Question

Problem III. Suppose that, in a market of a certain good, there are firms that are engaged in a Cournot competition. The inverse demand function is given by P(Q) = 120 − 6Q, where Q is the total supply of the good. All firms have the same cost function C(qi) = 30qi + 50.

Q7. What is the Cournot equilibrium price of the good when there are N firms in the market?

(a) (30N + 200)/(N + 1) 2

(b) (50N + 120)/(N + 1)

(c) (120N + 50)/N

(d) (30N + 120)/(N + 1)

(e) (120N + 30)/(N + 1)

Q8. What is the profit of each firm at the Cournot equilibrium when there are N firms in the market?

(a) 30[45/(N + 1)2 − 1]

(b) 50[27/(N + 1)2 − 1]

(c) 50[9/(N + 1)2 − 1]

(d) 30[50/(N + 1)2 − 1]

(e) 50[45/(N + 1)2 − 1]

Q9. When there is free entry in this market, what is the number of firms that will compete in this market?

(a) 7

(b) 5

(c) 6

(d) 4

(e) 8

Answer 1

Derivation of Cournot market with n firms is given below.

In that the price charged is P = (a + nm) / (1 + n) where demand is P = a - bQ and marginal cost is m

Then we have P = (120 + 30N)/(1 + N)

Also the quantity by one firm is q = (a - m)/b(1 + N) = (120 - 30)/(6*(1 + N) or 15/(1 + N)

Profit = revenue - cost = (120 + 30N)*15/(1 + N)^2 - 30*15/(1 + N) - 50

= 30*15(4 + N)/(1+N)^2 - 450/(1+N) - 50

= 50*((9(4 + N)/(1 + N)^2 - 9/(1+N) - 1)

= 50*(27/(1+N)^2 - 1)

Profit is 0 when there is free entry so we have

50*(27/(1+N)^2 - 1) = 0

27/(1+N)^2 = 1

(1 + N)^2 = 27

N = 6.

7) Option E is correct

8) Option B is correct

9) Option C is correct