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In: Economics
Q: A homogenous-good duopoly faces an inverse market demand function of P = 57 – Q....
Q:
A homogenous-good duopoly faces an inverse market demand function of P = 57 – Q. Firm 1 has a constant marginal cost of MC1 = 10. Firm 2’s constant marginal cost is MC2 = 6. Calculate the output of each firm, market output, and price for a Nash-Cournot equilibrium. Draw the best response functions for each firm.
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Solutions
Expert Solution
P = 57 - Q = 57 - Q1 - Q2 [Since Q = Q1 + Q2]
For firm 1,
Total revenue (TR1) = P x Q1 = 57Q1 - Q12 - Q1Q2
Marginal revenue (MR1) = 
TR1/Q1
= 57 - 2Q1 - Q2
Equating MR1 and MC1,
57 - 2Q1 - Q2 = 10
2Q1 + Q2 = 47......(1) [Best response, firm 1]
For firm 2,
TR2 = P x Q2 = 57Q2 - Q1Q2 - Q22
MR2 =
TR2/Q2
= 57 - Q1 - 2Q2
Equating MR2 and MC2,
57 - Q1 - 2Q2 = 6
Q1 + 2Q2 = 51........(2) [Best response, firm 2]
Cournot equilibrium is obtained by solving (1) and (2).
(2) x 2 yields:
2Q1 + 4Q2 = 102......(3)
2Q1 + Q2 = 47 ......(1)
(3) - (1) yields:
3Q2 = 55
Q2 = 18.33
Q1 = 51 - 2Q2 [From (2)] = 51 - (2 x 18.33) = 51 - 36.66 = 14.34
Q = 14.34 + 18.33 = 32.67
P = 57 - 32.67 = 24.33
From Best response function for firm 1,
When Q1 = 0, Q2 = 47 (Vertical intercept) and when Q2 = 0, Q1 = 47/2 = 23.5 (Horizontal intercept).
From Best response function for firm 2,
When Q1 = 0, Q2 = 51/2 = 25.5 (Vertical intercept) and when Q2 = 0, Q1 = 51 (Horizontal intercept).
In following graph, BR1 and BR2 are best response functions for firm 1 and 2 respectively.
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