Demand Elasticity

Chapter 4

Demand Analysis

• Where does demand come from? How can managers have any predictive tools about how

changes in certain variables will impact their demand?

• Demand ultimate derives from the individual consumer or firm. Consumers demand

products to satisfy their wants and needs. Firms demand products to produce other

goods and services to sell in a market.

• Consumers derive utility from their consumption of goods and services.

• Firms derive profit from their consumption of goods and services.

• It is thought that consumers seek to maximize the utility they gain from consumption

just as firms seek to maximize profit.

• Households are limited in their quest to maximize utility. They only have so much money

to spend on consumption and they face a lot of different wants/needs.

• Ultimately, the income and the prices of the goods determine a household’s choice set. The

income/prices combined with household preferences determine household consumption

bundle.

• The market demand curves are the aggregation of the individual household demand

curves.

• How much does quantity demanded change when there is a change in price (perhaps from

a supply-side influence), a change in income, other prices, population, etc.?

49

50 CHAPTER 4. DEMAND ANALYSIS

4.1 Elasticity

• Discussing how much quantity demanded changes due to a supply shock or some other

demand shock is difficult to do across markets because prices and other influences on

demand are measured in different units and take on different importance.

• Economists developed the concept of elasticity to get around these problems of units and

market differences.

• An elasticity measures the percentage change in one variable due to a percentage change

in another variable. Elasticity measures are unit-less and can be compared across markets

and time.

• Elasticity is mathematically defined as

%∆Y

%∆X

• A point elasticity is calculated at a particular point on the demand/supply curve and is

written as

εX =%∆Y

%∆X=

∂Y

∂X

X

Y

• For example, if εX = 3 then a 1% increase in X yields a 3% increase in Y (and vice-versa).

• Point elasticities are useful when analyzing a single point on a demand/supply curve.

• At other times we will be working with two different points on a demand/supply curve.

In these cases we use the arc elasticity:

εX =%∆Y

%∆X=

∂Y

∂X

X

Y

where X is the average X of the two points and Y is the average Y of the two points.

4.1.1 Price Elasticity of Demand

• The most common elasticity used in microeconomics is the own-price elasticity of demand.

• Own-price elasticity measures the percentage change in quantity demanded due to a

percentage change in price:

4.1. ELASTICITY 51

εPd =

%∆QD

%∆P=

∂QD

∂P

P

QD

• If ∆Q/∆P → 0 or ∆P/∆Q → −∞ then εpd → 0 (from below) or demand is relatively

inelastic.

• If ∆Q/∆P → −∞ or ∆P/∆Q → 0 then εpd → −∞ or demand is relatively elastic.

• If |∆Q/∆P | = |P/Q| then εpd = −1 or demand is unitary elastic.

• Because the demand curve is downward sloping, and therefore ∆Q/∆P is negative, it is

common to take the absolute value of the price elasticity of demand.

• An example: Assume Q = 60, 000− 5, 000P . Consider the following scenarios:

P = 8 εpd = −5000(8/20000) = −2 Relatively elastic

P = 10 εpd = −5000(10/10000) = −5 Relatively elastic

P = 2 εpd = −5000(2/50000) = −1/5 Relatively inelastic

P = 6 εpd = −5000(6/30000) = −1 Unitary elastic

• Generally, the higher the price the more elastic the demand. The lower the price, the

more inelastic (less elastic) the demand.

• How does the price elasticity help managers? Consider the following general results

If |εpd| > 1 Raising price will lower total revenue

If |εpd| = 1 Raising price will not change TR (TR is maximized)

If |εpd| < 1 Raising price will increase total revenue

Perfectly inelastic demand and perfectly elastic demand


• Total revenue can be written as TR = P ×Q which implies that

MR =∆P

∆QQ + P

• For most firms, ∆P/∆Q = 0 because the individual firm is very small. However, if we

consider the market for the product:

