h1answ08 - Department of Economics The Ohio State...

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Department of Economics The Ohio State University Econ 805—Homework #1 Questions and Answers Prof James Peck Winter 2008 1. A consumer’s o f er curve is the locus of tangencies between the budget lines and indi f erence curves as relative prices change. (i) For a two-person, Edgeworth Box economy, explain why any intersection of the two o f er curves is a competitive equilibrium. (ii) What is the equation for the following consumer’s o f er curve? u ( x 1 ,x 2 )= log ( x 1 )+ log ( x 2 ) , ( ω 1 2 )=(1 , 1) . Answer: (i) The absolute value of the slope of the line connecting the endowment point and the o f er curve is the price ratio for which the bundle demanded is the point on the o f er curve. If two o f er curves intersect in an Edgeworth Box diagram, then the intersection point represents a feasible allo- cation that is utility maximizing for the same price ratio. Thus, the allocation, along with the price ratio equal to the absolute value of the slope of the segment from the endowment to the o f er curve, together forms a competitive equilib- rium. (ii) Normalizing the price of good 2 to be one and the price of good one to be p , the (necessary and su cient) f rst order conditions of the utility maximization problem are x 2 x 1 = p and (1) px 1 + x 2 = p +1 . (2) We want to determine from (1) and (2) an equation in consumption space, so we must eliminate p . Substituting (1) into (2), we have x 2 x 1 x 1 + x 2 = x 2 x 1 , which simpli f es to x 2 = x 1 2 x 1 1 . Of course, since we are restricting ourselves to positive consumption, we must have x 1 > 1 2 . In other words, this is a hyperbola, and we restrict ourselves to the curve contained in the nonnegative orthant. 2. Provide a counterexample to the f rst welfare theorem (you can use a carefully labeled and explained Edgeworth box) when some consumer has a 1
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“bliss point” giving the highest possible utility. That is, f nd an example of a competitive equilibrium allocation that is not strongly Pareto optimal. Answer: Suppose consumer 1 and consumer 2 each have an endowment of ( 1 2 , 1 2 ) . Consumer 1 has a bliss point at ( 1 4 , 1 4 ) , so that his utility function is a “mountain” with a highest possible utility at ( 1 4 , 1 4 ) , and lower utility as we move away from that point. Consumer 2’s utility function is u 2 ( x 1 2 ,x 2 2 )= log ( x 1 2 )+ log ( x 2 2 ) . The following constitutes a competitive equilibrium that is not strongly Pareto optimal: p =( 1 ,
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This note was uploaded on 07/17/2008 for the course ECON 805 taught by Professor Peck during the Spring '08 term at Ohio State.

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h1answ08 - Department of Economics The Ohio State...

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