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Unformatted text preview: ECO 475 HW5 Solution Joon Song Oct 14, 2009 1. Small Open Economy Let b t +1 be an amount of bond purchased at time t that delivers Rb t +1 unit of consumption good at time t + 1 . Then, the sequential problem is the following. max 1 P t =0 & t log ( c t ) s.t. c t + k t +1 + b t +1 & p k t + (1 ) k t + Rb t c t k t +1 b t +1 k > and b given. (1) Recursive Problem State variables: k; b Control variables: c; k ; b Recursive Problem: V ( k; b ) = max c;k ;b [log c + &V ( k ; b )] s.t. c + k + b & p k + (1 ) k + Rb (1) c (2) k (3) b (4) Before we proceed, let us inspect constraints &rst. Constraint (1) becomes equality since periodutility func tion is strictly increasing in c . Constraint (2) becomes strict inequality since marginal utility of consumption goes to in&nity as c approaches to zero. Constraint (3) becomes strict inequality since marginal productivity 1 of capital goes to in&nity as k approaches to zero. In sum, the recursive problem turns out to be V ( k; b ) = max c;k ;b [log c + &V ( k ; b )] s.t. c + k + b = p k + (1 & ) k + Rb (5) c > (6) k > (7) b . (8) (2) Optimality Conditions Let ( 0) be a Lagrange multiplier for constraint (8) . By substituting c with p k +(1 & ) k + Rb & k & b , we get the following &rst order su cient and necessary (why?) conditions: k : 1 c = &V k ( k ; b ) b : 1 c = &V b ( k ; b ) + b complementary slackness Envelop conditions: V k ( k; b ) = 1 c & 1 2 p k + (1 & ) V b ( k; b ) = 1 c R Thus, Euler equations are the following: 1 c = 1 c & & 1 2 p k + (1 & ) 1 c = 1 c &R + with complementary slackness condition b , which leads to following two cases. Case1: > , b = 0 1 c = 1 c & & 1 2 p k + (1 & ) (9) 1 2 p k + (1 & ) > R 2 Case2: & = 0 , b > 1 c = 1 c & 1 2 p k + (1 & ) = 1 c R (10) 1 2 p k + (1 & ) = R Combining these with resource constraints (5) & (8) provides optimality conditions. The Euler equations are quite intuitive. Suppose you forgo one unit of consumption this period. Then, the loss in terms of current period utility is 1 c . With that forgone consumption in the current period, you can buy either one more unit of capital or one more unit of bond. If you buy capital, you can have additional 1 2 p k +(1 & ) unit of consumption good in the next period. Likewise if you buy bond, you can have additional R unit of consumption good in the next period. By comparing 1 2 p k + (1 & ) and R , you decide which asset you are going to buy. Evaluating these in term of the current period utility gives 1 c h 1 2 p k + (1 & ) i (buying more capital) or 1 c R (buying more bond). In the optimum, the loss and gain should be equalized. Therefore, the following holds....
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 Fall '07
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