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Managerial Economics &
Business Strategy
Chapter 3
Quantitative Demand Analysis
McGraw-Hill/IrwinMichael R. Baye, Managerial Economics andBusiness Strategy Copyright 2008 by the McGraw-Hill Companies, Inc. All rights reserved.
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Overview
I. The Elasticity Concept Own Price Elasticity
Elasticity and Total Revenue
Cross-Price Elasticity
Income Elasticity
II. Demand Functions Linear
Log-Linear
III. Regression Analysis
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The Elasticity Concept
How responsive is variable G to a change
in variable S
IfEG,S > 0, then S and G are directly related.
IfEG,S < 0, then S and G are inversely related.
S
GE SG
%
%,
IfEG,S = 0, then S and G are unrelated.
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The Elasticity Concept Using
Calculus An alternative way to measure the elasticity
of a function G = f(S) is
G
S
dS
dGE SG ,
IfEG,S > 0, then S and G are directly related.
IfEG,S < 0, then S and G are inversely related.
IfEG,S = 0, then S and G are unrelated.
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Own Price Elasticity of
Demand
Negativeaccording to the law of demand.
Elastic:Inelastic:
Unitary:
X
d
X
PQ
P
QE
XX
%
%,
1,
XX PQE
1, XX PQE
1, XX PQE
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Perfectly Elastic &
Inelastic Demand
)(ElasticPerfectly , XX PQE
D
Price
Quantity
D
Price
Quantity
)0(InelasticPerfectly , XX PQE
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Own-Price Elasticity
and Total Revenue Elastic
Increase (a decrease) in price leads to a decrease (anincrease) in total revenue.
Inelastic Increase (a decrease) in price leads to an increase (a
decrease) in total revenue.
Unitary Total revenue is maximized at the point where
demand is unitary elastic.
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Elasticity, Total Revenue and Linear
Demand
QQ
PTR
100
0 010 20 30 40 50
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Elasticity, Total Revenue and Linear
Demand
QQ
PTR
100
0 10 20 30 40 50
80
800
0 10 20 30 40 50
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Elasticity, Total Revenue and Linear
Demand
QQ
PTR
100
80
800
60 1200
0 10 20 30 40 500 10 20 30 40 50
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Elasticity, Total Revenue and Linear
Demand
QQ
PTR
100
80
800
60 1200
40
0 10 20 30 40 500 10 20 30 40 50
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Elasticity, Total Revenue and Linear
Demand
QQ
PTR
100
80
800
60 1200
40
20
0 10 20 30 40 500 10 20 30 40 50
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Elasticity, Total Revenue and Linear
Demand
QQ
PTR
100
80
800
60 1200
40
20
Elastic
Elastic
0 10 20 30 40 500 10 20 30 40 50
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Elasticity, Total Revenue and Linear
Demand
QQ
PTR
100
80
800
60 1200
40
20
Inelastic
Elastic
Elastic Inelastic
0 10 20 30 40 500 10 20 30 40 50
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Elasticity, Total Revenue and Linear
Demand
QQ
P TR
100
80
800
60 1200
40
20
Inelastic
Elastic
Elastic Inelastic
0 10 20 30 40 500 10 20 30 40 50
Unit elastic
Unit elastic
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Demand, Marginal Revenue (MR)
and Elasticity For a linear inverse
demand function,MR(Q) = a + 2bQ,where b < 0.
When MR > 0, demand is
elastic;
MR = 0, demand is unitelastic;
MR < 0, demand isinelastic.
Q
P
100
80
60
40
20
Inelastic
Elastic
0 10 20 40 50
Unit elastic
MR
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Factors Affecting
Own Price Elasticity
Available Substitutes
The more substitutes available for the good, the more
elastic the demand.
Time
Demand tends to be more inelastic in the short term than
in the long term.
Time allows consumers to seek out available substitutes.
Expenditure Share
Goods that comprise a small share of consumers
budgets tend to be more inelastic than goods for which
consumers spend a large portion of their incomes.
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Cross Price Elasticity of
Demand
IfEQX,PY
> 0, thenXand Yare substitutes.
IfEQX,PY< 0, thenXand Yare complements.
Y
d
X
PQ P
QE
YX
%
%
,
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Income Elasticity
IfEQX
,M> 0, thenXis a normal good.
IfEQX,M< 0, thenXis a inferior good.
