Solutions Problem Set 2 Macro II (14.452)

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1 Solutions Problem Set 2 Macro II (4.452) Francisco A. Gallego 4/22 We encourage you to work together, as long as you write your own solutions. Intertemporal Labor Supply Consider the following problem. The consumer problem is: Max fc tg;fn tg E s:t:! t=t X t f C t $ + 2 N t $2 g t=o A t+ = R(A t + N t W t C t ) Where C is consumption, N is labor supply, A is initial wealth, R = + r, and the greek letters are parameters. Only W is stochastic ). Derive and interpret the rst-order conditions for C t and N t. What are the intertemporal and intratemporal optimal conditions There are at least two (equivalent ways of setting up and solving this problem). I will implement both of them. Lagrangian method I: In this case the Lagrangian is: Max fc tg;fn tg;fa t+g $ = E F.O.C.! t=t X t f C t $ + 2 N t $2 t (A t+ R(A t + N t W t C t )g t=o s.t. A fc t t =, $ C $ t = t R ()

2 @$ fn t g : 2 $ 2 N t $2 = t W t R t The interpretation is more or less natural. The rst condition tells you that the marginal utility of C t has to be equal to the marginal utility of wealth (do not worry too much about R it is an implication of the fact that C in this case is realized at the beginning of the period t so the value of saving one unit of consumptions is times R). The second condition tells you that marginal disutility of work has to be equal to the marginal utility of wealth times how much consumption you can get from one our of work (the wage rate). Why no expectation? This is important, at time t, the only source of uncertainty is realized (W t ) and, roughly speaking, t is a function of expected future wages (see below). Combining both equations we get the intratemporal condition: 2 $ 2 N t $2 = W $ C $ t To get the intertemporal condition, we need t+ =, t t = t+ t+ R Using the FOC for consumption, we get our intertemporal condition: 2. Lagrangian method II: t $ C $ t = t+ RE( $ C $ = RE " Ct+ C t $ # t+ ) Notice that you can iterate the constraint and write the intertemporal budget constraint. TX R t C t = A + t= TX R t N t W t Two comments: () Why no expectation? This has to hold exactly. (2) what about A T +? Max fc tg;fn tg $ = E F.O.C. t=o t= t=t X t f C t $ + 2 N t $2 g ( TX T! X R t C t A R t N t W t ) t= t= 2

3 fc t t =, t $ C $ t = R t (2) fn t t =, t 2 $ 2 N $2 t = W R t (3) If you combine both equations, you get the same intratemporal condition as before. To get the intertemporal condition write the FOC for t + taking expectations at h fc t+ g : =, E t $ C $ t+ R t i = t+ Using the FOC for C t : h E t t+ $ C $ t+ t $ C $ t R t i = and you get the same as before: = RE " Ct+ C t $ # What is more interesting about this method is that can be interpreted as the su cient statistic you need to solve the problem. Notice that () and (2) imply that: t R = (R) t How can you interpret this condition? We can plug () and (3) into the budget constraint and we will get a decreasing implicit function of as a function of fw i g with i t and, of course, some constant terms (among them R). Thus, the only thing you need to compute is to know something about fw i g. Therefore, you will keep your constant in the future if the realizations of fw i g were more or less what you expected in the past. 2). What is the link between C t and N t in this model? What assumption(s) is (are) producing this result The only link between both variables is (or t ). This implies the intratemporal condition and the negative correlation between both variables if W is kept constant. And, the lack of correlation if W increases without a ecting. Assumption: () both terms are additively separable in the utility function and (2) perfect capital markets (what may happen if we introduce frictions as A t > ) 3

4 3). How can you analyze changes to wages that do not a ect the expected wealth of the consumers? (Hint: what is the Lagrange multiplier in this model?). Let s de ne the wealth-constant ln Nt of labor supply as ln W t when wealth is constant. What is in this model? Is it positive or negative? Why? Given our previous discussion, these are changes that do not a ect, therefore we can take logs in (3) and get: Then log( 2 $ 2 (R) t ) + ($ 2 ) log(n t ) = log W t = log(n t) = log W t ($ 2 ) Notice that >. There are several ways of understanding this. Here is one: First, notice that obviously > ; 2 < ; $ <. Why? People like C, dislike N, and marginal utility of consumption is decreasing in general. Therefore if the instantaneous utility function is to be concave (and, therefore, the solution we just proposed has sense), it has to be true 2 2 = 2$ 2 ($ 2 ) N $ 2 2 t < ) ($ 2 ) > 4). Take the FOCs and discuss the e ect of the following situations on fc t g and fn t g :. Cross-sectional di erences associated with permanent differences in human capital (assume W varies in the crosssection) Higher human capital will probably imply higher W. Therefore, #, C ". What about N? Assume all other parameters do not vary in the cross-section and take two individuals i and j log( N j N i ) = ($ 2 ) log W j W i + log j i So e ect is ambiguous. One of them is richer, but at the same time the alternative cost of leisure is higher. Life-cycle changes in wages (i.e. if Mincerian equations are correct, wages follow an inverted-u behavior over the lifecycle) Notice that these evolutionary changes were predictable at the beginning of the life-cycle and, therefore, as W changes, is constant. Thus C is also constant see (2). But given that > and is constant, N moves together with W. 4

