Answer to Question #49917 in Economics for nuraini

Question

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).

Answer 1

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