∆P

∆Q≤ 0 ⇒ MR ≤ P

If∆P

∆Q= 0 ⇒ MR = P

• When MR = 0 we have

P = −∆P

∆QQ

1 = −∆P

∆Q

Q

P

1 = −∆Q

∆P

P

Q

−1 = εPd

• When MR = 0, TR is maximized and price elasticity of demand is one in absolute value

4.1. ELASTICITY 53

• Elasticity measures can help firms determine optimal pricing so to maximize profit:

Π = TR− TC

mΠ = MR−MC

MR =∆TR

∆Q= P + Q

∆P

∆Q

= P

[1 +

Q

P

∆P

∆Q

]

Recall: εpd =

∆Q

∆P

P

Q


⇒ MR = P [1 +1

ε]

• From this, when the firm sets MR = MC it obtains

P [1 +1

ε] = MC (4.1)

P ∗ =MC

1 + 1ε

(4.2)

• Example: Let a 2% drop in price yield a 4% increase in sales:

εpd =

+4%

−2%= −2

If MC = $100 then

P ∗ =100

1 + 1−2

= $200

If MC = $90 then

P ∗ =90

1 + 1−2

= $180

• Price falls faster than marginal cost because demand is elastic.

• If demand is inelastic, then price would fall slower than marginal cost.

• As costs decline, it is profit enhancing for the firm to lower price and produce more.

• What if εPd = −20 and MC = $100?

P ∗ =$100

1 + 1−20

= $105.26

• What if εPd = −1000 and MC = $100?

P ∗ =$100

1 + 1−1000

= $100.10

In other words, with very elastic demand there is very little profit. In this example profit

is $0.10 on $100 of cost.

• The greater the demand elasticity, the closer is price to marginal cost.

4.1. ELASTICITY 55

• Why do firms want inelastic demand? Greater markups over marginal cost, for one thing.

• How to reduce price elasticity of demand?

1. Fewer substitutes

2. Greater quality

3. Brand loyalty

4. Infrequent purchases

5. Lower percentage of household budget

6. Expectations that prices will increase dramatically in the future

4.1.2 Examples from the real world

Major League Baseball

• Opening day 2002 for the Montreal Expos: All tickets to the Montreal Expos vs. the

Florida Marlins were $1 and attendance was 34,351.

• The next day, the same two teams played again: average ticket price was $9 and atten-

dance was 4,771.

• With these two games, the price elasticity for the Montreal Expos:

εPd =

∆Q

∆P

P

Q=−29580

+8

5

19561= −0.945

• In essence, the arc-elasticity between these two prices was negative one.

• Montreal Expos might have been revenue maximizers in their tickets (consistent with

price elasticity equal to one in absolute value).

• Question: Why not price at $1 and make a lot of money on beer?

Price of Oil

• What is the short-run price elasticity of gasoline?

• From the Energy Information Agency (www.eia.gov)


• Sales to end users, Millions of gallons per day from 2008:

Month Qty Price %∆Q %∆P 1 mo εPd 2 mo εP

d

Jan-08 45,099.00 3.095

Feb-08 46,935.10 3.078 0.041 -0.005 -7.412

Mar-08 46,666.30 3.293 -0.006 0.070 -0.082 0.551

Apr-08 47,662.40 3.507 0.021 0.065 0.328 0.118

May-08 48,092.00 3.815 0.009 0.088 0.103 0.205

Jun-08 47,963.30 4.105 -0.003 0.076 -0.035 0.040

Jul-08 46,912.60 4.114 -0.022 0.002 -9.992 -0.329

Aug-08 47,335.80 3.833 0.009 -0.068 -0.132 0.192

Sep-08 45,027.10 3.756 -0.049 -0.020 2.428 0.451

Oct-08 47,004.00 3.112 0.044 -0.171 -0.256 0.034

Nov-08 45,680.50 2.208 -0.028 -0.290 0.097 -0.028

Dec-08 45,237.20 1.745 -0.010 -0.210 0.046 0.068

Averages 0.001 -0.042 -0.015

• The price elasticity of demand between Feb and March 2008 was -0.08, or essentially zero.

