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CHAPTER 3
Quantitative Demand
Analysis
McGraw-Hill/Irwin
Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter Outline
Chapter Overview
• The elasticity concept
• Own price elasticity of demand
– Elasticity and total revenue
– Factors affecting the own price elasticity of demand
– Marginal revenue and the own price elasticity of demand
• Cross-price elasticity
– Revenue changes with multiple products
• Income elasticity
• Other Elasticities
– Linear demand functions
– Nonlinear demand functions
• Obtaining elasticities from demand functions
– Elasticities for linear demand functions
– Elasticities for nonlinear demand functions
• Regression Analysis
– Statistical significance of estimated coefficients
– Overall fit of regression line
– Regression for nonlinear functions and multiple regression
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Introduction
Chapter Overview
• Chapter 2 focused on interpreting demand
functions in qualitative terms:
– An increase in the price of a good leads quantity
demanded for that good to decline.
– A decrease in income leads demand for a normal
good to decline.
• This chapter examines the magnitude of
changes using the elasticity concept, and
introduces regression analysis to measure
different elasticities.
3-3
The Elasticity Concept
The Elasticity Concept
• Elasticity
– Measures the responsiveness of a percentage
change in one variable resulting from a
percentage change in another variable.
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The Elasticity Concept
The Elasticity Formula
• The elasticity between two variables, 𝐺 and 𝑆,
is mathematically expressed as:
%Δ𝐺
𝐸𝐺,𝑆 =
%Δ𝑆
• When a functional relationship exists, like 𝐺 =
𝑓 𝑆 , the elasticity is:
𝑑𝐺 𝑆
𝐸𝐺,𝑆 =
𝑑𝑆 𝐺
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The Elasticity Concept
Measurement Aspects of Elasticity
• Important aspects of the elasticity:
– Sign of the relationship:
• Positive.
• Negative.
– Absolute value of elasticity magnitude relative to
unity:
• 𝐸𝐺,𝑆 > 1 𝐺 is highly responsive to changes in 𝑆.
• 𝐸𝐺,𝑆 < 1 𝐺 is slightly responsive to changes in 𝑆.
3-6
Own Price Elasticity of Demand
Own Price Elasticity
• Own price elasticity of demand
– Measures the responsiveness of a percentage change
in the quantity demanded of good X to a percentage
change in its price.
𝑑
𝐸𝑄
𝑋
𝑑
,𝑃𝑋
%Δ𝑄𝑋
=
%Δ𝑃𝑋
– Sign: negative by law of demand.
– Magnitude of absolute value relative to unity:
• 𝐸𝑄
𝑋
• 𝐸𝑄
𝑋
• 𝐸𝑄
𝑋
𝑑
,𝑃𝑋
𝑑
,𝑃𝑋
𝑑
,𝑃𝑋
> 1: Elastic.
< 1: Inelastic.
= 1: Unitary elastic.
3-7
Own Price Elasticity of Demand
Linear Demand, Elasticity, and Revenue
Linear Inverse Demand: 𝑃 = 40 − 0.5𝑄
Demand: 𝑄 = 80 − 2𝑄
$30
$20 × 60
20
40 = $600
$800
• Revenue = $10
Price
$40
$35
$30
$20
$10
−3
−1
• Elasticity: −2 × 60
= −0.333
20
40
elastic.
unitary elastic.
• Conclusion: Demand is inelastic.
$30
$25
Observation: Elasticity
varies along a linear
(inverse) demand curve
$20
$15
$10
$5
Demand
0
10 20 30
40
50
60
70
80
Quantity
3-8
Own Price Elasticity of Demand
Total Revenue Test
• When demand is elastic:
– A price increase (decrease) leads to a decrease
(increase) in total revenue.
• When demand is inelastic:
– A price increase (decrease) leads to an increase
(decrease) in total revenue.
• When demand is unitary elastic:
– Total revenue is maximized.
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Own Price Elasticity of Demand
Extreme Elasticities
Price
Demand
𝐸𝑄𝑋 𝑑 ,𝑃𝑋 = 0
Perfectly
elastic
Demand
𝐸𝑄𝑋𝑑 ,𝑃𝑋 = −∞
Perfectly Inelastic
Quantity
3-10
Own Price Elasticity of Demand
Factors Affecting the Own Price Elasticity
• Three factors can impact the own price
elasticity of demand:
– Availability of consumption substitutes.
– Time/Duration of purchase horizon.
– Expenditure share of consumers’ budgets.
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Own Price Elasticity of Demand
Elasticity and Marginal Revenue
• The marginal revenue can be derived from a
market demand curve.
– Marginal revenue measures the additional revenue
due to a change in output.
• This link relates marginal revenue to the own
price elasticity of demand as follows:
1+𝐸
𝑀𝑅 = 𝑃
𝐸
– When −∞ < 𝐸 < −1 then, 𝑀𝑅 > 0.
– When 𝐸 = −1 then, 𝑀𝑅 = 0.
– When −1 < 𝐸 < 0 then, 𝑀𝑅 < 0.
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Own Price Elasticity of Demand
Demand and Marginal Revenue
Price
6
Unitary
𝑃
MR
Demand
0
1
3
6
Quantity
Marginal Revenue (MR)
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Cross-Price Elasticity
Cross-Price Elasticity
• Cross-price elasticity
– Measures responsiveness of a percent change in
demand for good X due to a percent change in the
price of good Y.
𝑑
𝐸𝑄
𝑋
– If 𝐸𝑄
𝑋 ,𝑃𝑌
– If 𝐸𝑄
𝑋
𝑑
𝑑
,𝑃𝑌
𝑑
,𝑃𝑌
%Δ𝑄𝑋
=
%Δ𝑃𝑌
> 0, then 𝑋 and 𝑌 are substitutes.
< 0, then 𝑋 and 𝑌 are complements.
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Cross-Price Elasticity
Cross-Price Elasticity in Action
• Suppose it is estimated that the cross-price
elasticity of demand between clothing and
food is -0.18. If the price of food is projected
to increase by 10 percent, by how much will
demand for clothing change?
%∆𝑄𝐶𝑙𝑜𝑡ℎ𝑖𝑛𝑔 𝑑
𝑑
−0.18 =
⇒ %∆𝑄𝐶𝑙𝑜𝑡ℎ𝑖𝑛𝑔 = −1.8
10
– That is, demand for clothing is expected to decline
by 1.8 percent when the price of food increases
10 percent.
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Cross-Price Elasticity
Cross-Price Elasticity
• Cross-price elasticity is important for firms
selling multiple products.
– Price changes for one product impact demand for
other products.
• Assessing the overall change in revenue from
a price change for one good when a firm sells
two goods is:
∆𝑅 = 𝑅𝑋 1 + 𝐸𝑄
𝑋
𝑑
,𝑃𝑋
+ 𝑅𝑌 𝐸𝑄
𝑌
𝑑
,𝑃𝑋
× %∆𝑃𝑋
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Cross-Price Elasticity
Cross-Price Elasticity in Action
• Suppose a restaurant earns $4,000 per week in
revenues from hamburger sales (X) and $2,000
per week from soda sales (Y). If the own price
elasticity for burgers is 𝐸𝑄𝑋 ,𝑃𝑋 = −1.5 and the
cross-price elasticity of demand between sodas
and hamburgers is 𝐸𝑄𝑌,𝑃𝑋 = −4.0, what would
happen to the firm’s total revenues if it reduced
the price of hamburgers by 1 percent?
∆𝑅 = $4,000 1 − 1.5 + $2,000 −4.0 −1%
= $100
– That is, lowering the price of hamburgers 1 percent
increases total revenue by $100.
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Income Elasticity
Income Elasticity
• Income elasticity
– Measures responsiveness of a percent change in
demand for good X due to a percent change in
income.
𝑑
𝐸𝑄
𝑋
– If 𝐸𝑄
– If 𝐸𝑄
𝑑
,𝑀
%Δ𝑄𝑋
=
%Δ𝑀
𝑑
> 0, then 𝑋 is a normal good.
𝑑
< 0, then 𝑋 is an inferior good.
𝑋 ,𝑀
𝑋 ,𝑀
3-18
Income Elasticity
Income Elasticity in Action
• Suppose that the income elasticity of demand for
transportation is estimated to be 1.80. If income
is projected to decrease by 15 percent,
• What is the impact on the demand for
transportation?
𝑑
%Δ𝑄𝑋
1.8 =
−15
– Demand for transportation will decline by 27 percent.
• Is transportation a normal or inferior good?
– Since demand decreases as income declines,
transportation is a normal good.
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Other Elasticities
Other Elasticities
• Own advertising elasticity of demand for good X
is the ratio of the percentage change in the
consumption of X to the percentage change in
advertising spent on X.
• Cross-advertising elasticity between goods X
and Y would measure the percentage change in
the consumption of X that results from a 1
percent change in advertising toward Y.
3-20
Obtaining Elasticities From Demand Functions
Elasticities for Linear Demand Functions
• From a linear demand function, we can easily
compute various elasticities.
• Given a linear demand function:
𝑑
𝑄𝑋 = 𝛼0 + 𝛼𝑋 𝑃𝑋 + 𝛼𝑌 𝑃𝑌 + 𝛼𝑀 𝑀 + 𝛼𝐻 𝑃𝐻
– Own price elasticity: 𝛼𝑋
𝑃𝑋
𝑄𝑋
– Cross price elasticity: 𝛼𝑌
– Income elasticity: 𝛼𝑀
𝑑
.
