9781118041581

the marginal profit at Q 3.0 is $12 thousand per lot.^4 In turn, we can imme-

diately determine the firm’s profit-maximizing level of output. Using Equation

2.7, we simply set M0 and solve for Q:

Therefore, we find that Q 132/40 3.3 lots. At 3.3 lots per week, the firm’s

marginal profit is zero. This is the output that maximizes profit.

Figure 2.7 graphs the firm’s total profit in part (a) as well as the firm’s mar-

ginal profit in part (b). Note that at the optimal output, Q 3.3 lots, total

profit reaches a peak in Figure 2.7a, whereas marginal profit is exactly zero

(i.e., the marginal profit graph just cuts the horizontal axis) in Figure 2.7b.

A complete solution to the firm’s decision problem requires two additional

steps. We know the optimal quantity is Q 3.3 lots. What price is required for

the firm to sell this quantity? The answer is found by substituting Q 3.3 into

Equation 2.2: P  170 (20)(3.3) $104 thousand. What is the firm’s final

profit from its optimal output and price decision? At this point, we can sepa-

rately compute total revenue and total cost. Alternatively, we can compute

profit directly from Equation 2.5 (with Q 3.3). Either way, we arrive at 

$117,800. This completes the algebraic solution.

M 132 40Q0.

42 Chapter 2 Optimal Decisions Using Marginal Analysis

(^4) The difference between Equation 2.7 and Table 2.1 is that the table lists marginal profit over small,
discrete intervals of output, whereas the equation lists marginal profit atparticular output levels.
When we use a very small interval, the discrete marginal profit between two output levels is a very
close approximation to marginal profit at either output. For example, with an interval of.01, the dis-
crete marginal profit at Q 3 is the slope of the line connecting the points Q 2.99 and Q 3.00.
This line is nearly identical to the tangent line (representing marginal profit) atQ 3. Thus,
Using a .01 interval: M$12,200
Via calculus (equation 2.7) M$12,000
Once again consider the inverse demand curve P  340 .8Q and the cost function
C  120 100Q. Derive the formula for Mas it depends on Q. Set M0 to find
the firm’s optimal output.

MARGINAL REVENUE AND MARGINAL COST

The concept of marginal profit yields two key dividends. The general concept

instructs the manager that optimal decisions are found by making small changes

in decisions, observing the resulting effect on profit, and always moving in the

direction of greater profit. A second virtue of the approach is that it provides an

efficient tool for calculating the firm’s optimal decision. The discussion in this

section underscores a third virtue: Marginal analysis is a powerful way to iden-

tify the factors that determine profits and, more important, profit changes. We

will look once again at the two components of profit, revenue and cost, and

highlight the key features of marginal revenueand marginal cost.

CHECK

STATION 4

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