pro…t out (k)

=

in (k)

=

a2 (N

2;

k + 2) a2

k (N

2:

k + 2)

(b) For a pool of k …rms to be stable, it must be that no pool member has an incentive to unilaterally leave the pool, which is equivalent to in

(k)

out

(k

1) ,

a2 k (N

a2 2

k + 2)

(N

2:

k + 3)

The latter condition can be rewritten as follows:

(N

(N

k + 3)

2

k (N

k) + 6 (N

k) + 9

k (N

(1

k) + (6

2

4k) (N

2

k) (N

2

k + 2) , 2

k) + 4k (N k) + (9

4k)

k) + 4k , 0:

It is easily seen that for k 3, which supposes N 3, all the terms of the above inequality are negative. This implies that the condition is always violated, meaning that whatever the size of the pool, each member has an incentive to leave it. It remains to check if a pool with 2 …rms is stable.

Evaluated at k = 2, the LHS of the latter condition becomes 2N N 2 +1, which is negative for N 3. So, a pool of two …rms is unstable as well, which proves our result. The intuition is the same as in the case of cartel formation (see Chapter 14 and note that the problem we consider here– price setting of perfect complements–is technically equivalent to Cournot competition over perfect substitutes): the incentive to free-ride on the price reduction induced by the formation of the pool is too strong and each …rm prefers to stay outside the pool, which, as a result, never forms.

14

Exercise 9 Optimal copyright length “The Adelphi Charter on Creativity, Innovation and Intellectual Property responds to one of the most profound challenges of the 21st century: How to ensure that everyone has access to ideas and knowledge, and that intellectual property laws do not become too restrictive. The Charter sets out new principles for copyrights and patents, and calls on governments to apply a new public interest test. It promotes a new, fair, user-friendly and e¢ cient way of handing out intellectual property rights in the 21st century. The Charter has been written by an international group of artists, scientists, lawyers, politicians, economists, academics and business experts.”

Here is what is written in the Adelphi Charter about the length of IP protection: “The ‘term’– the period over which a copyright or patent is e¤ective –has always been limited –and for very good reasons. Now, however, there is a disturbing trend towards lengthening terms, particularly in the area of copyright. [. . . ] The earliest ‘term’ of copyright, given as a right to authors, was 14 years, with the option of extending for an extra 14 years. The length of copyright term has been continually extended, particularly since the 1960s. In some jurisdictions, there is now pressure to lengthen terms to 90 or 120 years, and a continual pressure for all governments to follow suit and ‘level-up’. There are a number of problems associated with the lengthening copyright terms.” (www.adelphicharter.org) 1. Set out the economic problem that a regulator must solve when choosing the optimal length of IP protection (use either a mathematical or a graphical approach).

2. Do you concur with the authors of the Adelphi Charter when they claim that “there are a number of problems associated with the lengthening copyright terms”? Use the framework you have just outlined to express your opinion.

Solutions to Exercise 9 1. See Subsection 19.2.1 for an analytical approach. Figure 19.1 can be sued for the graphical approach.

2. Arguments can be found in Case 19.5.

Exercise 10 Formation of patent pools [included in 2nd edition of the book] Suppose that three …rms (noted i = 1; 2; 3) each own a patent that is essential to the production of a given …nal product. For simplicity, we assume that there is a competitive industry that produces this …nal product, buying and assembling the necessary components from each of these three …rms. We assume that the assembly …rms incur no other assembly cost in addition to (i) paying royalties 15

for the use of the three essential patents, (ii) incurring transaction costs when inquiring about the license fees. Regarding the latter transaction costs, it is assumed that they are inversely related to the number of di¤erent license fees that are set by the patent holders. Patent holders have indeed the possibility to form so-called “patent pools” whereby they coordinate their decisions to set a unique license fee that allows assembly …rms to access two (or three, if all …rms join) patents at once. This is modelled as follows.

The price of the …nal product is denoted by p. Demand for the …nal product is given by q = a p.

Competition in the assembly industry therefore ensures that p be equal to the the marginal cost of assembly, which depends on the patent pool that patent holders may have formed. In particular, three options are possible: 8 < r1 + r2 + r3 + 3 ; if no pool is formed, rij + rk + 2 ; if …rms i and j form a pool, p= : rp + ; if the three …rms form a pool,

where 0 < a=3 is the cost per transaction, ri is the license fee set by …rm i for accessing its patent, rij (resp., rp ) is the common fee set by the pool formed by …rms i and j (resp., all …rms) for accessing the bundle of patents i and j (resp., all patents).

1. Suppose that the three patent holders set their license fee independently and noncooperatively. Derive the Nash equilibrium in fees, compute (i) the price of the …nal product (denote it ps ) and (ii) the pro…t of each patent holder at equilibrium (denote it s ).

2. Repeat the previous analysis by assuming that …rm i and j coordinate their decisions to set a common fee rij that has to be paid for acquiring the right to use patents i and j. Firm k, on the other hand, still acts separately and sets its license fee rk . Compute again the price of the …nal product (denote it p2 ), as well as the equilibrium pro…ts of the …rms.

Denote the pro…t of …rms i and j in (supposing that they divide their joint pro…t equally) and the pro…t of …rm k, out .

3. Suppose now that the three …rms form a patent pool and choose a common license fee rp to maximize their joint pro…t. Assembly …rms that pay rp have access to the whole set of patents. Derive the optimal rp . Supposing that the pool’s pro…t is equally distributed among the three …rms, compute each individual …rm’s pro…t (denote it by p ). Compute also the price of the …nal product (denote it pp ).

4. Comparing your answers to the previous three questions, show that, even in the absence of transaction costs ( = 0), the more there are patents in the pool, the smaller the price of the …nal product, and the larger the sum of the patent holders’pro…ts. Discuss the economic intuition behind these results.

16

5. Consider now the following pool formation game. Before they set the license fees, patent holders simultaneously decide whether to join a patent pool or not (only one pool can form). Set a = 120 and characterize the Nash equilibrium of this game for all admissible values of the transaction cost (i.e., 0 < 40). Discuss the economic intuition behind your results.

Solutions to Exercise 10 1. Firm i’s maximization program is maxri ri (a

ri r i 3 ), where r i denotes the sum of the license fees of all …rms but i. The …rst-order condition yields the reaction function ri (r i ) = 21 (a r i 3 ). As …rms are symmetric, they all set the same fee, rs , at equilibrium. Substituting into the previous equation, we get rs = (1=2) (a 3 2rs ), which is equivalent to rs = (1=4) (a 3 ).

This implies that the price of the …nal product is ps = 3rs +3 = (3=4) (a + ).

Each …rm makes the following pro…t at equilibrium: 1 s = 4 (a

3 ) a

3 4 (a +

1 ) = 16 (a

2

3 ) :

2. The pool formed by …rms i and j and …rm k have, respectively, the following maximization programs are maxrij rij (a 2 rij rk ) and maxrk rk (a 2 The …rst-order conditions yield the following reaction functions: rij (rk ) = (1=2) (a 2 rk ) and rk (rij ) = (1=2) (a 2 rij ) : Solving this system of equations, one …nds: rij = rk = (1=3) (a 2 ), which implies that the price of the …nal product is p2 = rij + rk + 2 = (2=3) (a + ). Equilibrium pro…ts are as follows: in

=

out

=

1 1 2 3 (a 1 2 3 (a

2 3 (a +

2 ) a ) a

2 3 (a +

)

2

1 = 18 (a

) = 91 (a

2 ) ; 2

2 ) :

3. The pool’s maximization program is maxrp rp (a

rp ). From the …rst-order condition, it is easily found that the optimal price is rp = (1=2) (a ). It follows that the price of the …nal product is pp = rp + = (1=2) (a + ) and 2 that the pool’s pro…t is equal to (a ) =4, which implies that each …rm makes a pro…t of 1 p = 12 (a

2

) :

4. Comparing prices of the …nal product and total pro…ts, we easily conclude: pp = 21 (a + ) < p2 = 23 (a + ) < ps = 43 (a + ) ; tot p

=

tot 2

>

2 1 ) > tot 2 = 2 in + 4 (a 2 tot 3 3 ) ; s = 3 s = 16 (a

2 out = 9 (a

2

2 ) and

and all these inequalities hold for all 0 < a=3. The intuition is simple: each individual …rm ignores the positive e¤ect that a decrease in its own price

17

rij

rk ).

has on the demand for the other …rms’patents; in contrast, the pools internalize the complementarity between the patents and, hence, have a further incentive to decrease fees; also, the internalization of the externality allows …rms to reach a higher pro…t. The larger the pool (and the larger the transaction cost), the larger this e¤ect.

5. All three …rms join the pool i¤ 1 12 (120

2

out , which is equivalent to

p

2

2 ) , 13 2 1200 + 14400 p 120 5 2 3 ' 14:178: 13

1 9 (120

)

,

0

A pool with two …rms is an equilibrium if the …rm outside the pool does not want to join it (i.e., out p , which is the opposite of the latter condition), and no …rm within the pool wants to leave it, i.e., in s , which is equivalent to 1 18 (120

2

2

1 16 (120

2 )

,

120 49

3 ) , 49 2 2640 + 14400 p 11 6 2 ' 6:159:

0

In sum, the Nash equilibrium of the pool formation game is as follows: (i) for 0 < 6:159, no pool is formed; (ii) for 6:159 < 14:178, a pool of two patents is formed; (iii) for 14:178 < 40, a pool of three patents is formed.

The intuition is the same as in the case of cartel formation (see Chapter 14 and note that the problem we consider here–price setting of perfect complements–is technically equivalent to Cournot competition over perfect substitutes): when the transaction costs are small enough, the incentive to free-ride on the price reduction induced by the formation of the pool is too strong and each …rm prefers to stay outside the pool, which, as a result, never forms. However, as transaction costs increase, incentives to stay in the pool are stronger as leaving it would mean a substantial decrease in demand because assembly …rms would have to enquire about a larger number of fees and would, thereby, face higher assembly costs.

Exercise 11 Patent pools and mergers Consider a vertical market structure with 2 upstream …rms (A and B) and 2 downstream …rms (a and b). The downstream …rms require the input of each of the upstream …rms, who demand linear royalties (raA , rbA , raB , rbB ) charged for each unit the respective downstream …rm sells. Downstream …rms face the inverse demand P (q) = a bq; where q = qa + qb . Assume that the royalties accruing to the upstream …rms are the only costs that the downstream …rms face (ca;b = 0) and all costs of the upstream …rms are sunk.

1. Brie‡y explain in one or two sentences why perfectly complementary upstream products are a good way to describe a set of complementary patents.

18

2. Solve for the symmetric subgame-perfect Nash equilibrium in which the upstream …rms set non-discriminatory royalties (riI = rjI ) in the …rst stage and downstream …rms engage in Cournot-competition in the second stage.

3. Now assume that …rms A and a merge (vertical merger) and maximize joint pro…ts. Solve for a subgame-perfect Nash equilibrium in which the upstream …rms set royalties in the …rst stage and downstream …rms engage in Cournot-competition in the second stage. How does the merger a¤ect total royalties charged and quantities sold?

4. Starting from the original (separate) setup, now assume that …rms A and B merge (horizontal merger) and maximize joint pro…ts. Solve for a subgame-perfect Nash equilibrium in which the upstream …rm(s) set nondiscriminatory royalties (ra = rb ) in the …rst stage and downstream …rms engage in Cournot-competition in the second stage. How does the merger a¤ect total royalties charged and quantities sold?

Solutions to Exercise 11 1. If producing a certain product requires the use of multiple patented ideas, then the producer needs to obtain licenses from each of the patentees. In most cases, these licensing agreements follow a linear (in quantities) pricing scheme. Translating this into the model is equivalent to Leontie¤-type upstream products with linear pricing.

2. The downstream interactions are of standard Cournot-type. Downstream …rms individually maximize. In particular, a = qa (a b(qa + qb ) rA rB ). One obtains the reaction functions

qi (qj ) =

a

bqj

rA 2b

rB

:

Therefore, in equilibrium, the downstream …rms each choose quantity qi;j = a rA rB .

3b Anticipating this, the upstream …rms maximize I = rI (Q(rI ; rJ )) = rI ( a rAb rB ), which yields the reaction functions rI (rJ ) = a 2rJ . We therefore obtain rA = 2a rB = a3 . Total royalties in the market are 2a 3 , total output is 9b .

3. Denoting the pro…ts of the merged …rm as

A , we get the following pro…t functions (focusing on the internal solution; in the alternative possible solution A sets rA so high to price b out of the market, and we would have a downstream and an upstream monopoly): B = rB q , A = rA qb + qa (a bq rB ) and bq rA rB ). While rA does not a¤ect the output decision of b = qb (a downstream …rm a anymore, it does still a¤ect the choice of b (raising rivals cost e¤ect). On the other hand, as a increases qa , this does not only lower the price in the downstream market, it also lowers the amount of the good …rm b produces. In a Cournot equilibrium, a will not take this into account, as

19

it takes the amount produced by b as given (as opposed to, say, the Stackelberg case). The two …rms’reaction functions are: qb (qa ) = a bqa 2brA rB and

qa (qb ) = a bq2bb rB . Inserting these yields the equilibrium production levels A rB A rB and qb = a 2r3b . qa = a+r3b

A 2rB Anticipating this, upstream …rm B maximizes B = rB ( 2a r3b ) which 2a rA yields the reaction function rB (rA ) = .

