What effect, if any, does each of the following events have on the price elasticity of demand for corporate-owned jets? a)A decline in corporate earnings causes firms to cut their travel budgets, which in turn causes expenditures on corporate jet travel to become a larger fraction of total spending on corporate travel. b)A new, much more fuel-efficient corporate jet is introduced. c)Further deregulation of the commercial airlines industry substantially increases the variety of departure times and destinations offered by commercial airlines. d)The cost of manufacturing corporate jets rises. Assume that the demand for cosmetic or plastic surgery is price inelastic. Are the following statements true or false? Explain. a)When the price of plastic surgery increases, the number of operations decreases. b)The percentage change in the price of plastic surgery is less than the percentage change in quantity demanded. c)The marginal revenue of another operation is negative. d)Changes in the price of plastic surgery do not affect the number of operations. e)Quantity demanded is not very responsive to changes in price. f)If more plastic surgery is performed, expenditures on plastic surgery will decrease. Fill in the blanks: a.The price elasticity of demand for a firm’s product is equal to –1.75 over the range of prices being considered by the firm’s manager. If the manager increases the price of the product by 9 percent, the manager predicts the quantity demanded will ________ (increase, decrease) by ________ percent. b.The price elasticity of demand for an industry’s demand curve is equal to –1.75 for the range of prices over which supply decreases. If total industry output is expected to decrease by 14 percent as a result of the supply decrease, managers in this industry should expect the market price of the good to ________ (increase, decrease) by ________percent. Use the linear demand curve shown below to answer the following questions. a.The point elasticity of demand at a price of $500 is _________. b.The point elasticity of demand at a price of $175 is _________. c.Demand is unitary elastic at a price of $_________. d.As price falls, |E| __________________ (gets larger, gets smaller, stays the same) for a linear demand curve.4 Scholarly Sources pleaseSheet1

Managerial Economics

Complete Rubric – Unit 3

Segment

Points

Subject Matter Competency

Point Breakdown

45

2.5 points for each part

or fill-in-the-blank for each question.

Sources and APA

12

4 scholarly sources

Quality of writing and word count

18

1500 words

Total:

75

**Please see syllabus for explanation of each rubric line item.**

Page 1

Chapter

6

C

A

L

After reading this chapter,Vyou will be able to:

6.1 Define price elasticityEof demand and use it to predict changes in quantity

demanded and changes in the price of a good.

R

6.2 Explain the role price elasticity plays in determining how a change in price

T

affects total revenue.

, factors that affect price elasticity of demand.

6.3 List and explain several

Elasticity and Demand

6.4 Calculate price elasticity over an interval along a demand curve and at a point

on a demand curve. T

6.5 Relate marginal revenue to total revenue and demand elasticity and write the

E

marginal revenue equation for linear inverse demand functions.

R

6.6 Define and compute the income elasticity of demand and the cross-price

elasticity of demand.R

E

N

ost managers agree

C that the toughest decision they face is the decision

to raise or lower the price of their firms’ products. When Walt Disney

E to raise ticket prices at its theme parks in Anaheim,

Company decided

M

California, and Orlando, Florida, the price hike caused attendance at the Disney

parks to fall. The price increase was a success, however, because it boosted Disney’s

1 multiplied by the number of tickets sold. For Disney,

revenue: the price of a ticket

the higher ticket price more

8 than offset the smaller number of tickets purchased,

and revenue increased. You might be surprised to learn that price increases do not

5

always increase a firm’s revenue.

For example, suppose just one gasoline producer,

ExxonMobil, were to increase

the

price of its brand of gasoline while rival gasoline

9

producers left their gasoline prices unchanged. ExxonMobil would likely experiTthough it increased its price, because many ExxonMobil

ence falling revenue, even

customers would switch S

to one of the many other brands of gasoline. In this situation, the reduced amount of gasoline sold would more than offset the higher price

of gasoline, and ExxonMobil would find its revenue falling.

197

tho21901_ch06_197-235.indd 197

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198 C H A P T E R 6 Elasticity and Demand

When managers lower price to attract more buyers, revenues may either rise

or fall, again depending upon how responsive consumers are to a price reduction. For example, in an unsuccessful marketing strategy, called “Campaign 55,”

McDonald’s Corporation lowered the price of its Big Mac and Quarter Pounders

to 55 cents in an effort to increase revenue. The price reduction resulted in lower

revenue, and McDonald’s abandoned the low-price strategy for all but its breakfast meals—lower prices did increase breakfast revenues. Obviously, managers

need to know how a price increase or decrease is going to affect the quantity sold

C chapter, you will learn how to use the concept

and the revenue of the firm. In this

of price elasticity to predict how A

revenue will be affected by a change in the price

of the product. You can easily understand why managers of price-setting firms

L useful; they can use knowledge about demand

find this chapter to be particularly

elasticities to help them make better

V decisions about raising or lowering prices.

And, even for managers of price-taking firms (i.e., firms in competitive markets

where prices are determined by E

the intersection of market demand and supply),

knowledge of price elasticity of industry

demand can help managers predict the

R

effect of changes in market price on total industry sales and total consumer expenT

ditures in the industry.

Managers recognize that quantity

, demanded and price are inversely related.

When they are making pricing decisions, as you saw in the examples of Disney,

ExxonMobil, and McDonald’s, it is even more important for managers to know

Ta given change in price. A 10 percent decrease

by how much sales will change for

in price that leads to a 2 percentEincrease in quantity demanded differs greatly

in effect from a 10 percent decrease in price that causes a 50 percent increase in

R

quantity demanded. There is a substantial

difference in the effect on total revenue

between these two responses to aR

change in price. Certainly, when making pricing

decisions, managers should have a good idea about how responsive consumers

will be to any price changes and E

whether revenues will rise or fall.

The majority of this chapter is N

devoted to the concept of price elasticity of demand,

a measure of the responsiveness of quantity demanded to a change in price along

a demand curve and an indicatorC

of the effect of a price change on total consumer

expenditure on a product. The concept

E of price elasticity provides managers, economists, and policymakers with a framework for understanding why consumers in

some markets are extremely responsive to changes in price while consumers in other

markets are not. This understanding

1 is useful in many types of managerial decisions.

We will begin by defining the price elasticity of demand and then show how

8

to use price elasticities to find the percentage changes in price or quantity that re5 curve. Next, the relation between elasticity

sult from movements along a demand

and the total revenue received by9firms from the sale of a product is examined in

detail. Then we discuss three factors that determine the degree of responsiveness

T elasticity of demand. We also show how to

of consumers, and hence the price

compute the elasticity of demandSeither over an interval or at a point on demand.

Then we examine the concept of marginal revenue and demonstrate the relation

among demand, marginal revenue, and elasticity. The last section of this chapter

introduces two other important elasticities: income and cross-price elasticities.

tho21901_ch06_197-235.indd 198

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C H A P T E R 6 Elasticity and Demand 199

6.1 THE PRICE ELASTICITY OF DEMAND

price elasticity of

demand (E )

The percentage change

in quantity demanded,

divided by the percentage change in price.

