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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

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.

Suppose that a competitive firm has a total cost func\(\operatorname{tion} C(q)=450+15 q+2 q^{2}\) and a marginal cost function \(M C(q)=15+4 q .\) If the market price is \(P=\$ 115\) per unit, find the level of output produced by the firm. Find the level of profit and the level of producer surplus.

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?

A sales tax of 10 percent is placed on half the firms (the polluters) in a competitive industry. The revenue is paid to the remaining firms (the nonpolluters) as a 10 percent subsidy on the value of output sold. a. Assuming that all firms have identical constant long-run average costs before the sales tax-subsidy policy, what do you expect to happen (in both the short run and the long run), to the price of the product, the output of firms, and industry output? (Hint: How does price relate to industry input?) b. Can such a policy always be achieved with a balanced budget in which tax revenues are equal to subsidy payments? Why or why not? Explain.

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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