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ECON 510: Quiz 3 (take-home)
Student’s Name: Student’s I.D.: 1. This is an individual assignment. You are not allowed to discuss between each other. 2. You can use whatever material you want. 3. The deadline is: 6 PM 2 May (Saturday). 4. Submit via: https://forms.gle/BtsweZeHKJr3UF5s9
Problem 1. Screening: Consider the market for used cars. There are lemons (low quality cars) and cream puffs (high quality cars). You are provided with the following valuations of buyers and sellers of each type:
Value to Seller
Value to Buyer
Lemon
1
3
Cream Puff
7
13
The value to the buyer represents the maximum price she will pay for a car of the given type. Assume buyers are risk neutral. (e.g. if a car has a ½ probability of being a lemon and a ½ probability of being a cream puff the buyer would be willing to pay at most ½*3 + ½*13 = 8.) The seller’s valuation is the minimum price for which she will sell the car.
a. Suppose there is only one price for cars. Compute the maximum proportion of lemons that the market can sustain (i.e. so that cars of both types will be traded)?
b. Now car inspection is possible for a fee, and different prices are allowed for each type: P_cream puff=11, P_lemon=2. The inspector’s word is always taken as true. If the inspector says a car is a lemon, the buyer believes her. But the sellers know that the inspector is only right 90% of the time when inspecting a cream puff and 80% of the time when inspecting a lemon. What fee, if any, can be set that will allow the two types of cars to separate themselves (i.e. buyers willing to buy lemons at the lemon price, and sellers willing to sell at that price; and the same for cream puffs)?
Problem 2. Screening: In class, we considered price discrimination via quality. The same model also works for price discrimination via lump sum discount (buy more at a lower price per unit). Consider a small yogurt shop that sells fresh-made yogurt to customers. The shop owner is facing two types of customers and is currently designing a price menu and want to max. expected revenue by giving a larger discount to the customers who buy more.
Buyer’s utility function is given by: 𝑢(𝑣,𝑞,𝑡)=𝑣⋅𝑞−𝑝⋅𝑞
In which:
o Her willingness to pay for quality: 𝑣∈{1,2}.
o q: quantity of yogurt she is going to buy
o p: the price she has to pay for each unit she buys
Seller has cost of production for each customer: 𝑐(𝑞)=𝑞2
The profit is given by: 𝑝⋅𝑞− 𝑐(𝑞)
Seller does not know buyer’s willingness to pay but knows the distribution:
𝑃𝑟𝑜𝑏(𝑣=1)=45; 𝑃𝑟𝑜𝑏(𝑣=2)=15 .
Solve for the second degree price discrimination pricing strategy. [Hint: Seller wants the low type to participate and does not want high type to pretend to be low type.]
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Problem 3. Adverse Selection and Insurance:
In Clintonia, every citizen has a wealth of 16 and a utility represented by 𝑈(𝐼) =√𝐼 . Clintonians occasionally get sick, and when they do, they go to the hospital and are made immediately better. A hospital visit costs 16. There are two types of Clintonians and each has a different probability of getting sick (types are genetic and cannot be changed):
Type
Probability of Illness
Vegetarian
0.25
Smoker
0.75
a. In the absence of health insurance, find the expected utility of each type of Clintonian.
b. A private health plan, Green Cross, is set up. Membership is optional, but since everyone is risk averse, the managers assume that everybody will join (for an insurance with full coverage). Green Cross sets their premiums according to the chance that an average Clintonian gets sick (Green Cross cannot distinguish between types). Assume there are equal proportions of each type in the population. What is the actuarial fair premium (i.e., the insurance policy at which the insurance company breaks even)?
c. If Green Cross charges the actuarial fair rate, will every type of Clintonian join the plan? What is the expected utility for each type?
d. After the 1st year of operation, Green Cross finds they have lost money. Why? They commission a study to discover the probability that a member of Green Cross gets sick. What is the probability? They fix a new actuarial fair premium based on their study. What is the new rate? Who joins the plan in the 2nd year?
e. Suppose now that Green Cross gets smart and offers two plans. The first plan is based on the plan in (d). Call it Plan I. The second plan offers a co-payment system (Plan II). For 3, Plan II pays 5 if an individual gets sick and the individual pays the rest. Who buys which plan? Why? Are these plans sustainable (i.e. the firm is not losing money)? How does this strategy solve the asymmetric information problem?
h. Suppose now the government comes up with a plan that mandates health insurance. What rate will be charged (the rate will be the same for everyone and the government covers its costs)? Who gains and who loses under this plan versus the plan in (g) (i.e. who votes for the plan)?
j. Suppose now that Steven Segal invents a new method to quit smoking (“boot-to-the-head therapy”). This therapy is so effective that it overcomes all genetic predisposition to smoke. For a cost of only 10 you never become sick again (probability of getting sick becomes zero). If the private plan is in effect (people can choose Plan I or Plan II), will smokers undergo the Segal therapy? Suppose the government plan is in effect, will smokers undergo the therapy? Explain.
Problem 4. Stackelberg Competition
Suppose the market of fried chicken is dominated by two oligopolists: Macdonald and KFC.
The inverse demand function is: P=20-Q.
Production cost of Macdonald is: 2Qm2 .
Production cost of KFC is: Qk .
Suppose Macdonald is the first mover. Find the equilibrium of the Stackelberg game.
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Problem 5: Moral Hazard (Principal-Agent Problem)
Consider a firm that hires people to load and unload shipping trucks. The company cannot perfectly observe whether each individual worker works or shirks (i.e., not do their job), all the firm can observe has an insignificant effect on the overall outcome for the firm. The firm does however care about the effort level of their pool of workers so they do want to induce them to actually work. Due to the fact that the single worker has no measurable impact on the outcome for the firm, the company cannot make wage offers contingent upon the outcome. The only thing a firm can do is to offer a fixed wage and then fire any worker they observe to be shirking. Their observation is imperfect though and so the firm only observes a worker shirking with some probability. The firm also occasionally makes mistakes and even fires workers who were working. The structure of the game is that:
(1) the firm offers a wage contract,
(2) the worker choose to accept or reject (if they reject they receive some level of reservation utility 𝑢̅=10 and the firm receives 0),
(3) If the worker chose to shirk he is fired with probability p=0.5 and has no effort cost while if the worker did not shirk there is no chance that he will be fired, but he pays an effort cost e=5.
(4) If a worker gets fired, he does not receive his wage but instead receive his reservation utility of 𝑢̅=10. The utility for a worker in any state is simply the wage he receives less the cost of any effort.
Assume the firm’s revenues are such that they can profitably pay the wage required to induce high effort but they lose money if all their workers shirk.
a. Begin with a simpler case in which we will assume that the firm could actually contract on effort level. What wage contract would the firm offer to a worker if the firm wants to elicit high effort?
For the following parts, reassert all the original structure under the assumption that effort level is unobservable!
b. What are the conditions that must be satisfied to elicit high effort?
1. What is the lowest wage that satisfies IR condition?
2. What is the lowest wage that satisfies both conditions?
3. Do the wages you found in b.1 and b.2 differ. If yes, why? If no, why? (Briefly explain it).
c. (*Small bonus question) Besides firing a person who shirks, what else can the firm when designing contracts in order to elicit high effort? Be very brief.
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