Variations in Cournot Competition

Problem Set

Problem 1

1 point possible (graded)

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 =}

Problem 2

3points possible (graded)

Solove 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^* =

Problem PS7.2.3a

3 points possible (graded)

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

Problem 3

1 point possible (graded)

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

Problem 4

1 point possible (graded)

Now let's go back to the case where all firms have the same cost function. In class we saw the Cournot competition model for two firms. Now, we are going to get you through the Cournot model with three firms. Assume that demand for a good is given by P = a - bQ^d , and that there are 3 firms competing in quantity with a constant marginal cost c < a.

Write down the maximization problem of a representative firm in the market. Solve for the reaction function of this firm: how the optimal quantity produced depends on the quantity produced by the other two firms in the market.

To verify that you have found the correct reaction function, compute the optimal q₁ if q₂ = 40, q₃ =60, a = 4, b = 0.01, and c =2. (note that this is not necessarily and equilibrium.)

q₁ =

Problem PS7.2.5a

2 points possible (graded)

In the three-firm case, what will be the equilibrium price and the total quantity produced in the market?

To Verify that you have found the correct equilibrium, compute p and Q if a = 4, b = 0.01 and c = 2.

P =

Q =

Problem PS7.2.5b

1 point possible (graded)

How does this equilibrium compare to the perfectly competitive case?

oCournot competition results in an efficiency gain relative to perfect competition

oCournot competition results in an efficiency loss relative to perfect competition

Problem PS7.2.6

1 point possible (graded)

Intuitively, what will happen in this market when the number of firms goes to ?

oThe market converges to perfect competition

oNone of the above

oEfficiency losses become arbitrarily large

oCournot competition will become the more realistic model

oPrice will approach zero