The market for widgets consists of two firms that produce identical products. Co

Question

The market for widgets consists of two firms that produce identical products. Competition in the market is such that each of the firms independently produces a quantity of output, and these quantities are then sold in the market at a price that is determined by the total amount produced by the two firms. Firm 2 is known to have a cost advantage over Firm 1. A recent study found that the (inverse) market demand curve faced by the two firms is: P = 280 – 2(Q1 + Q2) and costs are: C1(Q1) = 3Q1 C2(Q2) = 2Q2. Please identify the price each firm will charge, output each firm will produce, and the profit/loss each firm will make.

Explanation / Answer

P = 280 – 2Q1 – 2Q2

So,

Total revenue of firm 1, TR1 = P x Q1 = 280Q1 – 2Q12 – 2Q1Q2

Total revenue of firm 2, TR2 = P x Q2 = 280Q2 – 2Q1Q2 – 2Q22

MC1 = dTC1 / dQ1 = 3

MC2 = dTC2 / dQ2 = 2

So,

Marginal revenue of firm 1, MR1 = dTR1 / dQ1 = 280 - 4Q1 - 2Q2

Equating with MC1:

280 - 4Q1 - 2Q2 = 3

4Q1 + 2Q2 = 277......(1) [Reaction function, firm 1]

Marginal revenue of firm 2, MR2 = dTR2 / dQ2 = 280 – 2Q1 – 4Q2

MC2 = 2

Equating MR2 = MC2,

280 – 2Q1 – 4Q2 = 2

Or,

2Q1 + 4Q2 = 278.......(2) [Reaction function, firm 2]

Equilibrium is obtained by solving (1) & (2).

4Q1 + 2Q2 = 277......(1)

(2) x 2:

4Q1 + 8Q2 = 556 .....(3)

(3) - (1): 6Q2 = 279

Q2 = 46.5

Q1 = (278 – 4Q2) / 2 = [278 – (4 x 46.5)] / 2 = (278 – 186) / 2 = 92 / 2 = 46

Q = Q1 + Q2 = 46 + 46.5 = 92.5

P = 280 - 2Q = 280 - (2 x 92.5) = 280 – 185 = 95

Profit of firm 1 = TR1 – TC1 = (P x Q1) – 3Q1 = Q1 x (P – 3) = 46 x (95 – 3)

= 46 x 92 = 4,232

Profit of firm 2 = TR2 – TC2 = (P x Q2) – 2Q2 = Q2 x (P – 2) = 46.5 x (95 – 2)

= 46.5 x 93 = 4,324.5

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