finalans03 - The Ohio State University Department of...

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The Ohio State University Department of Economics Econ 805 Final Exam Answers Professor Peck Winter 2003 Here is the distribution of scores on the …nal exam: 100 6 90-99 5 80-89 6 70-79 4 60-69 6 50-59 6 40-49 7 30-39 8 1. Consider a pure exchange economy, which we will call the n-consumer economy, with k goods and n consumers. Each consumer’s utility function and endowment vector satis…es assumption A: Assumption A: The utility function is continuous, strictly monotonic, and strictly quasi-concave. The endowment vector is strictly positive. Now consider the pure exchange economy in which an n + 1 st consumer, whose utility function and endowment also satis…es assumption A, joins the original n consumers. We call this economy the n+1-consumer economy. For the statements in (a) and (b), either sketch the argument for why the statement is true, or give a counterexample if the statement is false. (a) (15 points) For all i = 1 ; :::; n , consumer i receives at least as high a utility level at the competitive equilibrium of the n+1-consumer economy than at the competitive equilibrium of the n-consumer economy. (b) (15 points) Suppose that the initial endowment allocation of the n- consumer economy is Pareto optimal. Then for all i = 1 ; :::; n , consumer i receives at least as high a utility level at the competitive equilibrium of the n+1-consumer economy than at the competitive equilibrium of the n-consumer economy. Answer: (a) This statement is false. While it is possible for all of the …rst n consumers to be better o¤ at the C.E. of the n+1-consumer economy, we 1
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can …nd a counterexample. For example, let k=2 and n=2 hold, and consider the following utility functions and endowments: u 1 ( x 1 ) = log( x 1 1 ) + log( x 2 1 ) ; ! 1 = (2 ; 1) u 2 ( x 2 ) = log( x 1 2 ) + log( x 2 2 ) ; ! 2 = (1 ; 2) u 3 ( x 3 ) = log( x 1 3 ) + log( x 2 3 ) ; ! 3 = (2 ; 1) : It is easy to check that consumer 1 receives a higher utility, in the C.E. to the economy with consumers 1 and 2, than in the C.E. to the economy with all three consumers. Intuitively, the introduction of consumer 3 creates an unfavorable price shift, making the good that consumer 1 wants to buy more expensive relative to the good that consumer 1 wants to sell.
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