Firm A anticipates downstream 4 interactions and therefore maximizes

A = rA (

a

2rA 3b

rB

)+(

a + rA 3b

rB

)(a

b(

2a

2rB 3b

rA

)

rB );

3 which yields the reaction function rA (rB ) = 12 a 5 rB . Therefore, in equilib7a rium, the royalty rates chosen are rA = 32 and rB = 15a 32 . The total quantity 9a . Note that while the total royalty has produced downstream is therefore 32b increased, downstream production still increases, because a is not a¤ected by rA .

4. Now there is one upstream monopolist charging r. The downstream Cournot equilibrium gives us the quantities qi;j = a3br . Therefore, the upstream …rm maximizes A = r( 2a3b2r ), which (unsurprisingly) yields the royalty rate r = a 2 . This is clearly lower than in the original case which yields higher produced

quantities and, thus, (generally) a welfare gain.

Exercise 12 “Pay-for-delay” and generic drugs Under the “pay-for-delay” deal the patent holder of a drug pays a maker of generic drugs to delay its launch of a cheap copy.

1. Discuss the likely consequence for consumer welfare of such behavior.

2. Model the market for drugs as a homogeneous Cournot market with linear demand (it would be straightforward to include asymmetries between different pharma companies but symmetry makes the analysis particularly easy). Are there any contracts between the branded drugmaker and the generic drugmaker that lead to higher pro…ts of both …rms?

3. Some branded drugmakers have decided to o¤er lower-priced “authorized” generic version shortly before the patent expires. Note that the …rst (nonauthorized) generic producer entering the market is protected from further competition for a period of 6 months. In light of this regulation, discuss the conseqences of authorized generic version on entry and competition.

4. Some generic drugmakers now agree to delay their launch of their generic in return for a promise by the branded drugmaker not to launch an “authorized” generic version itself. Discuss the likely consequences of such agreements on consumer and total welfare.

20

Industrial Organization: Markets and Strategies Paul Belle‡amme and Martin Peitz published by Cambridge University Press

Part VIII. Networks, standards and systems Exercises & Solutions Exercise 1 Network e¤ ects and ful…lled expectations [included in 2nd edition of the book] Consider the market for a single network good and suppose that consumers di¤er in their valuation of both the stand-alone and the network bene…ts (it can indeed be argued that it is more plausible that a user who has a higher value for the stand-alone component of a technology also assigns more importance to the size of its network.) To capture this idea, write the consumer’s utility function for joining the network as U ( ) = (a + ne ), where a is the standalone bene…t, > 0 measures the network e¤ect, ne is the expected number of users joining the network, and is uniformly distributed on the unit interval.

1. Identify the indi¤erent consumer for a given price p and a given expected network size ne .

2. Express the willingness to pay for the nth unit of the good when ne units are expected to be sold; check that the ‘law of demand’ e¤ect con‡icts with the ‘network expansion’e¤ect.

3. Express the ful…lled-expectations demand curve and draw it. In particular, show that for a, the ful…lled-expectations demand is decreasing everywhere and there is a single equilibrium for all p a. On the other hand, for > a, show that the ful…lled-expectations demand has both an increasing and a decreasing portion; characterizes the range of prices for which two levels of demand satisfy the equilibrium condition.

Solutions to Exercise 1 1. The indi¤erent consumer is such that (a + ne ) p = 0, which is equivalent to = p= (a + ne ) ^. Note that there are buyers as long as ^ 1, which is equivalent to p a + ne .

2. As all consumers with a value of larger than ^ choose to buy, the mass of buyers is n = 1 ^. Replacing ^ by its value and solving for p, we …nd: p (ne ; n) = (a + ne )

(a + ne ) n:

(1)

We see that p (ne ; n) is a decreasing function of n (as a + ne > 0); hence, the law of demand applies; We also see that p (ne ; n) is an increasing function of ne (as @p=@ne = (1 n) 0), which demonstrates the network e¤ect.

1

3. To …nd the ful…lled-expectations demand curve, we set ne = n in (1): p (n; n) = (a + n) (1

n) :

We also note that any p a is consistent with ne = n = 0. Indeed, if ne = 0, then n = 1 ^ = 1 (p=a), and 1 (p=a) 0 for p a. We compute

dp (n; n) = dn

a

2 n:

Clearly, if < a, then the latter derivative is always negative. If > a, then the derivative is positive as long as n < ( a) = (2 ) and negative otherwise. The maximum price is thus reached for n = ( a) = (2 ), which is smaller than 2 unity. We compute p (n ; n ) = (a + ) = (4 ) and p (0; 0) = a. Hence, the range h of prices for which i two levels of demand satisfy the equilibrium condition 2

is a; (a + ) = (4 ) . Actually, there are even three levels of demand in that range as n = 0 is a ful…lled-expectations equilibrium for p

a.

Exercise 2 Network e¤ ects and coverage [included in 2nd edition of the book] Consider the situation of the exercise 20.1, where users are heterogeneous in terms of both network and stand-alone bene…ts. Demand is then given by p (n; n) = (a + n) (1 n).

1. Check that under perfect competition, there is a unique equilibrium network size for p = c < a, which is strictly lower than one unless c = 0.

2. Compute the monopolist’s pro…t-maximizing network size and show that it is strictly comprised between 0 and 1 for all a; > 0.

3. Compare the network size that is provided under perfect competition and monopoly. Which one is larger and why?

Solutions to Exercise 2 1. As the price is driven down to marginal cost under perfect competition, we have p (n; n) = (a + n) (1 n) = c. The latter equation de…nes a second-degree polynomial in n: n2 ( a) n (a c) = 0. The two roots are q 2 a n = 21 ( a) + 4 (a c) : As a c > 0, the small root is clearly negative and can thus be rejected. We therefore retain the large root, which can be rewritten as

nc =

1 2

a+

q

2

(a + )

4 c :

We observe that if c = 0, then nc = 1; otherwise, nc < 1. When the cost of providing the network good is positive, a perfectly competitive industry fails to internalize the positive network e¤ects among the consumers.

2

2. The monopolist’s program is maxn ((a + n) (1 condition for pro…t maximization is 3 n2 + 2 (

n) c) n. The …rst-order a) n + (a c) = 0. The

two roots of the second-degree polynomial are

n = 31

(

a)

q (

2

a) + 3 (a

c) :

The small root is negative and can thus be discarded (it corresponds to a minimum, as shown by the second-order condition). We retain the large root (which corresponds to a maximum); it can be rewritten as

n

m

= 31

q 2 a + (a + )

(a + 3c) :

We need to show that nm < 1; that is

q 2 (a + )

2

(a + 3c) < 2 + a () (a + )

2

(a + 2 ) <

(a + 3c) ;

which is clearly true as the LHS is negative while the RHS is positive (even for c = 0). Not only does the monopolist (like the perfectly competitive industry) fails to internalize all network e¤ects, but also the monopoly exerts market power by restricting demand.

3. We want to whow that nc > nm . This is so if q q 2 2 1 1 a + (a + ) 4 c > a + (a + ) 2 3 Note …rst that c < a implies that 4 c <

q 2 a + (a + )

4 c>

(a + 3c) :

(a + 3c). Hence, q 2 a + (a + ) (a + 3c):

The claim follows as 1= (2 ) > 1= (3 ). We have thus demonstrated that a perfectly competitive industry supplies a larger quantity of the network good than a monopoly. This is due to the market power that the monopoly exerts.

Exercise 3 Network e¤ ects in the Hotelling model1 [included in 2nd edition of the book] Consider the Hotelling model with linear transport costs where two …rms, 1 and 2, are located at the extreme points of the unit interval. There is a unit mass of consumers who are uniformly distributed over this interval. Suppose that the 1 This exercise draws from Grilo, I, Shy, O. and Thisse, J.-F. (2001). Price Competition when Consumer Behavior is Characterized by Conformity or Vanity, Journal of Public Economics 80, 385-408.

3

products o¤ered by the two …rms exhibit network e¤ects. This is modelled as follows: a consumer located at x 2 [0; 1] has utility r r

x p1 + ne1 (1 x) p2 + ne2

if she buys one unit of product 1, if she buys one unit of product 2,

(2)

where nei is the expected mass of consumers buying product i (i = 1; 2). We assume that r is large enough so that each consumer buys one or the other product; it follows that ne1 = 1 ne2 .

We look for the subgame perfect Nash equilibrium in pure strategies of the following two-stage game: in the …rst stage, …rms select their price pi ; in the second stage, given any pair of prices (p1 ; p2 ), consumers allocate themselves between the two products (an equilibrium at this stage is a partition of consumers between the two …rms, (n1 ; n2 ), such that no consumer with utility (1) is strictly better o¤ by switching products; this is equivalent to say that expectations must be ful…lled at equilibrium, i.e., nei = ni ). To simplify the computations, we assume that both …rms produce at zero marginal cost.

1. Characterize the equilibrium consumer partitions at the second stage of the game. In particular, establish and discuss the following two results: (a) if > , then a unique equilibrium consumer partition obtains for any pair of prices; (b) if < , then there exist pairs of prices for which mutliple equilibrium consumer partitions coexist.

2. For the case where > , solve for the equilibrium prices at the …rst stage of the game. Discuss the impact of the network e¤ects on the equilibrium prices and pro…ts.

Solutions to Exercise 3 1. The consumer indi¤erent between the two products is identi…ed as x ^ such that r

x ^

p1 + ne1 = r

(1

x ^)

p2 + ne2 :

As the market is covered, all consumers located at the left (resp. right) of x ^ buy from …rm 1 (resp. 2). Moreover, as expectations are ful…lled at equilibrium, we have that ne1 = x ^ and ne2 = 1 x ^. Plugging these values into the above equality and solving for x ^, we …nd

x ^=

p2 1 + 2 2(

p1 : )

Three equilibrium consumer partitions are possible: (CS) consumers split between the two …rms; (C1) all consumers buy product 1; (C2) all consumers buy product 2. For CS to be an equilibrium, we must have 0 < x ^ < 1. For C1 to be an equilibrium, the consumer located at 1 must be better o¤ with product 1

4

than with product 2 when n1 = 1. For C2 to be an equilibrium, the consumer located at 0 must be better o¤ with product 2 than with product 1 when n2 = 1.

We now develop these conditions for the two cases.

(a) Consider the case where

> . Here the di¤erentiation of the product (measured by ) is stronger than the network e¤ects (measured by ).

(CS)

x ^ > 0 , p2 x ^ < 1 , p2

p1 > p1 <

(

)

(3)

(C1)

r

p1 +

>r

p2 , p 2

p1 >

:

(4)

(C2)

r

p2 +

>r

p1 , p2

p1 <

(

):

(5)

We observe that conditions (2), (3) and (4) are exclusive, which proves that there is a unique equilibrium consumer partition for any pair of prices: C2 is the unique equilibrium partition for p2 p1 < ( ); CS is the unique equilibrium partition for ( ) < p 2 p1 < ; and C1 is the unique equilibrium partition for p2 p1 > .

(b) Consider the case where

< . Nothing changes for the conditions regarding C1 and C2 but the conditions for CS to be an equilibrium now become (CS)

x ^ > 0 , p2 x ^ < 1 , p2

p1 < p1 >

(

)

(6)

We observe that conditions (3), (4) and (5) are no longer exclusive, meaning that there can be multiple equilibrium consumer partitions for some pair of prices. We have the following: C2 is the unique equilibrium partition for p2 p1 < ( ); CS, C1 and C2 are equilibrium partitions for ( ) < p2 p1 < ; and C1 is the unique equilibrium partition for p2 p1 > . In the present case, the ‘concentration force’resulting from the network e¤ects outweighs the ‘dispersion force’ resulting from product di¤erentiation; as a consequence, any price pair induces at least one equilibrium partition such that all consumers adopt the same product and multiple equilibria may coexist when prices are su¢ ciently close.

2. Consider the case where

> and suppose that prices are such that conditions (2) are satis…ed so that the equilibrium consumer partition is CS. The pro…t function of …rm i can be written as i = pi

pj 1 + 2 2(

pi )

5

, with i 6= j 2 f1; 2g .

From the F.O.C., e …nd …rm i’s best-response function:

pi =

1 ( 2

+ pj ) :

Exploiting the symmetry of the model, we can set pi = pj p in the above equation and derive the common equilibrium price of the two …rms:

p =

> 0:

We check that these prices satisfy condition (2), which con…rms our initial assumption that CS is the prevailing second-stage equilibrium consumer partition.

As prices are equal, the indi¤erent consumer at equilibrium is located at 21 , which implies that equilibrium pro…ts are equal to

= 12 (

):

We observe that equilibrium prices and pro…ts decrease with the intensity of network e¤ects ( ). The intuition is simple: as network e¤ects increase, it becomes more pro…table for each …rm to attract additional consumers as each consumer increases the willingness to pay of other consumers; as a result, …rms compete more …ercely and equilibrium prices decrease.

Exercise 4 Adoption of network technologies in oligopolies2 [included in 2nd edition of the book] Consider the following two-stage game. In the …rst stage, a set of n …rms choose simultaneously whether to adopt or not a network technology. The technology has the e¤ect to reduce the …rms’marginal cost of production; moreover, because of network e¤ects, the cost reduction grows larger as more …rms adopt the technology. Then, in the second stage, …rms produce a homogeneous good and compete on the market in a Cournot fashion. Note that each …rm’s marginal cost at the second stage of the game depends on the size of the network, which means that it depends not only on this …rm’s …rst-stage adoption decision but also on its rivals’decisions.

