Question

Suppose that the potential customers for hair braiding in a city believe that all hair braiding is identical and that the market is perfectly competitive. Hair braiding requires special skills so the supply of workers in this industry is upward-sloping, and the wages earned by hair braiders increase as the industry output increases. (Total 12 Marks)

Firms in this market face the following total cost:

TC = Q3 ?8Q2 + 20Q + W

where Q is the number of hair braidings and W is the daily wage paid to workers. The wage, which depends on total industry output, equals W = 0.1NQ, where N is the number of ?rms. Market demand is:

QD = 500?20P

(a) How does average total cost for the ?rm change as industry output increases and what does this imply for industry’s long-run supply curve?

(b) Calculate the long-run equilibrium output for each ?rm.

(c) Explain how the long-run equilibrium price changes as the number of ?rms increases?

(d) Calculate the long-run equilibrium number of ?rms and total industry output.

(e) Calculate the long-run equilibrium price.

Answer 1

TC = Q³ - 8Q² + 20Q + 0.1NQ

QD = 500 - 2P

2P = 500 - QD

P = 250 - 0.5QD

(a) Average total cost (ATC) = TC/Q

ATC = Q² - 8Q + 20 + 0.1N

As output increases, ATC initially decreases, reaches a minimum and the starts increasing. So long run industry supply curve is U-shaped.

(b) In long run equilibrium, Price = ATC = MC

MC = dTC/dQ = 3Q² - 16Q + 20 + 0.1N

Equating MC and ATC,

3Q² - 16Q + 20 + 0.1N = Q² - 8Q + 20 + 0.1N

2Q² - 8Q = 0

2Q x (Q - 8) = 0

Assuming Q is non-zero, Q = 8

(b) When Q = 8, Price = AC = (8 x 8) - (8 x 8) + 20 + 0.1N = 20 + 0.1N

As nmber of firms (N) increases, market price increases.

(d) When Q = 8, Industry supply = QS = Q x N = 8N

In equilibrium, QD = QS

500 - 20 x (20 + 0.1N) = 8N

500 - 400 - 2N = 8N

10N = 100

N = 10

Industry output = 8 x N = 8 x 10 = 80

(e) Price = 20 + (0.1 x 10) = 20 + 1 = 21