Question

Suppose that the market for gourmet deli sandwiches is perfectly competitive and that the supply of workers in this industry is upward-sloping, so that wages increase as industry output increases. Delis in this market face the following total cost:

TC = q³ - 20 q² + 120 q + W

where,

Q = number of sandwiches

W = daily wages paid to workers

The wage, which depends on total industry output, equals: W = 0.3 Nq

where,

N = number of firms.

Assume that the market demand is:

Q^D = 900 - 10 P

1. What is the long-run equilibrium output for each firm?   

2. How much does the long-run equilibrium price change as the number of firms increases?

3. What is the long-run equilibrium number of firms?  

4. What is the total industry output?   

5. What is the long-run equilibrium price?

Answer 1

1. Long run equilibrium occurs at the minimum of ATC.
ATC = TC/q = (q3 - 20 q2 + 120 q + 0.3Nq)/q = q2 -20q + 120 + 0.3N
d(ATC)/dq = 2q - 20 = 0
So, 2q = 20
So, q = 20/2 = 10
Thus, q = 10

2. ATC = q2 -20q + 120 + 0.3N
So, as N increases by 1 unit, ATC will increase by 0.3 units. So, P will increase by 0.3 units.

3. P = minimum of ATC
Ong run equilibrium number of firms, N = Q/q = (900 - 10 P)/q
P = q2 -20q + 120 + 0.3N = 102 -20(10) + 120 + 0.3N = 100 - 200 + 120 + 03N = 20 + 0.3N
So, (900 - 10 P)/q = (900 - 10(20+0.3N))/10 = N
So, 900 - 200 - 3N = 10N
So, 10N + 3N = 13N = 700
So, N = 700/13
So, N = 53.85

4. P =  20 + 0.3N = 20 + 0.3(53.85) = 20 + 16.155 = 36.155
Q = 900 - 10 P = 900 - 10(36.155) = 900 - 361.55
So, Q = 538.45

5. P = 36.155