Deriving maximum profit with derivatives

Chapter 9: Limited Decision-Making in Perfect Competition

Two methods enable you to maximize total profit. First, you can maximize total profit by using total revenue and total cost. Alternatively, you can use marginal revenue and marginal cost to maximize profit.

Calculating economic profit

After you determine the profit-maximizing quantity of output, you want to determine how much profit you make. Using total revenue and total cost, your total profit is easily calculated by subtracting total revenue from total cost. However, determining the profit-maximizing quantity of output by using marginal revenue and marginal cost doesn’t directly provide you with a measure of total profit.

In this situation, total profit is determined by first calculating your profit per unit of output and then multiplying that amount by the profit-maximizing quantity of output. Profit per unit equals price minus average total cost, or

Total profit is determined by multiplying profit per unit by the quantity of output produced

In Figure 9-2, profit per unit is represented by the double-headed arrow between price and average total cost at the output level q0.

The total revenue and total cost equations from previous examples in this chapter are

and

Given these equations, you determine the profit-maximizing quantity is 700 units of the good given the market-determined price of $80. In order to determine the profit per unit and total profit, you take the following steps:

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1. Determine the average total cost equation.

Average total cost equals total cost divided by the quantity of output.

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

2. Substitute the profit-maximizing quantity of 700 for q to determine average total cost.

3. Calculate profit per unit.

or profit per unit equals $17.143.

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

or total profit equals $12,000.

Making the best of a bad situation by minimizing losses

To this point, I’ve used situations where the firm is earning positive profit. Of course it isn’t always the case that the firm’s profit is positive. Nevertheless, firms may continue producing in the short run in order to minimize losses. It’s important to remember that firms who shut down in the short run still have production costs — total fixed cost can’t be changed. Thus, if a firm loses less money than total fixed cost by producing in the short run, the firm should continue production in order to minimize losses.

Figure 9-3 illustrates a situation where a firm minimizes losses by producing in the short run. The profit-maximizing quantity of output is still determined by equating marginal revenue and marginal cost. The firm produces the profit-maximizing quantity of output q0 at that point. However, because price is less than average total cost at q0, the firm loses money. Its loss per unit equals price minus average total cost. This loss per unit is represented by the double-headed arrow in Figure 9-3.

If instead of producing q0 the firm shuts down, it loses total fixed cost. As I indicate in Chapter 8, total fixed cost equals average fixed cost multiplied by the quantity of output, and average fixed cost equals average total cost minus average variable cost. Thus, in Figure 9-3, the difference between average total cost, ATC0, and average variable cost, AVC0, represents the fixed cost per unit at q0. This difference between average total cost and average variable cost is clearly larger than the difference between price and average total cost. Thus, shutting down and losing your fixed costs is a greater loss than occurs if you produce q0.