Question

Suppose a firm engaged in the illegal copying of computer CDs has a daily short-run total cost function given by \\[ S T C=q^{2}+25 \\] a. If illegal computer CDs sell for \(\$ 20\), how many will the firm copy each day? What will its profits be? b. What is the firm's short-run producer surplus at \(P=\$ 20 ?\) c. Develop a general expression for this firm's producer surplus as a function of the price of illegal CDs.

Short answer

Short Answer: To maximize its profit, the firm engaged in illegal copying of computer CDs should produce 10 CDs per day, at which point its profit will be $75 per day. At the given price of $20 per CD, the short-run producer surplus for this firm is $100. The general expression for the firm's producer surplus as a function of the price of illegal CDs is: PS(P) = (P^2) / 4.

Step-by-step solution

Find the Marginal Cost (MC) to maximize profit.

To find the quantity of CDs that will maximize the firm's profit, we first find the marginal cost. The marginal cost is the derivative of the short-run total cost function (STC) with respect to the quantity of CDs, q: \\[ MC = \frac{d(STC)}{dq} = \frac{d(q^2 + 25)}{dq} = 2q \\]

Set marginal cost (MC) equal to the selling price and solve for the profit-maximizing quantity (q).

The firm will maximize its profit by producing a quantity of CDs such that its marginal cost is equal to the selling price of \(20. So, set MC = \)20 and solve for q: \\[ 2q = 20 \\] \\[ q = 10 \\] The firm should copy 10 CDs per day to maximize its profit.

Calculate the total revenue and total cost to find the profit.

Now that we know the profit-maximizing quantity of CDs, let's calculate the total revenue, total cost, and profit at this quantity. Total Revenue = (Price per CD) × (Number of CDs) Total Revenue = \(20 × 10 = \)200 Total Cost at 10 CDs: \\[ STC = q^2 + 25 = 10^2 + 25 = 100 + 25 = 125 \\] Profit = Total Revenue - Total Cost Profit = \(200 - \)125 = $75 The firm's profit will be $75 per day.

Find the short-run producer surplus at \(P = \$20\).

Producer surplus is the difference between the actual revenue received by a producer and the minimum amount they would be willing to accept to produce the good (represented by the area above the supply curve up to the given price). In this case, the supply curve is essentially the marginal cost curve (MC): Short-run Producer Surplus = Total Revenue - Total Variable Cost First, let's find the Total Variable Cost (TVC) by subtracting the fixed cost (FC) from the Short-run Total Cost (STC) function TVC = STC - FC Since the fixed cost of production is $25, the TVC will be: \\[ TVC = q^2 + 25 - 25 = q^2 \\] At a quantity of 10 CDs: \\[ TVC = 10^2 = 100 \\] Short-run Producer Surplus = Total Revenue - Total Variable Cost Short-run Producer Surplus = \(200 - \)100 = $100 The firm's short-run producer surplus at a price of \(20 is \)100.

Develop a general expression for the firm's producer surplus as a function of the price of illegal CDs.

Using the previous steps, we can now develop a general expression for the producer surplus. Let P be the price of illegal CDs. 1. Set MC = P \\[ 2q = P \\] 2. Solve for q \\[ q = \frac{P}{2} \\] 3. Calculate Total Revenue (TR) \\[ TR = Pq = P \cdot \frac{P}{2} = \frac{P^2}{2} \\] 4. Calculate Total Variable Cost (TVC) at a quantity of \(\frac{P}{2}\): \\[ TVC = \left(\frac{P}{2}\right)^2 = \frac{P^2}{4} \\] 5. Finally, calculate the Producer Surplus as a function of price: \\[ PS(P) = TR - TVC = \frac{P^2}{2} - \frac{P^2}{4} = \frac{P^2}{4} \\] The general expression for the firm's producer surplus as a function of the price of illegal CDs is: \\[ PS(P) = \frac{P^2}{4} \\]

Key concepts

Marginal Cost

Understanding the concept of marginal cost is crucial for economics students, as it represents the cost of producing one additional unit of a good. In the exercise, the firm calculates marginal cost to decide how many CDs to produce in order to maximize profits.

The marginal cost (MC) is derived from the short-run total cost function (\(STC = q^2 + 25\)), resulting in \(MC = 2q\). This formula shows that the marginal cost, in this case, increases linearly with the number of CDs produced. It's essential for a firm to set the price of its product equal to its marginal cost to achieve profit maximization, which leads us right into our next key topic.

Profit Maximization

The ultimate goal of most firms is profit maximization, a concept that appears central to our exercise. This term refers to the process by which a firm determines the price and output level that returns the greatest profit.

The condition for profit maximization requires that the marginal cost equals the price of the product, \(MC = P\). By using the marginal cost calculated previously, the firm can decide to produce 10 CDs a day, as solving \(2q = 20\) yields \(q = 10\). This volume of production maximizes the firm's profit, which can be calculated by subtracting the total cost from the total revenue generated by the sale of CDs.

Total Revenue

The concept of total revenue is directly tied to a firm's profitability. Total revenue (TR) is the income that a firm receives from selling its products or services, calculated as the unit price times the number of units sold.

In our scenario, the total revenue is determined by multiplying the selling price of the illegal CDs, which is \(\(20\), by the profit-maximizing quantity of 10 CDs. This results in a total revenue of \(TR = 20 \times 10 = \)200\). Being able to calculate total revenue allows for the next step of determining profitability: subtracting the total costs from the total revenue.

Short-run Total Cost

Closely related to the previous concepts, short-run total cost plays a vital role in the context of a firm's production process. It comprises all costs that a firm incurs in the production of goods or services in the short run, including both fixed and variable costs.

In our exercise, the short-run total cost (STC) function is given by \(STC = q^2 + 25\), where \(q\) is the number of CDs produced. The \(25\) in the STC function represents the fixed cost, which remains constant regardless of the quantity produced. Subtracting this fixed cost from the STC gives us the total variable cost (TVC). Knowing the STC allows the firm to deduce whether it's beneficial to produce additional goods or not.