Question

7. Suppose that the firm operates in a perfectly competitive market. The market price of his product is $4. The firm estimates its cost of production with the following cost function:

TC=50+20q-5q2+0.33q3

a. What level of output should the firm produce to maximize its profit?

b. Determine the level of profit at equilibrium.

c. What minimum price is required by the firm to stay in the market?

Answer 1

a. What level of output should the firm produce to maximize its profit?

The firm will maximize profits at the point where:

$$P = MC$$

From the total cost curve, the marginal cost is:

$$MC = \dfrac{\Delta TC}{\Delta q} = 20 - 10q + 0.99q^2$$

The market price is $p = \$4$ . Thus:

$$20 - 10q + 0.99q^2 = 4$$

$$0.99q^2 - 10q - 24 = 0$$

Solving for $q$ , we get:

$$q^* \approx 12.104, \quad q^* \approx -2.003$$

Ignoring the negative value of $q$ , the profit maximizing level of output is $q^* \approx 12.104$ .

b. Determine the level of profit at equilibrium.

The revenue collected by the firm is:

$$R = \$4\times 12.104 = \$48.416$$

The total cost is:

$$TC=50+20(12.104)-5(12.104)^2+0.33(12.104)^3 = \$144.741$$

$$\rm Profit = \$48.416 - \$144.741 = -\$96.325$$

c. What minimum price is required by the firm to stay in the market?

This happens at the point where the marginal cost is equal to the average cost.

$$MC = 20 - 10q + 0.99q^2$$

$$AC = \dfrac{TC}{q}=\dfrac{50+20q-5q^2+0.33q^3}{q}$$

$$AC =\dfrac{50}{q}+20-5q+0.33q^2$$

Equating the marginal cost to the average cost:

$$20 - 10q + 0.99q^2 = \dfrac{50}{q}+20-5q+0.33q^2$$

$$-5q + 0.66q^2 = \dfrac{50}{q}$$

$$-5q^2 + 0.66q^3 = 50$$

$$0.66q^3 -5q^2- 50 = 0$$

Solving for $q$ in the cubic equation above, we get:

$$0.66q^3 -5q^2- 50 = 0$$

$$q^* = 8.60$$

$$P = MC = 20 - 10q + 0.99q^2$$

$$P = 20 - 10(8.60) + 0.99(8.60)^2$$

$$\boxed{P = \$7.22}$$