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Exchange - Competitive Markets Econ 210C UCSB Pure Exchange...

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Competitive Markets Econ 210C UCSB May 11, 2011
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Pure Exchange Economies Let l denote the number of commodities. We assume 0 < l < . Consumer Characteristic: ( X i , i , ω i ) where X i ⊂ < l is the consumption set of consumer i , i is his preference relation over consumption bundles in X i , ω i X i is consumer i ’s initial endowment. Utility Representation: Preference relation i is represented by utility function u i : X i -→ < if x i i y i u i ( x i ) u i ( y i ) , for all x i , y i X i . Pure Exchange Economy: An n -person (pure) exchange economy is a list E = { X i , i , ω i } n i = 1 of n consumer characteristics for consumers i = 1, 2, · · · , n . Econ 210C UCSB Paper
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An allocation for E is an n -tuple x = ( x 1 , x 2 , · · · , x n ) of consumption bundles x i for consumers i = 1, 2, · · · , n . Feasible Allocations: Allocation x = ( x 1 , x 2 , · · · , x n ) is feasible for E if x i X i for all i and n i = 1 x i = n i = 1 ω i . Pareto Optimal Allocations: A feasible allocation ( x 1 , x 2 , · · · , x n ) is Pareto optimal for economy E , if it is not possible to make one consumer better off without making any other consumer worse off, equivalently, if there does not exist another feasible allocation y = ( y 1 , y 2 , · · · , y n ) such that y i i for all i and y j j x j for at least one j . Econ 210C UCSB Paper
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Edgeworth Box Assume l = 2 and n = 2. In this case, feasible allocations can be represented by points in Edgeworth box. Econ 210C UCSB Paper
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Example 1: X 1 = X 2 = R 2 + , ω 1 = ( 10, 0 ) , ω 2 = ( 0, 10 ) , and u i ( x i ) = min { x i 1 , x i 2 } for i = 1, 2. Example 2: X 1 = X 2 = R 2 + , ω 1 = ( 10, 0 ) , and ω 2 = ( 0, 10 ) , u i ( x i ) = x i 1 + x i 2 for i = 1, 2. (What if u 1 ( x 1 ) = 2 x 11 + x 12 ?) Example 3: X 1 = X 2 = < 2 + , ω 1 = ( 5, 0 ) , ω 2 = ( 0, 5 ) , u 1 ( x 1 ) = x 11 ( 5 + x 12 ) , and u 2 ( x 2 ) = x 11 x 12 . Econ 210C UCSB Paper
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Characterization of Pareto Optimal Allocations Assume that i is monotonically increasing and is representable by utility function u i . An allocation ¯ x = ( ¯ x 1 , ¯ x 2 , · · · , ¯ x n ) is Pareto optimal for E = { X i , i , ω i } n i = 1 if and only if it solves max x 1 , x 2 , ··· , x n u 1 ( x 1 ) u i ( x i ) = u i ( ¯ x i ) , i = 2, 3, · · · , n n i = 1 x i = n i = 1 ω i , x i X i , i . (1) By (1), if u i is differentiable for all i , then interior Pareto optimal allocations can be characterized by the following system of equations: MU 1 h ( x 1 ) MU 1 k ( x 1 ) = MU ih ( x i ) MU ik ( x i ) , i = 2, · · · n , h 6 = k (2) n i = 1 x i = n i = 1 ω i , x i X i , i (3) Econ 210C UCSB Paper
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Competitive Equilibrium Perfect competition: An idealized market environment in which every market participant is too small to affect the market price by acting on its own. Price-taking behavior is automatic (not assumed) under perfect competition.
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