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Unformatted text preview: ECON 610: Microeconomic Theory II Prof. Blume Problem Set 2 1 Problem Set 2 This is a solution key for problem set 2. Some intermediate steps are left to the reader. If you still have questions regarding any of the problems, please visit Prof. Blume or one of the TAs during office hours. Thanks, Liyan and Max Problem 1: Yet another coconut economy. The production function is f ( x ) = (3 / 2) x 2- (2 / 3) x 3 for 0 = x < 3 / 2 , and 9 / 8 for x > 3 / 2 . Suppose Rob is endowed with 3/2 unit of leisure. (a) Utility is of the form u ( c,l ) = c + al for a > . As a varies between and 8 , identify the Pareto optimal allocations. Which allocations cannot be supported as competitive equilibria? (b) Utility is of the form u ( c,l ) = minc,al . Describe the competitive equilibrium wage as a function of a for a > . Solution: Drawing the picture helps. The production function is S-shaped, with f (0) = f (3 / 2) = 0. (a) For the linear preference, there are three cases depending on whether a exceeds 27 / 32, the slope of the tangent line of f ( x ) through the origin. If a > 27 / 32, then x = 0; if a < 27 / 32, then x is determined by MRS = MRT , i.e., f ( x ) = a ; and if a = 27 / 32, then x = 0 or 9/8. It could be easily shown that all the Pareto efficient allocations could be supported by a CE. (b) For the Leontief preference, only when a (3 / 2)(9 / 8) 2- 2 / 3(9 / 8) 3 3 / 2- 9 / 8 = 81 32 , could a PE allocation be supported by a CE 1 . The wage rate is determined by the FOC in the profit maximization problem: w = f ( x ), where x is the solution to a (3 / 2- x ) = 3 / 2 x 2- 3 / 2 x 3 (this equation follows by market clearing conditions c = y and x + l = 3 / 2). Problem 2: Rob can not only produce coconuts, he can also catch fish. The production function for coconuts is f ( x ) = x , and that for fish is g ( x ) = x . His utility function over triples of coconuts, fish and leisure is u ( c,f,l ) = ln min c,f + ln l . He is endowed with L units of time. 1 By the first fundamental welfare theorem, any CE allocation should be a Pareto efficient. ECON 610: Microeconomic Theory II Prof. Blume Problem Set 2 2 (a) Describe all the optimal allocations. (b) Describe the competitive equilibrium allocations. You should think about how to answer this question if Robs utility function is Cobb-Douglas. Solution: (a) In Pareto efficient allocations, c = f. Hence, c = x c = f = x f , where x c and x f are the labor inputs in coconut and fish production respectively. Also, time would be used up since Rob likes leisure. Thus, x c + x f + l = L . So, the PE allocations could be found by max x f > log x f + log L- ( x 2 f + x f ) . FOC gives 1 x f = 2 x f + 1 L- ( x 2 f + x f ) x f = 1 + 3 L- 1 3 ....
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- Spring '05