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3 Marginal Revenue, Marginal Cost, and Profit Maximization

# 3 Marginal Revenue, Marginal Cost, and Profit Maximization

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CHAPTER 8 • Profit Maximization and Competitive Supply 285

Cost,

revenue,

profit

(dollars per year)

C(q)

R(q)

A

F IGURE 8.1

PROFIT MAXIMIZATION IN THE

SHORT RUN

B

A firm chooses output q*, so that profit, the

difference AB between revenue R and cost

C, is maximized. At that output, marginal

revenue (the slope of the revenue curve) is

equal to marginal cost (the slope of the cost

curve).

0

q0

q*

q1

π (q)

Output (units per year)

For the firm illustrated in Figure 8.1, profit is negative at low levels of output because revenue is insufficient to cover fixed and variable costs. As output

increases, revenue rises more rapidly than cost, so that profit eventually becomes

positive. Profit continues to increase until output reaches the level q*. At this

point, marginal revenue and marginal cost are equal, and the vertical distance

between revenue and cost, AB, is greatest. q* is the profit-maximizing output

level. Note that at output levels above q*, cost rises more rapidly than revenue—

i.e., marginal revenue is less than marginal cost. Thus, profit declines from its

maximum when output increases above q*.

The rule that profit is maximized when marginal revenue is equal to marginal

cost holds for all firms, whether competitive or not. This important rule can also

be derived algebraically. Profit, p = R - C, is maximized at the point at which

an additional increment to output leaves profit unchanged (i.e., ⌬p/⌬q = 0):

⌬p/⌬q = ⌬R/⌬q - ⌬C/⌬q = 0

⌬R/⌬q is marginal revenue MR and ⌬C/⌬q is marginal cost MC. Thus we conclude that profit is maximized when MR - MC = 0, so that

MR(q) = MC(q)

Demand and Marginal Revenue for a Competitive Firm

Because each firm in a competitive industry sells only a small fraction of the

entire industry output, how much output the firm decides to sell will have no effect

on the market price of the product. The market price is determined by the industry

demand and supply curves. Therefore, the competitive firm is a price taker. Recall

that price taking is one of the fundamental assumptions of perfect competition.

The price-taking firm knows that its production decision will have no effect on

the price of the product. For example, when a farmer is deciding how many acres

of wheat to plant in a given year, he can take the market price of wheat—say, \$4

per bushel—as given. That price will not be affected by his acreage decision.

286 PART 2 • Producers, Consumers, and Competitive Markets

Often we will want to distinguish between market demand curves and the

demand curves faced by individual firms. In this chapter we will denote market output and demand by capital letters (Q and D) and the firm’s output and

demand by lowercase letters (q and d).

Because it is a price taker, the demand curve d facing an individual competitive

firm is given by a horizontal line. In Figure 8.2(a), the farmer’s demand curve corresponds to a price of \$4 per bushel of wheat. The horizontal axis measures the

amount of wheat that the farmer can sell, and the vertical axis measures the

price.

Compare the demand curve facing the firm (in this case, the farmer) in

Figure 8.2(a) with the market demand curve D in Figure 8.2(b). The market

demand curve shows how much wheat all consumers will buy at each possible

price. It is downward sloping because consumers buy more wheat at a lower

price. The demand curve facing the firm, however, is horizontal because the

firm’s sales will have no effect on price. Suppose the firm increased its sales

from 100 to 200 bushels of wheat. This would have almost no effect on the market because industry output is 2,000 million bushels. Price is determined by the

interaction of all firms and consumers in the market, not by the output decision

of a single firm.

By the same token, when an individual firm faces a horizontal demand curve,

it can sell an additional unit of output without lowering price. As a result, when

it sells an additional unit, the firm’s total revenue increases by an amount equal to

the price: one bushel of wheat sold for \$4 yields additional revenue of \$4. Thus,

marginal revenue is constant at \$4. At the same time, average revenue received by

In §4.1, we explain how the

demand curve relates the

quantity of a good that a

price of that good.

