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Unformatted text preview: Viktor Tsyrennikov Graduate Macroeconomics Dynamic Programming In the notes on the recursive formulation of the saving problem we showed that the sequence problem (SP) max { c t ,a t +1 } ∞ summationdisplay t =0 β t u ( c t ) s.t. a t +1 = R t +1 ( a t + y c t ) , a t +1 greaterorequalslant B t +1 , ∀ t, (SP) and the recursive problem (RP) V ( a ) = max c,a ′ [ u ( c ) + βV ( a ′ )] s.t. a ′ = R ( a + y c ) , a ′ greaterorequalslant B, (RP) share the same solution when βR t +1 = 1 , ∀ t . We also showed that the value function V , computed analytically and by iteration, coincides with the opti mal lifetime utility in the sequence formulation. But let’s pause for a moment and think of what may go wrong with the recursive approach. First, Does the value function V that satisfies the (RP) always exist?” Second, ”Is the value function V that satisfies the (RP) unique?” Consider now the third formulation in which the planning horizon is finite: max { c t ,a t +1 } T summationdisplay t =0 β t u ( c t ) s.t. a t +1 = R t +1 ( a t + y c t ) , a t +1 greaterorequalslant B t +1 , ∀ t. (SP T ) We showed that, again when βR t +1 = 1 , ∀ t , the solution to a Tperiod saving problem converges to the RP solution when T → ∞ . (So, the iterative pro cedure could be nothing more than computing solutions to the finite horizon problem.) Mathematically, we found that max { c t ,a t +1 } ∞ summationdisplay t =0 β t u ( c t ) ≡ max { c t ,a t +1 } lim T →∞ T summationdisplay t =0 β t u ( c t ) = lim T →∞ max { c t ,a t +1 } T summationdisplay t =0 β t u ( c t ) . (1) The question is “Can we always exchange the limit and the max operators?” If yes, then we would know that the recursive approach would get us the same solution as in the sequence problem SP. In general, the solution to 1 Viktor Tsyrennikov Graduate Macroeconomics the sequence problem yields higher utility to the individual. Why? Because the individual could always choose to follow the solution to the recursive problem. 1 Mathematically, U ( c ∗ ; a ) = max c,a ∞ summationdisplay t =0 β t u ( c t ) = ∞ summationdisplay t =0 β t u ( c ∗ t ) greaterorequalslant V ( a ) , ∀ a , where { c ∗ t , a ∗ t +1 } denotes the optimal solution in the sequence formulation SP. Hence, the last question is “When are the two problems equivalent? That is, when do they share the same solution and provide the individual with the same level of utility?” Now with the three questions in mind let’s dive into dynamic programming. 1 Equivalence of SP and RP 1.1 Notation and assumptions First some notation. Let Γ describe the set of feasible choices: { c t , a t +1 } ∈ Γ( a t ). In the case of a saving problem Γ( a t ) = { ( c t , a t +1 ) : a t +1 lessorequalslant R ( a t + y c t ) } . Let Π( a ) denote the set of all feasible { c t , a t +1 } sequences that start from a . Mathematically, Π( a ) = {{ c t , a t +1 } ∞ t =0 : ( c t , a t +1 )...
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 Spring '11
 ViktorTsyrennikov
 Macroeconomics, Derivative, Recursion, Optimization, Metric space, Compact space, Viktor Tsyrennikov

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