ECON607 Fall 2010 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment 1 Suggested Solutions

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1 ECON607 Fall 200 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment Suggested Solutions The due date for this assignment is Thursday, Sep Consider an stochastic optimal growth model as in the lecture with utility function u(c) = ln c, production function f(k) = k, and resource constraint c t +k t+ e zt f(k t ), where z t is the i.i.d. productivity shock belongs to a lognormal distribution. (5 points) (a) Write the Bellman equation for the social planner s problem. Answer: v(k; z) = max k k 0 ) + E[v(k 0 ; z 0 )]g () 0k 0 e z k fln(ez (b) Use Guess and Verify method to slove the Bellman equation. v(k; z) = H + F ln k + Gz:) Answer: Let s guess value function takes form v(k; z) = H + F ln k + Gz (Hint: Guess What we need to do is to verify this is a right guess and determine the values of constant term H; F and G. Substituting our guess into the RHS of the Bellman equation (BE), we obtain v(k; z) = max k 0 fln(e z k k 0 ) + E[H + F ln k 0 + Gz 0 ]g = max k 0 fln(e z k k 0 ) + E[H + F ln k 0 ]g ( * E(z 0 ) = 0) (2) FOC w.r.t. k 0 implies e z k k + F = 0 0 k 0 From it we have the optimal policy function as following k 0 = Put this policy function back into equation (2), we have F + F ez k (3) V (k; z) = H + F ln k + Gz = fln(e z k F + F ez k F ) + H + F ln + F ez k g = ln( + F ) + F ln( F ) + H + ( + F ) ln k + ( + F )z + F

2 To match coe cients, we need to have H = ln( + F ) + F ln( F + F ) + H F = ( + F ) G = ( + F ) which implies F = H = H = [ln( ) + ln ]: 2. Prove that the budget constraint in sequential market equilibrium (SME) as in the lecture is as same as the one in ADE, i.e., BC in SME ) BC in ADE. (5 points) Answer: BC in SME is Let s start from period 0 BC c t + q t s t+ + i t = r t k t + w t n t + s t + t ; 8t (4) c 0 + q 0 s + i 0 = r 0 k 0 + w 0 n 0 + s Rewrite c 0 + i 0 = r 0 k 0 + w 0 n s 0 q 0 s (5) From the BC above, we know s = c + i + q s 2 (r k + w n + ) Substituting s into equation (5), we have (c 0 + i 0 ) + q 0 (c + i ) + q 0 q s 2 = r 0 k 0 + w 0 n 0 + q 0 (r k + w n ) q 0 + s 0 Then we can also substitute out s 2. Repeatly doing like this, we end up with Yt TY q s (c t + i t ) + q t s T + = s=0 Yt q s (r t k t + w t n t + t ) + s 0 s=0 Yt Let q s = p t and notice that s T + = 0 and s 0 = 0, we have s=0 p t (c t + i t ) = p t (r t k t + w t n t + t ) 2

3 Notice that Therefore, we have p t t = p t (c t + i t ) = p t (r t k t + w t n t ) + This is exactly is BC in ADE for nite horizon case. For the in nite horizon case, we need a transversality condition as following T! TY q t s T + = 0 to guarantee the identical BCs in SME and ADE. 3. (Two-period Emdowment Economy) Consider following two-period pure exchange economy. There is a single consumption good in each period. There are two HHs who have identical preferences over consumption given by u(c ; c 2 ) = ln c + ln c 2 HH has endowments given by (w ; 0) and HH 2 has endowments (0; w 2 ). (20 points) (a) De ne an ADE for this economy. Answer: An ADE is a set of prices fp ; p 2 g and an allocation fc ; c 2; c 2 ; c 2 2g such that (i). Given prices, the allocation solves HH s problem: HH s problem HH2 s problem (ii). Markets clear. p c + p 2 c 2 p w c ; c 2 0 p c 2 + p 2 c 2 2 p 2 w 2 c 2 ; c max ln c c + ln c 2 ;c 2 max ln c 2 c 2 + ln c 2 2 ;c2 2 s:t c + c 2 = w c 2 + c 2 2 = w 2 3

