Question #49917

Assume that two companies (A and B) are duopolists who produce identical products. Demand for the products is given by the following linear demand function:

p = 200 − Q_A-Q_B

where Q_A and Q_B are the quantities sold by the respective firms and P is the selling price. Total cost functions for the two companies are

TC_A=1500+55Q_A+〖Q^2〗_A

TC_B=1200+20Q_B+2〖Q^2〗_B

Assume that the firms act independently as in the Cournot model (i.e., each firm assumes that the other firm’s output will not change).

a. Determine the long-run equilibrium output and selling price for each firm.

b. Determine Firm A, Firm B, and total industry profits at the equilibrium solution found in Part (a).

p = 200 − Q_A-Q_B

where Q_A and Q_B are the quantities sold by the respective firms and P is the selling price. Total cost functions for the two companies are

TC_A=1500+55Q_A+〖Q^2〗_A

TC_B=1200+20Q_B+2〖Q^2〗_B

Assume that the firms act independently as in the Cournot model (i.e., each firm assumes that the other firm’s output will not change).

a. Determine the long-run equilibrium output and selling price for each firm.

b. Determine Firm A, Firm B, and total industry profits at the equilibrium solution found in Part (a).

Expert's answer

a) Demand function has been given as: P = 200 - Qa - Qb

To solve for the Cournot equilibrium, we first derive the reaction functions of the two firms by setting MR = MC.

Firm A:

MR = TR' = (P*Qa)' = 200 - 2Qa

MC = TC' = (1500 + 55Qa + Qa^2)' = 55 + 2Qa

200 - 2Qa = 55 + 2Qa

Qa = 145/4 = 36.25 units

Firm B:

MR = TR' = (P*Qb)' = 200 - 2Qb

MC = TC' = (1200 + 20Qb + 2Qb^2)' = 20 + 4Qb

200 - 2Qb = 20 + 4Qb

Qb = 180/6 = 30 units

P = 200 - 36.25 - 30 = $133.75

b) Total profits will be:

TPa = TR - TC = P*Qa - TCa = 133.75*36.25 - (1500 + 55*36.25 + 36.25^2) = $40.625

TPb = TR - TC = P*Qb - TCb = 133.75*30 - (1200 + 20*30 + 2*30^2) = $412.5

Total industry profits are: TP = TPa + TPb = 40.625 + 412.5 = $453.125

To solve for the Cournot equilibrium, we first derive the reaction functions of the two firms by setting MR = MC.

Firm A:

MR = TR' = (P*Qa)' = 200 - 2Qa

MC = TC' = (1500 + 55Qa + Qa^2)' = 55 + 2Qa

200 - 2Qa = 55 + 2Qa

Qa = 145/4 = 36.25 units

Firm B:

MR = TR' = (P*Qb)' = 200 - 2Qb

MC = TC' = (1200 + 20Qb + 2Qb^2)' = 20 + 4Qb

200 - 2Qb = 20 + 4Qb

Qb = 180/6 = 30 units

P = 200 - 36.25 - 30 = $133.75

b) Total profits will be:

TPa = TR - TC = P*Qa - TCa = 133.75*36.25 - (1500 + 55*36.25 + 36.25^2) = $40.625

TPb = TR - TC = P*Qb - TCb = 133.75*30 - (1200 + 20*30 + 2*30^2) = $412.5

Total industry profits are: TP = TPa + TPb = 40.625 + 412.5 = $453.125

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