Chapter 9: Problem 20

How can a monopolist identify the profitmaximizing level of output if it knows its total revenue and total cost curves?

Short Answer

Expert verified

A monopolist can identify the profit-maximizing level of output by following these steps: 1. Understand the concepts of profit maximization, total revenue, and total cost. 2. Calculate marginal revenue (MR) and marginal cost (MC) using the formulas \(MR = \frac{d(TR)}{dQ}\) and \(MC = \frac{d(TC)}{dQ}\), where TR is the total revenue function, TC is the total cost function, and Q is the quantity of output. 3. Set marginal revenue equal to marginal cost: \(\frac{d(TR)}{dQ} = \frac{d(TC)}{dQ}\). 4. Solve for the profit-maximizing quantity of output, Q. This may involve solving a differential equation or finding critical points using calculus. 5. Verify and interpret the results, considering the implications for the monopolist's pricing and output behavior, as well as consumer surplus and welfare.

Step by step solution

Understand the Concepts

To understand how a monopolist can identify the profit-maximizing level of output, we need to know the concepts of profit maximization, total revenue, and total cost. Profit maximization is when a firm decides on the level of output that maximizes its profit. Total revenue is the amount earned from selling a certain quantity of output, while total cost refers to the cost of producing that quantity of output.

Calculate Marginal Revenue and Marginal Cost

The profit-maximizing level of output is found when the marginal revenue (MR) is equal to the marginal cost (MC). Marginal revenue is the change in total revenue resulting from an increase in output, while marginal cost is the change in total cost resulting from an increase in output. The formulas for MR and MC are: \(MR = \frac{d(TR)}{dQ}\) and \(MC = \frac{d(TC)}{dQ}\) where TR is the total revenue function, TC is the total cost function and Q is the quantity of output.

Set Marginal Revenue Equal to Marginal Cost

To identify the profit-maximizing level of output, we need to set the marginal revenue equal to the marginal cost and solve for the quantity of output, Q: \(MR = MC\) \(\frac{d(TR)}{dQ} = \frac{d(TC)}{dQ}\)

Solve for the Profit-maximizing Quantity of Output

Solving the equation from step 3 for Q will give us the profit-maximizing quantity of output for the monopolist. This may involve solving a differential equation or finding the critical points of the equation using calculus.

Verify and Interpret Results

After finding the profit-maximizing level of output, verify that MR and MC are indeed equal at that level of output, and that the profit is maximized. To interpret the results, consider what this implies for the monopolist's pricing and output behavior, as well as the implications for consumer surplus and welfare. By following these steps, a monopolist can identify the profit-maximizing level of output when given its total revenue and total cost curves.