4 (b) Suppose HHs has incentive to smooth their consumption over time by borrowing and lending. De ne a SME for this economy. Show that the ADE and SME give identical allocations. Answer: A SME is a set of price r, an allocation fc i t; s i tg 2 t=; i = ; 2 such that (i). Given r, fc i t; s i tg 2 t=; i = ; 2 solves HH s problem. HH s problem HH2 s problem c + s w c 2 + s 2 ( + r)s c ; c 2 0 max ln c c + ln c 2 ;c 2 ;s ;s 2 (ii). Markets clear. c 2 + s 2 0 max ln c 2 c 2 + ln c 2 2 ;c2 2 ;s2 ;s2 2 c s 2 2 w 2 + ( + r)s 2 c 2 ; c c + c 2 = w c 2 + c 2 2 = w 2 s + s 2 = 0 s 2 + s 2 2 = 0 Now let s solve SME. First of all, notice that we should have s 2 = s 2 2 = 0. FOCs for HH s problem: c : c = 0 c 2 : c 2 2 = 0 Combining FOCs, we have EE s : + ( + r) 2 = 0 c 2 c = ( + r) (6) 4

5 Similarly, we have EE for HH2 c 2 2 c 2 = ( + r) (7) Combining (6) and (7), we get c 2 c = c2 2 c 2 Notice that as in problem 2, we can combine two period BCs into one BC through substituting out s in HH s problem and substituting out s 2 in HH2 s problem. Hence HH s BC becomes c + + r c 2 w Similarly HH2 s BC is c r c2 2 w Now let s turn to ADE. FOCs for HH in ADE are c : c p = 0 c 2 : c 2 p 2 = 0 Combining two FOCs, we obtain EE c 2 c = p p 2 Similarly EE for HH2 is Hence we also have Notice that it is easy to show c 2 2 c 2 c 2 c = p p 2 = c2 2 c 2 c 2 c = c2 2 c 2 = c 2 + c 2 2 c + c 2 = w 2 w Comparing EEs in ADE and SME, we conclude p p 2 = + r Normailze p =, then p 2 =, substituting these expression into BCs in ADE, +r we end up the same BCs as in SME. Thus, SME and ADE share the identical objective function, identical BCs and identical EEs, it implies the solotions to ADE and SME must be same. 5

6 c = c 2 =! + ; c2 =! +! 2 + ; c2 =! 2 + : (c) Suppose w = w 2 = and =, calculate the competitive equilibrium. Answer: c = c 2 = c 2 = c 2 2 = 2 s = 2 ; s2 = 2 p = p 2 = r = 0 (d) Suppose HHs cannot borrow or lend from each other. Try to de ne a competitive equilibrium in this case. Show that a policy which transfers goods from the rich to the poor in each period in Pareto-improving. Answer: A CE is a price system fp ; p 2 g, an allocation fc i tg 2 t=; i = ; 2 such that (i). Given prices, the allocation solves HH s problem: HH s problem HH2 s problem c w c 2 0 c ; c 2 0 max ln c c + ln c 2 ;c 2 (ii). Markets clear. c 2 0 c 2 2 w 2 c 2 ; c max ln c 2 c 2 + ln c 2 2 ;c2 2 c + c 2 = w c 2 + c 2 2 = w 2 Obviously the solution is an autarky equilibrium c = w ; c 2 = 0; c 2 = 0; c 2 2 = w 2 6

7 Now we show that any transfer mechanism will Pareto dominates the autarky equilibrium. Suppose in each period we trnasfer 0 < T < w from rich to poor. Now for HH, life time utility is ln(w T ) + ln T. Obviously we have Simiarly, we have for HH2 ln(w T ) + ln T > ln w + ln 0 = ln T + ln(w T ) > ln 0 + ln w = : 4. (When Robinson Crusoe meets Arrow-Debreu) Consider a Robinson-Crusoe economy. Crusoe as the single HH on the island has a utility function over consumption of sh c t and leisure ( l t ) as following t [ln c t + ln( l t )] where 2 (0; ) and > 0. Crusoe has a technology transforming labor into output of sh by y t = A t l t where 2 (0; ), fa t g T is a sequence of numbers which measures Crusoe s state of productivity. The sh can only be used for consumption (Robinson does not have a refrigerator). (25 points) (a) Set up the social planner s problem for this economy. Answer: Social planner s problem is max fc t;l tg T t [ln c t + ln( l t )] c t A t l t ; t = 0; ; 2; ::; T c t 0; 0 l t Solving this problem using FOC ends up with ) l t = l t lt = + c t = A t (lt ) = A t ( + ) : (b) One interpretation of the production technology is there is a xed supply of one boat (or catamaran, denoted by k) so that the technology is y t = A t kt lt : 7

