U.S. cigarette makers face enormous punitive damage
penalties after losing a series of class-action lawsuits that heaped penalties
amounting to several hundred billion dollars on the tobacco industry. In spite
of the huge penalties, The Wall Street Journal reported, “The damage (to
cigarette makers) is generally under control.” What action do you suppose
the cigarette companies took to avoid bankruptcy? Why did this action succeed?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?
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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
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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
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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
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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
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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​
__
P​R, 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 c​M​ 1 d​P​R. 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 c​M​1 d​P​R. 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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