E is always a negative

number because P and Q

are inversely related.

elastic

Segment of demand for

which |E | . 1.

inelastic

Segment of demand for

which |E | , 1.

unitary elastic

Segment of demand for

which |E | 5 1.

T A B L E 6.1

Price Elasticity of

Demand (E ),

%DQ

E 5 _____

%DP

As noted earlier, price elasticity of demand measures the responsiveness or sensitivity of consumers to changes in the price of a good or service. We will begin this

section by presenting a formal (mathematical) definition of price elasticity and

then show how price elasticity can be used to predict the change in sales when

price rises or falls or to predict the percentage reduction in price needed to stimulate sales by a given percentage amount.

Consumer responsiveness

C to a price change is measured by the price elasticity

of demand (E), defined as

A

%DP L

Percentage change in price

V

Because price and quantity demanded are inversely related by the law of demand,

the numerator and denominator

always have opposite algebraic signs, and the

E

price elasticity is always negative. The price elasticity is calculated for movements

R

along a given demand curve (or function) as price changes and all other factors

T are held constant. Suppose a 10 percent price deaffecting quantity demanded

crease (%DP 5 210%) causes consumers to increase their purchases by 30 percent

,

(%DQ 5 130%). The price elasticity is equal to 23 (5 130%y210%) in this case.

Percentage change in quantity demanded

%DQ _____________________________________

E 5 _____

5

In contrast, if the 10 percent decrease in price causes only a 5 percent increase in

sales, the price elasticityTwould equal 20.5 (5 15%y210%). Clearly, the smaller

(absolute) value of E indicates less sensitivity on the part of consumers to a change

E

in price.

When a change in price

R causes consumers to respond so strongly that the

percentage by which they adjust their consumption (in absolute value) exceeds the

R (in absolute value), demand is said to be elastic over

percentage change in price

that price interval. In mathematical

terms, demand is elastic when |%DQ| exceeds

E

|%DP|, and thus |E| is greater than 1. When a change in price causes consumers to

respond so weakly that N

the percentage by which they adjust their consumption

(in absolute value) is lessCthan the percentage change in price (in absolute value),

demand is said to be inelastic over that price interval. In other words, demand is

E

inelastic when the numerator (in absolute value) is smaller than the denominator

(in absolute value), and thus |E| is less than 1. In the special instance in which the

percentage change in quantity (in absolute value) just equals the percentage change

1

in price (in absolute value), demand is said to be unitary elastic, and |E| is equal

8 this discussion.

to 1. Table 6.1 summarizes

5

9

Elasticity

ElasticT

Unitary elastic

S

Inelastic

Responsiveness

|%DQ| . |%DP |

|%DQ| 5 |%DP |

|%DQ| , |%DP |

|E|

|E | . 1

|E | 5 1

|E | , 1

Note: The symbol “| |” denotes the absolute value.

tho21901_ch06_197-235.indd 199

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200 C H A P T E R 6 Elasticity and Demand

Predicting the Percentage Change in Quantity Demanded

Suppose a manager knows the price elasticity of demand for a company’s product

is equal to 22.5 over the range of prices currently being considered by the firm’s

marketing department. The manager is considering decreasing price by 8 percent

and wishes to predict the percentage by which quantity demanded will increase.

From the definition of price elasticity, it follows that

%DQ

_____

C22.5 5 28%

so, with a bit of algebraic manipulation,

%DQ 5 120% (5 22.5 3 28%). Thus, the

A

manager can increase sales by 20 percent by lowering price 8 percent. As we menL

tioned in the introduction, price elasticity information about industry demand

V make predictions about industry- or marketcan also help price-taking managers

level changes. For example, suppose

E an increase in industry supply is expected to

cause market price to fall by 8 percent, and the price elasticity of industry demand

is equal to 22.5 for the segmentR

of demand over which supply shifts. Using the

same algebraic steps just shown,Ttotal industry output is predicted to increase by

20 percent in this case.

,

Predicting the Percentage Change in Price

Suppose a manager of a different firm faces a price elasticity equal to 20.5 over

T

the range of prices the firm would consider charging for its product. This manager

wishes to stimulate sales by 15 E

percent. The manager is willing to lower price

to accomplish the increase in sales but needs to know the percentage amount by

R

which price must be lowered to obtain the 15 percent increase in sales. Again usR of demand, it follows that

ing the definition of price elasticity

E20.5 5 ______

115%

%DP

N

so, after some algebraic manipulation, %DP 5 230% (5 15%y20.5). Thus, this

manager must lower price by 30Cpercent to increase sales by 15 percent. As we

explained in the case of predicting

E percentage changes in quantity demanded,

Now try Technical

Problems 1–2.

tho21901_ch06_197-235.indd 200

elasticity of industry demand can also be used to make predictions about changes

in market-determined prices. For example, suppose an increase in industry supply

1 to rise by 15 percent, and the price elasticity

is expected to cause market output

of industry demand is equal to 20.5 for the portion of demand over which supply

8

shifts. Following the algebraic steps shown above, market price is predicted to fall

5 techniques for predicting percentage changes

by 30 percent. As you can see, the

in quantity demanded and price9can be applied to both individual firm demand

curves or industry demand curves.

As you can see, the concept of T

elasticity is rather simple. Price elasticity is nothing more than a mathematical measure

of how sensitive quantity demanded is

S

to changes in price. We will now apply the concept of price elasticity to a crucial

question facing managers. How does a change in the price of the firm’s product

affect the total revenue received?

8/11/15 4:54 PM

C H A P T E R 6 Elasticity and Demand 201

6.2 PRICE ELASTICITY AND TOTAL REVENUE

total revenue (TR)

The total amount paid to

producers for a good or

service (TR 5 P 3 Q).

Managers of firms, as well as industry analysts, government policymakers, and

academic researchers, are frequently interested in how total revenue changes

when there is a movement along the demand curve. Total revenue (TR), which

also equals the total expenditure by consumers on the commodity, is simply the

price of the commodity times quantity demanded, or

TR 5 P 3 Q

C

As we have emphasized, price and quantity demanded move in opposite direcA If price rises, quantity falls; if price falls, quantity

tions along a demand curve:

rises. The change in priceLand the change in quantity have opposite effects on total

revenue. The relative strengths of these two effects will determine the overall effect on TR. We will now V

examine these two effects, called the price effect and the

quantity effect, along with

E the price elasticity of demand to establish the relation

between changes in price and total revenue.

R

Price Elasticity and Changes

T in Total Revenue

, price of a product, the increase in price, by itself, would

When a manager raises the

price effect

The effect on total

revenue of changing

price, holding output

constant.

quantity effect

The effect on total

revenue of changing

output, holding price

constant.

increase total revenue if the quantity sold remained constant. Conversely, when

a manager lowers price, the decrease in price would decrease total revenue if the

T

quantity sold remained constant.

This effect on total revenue of changing price, for

a given level of output, isEcalled the price effect. When price changes, the quantity

sold does not remain constant; it moves in the opposite direction of price. When

R to a decrease in price, the increase in quantity, by

quantity increases in response

itself, would increase total

R revenue if the price of the product remained constant.

