Question

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

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

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.

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.

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.

Key concepts

Optimal Output

In a competitive market, the optimal output level is determined by setting the marginal cost (MC) of production equal to the market price. This is because a profit-maximizing firm aims to produce units until the cost of producing an extra unit (MC) is exactly equal to what they can earn from selling it. This ensures that no profit opportunity is wasted.

For the given problem, the MC function is provided as \(MC(q) = 3 + 2q\). The firm can sell its product at a market price of \(\$9\). To find the optimal output, we set \(MC(q) = 9\) and solve for \(q\):

• \(3 + 2q = 9\)
• \(2q = 6\)
• \(q = \frac{6}{2} = 3\)

Thus, the optimal output level is 3 units. By producing exactly 3 units, the firm optimizes its production and achieves potential profit maximization.

Producer Surplus

Producer surplus is a key concept in understanding producer profitability. It represents the difference between the amount a producer is willing to accept for a good versus what is actually received, typically measured as the area above the supply curve (or MC curve in competitive markets) and below the price line for the quantity produced.

In this exercise, to calculate the producer surplus, we integrate the difference between the market price and the marginal cost curve from 0 to the optimal output level, which is 3. Our integral becomes:

• \( \text{Producer Surplus} = \int_0^3 (9 - (3 + 2q))dq \)

Solving this integral:

• \(\int_0^3 (9 - (3 + 2q))dq = 9q - \frac{3q}{2} - q^2 \bigg|_0^3\)
• Breaking it down: \(9(3) - 6(3) - \frac{3^2}{2} \)
• Which simplifies to: \(27 - 18 - 4.5 = 4.5\)

In this example, the producer surplus is further simplified to \(\$9\), representing the benefit gained from participating in the market.

Short-run Profit

Short-run profit is an essential assessment of a firm's profitability over the near term without changing its scale of operation. In this context, profit calculation involves comparing total revenue (TR) and total cost (TC), where TR is the product of market price and quantity sold, and TC includes both average variable cost (AVC) and fixed costs (FC).

For this firm, we know:

• Market Price (P) = \\(9
• Quantity Produced (Q) = 3 units
• AVC given by \(AVC(q) = 3 + q\)
• Fixed Costs (FC) = \\)3

Calculating:

• TR = P * Q = \(9 * 3 = \\(27\)
• TC = (AVC * Q) + FC = \([3 + 3]*3 + 3 = \\)21\)

Subtracting total cost from total revenue, profit is determined as \(TR - TC = 27 - 21 = \$6\). Therefore, this firm achieves a positive profit in the short run, confirming efficient operation and financial viability.