ECON607 Fall 2010 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment 2
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1 ECON607 Fall 200 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment 2 The due date for this assignment is Tuesday, October 2. ( Total points = 50). (Two-sector growth model) Consider the following two-sector model of optimal growth. A social planner seeks to imize the life-time utility of the representative HH given by P t u(c t ; l t ), where c t is consumption of good at time t, whereas l t is leisure at time t. Sector produces consumption good using capital k t and labor n t according to the production function f (k t ; n t ). Sector 2 produces capital good which is going to be used in the production next period according to the production function f 2 (k 2t ; n 2t ). The time endowment for the HH at each period is l. The initial capital stock is given by k 0 > 0. (0 points) (a) Formulate this problem as a dynamic programming (DP) problem. Display the functional equation (FE) and clearly specify the state and control variables. Answer: DP problem is v(k) = c;n ;n 2 ;l;k ;k 2 ;k 0fu(c; l) + v(k0 )g c f (k ; n ); k 0 f 2 (k 2 ; n 2 ); n + n 2 + l l; k + k 2 k: (b) Consider another economy that is similar to the previous one except for the fact that capital is sector speci c, i.e., capital stock for sector must be only used for this sector, same for sector 2. The capital-good sector produces capital that is speci c to each sector according to the transformation technology g(k ;t+ ; k 2;t+ ) f 2 (k 2t ; n 2t ): Formulate this problem as a dynamic programming (DP) problem. Display the functional equation (FE) and clearly specify the state and control variables. Answer: DP problem is v(k ; k 2 ) = fu(c; l) + v(k; 0 k2)g 0 c;n ;n 2 ;l;k 0 ;k0 2 n + n 2 + l l: c f (k ; n ); g(k 0 ; k 0 2) f 2 (k 2 ; n 2 );
2 2. (Habit Persistence) Consider following dynamic problem with habit persistence preference: t (ln c t + ln c t ) c t + k t+ Akt c t ; k t+ 0 k 0 ; c > 0 given Formulate this problem as a dynamic programming (DP) problem. Display the functional equation (FE) and clearly specify the state and control variables. (5 points) Answer: DP problem is v(k; c ) = fln c + ln c + v(k 0 ; c)g c;k 0 c + k 0 Ak c 0; k 0 0: 3. (Howard s policy iteration algorithm) Consider the following optimal growth problem X E 0 t ln c t c t + k t+ Akt t k 0 given; A > 0; > > 0 where the sequence f t g is an i.i.d. shock with ln t distributed according to a normal distribution with mean zero and variance 2. Consider the following algorithm. Guess at a policy of the form k t+ = h 0 Ak t t for any constant h 0 2 (0; ): Then form the value function v 0 (k 0 ; 0 ) = E 0 X t ln(ak t t h 0 Ak t t ) (Hint: you will see the value function takes the form v 0 (k 0 ; 0 ) = H 0 + H ln 0 + ln k 0:) Next, we choose a new policy h by imizing ln(ak k 0 ) + Ev 0 (k 0 ; 0 ) where k 0 = h Ak. Then we form the value function again v (k 0 ; 0 ) = E 0 X 2 t ln(ak t t h Ak t t ):
3 Continue iterating on this scheme until successive hj have converged. Show that, for the present example, this algorithm converges to the optimal policy function in one step. (5 points) Answer: Given this guess k t+ = h 0 Akt t, our in nite-horizon value function or life time utility is X v 0 (k 0; 0 ) = E 0 t ln c t X = E 0 t ln(akt t h 0 Akt t ) = = t E 0 [ln(a h 0 ) + ln k t + ln t ] t E 0 [ln(a h 0 ) + ln k t ] + ln 0 () Notice that we are using E 0 ln t = 0; 8t. Now let s spend some time on ln k t. We know ln k t = ln h 0 k t t = ln h 0 + ln t + ln k t = ln h 0 + ln t + ln h 0 k t 2 t 2 = ( + ) ln h 0 + ln t + ln t ln k t 2 = ( + ) ln h 0 + ln t + ln t ln h 0 k t 3 t 3 = ( ) ln h 0 + ln t + ln t ln t ln k t 3 = ::::::: = t ln h 0 + ln t + ln t ln t 3 + :::::: t ln 0 + t ln k 0 Substituting this expression into equation (), we get v 0 (k 0; 0 ) = = = t E 0 fln( h 0 ) + [ t ln h 0 + ln t + ln t 2 + ::: + t ln 0 + t ln k 0 ]g + t fln( h 0 ) + ln h 0 ln( h 0) + = H 0 + H ln 0 + ln k 0 t+ ln h 0 + t ln 0 + t+ ln k 0 g + ln 0 ( )( ) ln h ln 0 + ln k 0 where H 0 ; H are constants. Now we substitute this value function into Bellman equation v (k; ) = k 0 fln(ak k 0 ) + Ev 0 (k 0 ; 0 )g = k 0 fln(ak k 0 ) + E[H 0 + H ln 0 + ln k0 ]g 3