Alternatively, when quantity falls after a price increase, the reduction in quantity,

by itself, would decreaseE

total revenue if product price remained constant. The effect on total revenue of changing

the quantity sold, for a given price level, is called

N

the quantity effect. The price and quantity effects always push total revenue in

opposite directions. TotalC

revenue moves in the direction of the stronger of the two

effects. If the two effects are

E equally strong, no change in total revenue can occur.

Suppose a manager increases price, causing quantity to decrease. The price effect, represented below by an upward arrow above P, and the quantity effect, represented by a downward1arrow above Q, show how the change in TR is affected

by opposing forces

8

5

TR 5 P 3 Q

9

To determine the direction of movement in TR, information about the relative

T and output effect must be known. The elasticity of

strengths of the price effect

demand tells a manager S

which effect, if either, is dominant.

If demand is elastic, |E| is greater than 1, the percentage change in Q (in absolute value) is greater than the percentage change in P (in absolute value), and the

quantity effect dominates the price effect. To better see how the dominance of the

tho21901_ch06_197-235.indd 201

8/11/15 4:54 PM

202 C H A P T E R 6 Elasticity and Demand

quantity effect determines the direction in which TR moves, you can represent

the dominance of the quantity effect by drawing the arrow above Q longer than

the arrow above P. The direction of the dominant effect—the quantity effect here—

tells a manager that TR will fall when price rises and demand is elastic:

TR 5 P 3 Q

C demand is elastic, the arrows in this diagram

If a manager decreases price when

reverse directions. The arrow above

A Q is still the longer arrow because the quantity effect always dominates the price effect when demand is elastic.

Now consider a price increaseLwhen demand is inelastic. When demand is inelastic, |E| is less than 1, the percentage

V change in Q (in absolute value) is less than

the percentage change in P (in absolute value), and the price effect dominates the

E effect can be represented by an upward arrow

quantity effect. The dominant price

above P that is longer than the downward

arrow above Q. The direction of the

R

dominant effect tells the manager that TR will rise when price rises and demand

T

is inelastic

,

TR

T 5P3Q

When a manager decreases price

E and demand is inelastic, the arrows in this

diagram would reverse directions. A downward arrow above P would be a long

R dominates the quantity effect when demand

arrow because the price effect always

is inelastic.

R

When demand is unitary elastic, |E| is equal to 1, and neither the price effect nor

E two effects exactly offset each other, so price

the quantity effect dominates. The

changes have no effect on total revenue

when demand is unitary elastic.

N

C

Relation The effect of a change in price on total revenue (TR 5 P 3 Q) is determined by the price

E (inelastic), the quantity (price) effect dominates. Total revenue

elasticity of demand. When demand is elastic

always moves in the same direction as the variable (P or Q) having the dominant effect. When demand is

unitary elastic, neither effect dominates, and changes in price leave total revenue unchanged.

T A B L E 6.2

Relations between

Price Elasticity and Total

Revenue (TR )

1

Table 6.2 summarizes the relation

8 between price changes and revenue changes

under the three price elasticity conditions.

5

9

Elastic

Unitary elastic

Inelastic

|%DQ| . |%DP | T

|%DQ| 5 |%DP |

|%DQ| , |%DP |

Q-effect dominates

No dominant effect

P-effect dominates

S

Price rises

TR falls

No change in TR

TR rises

Price falls

tho21901_ch06_197-235.indd 202

TR rises

No change in TR

TR falls

8/11/15 4:54 PM

C H A P T E R 6 Elasticity and Demand 203

Changing Price at Borderline Video Emporium: A Numerical Example

The manager at Borderline Video Emporium faces the demand curve for Blu-ray

DVD discs shown in Figure 6.1. At the current price of $18 per DVD Borderline

can sell 600 DVDs each week. The manager can lower price to $16 per DVD and

increase sales to 800 DVDs per week. In Panel A of Figure 6.1, over the interval a

to b on demand curve D the price elasticity is equal to 22.43. (You will learn how

to make this calculation in Section 6.4.) Because the demand for Blu-ray DVDs

is elastic over this rangeC

of prices (|22.43| . 1), the manager knows the quantity

effect dominates the price effect. Lowering price from $18 to $16 results in an increase in the quantity ofADVDs sold, so the manager knows that total revenue,

which always moves in the

L direction of the dominant effect, must increase.

To verify that revenue indeed rises when the manager at Borderline lowers the

V of demand, you can calculate total revenue at the two

price over an elastic region

prices, $18 and $16

E

R

Point a: TR 5 $18 3 600 5 $10,800

T

Point b: TR 5 $16 3 800 5 $12,800

,

Price per DVD (dollars)

24

18

16

13

11

0

Quantity effect dominates

a

E = 22.43

b

f

g

600 800 1,100 1,300

Quantity of DVDs per week

Panel A — An elastic region of demand

tho21901_ch06_197-235.indd 203

T

E

R

R

E

N

C

E

1

8

5

D

9

T

2,400

S

24

Price per DVD (dollars)

F I G U R E 6.1

Changes in Total Revenue of Borderline Video Emporium

18

16

9

7

Price effect dominates

a

b

c

E = 20.50

d

D

0

600 800

1,500 1,700

Quantity of DVDs per week

2,400

Panel B — An inelastic region of demand

8/11/15 4:54 PM

204 C H A P T E R 6 Elasticity and Demand

Total revenue rises by $2,000 (5 12,800 2 10,800) when price is reduced over this

elastic region of demand. Although Borderline earns less revenue on each DVD

sold, the number of DVDs sold each week rises enough to more than offset the

downward price effect, causing total revenue to rise.

Now suppose the manager at Borderline is charging just $9 per compact disc

and sells 1,500 DVDs per week (see Panel B). The manager can lower price to $7 per

disc and increase sales to 1,700 DVDs per week. Over the interval c to d on demand

curve D, the elasticity of demand equals 20.50. Over this range of prices for DVDs,

the demand is inelastic (|20.50| ,C1), and Borderline’s manager knows the price effect dominates the quantity effect.

AIf the manager lowers price from $9 to $7, total

revenue, which always moves in the direction of the dominant effect, must decrease.

L the manager at Borderline lowers price over

To verify that revenue falls when

an inelastic region of demand, you

V can calculate total revenue at the two prices,

$9 and $7

E

Point c: TR 5 $9 3 1,500 5 $13,500

R

Point d: TR 5 $7 3 1,700 5 $11,900

T

Now try Technical

Problems 3–5.

Total revenue falls by $1,600 (DTR 5 $11,900 2 $13,500 5 2$1,600). Total revenue

, over an inelastic region of demand. Borderline

always falls when price is reduced

again earns less revenue on each DVD sold, but the number of DVDs sold each week

does not increase enough to offset the downward price effect and total revenue falls.

If the manager decreases (orTincreases) the price of Blu-ray DVDs over a

unitary-elastic region of demand,Etotal revenue does not change. You should verify that demand is unitary elastic over the interval f to g in Panel A of Figure 6.1.

Ris elastic over the $16 to $18 price range but inNote in Figure 6.1 that demand

elastic over the $7 to $9 price range.