By June 2008, the elasticity had jumped to -10.11 but by November and December it was

essentially zero again.

• In November 2010, only 36,594.8 thousand gallons per day of gasoline were sold in the

United states and the price was 2.859. Compared to November 2008, this implies a

longer-run price elasticity of

εPd =

∆Q

∆P

P

Q=−9, 085.7

+0.658

2.208+2.8592

45,680.5+36,594.82

= −13, 808.05× 5.074

82.275.3= −0.860

• Could it be that today our response to changes in the price of gasoline are much more

elastic and therefore do not elicit the same angst as similar price changes did in 2008?

If price of gasoline goes back to $4 do we anticipate that the quantity demanded would

actually increase in equilibrium (in the short run)?

Increasing the Gasoline Tax

• In a 1996 article in the Energy Journal, authors Jonathan Haughton and Soumodip Sarkar

attempt to answer the question of what impact a $1 gasoline tax increase would have on

driving and accidents. They submit that with a gas tax of $1, miles driven would decrease

by up to 12% and fatalities by up to 18%.

4.1. ELASTICITY 57

• How do they get to these results? By estimating and using the own-price elasticity of

gasoline to calculate the impact on gasoline consumption.

• The retail price of gasoline in 1991 was $1.13, of which 28% or $0.32 was tax. Assuming

that the entire tax increase is applied to the price, this means a price increase of $0.68,

or 46%. The long run own-price elasticity of demand for gasoline over a ten-year period

was calculated to be in the range -0.23 to -0.35. Using this knowledge we can calculate

the change in consumption:

Low end: %∆Q/46% = −0.23

%∆Q = −0.23× 46% = −10.6%

High end: %∆Q/46% = −0.35

%∆Q = −0.35× 46% = −16.1%

• Without going into the effect on accidents, we can still determine that a gas tax of $1

would have been expected to decrease gas consumption by between 10.6% and 16.1%.

• Source: Haughton, J. and Sarkar, S. (1996), “Gasoline Tax as a Corrective Tax: Estimates

for the United States, 1970-1991,” Energy Journal, 17(2), pp. 103-26.

Escorts vs. Civics

• Ford and Honda cater to the subcompact segment of the automobile market with their

Escort and Civic models, respectively. Are Ford Escort buyers more or less price sensitive

than buyers of Honda Civics? One way to answer this question is to estimate the change

in quantity demanded from a $100 increase in the price of each make. But this does not

compare like with like.

• A consistent way of comparing the price sensitivity of Escort and Civic buyers is to use

the own-price elasticities of the demands. The own-price elasticities of the demands for

Escorts and Civics have been estimated to be both −3.4. For a 1% increase in price, both

groups would reduce purchases by 3.4%.

• Source: Pinelopi Koujianou-Goldberg (1995), “Product Differentiation and Oligopoly in

International Markets: the Case of the U.S. Automobile Industry,” Econometrica, 63(4),

pp. 891-951.


Higher Education

• From Eric Steger, East Central University. Consider the following table:

Tuition Number Semester Total

Price/ of Hours Semester Total

Hour Students Hours Revenue

$25 4000 15 60,000 60, 000× 25 = $1, 500, 000

$30 3900 15 58,500 58, 500× 30 = $1, 755, 000

• The elasticity in this example is then = −15005

30+2560,000+58,500

= −0.139

• Bezmen and Depken (1998, Economics of Education Review): estimate the short-run

tuition elasticity of enrollment in U.S. public colleges to be -0.02.

4.2 Other Elasticity Measures

4.2.1 Cross Price Elasticity

• This elasticity measures the relative response to the quantity demanded for Good A due

to a relative change in the price of Good X.

• Mathematically this looks like:

εxd =

%∆QDA

$∆PX

4.2. OTHER ELASTICITY MEASURES 59

• If εxd < 0: The price of Good X increases and the quantity demand of Good A decreases.

This implies that Good A and Good X are complements in consumption or are used

together.

