HW2_key - ECON 3102 Summer 2010 Homework 2 Answer Key 1...

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ECON 3102 Summer 2010 Homework 2 Answer Key 1 . Consider a one-period, closed-economy model where the representative consumer has util- ity function U ( C, ‘ ) = C 1 / 2 1 / 2 and has h available hours to divide between work and leisure. The representative firm has technology given by Y = zK 1 / 3 N 2 / 3 . There is a government that sets its expenditure level at a value G 0 0 . a. Define a CE for this economy. b. Find the CE values ( C * , ‘ * , N * , T * , Y * , w * ) given parameters h = 14 , z = 1 , K = 15 , 625 , and G 0 = 0 . Answer : (a) A competitive equilibrium is a set of ( C, N s , N d , T, Y, w ) such that given ( G 0 , z, K, h ): The representative consumer chooses C, ‘ to solve max C,‘ C 1 / 2 1 / 2 subject to C = w ( h - ) + π - T C 0 0 h, here N s = h - . The representative firm chooses N d to solve max N d zK 1 / 3 N 2 / 3 - wN d subject to N d 0 , where Y = zK 1 / 3 N 2 / 3 , π = Y - wN d = zK 1 / 3 N 2 / 3 - wN d . All markets clear: N d = N s . The government’s budget constraint is satisfied: G = T. (b) To find the competitive equilibrium, note first that the production function is given by Y = (1)(15 , 625) 1 / 3 N 2 / 3 = 25 N 2 / 3 . 1
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The equations characterize the CE are: C = (50 / 3) N - 1 / 3 C = 25 N 2 / 3 Considering N = h - = 14 - , we can equate them: (50 / 3) N - 1 / 3 = 25 N 2 / 3 = 1 . 5 N = 1 . 5(14 - ) + 1 . 5 = 21 2 . 5 = 21 so that * = 8 . 4. It follows that N * = 14 - * = 14 - 8 . 4 = 5 . 6, C * = 25 N 2 / 3 = 25(5 . 6) 2 / 3 = 78 . 8 and w * = (50 / 3) N - 1 / 3 = (50 / 3)(5 . 6) - 1 / 3 = 9 . 4. Since G 0 = 0, T = 0, and from the
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