Macroeconomics Preliminary Examination Part II Spring 2012 Instructions: Answer two (2) questions in Part II. Each question is worth fifty (50) points. PLEASE MAKE YOUR ANSWERS NEAT AND CONCISE. Make whatever assumptions you need to answer the questions. BE SURE TO STATE THEM CLEARLY. Part II Question I Ramsey Taxation Suppose that an infinite horizon setting in which there is a representative consumer and a representative firm as in the standard single sector growth model. The utility function of the representative consumer is given by ∞ ∑ β u ( c ,l , g ) , t t t t t=0 where gt is the amount of government goods and services produced in each period. The feasibility constraint for the firm is ct + xt + gt = F ( kt ,nt ) . There is no technological change. Investment is done at the household level, and the standard law of motion for capital is assumed to hold. Suppose that the government has at its disposal only labor and capital income taxes for financing expenditures, but can freely borrow and lend (i.e., it faces a present value budget constraint). Assume that the consumer takes gt , τ nt , and τ kt as given when making its decisions. (a) Set up and define a TDCE (Tax Distorted Competitive Equilibrium) for each fixed sequence ( gt , τ nt , τ kt ) . (b) Assuming interiority of the equilibrium, give the FOCs that characterize the equilibrium you defined in (a). (c) Set up the Ramsey problem for this economy. (d) If the government acts benevolently in choosing ( gt , τ nt , τ kt ) , will it be true that τ kt → 0 ? That is, does the Chamley-Judd characterization of the asymptotic behavior of Ramsey tax systems extend to this setting in which gt enters the utility function? Prove it. Assume that in the Ramsey allocation, all quantities converge to constant levels, ct → c∞ , etc. Question II Overlapping Generations Consider an overlapping generations economy in which there is one good in each period and each generation, except the initial one, lives for two periods. The representative consumer in generation t,t = 1,2,... , has the utility function t log ctt + log ct+1 ( ) t = ( 3,1) . The representative consumer in and the endowment wtt ,wt+1 generation 0 lives only in period 1, has the utility function log c10 and has the endowment w10 = 1. There is no fiat money. (a) Define an Arrow-Debreu equilibrium for this economy. Calculate the unique Arrow-Debreu equilibrium. (b) Define a sequential markets equilibrium for this economy. Calculate the unique sequential markets equilibrium. (c) Define a Pareto efficient allocation. Is the equilibrium allocation Pareto efficient? Prove it. Suppose now that the representative consumer in generation t,t = 1,2,... , is endowed with one unit of labor when young and none when old. The representative consumer in generation 0 is endowed also with k 1 units of capital. The resource constraint is ctt−1 + ctt + kt+1 − (1− δ ) kt = θ ktα nt1−α , where θ > 0,0 < α < 1, and 0 < δ ≤ 1 . (d) Define an Arrow-Debreu equilibrium for this economy. Characterize the Arrow-Debreu equilibrium. (Make all necessary assumptions.) Question III Dynamic Programming Consider the optimal growth problem ∞ max ∑ β t log ct t=0 s.t. ct + kt+1 ≤ θ ktα ct , kt ≥ 0 k0 = k 0 . Here 0 < α < 1,0 < β < 1, and θ > 0. (a) Write down the Euler conditions and the transversality condition for this problem. (b) Calculate the steady state values for c and k. (c) Write down the functional equation that defines the value function for this problem. Guess that the value function has the form a0 + a1 log k. Calculate the policy function. Consider now an overlapping generations economy in which there is one good in each period and each generation, except the initial one, lives for two periods. The representative consumer in generation t,t = 0,1,... , has the utility function t log ctt + log ct+1 and is endowed with one unit of labor when young and none when old. The representative consumer in generation -1 lives only in period 0, has the utility function log c0−1 and is endowed with k 0 units of capital and amount b 0 of nominal government assets. The resource constraint is ctt−1 + ctt + kt+1 − (1− δ ) kt = θ ktα , where θ > 0 , 0 < α < 1, and 0 < δ ≤ 1. (d) Define a sequential markets equilibrium. Calculate the two steady states. Make all necessary assumptions. (Hint: one steady state is when r b = 0 , where r b is the interest rate on bonds, and another one is when b = 0. )