Price

(dollars per

bushel)

Firm

Price

(dollars per

bushel)

d

\$4

Industry

\$4

D

100

200

(a)

2,000

q

Output (bushels)

(b)

Q

Output (millions

of bushels)

F IGURE 8.2

DEMAND CURVE FACED BY A COMPETITIVE FIRM

A competitive firm supplies only a small portion of the total output of all the firms in an industry.

Therefore, the firm takes the market price of the product as given, choosing its output on the

assumption that the price will be unaffected by the output choice. In (a) the demand curve facing

the firm is perfectly elastic, even though the market demand curve in (b) is downward sloping.

CHAPTER 8 • Profit Maximization and Competitive Supply 287

the firm is also \$4 because every bushel of wheat produced will be sold at \$4.

Therefore:

The demand curve d facing an individual firm in a competitive market is

both its average revenue curve and its marginal revenue curve. Along this

demand curve, marginal revenue, average revenue, and price are all equal.

Profit Maximization by a Competitive Firm

Because the demand curve facing a competitive firm is horizontal, so that

MR = P, the general rule for profit maximization that applies to any firm can

be simplified. A perfectly competitive firm should choose its output so that

marginal cost equals price:

MC(q) = MR = P

Note that because competitive firms take price as fixed, this is a rule for setting

output, not price.

The choice of the profit-maximizing output by a competitive firm is so important that we will devote most of the rest of this chapter to analyzing it. We begin

with the short-run output decision and then move to the long run.

8.4 Choosing Output in the Short Run

How much output should a firm produce over the short run, when its plant size

is fixed? In this section we show how a firm can use information about revenue

and cost to make a profit-maximizing output decision.

Short-Run Profit Maximization by a Competitive Firm

In the short run, a firm operates with a fixed amount of capital and must choose

the levels of its variable inputs (labor and materials) to maximize profit. Figure 8.3

shows the firm’s short-run decision. The average and marginal revenue curves are

drawn as a horizontal line at a price equal to \$40. In this figure, we have drawn

the average total cost curve ATC, the average variable cost curve AVC, and the

marginal cost curve MC so that we can see the firm’s profit more easily.

Profit is maximized at point A, where output is q* ϭ 8 and the price is \$40,

because marginal revenue is equal to marginal cost at this point. To see that

q* ϭ 8 is indeed the profit-maximizing output, note that at a lower output, say

q1 ϭ 7, marginal revenue is greater than marginal cost; profit could thus be

increased by increasing output. The shaded area between q1 ϭ 7 and q* shows

the lost profit associated with producing at q1. At a higher output, say q2, marginal cost is greater than marginal revenue; thus, reducing output saves a cost

that exceeds the reduction in revenue. The shaded area between q* and q2 ϭ 9

shows the lost profit associated with producing at q2. When output is q* ϭ 8,

profit is given by the area of rectangle ABCD.

The MR and MC curves cross at an output of q0 as well as q*. At q0, however,

profit is clearly not maximized. An increase in output beyond q0 increases

profit because marginal cost is well below marginal revenue. We can thus

Marginal, average, and total

cost are discussed in Đ7.1.

288 PART 2 Producers, Consumers, and Competitive Markets

MC

Price

(dollars per

unit)

60

50

Lost profit for

q1 < q*

D

40

Lost profit for

q2 > q*

A

AR = MR = P

ATC

C

AVC

B

30

20

10

0

1

q0

2

3

4

5

6

7

8

9

q1

q*

q2

10

11

Output

F IGURE 8.3

A COMPETITIVE FIRM MAKING A POSITIVE PROFIT

In the short run, the competitive firm maximizes its profit by choosing an output q* at which its

marginal cost MC is equal to the price P (or marginal revenue MR) of its product. The profit of the

firm is measured by the rectangle ABCD. Any change in output, whether lower at q1 or higher at q2,

state the condition for profit maximization as follows: Marginal revenue equals

marginal cost at a point at which the marginal cost curve is rising. This conclusion

is very important because it applies to the output decisions of firms in markets that may or may not be perfectly competitive. We can restate it as follows:

Output Rule: If a firm is producing any output, it should produce at the level

at which marginal revenue equals marginal cost.

