Consider a competitive industry with a large number of firms, all of which have

Question

Consider a competitive industry with a large number of firms, all of which have identical cost functions c(y)=y2+1 for y>0 and c(0)=0. Suppose that initially the demand curve for this industry is by D(p)=52-p. (The output of a firm does not have to be an integer number, but the number of firms does have to be an integer.)

1. What is the supply curve of an individual firm? If there are n firms in the industry, what will be the industry supply curve?

2. What is the smallest price at which the product can be sold?

3. What will be the LR equilibrium number of firms in the industry, equilibrium price, equilibrium output of each firm, equilibrium output of the industry?

Explanation / Answer

c(y) = y2 + 1

D(p) = Q = 52 - p

(1) Individual firm's supply curve is it marginal cost (MC) function.

MC = dc(y)/dy = 2y

Firm supply function: p = 2y

If there are n firms, then market supply (Q) = n x y

y = Q/n

p = 2 x (Q/n) = 2Q/n

Q = nP/2 [Industry supply function]

(2) Smallest price is the shutdown price of a firm, which is equal to the minimum average varable cost (AVC).

Total variable cost (TVC) = y2

AVC = TVC/y = y

AVC is minimum when y = 0. Therefore,

Smallest price = minimum AVC = 0

(3) In long-run equilibrium, Price = MC = Average total cost (ATC)

ATC = c(y)/y = y + (1/y)

Equating ATC and MC,

y + (1/y) = 2y

y = 1/y

y2 = 1

y = 1 (Firm output)

p = MC = 2y = 2 x 1 = 2 (Market price)

From market demand function: Q = 52 - 2 = 50 (Market output)

Number of firms = Market output / Firm output = 50/1 = 50

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