This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ECO 310  Fall 2007 Microeconomic Theory  A Mathematical Approach Problem Set 7  Answer Key Question 1: (a) There were two hard parts to this question: nasty algebra, and the intuition. Firm 1’s profit is Π 1 = ( p c ) q 1 = { [ q 1 + q 2 ] 3 / 2 c } q 1 . So its Cournot FONC (taking q 2 as given) is ∂ Π 1 /∂q 1 = [ q 1 + q 2 ] 3 / 2 c 3 2 [ q 1 + q 2 ] 5 / 2 q 1 = 0 Similarly for firm 2, ∂ Π 2 /∂q 2 = [ q 1 + q 2 ] 3 / 2 c 3 2 [ q 1 + q 2 ] 5 / 2 q 2 = 0 Subtracting the two FONCs, we have q 1 = q 2 . This is very useful, and something we might have been able to guess since the two firms are symmetric. Write q 1 + q 2 = Q , the industry output, so q 1 = q 2 = Q/ 2. Using this in either FONC, 0 = Q 3 / 2 c 3 2 Q 5 / 2 X 2 = 1 4 Q 3 / 2 c. So Q 3 / 2 = 4 c , that is, p = 4 c , and Q = (4 c ) 2 / 3 with each firm producing half of this. The total profit is Π = Π 1 + Π 2 = ( p c ) X = (3 c ) (4 c ) 2 / 3 = 3 × 4 2 / 3 c 1 / 3 = 1 . 1906 c 1 / 3 . (b) With Π ∝ c 1 / 3 , total profit increases as c increases. What is going on? The economic explanation is the gametheoretic interaction in the duopoly. In Cournot equilibrium, the firms are producing too much from the point of view of their joint profitmaximizing interest. It would be better for them to produce less and drive the price up. But under Cournot behavior they cannot credibly promise each other to keep outputs low and prices high. When the cost increases, they each produce less, which moves them in the right direction. This raise prices so much (since both are doing it) that profits increase despite the increase in costs and decrease in quantity produced....
View
Full Document
 Fall '08
 StephenE.Morris
 Game Theory, p1, Jen, best response

Click to edit the document details