Question

A competitive constant-cost industry is made of identical firms producing q. The short run cost function of a representative firm is C(q)=1/2q²-10q +200. Market demand is given by: Q_d=1500-50P.

a) For what level of q is average cost minimized?

b) What is the market equilibrium price that achieves the long-run equilibrium of zero-profit? How many units is each firm producing?

c) How many units clear the market at that price? how many firms are there in the market?

d) write an equation for the long-run supply curve for this industry.

Answer 1

(a)

AC(Average Cost) = C/q = ((1/2)q²-10q +200)/q = (1/2)q- 10 + 200/q

FOC:

d(AC)/dq = 0 => 1/2 -200/q² = 0 => q = 20

Hence AC is minimized when q = 20

(b)

Note in the long run a perfect competitive firm earns 0 profit.

According to profit maximizing condition a perfect competitive firm produces that quantity at which P = MC. Also profit = 0 when TR = TC => Pq = AC*q => P = AC. Hence we have the following condition in the long run:

P = MC = AC. MC = dC/dq = q - 10 and AC = (1/2)q- 10 + 200/q

Now MC = AC => q/2 = 200/q => q = 20.

Hence P = MC = 20 - 10 = 10.

Hence Long run equilibrium price = 10

(c) When P = 10. Market demand = 1500-50*10 = 1000

Each firm is supplying q = 20 units(from part (b))

Hence let there are N firms. then total supply = 20N and total demand = 1000

=> 20 N = 1000 => N = 50

Hence Number of firms = 50.

(d)

Supply curve for a perfect competitive market is given by;

P = MC => P = q - 10 => q = P + 10 and there are 50 firms => Supply curve is given by:

q = 50(p + 10)