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70207notes_week2 - Jan 30th, 2007 1 Existence and Pareto...

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Unformatted text preview: Jan 30th, 2007 1 Existence and Pareto Optimality in the Growth Model To support a Pareto Optimal allocation as a solution to the growth model presented before, we have to take care of certain issues that arise when we apply the SBWT to get our equilibrium. Those issues/solutions are listed below: What are the transfers of the conclusion of the SBWT in terms of the growth model? / we dont need transfers; agents are homogeneous, so even if they can act differently, they choose to do the same as everyone else. Do we have to worry about the Quasi part of the equilibrium? / If we can find a cheaper point in the feasible set, then the Quasi equilibrium is equivalent to the AD equilibrium representation of prices/ if we can check the conditions of the Prescott & Lucas Theorem, then we have a dot product representation of prices. 1.1 Characterization of the solution to the growth model The solution to the growth model is triplet of sequences { c * t , k * t +1 , q * t } t =0 . As you proved in the homeworks, you can use the Arrow-Debreu apparatus in order to argue that such an equilibrium exists. To characterize more carefully the equilibrium, we have to impose additional restrictions: u, f are C 2 (twice continuously differentiable) Inada conditions (see the Stockey and Lucas textbook for specifics) With these conditions, we can restrict our attention to interior solu- tions, which means that first order conditions are sufficient to characterize equilibria. Rewriting the growth model (replacing consumption in the utility func- tion using the budget constraint): max { k t +1 } t =0 u [ f ( k t )- k t +1 ] 1 Taking the FOC with respect to k t +1 and replacing for c t to ease notation, we get (note that we are using variables with * to denote that the following are equilibrium conditions)- t u [ c * t ] + t +1 u [ c * t +1 ] f ( k * t +1 ) = 0 rearranging terms u [ c * t ] u [ c * t +1 ] = f ( k * t +1 ) (1) Therefore, the solution to the growth model has to satisfy the condition in (1). Now, for prices, we can rewrite the budget equation from the AD setting (if the conditions of Prescott and Lucas are satisfied so that prices have a dot product representation) as follows p ( x ) X t =0 ( q 1 t x 1 t + q 2 t x 2 t + q 3 t x 3 t ) (2) Since c t + k t +1 = x 1 t , k t - x 2 t 0 and 1 - x 3 t 0, (2) becomes X t =0 ( q * 1 t ( c t + k t +1 )- q * 2 t k t- q * 3 t ) (3) Note that in (3), we have used the fact that there is no waste (agents rent their full capital and labor services) and that agents take the equilibrium prices as given. The maximization problem now can be set as a Lagrangian: max { c t ,k t +1 } t =0 = X t =0 t u [ c t ]- { X t =0 q * 1 t ( c t + k t +1 ) + q * 2 t k t + q * 3 t } (4) The first order conditions of this problem with respect to c t and k t +1 are respectively t u [ c * t ] q * 1 t = (5) q * 1 t- q * 2 ,t +1 = 0 (6)...
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This note was uploaded on 11/12/2010 for the course ECON 8108 taught by Professor Staff during the Spring '08 term at Minnesota.

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70207notes_week2 - Jan 30th, 2007 1 Existence and Pareto...

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