Speci…cally, supposeP that inverse demand on the …nal market is given by n p = 1 Q, where Q = i=1 qi is the total quantity produced by the n …rms.

The technology is described in the following way. Firm i’s marginal cost of production, ci , is equal to c if i does not adopt the technology, or to c k if i adopts the technology and the network size is k, with 0 < c < 1, 1 k n, and 0 < < c=n.

1. Derive the Cournot equilibrium at the second stage of the game. (a) Express the …rms’equilibrium pro…t when they have all adopted the technology at stage one (k = n). Denote this pro…t by in (n).

2 This exercise draws from Belle‡amme, P. (1998). Adoption of Network Technologies in Oligopolies. International Journal of Industrial Organization 16: 415-444.

6

(b) Express the …rms’equilibrium pro…t when none of them has adopted the technology at stage one (k = 0). Denote this pro…t by out (0). (c) Suppose 0 < k < n. Express the equilibrium pro…ts (i) for the …rms that adopted the technology at stage one, and (ii) for those that did not adopt the technology at stage one. Denote these pro…ts respectively by in (k) and out (k).

2. Suppose that k < n. Show that a technology adopter’s equilibrium pro…t increases when an additional …rm joins the network if and only if the network comprises less than half of the population of …rms (i.e., k < n=2).

Explain the intuition behind this result.

3. Derive the …rst-stage equilibrium. How many …rms adopt the network technology at the subgame-perfect equilibrium of this two-stage game?

4. Construct a numerical example (where you give values to the parameters n, c, and ) showing that the total pro…t (i.e., the sum of the pro…ts of the n …rms) is not maximum at the subgame-perfect equilibrium of the game. Discuss.

Solutions to Exercise 4 1. The standard analysis of a Cournot model with linear Pn demand and costs yields the second-stage equilibrium (where C stands for j=1 cj ): qi =

1 (1 n+1

2 i = (qi ) :

(n + 1) ci + C) and

(7)

We use this generic result to derive the second-stage pro…ts.

(a) If all …rms have adopted the technology, then ci = c n (c n). It follows that in (n) =

1

n (8i) and C =

2

c+ n n+1

:

(b) If no …rm has adopted the technology, then ci = c (8i) and C = nc. It follows that

out (0) =

1 c n+1

2

:

(c) Suppose that 0 < k < n. If …rm i has adopted the technology, its marginal cost is equal to ci = c k , while the sum of all …rms’marginal costs is easily computed as C = k (c k) + (n k) c = nc k 2 . Substituting into expression (7), we compute …rm i’s equilibrium pro…t as: in (k) =

2

1 n+1

(1

7

c + k (n

2

k + 1)) :

On the other hand, if …rm i did not adopt the technology, then its marginal cost is ci = c and its pro…t is then equal to 2

1 n+1

out (k) =

1

c

2

k2

:

2. We want to assess how a technology adopter’s equilibrium pro…t changes when an additional …rm joins the network. Supposing that k < n, simple computations establish that the sign of in (k + 1) in (k) is determined by the sign of (k + 1) (n k) k (n k + 1) = (n 2k). Hence, network e¤ects are positive (i.e., in (k + 1) in (k) > 0) if and only if the network comprises less than half of the population of …rms (i.e., k < n=2). As the network grows larger than this critical size, network e¤ects become negative (the equilibrium pro…t of each technology adopter decreases when an additional …rm joins the network).

The intuition for this result is simple. In such a setting, the enlargement of the network induces two simultaneous contrasting e¤ects on its members’ pro…ts: on the one hand, the members bene…t from the reduction in their own marginal cost, but on the other hand, they su¤er from increased competition due to the reduction in their competitors’marginal costs. As a result, there is a critical size after which the admission of new members in a network has an adverse e¤ect on the initial members’pro…t (implying that network e¤ects become negative once the critical network size is reached).

3. What is a …rm’s best response when it assumes that k …rms (with 0

k < n) adopt the technology? The best response is to adopt the technology as well if and only if in (k + 1) out (k) > 0. It is easily seen that the sign of the di¤erence (k) is determined by the sign of (k + 1) (n k) + k 2 , (k + 1) out in which is positive for all k . Therefore, we conclude that it is a dominant strategy for each …rm to adopt the technology, meaning that the network comprises all …rms at the subgame-perfect equilibrium of the game.

4. Total pro…t at the subgame-perfect equilibrium of the game is 1

tot (n) = n in (n) = n

If only n

c+ n n+1

2

:

1 …rms adopt the network technology, total pro…t is equal to tot (n

1) = (n

1)

in (n

1) +

out (n

1) :

We now construct an example where tot (n 1) > tot (n). Take n = 9 and c = 1=2. We impose thus that < 1=18 ' 0:056. We have tot (9)

= 9

tot (8)

=

1 2 +9

10

2

=

9 2 (1 + 18 ) 400 2

=

8 1 1 1 + 16 + 100 2 100 2 1 256 + 24 576 2 + 9 : 400 8

2

64

2

256 + 24 576 2 + 9 > 9 (1 + 18 ) , 4 (5415 17) > 0 , > 17=5415 ' 0:0031. For instance, take = 0:01.

Then tot (9) = 0:03133 < tot (8) = 0:03504.

It follows that

tot (8) >

tot (9) ,

Exercise 5 Standardization and variety3 [included in 2nd edition of the book] Suppose that two incompatible network technologies, 1 and 2, are competitively supplied at zero marginal cost (as a result, we can ignore pricing decisions). A unit mass of consumers is split into two homogeneous groups. The …rst group is of mass n1 , with 0 < n1 < 1, and prefers technology 1; the other group is of mass n2 = 1 n1 and prefers technology 2. The utility a consumer gets from adopting a network technology depends on two additively separable components: a network bene…t and a stand-alone bene…t. The network bene…t has the same form for the two technologies: x, where > 0 measures the strength of the network e¤ect and x is the mass of consumers who adopt the same technology. The stand-alone bene…t is equal to ai > 0 when a consumer of type i (i = 1; 2) adopts her preferred technology and to zero otherwise.

Consumers simultaneously choose which technology to adopt. We are interested in characterizing the equilibria of this game and in assessing their e¢ ciency. As consumers are identical within each group, three outcome can emerge at equilibrium.

Incompatibility: consumers of group i adopt technology i, thereby getting a utility of ai + ni ; social surplus in this case is equal to n1 (a1 + n1 ) + n2 (a2 + n2 ).

Standardization on technology 1 : all consumers adopt technology 1, thereby getting network bene…ts of (n1 + n2 ) = ; consumers of group 1 (resp. group 2) get a stand-alone bene…t of a1 (resp. 0); social surplus in this case is equal to + n1 a1 .

Standardization on technology 2 : all consumers adopt technology 2, thereby getting network bene…ts of (n1 + n2 ) = ; consumers of group 1 (resp. group 2) get a stand-alone bene…t of 0 (resp. a2 ); social surplus in this case is equal to + n2 a2 .

We make the following assumptions: (A1) (A2)

n1 > 21 ; n1 a1 > n2 a2 :

According to Assumption (A1), group 1 is larger than group 2. Assumption (A2) implies that standardization is socially more desirable on technology 1 than on technology 2.

3 This exercise draws from Farrell, J. and Saloner, G. (1986). Standardization and Variety.

Economics Letters 20: 71-74.

9

We say that an outcome is e¢ cient if it maximizes social surplus. We say that it is an equilibrium if no consumer would wish to deviate unilaterally to a di¤erent technology from the one she is meant to be getting. Note that each consumer is in…nitesimal in the market as a whole; as a result, the deviation by an individual consumer has no measurable impact on the size of the two networks.

1. Express the condition under which incompatibility is e¢ cient (express the condition by isolating and by using n2 = 1 n1 ).

2. Express the conditions for each outcome to be an equilibrium. Show that there can be multiple equilibria for some parameter constellations.

3. Show that if incompatibility is e¢ cient, then it is an equilibrium.

4. Show that if standardization is the unique equilibrium, then it is also e¢ cient. Discuss the intuition behind this result.

5. A consequence of the previous results is that when there are multiple equilibria, it is possible that one of them involves too much standardization.

Construct an example with an ine¢ cient standardization equilibrium. Discuss.

Solutions to Exercise 5 1. Incompatibility is e¢ cient if and only if n1 (a1 + n1 ) + n2 (a2 + n2 ) > Using n2 = 1

+ n1 a1

n1 , this is equivalent to (1

n1 ) (a2

2 n1 ) > 0 ,

<

a2 : 2n1

2. Equilibrium conditions (a) Incompatibility no deviation from group 1: a1 + n1 no deviation from group 2: a2 + n2

n2 ; n1 :

As n1 > 21 > n2 , the …rst condition is always met. The second condition can be rewritten as

a2

2n1

1

:

(b) Standardization on technology 1 no deviation from group 1: a1 + 0; no deviation from group 2: a2 : As the …rst condition is clearly met, standardization on technology 1 is an equilibrium i¤

a2 : 10

(c) Standardization on technology 2 no deviation from group 1: a1 ; no deviation from group 2: a2 + 0: Here, it is the second condition that is clearly met. So, standardization on technology 2 is an equilibrium i¤

a1 : (d) Clearly, for a2

a2 = (2n1 1), incompatibility and standardization on technology 1 are equilibria (and so is standardization on 2 if, for instance, a2 a1 ).

a2 and it is an equilibrium i¤ < 2n 1 a2 The proof of the claim follows from the fact that 2n < 2na12 1 .

1

3. Incompatibility is e¢ cient i¤

a2 2n1 1 .

4. This claim is a corollary of the previous one. Indeed if standardization is the unique equilibrium, then incompatibility is not an equilibrium. But, reversing the previous claim, we know that if incompatibility is not an equilibrium, then it is not e¢ cient, which means that standardization is. The intuition is simple.

Paraphrasing Farrell and Saloner (1986), if all consumers of group 2 (if they could coordinate their actions) would prefer to join the consumers of group 1 (the condition that incompatibility is not an equilibrium implies this), then group 1 would bene…t: the externality is positive. “Thus a defection by a whole group from the incompatible outcome is desirable if it will occur. And if one buyer would like to defect alone, then certainly he would like to if he could bring along all the buyers of his type, who of course share his preferences. It follows that if the incompatibile outcome is not an equilibrium, then it is not the optimum.” a2 < 2n , standardization on technology 2 is an equilibrium and is a2 1 ine¢ cient. As 2n1 < 2na12 1 , we also have that incompatibility is an equilibrium.

For instance, take n1 = 34 , a1 = 1, a2 = 2 and v = 1:2. We …rst check that n1 a1 = 34 > n2 a2 = 12 . We have two equilibria: incompatibility and standardization on technology 2: (i) incompatibility is an equilibrium as = a2 1:2 2n1 1 = 4; (ii) standardization on technology 2 is an equilibrium as = 1:2 a1 = 1; (iii) standardization on technology 1 is not an equilibrium as = 1:2 < a2 = 2. The levels of social surplus under incompatibility and

5. For a1

standardization on technology 2 are respectively equal to

n1 (a1 + n1 ) + n2 (a2 + n2 ) ' 2 + n2 a2 = 1:7; which proves our result.

11

Exercise 6 Infomediation in the e-tourism sector4 [included in 2nd edition of the book] Suppose there are two suppliers of B&B accommodation, noted 1 and 2.

The inverse demand function for the service provided by supplier i is given by pi = 1 + mi qi dqj (i 6= j 2 f1; 2g) where pi is the price for one unit of service (say, e.g., one night per person in the accommodation), qi and qj are the quantities of service provided respectively by suppliers i and j (these quantities can be thought of as the accommodation capacity, e.g., the number of rooms or of beds), d 2 (0; 1) is an inverse measure of product di¤erentiation between the services of the two suppliers, and mi indicates the “market exposure”(or brand recognition) of service i (the larger mi , the higher the willingness to pay of each consumer whatever the quantities qi and qj produced by the …rms).

Market exposure can be obtained either through self-promotion or by using the services of an intermediary. We analyze the following three-stage game: …rst, the intermediary sets a registration fee F ; second, the two B&B owners simultaneously decide whether or not to register with the intermediary; third, the two B&B owners compete à la Cournot on the product market.

Being listed on the intermediary’s website has the e¤ect of increasing the supplier’s market exposure (with respect to self-promotion via other means).

Moreover, infomediation generates network e¤ects insofar as each supplier’s exposure further increases when the other supplier also registers with the intermediary. Such network e¤ects can be justi…ed by scale and scope economies in promotional activities enjoyed by the intermediary, and because consumers are willing to pay more for the two accommodations when they are given the opportunity to compare them more easily. We translate this idea by assuming that mi = 0 when supplier i does not register with the intermediary, mi = m when supplier i is the only supplier who registers with the intermediary, and mi = M when both suppliers register with the intermediary, with 0 < m < M .

For simplicity, we assume that accommodation is supplied at zero marginal cost.

1. Solve for the Cournot equilibrium at the third stage of the game and express the equilibrium pro…ts of the two suppliers for any couple (mi ; mj ) resulting from the second-stage decisions.

2. Consider now the second-stage of the game and assume that 2 d dm > 0.

4 This exercise draws from Belle‡amme, P. and Neysen, N. (2010).

Coopetition in infomediation: General analysis and application to e-tourism. In A. Matias, P. Nijkamp and M. Sarmento (eds.) Advances in Modern Tourism, vol. II. Chapter 14. Berlin: Springer.

