Problem Set 9 Solutions. (a) Using the guess that the value function for this problem depends only on y t and not ε t and takes the specific form

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1 EC Math for Economists Samson Alva Department of Economics, Boston College December 9, 20 roblem Set 9 Solutions. Stochastic Linear-Quadratic Dnamic rogramming (a) Using the guess that the value function for this problem depends onl on t and not ε t and takes the specific form v( t, ε t ) = v( t ) = 2 t + d, where and d are unknown constants, the Bellman equation for this problem becomes t 2 + d = max Rt 2 + Qzt 2 + β E t [(A t + Bz t + ε t+ ) 2 ] + βd z t = max Rt 2 + Qzt 2 + β (A t + Bz t ) 2 + β σ 2 + βd z t (b) Using the expression on the last line from part (a), the first-order condition for z t becomes 2Qz t + 2βB (A t + Bz t ) = 0 and the envelope condition for t becomes 2 t = 2R t + 2βA (A t + Bz t ). Notabl, the introduction of uncertaint into the linear-quadratic problem does not alter the form of the optimalit conditions for z t and t. This special feature of the stochastic LQ model is often referred to as the propert of certaint equivalence, and it does not appl more generall for problems that do not have the linear-quadratic structure. (c) Rewrite the first-order condition as z t = ( ) βab Q + βb 2 t and substitute this result into the envelope condition to obtain ( ) β t = R t + βa 2 2 A 2 B 2 2 t Q + βb 2 t

2 or, after combining the last two terms and dividing through b t, = R + βa2 Q Q + βb 2. which is the same Riccati equation that helps characterize the problem s solution in the nonstochastic case, and therefore another reflection of the certainl equivalent propert. (d) Substituting the first-order condition for z t back into the Bellman equation ields ( ) 2 [ ( )] 2 βab βab t 2 +d = Rt 2 +Q 2 Q + βb 2 t +β A B Q + βb 2 t 2 +β σ 2 +βd. However, the Riccati equation implies that all of the terms involving t 2 out, leaving d = β σ 2 + βd, which can be solved for d = ( ) β σ 2. β cancel 2. Stochastic Growth (a) Using the guess that the value function for this problem takes the form v(k t, z t ) = E + F ln(k t ) + G ln(z t ), where E, F, and G are unknown constants, the Bellman equation for this problem can be written as E + F ln(k t ) + G ln(z t ) = max c t ln(c t ) + βe + βf ln(z t k α t c t ), where the term involving GE t [ln(z t+ )] drops out of the right-hand-side of the Bellman equation in light of the assumption that E t [ln(z t+ )] = 0. (b) Using the Bellman equation as it appears above, the first-order condition for c t is and the envelope condition for k t is (c) Rewrite the first-order condition as βf = 0 c t z t kt α c t F = αβf z tkt α. k t z t kt α c t c t = ( + βf 2 ) z t k α t

3 and substitute this expression into the envelope condition to obtain ( ) F z t kt α F z t kt α = αβf z t kt α, + βf which implies that + βf = αβ. In light of this last expression, the first-order condition for c t implies c t = ( αβ)z t k α t and the binding constraint for k t+ implies k t+ = αβz t k α t. These last two expressions confirm that the ke result from the perfect foresight case, that with complete depreciation is is optimal to consume the fixed fraction αβ of output and to save the remaining fraction αβ, carries over to the stochastic case as well. (d) The expression from above implies that + βf = αβ F = α αβ. Substituting this result, along with the solution for c t, back into the Bellman equation ields E + F ln(k t ) + G ln(z t ) = ln( αβ) + ln(z t ) + α ln(k t ) +βe + βf ln(αβ) + βf ln(z t ) + αβf ln(k t ). The solution for F implies that the terms involving k t drop out of this last expression, leaving E + G ln(z t ) = ln( αβ) + ln(z t ) + βe + βf ln(αβ) + βf ln(z t ). Since this last equation must hold for all possible realizations of z t, it requires that G ln(z t ) = ln(z t ) + βf ln(z t ) or, using the solution for F again, G = αβ. 3

4 Hence, the Bellman equation also requires that E = β E = ln( αβ) + βe + βf ln(αβ) or, using the solution for F one last time, [ ln( αβ) + 3. Saving with a Random Return ( ) ] αβ ln(αβ). αβ (a) Allowing for the possibilit of serial correlation in the random return on saving, the Bellman equation for this problem can be written as v(a t, R t ) = max s t u(a t s t ) + βe t [v(r t+ s t, R t+ )]. Write down the Bellman equation for this problem, using A t as the state variable, s t and the control variable, and allowing the value function for time t to depend on R t as well as A t. (b) Using the assumed form u(a t s t ) = (A t s t ) σ σ for the single-period utilit function, the additional assumptions that the random interest rate R t+ is independentl and identicall distributed with E t (R σ t+ ) = for all t = 0,, 2,..., and the guess that under these conditions, the value function depends onl on A t and takes the specific form v(a t ) = KA σ t σ, where K is an unknown constant, the Bellman equation can be rewritten as KA σ t σ = max (A t s t ) σ s t σ The first-order condition for s t then becomes (A t s t ) σ + βks σ t = 0 and the envelope condition for A t becomes KA σ t = (A t s t ) σ. + βks σ t σ. 4

5 (c) There are a number of was to derive this result, but one is to start b rewriting the first-order condition for s t as [ ] (βk) /σ s t = A + (βk) /σ t and substitute this result into the envelope condition to obtain which can be solved for [ KA σ t = + (βk) /σ ( ) σ K =. β /σ (d) Since the solution for K just derived implies that (βk) /σ = β/σ + β /σ, ] σ A σ t, the first-order condition for s t implies that the optimal choice for s t is given b Hence, the optimal choice for c t is s t = β /σ A t. c t = ( β /σ )A t. 5