8 Assume that the boat is owned by Crusoe and the technology is operated by a rm (Dole Fishing Co.) which rents capital k t and labor l t in a competitive market at factor prices r t and w t. Let p t denote the Arrow-Debreu price of one unit of consumption at time t.. Set up Crusoe s decison problem as a HH problem. Answer: Crusoe s problem is p t c t max t [ln c t + ln( l t )] fc t;l tg T p t (r t kt s + w t lt s ) 2. Set up the rm s problem. Answer: The rm s problem is c t 0; 0 l s t ; 0 k s t k max = fkt d;ld t gt p t [y t r t kt d w t lt d ] y t A t (k d t ) (l d t ) l d t 0; k d t 0 This is equivalent to a one period problem max t = p t [y t r t kt d w t lt d ] kt d;ld t y t A t (k d t ) (l d t ) l d t 0; k d t 0: 3. What are the resource constraint? Answer: c t A t kt lt : 4. De ne a competitive equilibrium. Answer: A CE is a set of prices fp t ; r t ; w t g T, an allocation fk d t ; l d t ; y t g T for the rm, and an allocation fc t ; k s t ; l s t g T for the HH, such that (i). Given prices, fc t ; k s t ; l s t g T solves HH s problem. (ii). Given prices, fk d t ; l d t ; y t g T solves the rm s problem. (iii). All markets clear. k d t = k s t = k l d t = l s t = l t c t = A t k l t : 8

9 5. Solve for the CE allocations. Answer: Firm s FOCs HH s EE r t = A t ( )k l t (8) w t = A t k l t (9) w t c t = l t (0) Substituting (9) into equation (0) to replace w t, and using resource constraint c t = A t k lt, we have lt e = + c t = A t (lt ) = A t ( + ) : 6. How does the CE change when A 0 rises? When A rises? Answer: When A 0 ", obviously c 0 ", but (c ; c 2 ; :::; c T ) are unchanged. (Because there is no storage technology to transform higher productivity today to smooth consumption.) r 0 and w 0 will also increase at the same rate as c 0. fl t g T remains unchanged. Same things happen to the rise in A. 5. (Irrelevance of capital ownership) Exercise 2.9 in SLP. (0 points) Answer: Under this new setup. the HH s problem is p t c t The rm s problem changes to max fc t;n s t g t U(c t ) p t w t n s t + c t 0; 0 n s t max = fy t;i t;n d t g p t [y t w t n d t i t ] i t = k t+ ( )k t y t F (k t ; n d t ) k t 0; n d t 0 FOC for the HH is t U 0 (c t ) = p t () 9

10 FOCs for the rm are Substituting () into (3), we obtain EE n t : w t = F n (k t ; n t ) (2) k t+ : p t = p t+ [F k (k t+ ; n t+ ) + ] (3) U 0 (c t ) = U 0 (c t+ )[F k (k t+ ; n t+ ) + ] This is the same EE as in the CE which HHs have the capital ownership. Besides, we can substitute the expression of rm s pro ts into HH s BC ) p t c t = p t w t n t + = = p t w t n s t + p t [y t i t ] p t [y t w t n d t i t ] p t (c t + i t ) = p t y t = p t [r t k t + w t n t ] Therefore, these two problems are identical. 6. (Continuity of contraction mapping) Exercise 3.8 in SLP. (5 points) Answer: The function T : S! S is uniformly continuous if 8" > 0, 9 > 0 such that for all x; y 2 S with jx yj < we have that jt x T yj < ". We already know that if T is a contraction, then for some 2 (0; ), we have jt x jx T yj yj <, forallx; y 2 S with x 6= y Now let " > 0, then for any arbitray " > 0, if jx yj <, then jt x T yj jx yj < " = ": 7. (An example of contraction mapping) Exercise 3.0 in SLP. (0 points) Answer: a). Since v is bounded, so the continuous function f is bounded, say by M, on [ kvk ; + kvk]. Then it is easy to show that 0 < jt v(s)j jcj + s kvk, which the right hand side is clearly positive and bounded. Therefore, 8t > 0, T v is bounded on [0; t]. The continuity of T V is straightforward. 0