12

(a) Use your answer to question 1 to …ll in the following matrix (where R stands for ‘register’and N for ‘not register’).

Register RR ; RR NR ; RN

Register Not register

Not register RN ; NR NN ; NN

(b) Show that although the infomediation services exhibit network effects, a supplier who registers with the infomediary may be worse o¤ when the other supplier registers too. Explain the intuition behind this result. (c) Set d = 1, M = 1 and suppose 3=5 < m < 1. Under these assumptions, characterize the equilibrium registration decisions.

3. Using your previous answers, determine the intermediary’s optimal registration fee at the …rst stage of the game.

Solutions to Exercise 6 1. Supplier i’s pro…t function can be written as

i = (1+mi qi dqj )qi . Setting to zero the derivative of pro…t with respect to qi , we derive supplier i’s reaction function: qi (qj ) = 21 (1 + mi dqj ). We proceed in the same way for the other supplier. Solving for the system of equations in two unknowns given by the two reaction functions, we …nd the Nash equilibrium quantities and pro…ts:

qi =

2

d + 2mi dmj ; 4 d2

2 i = (qi ) ;

i 6= j 2 f1; 2g :

(8)

2. In the above expression, the exact values of (mi ; mj ) depend on the registration decisions made by the suppliers at stage 2.

(a) There are three situations to consider. (1) If no supplier registers, then mi = mj = 0. Substituting these values into expression (8), we obtain the equilibrium pro…ts, which are the same for the two suppliers: i =

j =

NN

=

2

1 2+d

:

(2) If supplier i registers while supplier j does not, then mi = m and mj = 0. In that case, the suppliers’ equilibrium pro…ts di¤er and are respectively given by i =

RN

=

2

d + 2m 4 d2

2

F and

j =

NR

=

2

d 4

dm d2

2

:

(3) Finally, if both suppliers register, then mi = mj = M . We use again expression (8) to …nd the equilibrium pro…ts for the two suppliers as i =

j =

RR

13

=

1+M 2+d

2

F:

(b) A supplier does not welcome the registration of the other supplier if RR < RN , which is equivalent to 2 (M m) < dM . The LHS measures the bene…ts resulting from the other supplier’s registration: the increased exposure, (M m), a¤ects the Cournot equilibrium quantity by a factor 2.

The RHS measures the costs resulting from the other supplier’s registration: by registering, the rival supplier boosts his exposure (from 0 to M ) and thereby improves his competitive position; the impact of this e¤ect on the Cournot equilibrium quantity is proportional to d. We see that the costs are more likely to outweigh the bene…ts (i) the smaller the network e¤ects (measured by the increased exposure, M m) and (ii) the larger the intensity of competition between the two suppliers (measured by the degree of product substitutability, d).

(c) Supposing that supplier j does not register, supplier i prefers to register provided that RN

NN

,F

1+2m 2 3

1 2 = 49 m (1 + m) 3

F1 :

Supposing now that supplier j does register, supplier i prefers to register as well provided that RR

NR

,F

2 2 3

1 m 2 = 19 (3 3

m) (1 + m)

F2 :

We compute

F1

F2 = 91 (1 + m) (5m

3) > 0.

It follows that a supplier is willing to pay more for the intermediary’s services when it is the only one to use them. The Nash equilibrium of the second stage is then characterized as follows: for F F2 , 2 registrations; for F2 < F F1 , 1 registration; for F > F1 , no registration.Under our assumptions, any fee set by the intermediary generates a unique equilibrium.

3. The intermediary knows that to attract the two suppliers, it has to lower the registration fee from F1 to F2 (as each supplier is willing to pay less when the other supplier also registers). The intermediary …nds it pro…table to do so as long as 2F2 > F1 , which is equivalent to (2=9) (3 m) (1 + m) > (4=9) m (1 + m) , m < 1, which holds under our assumption. Hence, the intermediary’s optimal conduct is to set F = F2 and have the two suppliers register.

14

Industrial Organization: Markets and Strategies Paul Belle‡amme and Martin Peitz published by Cambridge University Press

Part IX. Market intermediation Exercises & Solutions Exercise 1 Dealer vs pure platform operator [included in 2nd edition of the book] Repeat the analysis of Section 21.1.2 under the following assumptions. There is a unit mass of sellers and a unit mass of buyers. Each seller produces a totally di¤erentiated good at a constant unit cost c, which is assumed to be uniformly distributed over [0; ]. Buyers have unit demand for each good; they buy if they are o¤ered a price below or equal to their reservation price v, which is also assumed to be uniformly distributed over [0; ].

The di¤erence with Section 21.1.2 is that we allow now for 6= 1 and/or 6= 1. Show that in spite of this di¤erence, the main result still holds, namely that the intermediary is indi¤erent between the roles of dealer and platform operator.

Solutions to Exercise 1 Suppose …rst that the intermediary acts as a dealer who makes take-it-or-leave-it o¤ers to both sides of the market. That is, the intermediary buys the goods from the sellers and sells them to the buyers. It seeks to maximize its pro…t by setting a retail price or “ask price” p for the buyers and a wholesale price or “bid price” (p P ) for the sellers. Here, the transaction fee P corresponds to the bid-ask spread. According to our previous notation, w = p P . We are considering the situation in which, at stage 1, the intermediary sets prices p and P and, at stage 2, buyers and sellers simultaneously decide which product to buy or to sell. Given these prices, the indi¤erent buyer is identi…ed by v = p and the indi¤erent seller by c = p P . Under dealer intermediation, all active sellers obtain the same retail price p. The dealer cannot distinguish between the di¤erent types of buyers and sellers because buyer and seller type are assumed to be private information. This leads, as we will see, to socially insu¢ cient trade since the intermediary exerts its monopoly power on both sides of the market.

Given prices p and P , we obtain the number of participating buyers and sellers; this is the non-strategic decision buyers and sellers take at stage 2. For the uniform distribution, they are given by nb = ( p) = and ns = (p P ) = . Therefore, the total quantity exchanged is equal to ns nb = ( p) (p P ) = ; the intermediary chooses p and P to maximize = P ns nb . The …rst-order conditions are d =dp = 0 , P (P 2p + ) = 0 and d =dP = 0 , ( p) (p 2P ) = 0. As P = 0 and/or p = yield zero pro…t, the pro…t-maximizing prices are easily found as (where the superscript D stands for ‘dealer’): pD = (2=3) , and P D = (1=3) .

Thus buyers pay the retail price pD = (2=3) and sellers receive the wholesale price D pD P D = (1=3) . It follows that nD b = 1=3, ns = = (3 ) (meaning that the

1

volume of trade is equal to = (9 )) and that the intermediary achieves a pro…t of D = 2 = (27 ).

Suppose now that the intermediary stops buying from sellers in order to resell to buyers, but lets buyers and sellers interact on its platform (or marketplace). Instead of …xing bid and ask prices, the intermediary then sets a transaction fee, P , on each unit exchanged on the platform. We assume, without any loss of generality, that the transaction fee is entirely borne by the sellers.1 The timing of the game is as follows: …rst the intermediary sets P ; next agents on one side of the market choose the retail price p for the goods and …nally, agents on the other side of the market decide to participate or not. There are thus two cases to consider according to whether the sellers or the buyers have the price-setting power.

1. Sellers set the price. At the last stage, buyers decide to purchase if their reservation price is larger than the price p set by the sellers. Therefore, each seller faces a demand q (p) = ( p) = for its product. Moving to the second stage, we have that a seller with unit cost c chooses its price p to maximize c P)( p) = . The pro…t-maximizing price is easily found as s = (p p (c) = (1=2) ( + P + c). Only the sellers who can make pro…t will participate to the market, i.e. the sellers such that p (c) c + P or, equivalently, c P = c. We can now compute the total quantity that will be exchanged on the market as:

Q=

Z c

1

p (c)) 1 dc =

(

Z

P 1 2

0

0

(

c

P ) dc = 4 1 (

2

P) ;

which is the same as in the dealer market (i.e., = (9 )) with P D = =3. Hence pro…t and trading volume must be the same in the two settings. To complete the analysis, we note that, at the …rst stage of the game, the intermediary’s 2 chooses P to maximize = P( P ) = (4 ). The optimal transaction fee and pro…t are (note that the superscript O stands for ‘platform operator’): P O = (1=3) and O = 2 = (27 ).

2. Buyers set the price. Only sellers such that p > c+P accept to participate. That is, each buyer faces a supply equal to p P and hence, chooses p to maximize ub = (v p) (p P ) under the constraint that p v . The uncontrained price is equal to p (v) = (v + P ) =2, which satis…es the constraint as long as v P. We can now compute the total quantity that will be exchanged on the market as

Q=

Z

P

1

(v

p (v)) 1 dv =

Z

P

1 2

(v

P ) dv = 4 1 (

2

P) ;

which is the same volume of trade as in the case where sellers have market power.

The intermediary’s problem is thus unchanged, and so are the intermediary’s optimal transaction fee and pro…t.

1 It is easy to show that the results are invariant to the speci…c split of the fee between buyers and sellers.

2

Comparing the previous results, we observe that, in the special case of the uniform distribution, the intermediary is indi¤erent between the two forms of intermediation as they both yield the same pro…t: D = O = 2 = (27 ). The optimal bid-ask spread is equal to the optimal transaction fee: P D = P O = (1=3) . Also the volume D O of trade in the two cases is the same, nD s nb = Q = = (9 ). In other words, the intermediary achieves the same pro…t whether he buys the sellers’ goods and resells them at a higher price to the buyers, or when he lets sellers and buyers freely interact on his platform but taxes the transactions.

Exercise 2 Price-setting by a monopoly intermediary on a two-sided platform Suppose that an intermediary faces a certain number of buyers and sellers. The intermediary sets usage prices Ps and Pb to be paid, respectively, by sellers and buyers whenever there is an interaction between a particular seller and buyer. Interaction can only take place on the platform owned by the intermediary. (The exact nature of this interaction is not speci…ed. It is simply postulated that a buyer’s gross surplus can be expressed in a reduced form that depends only on the number of sellers, and that a seller’s surplus can be expressed in a reduced form that depends only on the number of buyers; that is, positive indirect network e¤ects are present). Let nb and ns denote the number of buyers and sellers who decide to interact on the platform. Suppose that the intermediary does not incur any cost. Hence, the intermediary chooses Pb and Ps to maximize total revenues, R = nb ns (Pb + Ps ), where nb ns is the total number of transactions conducted on the platform and (Pb + Ps ) is the sum of usage fees paid per transaction.

1. Suppose …rst that there are 3 buyers and 3 sellers (so nb ; ns 2 f0; 1; 2; 3g) and that the net surplus of buyers and sellers are as follows: all buyers enjoy a net surplus of ub = (2 Pb ) ns ; seller i (i = 1; 2; 3) enjoys a net surplus of uis = (i Ps ) nb . (a) Find the price Pb that maximizes revenues on the buyers’side. (b) Given your answer at (a) and the corresponding buyers’participation, …nd the price Ps that maximizes revenues on the sellers’side. (c) Show that the intermediary can increase its revenues by setting a lower price than the one you found at (b) and …nd the prices Pb and Ps that maximize total revenues.

2. Suppose now that there are 6 buyers and 6 sellers, with the net surplus of buyers and sellers being given by: all buyers enjoy a net surplus of ub = (6 Pb ) ns ; seller i (i = 1; : : : ; 6) enjoys a net surplus of uis = (i 3 Ps ) nb . (a) Repeat steps (a) to (c) of the previous question and show that the intermediary …nds it optimal to subsidize sellers’participation in this case.

3

(b) Find the welfare-maximizing prices. (c) Using your answers, explain the intuition behind the next three statements: (i) The pro…t-maximizing price structure of an intermediary re‡ects price elasticities and the sizes of the indirect network e¤ects on the two sides of the market. (ii) A pro…t-maximizing intermediary may subsidize one side of the market so as to generate a higher volume of trade and thus, higher pro…ts on the other side of the market. (iii) While a pro…t-maximizing intermediary may decide to subsidize one side of the market, the subsidy is too low from a social point of view.

Solutions to Exercise 2 1. 3 buyers, 3 sellers (a) The intermediary sets the price on the buyer side equal to 2. Here, since buyers are homogeneous the intermediary can extract all surplus from buyers and still assure participation by all buyers.

(b) Given nb = 3, revenues on the sellers’side are Rs = 3ns (Ps ) Ps , with 8 0 if Ps > 3 > > < 1 if 2 < Ps 3 ns (Ps ) = 2 if 1 < Ps 2 > > : 3 if Ps 1.

Hence to maximize revenues on the sellers’ side, the intermediary should set Ps = 2, meaning that 2 sellers participate. There are thus 6 interactions and total revenues are equal to 6 (2 + 2) = 24.

(c) If the seller reduces the price sellers have to pay to Ps = 1, all sellers will join. Then there are 9 buyer-seller interactions. On the one hand, the intermediary’s revenues on the seller side decrease from 12 to 9 because of the lower price. On the other hand, because of the increased number of interactions, revenues on the buyer side increase from 12 to 18. Overall revenues increase from 24 to 27. In this example, Pb = 2 and Ps = 1 is indeed the pro…t-maximizing price structure. Here, it is worthwhile to encourage further participation on the seller side — this leads to a situation in which seller have to pay a lower access price than buyers.

