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Chapter 8: Problem 5

Suppose that a competitive firm's marginal cost of producing output \(q\) is given by \(\mathrm{MC}(q)=3+2 q\). Assume that the market price of the firm's product is \(\$ 9\) a. What level of output will the firm produce? b. What is the firm's producer surplus? c. Suppose that the average variable cost of the firm is given by \(\mathrm{AVC}(q)=3+q\). Suppose that the firm's fixed costs are known to be \(\$ 3\). Will the firm be earning a positive, negative, or zero profit in the short run?

Short Answer

Expert verified
The company will produce 3 units of the product. The company's producer surplus would be $9. In the short term, the company will make a positive profit.

Step by step solution

01

Find the optimal output level

Given the firm's MC function is \(MC(q) = 3 + 2q\), and the market price of the product is $9, to find the optimal output level, simply set \(MC(q)\) equal to the price and solve for \(q\): \[3 + 2q = 9 \rightarrow q = \frac{9 - 3}{2} = 3\] So, the optimal level of output that the firm will produce is 3 units.
02

Calculate the producer surplus

The producer surplus is the area between the market price and the MC curve from 0 to the quantity produced. To get this, integrate the market price minus MC from 0 to 3 where 3 is the quantity produced: \[ \text{Producer Surplus} = \int_0^3 (9 - (3 + 2q))dq = 3*(9 - 3 - 3) = \$9 \]. So, the producer surplus of the firm is $9.
03

Determine the firm's short-run profit situation

The average variable cost is given by \(AVC(q) = 3 + q\), and the fixed costs are $3. The total cost would then be \(TC = AVC \cdot Q + FC\) or \(TC = (3 + q)q + 3\). In this case, q=3, so \(TC = (3+3)3 + 3 = \$21\). The total revenue is the market price times the quantity, or \(TR = PQ = 9*3 = \$27\). Comparing the total cost with the total revenue, we can determine that the firm is making a positive profit since \$21 < \$27. Hence, the firm is earning a positive profit in the short run.

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Most popular questions from this chapter

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?

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.

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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?
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