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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. The firm should produce 4 units of output in the long run to maximize profit.

Step by step solution

01

Determine current profit situation

Compute the Marginal Revenue (MR) at the given conditions. In a competitive market, MR is equal to the price of the output commodity, which is $20. Compare this with the Marginal Cost (MC) at the current quantity of 5 units, using the given MC function, \(\mathrm{MC}=4+4 q\). Substituting \(q = 5\) into the equation gives \(\mathrm{MC} = 24\). The comparison shows that the MR is less than MC at the current level of production, indicating that the firm is not maximizing its profit.
02

Determine optimal quantity of output

The firm maximizes its profit when MC equals MR. Set the MC function equal to the price since in a competitive market, price equals MR. Solve the equation \(4+4q = 20\) to get the optimal quantity \(q = 4\). So, in long run, the firm should produce 4 units of output.

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Most popular questions from this chapter

Suppose you are the manager of a watchmaking firm operating in a competitive market. Your cost of production is given by \(C=200+2 q^{2}\), where \(q\) is the level of output and \(C\) is total cost. (The marginal cost of production is \(4 q ;\) the fixed cost is \(\$ 200 .\) ) a. If the price of watches is \(\$ 100,\) how many watches should you produce to maximize profit? b. What will the profit level be? c. At what minimum price will the firm produce a positive output?

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 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 \text { 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. d. 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 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 \(\mathrm{AVC}\) 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

Suppose that a competitive firm has a total cost function \(C(q)=450+15 q+2 q^{2}\) 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.
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