answer all the questions.

Consider the social planner's problem for a real business cycle model. The household makes consumption (C) and leisure (1 N, where N is hours worked) decisions to maximize lifetime utility: E0 X t=0 tu (Ct , 1 Nt) (1) Specific functional forms will be given below. Output is produced using capital K and labor N Yt = ZtK t N 1 t (2) Zt is a TFP shock and is governed by a discrete state Markov chain. Capital evolves: Kt+1 = (1 )Kt + It (3) but assume full depreciation so = 1. There is no trend growth. Finally, Yt = Ct + It First suppose that the utility function u is as follows: ln Ct N2 t 2 (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).

d) Compare the business cycle properties implied by these two models and explain why a TFP shock affects output, consumption, investment and hours worked. Some RBC modelers prefer preferences used in parts (a/b) to those in part (c), why this might be the case? Key points With the first set of preferences TFP shocks lead to an increase in output, consumption, hours worked and investment, as in the data. The answer should briefly touch on the main mechanisms outlined in the lecture notes, the answer keys to problem sets and the mid-term. What's important is that the first set of preferences do not feature a wealth effect on labor supply. This can be seen from the fact that consumption does not enter the first order condition for hours worked. In the second set of preferences, when = 1, the income and substitution effects cancel out and labor supply is constant. The second set of preferences deliver a result for hours worked that is inconsistent with the data. With the first set of preferences there is no wealth or income effect. This amplifies the effect of TFP shocks on hours worked. Even with = 1 hours worked now respond. This leads to a larger effect of TFP shocks on investment and consumption. Standard preferences typically require a high degree of labor supply to match the empirical volatility of hours in the data. RBC models typically lack sufficient amplification and therefore require large TFP shocks. The first set of preferences, by excluding an income effect (which offsets the effect of TFP shocks on hours worked), can lead to a larger effect of TFP shocks on hours worked. This is why some modellers prefer these preferences. e) If < 1 briefly explain how you would solve this model computationally using value function iteration. Give one advantage of this method.

Question 1 a) As usual, the BC is given by u = + q() = + 1+ . (1) b) Suppose we are in period t. The measure of workers who have been unemployed for exactly one period, are simply the workers who lost their job in t 1. In steady state, that number is equal to (1 u). The measure of workers who have been unemployed for two periods, are those who lost their job in t 2, and were unsuccessful in finding a job in t 1. That number is equal to (1 u)(1 q()). The measure of workers who have been unemployed for three periods, are those who lost their job in t3, and were unsuccessful in finding a job both in t2 and t1. That number is equal to (1 u)(1 q())2 , and so on. As a verification that this reasoning is correct, sum up all these workers to get (1 u) i=0 (1 q())i = (1 u) q() = q() = u. c) V = k + q()J + (1 q())V, (2) J = p w + (1 )J. (3) d) The typical worker in this model can be employed, unemployed with benefits, and unemployed without benefits. It is very important to distinguish between the last two states. For the employed worker we have W = w + (1 )W + Uz. (4) 1 For an unemployed worker who is eligible for benefits, we have Uz = z + q()W + (1 q())U, (5) and for the unemployed worker who is not eligible for benefits, we have U = q()W + (1 q())U. (6) Notice that the last two expressions immediately imply that Uz = z + U, (7) which will be useful later on. e) Exploiting the usual free entry argument, we can derive the following JC: w = p k(1 + )[1 (1 )] . f) We start from the Nash bargaining problem. One important thing to notice here is that the outside option of a worker is not the Uz value function, but the U value function: By definition, a worker who has met a firm and is bargaining with it, has already been unemployed for more than one period, and, hence, is no more eligible for unemployment benefits. The Nash bargaining problem will give us: (1 )(W U) = J. Using equations (3), (4), and (6), and exploiting the relationship between U, Uz (i.e., equation (7)), yields (after some algebra) the following wage curve: w = (p ) (1 )z + k. g) For a balanced government budget, we need: (1 u) = (1 u)z, where the left-hand side is the total funds generated by taxes, and the right-hand side is the total expenditure on unemployment benefits, i.e., z times the total number of workers who have been unemployed for one period. Clearly, this last expression implies that = z, which we can use in order to get rid of in the JC and WC expressions. 2 h) The JC curve is now given by: w = p z k(1 + )[1 (1 )] . Clearly, it will have the usual negative slope. To make sure that the JC curve lies in the positive orthant for some 's, we require that the value of w, evaluated at = 0 is positive. More precisely, we need p z k[1 (1 )] > 0. (8) This is likely to happen if , k, z are small and is large. i) The WC becomes: w = (p z) (1 )z + k. Clearly, it will have the usual positive slope. j) Given the discussion so far, we know that if an equilibrium exists, it will be unique. A sufficient condition for existence is that

