METHODS EXAMINATION - MAY MACROECONOMICS Meyer (chair), Conlin, Doblas Madrid, Wilson, Wooldridge

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1 METHODS EXAMINATION - MAY MACROECONOMICS Meyer (chair), Conlin, Doblas Madrid, Wilson, Wooldridge 1) Open Economy with Money in Capital Good Production (Total 40 Points). Consider the following model. There is a representative agent who derives utility from the consumption of a final good, from real money holdings and from leisure as follows: w s=t [ln C, + ln(z - L,) + In -1 where C is the consumption flow of the final good, z is the total endowment of time, L, is the amount of time spent working, and is the real money holdings. The final good can be produced using labor and capital as follows M3 Ps afo= where % > 0, < 0, lim~,-o = oo, lim~,-~ al, af0 = 0. 0, > 0, < 0, lim~~-~-0 = 00, lim~~-,-m ak,-l Capital (i.e. the stock of machines) evolves according to the following iaw of motion Ks = Ks-1(1-6) + AX,, (3) where 6 is the depreciation rate (0 < 6 < 1) and Ax,, is the flow of new machines in period s. The peculiar characteristic of this economy is that the flow of new machines depends on the amount of money holdings of the representative agent, besides the amount of final good Is the agent invests in producing machines (for example, agents can better organize the production of machines if they have more money). Precisely, the technology for producing machines is as follows Ms Ax,, = Is + AH(-) Ps (4) where, just like F(.), the function H(.) satisfies standard neoclassical properties, while A is an exogenous productivity parameter. (HInt: remember that in the standard case Ax,, = Is). The economy is small and open to the rest of the world. Except for money, the only financial asset is a bond (B) which yields a gross real interest rate of R each period. Answer all the questions below: i) Write down the budget constraint of the representative agent and derive the Euler conditions for all the choice variables. (12 Points) ii) Characterize the steady state of this economy. Write down the whole system that defines the steady state values of the endogenous variables. (12 Points)

2 iii) Characterize the effects of money growth on the steady state. (8 Points) iv) Discuss the effects of an exogenous improvement in the technology for the production of machines (i.e. an increase in A) on the steady state. (8 Points) 2) Nominal and Real Rigidity. Discuss the concepts of nominal and real rigidity and their role for the effectiveness of monetary policy. (Total 10 Points)

3 3. OVERLAPPIIVG GENERATIONS (25 points) Consider a basic overlapping generations model where time is discrete and indexed by t=1,2,... Agentsoftype i=1,2,... livefortwoperiods, t=i and t=i+l. Thereisa unit mass of identical and infinitesimal agents within each type. A type-i agent has consumption given by cl when young and c;+, when old. (Superscripts denote agents, subscripts the time when consumption takes place.) A type-i agent's utility U' is given by Ui (c,!, c,!+,) = 1n ci + 1n c;, for i=l,2,... There is also a unit mass of identical type-0 agents who are alive at time 1. Type-0 agents have utility u0(c;) = YC; with Y > 0. Endowments are given by y,! = 0.8 for all i = 1,2,... and yl!+, = 0.2 for all i = O,1,2,.... The initial old also own M > 0 units of fiat money. a. (4 points) Define a Pareto optimal allocation for this economy. b. (4 points) Define an equilibrium with sequential trading and valued fiat currency. c. (7 points) Find agent i's offer curve, for an arbitrary i 2 1. Plot the offer curve in a diagram with cl on the horizontal axis and c,!+, on the vertical axis. On the same diagram, depict the economy's feasibility constraint (with c: on the horizontal axis and c,!-' on the vertical axis). d. (7 points) Calculate a stationary equilibrium with valued fiat currency, that is, calculate the equilibrium prices and the equilibrium allocation. Are there nonstationary equilibria with valued fiat currency? If yes, briefly describe how to find such equilibria (you may use your graph from part c. to help your description.) e. (3 points) Does the first welfare theorem hold in this economy? Why or why not?

4 4. VALUE FUNCTION ITERATION ALGORITHM (25 points) Time is discrete and infinite with periods denoted by t = 0,1,.... Every period, a stochastic event s, E S = {1,2,3) is realized. This event is Markov with transition matrix where 4, > 0 is the probability of st+, = j conditional on s, = i, for i, j = 1,2,3. The initial realization so = 1 is given, and for each history s' = (so,...,st) E S', the probability that that history occurs is denoted by n,(sl). There is a continuum with measure one of identical households who are endowed with one unit of time, which they supply inelastically since they do not value leisure. The measure of households stays constant over time. Consumption at time t, history sf, is denoted by C,(st) and preferences over time- and history-contingent consumption plans are represented by the utility function Production utilizes capital K,(sr-') and, implicitly, inelastically supplied labor. Technology is Cobb-Douglas with output given by (st) = A, (st)[k, (st-l)]li3, where the TFP parameter A, (sf) is given by A, (s') = Feasible consumption and production 2 plans satisfl S (a) (5 points) Formulate the social planner's problem for this economy. Take first order conditions, and derive the Euler equation. (b) (4 points) Re-formulate the planner's problem in terms of a dynamic programming problem. Choose your state variable(s) carefully. (c) (7 points) Describe how you would implement a value function iteration algorithm to solve this optimization problem numerically. How would you choose a grid? How would you perform an iteration of the algorithm? When would you stop iterating?

5 (d) (4 points) Now suppose you have successfully implemented the algorithm, and observe that, at the solution: (for arbitrary t and st-') If K,(sl-')=12.95,and s,=l, K,+,(sl-',l) willbe If K,(st-I) =12.95, and s, =2, Kt+, (st-', 2) will also be If K,(sr-')=12.95,and s,=3, K,+,(sl-',3) willbe Given the above information, and supposing that K,(sl-') =12.95, give a formula for consumption C, (st-', s,) for each s, E {1,2,3} (you do not have to actually compute these values). (e) (5 points) By the second welfare theorem, the Pareto efficient allocation that solves the social planner's problem above, is also a competitive equilibrium allocation. In that equilibrium, write a formula for the price (expressed in terms of consumption at time t, history st = (st-', 2) with K, (sl-') = ) of a risk-free claim to 7 units of consumption at time t +l. Also, write a formula for the price (again, in terms of consumption at time t, history s' =(st-',2), with K,(s'-') = 12.95) of a claim to the wages earned by 1 unit of labor in the next period t +l, regardless of st+,. Again, just give formulas, you do not need to compute the values.