R In general, the elasticity of demand varies

along any particular demand curve, even one that is linear. It is usually incorrect

E

to say a demand curve is either elastic or inelastic. You can say only that a demand

curve is elastic or inelastic over aNparticular price range. For example, it is correct

to say that demand curve D in Figure 6.1 is elastic over the $16 to $18 price range

C

and inelastic over the $7 to $9 price range.

E

I L L U S T R AT I1

ON 6.1

P 3 Q Measures More Than Just Business’

Total Revenue

As you know from our explanation in Section 6.2,

demand elasticity provides the essential piece of information needed to predict how total revenue changes—

increases, decreases, or stays the same—when the price

of a good or service changes. We mention in that discussion that price multiplied by quantity can also mea-

tho21901_ch06_197-235.indd 204

8 the amount spent by consumers who buy Q units

sure

of the good at price P. In other words, total revenue for

5

a business is exactly equal to the total expenditure by

consumers.

9

While business owners and managers focus on

T Q as measuring their revenue for the purpose of

P3

computing

their business profit, politicians and govS

ernment policymakers frequently view P 3 Q as measuring the “burden” on consumers buying the good

8/11/15 4:54 PM

C H A P T E R 6 Elasticity and Demand 205

or service. And thus policymakers can use the price

elasticity of demand to predict how price changes are

likely to affect the total amount spent by consumers to

buy a product. For example, policymakers believe raising taxes on cigarettes causes cigarette prices to rise

and, by the law of demand, will reduce the quantity

of cigarettes purchased and improve smokers’ health.

Unfortunately, however, the demand for cigarettes

C

remains “stubbornly inelastic,” and this causes total

expenditure on cigarettes by smokers to rise substanA

tially with higher taxes on cigarettes. Critics of higher

cigarette taxes point out that, with the cigarette tax

L

increases, smokers’ health probably deteriorates even

V

more rapidly because smokers will only decrease the

number of cigarettes they smoke by a small amount;

E

they will simply spend more income to buy the cigarettes,

R

leaving less money for other, more healthful grocery

items. Policymakers sometimes defend the higher

T

taxes by (perversely and correctly) noting that further

cigarette price hikes will eventually move smokers into

,

the elastic region of their demand curves so that higher

prices would then cause significant declines in the

quantity demanded and reduce the amount spent T

on

cigarettes. Although we cannot dispute the analytical

E

conclusion that cigarette demand will become elastic

if only the price is high enough, getting to that price

R

point on cigarette demand is very likely to take a lot

R

more income out of smokers’ pockets before we see

any decline in spending on cigarettes.

6.3

Another example demonstrating the usefulness

of interpreting P 3 Q as a measure of total consumer

spending, rather than as a measure of total revenue, involves the “taxi cab” fare war going on in Manhattan.

The current price war in Manhattan was sparked by

the entry of new “car-service” firms such as Gett, Lyft,

and Uber that pick up riders who use their smartphones to “hail” cab rides.a Before these new competitors entered the market in Manhattan, taxi cab fares

were high enough to be positioned in the elastic region

of the demand for car rides. The price elasticity of demand is important in this situation because, for now,

drivers at the new companies are not complaining

about falling fares. Their incomes are rising, measured

by multiplying the cab fare times the number of rides

(i.e., P 3 Q) because demand is elastic at the current

fares. And, with rising incomes for their drivers, Gett,

Lyft, and Uber are able to expand the number of cars

servicing Manhattan. Of course, if cab fares continue

falling, eventually demand will become inelastic and

driver incomes, P 3 Q, will decline. At that point, car

drivers will not be so happy with the fare war!

These new car-service companies are not legally defined as

“taxi cabs” and therefore they cannot legally pick up riders on

the street who hail with a hand raised. Nonetheless, riders view

hailing one of these “app car-service” rides with their smartphones as nearly identical to hailing by hand a yellow taxi cab.

Source: Anne Kadet, “Car-App Car Services Compete for Passengers with Low Fares,” The Wall Street Journal, October 10, 2014.

a

E

N

C

FACTORS AFFECTING PRICE ELASTICITY OF DEMAND

E

Price elasticity of demand plays such an important role in business decision making

that managers should understand not only how to use the concept to obtain information about the demand

1 for the products they sell, but also how to recognize the

factors that affect price elasticity. We will now discuss the three factors that make

8 more elastic than the demand for other products.

the demand for some products

5

9

The availability of substitutes is by far the most important determinant of price

elasticity of demand. TheT

better the substitutes for a given good or service, the more

elastic the demand for that

Sgood or service. When the price of a good rises, consumAvailability of Substitutes

ers will substantially reduce consumption of that good if they perceive that close

substitutes are readily available. Naturally, consumers will be less responsive to a

price increase if they perceive that only poor substitutes are available.

tho21901_ch06_197-235.indd 205

8/11/15 4:54 PM

206 C H A P T E R 6 Elasticity and Demand

Some goods for which demand is rather elastic include fruit, corporate jets, and

life insurance. Alternatively, goods for which consumers perceive few or no good

substitutes have low price elasticities of demand. Wheat, salt, and gasoline tend to

have low price elasticities because there are only poor substitutes available—for

instance, corn, pepper, and diesel fuel, respectively.

The definition of the market for a good greatly affects the number of substitutes

and thus the good’s price elasticity of demand. For example, if all the grocery

stores in a city raised the price of milk by 50 cents per gallon, total sales of milk

C

would undoubtedly fall—but probably

not by much. If, on the other hand, only

the Food King chain of stores raised

price

by 50 cents, the sales of Food King milk

A

would probably fall substantially. There are many good substitutes for Food King

L substitutes for milk in general.

milk, but there are not nearly as many

V

E

The percentage of the consumer’s budget that is spent on the commodity is also

important in the determination ofRprice elasticity. All other things equal, we would

expect the price elasticity to be directly

related to the percentage of consumers’

T

budgets spent on the good. For example, the demand for refrigerators is probably

,

more price elastic than the demand for toasters because the expenditure required

Percentage of Consumer’s Budget

to purchase a refrigerator would make up a larger percentage of the budget of a

“typical” consumer.

T

E

Rin measuring the price elasticity affects the magThe length of the time period used

nitude of price elasticity. In general,

R the longer the time period of measurement, the

larger (the more elastic) the price elasticity will be (in absolute value). This relation

is the result of consumers’ havingEmore time to adjust to the price change.

Consider, again, the way consumers

N would adjust to an increase in the price of

milk. Suppose the dairy farmers’ association is able to convince all producers of milk

C by 15 percent. During the first week the price

nationwide to raise their milk prices

increase takes effect, consumers come

E to the stores with their grocery lists already

Time Period of Adjustment

made up. Shoppers notice the higher price of milk but have already planned their

meals for the week. Even though a few of the shoppers will react immediately to the

higher milk prices and reduce the amount

of milk they purchase, many shoppers will

1

go ahead and buy the same amount of milk as they purchased the week before. If the

8

dairy association collects sales data and measures the price elasticity of demand for

5hike, they will be happy to see that the 15 percent

milk after the first week of the price

increase in the price of milk caused

9only a modest reduction in milk sales.