2. 6 buyers, 6 sellers (a) On the buyer side the pro…t-maximizing price is obviously Pb = 6. Given that all buyers participate, the price that maximizes pro…ts on the seller side is Ps = 2. In this case, there are 12 buyer-seller pairs overall pro…ts are 6 12 + 2 12 = 96. However, this price does not maximize overall pro…ts. A lower price leads to more participation of sellers and thus to more transactions, which increases revenues on the buyer side. If the lowest

4

price the intermediary can set is 0, he would set this price on the seller side. There are then 24 buyer-seller pairs and all revenues are made on the buyer side. The pro…ts are 24 6 = 144. If the intermediary can subsidize one market side, he would actually do so and set Ps = 1. Then there are 30 buyer-seller pairs and pro…ts are 30 6 + 30 ( 1) = 150. Here the intermediary subsidizes the seller side to generate a high volume of transactions that generates pro…ts on the buyers side.

(b) The welfare maximizing solution is implemented through prices Pb 6 and Ps 2. Thus the socially optimal subsidization is even stronger than what is privately optimal. Hence, pricing below marginal costs may be socially desirable.

Exercise 3 Number of active two-sided platforms Suppose there are two buyers (b and ~b), two sellers (s and s~) and two intermediaries (1 and 2) operating separate platforms. Buyers and sellers can only be on one or the other platform (they cannot be on both at the same time, i.e., they singlehome). Hence, transactions can only take place between sellers and buyers who are present on the same platform.

Sellers do not incur any opportunity costs. The buyers’willingnesses to pay for the two products are indicated in Table 1. In each cell, the …rst entry tells how many monetary units a particular buyer is willing to pay for a particular product. The number in parentheses is the net gain from trade. It is assumed that any gains from trade are split evenly between buyer and seller. So, for instance, buyer b is willing to pay 4 for the product of seller s and 2 for the product of seller s~. If buyer b transacts with seller s, the net gain (gross of any charges set by the intermediary) is thus equal to 2; it is equal to 1 when buyer b transacts with seller s~. In this example each buyer has his preferred product, but the consumption of the other product also increases utility. Thus there are positive indirect network e¤ects as a buyer’s utility increases with the number of sellers who are active on her platform (and vice versa).

Buyer b Buyer ~b

Product of s 4 (2) 2 (1)

Product of s~ 2 (1) 4 (2)

Table 1: Willingnesses to pay and net gains from trade Let us consider the following two-stage interaction. First, the two intermediaries simultaneously set a uniform price that applies to all transactions and both market sides of the platform. This implies that we impose a particular price structure with the property that the intermediary cannot or does not bother to check the identity of the parties involved in a transaction. Second, buyers and sellers observe the prices Pb1 = Ps1 P 1 and Pb1 = Ps1 P 2 that are set at the platforms and decide to which platform to go.

5

1. Suppose that the two platforms are equally attractive for all buyers and sellers. (a) Show that there does not exist an equilibrium such that both platform have a positive volume of transactions (i.e., each platform attracts one buyer-seller pair) and that has the property to be robust against pairwise deviations (of one buyer and one seller). (b) Characterize the equilibrium where all buyers and sellers go to the same platform. Show that in this equilibrium, the active intermediary can make positive pro…ts (equal to 4) provided that all buyers and sellers cannot perfectly coordinate their actions.

2. Suppose now that platforms are di¤erentiated: buyer b and seller s prefer platform 1 while buyer ~b and seller s~ prefer platform 2; moreover, each market participant incurs a utility loss (or transportation cost) of t when going to their less preferred platform. (a) Consider …rst the case that the di¤ erentiation is not too strong; in particular, assume that t < 1. Show that it is still true that only one platform remains active in equilibrium. (b) Consider next the case where platforms are more di¤ erentiated (t > 1). Show that, in this case, both platforms are active at equilibrium.

(Remark. As often in models with discrete types–here on the buyer and seller side–, we cannot analyze Nash equilibria but strategy combinations that are stable in the following sense: …rst, no intermediary can lower its price such that it increases pro…t and second, intermediaries choose among strategies that are robust to such attempts).

Solutions to Exercise 3 1. Undi¤erentiated platforms (a) Note …rst that at any prices P 1 = P 2 < 1, it is optimal for each buyerseller pair to deviate to the other platform. Network e¤ects are of course the underlying reason: they lead to a utility gain of 2 in total for each transaction. We can also exclude prices P 1 = P 2 1 as a possible equilibrium with more than one active platform. Here, a unilateral deviation by one buyer or one seller can even be pro…table provided intermediaries have an incentive to undercut so as to attract all trade. Suppose that both platforms are active and set prices P 1 = P 2 = 2. They thus make a pro…t of 4 and buyers and sellers have a net gain of 0 after paying for the transaction to the platform. Suppose intermediary 1 contemplates to set a price slightly below 1. Then, given the behavior of sellers and the other buyer, the buyer who was supposed to buy via platform 2 has a strict incentive to go to platform 1. There he will make a less attractive transaction but the lower transaction fee overcompensates the lower utility. This reinforces

6

the incentive of the corresponding seller to migrate to platform 1. Thus, it is a dominant strategy for each buyer and seller to go to platform 1.

The pro…t of intermediary 1 would thus be close to 8, which is certainly an improvement.

(b) Suppose for instance that all buyers and sellers go to platform 1 and that they have to pay a price P 1 = 1=2 per transaction. Each buyer and seller then receives a net utility of 2 (i.e., (4 + 2) =2 2 (1=2)). Suppose that intermediary 2 tries to attract tra¢ c and sets the price P 2 = 0. Clearly, this is the lowest price it can set without making losses for any transaction that it processes. At this price, no single buyer-seller pair has an incentive to deviate and migrate to platform 2. The buyer and the seller doing so would obtain a maximum net utility of 2, which does not improve on trading via platform 1, given that the other buyer-seller pair stays on platform 1. Intermediary 1 can sustain positive pro…ts in equilibrium due to indirect network e¤ects; in our numerical example this pro…t is equal to 4. However, if all buyers and sellers can perfectly coordinate their actions, intermediary 1 cannot make positive pro…ts, because intermediaries e¤ectively become pure Bertrand competitors.

2. Di¤erentiated platforms (a) For t < 1, it is still true that only one platform remains active in equilibrium because indirect network e¤ects are su¢ ciently strong compared to the di¤erentiation between platforms. Because of the utility loss by some of the buyers and sellers, the active platform sets a lower transaction price than in the situation in which there is no di¤erentiation. For instance, if the utility loss is t = 1=2, the intermediary of the active platform will set a price of 1=4 and make pro…t of 2. If t increases the intermediary has to further lower its price to avoid the migration of some buyers and sellers to the other platform. (For t between 0 and 1, the equilibrium price is (1 t)=2. If buyer and seller can fully coordinate their decisions, the equilibrium price is determined as the maximum of t=2 and (1 t)=2.)

(b) Suppose that buyer b and seller s incur a utility loss greater than 1 if they visit platform 2. The same is supposed to hold for buyer ~ b and seller s~ if they visit platform 1. In this case, a single intermediary can no longer monopolize the market. Consider a situation in which intermediaries set prices P 1 = P 2 = P . We have to show that they do not gain by lowering their price even though they may attract additional buyers and sellers.

Suppose intermediary 1 sets a lower price P^ such that buyer ~ b and seller s~ gain from trading through platform 1, i.e., 3 t 2P^ > 2 P . Hence, he has to charge less than (P (t 1))=2. If he keeps the price P his pro…ts are 2P , whereas with a price P^ , his pro…ts are 8P^ < 4(P (t 1)).

Deviation pro…ts are lower if 2P 4(P (t 1)), which is equivalent to P 2(t 1). Take for instance the parameter value t = 3=2. Here the price per transaction that avoids pro…table monopolization is P 1 = P 2 = 1 and each intermediary makes pro…t of 2. Intermediary 1 could set

7

a low price P^ to attract both buyers and both sellers. This price would be less than 1=4 and hence the corresponding pro…t would be less than 2. This shows that su¢ cient di¤erentiation (in spite of positive indirect network e¤ects) allows for more than one active platform.

Exercise 4 Pricing by a monopoly platform [included in 2nd edition of the book] Consider a monopoly platform serving two distinct groups of users. Each group i = a; b comprises a unit mass of users who interact on the platform.

The platform charges (possibly di¤erent) membership fees for the two groups, Ma and Mb . The constant marginal cost of attracting users on the platform is normalized to zero. A user of group i enjoys the following net utility when interacting on the platform with users of the other group: Ui = ui +

i nj

Mi ;

where ui is the intrinsic value of being on the platform, i measures the indirect network e¤ect provided by an additional member of side j on each member of side i, nj is the number of members of side j on the platform. We assume that ui is drawn from a uniform distribution on [0; vi ]. As for indirect network e¤ects, we assume that they are positive on both sides ( a ; b > 0).

1. Derive the number of participating users on side i as a function of the number of participating users on the other side.

2. Solve for the system of equations that you derived in the previous question so as to express the number of participating users on the two sides as a function of the two membership fees. Why is it legitimate to assume that a b < 1? Discuss your answer.

3. Suppose now that a = (with 0 < < 1) and b = 0; that is, users on side a welcome more users on side b whereas users on side b are una¤ected by any change in participation on side a. To simplify the analysis, set va = vb = 1. Use your answers to question 2 to solve the pro…t-maximization problem of the monopoly platform. Express the membership fees as well as the platform’s pro…t at the optimum.

4. Interpret the results that you obtained at the previous question. Which side is charged the largest membership fee? Why? How do membership fees and the platform’s pro…t evolve with the strength of indirect network e¤ects? Explain the economic intuition.

Solution to Exercise 4 1. Facing a membership fee Mi and participation nj on the other side, a user of side i decides to join the platform if ui Mi i nj . Hence, the number of participating users on side i is computed as ni = vi + i nj Mi .

8

2. We have na = va + a nb for na and nb , we …nd na =

va

Ma + 1

Ma and nb = vb +

a (vb

Mb )

and nb =

b na

vb

a b

Mb . Solving this system Mb + 1

b (va

Ma )

:

a b

We see that participation on side i decreases with the membership fee on that side provided that a b < 1. It seems thus legitimate to make this assumption.

We also see that under this assumption and given positive indirect network e¤ects, participation on side i is also decreasing with the membership fee on the other side. The intuition is clear: other things being equal, if the membership fee increases on side j , fewer users will participate on this side, which will decrease the indirect network e¤ect that side j exerts on side i, thereby discouraging users of side i to join the platform. The property a b < 1 assures that there is a unique equilibrium and that the equilibrium is stable.

3. Under the new assumptions, the participation levels derived in question 2 can now be rewritten as na = 1 Ma + (1 Mb ) ; and nb = 1 Mb :Hence, the platform chooses Ma and Mb to maximize = Ma na + Mb nb . The two …rst-order conditions are 1 2Ma + Mb = 0 and 1 2Mb Ma = 0. ) and Mb = (1 ) = (2 ).

The solution is easily found as Ma = 1= (2 Plugging these values into the platform’s pro…t function, we obtain the optimal pro…t = 1= (2 ).

4. We clearly see that Ma > Mb , meaning that the platform charges a lower membership fee on side b. This is logical as side b exerts a positive indirect network e¤ect on side a, while the reverse does not hold; it pays the platform to lower the membership fee on side b as this generates more participation on that side, and through indirect network e¤ects, on the other side as well. If indirect network e¤ects become stronger (i.e., if increases), we see that Ma increases while Mb decreases. We also see that increases. The latter result is not surprising: as indirect network e¤ects increase, so does the utility of users on side a for any participation level on side b; this allows the monoply platform to generate larger pro…ts by capturing this extra utility. The monopoly platform does so by charging a higher fee on side a and a lower fee on side b; lowering the fee on side b makes sense as larger network e¤ects make any extra user on side b even more valuable; as a consequence, there are two reasons for charging more on side a: extra participation on side b and larger indirect network e¤ects.

Exercise 5 Pricing by a monopoly platform II2 Suppose that a unit mass of buyers and a unit mass of sellers have the possibility to interact on a monopoly platform. Sellers sell independent products and buyers have a demand for one unit from each seller. Each seller makes a 2 This is essentially the same exercise as the previous one.

9

pro…t per buyer of and each buyer derives utility u per seller. Then surplus of a seller who joins the platform is vs = nb

Ms + rs

s;

where nb is the number of buyers on the platform, Ms is the membership fee for sellers, rs is the stand-alone bene…t that a seller derives when using the platform, and s is the opportunity cost that a seller incurs when joining the platform. Sellers are heterogeneous with respect to the latter cost; we assume that s is uniformly distributed on the unit interval.

Similarly, the surplus of a buyer who joins the platform is vb = ns u

Mb + r b

b;

where ns is the number of buyers on the platform, Mb is the membership fee for buyers, rb is the stand-alone bene…t that a buyer derives when using the platform, and b is the opportunity cost that a buyer incurs when joining the platform, which is uniformly distributed on the unit interval.

We normalize to zero the utility of any buyer or any seller who stays out of the platform. We also assume that the platform does not incur any cost for running the platform and for registering buyers and sellers. Finally, we assume that (A1) u < 1.

1. Determine the number of buyers and sellers on the platform by the free entry condition. How are these numbers a¤ected by the two membership fees? Discuss your answer.

2. Find the pro…t-maximizing membership fees Mb and Ms . Compute the intermediary’s pro…t at these fees.

3. Under which condition do buyers pay a lower fee than sellers? Interpret this condition.

4. Set = 1 and show that for any value of u that satis…es (A1), buyers are subsidized (Mb < 0) if and only if they have a lower stand-alone bene…t than sellers (rb < rs ).