Examples: iPhones and iPhone docking stations; tennis balls and tennis rackets; cars and

tires; computers and printers.

• If εxd > 0: The price of Good X increases and the quantity demand of Good A increases.

This implies that Good A and Good X are substitutes in consumption or are used in place

of one another.

Examples: Hondas and Toyotas; Samsung and Sony televisions; Dell and HP computers;

• If εxd = 0: The price of Good X increases and the quantity demand of Good A does

not change. This implies that Good A and Good X are not related to each other in

consumption or are independent goods.

Examples: Corn and telephones; X-box games and allergy medicine.

There is a saying in America: “What does that have to do with the price of tea in China?”

This would be used in a situation where two things were completely independent of each

other.

• If cross price elasticity is high (positive or negative) then the two goods are strong sub-

stitutes or complements.

• If cross price elasticity is close to zero, then the two goods are weak substitutes or com-

plements.

Cross-price elasticity example: cars and gasoline

• In 2008 the price of gasoline spiked above $4 per gallon. This made a lot of people

reconsider the type of vehicles they would purchase and drive. A New York Times article

from May 2, 2008 provided the following data for select vehicles.

• The April 2007 to April 2008 percentage change in the price of gasoline was (3.458 −2.845)/2.845 = 21.5%)


Make Model April 08 Sales %Change from Apr-07 εxd

Ford F-series 44,813 -27.0 -1.25

Toyota Camry 40,016 -2.6 -0.12

Chevrolet Silverado 37,231 -30.5 -1.42

Honda Accord 35,075 11.9 0.55

Toyota Prius 21,757 53.80 2.50

4.2.2 Income Elasticity of Demand

• The income elasticity of demand reflects the percentage change in the quantity demanded

due to a percentage change in income.

• Mathematically this looks like

εMd =

%∆QD

%∆M

If εMd > 0 ⇒ Normal good

If εMd < 0 ⇒ Inferior good

If εMd > 1 ⇒ Luxury good

If εMd = 1 ⇒ Necessary good

If εMd < 1 ⇒ Vital good


• Firms are likely concerned about how the demand for their product will change with

exogenous changes in household income. In the current economic slowdown in the United

States, Walmart has experienced an increase in demand whereas Macy’s has experienced

a decline in demand.

• Here is the five year trend line of the stock price for Walmart (blue) and Macy’s (red):

• We can see when times were good, before the recession, Macy’s was doing much better

than Walmart. As times turned bad, Walmart outperformed Macy’s. This suggests that

Macy’s sells normal/luxury goods and Walmart sells inferior goods.

4.2.3 Other elasticities

• Can you think of other elasticities that might be useful to firms?

• Advertising elasticity of demand:

εAd =

%∆QD

%∆Adv

• We might expect advertising elasticity to be positive and, in equilibrium, less than one.

• Population elasticity of demand

εPOPd =

%∆QD

%∆POP

• As population increases we might expect to see quantity demanded increase, holding

technology fixed.


• If population elasticity is greater than one this would imply network externalities or what

are called relational goods/mob goods.

4.2.4 Advertising

• To explore the concepts of multivariate optimization and the optimal level of advertising,

consider a hypothetical multivariate product demand function for CSI, Inc., where the

demand Q is determined by the price charged, P , and the level of advertising, A:

Q = 5, 000− 10P + 40A + PA− 0.8A2 − 0.5P 2

where Q is measured in units, P is measured in price, A is measured in hundreds of

dollars.

• When analyzing multivariate relations such as these, one is interested in the marginal

effect of each independent variable on the quantity sold, the dependent variable. Opti-

mization requires an analysis of how a change in each independent variable affects the

dependent variable, holding constant the effect of all other independent variables. The

partial derivative concept is used in this type of marginal analysis.