5 Unexpected temporary shocks to wages If change is temporary, it shouldn t a ect by much, then C is constant, and N moves together with W. Unexpected permanent shocks to wages In this case should be a ected. Therefore, C moves in the same direction as W and the e ect on N is unclear. 2 Log-Linear RBC Consider the following problem. Consumers maximize 2 3 X max E t 4 b j U( C e t+j ; N t+j ) 5 j= Subject to ek t+ = R tkt e + W f t N t Ct e U( C e t ; N t ) = log ect + log( N t ) while rms solve nn t ; e K t o and we assume = arg max Z t F (A t N; K) e (R t + ) K e W f t N N; K e F (A t N; e K) = Y = Z e K A t+ = A t : (A t N) This is the standard RBC model. trending variables are those with a hat. ) De-trend the model De ne V = e V A.. Utility function: U(C t ; N t ) = log (C t A t )+ log( N t ) = log (A t ) {z } +log (C t )+ log( N t ) Does not matter 2. Consumers B.C.K t+ A t+ = R t K t A t + W t A t N t C t A t K t+ A t+ = R t K t A t + W t A t N t C t A t A t+, K t+ = R t K t + W t N t A {z t } C t, K t+ = R t K t + W t N t C t 5

6 3. Pro ts fn t ; K t g = arg max N;K Z t (K t A t ) (A t N) (R t + ) A t K t A t W t N t, fn t ; K t g = arg max N;K Z t (K t ) (N) (R t + ) K t W t N t 2)Solve for the F:O:C of the consumer problem, either using a Lagrangian or a Bellman Equation. You should nd two nal equations, the Euler Equation and the labor supply.. Bellman = V (K) = Max C;N flog (C) + log( N) + be [V (K )]g s:t: K = RK + W [V (K C + [V (K N [V (K E = (4) N + W [V (K E = (5) [V = = [V E (a) Euler Equation: Using (5) and (4), we get: Then, = R C C = b R E C = (6) (a) Labor supply (5) and (4) imply: N = C W (7) 6

7 2. Lagrangean: Max $ = E fc tg;fn tg! t=t X b t flog (C) + log( N) t (K t+ (R t K t + W t N t C t ))g t=o FOC From (5) and (4) we get: Take additional FOC: fc t g : fc t g : C = t (8) N = tw t (9) N = C W fk t+ g : t t = E t+ t+ R t+ Then, combining (6) and (), we get our Euler equation: t = E t+ R t+, = t C t = E R t+ C t C t+ C t C t+ () () 3)Solve the rm problem to nd the labor and capital demands. Give the law of motion for capital (re-write K e t+ = R tkt e + W f t N t Ct e to get the usual equation) This should be easy., fn t ; K t g = arg max = arg max Z t (K t ) (N) (R t + ) K t W t N t N;K N;K FOC: fn t g t =, Z t Kt N t = W t fk t g t =, ( ) Z t Nt K t = (R t + ) 4) We have a total of 5 equations. Discuss how would you proceed to log-linearize our system (you don t need to do all the math, just apply the method to one or two equations). There is no unique way of log-linearize things. I will use here my way (which I think is simpler and does not use any math trick you didn t know). Let go one by one: 7

8 . Consumers BC: K t+ = R t K t + W t N t C t, K t+ = ( )K t + Y t C t It is also true that (where X denotes the steady state of X): Then: K = ( )K + Y C K t+ K = ( ) K t K + (Y t Y ) C t C K t+ K = ( ) K t K + (Y t Y ) Y C t C C K K Y K C K De ne x t = log (X t ) log X (Xt X) X, k t+ = ( ) Y K k t + y t C K c t (2) But, take the production function (I ll skip some steps):, y t = z t + n t + ( ) k t Then (2) becomes: k t+ + + ( ) Y k t K 2. The wage equation: C K c t + Y K n t + Y K z t = It is also true that Kt Z t N t = W t Then Z t Z Z Kt K N K N N t = W = W t W Take logs and use small letters to denote log deviations from the steady state and you get: z t + ( )(k t n t ) = w t 8