Over the coming weeks, however, consumers begin looking for ways to consume less milk. They substitute T

foods that have similar nutritional composition

to milk; consumption of cheese,S

eggs, and yogurt all increase. Some consumers

will even switch to powdered milk for some of their less urgent milk needs—

perhaps to feed the cat or to use in cooking. Six months after the price increase,

the dairy association again measures the price elasticity of milk. Now the price

tho21901_ch06_197-235.indd 206

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C H A P T E R 6 Elasticity and Demand 207

Now try Technical

Problem 6.

elasticity of demand is probably much larger in absolute value (more elastic)

because it is measured over a six-month time period instead of a one-week

time period.

For most goods and services, given a longer time period to adjust, the demand

for the commodity exhibits more responsiveness to changes in price—the demand

becomes more elastic. Of course, we can treat the effect of time on elasticity within

the framework of the effect of available substitutes. The greater the time period

available for consumer adjustment, the more substitutes become available and

Cwe stressed earlier, the more available are substitutes,

economically feasible. As

the more elastic is demand.

A

L

V

As noted at the beginning of the chapter, the price elasticity of demand is equal to

E change in quantity demanded divided by the percentthe ratio of the percentage

age change in price. When

R calculating the value of E, it is convenient to avoid

computing percentage changes by using a simpler formula for computing elasticT

ity that can be obtained through

the following algebraic operations

,

DQ

____

6.4 CALCULATING PRICE ELASTICITY OF DEMAND

3 100

%DQ _________

Q

E 5 _____

5

%DP

DP

___

3 100

T

P

DQ __

E 5 ____

P

3

Q

DP

R

Thus, price elasticity can be calculated by multiplying the slope of demand

R

(DQyDP) times the ratio of price divided by quantity (PyQ), which avoids making

tedious percentage change

E computations. The computation of E, while involving

the rather simple mathematical formula derived here, is complicated somewhat

N

by the fact that elasticity can be measured either (1) over an interval (or arc) along

C point on the demand curve. In either case, E still meademand or (2) at a specific

sures the sensitivity of consumers

to changes in the price of the commodity.

E

interval (or arc)

elasticity

Price elasticity calculated

over an interval of a

demand curve:

Average P

E 5 ____

DQ 3 __________

DP Average Q

tho21901_ch06_197-235.indd 207

The choice of whether to measure demand elasticity at a point or over an

interval of demand depends on the length of demand over which E is measured.

If the change in price is1relatively small, a point measure is generally suitable.

Alternatively, when the price change spans a sizable arc along the demand

8

curve, the interval measurement

of elasticity provides a better measure of

consumer responsiveness

5 than the point measure. As you will see shortly, point

elasticities are more easily computed than interval elasticities. We begin with a

9 elasticity of demand over an interval.

discussion of how to calculate

T

S

Computation of Elasticity over an Interval

When elasticity is calculated over an interval of a demand curve (either a linear

or a curvilinear demand), the elasticity is called an interval (or arc) elasticity. To

measure E over an arc or interval of demand, the simplified formula presented

8/11/15 4:54 PM

208 C H A P T E R 6 Elasticity and Demand

earlier—slope of demand multiplied by the ratio of P divided by Q—needs to be

modified slightly. The modification only requires that the average values of P and Q

over the interval be used:

Average P

DQ

E 5 ____

3 __________

Average Q

DP

Recall from our previous discussion of Figure 6.1 that we did not show you how

to compute the two values of theCinterval elasticities given in Figure 6.1. You can

now make these computations for

A the intervals of demand ab and cd using the

above formula for interval price elasticities (notice that average values for P and Q

L

are used):

V

_____

Eab 5 E

1200

3 ____

17 5 22.43

22

700

1200

_____

_____

Ecd 5 R

3 8 5 20.5

22

1600

T

,

Relation When calculating the price elasticity

of demand over an interval of demand, use the interval or

arc elasticity formula:

Now try Technical

Problem 7.

Average P

DQ

E 5 ____

3 ________

DP

Average

Q

T

E

R

As we explained previously, it is appropriate to measure elasticity at a point on

R

a demand curve rather than over an interval when the price change covers only

E computed at a point on demand is called

a small interval of demand. Elasticity

point elasticity of demand. Computing

the price elasticity at a point on demand

N

is accomplished by multiplying the slope of demand (DQyDP), computed at the

point of measure, by the ratio PyQ, C

computed using the values of P and Q at the point

of measure. To show you how thisEis done, we can compute the point elasticities in

Computation of Elasticity at a Point

point elasticity

A measurement of

demand elasticity

calculated at a point on

a demand curve rather

than over an interval.

Figure 6.1 when Borderline Music Emporium charges $18 and $16 per compact

disc at points a and b, respectively. Notice that the value of DQyDP for the linear

demand in Figure 6.1 is 2100 (5112400y224) at every point along D, so the two

point elasticities are computed as

8

Ea 552100 3 ____

18 5 23

600

9

____

Eb 5 2100 3 16 5 22

800

T

S of demand at a point on demand, multiply the slope of

Relation When calculating the price elasticity

demand (DQyDP ), computed at the point of measure, by the ratio PyQ, computed using the values of P and

Q at the point of measure.

tho21901_ch06_197-235.indd 208

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C H A P T E R 6 Elasticity and Demand 209

I L L U S T R AT I O N 6 . 2

Texas Calculates Price Elasticity

In addition to its regular license plates, the state of

Texas, as do other states, sells personalized or “vanity” license plates. To raise additional revenue, the

state will sell a vehicle owner a license plate saying

C

whatever the owner wants as long as it uses six letters

(or numbers), no one else has the same license as the

A

one requested, and it isn’t obscene. For this service,

the state charges a higher price than the price for stanL

dard licenses.

V

Many people are willing to pay the higher price

rather than display a license of the standard form,

E

such as 387 BRC. For example, an ophthalmologist anR

nounces his practice with the license MYOPIA. Others

tell their personalities with COZY-1 and ALL MAN.

T

When Texas decided to increase the price for vanity plates from $25 to $75, a Houston newspaper ,reported that sales of these plates fell from 150,000 down

to 60,000 vanity plates. As it turned out, demand was

rather inelastic over this range. As you can calculate T

using the interval method, the price elasticity was 20.86.

E

The newspaper reported that vanity plate revenue rose

after the price increase ($3.75 million to $4.5 million),

R

which would be expected for a price increase when demand is inelastic.

R

But the newspaper quoted the assistant director of

E

the Texas Division of Motor Vehicles as saying, “Since

the demand droppeda the state didn’t make money

N

from the higher fees, so the price for next year’s personalized plates will be $40.” If the objective of C

the

state is to make money from these licenses and if the

E

numbers in the article are correct, this is the wrong

thing to do. It’s hard to see how the state lost money

by increasing the price from $25 to $75—the revenue

1

increased and the cost of producing plates must have

decreased because fewer were produced. So the move

from $25 to $75 was the right move.

Moreover, let’s suppose that the price elasticity between $75 and $40 is approximately equal to the value

calculated for the movement from $25 to $75 (20.86).

We can use this estimate to calculate what happens to

revenue if the state drops the price to $40. We must first

find what the new quantity demanded will be at $40.

