### Managerial Economics & Business Strategy

```Managerial Economics & Business
Strategy
Chapter 3
Quantitative
Demand Analysis
McGraw-Hill/Irwin
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”
EG , S
%G

%S
If EG,S > 0, then S and G are directly related.
If EG,S < 0, then S and G are inversely related.
If EG,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
EG , S
dG S

dS G
If EG,S > 0, then S and G are directly related.
If EG,S < 0, then S and G are inversely related.
If EG,S = 0, then S and G are unrelated.
3-4
Own Price Elasticity of Demand
EQX , PX
%QX

%PX
d
 Negative according to the “law of demand.”
Elastic:
EQ X , PX  1
Inelastic: EQ X , PX  1
Unitary:
EQ X , PX  1
3-5
Perfectly Elastic & Inelastic Demand
Price
Price
D
D
Quantity
PerfectlyElastic(EQX ,PX  )
Quantity
PerfectlyInelastic( EQX , PX  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
Elasticity, Total Revenue
and Linear Demand
P
100
TR
0
10
20
30
40
50
Q
0
Q
3-8
Elasticity, Total Revenue
and Linear Demand
P
100
TR
80
800
0
10
20
30
40
50
Q
0
10
20
30
40
50
Q
3-9
Elasticity, Total Revenue
and Linear Demand
P
100
TR
80
1200
60
800
0
10
20
30
40
50
Q
0
10
20
30
40
50
Q
3-10
Elasticity, Total Revenue
and Linear Demand
P
100
TR
80
1200
60
40
800
0
10
20
30
40
50
Q
0
10
20
30
40
50
Q
3-11
Elasticity, Total Revenue
and Linear Demand
P
100
TR
80
1200
60
40
800
20
0
10
20
30
40
50
Q
0
10
20
30
40
50
Q
3-12
Elasticity, Total Revenue
and Linear Demand
P
100
TR
Elastic
80
1200
60
40
800
20
0
10
20
30
40
50
Q
0
10
20
30
40
50
Q
Elastic
3-13
Elasticity, Total Revenue
and Linear Demand
P
100
TR
Elastic
80
1200
60
Inelastic
40
800
20
0
10
20
30
40
50
Q
0
10
Elastic
20
30
40
50
Q
Inelastic
3-14
Elasticity, Total Revenue
and Linear Demand
P
100
TR
Unit elastic
Elastic
Unit elastic
80
1200
60
Inelastic
40
800
20
0
10
20
30
40
50
Q
0
10
Elastic
20
30
40
50
Q
Inelastic
3-15
Demand, Marginal Revenue (MR)
and Elasticity
 For a linear
inverse demand
function, MR(Q) =
a + 2bQ, where b
< 0.
 When
P
100
Elastic
Unit elastic
80
60
Inelastic
40
20
0
10
20
40
MR
50
Q
– MR > 0, demand is
elastic;
– MR = 0, demand is
unit elastic;
– MR < 0, demand is
inelastic.
3-16
Total Revenue Test
 TRT can help
manage cash
flows.
 Should a
company
increase prices
to boost cash
flow or cut prices
and make it up in
volume?
EQX , PX
%QX

%PX
d
3-17
TRT
 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
Time
Expenditure Share
3-19
Mid-Point Formula
 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
EQX , PY
%QX

%PY
d
If EQX,PY > 0, then X and Y are substitutes.
If EQX,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  RX 1  EQX , PX  RY EQY ,PX  %PX
3-23
Example
 Suppose a diner earns \$5000/wk selling
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
3-24
Solution
 [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
EQX , M
%QX

%M
d
If EQX,M > 0, then X is a normal good.
If EQX,M < 0, then X is a inferior good.
3-26
Income Elasticities
 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.
 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
 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
Calls would increase by 25.92 percent!
EQX , PX
%QX
 8.64 
%PX
d
%QX
 8.64 
 3%
d
 3%   8.64  %QX
d
%QX  25.92%
d
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
AT&T’s demand would fall by 36.24 percent!
EQX , PY
%QX
 9.06 
%PY
d
%QX
9.06 
 4%
d
 4%  9.06  %QX
d
%QX  36.24%
d
3-34
Interpreting Demand Functions
 Mathematical representations of demand
curves.
 Example:
QX  10  2PX  3PY  2M
d
– 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).
3-35
Linear Demand Functions and
Elasticities
 General Linear Demand Function and
Elasticities:
QX  0   X PX  Y PY  M M  H H
d
P
EQX , PX   X X
QX
Own Price
Elasticity
EQX , PY
PY
 Y
QX
Cross Price
Elasticity
M
EQX , M   M
QX
Income
Elasticity
3-36
Example of Linear Demand




Qd = 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 QX d  0   X ln PX  Y ln PY  M ln M  H ln H
Own PriceElasticity:  X
Cross PriceElasticity:  Y
IncomeElasticity:
M
3-38
Example of Log-Linear Demand
 ln(Qd) = 10 - 2 ln(P).
 Own Price Elasticity: -2.
3-39
Graphical Representation of
Linear and Log-Linear Demand
P
P
D
Linear
D
Q
Log Linear
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:
Yˆ  aˆ  bˆX
– Confidence Intervals.
– t-statistic.
– R-square 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.
 R2 = Explained variation/total variation
 R2 = SSregression / SStotal
 Subjective measure of goodness of fit.
 Remember! degrees of freedom
 Adjusted R2 better indicator of GOF.
 AdjR2 = 1 – (1 – R2) [(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 Qx  0   x ln Px  e
3-46
Summary Output
Regression Statistics
Multiple R
0.41
R Square
0.17
0.15
Standard Error
0.68
Observations
41.00
ANOVA
df
Regression
Residual
Total
Intercept
ln(P)
SS
1.00
39.00
40.00
MS
F
3.65
18.13
21.78
Coefficients Standard Error
7.58
1.43
-0.84
0.30
3.65
0.46
t Stat
5.29
-2.80
Significance F
7.85
0.01
P-value
0.000005
0.007868
Lower 95%
Upper 95%
4.68
10.48
-1.44
-0.23
3-47
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.
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