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Q: A homogenous-good duopoly faces an inverse market demand function of P = 57 – 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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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.

Rahul Sunny answered 1 year ago

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