McGraw-Hill/Irwin
Managerial Economics & Business
Strategy
Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved.

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
3-2

The Elasticity Concept
 How responsive is variable “G” to a change in variable “S”
E
G , S

%

G
%

S
If E
G,S
> 0, then S and G are directly related.
If E
G,S
< 0, then S and G are inversely related.
If E
G,S
= 0, then S and G are unrelated.
3-3

The Elasticity Concept Using
Calculus
 An alternative way to measure the elasticity of a function G = f(S) is
E
G , S
 dG dS
S
G
If E
G,S
> 0, then S and G are directly related.
If E
G,S
< 0, then S and G are inversely related.
If E
G,S
= 0, then S and G are unrelated.
3-4

Own Price Elasticity of Demand
E
Q
X
, P
X

%

Q
X
%

P
X d
 Negative according to the “law of demand.”
Elastic:
Inelastic:
E
Q
X
, P
X
E
Q
X
, P
X

1

1
Unitary: E
Q
X
, P
X

1
3-5

Perfectly Elastic & Inelastic Demand
Price
Price
D
D
Quantity
Perfectly Elastic ( E
Q
X
, P
X
 
)
Quantity
Perfectly Inelastic ( E
Q
X
, P
X

0 )
3-6

Own-Price Elasticity and Total Revenue
 Elastic
– Increase (a decrease) in price leads to a decrease (an increase) 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.
3-7

P
100
Elasticity, Total Revenue and Linear Demand
TR
0 10 20 30 40 50 Q 0 Q
3-8

P
100
80
Elasticity, Total Revenue and Linear Demand
TR
0 10 20 30 40 50
Q
800
0 10 20 30 40 50 Q
3-9

Elasticity, Total Revenue and Linear Demand
TR
P
100
80
60 1200
0 10 20 30 40 50
Q
800
0 10 20 30 40 50 Q
3-10

Elasticity, Total Revenue and Linear Demand
TR
P
100
80
60
40
1200
0 10 20 30 40 50
Q
800
0 10 20 30 40 50 Q
3-11

Elasticity, Total Revenue and Linear Demand
P
100
80
60
40
20
0 10 20 30 40 50
Q
TR
1200
800
0 10 20 30 40 50 Q
3-12

Elasticity, Total Revenue and Linear Demand
P
100
80
60
40
20
Elastic
0 10 20 30 40 50
Q
TR
1200
800
0 10 20
Elastic
30 40 50 Q
3-13

Elasticity, Total Revenue and Linear Demand
P
100
80
60
40
20
Elastic
Inelastic
0 10 20 30 40 50
Q
TR
1200
800
0 10 20
Elastic
30 40 50 Q
Inelastic
3-14

Elasticity, Total Revenue and Linear Demand
P
100
80
60
40
20
Elastic
TR
Unit elastic
Inelastic
1200
800
0 10 20 30 40 50
Q
Unit elastic
0 10 20
Elastic
30 40 50 Q
Inelastic
3-15

Demand, Marginal Revenue (MR) and Elasticity
P
100
80
60
40
20
Elastic
0 10 20
Unit elastic
Inelastic
40 50
MR
Q
 For a linear inverse demand function, MR(Q) = a + 2bQ, where b
< 0.
 When
– MR > 0, demand is elastic;
– MR = 0, demand is unit elastic;
– MR < 0, demand is inelastic.
3-16

 TRT can help manage cash flows.
 Should a company increase prices to boost cash flow or cut prices and make it up in volume?
E
Q
X
, P
X

%

Q
X
%

P
X d
3-17

 If elasticity of Demand = -2.3
 Cut prices by 10%
 Will sales increase enough to increase revenues?
 Qd will increase by 23%.
 Since the % decrease in price is< % increase in Qd, TR will increase.
3-18

Factors Affecting the
Own-Price Elasticity
 Available Substitutes
 Broad or narrowly defined categories
 Time
 Expenditure Share
3-19

 For consistency when working from a function whether it is Demand or Supply an average approximation of elasticity is used.
 Ep = Q2-Q1/[(Q2+Q1/2]/P2-P1/[(P2+P1/2]
3-20

Cross-Price Elasticity of Demand
E
Q
X
, P
Y

%

Q
X
%

P
Y d
If E
QX,PY
> 0, then X and Y are substitutes.
If E
QX,PY
< 0, then X and Y are complements.
3-21

Cross-Price Elasticity Examples
 Transportation and recreation = -0.05
 Food and Recreation = 0.15
 Clothing and food = -0.18
3-22

Predicting Revenue Changes from Two Products
Suppose that a firm sells two related goods.
If the price of X changes, then total revenue will change by:

R


R
X

1

E
Q
X
, P
X


R
Y
E
Q
Y
, P
X


%

P
X
3-23

 Suppose a diner earns $5000/wk selling egg salad sandwiches and $3000/wk selling French fries. If own price elasticity for egg salad is -3.2 and cross price elasticity between egg salad and French fries is -0.5 what happens to the firms total revenue if it increased the price of egg salad sandwiches by 5%?
3-24

 [5000 x (1+(-3.2)) +((3000 x (-0.5))] x +5%
 [5000 x (-2.2) – (1500)) x +5%
 [-550 – 75] = -$ 625
3-25

Income Elasticity
E
Q
X
, M

%

Q
X
%

M d
If E
QX,M
> 0, then X is a normal good.
If E
QX,M
< 0, then X is a inferior good.
3-26

 Transportation 1.80
 Food 0.80
 Ground beef, non-fed -1.94
3-27

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!
3-28

Example 1: Pricing and Cash Flows
 According to an FTC Report by Michael
Ward, AT&T’s own price elasticity of demand for long distance services is -8.64.
 AT&T needs to boost revenues in order to meet it’s marketing goals.
 To accomplish this goal, should AT&T raise or lower it’s price?
3-29