Figure 8.3 also shows the competitive firm’s short-run profit. The distance AB

is the difference between price and average cost at the output level q*, which is

the average profit per unit of output. Segment BC measures the total number of

units produced. Rectangle ABCD, therefore, is the firm’s profit.

A firm need not always earn a profit in the short run, as Figure 8.4 shows.

The major difference from Figure 8.3 is a higher fixed cost of production. This

higher fixed cost raises average total cost but does not change the average

variable cost and marginal cost curves. At the profit-maximizing output q*,

CHAPTER 8 • Profit Maximization and Competitive Supply 289

Price

(dollars per

unit of

output)

C

D

MC

ATC

B

P = MR

A

AVC

F

E

q*

Output

F IGURE 8.4

A COMPETITIVE FIRM INCURRING LOSSES

A competitive firm should shut down if price is below AVC. The firm may produce in the

short run if price is greater than average variable cost.

the price P is less than average cost. Line segment AB, therefore, measures the

average loss from production. Likewise, the rectangle ABCD now measures the

firm’s total loss.

When Should the Firm Shut Down?

Suppose a firm is losing money. Should it shut down and leave the industry?

The answer depends in part on the firm’s expectations about its future business conditions. If it believes that conditions will improve and the business

will be profitable in the future, it might make sense to operate at a loss in the

short run. But let’s assume for the moment that the firm expects the price of

its product to remain the same for the foreseeable future. What, then, should

it do?

Note that the firm is losing money when its price is less than average total

cost at the profit-maximizing output q*. In that case, if there is little chance that

conditions will improve, it should shut down and leave the industry. This decision is appropriate even if price is greater than average variable cost, as shown

in Figure 8.4. If the firm continues to produce, the firm minimizes its losses at

output q*, but it will still have losses rather than profits because price is less

than average total cost. Note also that in Figure 8.4, because of the presence of

fixed costs, average total cost exceeds average variable cost, and average total

cost also exceeds price, so that the firm is indeed losing money. Recall that

290 PART 2 Producers, Consumers, and Competitive Markets

Remember from Đ7.1 that

a fixed cost is an ongoing

cost that does not change

with the level of output but

is eliminated if the firm shuts

down.

fixed costs do not change with the level of output, but they can be eliminated

if the firm shuts down. (Examples of fixed costs include the salaries of plant

managers and security personnel, and the electricity to keep the lights and

heat running.)

Will shutting down always be the sensible strategy? Not necessarily. The

firm might operate at a loss in the short run because it expects to become profitable again in the future, when the price of its product increases or the cost of

production falls. Operating at a loss might be painful, but it will keep open

the prospect of better times in the future. Moreover, by staying in business,

the firm retains the flexibility to change the amount of capital that it uses

and thereby reduce its average total cost. This alternative seems particularly

appealing if the price of the product is greater than the average variable cost

of production, since operating at q* will allow the firm to cover a portion of its

fixed costs.

Our example of a pizzeria in Chapter 7 (Example 7.2) provides a useful illustration. Recall that pizzerias have high fixed costs (the rent that must be paid,

the pizza ovens, and so on) and low variable costs (the ingredients and perhaps

some employee wages). Suppose the price that the pizzeria is charging its customers is below the average total cost of production.Then the pizzeria is losing

money by continuing to sell pizzas and it should shut down if it expects business conditions to remain unchanged in the future. But, should the owner sell

the store and go out of business? Not necessarily; that decision depends on the

owner’s expectation as to how the pizza business will fare in the future. Perhaps

adding jalapeno peppers, raising the price, and advertising the new spicy pizzas

will do the trick.

