• Question 1
  • Question 2
  • Question 3
  • Question 4

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

Question 1

Find the Cournot-Nash equilibrium output and profit for each firm.

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

Symmetrically, q2=32

Both countries 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

Question 2

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

Question 3

Find the price and output if the two were to compete on price instead of quantity.

In Bertrand competition, each firm sets:

P=MC200−2Q=8192−2Q=0192=2Q96=Q

Each firm produces 48, earning no profits.

Question 4

Suppose Firm 1 produces before Firm 2, who can see how much Firm 1 produced. Find the price, output, and profit for each firm.

Substitute follower’s reaction function into market (inverse) demand function

P=200−2q1−2q2The inverse market demand functionP=200−2q1−2(48−0.5q1)Plugging in Firm 2's reaction function forq2P=200−2q1−96+1q1Multiplying by âˆ’3P=104−q1Simplifying the right

  • Find MR for Firm 1 from market demand

MR1=104−2q1

MR=MCProfit-max condition104−2q1=8Plugging in104=8+2qsaAdding 2q1 to both sides96=2qsaSubtracting 20 from both sides48=q∗saDividing both sides by 2

  • Firm 2 will respond

q∗2=48−0.5q1q∗2=48−0.5(48)q∗2=48−24q∗2=24

  • With q∗1=48 and q∗2=24, this sets a market price of

P=200−2QP=200−2(72)P=56

  • Profit for Firm 1 is

π1=q1(P−c)π1=48(56−8)π1=$2,304

  • Profit for Firm 2 is

π2=q2(P−c)π2=24(56−8)π2=$1,152