Question

Consider the equilibrium of the Cournot model of duopoly.

a. Can the firms benefit from collusion in this instance if they happen to initially be at the intersection of the two reaction curves and the collusion is enforced? Explain your answer.

b. Show that each firm has an incentive to break any collusive agreement.

c. Has the experience of the international oil cartel, OPEC, supported your answers?

Answer 1

The equilibrium condition of cournot duopoly model is explained using the example -

Let two firms (Firm A and Firm B) facing constant marginal cost of 20 and industry demand curve is given as P = 140 - (Q_a+Q_b). So, cournot equilibrium using best response concept is computed as -

Profits of firm A = P X Q_a - 20 X Q_a

= (140 - (Q_a+Q_b)) X Q_a - 20 X Q_a-----(1)

Differentiation above equation with respect to Q_a gives -

140 - 2Qa - Q_b - 20 = 0 -----(2)

From (2) we get Best response of Firm A, that is, BR_A -

(120 - Q_b)/2 = Q_a -------(3)

Similarly for Firm B is computed as -

Profits of firm A = (140 - (Q_a+Q_b)) X Q_b - 20 X Q_b------(4)

Differentiation above equation with respect to Q_bgives -

140 - Qa - 2Q_b - 20 = 0 -------(5)

From (5) we get Best response of Firm B, that is, BR_B -

(120 - Q_a)/2 = Q_b ------(6)

Substituting (3) in (6), and then substituting back in equation (3), we get -

60 = 2Q_b - Q_b/2

Q^c_b= 40 = Q^c_a

And price is P = 140 - (40+40) = 60 and firms A and B profits are same and equal to 1600 after substituting value of P and Q's back in equation (1) and (4). So, joint profits are 3200.

(a) Yes firm can benefit from collusion this can be seen as -

In the case of cartel equilibrium is determined by joint profit maximising condition -

joint profit =(140-(Q_a+Q_b)) X (Q_a+Q_b) - 20 X (Q_a+Q_b) -----(7)

Let Q_a+Q_b = Q, then joint profits are -

joint profit =(140-(Q)) X (Q) - 20 X (Q) ------(8)

Differentiation equation (8) with respect to Q, and we get -

140 - 2Q - 20 = 0

Q^Cartel = 60 and P^Cartel= 140 - 60 = 80 and Joint profits = (140-60) X 60 -20 X 60 = 3600.

Q_a^Cartel = Q_b^Cartel = 60/2 = 30

The profits of cartel are 3600 more than the joint profits of cournot duopoly, that is 3200. Thus, this is how firms benefit from collusion.

(b) As each firm aim is to maximise profit. Thus, each firm has an incentive to deviate from cartel optimum quantity, that is, 30. This is shown below -

Suppose firm B is committed to cartel quantity, that is, 30. So, given Q_b= 30 the best response of firm A is derived using equation (3) -

Q_a = (120 - 30)/2 = 45 and P = 140 - (30+45) = 65

And joint profits are computed in this case is -Joint profits = 65 X (45+30) - 20 X (45 + 30) = 3375 And individual profit of firm A is = 65 X 45 - 20 X 45 = 2025. And profits of firm A in the case of collusion = 80 X 30 - 20 X 30 = 1800.
From above one can conclude that as the firms are facing the marginal cost and profits of firm A is higher in the case of deviation from collusion. Thus, both the firms clearly have an incentive to break the collusive agreement as profits are higher in this case.

(c) The cartels like OPEC are formed with the claim of higher profits to its cartel members and thus have incentive to cooperate. But still, cartel is not very stable. As they have the incentive to deviate from equilibrium quantities, price etc. which is decided by cartel and increase their market share, as shown in the above example increasing equilibrium quantity from 30 to 45. And increasing their individual profits as shown above.