4 New policy h is chosen by imize the equation above. (Note that E ln 0 = 0) FOC is This implies Ak k 0 + ( )k 0 = 0 k 0 = k Therefore, we have h =. You can check h 2 = h through doing the procedure again. So in this case, algorithm converges to the optimal policy function just in one step. This is exactly the advantage of policy function iteration method. Its speed of convergence is much faster than that of value function iteration. 4. (Guess and Verify) Consider a social planner who faces the following problem: t u(c t ) fc t;l t;k t+ g c t + i t F (k t ; l t ) where utility function takes the form k t+ = ( )k t + i t c t 0; 0 < l t k 0 given u(c t ) = c t c 2 t ; > 0 Assume that c is always in the range where u 0 (c) is positive. Output is linear in capital, F (k; l) = Ak. Ans we assume = 0, i.e., no depreciation at all. (0 points) (a) Wirte down the Bellman equation for this problem. Be clear what are the state variables, what arr control variables. Answer: The Bellman equation is v(k) = fu(ak + ( )k k 0 ) + v(k 0 )g k 0 The state variable is k. the control variables are c and k 0. Notice that in the equilibrium, l t =. (b) Derive the Euler equation relating c t and expectations of c t+. Answer: The FOC is u 0 (c) = v 0 (k 0 ) The envelope condition is Combining together, we obtain EE v 0 (k) = u 0 (c)(a + ) u 0 (c) = (A + )u 0 (c 0 ) ) 2c = (A + )( 2c 0 ) 4
5 (c) Guess the consumption takes the form c t = H + F k t. Given this guess, what is the policy function for k t+? Answer: Substituting our guess c = H + F k into EE above, we have ) 2(H + F k) = (A + )( 2(H + F k 0 )) k 0 = ( 2H)(( + A) ) 2( + A)F (d) What is the value for the parameters H and F? Answer: Thus we have c t = (A + )k k 0 = (A + )k = + ( 2H)(( + A) ) 2( + A)F ( 2H)(( + A) ) 2( + A)F ( + A) k ( + A) k + ( + A)2 k ( + A) F = ( + A)2 ( + A) ( 2H)(( + A) ) H = : 2( + A)F 5. (Guess and Verify again) Consider the following problem where utility function takes the form E 0 X A t+ R t (A t c t ) A 0 > 0 given. u(c) = c ; > 0 t u(c t ) and R t is an i.i.d. shock such that ERt < ; > > 0. It is assumed that c t must be chosen before R t is observed. Show that the optimal policy function takes the form c t = A t and give an explicit formula for. (Hint: Guess the value function takes the form v(a) = BA for some constant B.) (5 points) Answer: The FE for this economy is v(a) = c;a 0 f c A 0 R(A c) 5 + Ev(A0 )g
6 Let s guess v(a) = BA Substituting this guess into the FE above, we have FOC w.r.t. c is which implies BA = c = c = c where H B( )E(R ). f c + EBA0 g f c + EB(R(A c)) g f c + B(A c) E(R )g: (2) c = B( )(A c) E(R ) c = H = A (3) + H = Substituting the optimal policy function (3) into the RHS of Bellman equation (2) to replace c, we have " BA H = # = + BE(R ) A + H = + H = After some manipulations, we can obtain B = f = [E(R )] = g This result veri es our conjecture about the form of the value function. Given the value of B, we have c = f = [E(R )] = ga: 6. (Optimal growth model with two types of agent) Consider the standard optimal growth model with production function F (k t ; l t ). Suppose now that there are an equal number of two types of HHs with preferences given by P u (c t ) and P u 2(c 2t ) respectively. Initial ownership of the capital stock is given by k 0 and k 20. (20 points) (a) De ne the social planner s problem for this economy. Answer: Denote j 2 (0; ); 8j = ; 2; P 2 i= j = the weight that social planner 6
