Part III: Cournot Competitors 1. Suppose two firms are Cournot competitors with
ID: 1205582 • Letter: P
Question
Part III: Cournot Competitors 1. Suppose two firms are Cournot competitors with T C(q)=4q (where q is a given firm’s output) and demand curve p = 100 ! 4Q, where Q is total industry output (summing all firm’s quantities, as usual).
(a) Compute the Cournot equilibrium price, profits for each firm, and the overall consumer surplus.
(b) Suppose a third firm enters (with the same cost function). Recalculate the equilibrium. Are total industry profits (summing across all firms) higher in (a) or (b)? What about consumer surplus in (a) vs. (b)? What about CS+profits in (a) vs. (b)?
I just really need the total industry profits and consumer surpluses in (a) and (b). Thank you :)
Explanation / Answer
a. q1 = a-c/3b = 100-4/3*4 = 8
q2 = 8
P = a+2c/3 = 100+8/3 = 108/3 = 36
Firm 1 Profit = TR - TC = 36*8 - 4*8 = 256
By symetry, firm 2 profit = 256
Total Industry profit = profit of firm1 + profit of firm2
= 256 + 256 = 512
Consumer Surplus = 1/2(vertical intercept - price)*quantity
= 1/2(100 - 36)*8
= 256
b. Three firms
Industry profit with n firms = n/(n+1)^2*(a-c)^2/b
So, Industry profit for 3 firms = 3/(3+1)^2*(100-4)^2/4
= (3/16)*(9216/4)
= 432
Q for n firms = n/n+1*a-c/b
So, Q for 3 firms = 3/4*(100-4)/4
Q = 18
q1 = q2 = q3 = 6
P = 1/n+1*a + n/n+1*c
= 1/3+1*100 + 3/3+1*4
= 1/4*100 + 3/4*4
= 25 + 3
= 28
Consumer Surplus = 1/2(100-28)*18
= 648