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exchange - 13 Exchange economy Goal General equilibrium...

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13 Exchange economy Goal: General equilibrium model – all markets clear. Agent (consumer): i , w i = ( w 1 i , w 2 i , . . . , w k i )–endowment, x i –consumption vector. Allocation: x = ( x 1 , x 2 , . . . , x n ), feasible if P n i =1 x i P n i =1 w i . Individual problem: (given prices p = ( p 1 , p 2 , . . . , p k )): max x i u i ( x i ) s.t. px i = pw i . Answer: x i ( p , pw i )–consumer’s demand.
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13.1 Walrasian equilibrium Look for ( p , x ) such that all markets clear n X i =1 x i ( p , p w i ) n X i =1 w i . De fi ne the aggregate excess demand function z ( p ) = n X i =1 ( x i ( p , pw i ) w i ) . Properties: homog. of degree 0 in p , continuous (pro- vided x i are continuous). Walras’ law: For any price p , pz ( p ) 0 . Corollary (market clearing): If k 1 markets clear and p k > 0, then k th market clears as well. Also, z j ( p ) < 0 implies p j = 0. 13.2 Existence Normalize prices: p i = ˆ p i P k j =1 ˆ p j . De fi ne: S k 1 = n p R k + : P k i =1 p i = 1 o . Brower fi xed-point theorem: If f : S k 1 S k 1 is continuous then there exists x S k 1 , such that x = f ( x ).
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