11 b). Note that jt v(s) T w(s)j Z s 0 Z s 0 jf(v(z)) B jv(z) Bs kv wk : f(w(z))j dz w(z)j dz Choose = =B, where 2 (0; ), then 0 s implies that Bs kv kv wk. We then have wk Put the sup on both side, we have jt v(s) T w(s)j kv wk ; 8v; w 2 C[0; t] kt v(s) T w(s)k kv wk ; 8v; w 2 C[0; t]: c). Suppose the xed point is x 2 C[0; ], with hence for 0 s; s 0, Therefore, we have x(s) x(s 0 ) = x(s) = c + Z s s 0 Z s 0 f(x(z))dz f(x(z))dz; = f(x(^z))(s s 0 ); for some ^z 2 [s 0 ; s]: x(s) x(s 0 ) s s 0 = f(x(^z)) Let s 0! s, then ^z! s, so we have x 0 (s) = f(x(s)). The uniqueness comes directly from the Contraction Mapping Theorem. 8. (a) We start with the rst period s budget constraint: For t = we have: c + qb 2 = b + w c 0 + qb 2 = b 0 + w b = q (b 0 + w c 0 ) b 2 = q (b + w c ) = q (q (b 0 + w c 0 ) + w c ) = q 2 b 0 + q 2 w + q w q 2 c 0 q c

12 For t = 2 we have: c 2 + qb 3 = b 2 + w b 3 = q (b 2 + w c 2 ) and so on, until we have: = q (q 2 b 0 + q 2 w + q w q 2 c 0 q c + w c 2 ) = q 3 b 0 + q 3 w + q 2 w + q w q 3 c 0 q 2 c q c 2 b (t+) = q (t+) b 0 + w(q (t+) + ::: + q ) (q (t+) c 0 + q t c + ::: + q c t ) q (t+) b (t+) = b 0 + w( + ::: + q t ) (c 0 + qc + ::: + q t c t ) tx tx = b 0 + w q k q k c k k=0 taking the it gives us: k=0 t! q(t+) b (t+) = b 0 + w t! 0 = b 0 + w q tx k=0 q k q t c t t! tx q k c k where we used the no-ponzi game condition on the LHS. So the consumer s consolidated (or lifetime) budget constraint is (b) The transversality condition is: q t c t = b 0 + w q k=0 or the Euler equation is t! t u 0 (c t )b t = 0 t! t u 0 (b t qb (t+) + w)b t = 0 qu 0 (b t qb t+ + w) = u 0 (b t+ qb t+2 + w) We want to show q t b t = 0: Note that the Euler equation holds 8t qu 0 (b 0 qb + w) = u 0 (b qb 2 + w) qu 0 (b qb 2 + w) = u 0 (b 2 qb 3 + w) q 2 u 0 (b 0 qb + w) = 2 u 0 (b 2 qb 3 + w) 2

13 So we have q t u 0 (b 0 qb + w) = t u 0 (b t qb t+ + w) Multiplying both sides by b t and taking its: t! qt u 0 (b 0 qb + w)b t = t u 0 (b t qb t+ + w)b t t! u 0 (b 0 qb + w) q t b t = 0 (using the transversality condition) t! and so t! qt b t = 0 (c) We want to prove that a sequence {b t g = 0 that satis es the transversality condition and the Euler equation maximizes the consumer s objective function, subject to the sequence of budget constraints and the npg condition. Modi ed proof: Consider any alternative feasible sequence b = fb t g :We want to show that for any such sequence, t [u(b t qb t+ + w) u(b t qb t+ + w)] 0 De ne T! A T (b) = t [u(b t qb t+ + w) u(b t qb t+ + w)] 0 We will show that, as T goes to in nity A T (b) is bounded below by zero. By convavity of u, A T (b) t [u 0 (b t qb t+ + w)(b t b t ) qu 0 (b t qb t+ + w)(b t+ b t+ )] = [ t u 0 (b t qb t+ + w)(b t b t )] [ t qu 0 (b t qb t+ + w)(b t+ b t+ )] = u 0 (b 0 qb + w)(b 0 b 0 )] + T [ t u 0 (b t qb t+ + w)(b t b t ) t= X t qu 0 (b t qb t+ + w)(b t+ b t+ ) T qu 0 (b T qb T + + w)(b T + b T + )] XT = [u 0 (b 0 qb + w)(b 0 b 0 )] + t+ u 0 (b t+ qb t+2 + w)(b t+ b t+ ) T [ X t qu 0 (b t qb t+ + w)(b t+ b t+ ) T qu 0 (b T qb T + + w)(b T + b T + )] = u 0 (b 0 qb + w)(b 0 b 0 ) T qu 0 (b T qb T + + w)(b T + b T + ) + XT t (b t+ b t+ )[u 0 (b t+ qb t+2 + w) qu 0 (b t qb t+ + w)] 3

