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# simpleDP - E CON 714 M ACROECONOMIC T HEORY II TA T IM L EE...

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E CON 714: M ACROECONOMIC T HEORY II TA: T IM L EE F EBRUARY 5, 2010 Dynamic Programming Let’s use the simple growth example from class: W ( k 0 ) = max { c t , k t + 1 } t = 0 t = 0 β t log ( c t ) s.t. c t + k t + 1 Ak α t t . We already know from class how to use the “guess and verify" technique to solve for the value function instead: V ( k ) = max k 0 [ 0, Ak α ] log ( Ak α - k 0 ) + β V ( k 0 ) , with the guess of E + F log k , so that E = ( 1 - β ) - 1 log ( 1 - αβ ) A + αβ 1 - αβ log αβ A F = α 1 - αβ g ( k ) = αβ Ak α where k 0* = g ( k ) is the policy function. Note that we can’t say V ( k ) W ( k 0 ) yet, this is the subject of next week’s lectures. For now take for granted that it holds in this case, and I’ll use this example throughout to highlight some other facts. 1. Iterating the value function: Although in this case we can just solve out for the value and policy functions pretty easily, usually we will be exploiting the Contrac- tion Mapping Theorem to get an approximation to the value function. Suppose we use this technique using v 0 0 as an initial guess. The first iteration gives

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