Best Economic Analysis Homework Help

Best Economic Analysis Homework Help

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1. In your course packet you will find the case “Lesser Antilles Lines – The Island of San Huberto”. This case is useful for reviewing basic concepts of game theory and considering the implications of alternative decisions. Read the case carefully, then answer the following questions. Note: The contribution matrix in Exhibit 4 of the case is denominated in thousands of dollars. a. With reference to the game situation depicted in the matrix at the end of the case: i. What is a Nash Equilibrium? Are there any Nash equilibria in this game? ii.What is a Dominant Strategy? Is there a dominant strategy for LAL? Is there a dominant strategy for KL? iii.What is a maximin strategy? What would the maximin strategy for LAL be? And for KL? iv.What is the ‘best’ outcome for the firms?

[HINT : To answer this question, think about what LAL and KL would agree on if they could write a binding contract. Consider both the cases that the contract can dictate prices and that the contract can dictate prices as well as a transfer from one firm to the other.] v. If LAL could somehow (irreversibly) commit to a price, would it do so? And if so, to what price would it commit? If LAL could somehow commit that it would always charge the same as KL, would it do so? vi.What would you recommend that LAL do? Why? economicshomeworkhelper.com economicshomeworkhelper.com

b. What happens if – due to an increase of world trade following a sudden deregulation wave - the market size suddenly doubles? Would that change the Nash equilibrium described above? [HINT: You do not need to recalculate the payoff matrix.]

2. a. Suppose the airline industry consisted of only two firms : American and Texas Air Corp.

Let the two firms have identical cost functions, C(q)=40q. Assume the demand curve for the industry is given by P=100-Q and that each firm expects the other to behave as a Cournot competitor. i. Calculate the Cournot-Nash equilibrium for each firm, assuming that each chooses the output level that maximizes its profits when taking its rival’s output as given. What are the profits of each firm? ii. What would be the equilibrium if Texas Air had constant marginal and average costs of 25 and American had constant marginal and average costs of 40? iii.Suppose that Texas Air’s cost advantage in ii) is based on a patented business practice.

What is the minimum price that Texas air is willing to accept to license the patent to American. What is the maximum price that American wants to pay? Discuss. b. Determine analytically the Nash equilibrium of the LAL case by going through the following steps : i. Write the equations for gross profits (i.e. not considering fixed costs) for LAL ii. Using thand for KL. iii.e first-order condition (i.e. taking the derivative and setting it equal to zero), determine the optimal pLAL for a given pKL. Determine analogously the optimal pKL for a given pLAL. economicshomeworkhelper.com economicshomeworkhelper.com

3. Magna Company must decide whether to buy all or part of its aluminum from the Steele Corporation. At the same time, Steele must decide whether or not to provide prompt delivery, which is more costly. If Steele Corporation provides prompt delivery of the aluminum it sells Magna, then Magna will make $2 million if it buys all its aluminum from Steele and $1 million if it buys only part of its aluminum from Steele. But if Steele provides slow delivery, then Magna will lose $5 million if it buys all its aluminum from Steele and lose $1 million if it buys only part from Steele. If it receives an order for all of Magna’s aluminum requirements, then Steele will make $3 million if it provides prompt delivery and $4 million if it is slow. If it receives an order for part of Magna’s aluminum requirements, then Steele will make $1 million if it provides prompt delivery and $2 million if it is slow. a. Magna must decide whether to buy all or part of its aluminum from Steele, and Steele must decide whether or not to provide prompt delivery. What is the payoff matrix for this game? b. Does any player have a dominant strategy? If so, what is it? Explain. c. Does this interaction have a Nash equilibrium? If so, what is it? Explain.

4. Consider two firms: Chip and Dale. Each of them is considering whether to expand its capacity. The decision depends on the payoffs, which depend on what the other player does. Assume, for sake of simplicity, that each faces three possibilities: Do Nothing (DN), a Small expansion, or a Large expansion. The payoff matrix is given by (Dale’s Payoff, Chip’s Payoff): Chip economicshomeworkhelper.com economicshomeworkhelper.com

Dale DN Small Large

DN $36m, $36m $40m, $30m $36m, $18m

Small $30m, $40m $32m, $32m $24m, $16m

Large $18m, $36m $16m, $24m $0m, $0m

a. Because of the long lead times, assume that the two players’ actions are simultaneous. What is the equilibrium strategy in this game? Explain. b. If one player, say Dale, preempts the other by moving first, plot the resulting game tree.

What is the equilibrium strategy now? Explain. c. Does the equilibrium strategy change if Chip instead moves first? Explain how.

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1. a. Game analysis i. Nash Equilibrium A pair of actions is a Nash equilibrium when each firm does the best it can, given the other firm’s action. This game (as described in the matrix) has a unique Nash equilibrium with KL setting a price of 1400 and LAL a price of 1300. It can be seen from the matrix that with pKL=1400, a price of 1300 is indeed LAL’s best response.

The analogous argument holds for KL. For any other set of prices at least one firm would be better off changing its price. ii. Dominant Strategy A firm has a dominant strategy if its profits are maximized by one and the same price level, regardless of the other firm’s price. In this case, neither firm has such strategy available: there is no price level that LAL could choose thereby maximizing its profits, regardless of KL’s reaction and vice-versa. iii.Maximin Strategy A firm’s maximin strategy is the price that maximizes its minimum possible profit, i.e. that maximizes its profit assuming that the other firm does precisely the worst thing from your perspective. This means that KL would choose to price its product at 1100, because this level maximizes its minimum profit (which is in this case 254). Similarly, LAL would price at 1000, thereby maximizing its minimum payoff at 124. economicshomeworkhelper.com economicshomeworkhelper.com

The maximum set of strategies would therefore be 1100, 1000 and the profits 423 for KL and 310 for LAL. Their risk aversion has forced them to far from optimal pricing. iv. Contractual Collusion If the two firms can write a contract that specifies both their prices and a transfer, then any optimal contract must specify those prices that maximize the sum of their profits (since otherwise they could improve the contract by switching to the prices that maximize joint profits and choose the transfer so as to split the gain in profit equally). The transfer will then determine how that joint profit is actually split up.

This will be determined by bargaining and is difficult to predict.

If the two firms cannot include a transfer in their contract, then it is not necessarily the case that they will maximize joint profits. They might agree to sacrifice some profits to make the profit split more equitable. (Note that before issues of equity could always be dealt with by the transfer.) We can, however, seriously restrict the number of possible contracts in the following way. A set of prices cannot be optimal if there exists an alternative set of prices that gives both firms higher profits (i.e. in ‘econ-speak’ if there exists a set of prices that Pareto dominates). So the optimal contract must be one of the following :

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LAL 1900 1900 1900 1800 1700 1600

p

KL 1700 1800 1900 1900 1900 1900 p

v. Commitment LAL would indeed like to be able to commit to a price. In particular, if it could commit to setting a price of 1500, it would be better off than under the Nash equilibrium: the best response of KL is to also set a price of 1500, which gives LAL a (gross) profit of 1028, which is higher than its profit in Nash equilibrium. (Note, however, how its profit would be still higher if KL were the one who could commit. KL would then commit to 1700, to which LAL would respond with 1500 for a LAL profit of 1542).

If it could, LAL would instead prefer to commit to always match KL prices. In that case, KL’s optimal price is 1900, which gives a LAL profit of 1652). vi.What should LAL do?

There are several options available for LAL. Probably the best solutions would be to try to collude (by sending the signal that if KL decides to sell at 1900, LAL will match the price). This could be supported by a promise to match the best price offer of any competitor.

In case this solution turns out to be unsustainable – because KL cheats or LAL cannot economicshomeworkhelper.com economicshomeworkhelper.com

resist the temptation to cheat itself – the best alternative is to offer the Nash equilibrium pricing, as this is the only sustainable behavior for LAL. b.

Double market size If the market doubles, the whole payoff matrix will change linearly, as the contributions are calculated using the following two formulas: Contributions LAL = (PLAL - $841) (Market Size) (Market Share LAL) Contributions KL = (PKL - $883) (Market Size) (Market Share KL) While total payoffs change, the optimal pricing strategies and therefore the Nash Equilibrium do not change.

a. In order to solve this exercise we have to calculate the reaction curves that each firm faces while setting the optimal quantities produced in the Airline industry. i. When the cost functions are the same, the solution is as follows: Demand curve: P = 100 – Q Total quantity produced: Q = QAA + QTA Marginal Costs: MCAA = MCTA = 40 The problem for American Airlines therefore is to maximize profits, when: Revenues = R = P* QAA = (100 – Q ) QAA = (100 - QAA - QTA) QAA economicshomeworkhelper.com economicshomeworkhelper.com

= 100 QAA - QAA2 – QTA QAA Marginal Revenues therefore are: dR / dQAA = MRAA = 100 – 2 QAA – QTA The optimal quantity is reached when MR = MC. For American Airlines we have: 100 – 2 QAA – QTA = 40, or QAA = 30 – ½ QTA Considering that both companies face the same demand curve and that they have identical cost curves, the Reaction curve for Texas Airlines will be identical to the one of American Airlines: QTA = 30 – ½ QAA The Cournot-Nash equilibrium requires that both firm produce at their reaction curves, which means that the optimal quantities QAA and QTA will be obtained by setting the two reaction curves equal to each other: QAA = = 30 – 15 + ¼ QAA  .75 QAA = 15 30 – ½ QTA = 30 – ½ (30 – ½ QAA) QAA = 20 economicshomeworkhelper.com economicshomeworkhelper.com

And: QTA = 30 – ½ * 20 = 20 Therefore: Q = 20 + 20 = 40 P = 100 – 40 = 60 ΠAA = P*QAA – TC(QAA) = ΠTA = P*QTA – TC(QTA) = 60 * 20 – 40 * 20 = 400 ii. If Texas Airlines now has the benefit of a lower marginal cost (and American Airlines knows about it) the problem changes as Texas Airlines will have a new reaction curve, while American will still calculate its optimal output using the same equation (QAA = 30 – ½ QTA) Texas Airlines’ problem is to set the Marginal Revenues equal to the new Marginal Costs.

In numbers: MRTA = MCTA  100 – 2 QTA – QAA = 25  The new reaction curve is : QTA = 37.5 – ½ QAA The new Cournot-Nash equilibrium is therefore given by: QTA= 37.5 – ½ QAA = 37.5 – ½ (30 – ½ QTA) = 37.5 – 15 + ¼ QTA .75 QTA = 22.5 QTA = 30 and QAA = 30 – ½ * 30 = 15 Total output will be: Q = 30 + 15 = 45 P = 100 – 45 = 55 economicshomeworkhelper.com economicshomeworkhelper.com

ΠAA = P*QAA – TC(QAA) = 55 * 15 – 40 * 15 = 225 ΠTA = P*QTA – TC(QTA) = 55 * 30 – 25 * 30 = 900 iii. If TA licenses the process to AA, both firms have the above reaction curve Qx = 37.5 – Qy/2. The new Cournot-Nash equilibrium is therefore given by: QTA = 37.5 – ½ QAA = 37.5 – ½ (37.5 – ½ QTA) = 37.5 – 18.75 + ¼ QTA .75 QTA = 18.75 QTA = 25 and QAA = 30 – ½ * 30 = 25 Total output will be: Q = 25 + 25 = 50 P = 100 – 50 = 50 ΠAA = P*QAA – TC(QAA) = (50-25)*25 = 625 ΠTA = P*QTA – TC(QTA) = (50-25)*25 = 625 Texas Airline loses 275 in profits. That is the minimum price it is willing to accept to share its technology. American gains 400 in profits. That is the maximum price it is willing to pay. So the firms should agree on a license for a price somewhere between 275 and 400.

(Note that it is not always the casethat firms are willing to share technology. It is easy to see that with perfect Bertrand competition, a firm would never want to share its technology, although it would be beneficial to society as a whole.) economicshomeworkhelper.com economicshomeworkhelper.com

b. i. The gross profit equations are: LAL = 3.9 (pLAL - 841) (400+pKL-pLAL) KL = 3.9 (pKL - 883) (600+pLAL-pKL) pLAL = (400+841+pKL)/2 pKL = (600+883+pLAL)/2 ii. The first order conditions are: (400+pKL-pLAL) - (pLAL - 841) = 0 or (600+pLAL-pKL) (pKL - 883) = 0 or 3. Combining the two equations and

pKL = 1402.333

gives: pLAL = 1321.667 a. Here is the payoff matrix (in millions of dollars): Steele Magna Buy all Buy part Prompt Delivery Slow Delivery 2,3 -5,4 1,1 -1,2 b. Dominant strategies:  Magna: If Steele offers prompt delivery, Magma prefers buying all. If Steele does not offer prompt deliver, Magma prefers buying part. Thus, Magma does not have a dominant strategy economicshomeworkhelper.com economicshomeworkhelper.com

 Steele: If Magma orders all from Steele, then Steele prefers slow delivery. The same holds if Steele orders only part of the aluminum. Thus, Steele has a dominant strategy: to provide slow delivery. b. Given that Steele has a dominant strategy to provide slow delivery, Magna will only buy part of its aluminum from Steele. The Nash equilibrium will be (Buy Part, Slow Delivery), with a payoff of (-1,2).

4.

a.

Dale DN Small Large

DN $36m, $36m $40m, $30m $36m, $18m

Chip Small $30m, $40m $32m, $32m $24m, $16m

Large $18m, $36m $16m, $24m $0m, $0m

The Nash equilibrium in this game is that both Chip and Dale will conduct a small expansion. b. If Dale moves first, this is the resulting tree:

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Large (36,18)

DN (30,40)

DN Small Large

(36,36) (30,40) (18,36)

Small (32,32)

DN Small Large

(40,30) (32,32) (16,24)

Large (36,18)

DN Small Large

(36,18) (24,16) (0,0)

Dale will decide a large expansion and then Chip will choose not to expand. The resulting payoff will be (36,18). c. If Chip moves first, the result will be exactly the opposite (because the payoff matrix is symmetric). Therefore, Chip will decide a large expansion and Dale will choose not to expand.

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