Consider a duopoly in which both firms produce exactly the same product. Suppose the firms' total cost functions are given as follows:

[Math Processing Error]TC1(Q1)=10Q1

[Math Processing Error]TC2(Q2)=10Q2

The market demand is given by [Math Processing Error]P=20−12Q.

A. In a price competition game, what price should be optimally set by each firm? How much will be sold by each firm?

B. In a quantity competition game, what quantity should be set by each firm? What will be the market price?

C. If both firms decide to collude, how much should each firm optimally produce? What will be the market price?

D. In which case do firms enjoy a larger profit margin? Explain.

A.

If the game is price competition, the demand curve will be:

[Math Processing Error]P=20−0.5QAsMC1=MC2=10p1=p2q1=q2=Q2P=MC

As the firms are selling in the price competitive market, and have identical marginal costs, the firms will sell at the price equal to their marginal cost.

[Math Processing Error]P=MC20−0.5Q=1010=0.5QQ=20P=20−0.5(20)=10

Hence the optimal price set by each firm is 10 and the quantity sold by each is by market is 20.

The quantity sold by each is 10.

B.

In the quantity competition game, the output will be ([Math Processing Error]Q=q1+q2).

Hence for the firm 1,

[Math Processing Error]P=20−0.5Qp1=20−0.5q1−0.5q2p2=20−0.5q1−0.5q2TR1=20q1−0.5q12−0.5q1q2TC=10q1PF1=TR1−TC1=20q1−0.5q12−0.5q1q2−10q1dPF1dq1=20−q1−0.5q2−10=010−q1−0.5q2=0q1=10−0.5q2

Similarly, for the firm 2, the value of q2 is:

[Math Processing Error]q2=10−0.5q1

Solving the two equations,

[Math Processing Error]q1=10−0.5q2q1=10−0.5(10−0.5q1)=10−5+0.25q10.75q1=5q1=6.67q2=10−0.5(6.67)=6.665Q=6.67+6.67=13.34P=20−0.5(13.34)=13.33

The optimal quantity set by each firm is 6.67 and the market price is 13.33. Total market quantity is 13.34 units.

C.

If both the firms decide to collude, the firms act like monopoly. The optimal price and quantity is when MR=MC.

[Math Processing Error]P=20−0.5QMR=20−QMC=10MR=MC20−Q=10Q=10P=20−0.5(10)=15q1=q2=102=5

The optimal quantity set by each firm is 5, total market quantity is 10 and the market price is 15.

D.

In the case of collusion, the firm will enjoy larger profit margin. This is because, the firms are behaving like market monopoly in the collusion. The monopolist profit is always largest than the duopoly profit. The price in collusion is larger than the duopoly quantity or price competition, considering marginal cost (MC). Hence the profit in monopoly collusion is larger.