Question

Problem 2. Assume the market is perfectly competitive. There are 100 firms currently operating in the market. The cost function for each firm is given by c(q) = 25 + 2q + q 2 , and the demand function is Q = 1024−2p.

• Derive the current market price and total output in equilibrium. (Hint: Q = 100q)

• Given the above answer, is the market in the long-run equilibrium right now? If not, what are the equilibrium levels of price and total output in the long run, and how many firms will operate in the market in the long run?

Answer 1

(a) Individual firm's supply price is its Marginal cost (MC) function.

C(q) = 25 + 2q + q²

MC = dC(q) / dq = 2 + 2q

Therefore, firm supply function: p = 2 + 2q

Market output, Q = 100q

q = Q / 100

p = 2 + (2 x Q / 100) = 2 + (Q / 50)

Q / 50 = p - 2

Q = 50p - 100 [Market supply function]

Equating market demand and market supply,

1,024 - 2p = 50p - 100

52p = 1,124

p = 21.62

Q = 1,024 - (2 x 21.62) = 1,024 - 43.24 = 980.76

(b) In long run, p = MC = Average cost (AC)

When Q = 981, Firm output (q) = 980.76 / 100 = 9.81

MC = 2 + (2 x 9.81) = 2 + 19.62 = 21.62

AC = C(q) / q = (25 / q) + 2 + q = (25 / 9.81) + 2 + 9.81 = 2.55 + 11.81 = 14.36

Since MC does not equal AC, this is not long run equilibrium.

Equating MC and AC,

2 + 2q = (25 / q) + 2 + q

q = 25 / q

q² = 25

q = 5 (Long run firm output)

Long run price (p) = MC = 2 + (2 x 5) = 2 + 10 = 12

Long run market output (Q) = 1,024 - (2 x 12) = 1,024 - 24 = 1,000

Number of firms in long run = Q / q = 1,000 / 5 = 200