Using the arc elasticity formula and the price elasticity

of 20.86,

Average P

DQ

E 5 ____

3 __________

Average Q

DP

(75 1 40)/2

60,000 2 Q ______________

5 __________

3

5 20.86

75 2 40

(60,000 1 Q)/2

where Q is the new quantity demanded. Solving

this equation for Q, the estimated sales are 102,000

(rounded) at a price of $40. With this quantity demanded and price, total revenue would be $4,080,000,

representing a decrease of $420,000 from the revenue

at $75 a plate. If the state’s objective is to raise revenue

by selling vanity plates, it should increase rather than

decrease price.

This Illustration actually makes two points. First,

even decision makers in organizations that are not run

for profit, such as government agencies, should be able

to use economic analysis. Second, managers whose firms

are in business to make a profit should make an effort

to know (or at least have a good approximation for) the

elasticity of demand for the products they sell. Only with

this information will they know what price to charge.

It was, of course, quantity demanded that decreased, not

demand.

Source: Adapted from Barbara Boughton, “A License for

Vanity,” Houston Post, October 19, 1986, pp. 1G, 10G.

a

8

5

9

T

Point elasticity when demand is linear Consider a general linear demand

S

function of three variables—price

(P), income (M), and the price of a related

good (PR)

Q 5 a 1 bP 1 cM 1 dPR

tho21901_ch06_197-235.indd 209

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210 C H A P T E R 6 Elasticity and Demand

__

Suppose

income and the price of the related good take on specific values of M

and

__

PR, respectively. Recall from Chapter 2 when values of the d

emand determinants

(M and PR in this case) are held constant, they become part of the constant term in

the direct demand function:

__

Q 5 a9 1 bP

__

where a9 5 a 1 cM 1 dPR. The slope parameter b, of course, measures the rate of

change in quantity demanded per unit change in price: b 5 DQyDP. Thus price

C curve can be calculated as

elasticity at a point on a linear demand

A

P

E 5 b __

Q

L

where P and Q are the values ofV

price and quantity at the point of measure. For

example, let’s compute the elasticity of demand for Borderline Music at a price

of $9 per CD (see point c in PanelEB of Figure 6.1). You can verify for yourself that

the equation for the direct demand

R function is Q 5 2,400 2 100P, so b 5 2100

and

T

3

9 5 2 __

E 5 2100

, _____

5 5 20.6

1,500

Even though multiplying b by the ratio PyQ is rather simple, there happens to

T

be an even easier formula for computing

point price elasticities of demand. This

alternative point elasticity formula

is

E

R E 5 ______

P

P2A

R

where P is the price at the point on demand where elasticity is to be measured,

E 1 Note that, for the linear demand equation

and A is the price-intercept of demand.

Q 5 a9 1 bP, the price intercept ANis 2a9yb. In Figure 6.1, let us apply this alternative formula to calculate again the elasticity at point c (P 5 $9). In this case, the

C

price-intercept A is $24, so the elasticity

is

E 9

E 5 ______

5 20.6

9 2 24

which is exactly equal to the value

1 obtained previously by multiplying the slope

of demand by the ratio PyQ. We must

8 stress that, because the two formulas b__

P and

Q

______

5

P are mathematically equivalent,

they always yield identical values for point

P2A

9

price elasticities.

T

S

1

This alternative formula for computing price elasticity is derived in the mathematical appendix

for this chapter.

tho21901_ch06_197-235.indd 210

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C H A P T E R 6 Elasticity and Demand 211

Now try Technical

Problems 8–9.

Relation For linear demand functions Q 5 a9 1 bP, the price elasticity of demand can be computed

using either of two equivalent formulas:

E 5 b __

P 5 ______

P

Q

P2A

where P and Q are the values of price and quantity demanded at the point of measure on demand,

and A (5 2a9yb) is the price-intercept of demand.

C

Point elasticity when demand is curvilinear When demand is curvilinear, the

A

DQ

formula E 5 ____

3 __

P can be used for computing point elasticity simply by subQ L

DP

stituting the slope of the curved demand at the point of measure for the value of

V

DQyDP in the formula. This can be accomplished by measuring the slope of the

tangent line at the point E

of measure. Figure 6.2 illustrates this procedure.

In Figure 6.2, let us measure elasticity at a price of $100 on demand curve D.

R

We first construct the tangent line T at point R. By the “rise over run” method,

T (5 2140y105). Of course, because P is on the vertical

the slope of T equals 24y3

axis and Q is on the horizontal

axis, the slope of tangent line T gives DPyDQ not

,

DQyDP. This is easily fixed by taking the inverse of the slope of tangent line T to

get DQ/DP 5 23y4. At point R price elasticity is calculated using 23y4 for the

slope of demand and using

T $100 and 30 for P and Q, respectively

DQ

3 ____

100

EE

5 ____

3 __

P 5 2 __

R

4 3 30 5 22.5

Q

DP

F I G U R E 6.2

Calculating Point

Elasticity for Curvilinear

Demand

Price (dollars)

140

100

90

40

0

R

R

E

N

C

E

1

8

5

9

T

S

R

ER = –2.5

ES = –0.8

T

30

S

D

T’

105

Q

Quantity

tho21901_ch06_197-235.indd 211

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212 C H A P T E R 6 Elasticity and Demand

As it turns out, the alternative formula E 5 P/(P 2 A) for computing point elasticity on linear demands can also be used for computing point elasticities on curvilinear demands. To do so, the price-intercept of the tangent line T serves as the

value of A in the formula. As an example, we can recalculate elasticity at point R

in Figure 6.2 using the formula E 5 Py(P 2 A). The price-intercept of tangent line

T is $140

100

P 5 _________

ER 5 ______

CA 100 2 140 5 22.5

P2

A

As expected, 22.5 is the same value for ER obtained earlier.

Since the formula E 5 Py(P 2LA) doesn’t require the slope of demand or the

value of Q, it can be used to compute

E in situations like point S in Figure 6.2

V

where the available information is insufficient to be able to multiply slope by

Eprice-intercept of T9 (5 $90) into the formula

the PyQ ratio. Just substitute the

E 5 Py(P 2 A) to get the elasticity

Rat point S

T

ES 5 ______

,P 5 _______

40 5 20.8

P 2 A 40 2 90

Relation For curvilinear demand functions,

T the price elasticity at a point can be computed using either

of two equivalent formulas:

E

___

E 5 DQ

3 __

P 5 ______

P

RDP Q P 2 A

where DQyDP is the slope of the curved demand

R at the point of measure (which is the inverse of the slope

of the tangent line at the point of measure), P and Q are the values of price and quantity demanded at the

E of the tangent line extended to cross the price-axis.

point of measure, and A is the price-intercept

N

Now try Technical

Problem 10.

We have now established that both formulas for computing point elasticities

C

will give the same value for the price elasticity of demand whether demand is

E students frequently ask which formula is the

linear or curvilinear. Nonetheless,

“best” one. Because the two formulas give identical values for E, neither one is

better or more accurate than the other. We should remind you, however, that you

1 information to compute E both ways, so you

may not always have the required

should make sure you know both

8 methods. (Recall the situation in Figure 6.2 at

point S.) Of course, when it is possible to do so, we recommend computing the

elasticity using both formulas to5make sure your price elasticity calculation is

correct!

