Worksheet 7: Unilateral pricing

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

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 $π = d (q) * q - c (q) = (40 - 2 q) q - 5 q - \frac{1}{2} q^{2}$ . Differentiating and setting equal to 0 yields $- 2 q + 40 - 2 q - 5 - q = - 5 q + 35 = 0.$ 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.