Leading your rivals with the Stackelberg model

Chapter 10: Monopoly: Decision-Making Without Rivals

As is the case for any firm, a monopolist determines profit per unit by subtracting average total cost from price. In Figure 10-3, profit per unit is represented by the double-headed arrow labeled π/q. Total profit is determined by multiplying profit per unit by the number of units sold, q0.

Maximizing profit with calculus

Figure 10-3 indicates that profit is maximized at the quantity of output where marginal revenue equals marginal cost. Marginal revenue represents the change in total revenue associated with an additional unit of output, and marginal cost is the change in total cost for an additional unit of output. Therefore, both marginal revenue and marginal cost represent derivatives of the total revenue and total cost functions, respectively. You can use calculus to determine marginal revenue and marginal cost; setting them equal to one another maximizes total profit.

Earlier in this chapter, in the section “Deriving maximum profit with derivatives,” I noted that the monopolist’s demand curve

generated the total revenue equation.

Also assume your total cost equation is

Given these equations, the profit-maximizing quantity of output is determined through the following steps:

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1. Determine marginal revenue by taking the derivative of total revenue with respect to quantity.

2. Determine marginal cost by taking the derivative of total cost with respect to quantity.

3. Set marginal revenue equal to marginal cost and solve for q.

172 Part III: Market Structures and the Decision-Making Environment

4. Substituting 2,000 for q in the demand equation enables you to determine price.

Thus, the profit-maximizing quantity is 2,000 units and the price is $40 per unit.

The profit-maximizing quantity and price are the same whether you maximize the difference between total revenue and total cost or set marginal revenue equal to marginal cost.

Calculating economic profit and the profit-per-unit fallacy

Economic profit per unit equals price minus average total cost, or

In Figure 10-3, economic profit per unit is illustrated by the double-headed arrow labeled π/q.

Total profit equals profit per unit multiplied by the number of units sold, or

Using the same information in the previous example, the monopolist’s demand curve is

And the monopolist’s total cost equation is

Given this information, the profit-maximizing quantity is 2,000 units at a price of $40 per unit.

Chapter 10: Monopoly: Decision-Making Without Rivals

In order to determine the monopolist’s economic profit per unit and total profit, you take the following steps:

1. Determine the average total cost equation by dividing the total cost equation by the quantity of output q.

2. Substitute q equals 2,000 in order to determine average total cost at the profit-maximizing quantity of output.

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Thus, the average total cost is $31 at the profit-maximizing quantity of 2,000 units.

3. Calculate profit per unit.

Profit per unit equals $9.

4. Determine total profit by multiplying profit per unit by the profitmaximizing quantity of output.

Total profit equals $18,000.

Don’t confuse maximizing total profit with maximizing profit per unit. You’re willing to accept less profit per unit if you sell a lot more units. For example, if your profit per unit was $11 and you sold 1,265 units of output, your total profit equals $13,915.

On the other hand, if your profit per unit is only $9 but you’re now able to sell 2,000 units because you charge a lower price, your total profit equals $18,000.

Remember, your goal is always to maximize total profit.