Question

3: For each (identical) firm in a perfectly competitive market the long-run cost function is C(q) = q1.5 + 16q0.5 with long run marginal cost being LMC = 1.5q0.5 + 8q-0.5, where q = firm’s output. Market demand curve: Q = 1600 – 2p, where Q = total output of all firms, and p = price of output. (a) For the firm find the long run average cost curve , as well as the price of output and the amount of output produced by each firm in a long run equilibrium. (b) Find the number of firms in the long run equilibrium, and what happens in the long run if the market demand curve shifts to Q = 160 – 20p?

Answer 1

a)

Long run average cost=LRAC=C(q)/q=[q^1.5+16q^0.5]/q=q^0.5+16q^(-0.5)

Set LRAC=MC to determine the value of q where LRAC is minimum

q^0.5+16q^(-0.5)=1.5q^0.5+8q^(-0.5)

0.5q^0.5=8q^(-0.5)

q=16 (output produced by a firm in long run)

Minimum LRAC=LRAC at output of 16 units=q^0.5+16q^(-0.5)=16^0.5+16*16^(-0.5)=8

Long run price=Minimum LRAC=$8

b)

Let us determine the quantity demanded at a price of $8

Q=1600-2p=1600-2*8=1584

Number of firms in long run=Q/q=1584/16=99

If Q=160-20p

Let us determine the quantity demanded at a price of $8

Q=160-20p=160-20*8=0

There will not be any quantity demanded at this price. So,

No firm will be there in long run at this demand function.