handoutdpf06 - HANDOUT ON DYNAMIC PROGRAMMING 1 Example...

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HANDOUT ON DYNAMIC PROGRAMMING 1E x a m p l e We’ll start with a simple growth example where a planner chooses how much people should consume and how much to invest in a productive technology. Preferences are given by u ( c t )=log( c t ) and what is not eaten out of production today, say k t +1 , becomes productive tomorrow y t +1 = f ( k t +1 )= k α t +1 where α 1 . If people discount the future at rate β< 1 and there is a general time horizon represented by T (which could be ) ,thesequencevers iono fthe programming problem can be written as: v T ( k 0 m a x { c t ,k t +1 } T t =0 T X t =0 β t u ( c t ) s.t.c t + k t +1 = f ( k t ) , t c t 0 ,k t +1 0 0 given. One way to solve this problem is to write down the system of f rst order conditions for each t { 0 , ..., T } . An alternative approach is to use dynamic programming. First consider what happens if a person enters the last period with k T units of capital and acts optimally. In that case they solve: v T ( k T m a x c T 0 ,k T +1 0 log( c T ) s.t.c T + k T +1 k α T The solution is simple, k T +1 =0and c T = k α T . In that case, v T ( k T δ (0) log( k α T )+ γ (0) where δ (0) = 1 and γ (0) = 0 and the argument in the function denotes the number of periods remaining. Now consider the problem if a person enters the next to last period t = T 1 with k T 1 units of capital and acts optimally. The problem can be stated as: v T 1 ( k T 1 m a x c T 1 0 ,k T 0 log( c T 1 βv T ( k T ) s.t.c T 1 + k T k α T 1 Since utility is strictly increasing in consumption, the constraint will hold with equality and we can substitute out consumption to yield the problem v T 1 ( k T 1 )=max k T 0 log( k α T 1 k T T ( k T )( 1 ) 1
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Substituting v T ( k T )= α log( k T ) into the objective, the foc are given by 1 k α T 1 k T = αβ k T ⇐⇒ k T = αβk α T 1 (1 + αβ ) (2) Substituting the decision rule (2) back into the objective (1) yields v T 1 ( k T 1 )=l o g μ k α T 1 αβk α T 1 (1 + αβ ) + αβ log μ αβk α T 1 1+ αβ v T 1 ( k T 1 )=(1+ αβ )log ¡ k α T 1 ¢ (1 + αβ )log(1+ αβ )+ αβ log( αβ ) v T 1 ( k T 1 δ (1) α log ( k T 1 γ (1) (3) where the coe cients are obviously de f ned as δ (1) = (1 + αβ )and γ (1) = αβ log( αβ ) (1 + αβ αβ ) . By induction, let there be τ periods remaining (i.e. t = T τ ): v T τ ( k T τ )= max k T τ +1 0 log( k α T τ k T τ +1 βv T τ +1 ( k T τ +1 ) which upon substituting the τ period analogue of (3) given by v T τ +1 ( k T τ +1 δ ( τ 1) α log ( k T τ +1 γ ( τ 1) into the above yields v T τ ( k T τ k T τ +1 0 log( k α T τ k T τ +1 )+ β [ δ ( τ 1) α log ( k T τ +1 γ ( τ 1)] (4) The foc is 1 k α T τ k T τ +1 = αβδ ( τ 1) k T τ +1 k T τ +1 = αβδ ( τ 1) k α T τ (1 + αβδ ( τ 1)) (5) Substituting the decision rule (5) back into the objective (4) yields v T τ ( k T τ o g μ k α T τ αβδ ( τ 1) k α T τ (1 + αβδ ( τ 1)) + αβδ T τ +1 log μ αβδ ( τ 1) k α T
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handoutdpf06 - HANDOUT ON DYNAMIC PROGRAMMING 1 Example...

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