midans08 - Department of Economics The Ohio State...

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Department of Economics The Ohio State University Midterm Answers—Econ 805 Prof. Peck February 7, 2008 1. (35 points) Consider the following economy with two goods, two f rms, and one con- sumer. The consumer’s utility function is given by u ( x 1 ,x 2 )= A log( x 1 )+log( x 2 ) , where A is a strictly positive parameter. The consumer owns both f rms and has the initial endowment vector, ω =(1 , 1) . Firm 1 produces good 1 with good 2 as an input (that is, y 1 1 0 and y 2 1 0 ) ,andhasaproduct ionfunc t ionor boundary of the production set given by y 1 1 = 1 2 y 2 1 . Firm 2 produces good 2 with good 1 as an input (that is, y 2 2 0 and y 1 2 0 ), and has a production function or boundary of the production set given by y 2 2 = 1 2 y 1 2 . (a) (10 points) De f ne a competitive equilibrium for this economy. (b) (20 points) Normalize the price of good 2 to be one, so the price vector is given by ( p, 1) . Compute the competitive equilibrium price and allocation, as afunct iono ftheparameter A . (Hint: Depending on A ,i tisposs ib letha tone or both f rms do not produce.) (c) (5 points) For what values of A is the competitive equilibrium allocation Pareto optimal? Answer: (a) Note: because of strict monotonicity, we can write budget constraints and market clearing as equalities. A CE is a price vector ( p 1 ,p 2 ) and an allocation, ( x 1 2 ,y 1 1 2 1 1 2 2 2 ) , such that (i) ( x 1 2 ) solves max A log( x 1 x 2 ) subject to p 1 x 1 + p 2 x 2 = p 1 + p 2 + π 1 + π 2 x 0 , 1
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(ii) ( y 1 1 ,y 2 1 ) solves max p 1 y 1 1 + p 2 y 2 1 subject to y 1 1 = 1 2 y 2 1 y 1 1 0 2 1 0 , (iii) ( y 1 2 2 2 ) solves max p 1 y 1 2 + p 2 y 2 2 subject to y 2 2 = 1 2 y 1 2 y 1 2 0 2 2 0 , (iv) markets clear x 1 =1+ y 1 1 + y 1 2 x 2 y 2 1 + y 2 2 .
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midans08 - Department of Economics The Ohio State...

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