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Chapter 8: Problem 6

A firm produces a product in a competitive industry and has a total cost function \(C=50+4 q+2 q^{2}\) and a marginal cost function \(\mathrm{MC}=4+4 q\). At the given market price of \(\$ 20,\) the firm is producing 5 units of output. Is the firm maximizing its profit? What quantity of output should the firm produce in the long run?

Short Answer

Expert verified
No, the firm is not maximizing its profit by producing 5 units of output as the marginal cost (24) is more than the market price (20). The firm should produce 4 units of output in the long run to maximize its profit.

Step by step solution

01

Calculate the Marginal Cost for current production

The given marginal cost function is MC=4+4q. Substitute q=5 (current production level) into the MC function: MC = 4 + 4*5 = 24.
02

Check Profit Maximization

Compare the marginal cost with the market price. Here, MC (24) > Price (20), which means that the firm is not maximizing its profit. The firm is producing more than the profit maximizing quantity as the cost of producing an additional unit is more than what it can be sold for.
03

Determine the Optimal Output Level

Set the marginal cost equal to the market price to find the profit maximizing output level: 4 + 4q = 20. Solve this equation to find the value of q. From this, we get q = 4. So, the firm should produce 4 units in the long run to maximize its profit.

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Most popular questions from this chapter

A number of stores offer film developing as a service to their customers. Suppose that each store offering this service has a cost function \(C(q)=50+0.5 q+0.08 \eta^{2}\) and a marginal cost \(M C=0.5+0.16 \eta\) a. If the going rate for developing a roll of film is \(\$ 8.50\), is the industry in long-run equilibrium? If not, find the price associated with long- run equilibrium. b. Suppose now that a new technology is developed which will reduce the cost of film developing by 25 percent. Assuming that the industry is in long run equilibrium, how much would any one store be willing to pay to purchase this new technology?

A competitive firm has the following short-run cost function: \(C(q)=q^{3}-8 q^{2}+30 q+5\) a. Find \(\mathrm{MC}, \mathrm{AC}\), and AVC and sketch them on a graph. b. At what range of prices will the firm supply zero output? c. Identify the firm's supply curve on your graph. d. At what price would the firm supply exactly 6 units of output?

Suppose you are given the following information about a particular industry: \\[ \begin{array}{ll} Q^{D}=6500-100 P & \text { Market demand } \\ Q^{s}=1200 P & \text { Market supply } \end{array} \\] \(C(q)=722+\frac{q^{2}}{200} \quad\) Firm total cost function \\[ M C(q)=\frac{2 q}{200} \quad \text { Firm marginal cost function } \\] Assume that all firms are identical and that the market is characterized by perfect competition. a. Find the equilibrium price, the equilibrium quantity, the output supplied by the firm, and the profit of each firm. b. Would you expect to see entry into or exit from the industry in the long run? Explain. What effect will entry or exit have on market equilibrium? c. What is the lowest price at which each firm would sell its output in the long run? Is profit positive, negative, or zero at this price? Explain. What is the lowest price at which each firm would sell its output in the short run? Is profit positive, negative, or zero at this price? Explain.

A sales tax of \(\$ 1\) per unit of output is placed on a particular firm whose product sells for \(\$ 5\) in a competitive industry with many firms. a. How will this tax affect the cost curves for the firm? b. What will happen to the firm's price, output, and profit? c. Will there be entry or exit in the industry?

Suppose the same firm's cost function is \(C(q)=4 q^{2}+16\) a. Find variable cost, fixed cost, average cost, average variable cost, and average fixed cost. (Hint: Marginal cost is given by \(\mathrm{MC}=8 q\).) b. Show the average cost, marginal cost, and average variable cost curves on a graph. c. Find the output that minimizes average cost. d. At what range of prices will the firm produce a positive output? e. At what range of prices will the firm earn a negative profit? f. At what range of prices will the firm earn a positive profit?
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