Return to the example from lessons 2.2 and 2.3: 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. Suppose Firm 1 is the Leader and Firm 2 is the follower. Find the Stackelberg Nash Equilibrium quantity for each firm. Hint, the Cournot reaction functions you found before were:

q⋆1=48−0.5∗q2q⋆2=48−0.5∗q1

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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 −3P=104−q1Simplifying the right

• Find MR for Firm 1 from market demand

MR1=104−2q1

MR=MCProfit-max condition104−2q1=8Plugging in104=8+2q1Adding 2q1 to both sides96=2q1Subtracting 20 from both sides48=q∗1Dividing both sides by 2

Firm 2 will respond:

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

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2. Find the market price.

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

P=200−2QP=200−2(72)P=56

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3. Find the profit for each firm. Compare their profits under Stackelberg competition to their profits under Cournot competition (from lesson 2.2).

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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

Compared to the Cournot equilibrium where each firm produces 32, setting a market price of $72, and a profit of $2048 for each firm, under Stackelberg competition, Firm 1 produces more than Cournot and earns higher than Cournot profits, while Firm 2 produces less than Cournot and earns less profit.