• In light of the fact that the CSI demand function includes two independent variables, the

price of the product itself and advertising, it is possible to examine two partial derivatives:

the partial of Q with respect to price, or ∂Q/∂P , and the partial of Q with respect to

advertising expenditures, or ∂Q/∂A. In determining partial derivatives, all variables

except the one with respect to which the derivative is being taken remain unchanged. In

this instance, A is treated as a constant when the partial derivative of Q with respect to

P is analyzed; P is treated as a constant when the partial derivative of Q with respect to

A is evaluated. Therefore, the partial derivative of Q with respect to P is:

∂Q/∂P = 0− 10 + 0 + A− 0− P = −10 + A− P

The partial with respect to A is:

∂Q/∂A = 0− 0 + 40 + P − 1.6A− 0 = 40 + P − 1.6A

• Solving these two equations simultaneously yields the optimal price-output-advertising

combination.


• Because −10+A−P = 0, P = A−10. Substituting this value for P into 40+P−1.6A = 0,

gives 40 + (A− 10)− 1.6A = 0, which implies that 0.6A = 30 and A = 50 or $5,000.

• Given this value, P = A− 10, 10 = 50− 10 = $40.

• Inserting these values for P and A into the CSI demand function yields Q = 5, 800.

4.2.5 Problems and Questions

Q4.1 From Ralph T. Byrns:

Describe in words and provide a predicted sign for the following elasticities

1. The TV football game elasticity of divorce rates.

2. The snow elasticity of ski lift ticket sales;

3. The temperature elasticity of lemonade sales;

4. The homerun elasticity of beer sales at a ballpark;

5. The condom elasticity of STDs;

6. Name and describe three additional elasticities of your own.

P4.5 The demand for personal computers can be characterized by the following point elastic-

ities: price elasticity = 5, cross-price elasticity with software = 4, and income elasticity

= 2.5. Indicate whether each of the following statements is true or false, and explain your

answer.

A. A price reduction for personal computers will increase both the number of units

demanded and the total revenue of sellers.

B. The cross-price elasticity indicates that a 5% reduction in the price of personal

computers will cause a 20% increase in software demand.

C. Demand for personal computers is price elastic and computers are cyclical normal

goods.

D. Falling software prices will increase revenues received by sellers of both computers

and software.

E. A 2% price reduction would be necessary to overcome the effects of a 1% decline in

income.

SOLUTION


A. True. A price reduction always increases units sold, given a downward sloping de-

mand curve. The negative sign on the price elasticity indicates that this is indeed

the case here. The fact that price elasticity equals 5 indicates that demand is elastic

with respect to price, and that a price reduction will increase total revenues.

B. False. The cross-price elasticity indicates that a 5% decrease in the price of software

programs will have the effect of increasing personal computer demand by 20%.

C. True. Demand is price elastic (see part A). Since the income elasticity is positive,

personal computers are a normal good. Moreover, since the income elasticity is

greater than one, personal computer demand is also cyclical.

D. False. Negative cross-price elasticity indicates that personal computers and soft-

ware are compliments. Therefore, falling software prices will increase the demand

for computers and resulting revenues for sellers. However, there is no information

concerning the price elasticity of demand for software, and therefore, one does not

know the effect of falling software prices on software revenues.

E. False. A 2% reduction in price will cause a 10% increase in the quantity of personal

computers demanded. A 1% decline in income will cause a 2.5% fall in demand.

These changes will not be mutually offsetting.

P4.6 In an effort to reduce excess end-of-the-model-year inventory, Harrison Ford offered a 1%

discount off the average price of 4WD Escape Gas-Electric Hybrid SUVs sold during the

month of August. Customer response was wildly enthusiastic, with unit sales rising by

10% over the previous month’s level.

A. Calculate the point price elasticity of demand for Harrison Ford 4WD Escape Gas-

Electric Hybrid SUVs sold during the month of August.

B. Calculate the profit-maximizing price per unit if Harrison Ford has an average whole-

sale (invoice) cost of $23,500 and incurs marginal selling costs of $350 per unit.

SOLUTION

A.

ε =∆Q/Q

∆P/P

= 10%/− 1%

= −10(Highly elastic)


B. The profit-maximizing price can be found using the optimal price formula:

P ∗ = MC/(1 + 1/εP )

= ($23, 500 + $350)/[1 + 1/(−10)]

= $26, 500

P4.7 The South Beach Cafe recently reduced appetizer prices from $12 to $10 for afternoon

”early bird” customers and enjoyed a resulting increase in sales from 90 to 150 orders per

day. Beverage sales also increased from 300 to 600 units per day.

A. Calculate the arc price elasticity of demand for appetizers.

B. Calculate the arc cross-price elasticity of demand between beverage sales and appe-

tizer prices.

C. Holding all else equal, would you expect an additional appetizer price decrease to $8

to cause both appetizer and beverage revenues to rise? Explain.

SOLUTION

A.

εp =∆Q

∆P

P2 + P1

Q2 + Q1

=150− 90

10− 12

10 + 12

150 + 90= −2.75

B.

εx =∆Q

∆Px

Px2 + Px1

Q2 + Q1

=600− 300

10− 12

10 + 12

600 + 300= −3.67

C. Yes, the |εp| = 2.75 > 1 calculated in part A implies an elastic demand for appetizers

and that an additional price reduction will increase appetizer revenues. εx = −3.67 <

0 indicates that beverages and appetizers are complements. Therefore, a further

decrease in appetizer prices will cause a continued growth in beverage unit sales and

revenues.

Alternatively, If P = a + bQ, then $12 = a + b(90) and $10 = a + b(150). Solving

for the demand curve gives P = $15 − $0.033Q. At P = $12, total revenue is

$1, 080(= $1290). If P = $10, total revenue is $1, 500(= $10150). At P = $8,

total revenue is $1, 680(= $8210). In any case, to determine the profit effects of

appetizer price changes it is necessary to consider revenue and cost implications of

both appetizer and beverage sales.


P4.8 Ironside Industries, Inc., is a leading manufacturer of tufted carpeting under the Ironside

brand. Demand for Ironside’s products is closely tied to the overall pace of building and

remodeling activity and, therefore, is highly sensitive to changes in national income. The

carpet manufacturing industry is highly competitive, so Ironside’s demand is also very

price-sensitive. During the past year, Ironside sold 30 million square yards (units) of

carpeting at an average wholesale price of $15.50 per unit. This year, household income

is expected to surge from $55,500 to $58,500 per year in a booming economic recovery.

A. Without any price change, Ironside’s marketing director expects current-year sales

to soar to 50 million units because of rising income. Calculate the implied income

arc elasticity of demand.

B. Given the projected rise in income, the marketing director believes that a volume of

30 million units could be maintained despite an increase in price of $1 per unit. On

this basis, calculate the implied arc price elasticity of demand.

C. Holding all else equal, would a further increase in price result in higher or lower total

revenue?

SOLUTION

A.

εM =∆Q

∆M

M1 + M2

Q1 + Q2

=50− 30

58, 500− 55, 500

58, 500 + 55, 500

50 + 30= 9.5

B. Without a price increase, sales this year would total 50 million units. Therefore, it

is appropriate to estimate the arc price elasticity from a before-price-increase base

of 50 million units:

εp =∆Q

∆PP2 + P1Q1 + Q2 =

30− 50

16.50− 15.5016.50 + 15.5030 + 50 = −8

C. Lower. Since carpet demand is in the elastic range, εp = 8, an increase (decrease) in

price will result in lower (higher) total revenues.

P4.9 B. B. Lean is a catalog retailer of a wide variety of sporting goods and recreational

products. Although the market response to the company’s spring catalog was generally

good, sales of B. B. Lean’s $140 deluxe garment bag declined from 10,000 to 4,800 units.

During this period, a competitor offered a whopping $52 off their regular $137 price on

deluxe garment bags.


A. Calculate the arc cross-price elasticity of demand for B. B. Lean’s deluxe garment

bag.

B. B. B. Lean’s deluxe garment bag sales recovered from 4,800 units to 6,000 units

following a price reduction to $130 per unit. Calculate B. B. Lean’s arc price elasticity

of demand for this product.

