Question

1. All (identical) firms in a competitive industry have the following long-run total cost curve: C(q) = q3 – 10q2 + 369 where q is the output of the firm. a. Compute the long run equilibrium price. What does the long-run supply curve look like? b. Suppose the market demand is given by Q=111 - p. Determine the long-run equilibrium number of firms in the industry.

Answer 1

Cost function of a firm is given,

$\large C = q^3 - 10q^2 + 36q$

In the long run equilibrium the firms earn zero economic profit thus the price is equal to minimum ATC. We are required to minimize the ATC.

$\large ATC =\frac{C}{q} = \frac{q^3 - 10q^2 + 36q}{q}$

ATC = 92 – 109 + 36

Differentiating ATC wrt q we get

$\large ATC' = \frac{d(q^2 - 10q + 36)}{dq}$

$\large \implies ATC' = 2q - 10$

The first order condition of minimization is that the first derivative must be equal to zero.

$\large \implies ATC' = 2q - 10 = 0$

$\large \implies 2q = 10$

$\large \implies q = 5 \ units$

Again differentiating ATC wrt q

$\large ATC'' = \frac{d(2q - 10)}{dq}$

$\large ATC'' = 2$

The SOC of minimization. Is that the second derivative must be a positive number. As we can see both condition is satisfied. Hence, ATC will be minimum at q = 5 units.

Minimum ATC = $ 11 per unit

ATC = 92 – 109 + 36

$\large \implies ATC = 5^2 - 10\times 5 + 36$

$\large \implies ATC = 25 - 50+ 36$

$\large \implies ATC = \$ \ 11$

A. Long run equilibrium price = $ 11 per unit.

The long run supply curve is constant and parallel to the horizontal axis.

B. Given, Q = 111 - P

Q = 111 - 11 = 100

Market demand = 100 units.

Quantity supplied by each firm, q = 5 (calculated above)

$\large Number \ of \ firms = \frac{100}{5} = 20$

Number of firms in long run = 20.

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