M
QE
d
X
MQX
%
%,
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Uses of Elasticities
Pricing.
Managing cash flows.
Impact of changes in competitors prices.
Impact of economic booms and recessions.
Impact of advertising campaigns.
And lots more!
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Example 1: Pricing and Cash
Flows According to an FTC Report by Michael
Ward, AT&Ts own price elasticity of
demand for long distance services is -8.64. AT&T needs to boost revenues in order to
meet its marketing goals.
To accomplish this goal, should AT&Traise or lower its price?
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Answer: Lower price!
Since demand is elastic, a reduction in price
will increase quantity demanded by a
greater percentage than the price decline,resulting in more revenues for AT&T.
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Example 2: Quantifying the
Change IfAT&T lowered price by 3 percent, what
would happen to the volume of long
distance telephone calls routed throughAT&T?
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Answer
Calls would increase by 25.92 percent.
%92.25%
%64.8%3
%3
%64.8
%
%64.8,
d
X
dX
d
X
X
d
XPQ
Q
Q
Q
P
QE
XX
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Example 3: Impact of a change
in a competitors price According to an FTC Report by Michael
Ward, AT&Ts cross price elasticity of
demand for long distance services is 9.06.
Ifcompetitors reduced their prices by 4
percent, what would happen to the demand
for AT&T services?
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Answer
AT&Ts demand would fall by 36.24 percent.
%24.36%
%06.9%4
%4
%06.9
%
%06.9,
d
X
dX
d
X
Y
d
XPQ
Q
Q
Q
P
QE
YX
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Interpreting Demand Functions
Mathematical representations of demand curves.
Example:
MPPQ YXd
X 23210
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Law of demand holds (coefficient of PX is negative).
X and Y are substitutes (coefficient of PY is positive). X is an inferior good (coefficient of M is negative).
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Linear Demand Functions and
Elasticities General Linear Demand Function and Elasticities:
HMPPQ HMYYXXd
X 0
Own Price
Elasticity
Cross Price
Elasticity
Income
Elasticity
X
X
XPQ Q
PE
XX
,X
MMQ
Q
ME
X
,X
YYPQ
Q
PE
YX,
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Example of Linear Demand
Qd = 10 - 2P.
Own-Price Elasticity: (-2)P/Q.
IfP=1, Q=8 (since 10 - 2 = 8).
Own price elasticity at P=1, Q=8:
(-2)(1)/8= - 0.25.
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0ln ln ln ln ln
d
X X X Y Y M HQ P P M H
M
Y
X
:ElasticityIncome
:ElasticityPriceCross
:ElasticityPriceOwn
Log-Linear Demand
General Log-Linear Demand Function:
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Example of Log-Linear
Demand ln(Qd) = 10 - 2 ln(P).
Own Price Elasticity: -2.
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P
Q Q
D D
Linear Log Linear
Graphical Representation of
Linear and Log-Linear Demand
P
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Regression Analysis
One use is for estimating demand functions.
Important terminology and concepts:
Least Squares Regression model: Y = a + bX + e. Least Squares Regression line:
Confidence Intervals.
t-statistic.
R-square or Coefficient of Determination.
F-statistic.
XbaY
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An Example
Use a spreadsheet to estimate the following
log-linear demand function.
0ln lnx x xQ P e
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Summary Output
Regression Statistics
Multiple R 0.41
R Square 0.17
Adjusted R Square 0.15
Standard Error 0.68
Observations 41.00
ANOVA
df SS MS F Significance F
Regression 1.00 3.65 3.65 7.85 0.01
Residual 39.00 18.13 0.46
Total 40.00 21.78
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 7.58 1.43 5.29 0.000005 4.68 10.48
ln(P) -0.84 0.30 -2.80 0.007868 -1.44 -0.23
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Interpreting the Regression
Output The estimated log-linear demand function is:
ln(Qx)= 7.58 - 0.84 ln(Px).
Own price elasticity: -0.84 (inelastic). How good is our estimate?
t-statistics of 5.29 and -2.80 indicate that the estimated
coefficients are statistically different from zero.
R-square of 0.17 indicates the ln(PX) variable explains
only 17 percent of the variation in ln(Qx).
F-statistic significant at the 1 percent level.
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Conclusion
Elasticities are tools you can use to quantifythe impact of changes in prices, income, andadvertising on sales and revenues.
Given market or survey data, regressionanalysis 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.
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