C. Assuming the same arc price elasticity of demand calculated in Part B, determine

the further price reduction necessary for B. B. Lean to fully recover lost sales (i.e.,

regain a volume of 10,000 units).

SOLUTION

A.

εx =4, 800− 10, 000

85− 137

85 + 137

4, 800 + 10, 000= 1.5 (Substitutes)

B.

εp =6, 000− 4, 800

130− 140

130 + 140

6, 000 + 4, 800= −3 (Elastic)

C.

εp = −3 =10, 000− 6, 000

P2 − 130

P2 + 130

10, 000 + 6, 000

which implies that

−12P2 + 1, 560 = P2 + 130

13P2 = 1, 430

P2 = $110

This implies a further price reduction of $20 because:

∆P = $130− $110 = $20

P4.10 Enchantment Cosmetics, Inc., offers a line of cosmetic and perfume products marketed

through leading department stores. Product Manager Erica Kane recently raised the

suggested retail price on a popular line of mascara products from $9 to $12 following

increases in the costs of labor and materials. Unfortunately, sales dropped sharply from

16,200 to 9,000 units per month. In an effort to regain lost sales, Enchantment ran a

coupon promotion featuring $5 off the new regular price. Coupon printing and distribution

costs totaled $500 per month and represented a substantial increase over the typical


advertising budget of $3,250 per month. Despite these added costs, the promotion was

judged to be a success, as it proved to be highly popular with consumers. In the period

prior to expiration, coupons were used on 40% of all purchases and monthly sales rose to

15,000 units.

A. Calculate the arc price elasticity implied by the initial response to the Enchantment

price increase.

B. Calculate the effective price reduction resulting from the coupon promotion.

C. In light of the price reduction associated with the coupon promotion and assuming

no change in the price elasticity of demand, calculate Enchantment’s arc advertising

elasticity.

D. Why might the true arc advertising elasticity differ from that calculated in part C?

SOLUTION

A.

εp =∆Q

∆P

P1 + P2

Q1 + Q2

=9, 000− 16, 200

12− 9

12 + 9

9, 000 + 16, 200= −2

B. The effective price reduction is $2 since 40% of sales are accompanied by a coupon:

∆P = −$5(0.4) = −2 or P2 = $12− $5(0.4) = 10

∆P = $10− $12 = −$2

C. To calculate the arc advertising elasticity, the effect of the $2 price cut implicit in

the coupon promotion must first be reflected. With just a price cut, the quantity

demanded would rise to 13,000, because:

εp =Q∗ −Q1

P2 − P1

P2 + P1

Q∗ + Q1

−2 =Q∗ − 9, 000

10− 12

10 + 12

Q∗ + 9, 000

−2 =−11(Q∗ − 9, 000

Q∗ + 9, 000

−2(Q∗ + 9, 000) = −11(Q∗ − 9, 000)

9Q∗ = 117, 000

Q∗ = 13, 000


Then, the arc advertising elasticity can be calculated as:

εA =Q1 −Q∗

A2 − A1

A1 + A2

Q2 + Q∗

=15, 000− 13, 000

3, 750− 3, 250

3, 750 + 3, 250

15, 000 + 13, 000= 1

D. It is important to recognize that a coupon promotion can involve more than just

the independent effects of a price cut plus an increase in advertising as is implied in

Part C. Synergistic or interactive effects may increase advertising effectiveness when

the promotion is accompanied by a price cut. Similarly, price reductions can have a

much larger impact when advertised. In addition, a coupon is a price cut for only the

most price sensitive (coupon-using) customers, and may spur sales by much more

than a dollar equivalent across-the-board price cut. Synergy between advertising

and the implicit price reduction that accompanies a coupon promotion can cause

the estimate in Part C to overstate the true advertising elasticity. Similarly, this

advertising elasticity will be overstated to the extent that targeted price cuts have

a bigger influence on the quantity demanded than similar across-the-board price

reductions, as seems likely.

Demand Elasticity

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