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.
3-30

Example 2: Quantifying the Change
 If AT&T lowered price by 3 percent, what would happen to the volume of long distance telephone calls routed through
AT&T?
3-31

Answer: Calls Increase!
Calls would increase by 25.92 percent!
E
Q
X
, P
X
 
8 .
64

%

Q
%
X

P
X d


8 .
64
3 %


%

Q



3 %
8 .
64

X d

%

Q
X d
%

Q
X d 
25 .
92 %
3-32

Example 3: Impact of a Change in a Competitor’s Price
 According to an FTC Report by Michael
Ward, AT&T’s cross price elasticity of demand for long distance services is 9.06.
 If competitors reduced their prices by 4 percent, what would happen to the demand for AT&T services?
3-33

Answer: AT&T’s Demand Falls!
AT&T’s demand would fall by 36.24 percent!
E
Q
X
, P
Y

9 .
06

%

Q
%
X

P
Y d
9 .
06

%


Q
X
4 % d

4 %

9 .
06

%

Q
X d
%

Q
X d  
36 .
24 %
3-34

Interpreting Demand Functions
 Mathematical representations of demand curves.
 Example:
Q
X d 
10

2 P
X

3 P
Y

2 M
– Law of demand holds (coefficient of P
X is negative).
– X and Y are substitutes (coefficient of P
Y is positive).
– X is an inferior good (coefficient of M is negative).
3-35

Linear Demand Functions and
Elasticities
 General Linear Demand Function and
Elasticities:
Q
X d  
0
 
X
P
X
 
Y
P
Y
 
M
M
 
H
H
E
Q
X
, P
X
 
X
P
X
Own Price
Elasticity
Q
X
E
Q
X
, P
Y
 
Y
P
Y
Q
X
Cross Price
Elasticity
 
M
M
E
Q
X
, M
Income
Elasticity
Q
X
3-36

Example of Linear Demand
 Q d = 10 - 2P.
 Own-Price Elasticity: (-2)P/Q.
 If P=1, Q=8 (since 10 - 2 = 8).
 Own price elasticity at P=1, Q=8:
(-2)(1)/8= - 0.25.
3-37

Log-Linear Demand
 General Log-Linear Demand Function: ln Q
X d  
0
 
X ln P
X
 
Y ln P
Y
 
M ln M
 
H ln H
Own Price Elasticity :
Cross Price Elasticity :
Income Elasticity :

X

Y

M
3-38

Example of Log-Linear Demand
 ln(Q d ) = 10 - 2 ln(P).
 Own Price Elasticity: -2.
3-39

P
Graphical Representation of
Linear and Log-Linear Demand
P
Linear
D
Q
Log Linear
D
Q
3-40

Regression Analysis
 One use is for estimating demand functions.
 Econometrics – statistical analysis of economic phenomena
 Important terminology and concepts:
– Least Squares Regression model:
– Y = a + bX + e.
– Least Squares Regression line:
– Confidence Intervals.
– t -statistic.
Y
ˆ  a
 ˆ
X
– Rsquare or Coefficient of Determination.
– F -statistic.
– Causality versus Correlation
3-41

Regression Analysis
 Standard error is a measure of how much each estimated coefficient would vary in regressions based on the same underlying true demand relation, but with different observations.
 LSE are unbiased estimators of the true parameters whenever the errors have a zero mean and are iid.
 If that is the case then C.I.s can be constructed
3-42

Evaluating Statistical Significance
 Confidence intervals:
 90% C.I.  a +/- 1 SE of the estimate
 95% C.I.  a +/- 2 SE of the estimate
 99% C.I.  a +/- 3 SE of the estimate
 T statistic: ratio of the value of the parameter estimate to its SE.
 When the absolute value of the t-statistic is >2 one can be 95% confident that the true value of the underlying parameter is not zero.
3-43

Evaluating Statistical Significance
 R-squared – coefficient of determination.
Fraction of the total variation in the dependent variable explained by the regression.
 R 2 = Explained variation/total variation
 R 2 = SS regression
/ SS total
 Subjective measure of goodness of fit.
 Remember! degrees of freedom
 Adjusted R 2 better indicator of GOF.
 AdjR 2 = 1 – (1 – R 2 ) [(n-1)/(n-k)]
3-44

Evaluating Statistical Significance
 F statistic – alternative measure of GOF.
Provides a measure of total variation explained by the regression relative to the total unexplained variation.
 Larger the F-stat the better the overall fit of the regression line to the data.
3-45

An Example
 Use a spreadsheet to estimate the following log-linear demand function.
ln Q x
 
0
  x ln P x
 e
3-46

Summary Output
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.41
0.17
0.15
0.68
41.00
ANOVA
Regression
Residual
Total
Intercept ln(P) df
1.00
39.00
40.00
SS
3.65
18.13
21.78
Coefficients Standard Error
7.58
-0.84
1.43
0.30
MS
3.65
0.46
F
7.85
Significance F
0.01
t Stat
5.29
-2.80
P-value
0.000005
0.007868
Lower 95% Upper 95%
4.68
-1.44
10.48
-0.23
3-47

Interpreting the Regression Output
 The estimated log-linear demand function is:
– ln(Q x
) = 7.58 - 0.84 ln(P x
).
– 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(P
X
) variable explains only 17 percent of the variation in ln(Q x
).
– F -statistic significant at the 1 percent level.
3-48

Multiple Regression
 MR – regressions of a dependent variable on multiple independent variables.
 Caveat: beware of using regression indiscriminately.
 Issues: Heteroskedacity, Multi-colinearity, etc.
3-49

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.
3-50