Question

Consider two firms with the cost function TC(q) = 5q (constant average and marginal cost,of 5), facing the market demand curve Q = 53 – p (where Q is the total of the firms’ quantities, and p is market price).

a. What will be each firm’s output and profit if they make their quantity choices simultaneously (as Cournot duopolists)?

b. Now suppose Firm 1 is the Stackelberg leader (its decision is observed by Firm 2 prior to that firm’s decision). What will be each firm’s output and profit?

Answer 1

Market demand curve, P = 53 - (q1 + q2)

Total revenue for i firm, TRi = P*qi = 53qi - (qi2 + qiqj) , i,j = 1,2 and i is not equal to j

=> Marginal revenue, MRi = dTRi/dqi = 53 - 2qi - qj

And marginal cost for both the firms, MC = 5

For firm 1

MR1 = MC

53 - 2q1 - q2 = 5

=> q1 = 24 - 0.5q2 —> BR1

For firm 2

MR2 = MC

53 - 2q2 - q1 = 5

=> q2 = 24 - 0.5q1 —> BR2

Solving BR1 and BR2

q1 = q2 = 16 units

=> P = $21

Profit of firm 1 = Profit of firm 2 = P*q1 - TC1 = 21*16 - 5*16 = $256

b) Substituting BR2 in demand function,

P = 53 - q1 - 24 + 0.5q1

TR1 = Pq1 = 29q1 - 0.5q12

MR1 = dTR1/dq1 = 29 - q1

At equilibrium,

MR1 = MC

=> 29 - q1 = 5

=> q1 = 24 units

From BR2,

q2 = 12 units

=> P = $17

Profit of 1 = TR1 - 5q1 = $288

Profit of firm 2 = TR2 - 5q2 = $144

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