205microeconomics

3. The following matrix describes strict preferences of student interns and hospitals where the students might work. The rst numbers represent the preferences of students and the second numbers these of hospitals. Here, 1 represents the top choice, etc. Every hospital prefers any student to being unmatched and every student prefers any hospital to being unmatched. H1 He He 51 1,2 2,2 3,1 a. Dene a stable match in this case. b. Apply the student proposing deferred acceptance algorithm and the hospital proposing deferred acceptance algorithm and nd the resulting matches. o. Compare the outcomes in part b according to the preferences of hospitals and according to the preferences of the students. d. Is the match {SJ-I1? 32H3,53H3} stable? Prove that your answer is correct. 2. Consider an economy with one rm, two consumers and two goods: food and time for either labor or leisure. IConsumer 1 initially owns 4 units of time and gets utility 1.1031, c1) when E, 3 {I is its leisure consumption and c, 23 is its food consumption. The function u is twice continuously diferentiable, homogeneous of degree 1, and strictly quasiconcave, with strictly positive partial derivatives. Consumer 2 owns the rm but no goods initially; it gets utility equal to its food consumption as I: U. The rm producm t} 2 10sr s2 9 units of food when it uses a E [1, 5] units of labor input and He} = {i if i] E s s". 1. The feasible allocation with (8,, c1] = {2, 7) is the only i'areto eicient (PE) allocation with c; = ID. 3.. Compare f'{s) and sh's, the marginal and average products of the firm at c = 3,}? and :2: = 4 and use the comparisons to describe the local returns to scale of the rm at input levels near 2 and 4. b. Graph the set of feasible allocations, represented by the feasible consumption vectors for consumer 1. Show the PE consumption vector (2, T} for consumer I with a. possible indier ence curve through it. Prove that there is no price quasiequilibrimn with transfers with this PE allocation. {For this economy, in a price quasi-equilibrium with transfers, the rm max\" imises its prot and neither consumer prefers a cheaper consumption vector.) Find prices and transfers such that, in this allocation, both consumers maximise their utilities subject to budget constraints and the rm satises the rst order condition for prot maximisation. c. In the graph of part b, Show how the whole set of PE allocations with El 2::- U can look. Be sure your graph is compatible with all the assumptions above. Prove that for each level of input for the rm there is at most one PE allocation. Show that Cl ,' 1 *4: W2 in each PE allocation with 1 2':- l}. d. Find all the PE allocations with El :> 0 that are part of price equilibria with transfers. What can be said about the transfers? Which consumer gives and which consumer receives a transfer payment? e. Prove that this economy does not have a competitive [Walrusian} equilibrium. Explain why not. [The proof can use the answer to part d.] 3. A monopolist M has two units of a durable good that it wants to sell to two consumers, H and L. There are two periods and the monopolist posts price 331 in period 1 and if necessary posts price p; in period 2. There is no {additional} cost in providing the units so the monopolist attempts to maximise the total [undiscounted] revenue from the two periods. If H buys the good in the rst period at p1 {and so gets to use it in both periods}, his payoff is $9013 _. 101. If he buys it in the second period at 352, his payoff is $3 - pg. If he does not buy in either period, his payoff is 0. Consumer L gets payo $5M p1 if he buys the good in period 1, $205 P2 if he buys it in period 2, and $3 if he does not buy in either period. Additional continuiption of the durable good adds nothing to the commonersT payois. in period 1, the monopolist posts price 391. Consumers H and L make simultaneous decisions whether to buy the durable good (B) or not buy {N}. If both oonsumere buy in period 1, the game ends there. If at least one consumer does not buy in period 1, the game moves to the second period and the monopolist posts 502. Any consumer(s} who did not buy in period 1 mice {simultaneous} buy-or-nothuy decision{s) in period 2 after learning 39;. Assume that a consumer buys the good if he is indifferent between buying and not buying. Denote a period 1 history as pIBN if M chooses p1, H buys, and L does not buy in period 1 and similarly for other histories. Whenever possible. justify your answers and prove that they are correct. A rm manager has already offered to hire a worker. IThe worker knows that working for the rm is either pleasant or unpleasant and initially believes that it is pleasant with probability p E (El. 1}. The manager knows if thejob is pleasant or unpleasant and has no way to affect the pleasant ness of the job for the worker. Before the worker decides whether to accept or reject the job oer. the manager can give the worker a gift. Suppose rst that the manager does not give the gift. If the worlmr does not work for the rm._ then both the manager and worker get payon I} from the interaction. If the worker works for the rm. then both get payoff l.li|I if working for the rm is pleasant; if it is not._ then the manager gets payoff 5 and the worker gets payoff -1. If._ instead. the manager gives the gift. then it raises the worker's payo" by -1 units no matter what the worker decides. [The worker keeps the gift whether she accepts or rejects the job offer.) Giving the gift reduces the manager's payon by c : 1] no matter what. In other words. no matter what the worker decides. c is subtracted from the manager's payon specied above. Before deciding whether to accept the job. the worker knows if a gift has been given. The manager and worker are both rational expected payo' maximizers with common knowledge of the entire interaction described above. {In particular. the manager knows the value of P-l a. The above interaction can be represented as a sequential {extensive form] game. Draw a complete game tree representing this game. The tree should include all the possible choices of the manager and worker given their respective information. Dene all the notation you introduce in labeling the tree. b. How many pure strategies does the manager have'ir Give an example of one. c. How many pure strategies does the worker have\"? Give an example of one. d. Suppose that. for some values of the parameters 3:: and c. there is a pure strategy sequential equilibrium {SE} in which the manager never gives a gift under any circum stances. 'I'b formulate the SE. it is still necessary to specify the worker's beliefs in case the manager did give a gift. Show that sequential equilibrium places no restriction on those beliefs. c. Find all parameter 1values of the game in part a such that there is a pure strategy SE in which the manager never gives a gift under any circumstances. Describe the SE completely when it exists and show that it is SE. f. Find all parameter values of the game in part a such that there is a pure strategy SE in which the manager gives a gift if and only if working for the rm is pleasant. Describe the SE completely when it exists and show that it is SE. Explain why the equilibrium strategies of manager and worker are equilibrium strategies