9781118041581

How were we able to graph the profit curve in Figure 2.5 so precisely? The

graph was constructed from the following basic profit equation, often called the

profit function:

[2.5]

In the second line, we have substituted the right-hand sides of the revenue and

cost equations to express profit in terms of Q, the firm’s decision variable. In

the third line, we have collected terms. The important point about the profit

equation is that it provides a numerical prediction of profit for any given quan-

tity Q. To check that the equation is correct, simply substitute in a value for Q,

say, two lots, and calculate profit:  100 (132)(2) (20)(4) $84 thou-

sand, the same result as in Figure 2.5.

 100 132Q20Q^2.

(170Q20Q^2 )(10038Q)

RC

38 Chapter 2 Optimal Decisions Using Marginal Analysis

CHECK

STATION 2

Suppose the inverse demand function is P  340 .8Q and the cost function is C  120 

100Q. Write down the profit function.

MARGINAL ANALYSIS

Consider the problem of finding the output level that will maximize the firm’s

profit. One approach is to use the preceding profit formula and solve the prob-

lem by enumeration,that is, by calculating the profits associated with a range of

outputs and identifying the one with the greatest profit. Enumeration is a viable

approach if there are only a few output levels to test. However, when the num-

ber of options is large, enumeration (and the numerous calculations it

requires) is not practical. Instead, we will use the method of marginal analysis

to find the “optimal” output level.

Marginal analysislooks at the change in profit that results from making

a small change in a decision variable. To illustrate, suppose the firm first con-

siders producing 3 lots, forecasting its resulting profit to be $116,000 as in

Figure 2.5. Could it do better than this? To answer this question, the firm

considers increasing production slightly, to, say, 3.1 lots. (One-tenth of a lot

qualifies as a “small” change. The exact size of the change does not matter as

long as it is small.) By substituting Q 3.1 into Equation 2.5, we see that the

new profit is $117,000. Thus, profit has increased by $1,000. The rateat which

profit has changed is a $1,000 increase per .1 lot increase, or 1,000/.1 

$10,000 per lot.

Here is a useful definition: Marginal profitis the change in profit resulting

from a small increase in any managerial decision variable. Thus, we say that

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