midans08

# midans08 - Department of Economics The Ohio State...

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(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 .
This is the end of the preview. Sign up to access the rest of the document.

## midans08 - Department of Economics The Ohio State...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online