Question

Suppose a perfectly competitive firm with total cost function given as:
TC= 400+20Q-2Q2+2
3Q
3.
A. Find profit maximizing level of output and the maximum profit if the average revenue
equals $180.
B. Calculate the shutdown level of output and price.

Answer 1

$TC=400 + 20Q - 2Q^2 + 23Q^3$

A. For a perfectly competitive firm average revenue is also equal to marginal revenue and thus price.

$P= AR = \$180$

Equilibrium condition in a perfectly competitive market;

$P = MC$

Marginal cost,

$MC = \frac{dTC}{dQ}$

$MC = 20 - 4Q + 69Q^2$

So,

$180 = 20 - 4Q + 69Q^2\\ 69Q^2 - 4Q - 160 = 0$

Solving for Q

$Q = 1.5$

B. The shutdown rule states that in the short run a firm should continue to operate if price exceeds average variable costs.

$VC = 20Q - 2Q^2 + 23Q^3\\ AVC = \frac{VC}{Q}\\ AVC = 20 - 2Q + 23Q^2$

So, shut down level will be where;

$P = AVC\\ 180 = 20 - 2Q + 23Q^2\\ 23Q^2 - 2Q - 160 = 0\\$

Solving for Q

$Q = 2.7\\ P = 180$