9

T

S

Elasticity (Generally) Varies along a Demand Curve

In general, different intervals or points along the same demand curve have

differing elasticities of demand, even when the demand curve is linear. When

demand is linear, the slope of the demand curve is constant. Even though the

tho21901_ch06_197-235.indd 212

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C H A P T E R 6 Elasticity and Demand 213

absolute rate at which quantity demanded changes as price changes (DQyDP)

remains constant, the proportional rate of change in Q as P changes (%DQy%DP)

varies along a linear demand curve. To see why, we can examine the basic

DQ

P . Moving along a linear demand does not

formula for elasticity, E 5 ____ 3 __

Q

DP

cause the term DQyDP to change, but elasticity does vary because the ratio PyQ

changes. Moving down demand, by reducing price and selling more output,

causes the term PyQ to decrease which reduces the absolute value of E. And, of

C demand, by increasing price and selling less output,

course, moving up a linear

causes PyQ and |E| to increase.

Thus, P and |E| vary directly along a linear deA

mand curve.

For movements alongLa curved demand, both the slope and the ratio PyQ vary

continuously along demand.

V For this reason, elasticity generally varies along curvilinear demands, but there is no general rule about the relation between price and

E demand.

elasticity as there is for linear

As it turns out, there R

is an exception to the general rule that elasticity varies

along curvilinear demands. A special kind of curvilinear demand function exists

T

for which the demand elasticity is constant for all points on demand. When demand takes the form Q ,5 aPb, the elasticity is constant along the demand curve

and equal to b.2 Consequently, no calculation of elasticity is required, and the price

elasticity is simply the value of the exponent on price, b. The absolute value of b

can be greater than, lessTthan, or equal to 1, so that this form of demand can be

elastic, inelastic, or unitary

E elastic at all points on the demand curve. As we will

show you in the next chapter, this kind of demand function can be useful in statistical demand estimation R

and forecasting.

Figure 6.3 shows a constant

elasticity of demand function, Q 5 aPb, with

R

the values of a and b equal to 100,000 and 21.5, respectively. Notice that price

Eboth points U and V where prices are $20 and $40,

elasticity equals 21.5 at

respectively

N

20

C 5 ______

E

P 5 __________

5 21.5

U

P 2 A 20 2 33.33

E

40

EV 5 ______

P 5 __________

5 21.5

P 2 A 40 2 66.67

Clearly, you never need

1 to compute the price elasticity of demand for this kind

of demand curve because E is the value of the exponent on price (b).

Now try Technical

Problem 11.

8

5

Relation In general, the price elasticity of demand varies along a demand curve. For linear demand

curves, price and |E| vary directly:

9 The higher (lower) the price, the more (less) elastic is demand. For a curvilinear demand, there is no general rule about the relation between price and elasticity, except for the

special case of Q 5 aP , whichT

has a constant price elasticity (equal to b) for all prices.

S

b

2

tho21901_ch06_197-235.indd 213

See the appendix at the end of this chapter for a mathematical proof of this result.

8/11/15 4:54 PM

214 C H A P T E R 6 Elasticity and Demand

F I G U R E 6.3

Constant Elasticity

of Demand

80

D : Q = 100,000P

–1.5

66.67

Price (dollars)

60

40

33.33

20

0

C

A

L

U: Eu = –1.5

V

E

R

2,000

T1,000 Quantity

,

V: Ev = –1.5

3,000

T

6.5 MARGINAL REVENUE, DEMAND, AND PRICE ELASTICITY

marginal revenue (MR)

The addition to total

revenue attributable to

selling one additional

unit of output; the slope

of total revenue.

Eto changes in the price of a good must be conThe responsiveness of consumers

sidered by managers of price-setting

R firms when making pricing and output decisions. The price elasticity of demand gives managers essential information about

Rby a change in price. As it turns out, an equally

how total revenue will be affected

important concept for pricing and

E output decisions is marginal revenue. Marginal

revenue (MR) is the addition to total revenue attributable to selling one additional

N

unit of output:

C

MR 5 DTRyDQ

E

Because marginal revenue measures the rate of change in total revenue as quan-

tity changes, MR is the slope of the TR curve. Marginal revenue is related to price

elasticity because marginal revenue, like price elasticity, involves changes in total

1

revenue caused by movements along a demand curve.

8

Marginal Revenue and Demand

5

As noted, marginal revenue is related

9 to the way changes in price and output affect total revenue along a demand curve. To see the relation between marginal

T

revenue and price, consider the following

numerical example. The demand schedule for a product is presented in columns

1

and 2 of Table 6.3. Price times quantity

S

gives the total revenue obtainable at each level of sales, shown in column 3.

Marginal revenue, shown in column 4, indicates the change in total revenue

from an additional unit of sales. Note that marginal revenue equals price only

tho21901_ch06_197-235.indd 214

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C H A P T E R 6 Elasticity and Demand 215

T A B L E 6.3

Demand and Marginal

Revenue

(1)

(2)

(3)

Unit sales

Price

Total revenue

(4)

Marginal revenue

(DTR/DQ)

0

1

2

3

4

5

6

$4.50

4.00

3.50

3.10

2.80

2.40

2.00

$

0

4.00

7.00

9.30

11.20

12.00

12.00

—

$4.00

3.00

2.30

1.90

0.80

0

C

A

7

1.50

10.50

21.50

L

V

E

for the first unit sold. For the first unit sold, total revenue is the demand price for

R

1 unit. The first unit sold adds $4—the price of the first unit—to total revenue, and

the marginal revenue of T

the first unit sold equals $4; that is, MR 5 P for the first

unit. If 2 units are sold, the second unit should contribute $3.50 (the price of the

,

second unit) to total revenue. But total revenue for 2 units is only $7, indicating

inframarginal units

Units of output that

could have been sold at

a higher price had a firm

not lowered its price to

sell the marginal unit.

that the second unit adds only $3 (5 $7 2 $4) to total revenue. Thus the marginal

revenue of the second unit

T is not equal to price, as it was for the first unit. Indeed,

examining columns 2 and 4 in Table 6.3 indicates that MR , P for all but the first

E

unit sold.

Marginal revenue is less

R than price (MR , P) for all but the first unit sold because price must be lowered in order to sell more units. Not only is price lowered

R unit sold, but price is also lowered for all the inframaron the marginal (additional)

ginal units sold. The inframarginal

units are those units that could have been sold

E

at a higher price had the firm not lowered price to sell the marginal unit. Marginal

N can be expressed as

revenue for any output level

C

Revenue lost by lowering price

MR 5EPrice 2

on the inframarginal units

The second unit of output sells for $3.50. By itself, the second unit contributes

1 marginal revenue is not equal to $3.50 for the second

$3.50 to total revenue. But

unit because to sell the second unit, price on the first unit is lowered from $4 to

8

$3.50. In other words, the first unit is an inframarginal unit, and the $0.50 lost on

5

the first unit must be subtracted

from the price. The net effect on total revenue of

selling the second unit is 9

$3 (5 $3.50 2 $0.50), the same value as shown in c olumn 4

of Table 6.3.

