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Unformatted text preview: ECO 475 PS1 Solution Yu LIU October 30, 2010 1 Complete Market and Representative Consumer (a) ArrowDebrew 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 β t π ( 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 λ i be the Lagrangian for consumer i 's budget constraint. FOC wrt c i,t ( s t ) : β t π ( s t ) u i ( c i,t ( s t )) = λ i q t ( s t ) So, π ( s t ) u i ( c i,t ( s t )) π ˜ s t u i c i,t ˜ s t = q t ( s t ) q t ˜ s t , ∀ i,s t , ˜ s t βπ ( 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 )) = λ i λ 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 α i β 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 α i = 1 . Let μ t ( s t ) be the Lagrangian for the resource constraint in period t with history s t . FOC wrt c i,t ( s t ) : α i β t π ( s t ) u i ( c i,t ( s t )) = μ t ( s t ) So, π ( s t ) u i ( c i,t ( s t )) π ˜ s t u i c i,t ˜ s t = μ t ( s t ) μ t ˜ s t , ∀ i,s t , ˜ s t βπ ( s t +1  s t ) u i ( c i,t +1 ( s t +1 )) u i ( c i,t ( s t )) = μ t +1 ( s t +1 ) μ t ( s t ) , ∀ i,s t ,s t +1 α i u i ( c i,t ( s t )) α 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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 Fall '07
 Hong
 Trigraph, cτ

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