All of the questions assume an economy with two dates, t = 0,1, and two states o
ID: 1119957 • Letter: A
Question
All of the questions assume an economy with two dates, t = 0,1, and two states of nature, s- 1,2. There is no consumption at date 0 and there is a single consumption good in each state at date 1·The probability of state s = 1,2 is , at date 0; the true state is revealed at date 1. The two consumers, A and B, have endowments wA-(wl,w2) and w" = (uf,w ), respectively, and maximize expected utility. The set of markets in which consumers trade at date 0 varies from question to question. Question 1 Suppose the two states are equally likely, that is, --, the consumers, endowments are w(2,1) and wB (4,2), and the consumers have identical VNM utility functions UA (z) = UB(z) = lnz. At date 0, there are markets in which two assets are traded, a safe asset that pays one unit in each state, and a risk asset that pays one unit in state 1 and two units in state 2. The assets are in zero net supply (neither consumer owns any of these assets to start with) and only these assets can be traded. What are the equilibrium asset prices? Hint: Start by showing that this economy has a representative consumer.] Question 2 (i) Suppose the two states have probabilities 1 and 2 and the consumers endowments are wA(4,2) and (2, 1), respectively. Consumer A has a VNM utility UA (z) = In z and consumer B has a VNM utility UB (z-z. Consumer A is risk averse and consumer B is risk neutral. At date 0 the consumers can trade contingent commodities. Does consumer A bear any risk in a competitive equilibrium. [Hint: Solve for the competitive equilibrium assuming that consumer A bears no risk and check that it is feasible.] (ii) Suppose the endowments were reversed, that is, w (2,1) and wB (4,2), but everything else remains the same. Would consumer A bear any risk in a competitive equi- librium of this economy? Question 3 Suppose that each consumer i A, B has one unit of an asset that produces a payoff win each state s = i 2. Consumers can trade the assets and contingent oommodities at date 0, Suppose the vector of asset prices at date 0 is q. = (1,2) and the payoffs are w(1,3) and B (2,2). What is the cost of a single unit of the contingent commodity in state 2? Hint: Find the portfolio of assets that yields a payoff equal to 0 in state 1 and 1 in state 2. Then explain why the price of the contingent commodity would equal the value of this portfolio.]Explanation / Answer
Cost of a single unit of the contingent commodity in state 2=1*1+1*2+2*3+2*2=1+2+6+4=13