Macroeconomics assignment.?
3. Consider a two-period world with uncertainty. Utility for the representative agent is given by U = " (CI) + Bu(C2) 8 0 Agents believe that they can save and dissave using capital and risk-free bonds, denoted by B (view B as an asset). Risk-free bonds earn an endogenous interest rate of r. Assume that capital depreciates at rate o and that the agent begins period I with no risk-free bonds. (a) Write equations for the agent's period 1 and period 2 budget constraints, where the agent knows there is no period 3. Assume that the agent chooses Ke and By to maximize expected utility and write Euler equations, one using bonds, and one using capital. (b) Consider four alternative specifications of utility: Linear: (C) = aC Quadratic: u(0) = 00 - 203 for C 0 Define certainty equivalence. Which utility functions exhibit certainty equivalence? Which bond Euler equations have certainty equivalence? (c) Linearize the marginal utility of consumption in the CES case about C2 = C1. Evaluate whether the linearized bond Euler equation has certainty equivalence. What are the implications of your answer for whether or not the linearized RBC model has certainty equivalence? (d) Define risk aversion and evaluate which utility functions are characterized by risk aver- sion. What characteristic of the utility function characterizes risk aversion? (e) Define precautionary saving and evaluate which utility functions are characterized by precautionary savings. How does precautionary savings differ from risk aversion? (f) Write an expression for the risk-premium on capital (relative to the risk-free interest rate) using the bond and capital Euler equations with generic utility (U (C)). Define the gross return on capital as 1 + R =1+- K2 - 5, and the risk premium as ER- r. Use your expression to explain the sign of the risk premium. Do all of the utility functions above have a positive risk premium?Question 3 (20 points) Consider the social planner's problem for a real business cycle model. The house- hold makes consumption (C) and leisure (1 - N, where N is hours worked ) decisions to maximize lifetime utility: E S'u (C,1 - N.) (1 ) 1=0 Specific functional forms will be given below. Output is produced using capital K and labor N (2) Z, is a TFP shock and is governed by a discrete state Markov chain. Capital evolves: KitI = (1 - 6) K, + 1, (3) but assume full depreciation so 6 = 1. There is no trend growth. Finally, First suppose that the utility function u is as follows: In C, - ME (4) a) Write down the recursive formulation of planner's problem and derive the first order conditions. b) Using guess and verify, find the policy functions for investment, consumption and hours worked (Hint: first consider the equilibrium condition for hours worked and guess that investment is a constant share of output). Now suppose the utility function is given by: In C, - NE (5) c) Repeat parts (a) and (b) using these new preferences. d) Compare the business cycle properties implied by these two models and explain how and why a TFP shock might affect output, consumption, investment and hours worked. Some RBC modelers prefer preferences used in parts (a/b) to those in part (c), why might this be the case? e) If 0 0. For simplicity, we assume that consumption of farmers can be negative, that is c e R. Moreover, farmers can save (or borrow if negative) in a non-state contingent and non-defaultable asset a, which has a rate of return r. Farmers face the following "No-Ponzi" condition on assets to (1 + r ): 20. (22) Moreover, farmers can produce and hold capital, k, which is rented to the representative firm in competitive markets (1 unit of final consumption good produces 1 unit of capital). Let r* be the rental rate of capital and o the depreciation rate. Unlike farmers, workers don't own land, and they use their available time to work in the representative firm. Assume each worker is endowed with one unit of time. They cannot trade the asset a, but they can produce and hold capital, k (with the same technology as farmers). Their per-period utility is given by u(c), with u'(c) > 0, u"(c)
