Question

A perfectly competitive firm facing the following TVC function: TVC= 150Q - 20Q² + Q³, where q= quantity.

Below what price should the firm shutdown it's operations

Answer 1

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The firm should shut down operations if price falls below $\$50.$

Explanation

The shut-down point of a company is a point whereby the marginal cost ($MC$ ) of a firm is equal to the average variable cost ($AVC$ ).

Marginal cost of a firm is a derivative of the total cost with respect to the quantity.

Therefore;

$MC=\frac{dTVC}{dQ}\\ MC =\frac{150Q - 20Q^2 + Q^3}{Q}\\ MC =150 -40Q +3Q^2$

According to profit maximization theory, $P=MC$ , thus

$P= 150 -40Q +3Q^2$

Average variable equation is given by;

$AVC = \frac{TVC}{Q}\\ AVC =\frac{150Q – 20Q2 +Q3)} { Q}\\ AVC = 150 -20Q +Q2$

Equate the two equations, that is $MC$ and $AVC$

$MC=AVC$

 $150 -40Q +3Q^2 =150 -20Q +Q^2\\ 2Q^2-20Q \ or\\ 2Q (Q-10) =0$

Thus, $Q = 0$ and $Q = 10$

substitute $Q=10$ in the marginal cost equation

$MC=150 -40Q +3Q^2\\ MC=150-40(10) +3(100)\\ MC=150-400+300\\ MC=50$

Therefore, $P=\$50.$

Also substitute $Q=0$ in the marginal equation

$MC= 150 -40Q +3Q^2\\ MC=150-40(0) +0\\ MC=150$

Therefore, $P=\$150$

A firm will shut down operations when $P < AVC$

Thus, in this case the firm will shut down when the price falls below $\$50.$