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# HW_6 - Fall 2009 Truman Bewley Economics 359a...

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Unformatted text preview: Fall 2009 Truman Bewley Economics 359a Homework #6 (Due Thursday, October 22) In chapter 5, do problems 1 and 2 on page 191 and problem 5 on page192. There is a condition on market excess demand left out of the deﬁnition of equilibrium in probiem 5. The problem shoulci read as follows. 5) Consider an economy with two consumer, two goods, and no firms. The endowment vectors of consumers A and B are eA and e ,respectively, where e >> 0 and eg >> 0. Assume that the B A utility of each consumer depends on the consumption of the other consumer as well as on his or her own consumption. That is, each consumer cares about what the other consumes, out of sympathy, envy, or because the other’s consumption interferes with or helps his or her own iiie. For instance, each neighbor might want the other to paint his or her house, but dislike smoke from his or her barbecue. Such effects are known as consumption externalities. More formally, if xA and x are the consumption bundles of consumers A and B, respectively, then B their utiiities are uA(x , x ) and uB(xB, xA), respectively. Assume that u and u are E . A A 8 continuous and that for i = A and B, u. is strictly increasing and strictly concave with respect to x'. (That is, uA( xA, x8) is both strictly increasing and strictiy concave with respect to xA, but I not necessariiy so with respect to x , and the symmetric statement applies to uB.) In an B equilibrium, consumer A chooses x so as to solve the problem A maxu(x,x) A xeFt2 B + s.t. p.x s p.e , A That is, consumer A holds x fixed when considering how to choose it . Similarly, consumer B B A chooses x so as to solve the problem B maxu(x ,x) XER2 B A 's.t. p.x s p.eB. In addition, the market excess demand for each good is non-positive and the price of any good in excess suppiy is zero. a) Prove that an equiiibrium exists. b) Is an equilibrium allocation Pareto optimal? Give a proof or counter example. ...
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