Question

Two firms Alphabet and Beta are in an industry. These firms will operate as a duopoly, the cost functions of the two firms are: C1(q1) = 60+30q1 and C2(q2)= 30+ 36q2. The firms have a linear market demand curve P(Q) = 150-15Q where Q=q1+q2 is the total output level.

Find and draw the firms reaction, best response curves and find and show the Nash equilibrium of the Cournot game when the firms compete simultaneously. What is each firm's equilibrium output and profit in the Cournot model? Compare the equilibrium outputs and profits with the equilibrium outputs and profits of Stackelberg's duopoly.

Answer 1

Two firms in industry,

First firm alphabet's cost function :-

C1(q1) = 60 + 30q1

Second firm's cost function :-

C2(q2) = 30 + 36q2

Market demand curve;P(Q) = 150 - 15Q where Q = q1 + q2

So, P(q1, q2) = 150 - 15q1 - 15q2.

Total revenue of 1st firm, TR1 = (150 -15q1 - 15q2)q1 = 150q1 - 15q1^2 - 15q1*q2

TR2 = (150 - 15q1 - 15q2)q2 = 150q2 - 15q1*q2 - 15q2^2

MR2 = delTR2/delq2 = 150 - 15q1 - 30q2

Marginal revenue of 1st firm(MR1) = delTR1/delq1 = 150 - 30q1 - 15q2.

Marginal cost of first firm(MC1) = 30.

Marginal cost of 2nd firm(MC2) = 36.

We know the condition of Maximum profit,

MR1 = MC1 ; 150 - 15q2 -30q1 = 30.

30q1 + 15q2 = 150 - 30 = 120

30q1 = 120 - 15q2 ;

q1 = (120 - 15q2)/30 = 4 - q2/2

Best response 1(BR1)(q2) = 0 if q2>=8 or,

= 40 - q2/2 if q2<8.

For firm 2, MR2 = MC2 ; 150 - 15q1 - 30q2 = 36.

15q1 + 30q2 = 124 ; 30q2 = 124 - 15q1

So, q2 = (124 - 15q1)/30.

BR2(q1) = 0 if q1>=8.26 or,

= (124 - 15q1)/30 if q1 <8.26

Red lind is for first firm and green for 2nd firm.

In this image point A is Nash equilibrium point.

So, duopoly profit at, (q1, q2) = (2.58, 2.84)

Profit of first firm = TR1 - TC1 = (150 - 15(2.58+2.84))*2.58 - 60 - 30*2.58

=(150 - 81.3)*2.58 -60 - 30*2.58 = 39.846.

Profit of2nd firm, = (150-81.3)*2.84 - 30 - 36*2.84 = 62.868.

For Stackelberg's duopoly,

Firm 1 is the first mover.

Firm 2's reaction function :- q2 = (124 - 15q1)/30

The demand curve, putting q2,

P = 150 - 15q1 - 15(124 - 15q1)/30 =150 - 15q1 - 62 +15q1/2 = 88 - 15q1/2

So MR1 = 88 - 15q1 = MC1 = 30

15q1 = 88 - 30 = 58 So, q1 = 58/15 = 1.93

Putting it in q2 = (124 - 15*58/15)/30 = (124 - 58)/30 = 66/30 = 2.2

Profit in stackelberg,

Profir1 =( 150 - 15(1.92+2.2))*1.92 - 60 - 30*1.92 = 51.744

Profit of firm 2 = (150 - 15(1.92 +2.2))*2.2 - 30 - 36*2.2 = 84.84.

(q1, q2) in cournot = (2.58, 2.84)

While (q1, q2) in stackelberg =(1.92, 2.2)

And (profit1, profit2) in cournot = (39.84, 62.86)

And in stackelberg's (profit1, profit2) = (51.74, 84.84)