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Unformatted text preview: Econ 51: Midterm Exam Solutions 1 Short Questions (28 points, 7 points each) (1) Let there be two states of the world s = 1 , 2, and let x and y be the consumptions in two states 1 and 2, respectively. Does the utility function u ( x,y ) = 6 + 5 x 2 + 7 y 2 +14 y +10 x represent a vNM preferences? Why or why not? If it does represent a vNM preferences, what are the implied probabilities of each state? Answer: The function represent a vNM preferences because, by rearranging, u ( x,y ) = 5 12 v ( x ) + 7 12 v ( y ) where v ( x ) = 12 x 2 + 24 x + 6, which is of the vNM utility function. The implied probabilities for states 1 and 2 are 5/12 and 7/12, respectively. (2) There are three states s = 1 , 2 with probabilities and 1 . The agents vNM utility function is u ( x,y ) = ( x 3 ) + (1 )( y 3 ), where x 0 and y 0 are consumptions in states 1 and 2, respectively. Is this agent riskaverse? Why or why not? Answer: This agent is riskaverse. To see this, let her utility function be v ( x ) = x 3 and note that v ( x ) = 3 x 4 , v 00 ( x ) = 12 x 5 < . (3) Suppose that there are two kinds of goods, xylophones ( x ) and yogurt ( y ), and two individuals, A and B. Utility functions of the individuals are given by u A ( x,y ) = 2 x + y and u B ( x,y ) = x + 2 y , respectively. There are 7 xylophones and 16 yogurts in the economy. Draw the contract curve in the Edgeworth box. Answer: The contract curve in this economy is composed of the two edges the bottom and the right. See Figure 1. (4) Country A can produce 4 apples or 9 bananas with one unit of labor while country B can produce 2 apples or 2 bananas with one unit of labor. Goods are freely traded between countries A and B, although immigration is impossible, so people in each country can work only in their home country. Now consider the following statement: Because A has a better technology than B for producing both apples and bananas, 1 Figure 1: Contract curve for question 1 (3). in Walrasian equilibrium only country A will produce goods. Is this statement true or false? Why or why not? Answer: False. Even if country A has better technology in producing everything (i.e., A has absolute advantage), country B will still specialize in producing a good that it has comparative advantage and trade with country A in equilibrium. In this case we expect that country B produces apples and export some apples to country A while importing bananas from country A. 2 2 Decision under uncertainty (30 points) (a) Insurance Suppose that an individual has a wealth W > 0. Her utility function for money is v ( x ) = log( x ). There are two states of the world, s = 1 and s = 2. In state s = 1, there is an earthquake and she needs to repair her house, paying W/ 3. In state s = 2, nothing happens. States 1 and 2 occur with probabilities 0 . 2 and 0 . 8, respectively....
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 Winter '07
 Tendall,M
 Utility

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