McGraw-Hill/Irwin

Managerial Economics & Business Strategy

Chapter 3 Quantitative Demand Analysis

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

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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.

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 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.

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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

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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 )

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 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.

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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

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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

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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

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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 Elastic 20 30 40 50

Q

Inelastic

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Elasticity, Total Revenue and Linear Demand

P

100 80 60 40 20 Elastic

TR

Unit elastic Inelastic 1200 0 10 20 30 40 50

Q

800 Unit elastic 0 10 Elastic 20 30 40 50

Q

Inelastic

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Demand, Marginal Revenue (MR) and Elasticity

P

100 80 60 40 20 0 10 Elastic 20 Unit elastic 40 MR Inelastic 50

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.

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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?

E Q X

,

P X

 % 

Q X

% 

P X 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.

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

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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]

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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.

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Cross-Price Elasticity Examples    Transportation and recreation = -0.05

Food and Recreation = 0.15

Clothing and food = -0.18

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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

Example

 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

Solution

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

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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.

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Income Elasticities

   Transportation 1.80

Food 0.80

Ground beef, non-fed -1.94

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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!

3-28

Example 1: Pricing and Cash Flows   AT&T needs to boost revenues in order to meet it’s marketing goals.

 According to an FTC Report by Michael Ward, AT&T’s own price elasticity of demand for long distance services is -8.64. To accomplish this goal, should AT&T raise or lower it’s 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.

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 X

 3 % 8 .

64  

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

E Q X

,

M

 

M M

Income Elasticity

Q X

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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.

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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.

– – –

R F

square or Coefficient of Determination.

-statistic.

Causality versus Correlation

Y

ˆ   ˆ

X

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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

95% C.I. 

a

99% C.I. 

a

+/- 1 SE of the estimate +/- 2 SE of the estimate +/- 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)]

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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

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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%

4.68

-1.44

Upper 95%

10.48

-0.23

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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.

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Multiple Regression  MR – regressions of a dependent variable on multiple independent variables. 

Caveat:

beware of using regression indiscriminately.

 Issues: Heteroskedacity, Multi-colinearity, etc.

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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.

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