Solution to Exercise 5 1. The seller who is indi¤erent between joining the platform or not is identi…ed by ^s = nb + rs Ms . As all sellers with a lower opportunity cost than ^s strictly prefer to join the platform, we have that ns = nb + rs Ms .

Similarly, the indi¤erent buyer is identi…ed by ^b = uns + rb Mb and hence, nb = uns + rb Mb . Substituting for the value of ns into the latter equation, we …nd nb = u ( nb + rs Ms ) + rb Mb , which is equivalent to nb =

(rb

Mb ) + u (rs 1 u 10

Ms )

:

As u < 1, we observe that nb decreases with both Mb and Ms : the …rst result is obvious; the second result comes from the two-sided nature of the market (an increase in Ms drives sellers away from the platform, which makes it less attractive for buyers). Solving for the number of sellers, we …nd a similar expression:

(rs

ns =

Ms ) + (rb 1 u

Mb )

:

2. The intermediary’s pro…t-maximization program is max Mb

(rb

Mb ;Ms

Mb ) + u (rs 1 u

Ms )

+ Ms

(rs

Ms ) + (rb 1 u

Mb )

:

The FOC yield

d dMb d dMs

(rb

=

2Mb ) + urs 1 u 2Ms ) + rb 1 u

(rs

=

(u + ) Ms

=0

(u + ) Mb

=0

We check that the SOC are satis…ed:

d2 dMb2

=

d2 dMb dMs d2 dMs2

d dMb2 d2 dMb dMs

2 u

1

<0

2 1 u ( +u) 1 u

=

( +u) 1 u 2 1 u

2

4

( + u) (1

u )

2

>0

u) rb + 2

u

u 2 rs

4

2

=

We can now solve the system of the two FOC

2Mb + (u + ) Ms = rb + urs ; (u + ) Mb + 2Ms = rs + rb : The solution is

Mb =

(u

) rs + 2 4

u

2

rb

2

( + u)

(

, Ms =

( + u)

It follows that the number of buyers and sellers are respectively

nb =

2rb + ( + u) rs 4

2

( + u)

, ns =

2rs + ( + u) rb 4

2

( + u)

Substituting into the intermediary’s pro…t function, we …nd

=

( + u) rb rs + rb2 + rs2 4

11

2

( + u)

:

:

:

3. Buyers pay a lower fee than sellers (i.e., Mb < Ms ) if and only if (u

) rs + 2 2

u (1

2

u

< (

u) rb + 2

+ u rb ) ( + u + 2) rb (1 ) rb

< 2 < (1 < (1

u u 2 u + rs , u) ( + u + 2) rs , u) rs :

2

u

u 2 rs ,

rb

To interpret this condition, suppose …rst that buyers and sellers have the same stand-alone bene…ts from joining the platform: rb = rs . Then, buyers pay less if u < , i.e. if the externality that an extra buyer generates on the sellers side ( ) is larger than the externality that an extra seller generates on the buyers side (u). Second, suppose that indirect network e¤ects are the same on both sides: u = < 1. Then, buyers pay less if rb < rs , i.e., if their stand-alone bene…t (which measures their willingness to pay) is lower.

4. With

= 1, we must have u < 1. On the other hand, Mb < 0 is equivalent to (u

1) rs + (1

u) rb < 0 , rb < rs ;

which completes the proof.

Exercise 6 Does advertising lower the price of newspapers to consumers?3 [included in 2nd edition of the book] A monopoly editor sells a newspaper to two groups of agents: readers and advertisers. Readers are of di¤erent types. Readers of type t have a willingness to pay for the newspaper equal to 1 t, with t uniformly distributed on the unit interval. At each point t of [0; 1], there is unit mass of readers that divides into two subsets: 5=6 of the readers are ‘advertising-lovers’ and 1=6 of the readers are ‘advertising-avoiders’. Advertising-lovers (resp. avoiders) gain (resp. loose) in utility when the editor sells a larger surface of the newspaper to advertisers.

In particular, the utility of a reader of type t when buying the magazine at price p is 1 t p + d if the reader is an advertising-lover, or 1 t p d if the reader is an advertising-avoider, where 0 d 1 is the share of the newspaper devoted to ad spots and where measures the intensity of ad-attraction when a reader is ad-lover or of ad-repulsion when he is ad-avoider. In what follows, we assume that 0 < < 15=16.

As for advertisers, there is a unit mass of them with unit demand (i.e., they buy a single ad or do not buy). As advertisers are interested in eyeballs, it is assumed that the utility of buying an ad in the newspaper increases proportionately with the size of its readership. More precisely, an advertiser of type , with uniformly distributed on the unit interval, has utility of buying an ad in the newspaper at a rate s given by D s, where D is the readership of the editor (i.e., the demand for the newspaper on the reading side).

3 This exercise draws from Gabszewicz, J., Laussel, D. and Sonnac, N. (2005). Does Advertising Lower the Price of Newspapers to Consumers? A Theoretical Appraisal. Economics Letters 87, 127-134.

12

1. Show that the demand on the reading side has the following form: 8 if 0 p 32 d; < 1 2 1 p + 3d if 23 d p 1 d; D (p; d) = : 5 p + d) if p 1 d: 6 (1

2. Derive the demand function d (s; D) on the advertising side. Suppose that the editor faces zero costs on each side of the market. Then, the editor’s objective is to choose p and s so as to maximize the following revenue function: R (p; s) = pD (p; d) + sd (s; D). Show that given a readership D, the revenue-maximizing rate is s = D=2, which implies a proportion d = 1=2 of the newspaper devoted to ad spots.

3. Using the expression of D (p; d) and the previous …ndings that d = 1=2 and s d (s ; D) = D=4, rewrite the revenue function as a function of p only.

4. Show that the optimal price is p = (4 + 9) =24.

5. Compare the optimal price p with the price p0 that the editor would choose if she was not operating in the advertising market. Show that p < p0 for 0 < < 3=4 and p p0 for 3=4 < 15=16, meaning that advertising serves as a subsidy to readers as long as the ad-attraction parameter is not too large. Discuss the intuition behind this result.

Solution to Exercise 6 1. De…ned by t (p; d) and t (p; d) the consumer-type for which, respectively, the ad-lovers and ad-avoiders of this type are indi¤erent between buying the newspaper or not at price p, when the proportion of the newspaper’s surface devoted to advertising is equal to d. We have that t (p; d) = 1 p + d and t (p; d) = 1 p d. Note that t (p; d) 0 for p 1 d. For such prices, only ad-lovers buy the newspaper and there is a mass (5=6) t (p; d) = (5=6) (1 p + d) of them. This gives us the third line of the demand function.

For prices below 1 d, some ad-avoiders also buy the newspaper. The mass of readers is then 5 6t

(p; d) + 16 t (p; d) = 1

p + 23 d ,

which is lower than one (the total mass of readers) as long as p is above 23 d ; otherwise, the market for readers is fully covered and D (p; d) = 1. This gives us the …rst two lines of the demand function.

2. The advertiser who is indi¤erent between buying an ad at rate s or not buying is (s) = s=D. As the unit mass of advertisers is uniformly distributed on [0; 1], it follows that d (s; D) = 1 s=D . The rate s that maximizes sd (s; D) = s (1 s=D) is easily found as s = D=2. It follows that d = 1 s =D = 1=2.

13

3. We need to consider separately the three segments of the demand on the reading side.

2 1 (a) For low prices, p 3 d = 3 , the reading market is fully covered: D = 1. Therefore, s d (s ; 1) = 14 and, consequently, R1 (p) = p + 41 .

(note that < 15 16 makes sure that such prices exist), both ad-lovers and ad-avoiders buy: D p; 12 = 1 p+ 13 . Here, R2 (p) = pD+D=4 = p + 14 1 p + 13 .

(b) For intermediate prices, 13

p

1

d =1

1 2

1 1 (c) Finally, for high prices, p 1 = 2 , only ad-lovers buy: D p; 2 5 1 1 5 1 1 p + and R (p) = pD + D=4 = p + 1 p + .

3 6 2 4 6 2

4. We have to examine separately the three segments of the revenue function. (a) On the …rst segment, R1 (p) = p + 41 increases with p. We select thus the largest possible p, i.e., p1 = 18 , which yields a revenue of R1 = 38 . (b) On the second segment, R2 (p) = p + 14 1 p + 13 . The F.O.C. for revenue maximization yields 4 + 9 24p = 0. Solving for p, we …nd p2 = 4 24+9 . We check that < 15 16 guarantees that the constraints are satis…ed: 31 p2 1 21 . The optimal revenue is computed as R2 = 1 5 2 6 + 8 . (c) On the third segment, R3 (p) = p + 41 56 1 p + 12 . The F.O.C. for revenue maximization yields 2 + 3 8p = 0. Solving for p, we …nd p = 2 8+3 , which does not satisfy the constraint that p 1 21 . We 1 3 choose thus p3 = 1 2 , with corresponding revenue R3 = 16 (5 2 ). (d) It is easily checked that R2 > R1 and R2 > R3 , which establishes that +9 the optimal price is p = 4 24 .

5. If the editor does not operate on the advertising market, we have d = 0 and all readers of type t have the same utility 1 t p. Revenue is then given by R (p) = p (1 p), which is maximized at p0 = 1=2. Comparing the two prices, we …nd

p0

p = 12

4 +9 1 24 = 24 (3

4 )>0,

< 34 ;

which establishes our result. The intuition is explained by Gabszewicz et al.

(2005) for the case where, like here, a majority of the readers’ population is advertising-lover. “With a su¢ ciently large value of the ad-attraction parameter, the monopoly power of the editor accordingly expands and allows him to quote a price for the magazine exceeding the monopoly price without advertising. Of course, those who pay for this increase of market power are those readers who belong to the minority of ad-avoiders, who not only have to tolerate the existence of ads in their magazine, but, on the top of that, have to pay their magazine at a higher price.”

Exercise 7 Platform competition with multihoming sellers

14

Consider the model of platform competition of Section 22.3.2. There are two sides of the market, the buyer side and the seller side. Suppose that each side is of mass 1, so that the total number of buyers adds up to 1, n1b + n2b = 1, and also the total number of sellers adds up to one, n1s + n2s = 1. A buyer at platform i buys one unit from each seller at the same platform. Hence, buyer and seller surplus gross of any opportunity cost of visiting a platform are vsi = nib

Msi and vbi = nis u

Mbi ;

where Mbi and Msi are the membership fees set by intermediary i. Suppose sellers and buyers are uniformly distributed on the unit interval and that platforms are located at the extreme point of the unit interval. Sellers and buyers are assumed to incur an opportunity cost of visiting a platform that increases linearly in distance at rates b and s , respectively. We furthermore assume that participation is su¢ ciently attractive such that all buyers and sellers participate in the market.

In Section 22.3.2, we assumed that each seller and each buyer could only go to either one of the two platforms 1 and 2; i.e., both sides were supposed to multihome. Here, we assume that sellers have the possibility to multihome (i.e., to visit both platforms), while consumers continue to singlehome.

On the buyer side of the market, we still have that the numbers of buyers visiting the two platforms correspond to the standard Hotelling speci…cation.

On the seller side, let s10 (resp. s20 ) denote the seller who is indi¤erent between visiting platform 1 (resp. 2) and not visiting any platform (thereby getting a utility of zero):

vs2

vs1

s s10

s (1

s20 )

= =

0 , s10 =

vs1

0 , s20 = 1

;

s

vs2

:

s

Because sellers now have the possibility to multihome, they are divided into three groups: those located between 0 and s20 visit platform 1 only, those located between s20 and s10 visit both platforms, and those located between s10 and 1 visit platform 2 only.

Hence, we derive the number of buyers and sellers visiting each platform respectively as 1 v i vbj vi nib = + b and nis = s , 2 2 b s or equivalently, using the expressions for buyer and seller surplus, as 1 nib = + 2

u nis

njs

Mbi 2 b

Mbj

and nis =

nib

Msi

.

s

1. Solve this system of four equations in four unknowns to derive the buyers’ and sellers’ demands for access to the two platforms as functions of the membership fees: nib (Mbi ; Mbj ; Msi ; Msj ) and nis (Mbi ; Mbj ; Msi ; Msj ).

15

2. Set to zero the platforms’cost per buyer and per seller. Each platform i solves the problem maxMbi ;Msi i where i

= Mbi nib (Mbi ; Mbj ; Msi ; Msj ) + Msi nis (Mbi ; Mbj ; Msi ; Msj ):

Assuming that 8 b s > Nash equilibrium are

2

+u2 +6 u, show that the fees at the symmetric

Mb

Mb1 = Mb2 =

Ms

Ms1

= Ms2

4 s (3u +

b

= 14 (

);

u):

Comment on these prices.

Solution to Exercise 7 1. The solution to the system of equations is nib nis

=

Msi ) + 2( b s

1 u(Msj + 2

= s

1 u(Msj + 2

j s (Mb

Mbi )

u )

Msi ) + 2( b s

j s (Mb

; Mbi )

u )

!

