A perfectly competitive painted necktie industry has a large number of potential

Question

A perfectly competitive painted necktie industry has a large number of potential entrants. Each firm has an identical cost structure such that long-run average cost is minimized at an output of 20 units (qi=20). The minimum average cost is $10 per unit. Total market demand is given by Q=1,500-50P.

A) What is the industry's long-run supply schedule

B) What is the long-run equilibrium price (P*)? The total industry output (Q*)? The output of each firm (q*i)? The number of firms? The profits of each firm?

C) The short-run total cost curve associated with each firm's long-run equilibrium output is given by: STC=0.5q^2-10q+200 where SMC=q-10 calculate the firm's short-run average and marginal cost curves. At what necktie output level does short-run average cost reach a minimum?

D) Calculate the short-run supply curve for each firm and the industry supply curve.

E) Suppose now painted neckties become more fashionable and the market deman function shifts to Q=2,000-50P. Using this demand curve, answer part B for the very short run when firms cannot change their output.

F) In the short-run use the industry short-run supply curve to recaclulate the answers to part B.

G) What is the new long-run equilibrium for this industry?

Explanation / Answer

(a) In perfect competition, long run supply is at the point where price equals minimum average cost.

Market demand is: Q = 1,500 - 50P

50P = 1,500 - Q

P = 30 - 0.02Q

So, long run supply function is:

30 - 0.02Q = 10

(B) Solving the long run supply function, we get:

0.02Q = 30 - 10 = 20

Q* = 1,000

P* = 30 - 0.02Q = 30 - (0.02 x 1,000) = 30 - 20 = 10

q* = 20 (given)

Number of firms = Q* / q* = 1,000 / 20 = 50

Profit of each firm = q* x (P - AC) = q* x 0 = 0, since in long run, P = AC for each firm.

(C)

STC = 0.5q2 - 10q + 200

SMC = dSTC / dq = q - 10

SAC = STC / q = 0.5q - 10 + (200 / q)

SAC is minimum when dSAC / dq = 0

0.5 - (200 / q2) = 0

(200 / q2) = 0.5

q2 = 200 / 0.5 = 400

q = 20

(D) Short run supply curve for a perfectly competitive firm is its SMC curve. So, short run supply function of a firm:

P = q - 10

Industry supply is the sum of individual market supply.

Q = 50 x q, so

q = Q / 50

Industry supply function is:

P = (Q / 50) - 10

50P = Q - 500

Note: First 4 sub-parts are answered.

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