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Econ501aL8 - Eciency and Trade Previously we treated prices...

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E±ciency and Trade Previously, we treated prices and income as param- eters, and saw how demand depended on these pa- rameters. Now, we ask what prices might emerge as consumers make mutually bene¯cial trades. Suppose that there are two consumers, and ¯x the total amount of goods x and y to be allocated: x and y . The Edgeworth Box {Consider a rectangle of size x £ y . Think of consumer 1 as being located at the southwest corner and consumer 2 as being located at the northeast corner (upside down). Then any point in the box represents an allocation of the available goods across the two consumers.
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0 0.2 0.4 0.6 0.8 1 y 0.2 0.4 0.6 0.8 1 x An Edgeworth Box
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Preferred bundles for consumer 1 are to the north- east, and preferred bundles for consumer 2 are to the southwest. 0 0.2 0.4 0.6 0.8 1 y 0.2 0.4 0.6 0.8 1 x Indi®erence Curves in the Edgeworth Box
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By ¯nding the bundles that both consumers prefer to an \initial" bundle, we can determine the gains from trade. 0 0.2 0.4 0.6 0.8 1 y 0.2 0.4 0.6 0.8 1 x Gains From Trade
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Pareto optimal allocations are ones for which there are no gains from trade. It is impossible to make one consumer better o® without hurting the other con- sumer. 0 0.2 0.4 0.6 0.8 1 y 0.2 0.4 0.6 0.8 1 x A Pareto Optimal Allocation
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The set of Pareto Optimal Allocations is called the Contract Curve. Indi®erence curves are tangent, so marginal rates of substitution are equal. 0 0.2 0.4 0.6 0.8 1 y 0.2 0.4 0.6 0.8 1 x The Contract Curve
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Mathematical Derivation of the Contract Curve Allocate resources to maximize consumer 1's utility, subject to the constraint that we are on a particular indi®erence curve of consumer 2. max u 1 ( x 1 ; y 1 ) subject to u 2 = u 2 ( x 2 ; y 2 ) x 1 + x 2 = x y 1 + y 2 = y : This can be solved with three multipliers, but it is simpler to use the resource constraints to eliminate x 2 and y 2 : max u 1 ( x 1 ; y 1 ) (1) subject to u 2 = u 2 ( x ¡ x 1 ; y ¡ y 1 )
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Set up the Lagrangean expression, L = u 1 ( x 1 ; y 1 ) + ¸ [ u 2 ( x ¡ x 1 ; y ¡ y 1 ) ¡ u 2 ] : The ¯rst order conditions are that the term in brackets is zero, and @u 1 @x 1 + ¸ @u 2 @x 2 ( ¡ 1) = 0 ; (2) @u 1 @y
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