Msi

:

s

2. Firm’s best responses are implicitly de…ned by the …rst-order conditions, which can be expressed as

Mb1

=

Ms1

=

(u + ) Ms1 + uMs2 + s Mb2 u+ b s ; 2 s 2 u+ (u + ) s Mb1 + u Ms2 + s Mb2 2 (2 b s u )

b s

:

Second-order conditions require that 8 b s > 2 + u2 + 6 u. This condition is also su¢ cient to have a unique and stable interior equilibrium. To …nd the symmetric Nash equilibrium, we set Mb1 = Mb2 = Mb and Ms1 = Ms2 = Ms in the above …rst-order conditions. Rearranging, we …nd: s Mb +

u s Mb + (4 b s

Ms 3u ) Ms

= =

b s

( b s

u; u) :

Solving this system of two equations in (Mb ; Ms ), we …nd:

Mb Ms

=

b

=

1 4(

4 s (3u +

);

u):

On the seller side, platforms have monopoly power. If the intermediary focused only on sellers, he would charge a monopoly price equal to =4 (assuming that each seller would have access to half of the buyers and, therefore, would have a

16

gross willingness to pay equal to =2). We observe that this price is adjusted downward by u=4 when the indirect network e¤ect that sellers exert on the buyer side is taken into account. Sellers are subsidized if u > , i.e. the indirect network e¤ect is stronger on the buyer than on the seller side. Similarly, on the buyer side, platforms charge the Hotelling price, b , less a term that depends on the size of the indirect network e¤ects.

Exercise 8 Platform competition and piracy4 Consider the following market of two competing software platforms: Platform 1 is located at 0, platform 2 at 1 on the Hotelling line. Consumers are uniformly distributed on the [0; 1] interval. Consumers derive utility from the services o¤ered by the platform r as well as the number of applications ni that are o¤ered on platform i. The utility a consumer x 2 [0; 1] derives from buying access to platform 1 is r + un1 Mb1 x. Here u is the net bene…t consumers derive per software application, Mb1 is the price the consumer has to pay to access platform 1 and is the standard disutility parameter in the Hotelling (1 x). model. The corresponding utility for platform 2 is r + un2 Mb2 Developers decide whether to be active on none, one, or both platforms.

Suppose that developers make a net pro…t per consumer. A developer’s pro…t on platform i is si Msi f where si is the share of consumers subscribed to platform i, Msi is the price the developer has to pay to access the platform, and f is a …xed cost which constitutes the type of the developer. We assume that there are potential developers of mass F with …xed costs f uniformly distributed on [0; F ] with F su¢ ciently large such that always some developers do not become active. Consider only situations such that there is full market coverage on the consumer side. b and 1. Given prices Ms1 ; Ms2 ; Mb1 ; Mb2 , determine the indi¤erent consumer x the numbers of developers on each platform, ni , as a function of these prices.

2. In the symmetric equilibrium, the fees charged by the platforms are given by Mb1 = Mb2 = Mb = (u + Ms ) and Ms1 = Ms2 = Ms =

uMb u 2 : 4 3 u

Determine equilibrium prices Mb ; Ms and equilibrium pro…ts of platforms . (In particular, the expression you obtain for Ms should be very simple and only depend on u.)

3. Suppose now that product piracy a¤ects the industry. The level of piracy is parametrized by 2 [0; 1]. As a consequence of piracy retail prices fall 4 This exercise is motivated by A. Rasch and T. Wenzel (2011), Piracy in a Two-Sided Software Market, mimeo.

17

even though consumers copying costs may be above the software developers’marginal costs and the quality of a pirated product may be lower.

However, suppose that the social surplus generated by an active developer increases in the level of piracy, i.e. ( ) + u( ) is increasing in . Piracy also a¤ects the rent distribution between developer and consumer: ( ) is decreasing in while u( ) is increasing in . Given your results in (2), analyze whether or not software platforms gain from an increased level in piracy. Explain your …ndings. (Note that due to the reduced form in u and you may obtain ambiguous results and may not be able to express your results as restrictions on parameters.)

4. Using the results obtained in (2) analyze whether or not developers gain from piracy. Discuss your …ndings. The same remarks as in (3) apply.

5. In the public debate content providers and platform providers sometimes appear to express di¤erent views on the e¤ects of piracy. Can the present model contribute to this debate? Discuss!

Solution to Exercise 8 1.

1 Mb2 + 2 2(

x b =

ni

=

+

2

Mb1 u(Ms2 + u) 2(

(Mbj 2(

Ms1 ) u)

Mbi ) u)

Msi +

3 u 4

1 4

u(Msj 2(

Msi ) u)

2.

Mb

=

Ms

= =

1 ( 4 2

2

u) 1 16

2

1 2 u 16

3 u 8

3. We can rewrite platform pro…ts as ( )=

2

1 ( ( ) + u( ))2 16

Taking the derivative with respect to

d

( ) d

=

d d

1 ( )u( ): 4

one obtains

1 ( ( ) + u( ))2 16

1 0 [ ( )u( ) + ( )u0 ( )] 4

The derivative of the …rst term in square brackets is positive, the second is of ambiguous sign. If u( ) and ( ) are of similar magnitude the overall e¤ect is negative, i.e. a higher level of piracy hurts software platforms. One reason for lower pro…ts is that overall network e¤ects are stronger thus leading to more intense competition between platforms.

18

4. In equilibrium, a developer’s pro…t gross of its …xed cost when joining a platform is

( ) 2

Ms

( ) 1 ( ( ) 2 4 ( ) + u( ) : 4

= =

u( ))

Clearly, by our assumption that ( ) + u( ) is increasing in , the developers’ pro…ts is increasing in the level of piracy. We observe that the direct e¤ect of piracy is negative since is decreasing in . However, the indirect e¤ect due to a change in platform pricing overcompensates the direct e¤ect.

5. In our setting developers always gain from piracy, while the e¤ect on platform pro…ts is ambiguous (it is negative if u and are of similar size). Thus these parties have con‡icting interests as to public policy towards piracy. Interestingly, the more society bene…ts from piracy (measures by the increase in u + ) the more platforms su¤er from piracy and thus are opposed to it. Again, this is due to the fact that stronger indirect network e¤ects tend to lead to more intense competition.

Exercise 9 Strategies in platform markets Refer to Section 22.3 to analyze the strategies that have been deployed in the video-game industry in 2007, as described in the following two texts. From the observed strategies, what can you infer about the nature of this two-sided market? Is there singlehoming or multihoming on each side of the market?

Which side do you think (i) is the most sensitive to price, (ii) exerts the strongest indirect network e¤ects on the other side? Discuss.

“In the competition among the makers of video-game consoles, momentum for the Wii from Nintendo is building among crucial allies: game developers and publishers. Inspired by the early success of the Wii, the companies that create and distribute games are beginning to shift resources and personnel toward building more Wii games, in some cases at the expense of the competing systems, the PlayStation 3 from Sony and Xbox 360 from Microsoft. The shift is closely watched because consumers tend to favor systems that have the biggest choice of games. More resources diverted to the Wii would mean more games, and that would translate into more consumers buying Wii consoles. (...) The interest in the Wii follows a period of uncertainty about the console among developers and publishers. They were initially cautious because the Wii was less technologically sophisticated. They worried that consumers would not take to its unorthodox game play, which uses a motion-controlled wand that players move to direct action on the screen. For example, to serve balls in the tennis game, players circle their arms overhead as they would in real tennis. (...) While the growing size of the 19

Wii’s customer base is attractive, developers are favoring Wii for other reasons. Developers are able to create games in less time than is needed for its rivals, because of the Wii’s less-complex graphics.

Colin Sebastian, a video-game industry analyst with Lazard Capital Markets, said that in rough terms, it costs around $5 million to develop a game for the Wii, compared with $10 million to $20 million to make a game for the Xbox 360 or PS3. Sebastian said that given the cost di¤erences, a developer would need to sell 300,000 copies of a Wii game to break even, compared with 600,000 of a game for the PS3 or Xbox 360.” (Nintendo’s Wii is winning battle of the game builders. By Matt Richtel and Eric A. Taub. New York Times, July 16, 2007.)

"Hoping to encourage the creation of more games for the PlayStation 3, Sony said Monday that it had cut the price of a software development kit for the struggling console in half. (...) Although its predecessor, the PlayStation 2, dominated the gaming market, the PS3 has struggled against the Nintendo Wii console. (...) It is generally more di¢ cult and expensive to create games for sophisticated machines like the PS3, which is packed with cutting-edge technology and is driven by the Cell microprocessor. But the consoles will not sell unless a variety of games can be played on them. Game developers who previously designed products for the PlayStation 2 are now increasingly making Wii versions of the games (...) The PS3 has also lagged behind the Xbox 360 machine from Microsoft, which has sold 13.4 million Xbox 360 consoles over the past two years." (Sony cuts price for PS3 developers. New York Times, November 19, 2007) Exercise 10 Experts and credence goods5 [included in 2nd edition of the book] Consider a unit mass of consumers. Each consumer has a problem that can be major or minor. Two treatments are available: a minor treatment can only solve a minor problem while a major treatment can solve both types of problem.

Let v denote the gross gain of a consumer when his problem is solved; otherwise he gets 0. The consumer knows that he has a problem but he does not know the type of this problem. Ex ante, each consumer expects that his problem is major with a probability h and minor with a probability (1 h).

An expert can detect the true type of the problem only by conducting a proper diagnosis. Without diagnosis, the expert cannot supply an appropriate treatment and can only choose to always supply a minor treatment or a major one. The cost of a major treatment is c, and the cost of a minor treatment is c, with c > c. If a diagnosis is performed, the expert bears a cost d that is charged to the consumers.

5 This exercice draws from Bonroy, O., Lemarié, S. and Tropéano, J.P. (2013). Credence Goods, Experts and Risk Aversion. Economic Letters 120: 464-467.

20

In the …rst period of the game, the expert posts prices p and p respectively for a major and a minor treatment, and commits to conducting a diagnosis or not. Consumers observe these actions and decide, in the second period, whether to visit the expert or not. In the third period, nature determines the type of the consumer’s problem (major or minor). In the fourth period, the expert conducts a diagnosis or not, recommends a treatment, charges for it and provides it. The action of making a diagnosis is observed by the client but the result of this diagnosis is not.

1. Show that the equilibrium prices (p; p) satisfy: (a) p c = p c with p = v h(v (c c))g, (b) p

c>p

d + (1

c with p = v for d

h)(c (1

(c) p c < p c with p = (1 h)v for d

c) for d h)(c

minf(1

c) and v

h(v (c c)) and v

h)(c

(c

c);

c) =h, (c

c) =h.

2. Does the asymmetric information lead the expert to bias his behavior?

3. Suppose now that consumers are risk-averse. In particular, when the expert posts prices (p; p) that ensure p c = p c, the consumer incurs a positive risk premium (due to di¤erentiated prices according to the treatment), and when the expert posts prices that always lead to a minor treatment (i.e. p c < p c), the consumer incurs a positive risk premium (due to the risk of an insu¢ cient treatment). Show that with risk-averse consumers the asymmetric information leads the expert to bias his behavior towards an ine¢ cient treatment for: d 2 min

(1 h(v

h)(c c); (c c)) +

; min

(1 h(v

h)(c c); (c c)) +

:

4. Based on the previous questions, explain why full insurance and information disclosure are not compatible strategies for the expert.

Solution to Exercise 10 1. Under equal markup prices (i.e., p

c = p c) and diagnosis committed, the consumer is provided honestly and its expected utility is: h u(v p d) + (1 h) u(v p d). Under markup prices more important for the major problem (resp. the minor problem), the consumer’s utility is u(v p) (resp. h u( p) + (1 h) u(v p)) because the expert is dishonest: he provides an overtreatment (resp. undertreatment) and has no interest in conducting a diagnosis.

The maximal pro…t per customer for a monopolist is: AT v d c h (c c) under equal markup (‘appropriate treatment’), O v c under overtreatment, and U (1 h) v c under undertreatment. It is easy to see that (1) AT O U i¤ d (1 h) (c c), (2) AT i¤ d h(v (c c)), and O U (3) i¤ v (c c) =h.

21

2. The consumers are e¢ ciently served when they are served as in an environment with symmetric information on the diagnostic outcome. The expert that provides an overtreatment (resp. an undertreatment) does not conduct a diagnosis, so he charges the same prices and has the same pro…t what the diagnosis result (1 h) v c), where is common knowledge or not ( O v c and U the superscript indicates an e¢ cient environment, i.e., with symmetric information. With symmetric information, the expert that provides an appropriate treatment maximizes his pro…t given by AT h(p c)+(1 h)(p c) under the consumer participation constraint h u(v p d)+(1 h) u(v p d) 0.

The expert charges p = p = v d and has the same pro…t as in the environment with asymmetric information ( AT v d c h(c c)). So the treatment provided with asymmetric information is e¢ cient whatever the value of parameters considered.

3. Since consumers are risk-averse, the maximal pro…t per customer for a monopv c olist is: AT v d c h (c c) under equal markup, O U under overtreatment, and (1 h) v c under undertreatment. It O U is easy to see that (1) AT i¤ d (1 h) (c c) , (2) AT O U i¤ d h(v (c c)) + , and (3) i¤ v (c c ) =h.

With symmetric information, the expert that provides an appropriate treatment maximizes his pro…t by charging a risk-free tari¤: p = p = v d.

