Handout_Bellman

Handout_Bellman - Handout: Dynamic Programming 1. The Basic...

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Unformatted text preview: Handout: Dynamic Programming 1. The Basic Growth Model 1 In the following pages I outline the solution to the basic growth model in dis- crete time. For the sake of comparison, I provide both the Bellman’s equation method (dynamic programming) and the Lagrangean method (which you can directly compare with the 2-period case). Later in the course you will probably solve the model in continuous time through the Hamiltonian method. 1.1 The Model Consider the “planner’s problem” of the basic growth model. max c t ,k t +1 + ∞ X t =0 β t u ( c t ) (1) k given (2) c t + i t + g ≤ f ( k t ) (3) k t +1 ≤ (1- δ ) k t + i t (4) where i is investment, g is government spending assumed constant over time and δ is the depreciation rate. We are looking, of course, for the sequence { c * t , k * t +1 } which gives optimum consumption, optimum investment, optimum savings, equilibrium interest rate. By assuming an interior solution (i.e. (3)- (4) will hold as equalites at the optimum) and eliminating i , (3)-(4) can be simplified as c t = f ( k t ) + (1- δ ) k t- k t +1- g (5) We can derive the solution sequentially. Let the value function at period 0 be V ( k ). 2 Then, V ( k ) = max c t ,k t +1 + ∞ X t =0 β t u ( c t ) k given c t = f ( k t ) + (1- δ ) k t- k t +1- g We can modify it as two sub-problems V(k ) = max c ,k 1 u ( c ) + β max ( c 1 ,k 2 )( c 2 ,k 3 ) ... [ u ( c 1 ) + βu ( c 2 ) + ... ] s.t. k given s.t. k 1 given c = f ( k ) + (1- δ ) k- k 1- g c t = f ( k t ) + (1- δ ) k t- k t +1- g 1 Let me know of any typos and/or errors....
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Handout_Bellman - Handout: Dynamic Programming 1. The Basic...

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