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Econ805Finalanspart2.02

# Econ805Finalanspart2.02 - Department of Economics The Ohio...

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Department of Economics The Ohio State University Final Exam Answers–Econ 805 Questions 2,3, and 4 Profs. Levin and Peck March 19, 2002 2. (20 points) Suppose that an economy with two commodities has the excess demand function given by z ( p 1 ; p 2 ) = ( ¡ 1 ; p 1 p 2 ) for p 1 ¸ 0 and p 2 > 0 . [If we have p 2 = 0 , then excess demand is not well- de…ned.] (a) Is this excess demand function consistent with Walras’ law? That is, does p ¢ z ( p ) = 0 for all prices for which excess demand is well-de…ned? (b) Could this excess demand function have been derived from each consumer maximizing utility subject to a budget constraint, where each utility function is continuous, strictly monotonic, and strictly quasi-concave, and where all initial endowments are strictly positive? Explain carefully. Answer: (a) Yes, this is consistent with Walras’ Law. Evaluating p ¢ z ( p ) , we have p 1 ( ¡ 1) + p 2 ( p 1 p 2 ) = 0 : (b) This is impossible. Under all these assumptions, our existence theo- rem guarantees that a CE exists. Therefore, there is a price vector such that z ( p ) · 0 . However, z 2 ( p ) · 0 can only hold if p 1 = 0 , which violates strict monotonicity. (If a price was zero, there would necessarily be excess demand for that good.) 3. (20 points) Consider the following economy, with two consumers and two commodities. The economy’s aggregate resources are given by the vector, (1 ; 1) : Consumer 1 has the utility function, u 1 ( x 1 ) = 2 log( x 1 1 ) + log( x 2 1 ) ;

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