If the firm is currentlyTselling 2 units and wishes to sell 3 units, it must lower

price from $3.50 to $3.10.SThe third unit increases total revenue by its price, $3.10.

To sell the third unit, the firm must lower price on the 2 units that could have

been sold for $3.50 if only 2 units were offered for sale. The revenue lost on the 2

inframarginal units is $0.80 (5 $0.40 3 2). Thus the marginal revenue of the third

tho21901_ch06_197-235.indd 215

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216 C H A P T E R 6 Elasticity and Demand

unit is $2.30 (5 $3.10 2 $0.80), and marginal revenue is less than the price of the

third unit.

It is now easy to see why P 5 MR for the first unit sold. For the first unit sold,

price is not lowered on any inframarginal units. Because price must fall in order to

sell additional units, marginal revenue must be less than price at every other level

of sales (output).

As shown in column 4, marginal revenue declines for each additional unit sold.

Notice that it is positive for each of the first 5 units sold. However, marginal revenue is 0 for the sixth unit sold,C

and it becomes negative thereafter. That is, the

seventh unit sold actually causes

Atotal revenue to decline. Marginal revenue is

positive when the effect of lowering price on the inframarginal units is less than

L sales at the lower price. Marginal revenue

the revenue contributed by the added

is negative when the effect of lowering

price on the inframarginal units is greater

V

than the revenue contributed by the added sales at the lower price.

E

R than price for all units sold after the first, because the price

Relation Marginal revenue must be less

must be lowered to sell more units. When marginal revenue is positive, total revenue increases when quanT

tity increases. When marginal revenue is negative, total revenue decreases when quantity increases. Marginal revenue is equal to 0 when total revenue

, is maximized.

Figure 6.4 shows graphically the relations among demand, marginal revenue,

T schedule in Table 6.3. As noted, MR is below

and total revenue for the demand

price (in Panel A) at every level of

E output except the first. When total revenue (in

Panel B) begins to decrease, marginal revenue becomes negative. Demand and

R sloped.

marginal revenue are both negatively

Sometimes the interval over R

which marginal revenue is measured is greater

than one unit of output. After all, managers don’t necessarily increase output by

just one unit at a time. Suppose inETable 6.3 that we want to compute marginal revenue when output increases fromN2 units to 5 units. Over the interval, the change

in total revenue is $5 (5 $12 2 $7), and the change in output is 3 units. Marginal

C per unit change in output; that is, each of the

revenue is $1.67 (5 DTRyDQ 5 $5y3)

3 units contributes (on average) $1.67

E to total revenue. As a general rule, whenever

the interval over which marginal revenue is being measured is more than a single

unit, divide DTR by DQ to obtain the marginal revenue for each of the units of

1

output in the interval.

As mentioned in Chapter 2 and as you will see in Chapter 7, linear demand

8

equations are frequently employed for purposes of empirical demand estimation

5 between a linear demand equation and its

and demand forecasting. The relation

marginal revenue function is no different

from that set forth in the preceding rela9

tion. The case of a linear demand is special because the relation between demand

T

and marginal revenue has some additional

properties that do not hold for nonlinear demand curves.

S

When demand is linear, marginal revenue is linear and lies halfway between

demand and the vertical (price) axis. This implies that marginal revenue must

be twice as steep as demand, and demand and marginal revenue share the same

tho21901_ch06_197-235.indd 216

8/11/15 4:54 PM

C H A P T E R 6 Elasticity and Demand 217

F I G U R E 6.4

Demand, Marginal Revenue, and Total Revenue

4.00

3.50

12.00

2.50

2.00

D

1.50

1.00

0.50

0

–0.50

–1.00

–1.50

1

2

3

4

Quantity

5

6

7

C

A

L

V

E

R

T

,

Total revenue (dollars)

Price and marginal revenue (dollars)

3.00

Q

10.00

TR

8.00

6.00

4.00

2.00

Q

0

1

2

3

4

5

6

7

T

MR

Quantity

E

Panel A

Panel B

R

R

E We can explain these additional properties and show

intercept on the vertical axis.

how to apply them by returning

to the simplified linear demand function (Q 5

N

a 1 bP 1 cM 1 dPR) examined earlier in this chapter (and in Chapter 2). Again we

Cand the price of the related good R constant at the spehold the values

income

__ of __

cific values M

and P

R, respectively.

E

__

__ This produces the linear demand equation Q 5

a9 1 bP, where a9 5 a 1 cM1 dPR. Next, we find the inverse demand equation by

solving for P 5 f(Q) as explained in Chapter 2 (you may wish to review Technical

Problem 2 in Chapter 2) 1

8

a9 1 __

P 5 2 __

1 Q

b

b

5

5 A 1 BQ

9

where A 5 2a9yb and B 5 1yb. Since a9 is always positive and b is always negaT it follows that A is always positive and B is always

tive (by the law of demand),

negative: A . 0 and B , S

0. Using the values of A and B from inverse demand, the

equation for marginal revenue is MR 5 A 1 2BQ. Thus marginal revenue is linear,

has the same vertical intercept as inverse demand (A), and is twice as steep as inverse demand (DMR/DQ 5 2B).

tho21901_ch06_197-235.indd 217

8/11/15 4:54 PM

218 C H A P T E R 6 Elasticity and Demand

Relation When inverse demand is linear, P 5 A 1 BQ (A . 0, B , 0), marginal revenue is also linear,

intersects the vertical (price) axis at the same point demand does, and is twice as steep as the inverse demand function. The equation of the linear marginal revenue curve is MR 5 A 1 2BQ.

Figure 6.5 shows the linear inverse demand curve P 5 6 2 0.05Q. (Remember

that B is negative because P and Q are inversely related.) The associated marginal

revenue curve is also linear, intersects the price axis at $6, and is twice as steep as

the demand curve. Because it is twice as steep, marginal revenue intersects the

C

quantity axis at 60 units, which is half the output level for which demand interA for marginal revenue has the same vertical

sects the quantity axis. The equation

intercept but twice the slope: MRL5 6 2 0.10Q.

V

Marginal Revenue and Price Elasticity

E the relation of price elasticity to demand and

Using Figure 6.5, we now examine

marginal revenue. Recall that if total

R revenue increases when price falls and quantity rises, demand is elastic; if total revenue decreases when price falls and quanT marginal revenue is positive in Panel A, from

tity rises, demand is inelastic. When

a quantity of 0 to 60, total revenue

, increases as price declines in Panel B; thus demand is elastic over this range. Conversely, when marginal revenue is negative, at

any quantity greater than 60, total revenue declines when price falls; thus demand

must be inelastic over this range.TFinally, if marginal revenue is 0, at a quantity of

60, total revenue does not change when quantity changes, so the price elasticity of

E

demand is unitary at 60.

Total revenue (dollars)

Price and marginal revenue (dollars)

R

F I G U R E 6.5

R

Linear Demand, Marginal Revenue, and Elasticity (Q 5 120 2 20P )

E

N

C

Inverse D: P = 6 – 0.05Q

6

E >1

E

E =1

4

3

E 1

E =1

TR = P 3Q = 6Q – 0.05Q2

180

1

8

5

9

0

T

S

E

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