Question

Consider Cournot model of quantity competition between two firms, firm 1 and firm 2. Suppose the inverse demand for the firms product is given by ?=40(?+2)−(?+1)(?1+?2)p=40(A+2)−(B+1)(q1+q2), where ??qi denotes the quantity of firm ?i, ?=1,2i=1,2, ?A is 6 and ?B is 9. Each firm's average cost is equal to ?+2c+2, where ?c is 4.
a) Derive and accurately plot each firm's best response function.
b) Find the (Nash) equilibrium quantities, price, profits and consumer surplus.
c) Suppose next that each firm has a fixed cost equal to ?=20(?+1)f=20(A+1) where A is 6. Would the allocation you found in part (b) be an equilibrium? Explain.

Answer 1

Inverse demand function: p=40(A+2)-(B+1)(q1+q2)

Since A=6,B=9 Hence, => p=320-10(q1+q2)

AC= c+2= 6 since c=4

a) Revenue for F1= p*q1= 320q1-10q1(q1+q2)= 320q1-10(q1)^2-10q1q2

Marginal revenue= 320-20q1-10q2

MC=6

MR=MC, hence 320-20q1-10q2=6

Best response function of F1 given q2 => q1= 15.7-0.5q2

Revenue for F2= p*q2= 320q2-10q2(q1+q2)= 320q2-10q1q2-10(q2)^2

Marginal revenue= 320-10q1-20q2

MC=6

MR=MC, hence 320-20q2-10q1=6

Best response function of F2 given q1 => q2= 15.7-0.5q1

b) Solving for Nash equilibrium,

Equate Best response functions of both the firms

That means 1.5q= 15.7

Hence,q1=q2=10.47

p=320-10*2*10.47= 110.67 (identical for Firm 1 and Firm 2)

Profit= (p-MC)*q=(110.67-6)*10.47= 1095.51 (identical for Firm 1 and Firm 2)

Consumer surplus= (31.4-10.47)*10.47= 219.10 (identical for Firm 1 and Firm 2)

c) Now if the fixed cost is ?=20(?+1) where A is 6

f= 140

Total cost= 6c-140

MC=6

The equilibrium quanity will still remain the same since MR is equated to MC and not total cost.