# Question #18c40

Jul 29, 2015

Profit maximization occurs when $M R = M C$. Supposing an imperfect competition market, you can use derivatives to find that each company will produce 2,500 units to maximize its profit.

#### Explanation:

If this is not a perfect competition market, then $P = A R \ne M R$ . Presuming that x is the quantities the company produces and sells, then:
$P = \frac{50}{\sqrt{q}} = M R$
Since the total revenue is $T R = P \cdot q$, we have:
$T R = 50 \sqrt{q}$
The marginal revenue is the derivative of the total revenue function with respect to with respect to q:
$M R = \frac{\partial T R}{\partial q} = \frac{25}{\sqrt{q}}$
The marginal cost is the derivative of the total cost function with respect to q:
$M C = \frac{\partial T C}{\partial q} = 0.5$
Equaling Marginal revenue and marginal cost:
$\frac{25}{\sqrt{q}} = 0.5$
Isolate q and solve:
$\sqrt{q} = 50$
$q = 2 , 500$
The profit function for this problem would have this graph:
graph{50/sqrt(x)*x-(0.5x+500) [, 5000, -500, 1000]}