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# Econ 157 Midterm Solution

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ECON 157 Midterm Solution

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Question 1 (45 points)
Consider a situation where a population of consumers is deciding whether or not to buy a street drug called Bearoin. A researcher observes these consumers’ purchases over two years, 1 and 2. The price of Bearoin changes from year 1 to year 2, as a result of an increased number of sellers in the market. The researcher observes the following prices of the drug, and quantities sold, in year 1 and year 2:
The year 1 price of the drug for consumers is P1 = \$80. The year 1 quantity consumed at that price is Q1 = 400
1. (5 points) Write down the formula for arc price elasticity of demand in terms of P1, Q1, P2, and Q2.
2. (10 points) What is the arc price elasticity of demand for Beroin?
3. (10 points) Now consider a second, new population of consumers buying the same drug in another location. In this new market:
The year 1 price of the drug for consumers is P1 = \$80. The year 1 quantity consumed at that price is Q1 = 240
The year 2 price of the drug for consumers is P2 = \$20. The year 2 quantity consumed at that price is Q2 = 1000
What is the arc price elasticity of demand for consumers in this second market?
4. (10 points) Now, there is a music festival where both of these populations come together and form one population. They have the same demand for Bearoin at the prices listed above in each market. What is the arc elasticity of demand for this combined population of consumers?
5. (10 points) The researcher now wants to use the arc elasticity computed for this combined population to determine what the quantity sold of the drug would be if she could enact a government crackdown so that the price of the drug at the festival rose from \$20 to \$120.
Using the arc elasticity computed above, and the quantities given above for P = 20, compute the quantity that will be purchased under this new price.
Question 2 (35 points)
1. (5 points) In the Grossman model, consumers face a time budget constraint as well as the usual financial budget constraint. Write out the time budget constraint for one day, and clearly list each type of time use that enters the model.
2. (10 points) Assume the consumer can do at most one of these exercises, and could also choose to do none of them (in which case there is no change to H or Z relative to baseline). If the consumer only considers his/her current period utility, which activity will she choose, if any?
3. (10 points) Now, assume that that consumer has interactions between health and the home good in their utility function, because being healthier specifically allows him / her to better enjoy other things being consumed. The new utility function is:
U(H,Z) = 3Z + 5H + 3(H ⇤ Z)
Assume that the consumer would have H = 5 and Z = 5 if they chose not to do any exercise.
Assume that the changes from any exercise chosen for H and Z are made relative to this baseline. Now which exercise will they choose, if any (5 points)? What is his/her utility for the current period, after making that choice (5 points)?
4. (10 points) Now, assume that instead of caring about static utility, consumers care about utility for the current period and the next period. The period-by-period utility function is the same in each period, and is as specified in the previous part of this question (part 3, with interactions). Assume that both periods are valued the same, i.e. there is no time discounting
( = 1).
The consumer chooses which exercise to pick (if any) in period 1 only, and there is no choice made for period 2 (i.e. no exercise is done in period 2). Assume that base consumption is
H = 5 and Z = 5 in each period (i.e. this is what happens if no exercise is chosen in period
1, and this is the base o↵ which changes to H and Z from exercise are made relative to).
Assume that the benefits from choosing an exercise for period 1 H and Z are as specified at the beginning of the question. Assume that there is no benefit from exercising in period 1 for the period 2 level of Z. Assume that the health benefits of exercising have some persistence for period 2, where they equal 50% of the incremental health benefits specified for period 1.
Which exercise choice do consumers make (if any) (5 points)? What is their combined utility over both periods with that choice (don’t forget to account for their base consumption each period) (5 points)?
Question 3 (50 points)
Consider a consumer with the following utility function for income I:
U(I) = pI
Think about the expected utility model that we studied in class and in chapter 7. This consumer has a probability of p = 0.25 of becoming sick and (1-p) = 0.75 of being healthy. If the consumer is uninsured, when sick she has income of IS = 16. If uninsured, but healthy, the consumer has income
IH = 144.
1. (5 points) Write down the formula for expected utility, in terms of p, 1−p, IS, and IH. Don’t insert the specific numbers yet, just give the general formula given these variables
2. (10 points) Using the exact numbers given in this problem for these variables, compute this consumer’s expected utility if uninsured.
3. (10 points) Using the exact numbers given in this problem, compute the utility gain to consumers of a fair and full insurance contract, relative to being uninsured. This means: compute the utility this consumer has under a fair and full contract, and subtract from this their expected utility if uninsured.
4. (10 points) Assume that a monopolist insurer o↵ers the consumer a full insurance contract, but as an unfair contract that extracts the most possible money from the consumer. What is the premium of this insurance contract?
5. (5 points) If no insurance contracts are available, i.e the consumer will remain uninsured apart from the government subsidy, what is the increase in utility provided by the subsidy for this consumer? 6. (10 points) Assume that an insurer can o↵er a partial insurance contract with a payout of
IH−Is
2 . Assume that insurance payouts occur before the government allocates subsidies to consumers: if your income with the insurance contract considered is greater than 36, you get no subsidy. If the partial insurance contract is fairly priced, what is total consumer expected utility if they buy this contract (5 points)? Will they buy the contract in this environment with the government subsidy (5 points)?
Question 4 (50 points)
Consider the Akerlof ’Lemons’ model discussed in class and discussed in Chapter 8 of the textbook.
Make the standard assumptions in that model that (i) sellers have private information and know the quality of their cars and (ii) buyers don’t know the quality of any specific car, and just know the probability distribution of car quality overall.
Buyers (who are risk-neutral, as in the standard model) have the following utility functions over quality Xj and money M:
1. (5 points) Consider a potential price in the market of P = 150. Don’t worry about how this price arises, just assume this is the price. What is the range of car quality that sellers will o↵er up for sale given this price? I.e. write down the lowest quality XL and highest quality
XH that sellers will o↵er for sale.
2. (10 points) Given this same potential price, P = 150, and the cars that sellers o↵er in this market, will buyers buy any cars at this price? Answer this question by providing (i) buyer expected utility for buying a car in this market given this price (5 points) and (ii) a yes or no answer about whether they will purchase any cars (5 points).
3. (10 points) Assume that a regulator is considering a quality standards policy that removes cars of quality less than some value X⇤ from the market. What is the minimum value of this minimum quality standard X⇤ such that any cars will be sold in the market if the price (for whatever reason) is P = 180?
4. (15 points) Assume that for whatever reason, the price in the market with these sellers is
P = 180. Specify the quality range(s) of cars that sellers will o↵er for sale in the market (10 points). What is the average quality of cars sold in the market at this price (5 points)?
5. (10 points, HARD) Assume that the cars not sold in the last part of this question (two types of sellers, P = 180) are able to break away from that market and form their own distinct market. These cars not sold in the overall market are the only ones that break away and the only ones that can be o↵ered for sale in this new market. Assume that seller utility functions remain the same as they were in the two-type market, and the buyer utility functions remain as have been throughout the question.
What is the minimum price at which at least half of the cars in this breakaway market are sold
(5 points)? What is the average surplus for buyers who actually buy cars in this breakaway market, at this minimum price where at least half of the cars are sold (5 points)?

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