Consider the following game. Firm 1, the leader, selects an output q1, after whi

Question

Consider the following game. Firm 1, the leader, selects an output q1, after which firm 2, the follower, observes the choice of q1 and then selects its own output q2. The resulting price is one satisfying the industry demand curve P = 200 q1 q2. Both firms have zero fixed costs and a constant marginal cost of 10. (A). Derive the equation for the follower firm’s best response function. Draw this equation on a graph with q2 on the vertical axis and q1 on the horizontal axis. Indicate the vertical intercept, horizontal intercept, and slope of the best response function. (B). Determine the equilibrium output of each firm in the leader-follower game. What are each firm’s profits in the equilibrium? (C). Now let the two firms choose their outputs simultaneously. Compute the Cournot equilibrium outputs and industry price. Who loses and who gains when the firms play a Cournot game instead of the Stackelberg one?

Explanation / Answer

P=200-q1-q2

where q1and q2 are the outputs of two firms

MR1=200-q2-2q1

thus to attin profit firm 1 will equate MR=MC

200-q2-2q1=10

190=q2+2q1

q1=(190-q2)*1/2

this is the response function of firm 1

q2=(200-10-q1)*1/2

=(190-q1)*1/2

this is firm 2 response function

b) TR=P*Q

TR1=(200-q1-q2)q1

=200q1-q12-q2q1

MR1=200-q1-95-1/2q1

=105-q1

MR=MC

105- q1=10

q1=125

q2=95-125/2

=95-62.5

=32.5

price=200-32.5-125

=42.5

profits

firm 1=125*42.5-10*125

= 5312.5-1250=4062.5

firm 2=32.5*42.5-10*32.5

=1381.25=1056.25

answer c)MR1=95-1/2q2

MR2=95-1/2q1

q1=95-1/2(95-1/2q1)

=95-47.5+1/4q1

q1=38

q2=95-1/238

=76

thus here firm 2 gains and in stackelberg firm 1 gains

industry price=200-38-76

=86

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