𝑃𝑌
𝑄𝑋
𝑀
𝑄𝑋 𝑑
𝑑
.
.
3-21
Obtaining Elasticities From Demand Functions
Elasticities for Linear Demand Functions In Action
• The daily demand for Invigorated PED shoes is estimated to
be
𝑄𝑋 𝑑 = 100 − 3𝑃𝑋 + 4𝑃𝑌 − 0.01𝑀 + 2𝑃𝐴𝑋
Suppose good X sells at $25 a pair, good Y sells at $35, the
company utilizes 50 units of advertising, and average
consumer income is $20,000. Calculate the own price,
cross-price and income elasticities of demand.
– 𝑄𝑋 𝑑 = 100 − 3 $25 + 4 $35 − 0.01 $20,000 + 2 50 =
65 units.
25
= −1.15.
65
35
Cross-price elasticity: 4 = 2.15.
65
20,000
Income elasticity: −0.01
= −3.08.
65
– Own price elasticity: −3
–
–
3-22
Obtaining Elasticities From Demand Functions
Elasticities for Nonlinear Demand Functions
• One non-linear demand function is the loglinear demand function:
ln 𝑄𝑋 𝑑
= 𝛽0 + 𝛽𝑋 ln 𝑃𝑋 + 𝛽𝑌 ln 𝑃𝑌 + 𝛽𝑀 ln 𝑀 + 𝛽𝐻 ln 𝐻
– Own price elasticity: 𝛽𝑋 .
– Cross price elasticity: 𝛽𝑌 .
– Income elasticity: 𝛽𝑀 .
3-23
Obtaining Elasticities From Demand Functions
Elasticities for Nonlinear Demand Functions
In Action
• An analyst for a major apparel company estimates that
the demand for its raincoats is given by
𝑙𝑛 𝑄𝑋 𝑑 = 10 − 1.2 ln 𝑃𝑋 + 3 ln 𝑅 − 2 ln 𝐴𝑌
where 𝑅 denotes the daily amount of rainfall and 𝐴𝑌
the level of advertising on good Y. What would be the
impact on demand of a 10 percent increase in the daily
amount of rainfall?
𝐸𝑄
𝑋
𝑑
,𝑅
= 𝛽𝑅 = 3. So, 𝐸𝑄
𝑋
𝑑
,𝑅
=
%∆𝑄𝑋 𝑑
%∆𝑅
⇒3=
%∆𝑄𝑋 𝑑
.
10
A 10 percent increase in rainfall will lead to a 30
percent increase in the demand for raincoats.
3-24
Regression Analysis
Regression Analysis
• How does one obtain information on the
demand function?
– Published studies.
– Hire consultant.
– Statistical technique called regression analysis
using data on quantity, price, income and other
important variables.
3-25
Regression Analysis
Regression Line and Least Squares Regression
• True (or population) regression model
𝑌 = 𝑎 + 𝑏𝑋 + 𝑒
– 𝑎 unknown population intercept parameter.
– 𝑏 unknown population slope parameter.
– 𝑒 random error term with mean zero and standard
deviation 𝜎.
• Least squares regression line
𝑌 = 𝑎 + 𝑏𝑋
– 𝑎 least squares estimate of the unknown parameter 𝑎.
– 𝑏 least squares estimate of the unknown parameter 𝑏.
• The parameter estimates 𝑎 and 𝑏, represent the values
of 𝑎 and 𝑏 that result in the smallest sum of squared
errors between a line and the actual data.
3-26
Regression Analysis
Excel and Least Squares Estimates
SUMMARY
OUTPUT
Regression Statistics
Multiple R
0.87
R Square
0.75
Adjusted R Square
0.72
Standard Error
112.22
Observations
10.00
Estimated Demand:
𝑄 = 1631.47 − 2.60𝑃𝑅𝐼𝐶𝐸
𝑎 = 1631.47
𝑏 = −2.60
ANOVA
Df
Regression
Residual
Total
Intercept
Price
1
8
9
SS
301470.89
100751.61
402222.50
Coefficients Standard Error
1631.47
243.97
-2.60
0.53
MS
301470.89
12593.95
F
Significance F
23.94
0.0012
t Stat
P-value Lower 95% Upper 95%
6.69 0.0002
1068.87 2194.07
-4.89 0.0012
-3.82
-1.37
3-27
Regression Analysis
Evaluating Statistical Significance
• Standard error
– Measure of how much each estimated estimate
varies in regressions based on the same true
demand model using different data.
• Confidence interval rule of thumb
– 𝑎 ± 2𝜎𝑎
– 𝑏 ± 2𝜎𝑏
• t-statistics rule of thumb
– When 𝑡 > 2, we are 95 percent confident the
true parameter is in the regression is not zero.
3-28
Regression Analysis
Excel and Least Squares Estimates
SUMMARY
OUTPUT
Regression Statistics
Multiple R
0.87
R Square
0.75
Adjusted R Square
0.72
Standard Error
112.22
Observations
10.00
𝑠𝑒 (𝑎) = 243.97
𝑠𝑒(𝑏) = 0.53
𝑡𝑎 = 6.69 > 2, the intercept is different
from zero.
𝑡𝑏 = −4.89 < 2, the intercept is different
from zero.
ANOVA
Df
Regression
Residual
Total
Intercept
Price
1
8
9
SS
301470.89
100751.61
402222.50
Coefficients Standard Error
1631.47
243.97
-2.60
0.53
MS
301470.89
12593.95
F
Significance F
23.94
0.0012
t Stat
P-value Lower 95% Upper 95%
6.69 0.0002
1068.87 2194.07
-4.89 0.0012
-3.82
-1.37
3-29
Regression Analysis
Evaluating Overall Regression Line Fit: R- Square
• R-Square
– Also called the coefficient of determination.
– Fraction of the total variation in the dependent
variable that is explained by the regression.
𝐸𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑆𝑆𝑅𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛
𝑅 =
=
𝑇𝑜𝑡𝑎𝑙 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛
𝑆𝑆𝑇𝑜𝑡𝑎𝑙
2
– Ranges between 0 and 1.
• Values closer to 1 indicate “better” fit.
3-30
Regression Analysis
Evaluating Overall Regression Line Fit:
Adjusted R-Square
• Adjusted R-Square
– A version of the R-Square that penalize
researchers for having few degrees of freedom.
𝑛−1
2
2
𝑅 =1− 1−𝑅
𝑛−𝑘
– 𝑛 is total observations.
– 𝑘 is the number of estimated coefficients.
– 𝑛 − 𝑘 is the degrees of freedom for the
regression.
3-31
Regression Analysis
Evaluating Overall Regression Line Fit: F-Statistic
• A measure of the total variation explained by
the regression relative to the total unexplained
variation.
– The greater the F-statistic, the better the overall
regression fit.
– Equivalently, the P-value is another measure of the
F-statistic.
• Lower p-values are associated with better overall
regression fit.
3-32
Regression Analysis
Excel and Least Squares Estimates
SUMMARY
OUTPUT
Regression Statistics
Multiple R
0.87
R Square
0.75
Adjusted R Square
0.72
Standard Error
112.22
Observations
10.00
ANOVA
Df
Regression
Residual
Total
Intercept
Price
1
8
9
SS
301470.89
100751.61
402222.50
Coefficients Standard Error
1631.47
243.97
-2.60
0.53
MS
301470.89
12593.95
F
Significance F
23.94
0.0012
t Stat
P-value Lower 95% Upper 95%
6.69 0.0002
1068.87 2194.07
-4.89 0.0012
-3.82
-1.37
3-33
Regression Analysis
Regression for Nonlinear Functions
and Multiple Regression
• Regression techniques can also be applied to
the following settings:
– Nonlinear functional relationships:
• Nonlinear regression example:
ln 𝑄 = 𝛽0 + 𝛽𝑝 ln 𝑃 + 𝑒
– Functional relationships with multiple variables:
• Multiple regression example:
𝑄𝑋 𝑑 = 𝛼0 + 𝛼𝑋 𝑃𝑋 + 𝛼𝑀 𝑀 + 𝛼𝐻 𝑃𝐻 + 𝑒
or
ln 𝑄𝑋 𝑑 = 𝛽0 + 𝛽𝑋 ln 𝑃𝑋 + 𝛽𝑀 ln 𝑀 + 𝛽𝐻 ln 𝑃𝐻 + 𝑒
3-34
Regression Analysis
Excel and Least Squares Estimates
SUMMARY
OUTPUT
Regression Statistics
Multiple R
0.89
R Square
0.79
Adjusted R Square
0.69
Standard Error
9.18
Observations
10.00
ANOVA
Df
Regression
Residual
Total
Intercept
Price
Advertising
Distance
SS
1920.99
505.91
2426.90
MS
640.33
84.32
Coefficients Standard Error
135.15
20.65
-0.14
0.06
0.54
0.64
-5.78
1.26
t Stat
6.54
-2.41
0.85
-4.61
3
6
9
F
Significance F
7.59
0.182
P-value Lower 95% Upper 95%
84.61
185.68
0.0006
0.0500
-0.29
0.00
0.4296
-1.02
2.09
0.0037
-8.86
-2.71
3-35
Conclusion
• Elasticities are tools you can use to quantify
the impact of changes in prices, income, and
advertising on sales and revenues.
• Given market or survey data, regression
analysis can be used to estimate:
– Demand functions.
– Elasticities.
– A host of other things, including cost functions.
• Managers can quantify the impact of changes
in prices, income, advertising, etc.
3-36