Question

In a Stackelberg duopoly, one firm is a 'leader' and one is a 'follower'. Both firms know each other's costs and market demand. The follower takes the leader's output as given and picks his own output accordingly (i.e., the follower acts like a Cournot competitor). The leader takes the follower's reactions as given and picks his own output accordingly. Suppose that firms 1 and 2 face market demand, \( p = 100 - (q_{1} + q_{2}) \). Firm costs are \( c_{1} = 10 q_{1} \) and \( c_{2} = q_{2}^{2} \). (a) Calculate market price and each firm's profit assuming that firm 1 is the leader and firm 2 the follower. (b) Do the same assuming that firm 2 is the leader and firm 1 is the follower. (c) Given your answers in parts (a) and (b), who would firm 1 want to be the leader in the market? Who would firm 2 want to be the leader? (d) If each firm assumes what it wants to be the case in part (c), what are the equilibrium market price and firm profits? How does this compare with the Cournot-Nash equilibrium in this market?

          In a Stackelberg duopoly, one firm is a 'leader' and one is a 'follower'. Both firms know each other's costs and market demand. The follower takes the leader's output as given and picks his own output accordingly (i.e., the follower acts like a Cournot competitor). The leader takes the follower's reactions as given and picks his own output accordingly. Suppose that firms 1 and 2 face market demand, \( p = 100 - (q_{1} + q_{2}) \). Firm costs are \( c_{1} = 10 q_{1} \) and \( c_{2} = q_{2}^{2} \).
	
	(a) Calculate market price and each firm's profit assuming that firm 1 is the leader and firm 2 the follower.
	
	(b) Do the same assuming that firm 2 is the leader and firm 1 is the follower.
	
	(c) Given your answers in parts (a) and (b), who would firm 1 want to be the leader in the market? Who would firm 2 want to be the leader?
	
	(d) If each firm assumes what it wants to be the case in part (c), what are the equilibrium market price and firm profits? How does this compare with the Cournot-Nash equilibrium in this market?

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Principles of Economics

Gregory Mankiw 8th Edition

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Solved by Expert Lainey Roebuck

12/10/2024

Step 1

- Market demand: \( p = 100 - (q_1 + q_2) \) - Cost for firm 1: \( c_1 = 10q_1 \) - Cost for firm 2: \( c_2 = q_2^2 \) Show more…

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In a Stackelberg duopoly, one firm is a 'leader' and one is a 'follower'. Both firms know each other's costs and market demand. The follower takes the leader's output as given and picks his own output accordingly (i.e., the follower acts like a Cournot competitor). The leader takes the follower's reactions as given and picks his own output accordingly. Suppose that firms 1 and 2 face market demand, \( p = 100 - (q_{1} + q_{2}) \). Firm costs are \( c_{1} = 10 q_{1} \) and \( c_{2} = q_{2}^{2} \). (a) Calculate market price and each firm's profit assuming that firm 1 is the leader and firm 2 the follower. (b) Do the same assuming that firm 2 is the leader and firm 1 is the follower. (c) Given your answers in parts (a) and (b), who would firm 1 want to be the leader in the market? Who would firm 2 want to be the leader? (d) If each firm assumes what it wants to be the case in part (c), what are the equilibrium market price and firm profits? How does this compare with the Cournot-Nash equilibrium in this market?

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Transcript

00:02 So we start off with a, which is where firm 1 is the leader, and firm 2 is the follower.

00:09 The profit for firm 2 is given by pi 2 is equal to pq 2 minus c2, which is 100 minus q2 plus q2 times q2 minus q2 squared.

00:25 So we're going to substitute the demand function, and you're going to get 100 minus q1 minus q2 times q2.

00:34 Minus q2 squared, and that is 100, q2, minus q1, q2, minus q2 squared, minus q2 squared, and then you simplify that, and you get pi 2 is equal to 100, q2 minus q2, minus 2 q2 squared.

00:58 So we need to take the derivative of this with respect to q2 and set it equal to 0.

01:03 So d pi 2 over dq2 is equal to 100 minus q1 minus 4 q2's equals 0.

01:17 If we solve for q2, that is 100 minus q1 over 4.

01:26 And if we do the same thing for firm 1, we have pi 1 is equal to pq1 minus c1, which is 100 minus q1.

01:44 Plus q2 times q1 minus 10 q1s.

01:49 So then we need to simplify and take the derivative.

01:57 And so q1 is 2 times 65 over 3, which gives you 43 .33.

02:05 So now we need to calculate q2.

02:07 So q2 is 100 minus 43 .33 divided by 4.

02:19 That leaves you with 14 .17.

02:26 So the market price is p equals 100 minus q1 plus q2, that's 100 minus 57 .5, which gives you 42 .5.

02:42 And so then the profits for firm 1 is pq1 minus c1, which would be 42 .5 times 43 .33 minus 10, times 43 .33, and that gives you 14 .10 .95.

03:04 And then for firm 2, you get 42 .5 times 14 .17 minus 14 .17 squared.

03:20 And you get 400 .45...

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