Variations in Cournot Competition

Problem I

In lecture we saw the Cournot competition model for two firms with the same cost function. Now, we are going to consider asymmetric cost functions. Assume that demand for a good is given by P = a - bQ^d (Q^d is quantity demanded), and that there are 2 firms competing in quantities. Both have no fixed costs and a constant marginal cost. Firm 1 has a marginal cost C₁, and firm 2 hasa marginal cost C₂. We have that a > C₁ > C₂.

Find the reaction functions of firms 1 and 2 in this market: how the optimal quantity produced depends on the quantity produced by the other firm.

To verify that you have found the correct reaction functions, compute the optimal q₁ if q₂ = 100, a =4, b = 0.01, C₁ =2, and C₂ =1. (Note that this is not necessarily an equilibrium.)

q_{1 = 50}

II. Solve for the quantity produced by each firm and the equilibrium price.

To verify that you have found the correct equilibrium, compute q₁^*, q₂^*, and p^* if a = 4, b = 0.01, c₁ = 2, and c₂ = 1.

q₁^* =

q₂^* =

p^* =

III. Find the equilibrium price and the quantity produced by each ifrm if they compete in prices (Bertrand competition). (Assume the parameters given above.)

P is close to what value?

oC₁

oNone of the above

oC₂

o^C1+C2/₂

o0

q₁ is close to what value?

o0

oNone of the above

o^a-c1-c2/_b

o^a-c1/_b

o^a-c2/_b

oNone of the above

o0

IV. How does this equilibrium compare to the perfectly competitive case (if firms sold at their marginal cost as through they faced perfect competition)?

oBertrand competition results in an efficiency gain relative to perfect competition

oBertrand competition results in an efficiency loss relative to perfect competition