Question

Assume a firm is a small business and act as a price-taker in the market, the market price of the firm’s product is 20. The firm’s cost function is:

C(q) = 0.5q²+5q+100.

1. What is the firm’s optimal output level?

2. What’s the firm’s highest profit?

Answer 1

1. Optimal output level

Firm’s cost function is:

$C(q) = 0.5q^2+5q+100$

Under perfect competition, a firm is a price taker of its good since none of the firms can individually influence the price of the good.

To calculate the optimal level of output at which its;

Marginal Cost (MC) = Market Price (P)

$MC=\frac{d(TC)}{dQ}$

$MC=\frac{d(0.5q^2+5q+100)}{dQ}=q+5$

$MC=P$

$q+5=20$

$q=15$

2. Firm's profit

$Profit = Revenue - TC$

$Revenue =PQ=20\times15=300$

$TC=0.5\times(15)^2 +(5\times15)+100=287.5$

$Profit = 300-287.5=12.5$