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midans08

# 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 fi 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 fi 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 ), and has a production function or 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 fi 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 a function of the parameter A . (Hint: Depending on A , it is possible that one or both fi 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 , x 2 , y 1 1 , y 2 1 , y 1 2 , y 2 2 ) , such that (i) ( x 1 , x 2 ) solves max A log( x 1 ) + log( 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 , y 2 1 0 , (iii) ( y 1 2 , y 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 , y 2 2 0 , (iv) markets clear x 1 = 1 + y 1 1 + y 1 2 x 2 = 1 + y 2 1 + y 2 2 .
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