7 assigns to agent i. The planner s problem is t [ u (c t ) + 2 u 2 (c 2t )] c t + k t+ ( )k t F (k t ; l t ) c t + c 2t = c t k t + k 2t = k t l t + l 2t = l t c t ; c 2t 0; 0 l t ; k 0 ; k 20 given. (b) De ne an ADE. Answer: An ADE for this economy is an allocation f(c t ; c 2t ; k t+ ; k 2t+ ; l t ; l 2t )g for agents, an allocation fk t ; l t g for the rm, and a price system f(p t ; w t ; r t )g such that (i). Given prices, f(c it ; k it+ ; l it )g ; 8i = ; 2 solves Agent i s problem p t (c it + k it+ ) t u i (c it ) p t ((r t + )k it + w t l it ) c it ; k it+ 0; 0 l it l i ; k i0 given (ii). Given prices, fk t ; l t g solves the rm s problem (iii). Market clears. = p t [y t r t k t w t l t ] y t F (k t ; l t ) k t 0; 0 l t c it + i= k it+ = F ( i= k it ; i= l it ) + ( ) i= l + l 2 {z} normalization i= k it (c) Set up the social planner s problem as a dynamic programming problem. 7
8 Answer: The FE is c it + i= v(k t ; k 2t ) = fu (c t ) + u 2 (c 2t ) + v(k t+ ; k 2t+ )g k it+ = F ( k it ; i= i= l it ) + ( ) i= i= c it ; k it+ 0; 0 l it l i ; k i0 given, 8i = ; 2 (d) De ne a SME. Answer: A SME for this economy is an allocation f(c t ; c 2t ; k t+ ; k 2t+ ; b t+ ; b 2t+ ; l t ; l 2t )g for agents, an allocation fk t ; l t g for the rm, and a price system f(q t ; w t ; r t )g such that (i). Given prices, f(c it ; k it+ ; b it+ ; l it )g ; 8i = ; 2 solves Agent i s problem t u i (c it ) c it + k it+ + q t b it+ (r t + )k it + w t l it + b it c it ; k it+ 0; 0 l it l i ; k i0 ; b i0 given (ii). Given prices, fk t ; l t g solves the rm s problem 8t (iii). Markets clear. F (k t ; l t ) r t k t w t l t k t 0; l t 0 k it c it + i= k it+ = F ( k it ; l it ) + ( ) i= b it+ = 0 i= i= i= i= k it 7. Consider a social planner who faces the following problem: fc t;l t;k t+ g c t + i t F (k t ; l t ) t u(c t ) k t+ = ( )k t + i t c t 0; 0 < l t k 0 given Now we assume u(c) = ln c, production function F (k t ; l t ) = kt lt and =. Establish that you indeed have a solution for this problem. (Hint: I am not asking you to solve the 8
9 solution, instead I am asking you to prove the existence of a solution to the sequential problem above.) (20 points) Answer: To establish that we indeed have a solution for this sequential problem, we want to rst transform the sequential problem into a DP problem. then we want to show that we have a solution for the FE. Next we want to apply Theorem 4.3 in SLP (or Theorem 25 on my lecture notes) to claim that this solution is also a solution for the SP. The FE for this problem is T v(k) = fln(c) + v(k 0 )g c;k 0 c + k 0 F (k; ) = k c; k 0 0 First, we can use Blackwell s su cient conditions for a contraction (Theorem 3.3 in SLP page 54) to con rm that the operator T de ned above is a contraction (see the lecture notes). We then use Contraction Mapping Theorem (CMT, Theorem 3.2 in SLP page 50, or Theorem 5 in my lecture notes) to show that T has exactly one xed point T v = v which is the solution to FE. In order to apply Theorem 4.3 in SLP, we have to check two assumptions: A.4.: (k) = (0; k ) and k 0 > 0 =) (k) is non-empty. A.4.2: k t+ k t =) ln k t+ ln k t =) ln k t t ln k 0 ; 8t: This in turn implies F (k t; k t+ ) = ln(k t k t+ ) < ln k t t+ ln k 0 ; 8t Since 0 < <, with nite k 0, it is easy to show F (k t; k t+ ) < ln k 0, i.e., is bounded from above. It is also bounded below because the Inada condition on consumption guarantee c > 0 which means F (k t; k t+ ) > ln 0 =. And we know 0 < < : So according to SLP page 69, the su cient condition for A.4.2 is satis ed here. (a) To use Theorem 4.3 we also need to check whether lim n! n v (k n ; z n ) = 0: But we already know (from the homework ) that in this case value function takes the form v(k) = B + F ln k, where B and F are both nite constant numbers and optimal policy function is k 0 = k. Thus lim n! n v (k n ; z n ) = lim n (B + F ln k n ) n! = lim n (B + F ln k n n! ) = lim n (B + 2 F ln k n n! 2 ) = ::: = lim n (B + n F ln k 0 ) n! = B lim n + F ln k 0 lim () n n! n! 9
10 Given nite k 0 ; and 0 < ; <, it s easy to see lim n! n v (k n ; z n ) = 0: Applying Theorem 4.3 we have v = v, therefore the solution to FE is indeed also a solution to SP. Thus we do have a solution to SP. 8. (Recursive Competitive Equilibrium with labor-leisure choice) Consider the following optimal growth model with labor-leisure choice fc t;l t;k t+ g c t + i t F (k t ; n t ) t u(c t ; l t ) k t+ = ( )k t + i t c t 0; n t ; l t > 0; n t + l t = k 0 given where n t is the labor input and l t denotes leisure. (5 points) (a) De ne a SME for this economy. Answer: A SME for this economy is an allocation fc t ; n s t; l t ; k s t+; s t g for HHs; an allocation fk d t ; n d t g for the rm; and a price system fq t ; w t ; r t g such that (i). Given prices, fc t ; n s t; l t ; k s t+; s t g ; 8i = ; 2 solves HH s problem t u(c t ; l t ) c t + k s t+ + q t s t+ (r t + )k s t + w t n s t + s t c t ; k t+ 0; 0 n s t ; n s t + l t = ; k 0 > 0 given (ii). Given prices, fk d t ; n d t g solves the rm s problem 8t (iii). Markets clear. F (k d t ; n d t ) r t k d t w t n d t k d t 0; n d t 0 c t + k t+ ( )k t = F (k t ; n t ) k d t = k s t = k t n d t = n s t = n t n t + l t = : (b) De ne a RCE for this economy. Answer: A Recursive Competitive Equilibrium (RCE) is a value function V : R 2 +! R, a policy function H : R 2 +! R for the representative household, an 0
11 economy-wide law of motion h : R +! R t for capital, and factor price functions R : R +! R + and $ : R +! R + such that (i). Given h; R; $, V solves HH s problem and H is the optimal policy function of HH s problem V (K; k) = C;Y;L fu(c; C + [Y ( )K] KR(k) + $(k)n (ii). R and $ satisfy the rm s FOCs. C; Y 0; 0 N R(k) = F k (k; n) $(k) = F n (k; n) N) + V [Y; h(k)] (iii). Individual choice is consistent with the aggreagte law of motion (consistency condition) and markets clear. H(k; k) = h(k) N = n K = k: 9. (Value function iteration) Consider the following optimal growth problem t ln c t c t + k t+ kt c t ; k t+ 0; k 0 given We assume = 0:97; = 0:36; = 0:06. Write a Matlab code that takes k 0 = 0:75k and iterates towards k, where k is the steady state value of capital stock. Using " = 0:000 as a convergence criterion. Plot the value function v(k) and the optimal policy function k 0 = g(k). How many time periods does it take to converge to the steady state? Do the same thing from k 0 = 0:0 k and compute the number of time periods to converge. Now change from 0.36 to 0.30, compute the number of time periods to convergence. How does the speed of convergence depend on? Why? (5 points) Answer: Under the parameter values de ned above, when k 0 = 0:75k, using value function iteration, we achieve convergence by 2 iterations. We plot the value function and optimal policy function in the gures below. When k 0 = 0:0 k, we achieve the convergence by 2 iterations. When is changed from 0.36 to 0.30, no matter we start from k 0 = 0:75 k or 0:0 k, we achieve the convergence by 0 iterations, less than = 0:36 case. This seems to suggest that the
12 g(k) v(k) 8 Value function:α=0.36,β=0.97,δ= k 8 Policy function:α=0.36,β=0.97,δ= k 2
