W9: Unilateral Pricing

Consider the firm’s inverse demand curve, \(d=40-2q\), and cost curve, \(c(q)=5q+\frac{1}{2}q^{2}\).

Question 1

What is the firm’s profit maximizing price and quantity?

To solve this, we just need to maximize the firm’s profit with respect to quantity, \(q\). Then we substitute the optimal quantity into the demand function to find the optimal \(p\).

The profit function in this case is \(\pi=d(q)*q - c(q) = (40-2q)q - 5q - \frac{1}{2}q^{2}\). Differentiating and setting equal to 0 yields \[-2q + 40 - 2q - 5 - q = -5q + 35 = 0.\] This holds for \(q=7\).

Plugging \(q=7\) into the demand function then yields \(p=40-14=26\). So the firm optimally sets a price of \(p=26\) and sells a quantity of \(q=7\).

Question 2

What is the firm’s markup over marginal cost?

We can find the marginal cost at \(q=7\) by plugging this quantity into the marginal cost function, \(c'(q) = 5 + q = 12\). Since the firm’s price is \(p=26\), we know the markup over marginal cost is 14. We usually think of this in percentage terms, where we calculate the markup relative to marginal cost, \(\frac{p-c}{c} = \frac{14}{12}\). So the markup is more than 100% of the cost.