9 3. Labor supply Then, Take logs N t = C t W t N = C W N N t = CW t C t W log( N ) = w t c t (3) N t Let s de ne L = N and l = log(l) log(l): Then, the LHS can be written as: log( N ) = N t l N N N N nn = n N N Then, (3) becomes: n N N = w t c t So there is a mistake in the log-linearization that appears in the program. 4. Price of capital As with the wage equation: Nt ( ) Z t = (R t + ) K t Then (after some steps) Take logs: Z t Z Nt N K = (R t + ) R + K t z t +n t k t (R t + ) R + R + = R t R R + R R + r t 9

10 5. Euler equation b C t = E R t+! C t+ b C = C R Then, Ct C R t+ = E = E [( + c t ) ( C C t+ R c t+ ) ( + r t+ )] E [ + c t c t+ + r t+ ] 5) Our log-linearized system is the following (small variables denote logs): The law of motion and the market clearing conditions: k t+ + + ( ) Y C k t K K c t + Y K n t + Y K z t = (k t n t ) w t + z t = N N c t + n t N N w t = R k t + R + r t n t z t = The Euler Equation (or any other asset pricing equation): We assume E t [r t+ c t+ ] + c t = : z t = z t + " t : We want to write the system of six equations in order to solve it numerically using the process in Uhlig(997). Let s t be the vector of endogenous state variables (capital stock), x t = [c t ; r t ; n t ; w t ] the endogenous control variables and z t the stochastic processes. To avoid confusion, matrices are named using two capitalized letters. There are two conceptually di erent sets of equations. The rst set describes the law of motion of the economy, together with the market equilibrium conditions: I use M_ for these matrices (Motion and Market). In the second set, the equations are forward looking and they involve the expectations of the agents: I use F _ for these equations (Forward). Finally, we have the equation for the stochastic process z t. Write the system in the following way: MS s t+ + MS s t + MX x t + MZ z t = E t [F S 2 s t+2 + F S s t+ + F Ss t + F X x t+ + F Xx t + F Z z t+ + F Zz t ] = z t+ ZZ z t " t+ = :

11 In other words, what does each matrix contain? Just by inspecting the log-linearize system we get: MS = MX = ; MS = ( ) Y K ( ) 3 C K C K ( ) N N N N R R ; 7 5 ; MZ = Y K F S 2 = ;F S = ;F S = ;F X = ; F X = ; F Z = ;F Z = ) We want to nd numerically a solution of the type s t+ = b S s t + c SZ z t x t = b X s t + d XZ z t ; because we are looking at small deviations from steady state. You have to solve that using matlab. I give you a program, linear, containing the function linear which does the solution of the system, so basically what you have to do is create a program containing the following: a) Values of the parameters of the model. We will assume = :25; = :4, = 2 ; = ; = :979; 3 R = :63; Y=K = =2=4; K = ; C = Y 68=; N = :2; e = :72 Before solving the problem, notice that we need a few clari cations here: There are some conditions that the steady state values have to satisfy: ( ) Z N Y = ( ) K K Y ) = R + K ( ) = R + = 3 (:63 + :25) 8:7

12 Y So is not a parameter that you can change. K K = ( )K + Y C, C K = ( ) + Y K ) C Y = ( )K Y + 8:7( :25 :4) + :7629 So the value that is in the PS is inconsistent with the budget constraint. Sorry about that. Moreover, N = C W = C = Y N C Y N, N = + C Y :38 So, if we want to have N = :2, we need 6=! Fortunately, this does not a ect the simulations Olivier showed in class because it is the sum of two mistakes that cancel out each other. In any case, keep in mind that if you change N you have to change : This is completely intuitive because is a measure of how much you value leisure. Notice that if =, then N = : b) The matrices derived in 5). c)then you have to call the function linear. 7) Now we want to do impulse-response functions. For that, attach to your program (copy and paste after part c)) the program linear 2. This programs calls two other functions, impulse and impulse-plot. The rst nds the reduced form model and the second hits it with a shock. That should be the end of your program. Explain the results. How do they depend on the values assumed in a)? (run the program for di erent values of at least two parameters and compare results). Interpret. Several exercises can be implemented: Base case (Figure ) Minor di erences with the gure Olivier showed in class, same intuition. 2

13 Baseline case Change to.999 and.3 (Figures 2 and 3) = :999 3

14 = :9 Change N = :8 (Figure 4). N = :8 R = : (Figure 5) 4

15 R = : = : (Figure 6) = : 3 (Optional) Discrete Time Dynamic Program- 5

16 ming Use the discrete time setup of Philippon and Segura-Cayuela (23) (up to section 5) and the MATLAB programs contained in the folder DynamicP rogramming. Read tutorial:doc and the comments in DP:m. ) Derive the Euler equation in discrete time in the model with exogenous labor, rst using dynamic programming and then using the Lagrange multiplier method. Compare it to the continuous time Euler equation. The problem is: t= Max $ = X fc tg t=o s:t: b t ek t+ = R t e Kt + f W t e Ct A t+ = A t ect Divide all the trending variables by A t :De ne V = e V A and notice that A t = A t Utility function: t= X MaxV = Max t=o b t = Max A t=t X = Max t=o t C ta t t= X t=o The budget constraint (same as before): b {z } (C t) (C t) t W t K t+ = R t K t + W t t=t X Thus, the problem becomes Max C t Bellman equation C t t (C t) t=o s.t. K t+ = R t K t + This exercise is optional in the sense that you could skip it if you feel comfortable with stochastic DP, and do it if you want to get more training. 6

17 t V (K) = Max C s:t: (C) K = RK + W N C + E [V (K = C C [V (K + = C C [V E = (4) [V = @C Using (5) and (4), we get: Then, = RC = R 2 3 = e C E 4R C 5 C e [V (K (5) But notice that: Then = b ec 2 3 e C = be 4R C 5 C Lagrangean: t= Max $ = X t fc tg (C t) t (K t+ (R t K t + W t N t C t )) FOC t=o fc t g : C t = t (6) fk t+ g : t t = E t+ t+ R t+ (7) Then, combining (6) and (), we get our Euler equation: t C t = E " i # h t+ C t+ R Ct t+, = E Rt+ C t+ 7

18 Continuous time Max fc tg Z ect exp ( s:t: ek t = (R t ) K e t + W f t Ct e t) dt A t+ = A t Detrending: Max Z = MaxA ect exp ( Z Notice that () + : Then MaxA = Max = Max Z t) dt = Max (C t) () t exp ( t) dt = exp log ( Z ) = exp (C t) () t exp ( t) dt (C t) exp with The budget constraint becomes Z (C t) exp ( t) dt Z (C ta t ) exp ( t) dt log () exp, K t = (R t )K t + W t C t t dt Then the Euler equation will be: C t = (R t ) = (R t C t = (R t ) ) 8

19 2) Numerical analysis using DP:m. Describe the dynamics of the economy when it starts far away from its steady state (say 3% below). How does the recovery depend on the evolution of productivity? What does it tell you about the rapid growth in Europe after the second world war? (Hint: for the question on the evolution of productivity you may want to replicate the model, say times, and see what happens with the recovery path) See gures Evolution of output after the recovery (Figure 7): Recovery Mean and median and max and min in simulations (Figures 8 and 9) 9

20 Mean and median ( recoveries) Fastest and Lowest Recovery ( simulations) 3)DP:m produces 5 plots. Describe brie y each of them (what they represent and how they are computed. I am not asking for the details of the computations. 2

21 Simply show me that you understand the architecture of the program). Figure : 3D graph showing optimal consumption as a function of realizations of Z and initial capital stock. By-product of the search for the policy rule. Figure 2: Policy rule. K t+ as a function of K t. Capital in the steady state goes from 4.6 to around 5.4 Figure 3: Evolution of consumption in a stochastic simulation after starting from some initial capital that is below the steady state value. Figure 4: Evolution of consumption in the same simulation. Figure 5: stationary distribution of capital in the steady state. 5) Numerical analysis using DP:m. We have seen in the continuous time setup that the reaction of consumption to a positive technology shock depends on the elasticity. Use the policy function estimated by DP:m to illustrate this nding. (note: in the program, they de ne gamma, the CRRA instead of. How are they related? Why?) Hint: DP:m produces 5 plots. The rst one is called consumption, and it is 3 dimensional. Consumption is measured on the vertical axis. The two horizontal axis are for K and Z. If you do not see well, you can rotate the picture with the mouse. Notice that CRRA = Di erent = f:; ; g Figures to 2. 2

22 = : = 22

23 = 6) Show numerically how the behavior of consumption is a ected by the persistence of the shocks. Interpret. Di erent = f:5; :99g = :5 23

24 = :99 24

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