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

Answer: At a price of $20 for illegal CDs, the firm's daily profit is $75 and the short-run producer surplus is $100.

Step-by-step solution

Find the marginal cost function

Differentiate the short-run total cost function with respect to q to find the marginal cost function. \\[ \frac{d(STC)}{dq} = \frac{d(q^2 + 25)}{dq} = 2q \\]

Set the marginal cost equal to the given price and solve for q

At the given price of $20, we can set the marginal cost equal to the price and solve for q: \\[ 20 = 2q \\ q = 10 \\]

Calculate the daily profit

The total revenue function is given by: \(TR = Pq\). Use the price, P = $20, and the quantity, q = 10, to calculate the total revenue: \\[ TR = 20(10) = 200 \\] Now, we can find the total cost at q = 10: \\[ STC = q^2 + 25 = (10)^2 + 25 = 125 \\] The profit can be calculated by subtracting the total cost from the total revenue: \\[ \text{Profit} = TR - STC = 200 - 125 = \$75 \\]

Calculate the short-run producer surplus at P = $20

Variable cost can be found by subtracting the fixed cost from the short-run total cost: \\[ VC = STC - FC = q^2 + 25 - 25 = q^2 \\] At P = $20 and q = 10: \\[ VC = (10)^2 = 100 \\] Producer surplus can be found by subtracting the variable cost from the total revenue: \\[ \text{Producer Surplus} = TR - VC = 200 - 100 = \$100 \\]

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

At a general price P, we can first find the new quantity, q': \\[ P = 2q' \\ q' = \frac{P}{2} \\] Now, we can substitute P for the price in the total revenue function, TR': \\[ TR' = Pq' = P \cdot \frac{P}{2} = \frac{P^2}{2} \\] The variable cost at q' is given by: \\[ VC' = (q')^2 = (\frac{P}{2})^2 = \frac{P^2}{4} \\] The producer surplus at the general price P can be calculated by subtracting VC' from TR': \\[ \text{Producer Surplus as a function of P} = \frac{P^2}{2} - \frac{P^2}{4} = \frac{P^2}{4} \\] This is the general expression for the firm's producer surplus as a function of the price of illegal CDs.