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Final-2006-150

# Final-2006-150 - Econ 6202 Fall 2006 Dmitry Shapiro Final...

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Econ 6202, Fall 2006 Dmitry Shapiro Final Exam December 12, 2006 ATTENTION! THIS IS 150 MINUTE EXAM! There are FIVE questions in the final. You need to do ALL of them. The exam is 150 minutes long and has 150 points. Each question has 30 points assigned. Thus it would make sense for you to start with easier questions. The exam is closed-book. Please turn off your cellphones and other sound-making devices. Good luck! Remember there are two exam options for you. The first one is to take 150 minute exam today (December 12). The second option is to take 2 seventy-five minute exams: 75 minutes today and 75 minutes on Thursday . You can pick any that suits you best. This is 150 minute exam! IMPORTANT: When you turn this page you will lose the option of taking two 75 minute exams. 1

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1. [Producer Theory and Monopoly - 30 points] Consider the following economy. There are two consumers, 1 and 2; who derive utility from a certain good y and from money m . The consumers utility functions are u 1 ( x, m ) = 36 y + m, u 2 ( x, m ) = 144 y + m. A monopolist produces the good y by using two inputs: x 1 and x 2 . The production function is y = x 1 + x 2 . Let w i denote the price of input i = 1 , 2. (a) Compute the monopolist’s cost function. (b) Suppose w 1 = 6, w 2 = 3. Suppose also that the monopolist observes the consumers utility and makes take-it-or-leave-it offers ( r i , y i ), i = 1 , 2 ( r i denotes the payment that consumer i has to make to receive y i units of the good). Compute the optimal offers (i.e. the offers that maximize the monopolist’s profits). 2. [General Equilibrium - 30 points] In a two-consumer, two-good exchange economy the consumers have the following preferences: u 1 ( x 1 , x 2 ) = min { x 1 , 2 x 2 } , u 2 ( x 1 , x 2 ) = x 1 / 2 1 x 1 / 2 2 . (1) The endowments of consumer 1 and 2 are e 1 = (5 , 9) and e 2 = (12 , 1), respectively. Suppose that the government of this economy can redistribute wealth between the two con- sumers by choosing monetary transfers. Let T denote the transfer from consumer 1 to con- sumer 2. If T > 0 then consumer 1 gives T dollars to consumer 2. If T < 0 then consumer 2 gives T dollars to consumer 1. Given a transfer T , let E ( T ) denote the following economy. Consumer 1 owns the bundle e 1 and - T dollars. Consumer 2 owns the bundle e 2 and T dollars. Consumers’ preferences are described by (1). (a) Consider the allocation x = ( x 1 , x 2 ) = ((8 , 4) , (9 , 6)). Can you find a transfer T such that x is the Walrasian equilibrium allocation of the economy E ( T )? If your answer is affirmative, compute the value of T and of the Walrasian equilibrium prices (fix p 1 = 1). If your answer is negative, explain why such a transfer does not exist. (b) Redo question 2a with the allocation ¯ x = ((7 , 3) , (10 , 7)). 3. [General Equilibrium II - 30 points] An exchange economy has two consumers, A and B , and two gods a and b . Consumers’ utility is given by U A ( a, b ) = a, U B ( a, b ) = b.
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