This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 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....
View Full
Document
 Fall '07
 Hong

Click to edit the document details