Question

Supply in a Competitive Market

Suppose that the potential customers for hair braiding in a city consider hair braiding to be 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. Firms in this market face the following total cost:

TC = Q³ – 8 Q² + 20Q + W

where, Q = number of hair braidings,

W = daily wage paid to workers, which depends on total industry output, equals W = 0.1NQ, where N = total # of firms.

Market demand is:

Q^D = 500 – 20P.

1. Find the long-run equilibrium output for each firm. (3 marks)

2. Find the long-run equilibrium number of firms (N), total industry output (NQ), and price (P).
   (6 marks)

3. Suppose demand increases to Q^D = 1,000 – 10P. Find the new long-run competitive equilibrium.
   (6 marks)

Answer 1

TC = Q3 – 8 Q2 + 20Q + W = Q3 – 8 Q2 + 20Q + 0.1NQ

a. In long run, P = minimum of ATC for a perfectly competitive market
ATC = (Q3 – 8 Q2 + 20Q + 0.1NQ)/Q = Q3/Q – 8 Q2/Q + 20Q/Q + 0.1NQ/Q = Q2 – 8Q + 20 + 0.1N
So, d(ATC)/dQ = 2Q - 8 = 0
So, 2Q = 8
So, Q = 8/2
So, Q = 4
The long-run equilibrium output for each firm is 4 units

b. P = minimum of ATC = Q2 – 8Q + 20 + 0.1N = 42 – 8(4) + 20 + 0.1N = 16 - 32 + 20 + 0.1N = 4 + 0.1N
So, QD = 500 – 20P = 500 – 20(4 + 0.1N) = 500 - 80 - 2N = 420 - 2N
N = QD/Q = (420 - 2N)/4
So, 4N = 420 - 2N
So, 4N + 2N = 420
So, 6N = 420
So, N = 420/6
So, N = 70

NQ = 70*4 = 280
So, NQ = 280

P = 4 + 0.1N = 4 + 0.1(70) = 4 + 7 = 11
So, P = 11

c. P = 4 + 0.1N
So, QD = 1000 – 10P = 1000 – 10(4 + 0.1N) = 1000 - 40 - N = 960 - N
N = QD/Q = (960 - N)/4
So, 4N = 960 - N
So, 4N + N = 960
So, 5N = 960
So, N = 960/5
So, N = 192

NQ = 192*4 = 768
So, NQ = 768

P = 4 + 0.1N = 4 + 0.1(192) = 4 + 19.2 = 23.2
So, P = 23.2