a) This part is very standard (as in the lecture notes, except that there are two types of money). The main point is to point out that both value functions will be linear in m1, m2. b) Consider a type-1 meeting, i.e., a meeting where the seller accepts only local currency, and let m1 denote the money-1 holdings of the buyer. Let q1, d1 denote the amount of good traded and the units of money 1 exchanged, respectively. The bargaining solution is as follows: If m1 q /1, then q1 = q and d1 = q /1, where 1 is the real value of one unit of money 1. If, on the other hand, m1 < q/1, then q1 = 1m1 and d1 = m1. Next, consider a type-2 meeting, i.e., a meeting where the seller accepts both currencies. Let (m1, m2) denote the buyer's portfolio. Let q2, d1, d2 denote the amount of good, the units of money 1, and the units of money 2 exchanged, respectively. The bargaining solution is as follows: If 1m1 + 2m2 q , then q2 = q and 1 d1 + 2 d2 = q . If, on the other hand, 1m1 + 2m2 < q , then q2 = 1m1 + 2m2, d1 = m1, and d2 = m2. c) While you should feel free to do all the work that leads to the objective function, here it is quite easy to guess that this function will be as follows: J(m 1 , m 2 ) = (1 + 1 )m 1 + (2 + 2 )m 2 + [u(q1(m 1 )) 1d1(m 1 )] + (1 )[u(q2(m 1 , m 2 )) 1 d1(m 1 , m 2 ) 2 d2(m 1 , m 2 )], (9) where the expressions q1(.), q2(.), d1(.), d1(.), d2(.) have all been described in part (b). Moreover, as is usual in these models, we know that the agent will never bring more money than she needs in order to get the first best (bringing more money has a weakly positive cost, and absolutely no benefit, since the first-best has already been achieved). This argument allows us to simplify the J function even further as: J(m 1 , m 2 ) = (1 + 1 )m 1 + (2 + 2 )m 2 + [u( 1m 1 )) 1m 1 ] + (1 )[u( 1m 1 + 2m 2 ) 1m 1 2m 2 ]. (10) d) For now, I simply wanted you to describe the equilibrium as a system of 4 equations whose solution will yield the equilibrium values for our 4 endogenous variables 4 {q1, q2, z1, z2}. These four equations will consists of two Euler equations in steady state (one for each type of money), and the bargaining solutions, also evaluated at steady state. More precisely, we have 1 + 1 = [u (q1) 1] + (1 )[u (q2) 1], (11) 1 + 2 = (1 )[u (q2) 1], (12) q1 = z1, (13) q2 = z1 + z2. (14) e) With the specific functional form in hand, we can be even more precise about how exactly the equilibrium will look like. I will describe the equilibrium for all possible values that the parameters 1, 2 could obtain. I was not asking for so much detail in the exam. (Basically, I gave full marks to everyone who gave a roughly accurate description of cases 3 and 4). Case 1: 1 = 1 and 2 1. The point here is that the money with the liquidity advantage is also completely free (because authority 1 is running the Friedman rule). In this case, regardless of the value of 2, we must have that q1 = q2 = q = , i.e., the first-best is always traded in all types of meetings. Case 2: 1 > 1 and 2 = 1. In this case, money 2 is free to carry over time, but money 2 has the liquidity disadvantage. It is easy to verify that in this case we have q2 = q = and q1 = 1 + 1 . Clearly, the only way to get q1 q = is to set 1 1. Case 3: 1 > 1 and 2 (1 )1 (1 ). This is the case in which z2 = 0, or, alternatively, q1 = q2. Since money 2 has a disadvantage, agents will hold it only if its rate of return is significantly better than the one on money one. How much better? The answer is given above. Agents will hold money 2 only if 2 < (1 )1 (1 ) (and not if 2 < 1, as some of you guessed without solving the model carefully.) As long as we are in this case, it is easy to see that q1 = q2 = 1 + 1 . Naturally, it is only the policy parameter 1 that matters for equilibrium outcomes, since money 2 is not circulating. (Of course, in the back round 2 is also relevant, since everything we are discussing here holds true only for 2 (1 )1 (1 )). 5 Case 4: 1 > 1 and 2 ( 1,(1 )1 (1 )). This is the case where money 2 circulates together with money 1. Here z2 > 0, which, of course, implies q2 > q1. More precisely, in this type of equilibrium, we have: