Practice Exercises 5
Spring 2021
微观经济作业代写 If a seller of type θ sells his car (of quality θ) for a price of p, his utility is us(p, θ). If he does not sell his car, then his utility is 0.
1.Thetable below summarizes the conditional probabilities generated from the bilat- eral relationship between an agent (A) and a principal (P)
x1 = 40 x2 = 100
e1 0.5 0.5
e2 0.2 0.8
Suppose P is risk-neutral, with B(x − w) = x − w, while A’s utility is given by w0.5 − c(e) where c(e1) = 0 and c(e2) = c > 0. 微观经济作业代写
(a)Find the optimal contract for the full information benchmark case.
(b)Findthe optimal contract for the case in which e is not verifiable (assume w ≥ 0).
2.Considera salesman with utility function u(w, e) = w0.5 − e2, where w denotes the wage and e is his effort He can choose between e = 0 and e = 3, and his reservation utility is 21. There are three possible levels of sales, x: 0; 1000 and 2500. The conditional probabilities of each outcome are summarized in the table below:
The owner of the firm maximizes her expected benefits (sales revenue minus wage). The preferences of the two participants satisfy the assumptions of von Neuman- Morgenstern.
(a)Findthe optimal contract in the full information case.
(b)Findthe least cost contract for implementing e = 3 when effort is not observable to the owner (but where sales are observable and verifiable).
(c)Whatis the optimal contract when effort is not observable?
3.Consider the following market for used cars. There are many sellers of used Each seller has exactly one used car to sell and is characterized by the quality of the used car he wishes to sell.
Let θ ∈ [0, 1] index the quality of a used car and assume that θ is unformly distributed on [0, 1]. If a seller of type θ sells his car (of quality θ) for a price of p, his utility is us(p, θ). If he does not sell his car, then his utility is 0. Buyers of used cars receive utility θ − p if they buy a car of quality θ at price p and receive utility 0 if they do not purchase a car. 微观经济作业代写
There is asymmetric information regarding the quality of used cars. Sellers know the quality of the car they are selling, but buyers do not know its quality. Assume that there are not enough cars to supply all potential buyers.
(a)Argue that in a competitive equilibrium under asymmetric information, wemust have E(θ|p) = p. (Consider the shortage of cars relative to demand and suppose
that a used car is traded at a price p < E(θ|p) or at p > E(θ|p)).
(b)Showthat if us(p, θ) = p − 
, then every p ∈ (0, 0.5] is an equilibrium price.
(c)Findthe equilibrium price when us(p, θ) = p − 
. Describe the equilibrium in words. In particular, what types of cars are traded in equilibrium?
(d)Findan equilibrium price when us(p, θ) = p − θ3. How many equilibria are there in this case?
4.Consider an insurance model with moral hazard. Suppose the consumer’s utility functionover final wealth, w, is u(w) = w0.5 and let his initial wealth be w0 = $100. Suppose that there are two potential loss levels, l = 0 and l = 51, so that without insurance, final wealth may end up being w = w0 − l = $100 or w = w0 − l = $49. 微观经济作业代写
The consumer can choose two effort levels, e ∈ {0, 1}. The consumer’s disutility of effort is given by the function d(e) where d(0) = 0 and d(1) = 
.
The insurance company offers a contract (p, Bl=0, Bl=51) that specifies the price p paid by the consumer to the insurance company and the benefit paid by the insurance company to the consumer contingent on the loss level; i.e. Bl=0 is how much the insurance company will pay the consumer if l = 0, while Bl=51 is how much they will pay the consumer if l = 51. The insurance company seeks to maximize expected profit, defined as p − E(Bl). For a given effort level, e, the expected profit is p − πe(l = 0)Bl=0 − πe(l = 51)Bl=51, where πe(l) is the probability of loss level l when the consumer’s effort choice is e.
Finally, suppose that the probabilities of each loss level depend on which effort level the consumer chooses as follows. If e = 0, then l = 0 with probability π0(l = 0) = and l = 51 with probability π0(l = 51) = 
. Meanwhile, if e = 1, then l = 0 with probability π1(0) = and l = 51 with probability π1(51) = . That is, higher effort
lowers the chance of suffering the large loss.
(a)Suppose the consumer chooses no insurance (this is his outside option) but can choose which effort level to exert. What will be his expected utility from behavingoptimally in this case? (Note: the different effort levels induce different lotteries over final wealth for the consumer; which one would the consumer prefer and what is his expected utility from this effort?) 微观经济作业代写
(b)Show that if information is symmetric, then it is optimal for the insurance company(who seeks to maximize expected profit) to induce high effort by the consumer.
(c)Showthat the optimal policy in part (b) will not induce high effort if information is
(d)Findthe optimal policy when information is asymmetric.
(e)Showthat both the insurance company and the consumer obtain lower utility (strictly lower profits in the insurance company’s case, identical utility in the consumer’s case) under asymmetric information than under symmetric informa-tion.
5.Consideran auction environment in which there are just two bidders (n = 2). In a second-price all pay auction, the two bidders simultaneously submit sealed The highest bidder wins the object and both bidders pay the second-highest bid (i.e., the loser pays her own bid, while the winner also pays the loser’s bid). 微观经济作业代写
Suppose values are independently distributed on [0, 1] according to the same distri- bution, F (·) with density f (·) > 0 on [0, 1].
(a)Findthe unique symmetric equilibrium bidding function for the two-bidder case, and give an interpretation.
(b)Dobidders bid higher or lower than in a first-price, all-pay auction (i.e., everyone pays their own bid).
(c)Derive the seller’s expected revenue.
(d)Howdoes the seller’s expected revenue compare with the expected revenue in a first-price winner-pay auction?
6.Considerthe following hidden action model with three possible actions, {e1, e2, e3}. There are two possible outcomes, x1 = 10 and x2 = 0. 微观经济作业代写
The probabilities of x1 conditional on the three effort levels are p(x1|e1) = , p(x1|e2) = 
, and p(x1|e3) = .The agent has effort costs given by c(e1) = 
, c(e2) = 
, and c(e3) = 
.
Suppose the agent has utility function u(w, e) = √w − c(e) and let u = 0.
(a)Whatis the optimal contract when effort is observable?
(b)Show that if effort is not observable, the principal cannot induce the agentto choose e2. For what levels of c(e2) would the effort level e2 be implementable?
(c)Whatis the optimal contract when effort is not observable?
7.Consider the following interaction between a seller and a buyer. The seller can producea good of varying quality, q ∈ [0, 1] at a cost of c(q) = q2. The seller’s profit from the sale of a good with quality q at price p is given by p − c(q). 微观经济作业代写
The profit from not making a sale is 0. The buyer’s utility if she buys a good with quality q at pricep is equal to θq − p, where θ is the buyer’s type. Her utility if she does not buy the good is 0, as is her reservation utility. Assume that θ is drawn from a (commonly
known) Uniform distribution on [0, 1], but is privately observed by the buyer.
(a)Supposethe seller can only offer a single quality of the good, and a single price, so that the seller makes a take-it-or-leave-it offer to the buyer, (q, p).
(i)Writedown the firm’s expected profit as a function of p and q.
(ii)Forany given quality q, what is the price, p∗(q) that maximizes the seller’s expected profit?
(iii)Puttingthese together, solve for the quality level q∗, and the corresponding price p∗(q∗) that maximize the seller’s expected profifit.
(b)Nowsuppose the seller can offer a menu of contracts rather than a single price and quality By the revelation principle, the seller can use a direct mechanism (q(·), t(·)). Let v(θˆ, θ) denote the utility for a buyer of type θ from reporting θˆ to the seller, and let U (θ) = v(θ, θ) denote the payoff to type θ from reporting truthfully. 微观经济作业代写https://essay2u.com/%e5%be%ae%e8%a7%82%e7%bb%8f%e6%b5%8e%e4%bd%9c%e4%b8%9a%e4%bb%a3%e5%86%99/
(i)Write down what it means for the mechanism (q(), t(·)) to be Bayesian incentive compatible. Then prove that if the direct mechanism is Bayesian incentivecompatible,the quality function q(·) is nondecreasing in θ.
(ii)Next, prove that if (i) q() is nondecreasing in θ and (ii) U (θ) = U (0)+
q(x)dx for all θ ∈ [0, 1], then the direct mechanism (q(·), t(·)) is incen-tive compatible.
(iii)Inthe expression, U (θ) = U (0) +q(x)dx, give an interpretation for the second term,q(x)dx.
(c)Solve for the optimal direct mechanism (menu of contracts) in the setting of part(b).
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