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fin-401-08wa

# fin-401-08wa - 1 14th April 2008 Instructions Answer 6 of...

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1 14th April 2008 Instructions : Answer 6 of the following 8 questions. All questions are of equal weight. Indicate clearly on the fi rst page which questions you want marked. 1. Consider a two-good pure exchange economy in which there is one person of type A and one person of type B. Suppose A has endowments ± e A 1 , e A 2 ² and B has endowments ± e B 1 , e B 2 ² . Assume U A ± x A 1 , x A 2 ² = ± x A 1 ² a ± x A 2 ² 1 a , 0 < a < 1 U B ± x B 1 , x B 2 ² = ± x B 1 ² a ± x B 2 ² 1 a e 1 e A 1 + e B 1 e 2 e A 2 + e B 2 Describe as precisely as you can the Pareto e cient allocations of this exchange economy where neither person is worse o ff than she/he is at her/his endowment point, and write down, precisely, the competitive equilibrium. ANSWER With Cobb-Douglas utility functions, the Pareto e cient allocations where neither person is worse o ff than she/he is at her/his endowment point are described by MRS A 12 = MRS B 12 U A ± x A 1 , x A 2 ² U A ± e A 1 , e A 2 ² U B ± x B 1 , x B 2 ² U B ± e B 1 , e B 2 ² The ( fi rst) equation implies ax A 2 (1 a ) x A 1 = ax B 2 (1 a ) x B 1 or x A 1 x A 2 = x B 1 x B 2 = e 1 x A 1 e 2 x A 2 or x A 1 x A 2 = x B 1 x B 2 = e 1 e 2

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2 The fi rst inequality implies ± x A 1 ² a ± x A 2 ² 1 a ± e A 1 ² a ± e A 2 ² 1 a . Using the equation ± x A 1 ² a # e 2 x A 1 e 1 \$ 1 a ± e A 1 ² a ± e A 2 ² 1 a or x A 1 ( e 1 ) 1 a ( e 2 ) a 1 ± e A 1 ² a ± e A 2 ² 1 a By symmetry between A and B, the second inequality imples x B 1 ( e 1 ) 1 a ( e 2 ) a 1 ± e B 1 ² a ± e B 2 ² 1 a or e 1 x A 1 ( e 1 ) 1 a ( e 2 ) a 1 ± e B 1 ² a ± e B 2 ² 1 a or e 1 e 1 a 1 ( e 2 ) a 1 ± e B 1 ² a ± e B 2 ² 1 a x A 1 Assembling the pieces we see that the desired PE allocations are described by x A 1 x A 2 = e 1 e 2 e 1 e 1 a 1 ( e 2 ) a 1 ± e B 1 ² a ± e B 2 ² 1 a x A 1 ( e 1 ) 1 a ( e 2 ) a 1 ± e A 1 ² a ± e A 2 ² 1 a . The competitive equilibrium may be found by setting demand equal to supply for good 1. a ± p 1 e A 1 + p 2 e A 2 ² p 1 + a ± p 1 e B 1 + p 2 e B 2 ² p 1 = e 1 or p 1 p 2 = a 1 a e 2 e 1 or x A 1 = a # e A 1 + p 2 p 1 e A 2 \$ or x A 1 = ae 2 e A 1 + (1 a ) e 1 e A 2 e 2 .Then x A 2 = e 2 x A 1 e 1 = ae 2 e A 1 + (1 a ) e 1 e A 2 e 1 . And then, by symmetry,
3 x B 1 = ae 2 e B 1 + (1 a ) e 1 e B 2 e 2 x B 2 = ae 2 e B 1 + (1 a ) e 1 e B 2 e 1 2. Consider a risk-averse individual with initial wealth w 0 and VNM utility func- tion u ( · ) who must decide whether and for how much to insure his car. Assume the probability that he will not have an accident is π . In the event of an accident, he incurs a loss of \$ L in damages. Suppose that insurance is available at an actuarially fair price, that is, one that yields insurance companies zero expected pro fi ts; denote the price of \$1 worth of insurance coverage by p.

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