Question

Consider a competitive industry where all firms have identical cost functions C(y) = y^2 +1 if y > 0 and C(0) = 0. Suppose that the demand curve for this industry is given by D(p) = 52 − p.

1. (a) Find the individual supply curve of each firm.

2. (b) Suppose there are n firms in the industry. Find the industry supply curve.

3. (c) Find the price and quantity corresponding to a short-run market equilibrium assuming there are n firms in the market.

4. (d) Use the zero-profit condition to find the price in the long-run equilibrium.

5. (e) Find the number of firms and equilibrium quantity in the long-run.

Answer 1

(a)

Individual Supply curve is determined where P = MC
C(y) = Y^2+1

MC = derivative of C(y) wrt y

MC = 2y

So, P = 2y

y = P / 2 this is the individual firm supply curve

(b)

Industry supply curve with n firms = ny = n*(P/2)

(c)

at equilibrium, Demand = Supply

52 - P = n*(P/2)

P = 104/(n+2)

Q = 52 - P

Q = 52 - 104/(n+2)

Q = 52n / (n+2)

where n = # of firms

(d)

In long run, P = MC = AC

AC = c(y) / y

AC = y + (1/y)

So, AC = MC

y + (1/y) = 2y

implies, y = +-1

So, y = 1 (long run equilibrium quantity of each firm)

P = MC = 2y = 2 (Long run equilibrium price)

(e)

Market Demand Q = 52 - P

Q = 52 - 2 = 50 ; market output

Individual firm output y = 1 (calculated in pard d)

So, # of firms = Market output / individual firm output

= 50/1 = 50

So, # of firms = 50