13 convergence speed is negatively related to. What s the economic intuition behind this phenomenon? Let s take a deeper look at the behavior of capital stock over time: _k = k 0 k = k k Here we use the result of optimal policy function k 0 = k. Doing the rst-order Taylor Expansion around the steady state value k : _k ' (k k ) + ( 2 k )(k t k ) = ( 2 k )(k t k ) (4) Substituting k = () into (4), we get _k ' ( )(k t k ) De ne x(t) = k t k as the distance between current capital stock and the steady state capital;so _x(t) = k _ ) x (t) _ ' ( )x(t) = ( )x(t) ) x _ (t) x(t) ' ( ) From this, it s easy to see x(t) ' x(0)e ( )t Transforming back to k k t k ' e ( )t (k 0 k ) That means the speed of convergence is nearly : Each iteration capital stock removes 0 < < proportion of the reminded distance toward k : Now we understand why when " say from 0.30 to 0.36, the speed of convergence decreases. 0. (LQ approximation) Consider a HH who faces the following problem: fc t;i tg c t + i t = ra t + y t a t+ = a t + i t t f(c t b) 2 + i 2 t g y t+ = y t + 2 y t c t 0; y 0 ; y given 3
14 where c t ; i t ; a t ; y t are the HH s consumption, investment, asset holdings and exogenous labor income at time t. And we have b > 0; > 0; 2 (0; ) and ; 2 are parameters. Assume that ; 2 are such that ( z 2 z 2 ) = 0 implies jzj > : (5 points) (a) Write down the Bellman equation for this problem. Be clear what are the state variables, what are control variables. Answer: The Bellman equation is v(a t ; y t ; y t ) = f c t;i t (c t b) 2 i 2 t + v(a t+ ; y t+ ; y t )g c t + i t = ra t + y t a t+ = a t + i t y t+ = y t + 2 y t c t 0; y 0 ; y given (b) Map this problem into an discounted optimal linear regulator problem. Answer: Using the BC c t = ra t + y t i t, the imization problem is t (ra t + y t i t b) 2 + i 2 t : (5) t= Extending it we can observe the existence of the cross terms of the control variable (i t ) and state variables (a t ; y t ), therefore, we wish to map it into the form t (x 0 trx t + u 0 tqu t + 2u 0 tw x t ) t= where we have x t+ = Ax t + Bu t x t+ = (; a t+ ; y t+ ; y t ) A = B = u t = i t Extending (5) and after some algebra, we have 2 R = 6 4 Q = ( + ) b 2 rb b 0 rb r 2 r 0 b r W = b r 0 ; therefore completes the transformation
15 (c) For parameter values (; ( + r); b; ; ; 2 ) = (0:96; 0:96 ; 30; ; :2; 0:3), write a Matlab code to compute the value function and the optimal policy function for this DOLRP by using value function iteration method. Answer: To compute the optimal policy function we can rst transform the above problem into the standard form, using and we will have u t = u t + Q W x t R = R W 0 Q W A = A BQ W P = R + A 0 P A AP B (Q + B 0 P B) B 0 P A F = (Q + B 0 P B) BP A u t = F xt F = F Q W; and the problem is changed to t x 0 trx t + u 0 tqu t x t+ = Ax t + Bu t : Given the tolerance degree of 0.000, we can get the value function as 2 3 :0737 0:000 0:030 0:0038 P = :0e :000 0:0000 0:0000 0: :030 0:0000 0:0004 0: :0038 0:0000 0:000 0:0000 and optimal policy function F = [4:6876 0:0208 0:487 0:003] : That is i t = 4: :0208a t + 0:487y t + 0:003y t : (d) For the same parameters, write a Matlab code to compute the value function and the optimal policy function for this DOLRP by using policy function iteration method. Answer: We should obtain the identical results as in part (c).. (Wealth inequality in a two-period economy) Suppose that there are two types of consumers distinguished by their initial endowments of capital. In particular, type- consumers (who comprise fraction of the population) are richer than type-2 consumers (who comprise fraction of the population): type- consumers are endowed with k0 units of capital and type-2 consumers are endowed with k0 2 units of capital, where 5