14 Note that: (b 0 b 0 ) = 0 =) u 0 (b 0 qb + w)(b 0 b 0 ) = 0 u 0 (b t+ qb (t+2) + w) qu 0 (b t qb t+ + w) = 0 (Euler Equation): And so we have Taking its we have Recalling that A T (b) T qu 0 (b T qb T + + w)(b T + b T + ) = T qu 0 (b T qb T + + w)( b T + + b T + ) = T u 0 (b T + qb T +2 + w)( b T + + b T + ) A T (b) T! T! T + u 0 (b T + qb T +2 + w)( b T + + b T +) q t u 0 (b 0 qb + w) = t u 0 (b t qb t+ + w) and We have u 0 (b 0 qb + w) = t q t u0 (b t qb t+ + w) t! qt b t = 0 t! t u 0 (c t )b t = 0 A T (b) T! T! T + u 0 (b T + qb T +2 + w)( b T + + b T +) = T! T + q T + qt + u 0 (b T + qb T +2 + w)b T + T! T + u 0 (b T + qb T +2 + w)b T + = T! u0 (b 0 qb + w)q T + b T + = 0 9. (a) Carefully de ne a sequential competitive equilibrium for this economy. A sequential competitive equilibrium for the economy {u A ; u B ; wg; is a sequence fc itg ; fa i;t+g ; fq t g (where q t means price of Arrow security) for i = A; B such that. For i = A; B; c it ; a i;t+ = arg max c it + qt a i;t+ = a i;t +!! Y a i;t+ q t 0 t! a i;0 = 0; c it 0 4 t i log (c it )

15 2. c At + ( )c Bt = w for t = 0; ; 2::: 3. a A;t+ + ( )a B;t+ = 0 for t = 0; ; 2::: (b) Carefully de ne a recursive competitive equlibrium for this economy. A Recursive Competitive Equilibrium (RCE) for the economy fu A ; u B; wg is a set of functions: price funtion: q(k); policy function: a 0 i = g i (a i ; K); value function: v i (a i; K); transition function: K 0 = G(K) (K t = P a A;t is the aggregate asset holdings at time t by type-a agents; a i is the individual asset holdings for the representative agent in each type i) such that:. For i = A; B; a 0 i = g i (a i; K) and v i (a i ; K) solve 2. Consistency v i (a i; K) = max fc i; a 0 i g log(c i ) + i v i (a 0 i; K 0 ) c i + q(k)a 0 i = a i + w K 0 = G(K) G(K) = g A (K; K) G(K) = g B( K; K) (c) Show that this economy has steady state: in particular, show that the type-b agents become poorer and poorer over time and consume zero in the it. To solve this problem, we rst get Euler equation. In recursive formulation, consumers solve v i (a i ; K) = max fc i ;a 0 i ) log(a i + w q(k)a 0 i) + i v i (a 0 i; K 0 ) K 0 = G(K) Solve for F.O.C. and use envelope condition, we get the euler equation i u 0 (c i;t+ ) u 0 (c i;t ) = q(k) or equivalently, A u 0 (c A;t+ ) u 0 (c A;t ) = B u 0 (c B;t+ ) u 0 (c B;t ) Now we can see that c i;t+ 6= c i;t (8i; 8t): Suppose not, without loss of generality let c A;t+ = c A;t : By feasibility condition, we know that c B;t+ = c B;t. Plug into the equation we get A = B, a contradiction. As a result, there cannot be any steady state in this economy. We start to prove the convergence property of consumption path. First, we want to show that [C At ] (fc Btg ) is an increasing (decreasing) sequence. We already 5

16 know that c A;t+ 6= c A;t (8t): Now suppose that c A;t+ < c A;t for some t. By the feasibility condition, we know that c B;t+ > c B;t. From the strict concavity of felicity function, we have ) u 0 (c A;t+ ) u 0 (c A;t ) A u 0 (c A;t+ ) u 0 (c A;t ) > > u0 (c B;t+ ) u 0 (c B;t ) > B u 0 (c B;t+ ) u 0 (c B;t ) which contradicts Euler equation. Since bounded montone sequence has a it, we have c At! c for t!. But we have shown that the economy has no steady state, so c At can converge to nowhere but the boundary, i.e. c At! w and c Bt! 0: 6