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Question 3 (20 points) Consider the following decentralized real business cycle model. The representative household has preferences over consumption and leisure. Expected lifetime utility is: Eo 1 - 0 I+ 1 (1) where C is consumption and /V is hours worked. These preferences include a form of habit formation and the "habit stock", He, is simply equal to consumption in t - 1: He = Ci-1. h governs the importance of habits and 0 Sh 1 (rather than 1. Government spending, gr, follows an AR(1) process 9t = pgt-1 + et (10) where et is i.i.d. and 0 S p 1 in period two. For example, an individual that invests an amount I will receive RI in period two, and has 1 - I stuffed under the mattress. In t = 1, individuals have the option of liquidating the long-term project at a penalty. If they liquidate, they only receive a return L - 1 (per unit invested) in period 1, rather than the return R in period 2. At time t = 1, a fraction # = 1/2 of the individuals receive a liquidity shock. These individuals are "impatient" and only value consumption in period one. The fraction 1 - # individuals that do not receive a liquidity shock are "patient" and only value consumption in period two. At time t = 0, all individuals have the same chance of being hit by the liquidity shock. Assume that individuals do not discount the future, so that their ex-ante expected utility is given by U = TU(C1) + (1 - #)u(c2), where c, and c, are the consumption in period 1 and 2, respectively, and u(c) = _", with o > 0. a) Assume there are no financial markets available, so that individuals must simply invest on their own. Given that an individual has invested an amount / at time t = 0, what will be the optimal levels of consumption, C1, C2, if: (i) the individual receives a liquidity shock (i.e. is impatient); (ii) the individual does not receive a liquidity shock (i.c. is patient). Let & and 2 denote the consumption of an impatient individual in period 1 and of a patient individual in period 2, respectively. b) What is the optimal level of investment when individuals have to invest on their own? Denote this level by I. Hint: Show that there exists L, LE [0, 1] such that if L 2 L, the optimal level of investment is equal to 1, and if L S L, the optimal level of investment is zero c) Suppose that when types are realized in period 1, this information is publicly observable. Suppose there exists a social planner that individuals entrust all of their endowment to at time 0. The social planner will pay impatient individuals cj in period 1 and patient individuals c; in period 2 (and zero otherwise). Solving the social planner's OQuestion 6 (20 points) Consider an economy with 2-period lived overlapping generations of agents. Popu- lation is constant and normalized to one. When young, agents have a unit endowment of labor, which they supply inclastically on the labor market at the wage w. They consume cut and save w - Ct,. For the moment, assume all their savings go into phys- ical capital ke+1, which fully depreciates after use. When old, they rent capital at the rate n+1 and consume cat+1 = reiki+1. Their preferences are u(Ct,t) + Bu(ct,+1) where u(c) = log c. The production function is Cobb-Douglas yo = kall-a a) Find the optimal savings decision of the consumer born at time t, taking as given the prices wt and rt+1. b) Solve the problem of the representative firm and use market clearing in the labor market to derive expressions for w, and r, as functions of ki. c) Obtain a law of motion for equilibrium ke+1- d) Find a steady state with constant capital stock ke = kiss. Show that if a 1+8 1 - Q B , and hence a planner would be able to make all agents better off by reducing the capital stock in all periods. f) Suppose now that the agents are allowed to trade a useless, non-reproducible asset in fixed unit supply, which trades at the price p. We call this asset a "bubble". Argue that if p. > 0 and ke+1 > 0 the agent must be indifferent between holding capital and the bubble asset, and derive the associated arbitrage condition. g) Show that if (14) holds, there exists a steady state equilibrium with p = pss > 0