Two identical firms compete simultaneously as a Cournot duopoly. The market dema

Question

Two identical firms compete simultaneously as a Cournot duopoly. The market demand is
P = 200 - 2~ where Q stands for the combined output of the two firms, Q = q, + q2. The total
cost function for firm 1 is C1 = 60 + 16ql. The total cost function for firm 2 is C2 = 50 + 24q2.

a) Derive the best-response functions for these firms expressing what ql and q2 should
be in this Cournot oligopoly.

b) Find the optimal quantity for each firm, the market price, and the profit each firm
earns in this Cournot oligopoly.

c) Now suppose that with the same market demand and cost conditions as above, Firm 1
may bring its product to market earlier and choose its output level earlier than Firm 2 by
spending an additional investment of 200. Explain carefully if Firm 1 will benefit from being first
to market.

d) Should firm 2 try to undercut Firm 1 on price and try to gain additional sales by
reducing its price below Firm l's level? Explain.

Explanation / Answer

P = 200 - 2Q = 200 - 2(q1 + q2) = 200 - 2q1 - 2q2

C1 = 60 + 16q1 , MC1 = 16

C2 = 50 + 24q2 , MC2 = 24

MR1 = 200 - 4q1 - 2q2

MR2 = 200 - 2q1 - 4q2

Equating marginal revenues with marginal cost, we get

200 - 4q1 - 2q2 = 16

=> q1 = (184 - 2q2) / 4 = (92 - q2) / 2

Similarly,

200 - 2q1 - 4q2 = 24

q2 = (176 - 2q1) / 4 = (88 - q1) / 2

b. Now solving the two reaction function we get

q1 = [92 - {(88 - q1) / 2}] / 2 = [184 - (88 - q1)] / 4

4q1 = 96 + q1

q1 = 96/3 = 32

q2 = (88 - 32) / 2

= 28

Thus P = 200 - 2(32+28) = 200 - 120 = 80

c. when firm 1 enters first then it will include the value of q2 from reaction function in its TR function

TR1 = 200q1 - 2(q1)^2 - 2q1q2

Substituting the value of q2 = (88 - q1) / 2 in above function, we get

TR1 = 200q1 - 2(q1)^2 - 2q1(88 - q1) / 2

= 200q1 - 2(q1)^2 - 88q1 + (q1)^2

= 112q1 - (q1)^2

thus MR1 = 112 - 2q1

equating it with MC1, we get

112 - 2q1 = 16

q1 = 96/2 = 48

q2 = (88 - 48) / 2 = 20

Thus firm 1 will be able to produce more if it enters the market first

P = 200 - 2 (48+20) = 68

d. Total variable cost for firm 2 = 24*20 = 480

TR2 = 200q2 - 2q1q2 - 2(q2)^2 (where q1 = 48 and q2 = 20)

Solving it we get

TR2 = 200(20) - 2(48)(20) - 2(20)^2

= 4000 - 1920 - 800 = 1280

Thus the revenues of firm 2 exceeds its variable cost thus it can reduce its prices till it can cover its variable cost. But under oligopoly, if one firm reduces its price other will follow the suit. Thus firm 2 has the liberty to reduce price but the result to increase sales by reducing its prices will depend on the decision of firm 1 to reduce its prices too.

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