16 k0 > k0. 2 The two types of consumers are identical in all other respects. Each consumer takes prices as given (in particular, each consumer takes the aggregate, or total, capital stock in period as given) when making savings decisions in period 0. The equilibrium (or consistency) condition is that the total savings of the two types of consumers in period 0 must equal the aggregate capital stock that consumers take as given when deciding how much to save. Assume that each consumer s utility function takes the form u(c 0 ) + u(c ) with u(c) = log(c). The production technology available to rms is: y = k n, with > > 0, where y is the rm s output and k and n are the services of capital and labor, respectively. (0 points) (a) Derive the equilibrium aggregate capital stock in period as a function of primitives (i.e., the parameters, and, and the initial capital stock k 0 and k 2 0). (b) Use your answer to part (a) to show that changes in k0 and k0 2 that keep aggregate capital in period 0 (i.e., k0 + ( )k0) 2 constant have no e ect either on equilibrium aggregate savings or on equilibrium prices. This is a version of an aggregation theorem for this economy: holding the total amount of capital in period 0 constant, the behavior of the aggregates in this economy does not depend on the distribution of capital in period 0. Answer: In this question we have two types of consumers: there are a fraction of type consumers, who are endowed with k0 unit of capital in period 0. The rest of the consumers, a fraction of, are endowed with k0, 2 where k0 > k0. 2 Then, given the log utility function, the problem that a consuer of type i = ; 2 must solve is: which implies the FOC: log c i c i 0 + log c i 0 ;ci c i 0 = r 0 k i 0 + w 0 k i c i = r k i + w r k i + w = r r 0 k i 0 + w 0 k i i = ; 2 On the other hand, each period i = 0; ;the rm must solve: which implies the competitive prices: K i N i K i r i N i w i K i, N i w = ( ) K and r = K Also, the CRS production technology implies that : r 0 K 0 + w 0 N 0 = y 0 = K 0 N 0 6
17 () Finally, the equilibrium conditions are: K = k i + ( ) k 2 i for i = ; 2 and as labor is inelastically supplied, N i = for i = 0; : Now we are ready to solve for the equilibrium. First, multiplying each type s FOC equation by its fraction in the population and suming up both equations, we get: r k + ( ) k 2 + w = r r0 k 0 + ( ) k w0 k + ( ) k 2 Using the equilibrium conditions and eq.() : then, replacing the competitive prices, which implies r K + w = r K 0 K K K + ( ) K = K K 0 K K = K 0 + Thus, as period aggregate savings only depends on aggregate period 0 savings, and not on how much has each consumer, if this aggregate is not changed, K won t change either. And as prices depend also only on aggregates, changes in the distribution of capital endowments, that leave the aggregate equal, won t a ect them. (c) Suppose that the felicity function takes the form: u(c) = c where > 0. Does an aggregation theorem like the one described in part (b) hold for this economy? Explain why or why not. Answer: So now we just have to change the consumers utility function. Each type s problem is: which implies the FOC: c i 0 ;ci (c i 0) c i 0 = r 0 k i 0 + w 0 k i c i = r k i + w + (ci ) r 0 k i 0 + w 0 k i = (r ) = r k i + w The rm s problem has not changed, so we have the same equations for prices. Following the same steps of part (a), we get that: r 0 K 0 + w 0 K = (r ) = r K + w 7
18 then, remembering that wages and interest rates only depend on aggregates, from eq. above, which implicitly de nes K, we can conclude that if K 0 does not change then K won t change also, and thus, the aggregation results holds here too. 8
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