econ 113

# econ 113 - Economics 113 Mr Troy Kravitz UCSD Prof R Starr...

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Lecture Notes, January 7 & 12 , 2010 The Edgeworth Box 2 person, 2 good, pure exchange economy Fixed positive quantities of X and Y, and two households, 1 and 2. Household 1 is endowed with of good X and of good Y, utility function X 1 Y 1 U 1 (X 1 , Y 1 ) . Household 2 is endowed with of good X and of good Y, utility X 2 Y 2 function U 2 (X 2 , Y 2 ) X 1 + X 2 = X 1 X 2 X Y 1 + Y 2 = . Y 1 Y 2 Y Each point in the Edgeworth box represents an attainable choice of X 1 and X 2 , Y 1 and Y 2 . 1's origin is at the southwest corner; 1's consumption increases as the allocation point moves in a northeast direction; 2's increases as the allocation point moves in a southwest direction. Superimpose indifference curves on the Edgeworth Box. Competitive Equilibrium (p o x , p o y ) so that (X o1 , Y o1 ) maximizes U 1 (X 1 , Y 1 ) subject to (p o x , p o y )(X 1 , Y 1 ) (p o x , p o y ) and  X 1 , Y 1 (X o2 , Y o2 ) maximizes U 2 (X 2 , Y 2 ) subject to (p o x , p o y 1 , Y 1 p o x , p o y ) , and  X 2 , Y 2 (X o1 , Y o1 ) + (X o2 , Y o2 ) = X 1 , Y 1  X 2 , Y 2 or (X o1 , Y o1 ) + (X o2 , Y o2 ) , where the inequality holds  X 1 , Y 1  X 2 , Y 2 co-ordinatewise and any good for which there is a strict inequality has a price of 0. Pareto efficiency: An allocation is Pareto efficient if all of the opportunities for mutually desirable reallocation have been fully used. The allocation is Pareto efficient if there is no available reallocation that can improve the utility level of one household while not reducing the utility of any household. Tangency of 1 and 2's indifference curves : Pareto efficient allocations. Pareto efficient allocation: Economics 113 Prof. R. Starr Mr. Troy Kravitz, UCSD Winter 2010 January 7 & 12, 2010 1

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(X o1 , Y o1 ), (X o2 , Y o2 ) maximizes U 1 (X 1 , Y 1 ) subject to U 2 (X 2 , Y 2 ) U o2 (typically equality will hold and U o2 =U 2 (X o2 , Y o2 ) ) and subject to the resource constraints X 1 + X 2 = X 1 X 2 X Y 1 + Y 2 = . Y 1 Y 2 Y Equivalently, X 2 - X 1 , Y 2 - Y 1 X Y Solving for Pareto efficiency (Assuming differentiability and an interior solution): Lagrangian L U 1 (X 1 , Y 1 ) + [U 2 X 1 - Y 1 ) - U o2 ] X Y L X 1 U 1 X 1  U 2 X 2 0 L Y 1 U 1 Y 1 U 2 Y 2 0 2 (X 2 , Y 2 ) - U o2 = 0 L  This gives us then the condition MRS 1 xy == M R S 2 xy or equivalently U 1 X 1 U 1 Y 1 U 2 X 2 U 2 Y 2 MRS 1 xy M R S 2 xy Y 1 X 1 U 1 constant Y 2 X 2 U 2 constant Pareto efficient allocation in the Edgeworth box: the slope of 2's indifference curve at an efficient allocation will equal the slope of 1's indifference curve; the points of tangency of the two curves. contract curve = individually rational Pareto efficient points Economics 113 Prof. R. Starr Mr. Troy Kravitz, UCSD Winter 2010 January 7 & 12, 2010 2
Market allocation p x , p y Household 1:Choose X 1 , Y 1 , to maximize U 1 (X 1 ,Y 1 ) subject to p x X 1 + p y Y 1 = p x + p y = B 1 X 1 Y 1 budget constraint is a straight line passing through the endowment point ( X 1 , Y 1 with slope .

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## This note was uploaded on 09/30/2011 for the course ECON 113 taught by Professor Starr,r during the Fall '08 term at UCSD.

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econ 113 - Economics 113 Mr Troy Kravitz UCSD Prof R Starr...

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