E XA MPLE 8.2 THE SHORT-RUN OUTPUT DECISION OF AN ALUMINUM

SMELTING PLANT

How should the manager of an

aluminum smelting plant determine the plant’s profit-maximizing

output? Recall from Example 7.3

(page 240) that the smelting plant’s

short-run marginal cost of production depends on whether it is running two or three shifts per day. As

shown in Figure 8.5, marginal cost

is \$1140 per ton for output levels

up to 600 tons per day and \$1300

per ton for output levels between

600 and 900 tons per day.

Suppose that the price of aluminum is initially P1 ϭ \$1250 per

ton. In that case, the profit-maximizing output is 600 tons; the firm

can make a profit above its variable cost of \$110

per ton by employing workers for two shifts a

day. Running a third shift would

involve overtime, and the price

of the aluminum is insufficient

profitable. Suppose, however,

that the price of aluminum were

to increase to P 2 ϭ \$1360 per

ton. This price is greater than

the \$1300 marginal cost of the

third shift, making it profitable

to increase output to 900 tons

per day.

Finally, suppose the price drops

to only \$1100 per ton. In this case,

the firm should stop producing,

but it should probably stay in business. By taking this step, it could

resume producing in the future should the price

increase.

CHAPTER 8 • Profit Maximization and Competitive Supply 291

Cost

(dollars per ton)

MC

1400

P2

1300

P1

1200

1140

1100

0

300

600

900

Output (tons per day)

F IGURE 8.5

THE SHORT-RUN OUTPUT OF AN ALUMINUM SMELTING PLANT

In the short run, the plant should produce 600 tons per day if price is above

\$1140 per ton but less than \$1300 per ton. If price is greater than \$1300 per

ton, it should run an overtime shift and produce 900 tons per day. If price drops

below \$1140 per ton, the firm should stop producing, but it should probably

stay in business because the price may rise in the future.

EX AMPLE 8. 3 SOME COST CONSIDERATIONS FOR MANAGERS

The application of the rule that marginal revenue

should equal marginal cost depends on a manager’s ability to estimate marginal cost.3 To obtain

useful measures of cost, managers should keep

three guidelines in mind.

First, except under limited circumstances, average variable cost should not be used as a substitute

for marginal cost. When marginal and average variable cost are nearly constant, there is little difference

between them. However, if both marginal and average cost are increasing sharply, the use of average

variable cost can be misleading in deciding how much

to produce. Suppose for example, that a company

has the following cost information:

Current output

100 units per day, 80 of which are produced during the regular shift and

20 of which are produced during overtime

Materials cost

\$8 per unit for all output

Labor cost

\$30 per unit for the regular shift; \$50 per unit for the overtime shift

3

This example draws on the discussion of costs and managerial decision making in Thomas Nagle and

Reed Holden, The Strategy and Tactics of Pricing, 5th ed. (Upper Saddle River, NJ: Prentice Hall, 2010), ch. 2.

292 PART 2 • Producers, Consumers, and Competitive Markets

Let’s calculate average variable cost and marginal cost

for the first 80 units of output and then see how both

cost measures change when we include the additional

20 units produced with overtime labor. For the first 80

units, average variable cost is simply the labor cost

(\$2400 ϭ \$30 per unit ϫ 80 units) plus the materials

cost (\$640 ϭ \$8 per unit ϫ 80 units) divided by the

80 units—(\$2400 ϩ \$640)/80 ϭ \$38 per unit. Because

average variable cost is the same for each unit of output, marginal cost is also equal to \$38 per unit.

When output increases to 100 units per day, both

average variable cost and marginal cost change.

The variable cost has now increased; it includes

the additional materials cost of \$160 (20 units ϫ

\$8 per unit) and the additional labor cost of \$1000

(20 units ϫ \$50 per unit). Average variable cost is

therefore the total labor cost plus the materials cost

(\$2400 ϩ \$1000 ϩ \$640 ϩ \$160) divided by the

100 units of output, or \$42 per unit.

