h6answ03 - The Ohio State University Department of...

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The Ohio State University Department of Economics Econ 805–Extra Problems on Production and Uncertainty: Questions and Answers Winter 2003 Prof. Peck (1) In the following economy, there are two consumers, two …rms, and two goods (labor/leisure and food). For i = 1,2, consumer i is endowed with zero units of food and 1 unit of leisure, ! i =(0 ; 1) .L e t t i n g x i denote consumer i’s consumption of food and ` i denote consumer i’s consumption of leisure, the utility function is: log( x i ) + log( ` i ) . Let y 1 denote …rm 1’s output of food and L 1 denote …rm 1’s labor input (so that L 1 must be nonnegative). Then …rm 1’s production function, the frontier of its production set, is given by: y 1 = AL 1 , where the parameter A is a positive real number. Firm 1 is owned by consumer 1. Let y 2 denote …rm 2’s output of food and L 2 denote …rm 2’s labor input (so that L 2 must be nonnegative). Then …rm 2’s production function, the frontier of its production set, is given by: y 2 =( L 2 ) 1 = 2 .F i rm2i sown edbycon sum e r 2. (a) De…ne a competitive equilibrium for this economy. (b) Calculate the competitive equilibrium price vector and allocation, as a function of the parameter, A. Assume that we have an interior solution, where both …rms produce output. (c) For what values of the parameter, A, will we have a corner solution, where one of the …rms produces zero output? Answer: (a) Normalizing the price of food to be 1 and denoting the price of labor as p , a Competitive Equilibrium is a price vector, (1 ;p ) , and an allocation, ( x 1 ;` 1 ;x 2 2 ;y 1 ;L 1 2 2 ) , such that: (i) ( x 1 1 ) solves: maxlog( x 1 )+log( ` 1 ) subject to x 1 + p` 1 = p ( x 1 1 ) ¸ 0 : This relies on the fact that utility is monotonic and …rm 1 has CRS and receives zero pro…ts at the CE. (ii) ( x 2 2 ) solves maxlog( x 2 ` 2 ) subject to x 2 + p` 2 = p + ¼ 2 ( x 2 2 ) ¸ 0 : 1
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(iii) ( y 1 ;L 1 ) solves max y 1 ¡ pL 1 subject to y 1 = AL 1 L 1 ¸ 0 : (iv) ( y 2 2 ) solves max y 2 ¡ pL 2 subject to y 2 =( L 2 ) 1 = 2 L 2 ¸ 0 : (v) x 1 + x 2 = y 1 + y 2 ` 1 + ` 2 + L 1 + L 2 =2 : (Equalities follow from strict monotonicity of utility.) (b) Starting with the pro…t maximization conditions, for an interior solution
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This note was uploaded on 07/17/2008 for the course ECON 805 taught by Professor Peck during the Spring '08 term at Ohio State.

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h6answ03 - The Ohio State University Department of...

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