Answer to Question #322562 in Microeconomics for Komal

Question

Question : Consider a market structure comprising two identical firms (A and B), each with the cost function given by

Ci = 30Qi , where Qi fori = {A, B} is output produced by each firm.

Market demand is given by

P = 210 − 1.5Q, where Q = QA + QB

(i) Find Cournot equilibrium.

(ii) What will be the outcome if the firms decide to collude? Compare it with the results

under the Cournot equilibrium.

Answer 1

(i) Given

$C_i=30Q_i$

$P=210-1.5Q$

Where

$Q=Q_A+Q_B$

Profit is given as

$\pi=TR-TC$

$TC=30Q_A+30Q_B$

$TR_A=P×Q_A$

$TR_A=210Q_A-1.5Q_A^2-1.5Q_AQ_B$

$\pi_A=210Q_A-1.5Q^2_A-1.5Q_AQ_B-30Q_A-30Q_B$

$\pi_A=180Q_A-30Q_B-1.5Q_AQ_B-1.5Q_A^2$

$\pi_A'=180-1.5Q_B-3Q_A=0$

The cournot equilibriums are therefore:

$Q_A^*=60-0.5Q_B$

$Q_B^*=60-0.5Q_A$

(ii)if the firms collude, then

$\pi=TR-TC$

$TR=210Q-1.5Q^2$

$\pi=210Q-1.5Q^2-30Q$

$\pi=180Q-1.5Q^2$

$\pi'=180-3Q=0$

$Q^*=60$

$Q_A=Q_B=30$

$P^*=210-1.5(60)$

$P^*=120$

When they do not collude:

$Q_A=60-0.5(60-0.5Q_A)$

$Q_A=30+0.25Q_A$

$0.75Q_A=30$

$Q_A^*=40$

$Q_B^*=40$

$P^*=90$

Profits when the firms collude is higher for each firm as both get to split the total profit of 5400 equally between themselves (2700 each). When they do not collude, profits drop to 1200 for each firm.