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final-solution02 - Econ 51: Final Exam Solution Friday,...

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Econ 51: Final Exam Solution Friday, March 18, 2011 1 Uncertainty and GE (16 points) (1) (4 points) Someone offers you a lottery ticket for $100. The lottery pays $100 with probability 0.99, and pays nothing with probability 0.01. If your utility function is strictly increasing in money and you are sufficiently risk loving, you should buy the ticket. True of false? Explain using at most two sentences. Answer. False. Buying the lottery ticket implies non-positive payoffs in any state of the world, so as long as v ( x ) is increasing in x , you should not buy the ticket, regardless of your risk preferences. (2) (4 points) Consider a gamble to flip a fair coin (it will result in Head and Tail with probability 1 / 2 each), in which you win A dollars if Head and lose A dollars if Tail. Suppose utility is v ( x ) = 5 + 2 log( x ), and the initial wealth is W > A . Derive the risk premium R analytically. Answer. Following the definition of the risk premium, R should satisfy v ( W - R ) = 1 2 v ( W + A ) + 1 2 v ( W - A ) ⇐⇒ 5 + 2 log( W - R ) = 1 2 [5 + 2 log( W + A )] + 1 2 [5 + 2 log( W - A )] ⇐⇒ log( W - R ) = 1 2 log( W + A ) + 1 2 log( W - A ) ⇐⇒ log( W - R ) = log h ( W + A ) 1 2 ( W - A ) 1 2 i ⇐⇒ W - R = ( W + A ) 1 2 ( W - A ) 1 2 ⇐⇒ R = W - ( W + A ) 1 2 ( W - A ) 1 2 ⇐⇒ R = W - W 2 - A 2 . (3) (4 points) 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 ) = x + 2 y and u B ( x,y ) = x + 3 y , respectively. There are 7 xylophones and 1
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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 Figure 1: Contract curve for question 1 (3). bottom and the right. See Figure 1. (4) (4 points) There are apples (a), bananas (b), and cucumbers (c). Suppose that Y = { ( a,b,c ) : a 0 ,b = 2 a,c = - 3 b } is a firms production set. This firm can transform apples into both bananas and cucumbers. True or false? Answer. False. Both apples and bananas are inputs. The firm transforms apples and bananas into cucumbers. 2
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2 Externalities (20 points) General Motors can produce cars and sell them across the country for a price p c = 20 (assume that there are many firms so that this market is competitive and GM cannot affect this price). The cost of producing c cars is c 2 (for simplicity, assume cars are divisible goods). However, producing one car also creates one unit of carbon emissions ( E ). Michigan has one resident who has preferences over money ( m ) and GM’s carbon emissions. She is endowed with e m = 100 units of money (normalize the price of money to 1). The resident dislikes carbon emissions as they may contribute to climate change.
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final-solution02 - Econ 51: Final Exam Solution Friday,...

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