What about marginal cost? While the materials cost

per unit has remained unchanged at \$8 per unit, the

marginal cost of labor has now increased to \$50 per

unit, so that the marginal cost of each unit of overtime

output is \$58 per day. Because marginal cost is higher

than average variable cost, a manager who relies on

average variable cost will produce too much.

Second, a single item on a firm’s accounting ledger may have two components, only one of which

involves marginal cost. Suppose that a manager

is trying to cut back production. She reduces the

number of hours that some employees work and

lays off others. But the salary of an employee who

is laid off may not be an accurate measure of the

marginal cost of production when cuts are made.

Union contracts, for example, often require the firm

to pay laid-off employees part of their salaries. In

this case, the marginal cost of increasing production is not the same as the savings in marginal cost

when production is decreased. The savings is the

labor cost after the required layoff salary has been

subtracted.

Third, all opportunity costs should be included in

determining marginal cost. Suppose a department

store wants to sell children’s furniture. Instead of

building a new selling area, the manager decides

to use part of the third floor, which had been used

for appliances, for the furniture. The marginal cost

of this space is the \$90 per square foot per day in

profit that would have been earned had the store

continued to sell appliances there. This opportunity

cost measure may be much greater than what the

store actually paid for that part of the building.

These three guidelines can help a manager to

measure marginal cost correctly. Failure to do so

can cause production to be too high or too low and

thereby reduce profit.

8.5 The Competitive Firm’s Short-Run

Supply Curve

A supply curve for a firm tells us how much output it will produce at every possible price. We have seen that competitive firms will increase output to the point

at which price is equal to marginal cost, but will shut down if price is below

average variable cost. Therefore, the firm’s supply curve is the portion of the marginal cost curve for which marginal cost is greater than average variable cost.

Figure 8.6 illustrates the short-run supply curve. Note that for any P greater

than minimum AVC, the profit-maximizing output can be read directly from

the graph. At a price P1, for example, the quantity supplied will be q1; and at P2,

it will be q2. For P less than (or equal to) minimum AVC, the profit-maximizing

output is equal to zero. In Figure 8.6 the entire short-run supply curve consists

of the crosshatched portion of the vertical axis plus the marginal cost curve

above the point of minimum average variable cost.

Short-run supply curves for competitive firms slope upward for the same reason that marginal cost increases—the presence of diminishing marginal returns to

one or more factors of production. As a result, an increase in the market price will

induce those firms already in the market to increase the quantities they produce.

CHAPTER 8 • Profit Maximization and Competitive Supply 293

Price

(dollars per

unit)

MC

F IGURE 8.6

P2

AC

AVC

P1

THE SHORT-RUN

SUPPLY CURVE FOR

A COMPETITIVE FIRM

In the short run, the firm

chooses its output so that

marginal cost MC is equal to

price as long as the firm covers its average variable cost.

The short-run supply curve

is given by the crosshatched

portion of the marginal cost

curve.

P = AVC

q1

0

q2

Output

The higher price not only makes the additional production profitable, but also

increases the firm’s total profit because it applies to all units that the firm produces.

The Firm’s Response to an Input Price Change

When the price of its product changes, the firm changes its output level to

ensure that marginal cost of production remains equal to price. Often, however,

the product price changes at the same time that the prices of inputs change. In

this section we show how the firm’s output decision changes in response to a

change in the price of one of its inputs.

Figure 8.7 shows a firm’s marginal cost curve that is initially given by MC1 when

the firm faces a price of \$5 for its product. The firm maximizes profit by producing

Price, cost

(dollars per unit)

In §6.2, we explain that

diminishing marginal returns

increase in an input results

in a smaller and smaller

increase in output.

MC 2

MC 1

F IGURE 8.7

THE RESPONSE OF A FIRM TO A CHANGE

IN INPUT PRICE

\$5

When the marginal cost of production for a firm

increases (from MC1 to MC2), the level of output that

maximizes profit falls (from q1 to q2).

q2

q1

Output

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