ps1_sol - ECO 475 PS1 Solution Yu LIU October 30, 2010 1...

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Unformatted text preview: ECO 475 PS1 Solution Yu LIU October 30, 2010 1 Complete Market and Representative Consumer (a) Arrow-Debrew equilirium: An ADE is a set of price q t ( s t ) and allocation { c i,t ( s t ) } ( i = 1 , ,I ) sequences such that 1. Consumer's Problem Given q t ( s t ) , for i = 1 , ,I , consumer solves the following by { c i,t ( s t ) } max { c i,t ( s t ) } X t =0 X s t t ( s t ) u i ( c i,t ( s t )) subject to X t =0 X s t q t ( s t ) c i,t ( s t ) = X t =0 X s t q t ( s t ) y i,t ( s t ) . c i,t ( s t ) , t,s t . 2. Market clearing I X i =1 c i,t ( s t ) = I X i =1 y i,t ( s t ) , t,s t . Let i be the Lagrangian for consumer i 's budget constraint. FOC wrt c i,t ( s t ) : t ( s t ) u i ( c i,t ( s t )) = i q t ( s t ) So, ( s t ) u i ( c i,t ( s t )) s t u i c i,t s t = q t ( s t ) q t s t , i,s t , s t ( s t +1 | s t ) u i ( c i,t +1 ( s t +1 )) u i ( c i,t ( s t )) = q t +1 ( s t +1 ) q t ( s t ) , i,s t ,s t +1 u i ( c i,t ( s t )) u j ( c j,t ( s t )) = i j , i,j,s t (1) Combine with the budget constraints and market clearing, we can get the allocations and prices. Social planner's problem: 1 max { c i,t ( s t ) } I X i =1 X t =0 X s t i t u i ( c i,t ( s t )) subject to I X i =1 c i,t ( s t ) I X i =1 y i,t ( s t ) , t,s t . c i,t ( s t ) , i,t,s t . where I i =1 i = 1 . Let t ( s t ) be the Lagrangian for the resource constraint in period t with history s t . FOC wrt c i,t ( s t ) : i t ( s t ) u i ( c i,t ( s t )) = t ( s t ) So, ( s t ) u i ( c i,t ( s t )) s t u i c i,t s t = t ( s t ) t s t , i,s t , s t ( s t +1 | s t ) u i ( c i,t +1 ( s t +1 )) u i ( c i,t ( s t )) = t +1 ( s t +1 ) t ( s t ) , i,s t ,s t +1 i u i ( c i,t ( s t )) j u j ( c j,t ( s t )) = 1 , i,j,s t (2) Combine with the resource constraints, we can get the allocations and prices.Combine with the resource constraints, we can get the allocations and prices....
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ps1_sol - ECO 475 PS1 Solution Yu LIU October 30, 2010 1...

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