examples_DP - Extensions of the Cake-Eating Problem We...

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Extensions of the Cake-Eating Problem We consider the basic framework of the cake-eating problem (as stated in the midterm). The two examples below provide some further elements to this basic structure. 1. Utility in period t is now given by u ( c t ,c t - 1 ) . (i) Solve a T -period problem using these preferences. (ii) Interpret the rst-order conditions. (iii) Formulate the Bellman equation for the in nite horizon version of this problem. We consider the nite horizon problem with nal period T < . And we want to write the recursive form of the optimization of the overall utility (over periods 1 to T ). V t ( W t ,c t - 1 ) = max 0 c t W t { u ( c t ,c t - 1 ) + βV t +1 ( W t +1 ,c t ) } s.t. W t +1 = W t - c t . Comments: - the state contains 2 variables, the current size of the cake and the previous level of consumption. This is all the information I need to nd the optimal path of consumption. - the action is still the current consumption. - the restriction put on the action is still c [0 ,W ] (cannot consume more than the whole cake of size W currently available) - the transition is still: W 0 = W - c . FOC of the above problem (di erentiate wrt c t ): u 0 1 ,t - βV 0 t +1 + βV 0 2 ,t +1 NB: here I use the following notations: f 0 1 ( x,y ) = ∂f ( x,y ) ∂x and f 0 2 ( x,y ) = ∂f ( x,y ) ∂y u t = u ( c t ,c t - 1 ) V t = V ( W t ,c t - 1 ) 1
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Envelop theorem: V 0 1 ,t = βV 0 1 ,t +1 V 0 2 ,t = u 0 2 ,t NB: back to your undergrad. optimization book if you have forgotten this theorem. Alternatively, if you want to derive it the hard way: assume the policy function is of
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examples_DP - Extensions of the Cake-Eating Problem We...

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