Suppose Firm 1 and Firm 2 have a constant MC=AC=8. The market (inverse) demand curve is given by:

P=200−2QQ=q1+q2

  1. Find the Cournot-Nash equilibrium output and profit for each firm if they compete on quantity.

Break demand apart into both firms’ output.

P=200−2QP=200−2(q1+q2)P=200−2q1−2q2

Solving for Firm 1, recalling that MR is twice the slope of the inverse demand curve: MR1=200−4q1−2q2

Firm 1 maximizes profit at q∗ where MR=MC:

MR1=MCProfit-max condition200−4q1−2q2=8Plugging in192−4q1−2q2=0Subtracting 8 from both sides192−2q2=4q1Adding 4q1 to both sides48−0.5q1=q∗1Dividing both sides by 4

Since Firm 2 is identical, its q∗ is: q∗2=48−0.5q1

Find Nash equilibrium algebraically by plugging in one firm’s reaction curve into the other’s

q1=48−0.5q2Firm 1's reaction functionq1=48−0.5(\textcolorblue48−0.5q1)Plugging in Firm 2's reaction functionq1=48−24+0.25q1Distributing the −0.5q1=24+0.25q1Subtracting0.75q1=24Subtracting 0.25q1 to both sidesq1=32Dividing by 0.75

Symmetrically, q2=32

Both firms produce 32:

P=200−2(32)−2(32)P=$72

We can find the profit for each firm:

π1=q1(P−c)π1=32(72−8)π1=$2,048

  1. Find the output and profit for each firm if the two were to collude.

Suppose now both firms collude and act like a single monopolist, who sets:

MR=MCProfit-max condition200−4Q=8Plugging in192−4Q=0Subtracting 8 from both sides192=4QAdding 4Q to both sides48=QDividing both sides by 4

So each firm will produce 24.

The monopoly price will then be P=200−2QP=200−2(48)P=$104

Total profit will then be: Π=Q(P−c)Π=48(104−8)Π=$4,608

with $2,304 going to each firm.

  1. Find the Nash equilibrium price each firm charges if they compete on price.

We know the Bertrand-Nash equilibrium is the perfectly competitive one, i.e. where p=MC.

P=MC200−2Q=8192−2Q=0192=2Q96=Q

Each firm produces 48.

  1. Find the industry price, output, and profits under this equilibrium (in part 3).

The price should be marginal cost, and profits should be zero, but we can confirm:

P=200−2QP=200−2(96)P=$8

Total profit will then be: Π=Q(P−c)Π=96(8−8)Π=$0

Comparing the different equilibria: