Question

Assume that two companies (A and B) are Cournot duopolists that produce identical products. Demand for the products is given by the following linear demand function: ? = 200 − ?A − ?B where ?c and ?d are the quantities sold by the respective firms and P is the price. Total cost functions for the two companies are ??A = 1,500 + 55?A + ?A2 ??B = 1,200 + 20?B + 2?B2 a. Determine the equilibrium price and quantities sold by each firm. Cournot Equilibrium Price output profit Company A Price 145 Output 30 Profit 300 Company B Price 145 Output 25 Profit 675 Total Industry Output 55 Profit $975 b. Determine the profits for the market as well as each firm.

Answer 1

MCA = dTCA/dQA = 55 + 2QA

MCB = dTCB/dQB = 20 + 4QB

P = 200 - QA - QB [Since Q = QA + QB]

For firm A,

Total revenue (TRA) = P x QA = 200QA - QA² - QA.QB

Marginal revenue (MRA) = TRA/QA = 200 - 2QA - QB

Equating MRA and MCA,

200 - 2QA - QB = 55 + 2QA

4QA + QB = 145........(1) (Best response, firm A)

For firm B,

Total revenue (TRB) = P x QB = 200QB - QA.QB - QB²

Marginal revenue (MRB) = TRB/QB = 200 - QA - 2QB

Equating MRB and MCB,

200 - QA - 2QB = 20 + 4QB

QA + 6QB = 180........(2) (Best response, firm B)

Cournot equilibrium is obtained by solving (1) and (2). Multiplying (2) by 4,

4QA + 24QB = 720.........(3)

4QA + QB = 145 ........(1)

(3) - (1) yields: 23QB = 575

QB = 25

QA = 180 - 6QB [from (2)] = 180 - (6 x 25) = 180 - 150 = 30

Q = 30 + 25 = 55

P = 200 - 30 - 25 = 145

Aggregate revenue (R) = P x Q = 145 x 55 = 7975

Aggregate cost (C) = TCA + TCB = [1500 + (55 x 30) + (30 x 30)] + [1200 + (20 x 25) + (2 x 25 x 25)]

= 2700 + 1650 + 900 + 500 + 1250 = 7000

Industry profit = R - C = 7975 - 7000 = 975

Firm A profit = TRA - TCA = (P x QA) - [1500 + (55 x 30) + (30 x 30)] = (145 x 30) - 4050 = 4350 - 4050 = 300

Firm B profit = TRB - TCB = (P x QB) - [1200 + (20 x 25) + (2 x 25 x 25)] = (145 x 25) - 2950 = 3625 - 2950 = 675