His pro…t is superior than in the environment with asymmetric information: AT v d c h (c c). The v d c h(c c) > AT expert that provides an overtreatment (resp. an undertreatment) does not conduct a diagnosis, so he charges the same prices and has the same pro…t whether the diagnosis result is common knowledge or not ( O v c and U O (1 h) v c). It is indeed easy to see that (1) AT i¤ AT U i¤ d h(v (c c) + , and (3) d (1 h) (c c), (2) O U i¤ v (c c ) =h. By comparing the environments with asymmetric and symmetric information, we …nd that asymmetric information leads the expert to bias his behavior towards an ine¢ cient overtreatment (resp. undertreatment) whenever (1 h) (c c) < d < (1 h) (c c) (resp. h(v (c c)) + < d < h(v (c c)) + ).

4. With symmetric information on the diagnostic outcome, if the expert undertakes the diagnosis, he chooses the same price for both treatments and then provides the appropriate treatment. The information symmetry on the diagnostic outcome allows the combination of a risk-free tari¤ and the completion of the appropriate treatment. With asymmetric information, in order to induce a truthful disclosure of the diagnosis result, the expert is constrained to di¤erentiate the price according to the treatment proposed. In other words, full insurance and information disclosure are no longer compatible. Thus, under symmetric information, the full insurance allows the expert to capture the risk premium while under asymmetric information the expert is constrained to leave that risk premium to the consumer. If the expert provides instead an overtreatment, there is no risk since the consumer always pays for the major

22

treatment, and the diagnostic cost is saved. This choice is ine¢ cient as long as the diagnostic cost is not too high but could be preferred by the expert that is no longer constrained to leave the risk premium to the consumer. The expert could also save the diagnostic cost and choose the undertreatment. As above, information asymmetry increases the risk incurred under the appropriate treatment and thus biases the expert choice between undertreatment and appropriate treatment, towards undertreatment.

Exercise 11 Competition in search markets6 [included in 2nd edition of the book] In Section 23.1.2, we developed a simpli…ed version of Baye and Morgan (2001).7 Let us recall the main assumptions of the model. Suppose there are two local markets. On each market, there is a single …rm and a unit mass of consumers. The two …rms sell identical products at a constant marginal cost which, for simplicity, is supposed to equal zero (the cost of delivering goods to consumers is also zero). Each consumer has a demand function q (p) = 2 p. The two local markets are completely segmented: consumers in local market i only have access to …rm i. Therefore, the expected pro…ts of …rm i when it charges a price p to consumers in its local market is (p) = p (2 p); the monopoly price is easily computed as pm = 1. It costs a consumer 0 < z < 12 to visit a local store. The assumption that z < 12 ensures that a consumer who is charged the monopoly price pm obtains su¢ cient surplus to make a visit worthwhile.

2 The consumer surplus at some price p is indeed computed as v (p) = 12 (2 p) ; 1 hence, v (pm ) z = 2 z > 0. The internet makes it possible for an intermediary to open a virtual marketplace and, thereby, eliminate geographic boundaries between the two local markets. In the absence of such a virtual marketplace, each …rm simply charges the monopoly price to all of its local consumers to earn pro…ts of (1) = 1. In contrast, the creation of a virtual marketplace allows …rms and consumers to globally transmit and access price information. The intermediary runs the ‡ows of information by charging an access fee, Ms 0, to …rms posting their price on the website, and a subscription fee, Mb 0, to consumers accessing price information from the website.

In Section 23.1.2, we assumed that …rms were not able to price discriminate among consumers. In particular, …rms had to charge the same price to all customers regardless of whether they purchase through the intermediary. There exist, however, situations where sellers can price discriminate between consumers who visit them directly and those who visit them through an intermediary.

We take this alternative assumption in this exercise and we analyze the following game: in the …rst stage, the intermediary announces the fees Ms and Mb ; in the second stage, given the fees, consumers decide whether or not to 6 This exercise draws from Nahm, J. (2003). The Gatekeeper’s Optimal Fee Structure When Sellers can Price Discriminate. Economics Letters 80, 9-14.

7 Baye, M. and Morgan, J. (2001). Information Gatekeepers on the Internet and the Competitiveness of Homogeneous Product Markets. American Economic Review 91: 454-474.

23

subscribe to the website; …rms choose their prices for the product and decide whether or not to post a price on the website; …nally, consumers shop. The game is solved for its symmetric subgame perfect equilibria.

1. Show that the following constitutes a subgame perfect equilibrium of the game: (a) The intermediary sets Mb = v (0) v (pm ) = 3:5 and Ms = 0. (b) All consumers subscribe to the intermediary. (c) Both …rms post their price on the website with probability one. (d) Both …rms advertise a price of c = 0 on the website and charge the monopoly price pm = 1 in their local market.

2. Show that this full participation equilibrium entails not only the socially optimal allocation but also the highest pro…t for the intermediary.

Solution to Exercise 11 1. (We adapt here the proof of Proposition 1 in Nahm, 2003). When Ms is zero, it is a weakly dominant strategy for sellers to post a price on the intermediary’s website. Suppose that all consumers subscribe to the intermediary. Then, the two …rms engage in Bertrand competition, which drives the prices posted on the website down to marginal cost, i.e., zero. Supposing then that both sellers sell their product at c = 0 through the intermediary, a consumer can get a surplus equal to v (0) = 4 when subscribing to the intermediary. On the other hand, if the consumer does not subscribe, she has to pay the monopoly price on the local market and her surplus is v (pm ) = 1=2. She is thus willing to subscribe to the intermediary as long as Mb 3:5. Hence, when Ms = 0 and Mb = 3:5, we have indeed a subgame-perfect equilibrium where all consumers and …rms subscribe to the intermediary and where …rms charge prices of zero on the website and of 1 in their local market.

2. (This is Proposition 2 in Nahm, 2003). Note …rst that Mb and Ms are pure transfers from buyers and sellers to the intermediary. Social optimality occurs when all transactions in the product market take place through the intermediary at marginal cost. This is what happens when the intermediary sets Mb = v (0) v (pm ) and Ms = 0, meaning that the socially optimal allocation is attained. Since consumers pay Mb = v (0) v (pm ) for subscribing to the intermediary, the net consumer surplus (after the payment to the intermediary) is zero. Also, the net pro…t for sellers is zero. Thus, the maximized total pie (social welfare) goes to the gatekeeper. This implies in turn that the gatekeeper cannot improve its pro…t by deviating from Mb = v (0) v (pm ) and Ms = 0.

Exercise 12 Search engine pricing8 [included in 2nd edition of the book] 8 This exercise is drawn from Eliaz, K. and Spiegler, R. (2011). A Simple Model of Search Engine Pricing. Economic Journal 121: F329-F339.

24

Consider a market consisting of a continuum of consumers, a continuum of …rms and a monopolistic search engine. A …rm is identi…ed by a parameter q that measures its ‘relevance’ for the consumers. That is, when a consumer is matched with a …rm of type q, there is a probability q that the match delivers a positive value to the consumer, i.e., that the consumer is willing to pay an amount v 0 for the …rm’s product. We assume that v is randomly drawn (independently across all matches) from a uniform distribution over the unit interval.

We analyze the following two-stage game. In the …rst stage, the search engine posts a ‘price-per-click’, r, which …rms have to pay to the search engine each time a consumer visits the …rm (whether or not the consumer eventually transacts with the …rm); only …rms that accept the posted price-per-click are admitted into the search pool. In the second stage, market interaction takes place. As for …rms, they simultaneously decide whether or not to pay the priceper-click r and if so, which price to set for their product. As for consumers, they correctly anticipate the set of …rms that enter the search pool and form a belief about the distribution of prices in the market; they follow a sequential-search process with a search cost of s per round (i.e., a consumer samples a …rst …rm for free, learns the value of the match and this …rm’s price, and then decides whether or not to pay the search cost s and draw a new sample).

1. Consider …rst the market interaction at the second-stage of the game.

Let E (q) denote the expectation of q with respect to the population of …rms in the search pool (which is given at this stage). Characterise a uniform-price equilibrium, which is de…ned as a price p and a stopping rule for consumers, which satisfy the following properties: (i) given that all …rms charge p , the consumers’stopping rule is optimal; (ii) given the consumers’stopping rule and the belief that all …rms charge p , no …rm has an incentive to deviate to a di¤erent price. Note that a stopping rule is a function that speci…es the realised match values and prices, (v ; p ), for which the consumer stops searching.

2. Express the equilibrium ‘conversion rate’for a …rm of type q (i.e., the equilibrium probability that a consumer who clicks on this …rm will buy from it); denote it CR (q). From there, express the equilibrium gross pro…tper-click that a …rm of type q earns at equilibrium as (q) = p CR (q).

Express also the consumer’s ex ante expected surplus from searching in the pool (i.e., the expected value of the item that will ultimately be purchased, minus its equilibrium price minus the expected search costs; denote it S ). Under which condition do consumers …nd it optimal to enter the market and face the uniform-price equilibrium (i.e., what is the condition for S 0)?

3. Assume that the search engine incurs no cost. The search engine then maximizes its revenue, which is de…ned as the price-per-click r multiplied by the expected number of clicks (i.e., the expected number of samples

25

that consumers draw in the uniform-price equilibrium induced by r, or equivalently the inverse of the expected conversion rate). (a) Determine q as the type of the …rm that is just indi¤erent between entering the search pool or not for a given price-per-click r. That is, (q ) = r. Using this de…nition and the results obtained in (2), show that the search engine’s pro…t can be expressed as =

p

2s

q 3=2

[E(q)]

;

where E (q), the average relevance in the pool, should now be written more precisely as EG (q j q q ), with G being the cumulative distribution function according to which q is distributed. (b) Suppose that q is distributed as follows: q = qH = 1 with probability and q = qL < 1 with probability 1 . Establish under which condition the search engine …nds it optimal to contaminate the search pool with …rms of relatively low relevance; that is, the search engine sets r so that q = qL and all low-type …rms enter. Discuss this result.

Solution to Exercise 12 1. We start with the consumers’stopping rule. Their stopping obeys a cuto¤ rule because of the stationary environment that they face (as all …rms charge the same price p ). In other words, there exist v in [0; 1] such that in equilibrium, consumers stop searching if and only if the current match value v is at least as large as v . The cuto¤ v is found by equating the extra expected bene…t and the extra cost of an additional sampling, i.e.,

E (q)

Z 1

(v

v ) f (v) dv = s:

v

As v is assumed to be uniformly distributed p on [0; 1], this equation is equivalent 2 2s=E (q). Now consider the pricing to 21 E (q) (1 v ) = s or v = 1 decision of a …rm of type q . If the …rm deviates from the equilibrium price p to another price p, a consumer who samples the …rm and learns that the match value is v > 0 will buy the …rm’s product if v p > v p (where the RHS represents the consumer’s reservation surplus conditional on a positive-value match). From the uniform distribution of v , the probability that the consumer will buy at p is 1 (v p + p). Therefore, the …rm will choose p to maximise p + p)]. The …rst-order condition yields 1 v + p 2p = 0. p [1 (v Hence, p is thep value of p that solves the latter equation: 1 v p = 0 or p = 1 v = 2s=E (q).

2. We …nd that CR (q) = q [1 F (v )]. As v is uniformly distributed on [0; 1], we have CR (q) = q (1 v ) and (q) = p q (1 v ) = 2sq=E (q). The 26

consumer’s ex ante expected surplus from searching in the pool is given by

S = E (v j v

v )

s

p

E (q) [1

F (v )]

;

which, p . That is S = 1 p by the de…nition of v , is simply equal to v 2 2s=E (q). We easily …nd that S 0 if and only if s E (q) =8.

(a) Developping (q ) = r, we …nd q = rE (q) = (2s), which is equivalent to r = 2sqp=E (q). The expected conversion rate is CRe (q) = E (q) (1 v ) = 2sE (q). The search engine’s pro…t is de…ned as =

p r q 2sq 1 p = 2s = : e 3=2 CR (q) E (q) 2sE (q) [E (q)]

(b) Given the distribution of q , the search engine has two possibilities. The …rst possibility is to set r so that the cuto¤ q is arbitrarily close to 1, meaning that the search engine admits only the p highest quality …rms. In that case, the search engine’s pro…t is 1 = 2s. The second possibility is to set r so that q = qL . In that case, pro…t is equal to 2 =

p

2s

qL [ + (1

The second option is more pro…table, 2=3

or qL

(1

2

3=2

) qL ]

:

1 , as long as qL

[ + (1

qL ) + qL , or 2=3

qL 1

qL : qL p 3

For instance, with qL = 1=2, the condition is 2 1 = 0:260. To understand this result, let us decompose the e¤ects on the search engine’s pro…t of setting r in a way that e¤ectively reduces the search pool’s marginal relevance q . Recall that the search engine’s pro…t is the product of the price-per-click and the expected number of clicks. First, the effect on the the price-per-click (which is equal to the marginal …rm’s gross pro…t per click) is ambiguous. To see this, note that as q goes down, so does the average relevance E (q) EG (q j q q ) (note that this is true in this example but this may not be true for di¤erent shapes of G). A …rst consequence is that the equilibrium product price p increases (because a lower-quality search pool creates a less competitive environment). A second consequence is that v goes down and therefore (1 v ) increases. It follows that the net e¤ect on the the marginal …rm’s gross pro…t per click (q ) = p q (1 v ) is ambiguous. Second, the expected number of clicks, which is the inverse of the expected conversion rate, CRe (q) = E (q) (1 v ), can also go either way as E (q) goes down while (1 v ) goes up. Now, in the present example, when is

27

3=2

) qL ]

small enough, choosing q = qL reduces E (q) quite a lot with respect to the case where q = 1, which explains why it may be the best option for the search engine.

28