CHAPTER 3 Quantitative Demand Analysis McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved

CHAPTER 3

Quantitative Demand Analysis

McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.

Chapter Outline• 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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Chapter Overview

Introduction• 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.

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Chapter Overview

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, and , is

mathematically expressed as:

• When a functional relationship exists, like , the elasticity is:

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Measurement Aspects of Elasticity• Important aspects of the elasticity:– Sign of the relationship: • Positive. • Negative.

– Absolute value of elasticity magnitude relative to unity:• is highly responsive to changes in .• is slightly responsive to changes in .

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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.

– Sign: negative by law of demand.– Magnitude of absolute value relative to unity:

• : Elastic.• : Inelastic.• : Unitary elastic.

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Own Price Elasticity of Demand

Linear Demand, Elasticity, and Revenue

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Quantity

Price

Demand

$40

0

$20

$10

20 30

$5

40

$15

$30

$25

$35

10 50 60 70 80

Linear Inverse Demand: Demand:

• Revenue = $• Elasticity: • Conclusion: Demand is elastic.

• Revenue = $• Elasticity: • Conclusion: Demand is unitary elastic.

• Revenue = $• Elasticity: • Conclusion: Demand is inelastic.

Observation: Elasticity varies along a linear (inverse) demand curve


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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Extreme Elasticities

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Quantity

DemandPrice

Perfectly Inelastic

𝐸𝑄𝑋𝑑 ,𝑃 𝑋

=0

Demand𝐸𝑄𝑋

𝑑 ,𝑃 𝑋=−∞

Perfectly elastic


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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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:

– When then, .– When then, .– When then, .

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Demand and Marginal Revenue

Quantity0

𝑃

MR

3

Price

6

Elastic

Demand


1

6

Inelastic

Unitary

Marginal Revenue (MR)

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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.

– If , then and are substitutes.– If , then and 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?

– 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 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:

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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 and the cross-price elasticity of demand between sodas and hamburgers is , what would happen to the firm’s total revenues if it reduced the price of hamburgers by 1 percent?

– That is, lowering the price of hamburgers 1 percent increases total revenue by $100.

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Income Elasticity• Income elasticity– Measures responsiveness of a percent change in

demand for good X due to a percent change in income.

– If , then is a normal good.– If , then is an inferior good.

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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?

– 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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Income Elasticity

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.

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Other Elasticities

Elasticities for Linear Demand Functions• From a linear demand function, we can easily

compute various elasticities.• Given a linear demand function:

– Own price elasticity: .– Cross price elasticity: .– Income elasticity: .

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Obtaining Elasticities From Demand Functions

Elasticities for Linear Demand Functions In Action• The daily demand for Invigorated PED shoes is

estimated to be

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.– units.– Own price elasticity: .– