Health economics 1st edition bhattacharya solutions manual download

Health Economics 1st Edition

Bhattacharya Solutions Manual

Full download at link: https://testbankpack.com/p/solutionmanual-for-health-economics-1st-edition-bhattacharya113702996x-9781137029966/

8

Adverse Selection: Akerlof’s Market for Lemons

Review the basic assumptions of the Akerlof model before answering these questions. Many exercises will refer to these basic assumptions.

Comprehension Questions

Indicate whether the statement is true or false, and justify your answer Be sure to state any additional assumptions you may need.

1. In the Akerlof model, suppose that the price of used cars is P and the quality of used cars (X) held by sellers varies between 0 and 100. Suppose further that sellers’ utility is given by

US = M + a

where M is the number of units of video games, which sell at $1 per game, and a is a utility function parameter that is strictly less than one (a < 1). Then sellers will offer cars with quality Xi = P on the market.

TRUE. If a < 1, then sellers get more utility from P dollars than from a car with quality P. Therefore, they will offer cars of quality P for sale.

2. In the model, buyers know the utility function of sellers, but do not know anything about the general quality of cars for sale.

FALSE. Buyers do not know anything about the quality of a particular car, but they know how the quality distribution of all cars.

3. If buyers care sufficiently more about cars than do sellers, then there are prices at which transactions can occur. In that scenario, there is no longer any adverse selection (although there still may be some information asymmetry).

FALSE. There are prices at which transactions can occur under these conditions, but there is still adverse selection because worse cars are offered for sale and better cars are not.

4. The Akerlof model indicates that government intervention is the only way to solve the adverse selection problem.

FALSE Adverse selection will occur as long as an information asymmetry persists. If a government or other organization – like a consumer watchdog group or even a private company – dismantles the information asymmetry, adverse selection will cease.

5. If the quality of cars is normally distributed rather than uniformly distributed, the market will not unravel.

FALSE The market may still unravel under these conditions.

6. Ultimately, the market unravels because buyers are risk averse. If buyers were risk neutral, there would always be prices at which cars would sell.

FALSE We are already assuming that buyers are risk neutral. It is true that the market would be more likely to unravel with risk-averse buyers, because uncertainty about car quality is more likely to dissuade them from buying cars.

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

Analytical Problems

7. Review the basic assumptions of the Akerlof model. Assume that, in this market, the quality of cars Xi is distributed as follows:

Xi ∼ Uniform[q1, q2]

Note that in the discussion above, we analyzed the version of the Akerlof model where q1 = 0 and q2 = 100.

(a) Let q1 = 0 and q2 = 50 Will any cars sell in this market? Explain your reasoning carefully.

Assume price 0 ≤ P ≤ 50. The expected quality of cars placed on the market is

The utility to buyers of that average car is:

Because this value is less than P, buyers expect to lose utility when they purchase a car and will not purchase cars.

(b) Let q1 = 0 and q1 = 200 Will any cars sell in this market? Explain your reasoning carefully. Does raising the maximum quality of cars that sellers possess have any effect on predictions of the model? Explain why or why not.

Assume price 0 ≤ P ≤ 200 The expected quality of cars placed on the market is

The utility to buyers of that average car is:

Because this value is less than P, buyers expect to lose utility when they purchase a car and will not purchase cars. This is exactly the same calculation as in the previous problem. Changing the value of q2 does not affect this calculation, so it cannot have any effect on whether cars sell.

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

(c) Let q1 = 50 and q2 = 100. Will any cars sell in this market? Explain your reasoning carefully. Does raising the minimum quality of cars that sellers possess have any effect on predictions of the model? Explain why

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

or why not.

Assume price 50 ≤ P ≤ 100. The expected quality of cars placed on the market is

The utility to buyers of that average car is:

Compare this utility gain to the utility loss from losing P dollars. We know that P ≤ 100, so 1

25

37.5 This means

The utility from purchasing a car outweighs the cost, so buyers will purchase cars. Raising the minimum quality of cars does effect buyers’ willingness to buy cars, because there are fewer bad cars on the market at any price. This raises the expected utility of buying a car.

8. In Section 8.4 we studied the case of a government-mandated ban on the sale of low-quality cars. Review the basic assumptions of the Akerlof model, assume that car quality Xi is distributed uniformly from 0 to 100, and assume that the buyer and seller utility functions are as originally supposed.

(a) Let the government-mandated minimum quality be denoted as B. If B = 50, this means car i can only be offered on the market if Xi ≥ 50 What is the range of prices, if any, that would allow transactions if B = 50?

We already showed than any price between 50 and 100 will lead to transactions. Now we explore prices above and below this range. Assume price P > $100. The expected quality of cars placed on the market is

= 100 + 50 =

The utility to buyers of that average car is:

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

As long as P ≤ $112.5, buyers will purchase cars.

Now assume price P < 50. We know that no sellers will offer cars if the price is this low So the range of prices at which transactions will occur is: 50 ≤ P ≤ 112.5.

(b) What if B = 90? B = 5?

A very similar calculation finds that, for B = 90: 90 ≤ P ≤ 142.5 and for B = 5: 5 ≤ P ≤ 15.

(c) Find the smallest B for which you can still find prices that will allow transactions to occur. If there is no minimum, explain why not.

There is no minimum B at which cars will sell. If B = 0, cars will not sell (this is the basic version of the Akerlof model). But for any small B (0 < B < 1 for example), cars will sell at price P = 3B.

9. Consider again the example where buyers value cars much more than sellers. We assume again that

Recall that, under these assumptions, any price P such that $0 < P ≤ $100 induced at least some sales.

(a) Will there be any transactions if P = $150? Why or why not?

The expected quality of cars placed on the market is:

The utility to buyers of that average car is:

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

Cars will not sell because 125 < 150.

(b) What is the maximum price P at which at least some transactions will occur?

Based on our calculations in the last question, we know that cars will sell as long as P ≤ 125.

(c) Suppose instead that the buyer and seller utility functions are given by

where h reflects how much more buyers value cars than sellers. What is the maximum price at which at least some transactions will occur, in terms of h?

We know that this maximum price Pmax is greater than 100. The expected quality of cars placed on the market is

The utility to buyers of that average car is:

Cars will still sell whenever 50h

(d) Assume further that the utility functions are still given by Equation 8.16, but that car quality X is distributed as follows: Xi ∼ Uniform[0, G]

where G > 100 is a distribution parameter Now, what is the maximum price at which at least some transactions will occur, in terms of h and G?

We know that this maximum price Pmax is greater than G. The expected quality of cars placed on the market is

The utility to buyers of that average car is:

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

(e) Interpret your findings about how the market’s functioning changes with h and G in non-mathematical terms.

For a given car distribution, transactions will occur at higher prices if buyers value cars more (higher h). For a given buyer valuation of cars, transactions will occur at higher prices if the overall car distribution includes more high-end cars (higher G).

10. Assume that instead of a uniform distribution, in this market, the quality of cars X follows a triangle distribution from 0 to 100 as depicted in the figure below.

Given this new distribution, the formula for expected value of X conditional on an upper bound P is:

We assume the original seller and buyer utility specifications:

(a) First assume P = $50. Will any cars sell in this market?

The expected quality of cars placed on the market is:

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

The utility to buyers of that average car is:

Because this value is equal to P, buyers will purchase cars.

(b) What is the range of prices for which cars will sell?

An examination of the calculation in the last exercise reveals that any P < 100 yields the same result. So cars will sell for any P between 0 and 100.

(c) Describe in qualitative terms how the triangle distribution is different from the uniform distribution we assumed earlier. Do the sellers possess more high-quality cars or low-quality cars? How does this allow the market to function even in the face of asymmetric information?

In this distribution, there are relatively more high-quality cars and relatively fewer low-quality cars. That means that adverse selection is not as harmful, because the distribution of cars on the market is skewed towards high quality.

(d) Use the formula for conditional expectation below to derive the conditional expectation from exercise 10. Note that f(x) is the probability density function and FX(x) is the cumulative distribution function for the triangle distribution above.

To determine FX(x), we have to determine how much area under the curve is to the left of any given x. This question translates to: what fraction of the cars are under quality x? Using the formula for the area of a triangle:

Next we plug P into the FX function:

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

and plug it all into the integral:

After some canceling and rearranging, we find:

(e) Now assume a different triangle distribution for car quality X as depicted in the figure below Without doing any calculations, predict whether Alternative triangle distribution.

the market will unravel. Justify your answer.

Unlike the previous distribution, this distribution is skewed toward bad cars. At any price, there will be more bad cars on the market and very few goods cars to balance them out. This market is likely to unravel.

11. Suppose a local car dealership is offering an inspection service that can perfectly determine the quality of any car on the market. The dealership is trying to determine how much it will be able to charge for this procedure. Review our original assumptions about the Akerlof model. Specifically, we assume the buyer and seller utilities from Equation 8.1, that car quality X is distributed uniformly from 0 to 100, and that the prevailing price P = 50

(a) Due to adverse selection, only some of the cars remain on the market (Ω(P)). How are these cars distributed, and what is their average quality?

c Bhattacharya, Hyde & Tu 2013 c Bhattacharya, Hyde & Tu 2013

These cars are distributed uniformly from 0 to P = 50. Average quality is 25.

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

(b) Suppose a buyer picks a car i at random from the cars still available on the market, and thinks about whether to purchase it. What is her expected change in utility from this transaction?

The utility to the buyers of that average car is: 3 3

so her expected change in utility is 37.5 50 = 12.5

(c) Even though her expected change in utility is negative, there is a possibility that she picked a relatively high-quality car and that her actual change in utility would be positive. What is the lowest value of Xi such that this will happen, and what is the probability that Xi is at least this high?

To find this minimum X value, we set the cost and benefits of the transaction equal:

3 50 = X 2 33.333 = X

To find the chance that a randomly-picked car has quality of at least 33.333, we find the fraction of cars on the market that are of this quality level: 50 −33.333 1

(d) Now suppose the local car dealership decides to offer this buyer its inspection service for free, before she decides whether to purchase the car. She resolves to purchase car i if it will increase her utility, and refrain from purchasing it if it will not increase her utility. Now, what is her expected change in utility from this transaction, considering that we do not yet know whether the car will be good enough to purchase? How does this differ from your answer in exercise 11(b)?

There is a two thirds chance that the car will be unsatisfactory and she will not purchase it, so there is a two thirds chance that her utility change is zero. But there is a one third chance that the car will be satisfactory and she will decide to purchase it for 50. Her expected utility in this case is:

3 50 + 33.333 3

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

So her overall expected utility change is:

This is a higher expected change in utility than in exercise 11(b), and it is positive.

(e) How much will the dealership be able to charge for the inspection service, assuming the price remains at $50 and that all other assumptions are constant? Justify your answer.

In the absence of the service, buyers do not purchase any cars and have no utility change. With the service, they enjoy an average utility gain of 4.167 Therefore, they should be willing to pay $4.167 for each inspection.

(f) Imagine that word about the inspection service gets out to everyone in the market, including sellers. How will sellers with high-quality cars react? How will sellers with low-quality cars react? Will there be any adverse selection in the market?

Sellers with high-quality cars will advertise their cars’ high quality, and they may even offer to pay for inspections to prove the worth of their cars. Sellers with low-quality cars will have a much harder time pretending their cars are as good, because they will be worried that buyers will take their car to the inspectors. Adverse selection may decrease or disappear when buyers have a way to get information cheaply about the cars on the market.

12. We assume again the original assumptions of the Akerlof model, but now alter the information assumptions in this exercise. Now assume that neither buyers nor sellers have information about specific car quality, although each group knows the distribution of Xi as before.

(a) Assume that there is a market equilibrium price of P = 80 What set of cars will be offered in this market? In other words, what is Ω(80) under these assumptions?

Every seller will offer his car, because each seller estimates his car’s worth at 50, which is less than the price.

(b) Does adverse selection occur in this situation?

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

No. Every car – bad and good – is equally likely to be offered up at market.

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

(c) Derive a more general expression for Ω(P) For any given P, what is the set of cars that will be offered?

If P ≥ 50, all cars are offered. Otherwise, none are offered.

(d) What is the range of P for which at least some cars will be offered and at least some cars will be purchased? Remember that sales do happen when buyers and sellers are indifferent.

Transactions will not occur if P < 50, because sellers will not offer their cars. Transactions will not occur if P > 66.667, because then the price will be greater than the buyers’ expected utility from purchasing a randomly-selected car. Therefore, the range where sales occur is 50 ≤ P ≤ 66.667

13. Thirdhand car markets (Question courtesy of Kyna Fong). Consider now two sequential used-car markets. The first is a market for secondhand cars (one previous owner). The second is a market for thirdhand cars (two previous owners).

In the first market, there are two types of traders: Original Owners and Original Buyers. Original Owners own all the cars and have the following utility function:

where M is the amount of non-car goods consumed, xj is the quality level of the jth car, and a is a parameter in the utility function.

Original Buyers own no cars and have the following utility function:

where again M is the amount of non-car goods consumed, xj is the quality level of the jth car, and b is a parameter in the utility function. There is a uniform distribution over the quality of all cars, with

Let P1 be the price of used cars put up for sale by Original Owners in equilibrium. Original Owners know the quality of the cars they are selling, but

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

Original Buyers only know the average quality of the cars on the market.

Original Buyers know the utility function of the Original Owners.

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

(a) What will be the average quality of cars offered on the market by Original Owners under these conditions, as a function of P1 and a?

Notice that Original Sellers offer cars only if their utility from the car is less than the price:

. The expected quality of cars placed on the market is:

This means cars are only offered if

(b) For what values of b will Original Buyers be willing to buy the cars, as a function a?

Original Buyers are willing to buy if the utility from the average car is greater than the price:

Plugging in the result from the previous problem, we find:

(c) Suppose b satisfies the condition you found in the previous question, and some cars are sold from the Original Owners to the Original Buyers. Now the second market takes place. The Original Buyers become Secondhand Owners and they are in a thirdhand car market with new buyers, which we call Secondhand Buyers. The Secondhand Buyers know all about the first market for secondhand cars.

The Secondhand Owners, after driving around their new cars, have learned the quality of the cars, but the Secondhand Buyers only know the average quality of the cars on the market. The Secondhand Buyers own no cars and have the following utility function:

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

where c is a parameter in the utility function. Let P2 be the price of used cars put up for sale by Secondhand Owners in equilibrium. For what values of c will Secondhand Buyers be willing to buy the cars, as a function of b?

First recognize that the cars that the Original Buyers/Secondhand Owners own are distributed from 0 to P1/a, because that was the distribution of cars offered on the secondhand market. Now, Secondhand Sellers offer cars only if their utility from the car is less than the price:

This means cars are only offered if

. The expected quality of cars placed on the market is:

Secondhand Buyers are willing to buy if the utility from the average car is greater than the price:

Plugging in the result from the first part of this problem, we find:

(d) Suppose b = 3. For what price P2 will the cars available on the thirdhand car market be uniformly distributed between 0 and 50?

The top end of the distribution on the thirdhand market was P2/b, so:

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

(e) Suppose also that c = 5 Will the Secondhand Buyers buy any cars in equilibrium if car quality is distributed uniformly between 0 and 100?

No, c would have to be at least 6. Remember that we showed cars will only sell in the thirdhand market if c > 2b.

Essay Questions

14. Assume that the market for lemons has unraveled, as it did in several of the above examples. Who is harmed by the existence of asymmetric information? Who is helped?

Anyone who would have found a partner to transact with if the same market had occurred with perfect information is harmed, because now those transactions cannot occur. No one is helped – not even the sellers with bad cars.

15. How can we be sure that the price P is the same for all vehicles? Why can’t sellers with excellent cars not simply advertise their high car quality and charge higher prices?

Sellers with excellent cars may try to do that, buy buyers have no way of verifying this information. Given this, sellers with terrible cars will lie and pretend to have great cars as well. If buyers cannot tell the difference between any pair of cars, they cannot possibly sell at different prices.

16. The Akerlof model can be used to model the health insurance market. In this market, which party is analogous to car buyers? which party is analogous to car sellers? What would it mean for the health insurance market to unravel?

The insurance firms are analogous to car buyers, and the insurance customers are analogous to sellers. The health insurance market unravels if no insurance company is willing to offer an insurance contract at any premium for fear of attracting the sickest customers. This is analogous to buyers refusing to buy cars at any price for fear of buying the worst cars. See section 8.3 for details.

17. What happens when we reverse the information assumptions in the Akerlof model? Let us assume that buyers have perfect information about car quality, and that sellers have no information about the quality of any specific car (although they do know the distribution of car quality). Assume that all other

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c Bhattacharya, Hyde & Tu 2013

basic assumptions apply as usual, including the buyer and seller utility functions.

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013

(a) Explain how these information assumptions might be possible in certain circumstances. What sorts of goods would likely have markets that feature these counterintuitive assumptions?

Consider a market in valuable or rare goods, like paintings or antique furniture. The people who own these things may not realize their true value, but expert buyers in these markets do know their value.

(b) Imagine that you are a car seller who owns car i with quality Xi (unknown to you). What strategy could you pursue to sell the car in such a way that your utility increases?

You could auction off your car That way, you do not have to worry about being ripped off – if someone gives you a bad offer, another buyer will step in and make a better offer. You do not need to know the value of your car; you simply rely on the buyers to compete with each other and bid up the price until the price is commensurate with quality.

(c) Does adverse selection occur in this market?

Adverse selection would occur if sellers offered their cars for fixed prices. Then savvy buyers would buy only the good cars, leaving the bad cars in the hands of the sellers.

c Bhattacharya, Hyde & Tu 2013

c Bhattacharya, Hyde & Tu 2013