Macroeconomics

1. First Welfare Theorem Let gpure = ({Xi, Zi, ei) -) be the standard pure exchange economy with free disposal, where X, = RL and > is locally nonsatiated for every i e I. Answer the following questions. (a) Define Walrasian equilibrium and Pareto efficient allocation in this econ- omy. (b) Prove that every Walrasian equilibrium allocation is Pareto efficient. (c) Suppose that I = {1, 2,3}. Suppose that consumer 1 and consumer 2 decide to trade exclusively with each other, effectively excluding consumer 3 from any exchange. Consumer 1 and 2 negotiate to come up with a pair of consumption vectors x],a, e Ry such that aft, s ente,. Of course consumer 3 just consumes her endowment es (or a part of it). Let (x*, p* ) ERY x R be any Walrasian equilibrium that would have realized if every consumer can participate in the market. Clearly consumer 3 is always (weakly) worse off by consuming es rather than a;. But is it possible that consumer 1 and 2 are better off negotiating with each other, i.e. a; Zix; for i = 1, 2 and > >1 2; for i = 1 or 2? If so, find such an example. If not, explain why.\f3. Subgame Perfect Equilibrium The diagram below shows a 4-player game. (a) Find all the subgame perfect equilibria in pure strategies. (b) Find all the subgame perfect equilibria (if any) in which player I plays a completely mixed strategy. (9/2,0,0,0) II (2,2/3,0,0) III (0,0,2,0) IV (3,2,3,4)4 Public Good Provision The government must decide whether to build a project that is of potential value to two rms. The cost of the project is c; the value to rm 1 is either 1 or I}, the value to rm 2 is either 2 or D; in each case the probability of a positive value is p [where ll :2\". 3? a: 1] and the probabilities are independent. Whenever the government decides to build the project it will divide the cost c between the rms but will never make a. prot or pro-mitts a subsidy. The government wants to use a socially efficient mechanism: that is: a mech anism that causes the project to be built if and only if the cost is less than the total value to the rms. [To avoid complications we will ignore cases where the cost might be exactly equal to the total value to the rms.) Notice that this is not a symmetric problem, so the mechanism(s} need not be symmetric either- [a] If 2 s: c s: 3, nd a socially efficient mechanism that is incentive oom patible and (interim) individually rational for the rms. [That is: the rms are willing to participate in the mechanism after they know their true values.) [b] If 1 c: c s: 2, nd a socially efcient mechanism that is incentive oom- patible and {interim} individually rational for the rms- {c} If D s: c 4:: 1, nd the region in cost c and probability p space for which there is a a socially e'icient mechanism that is incentive compatible and (in terim) individually rational for the rms- In that region nd such a mechanism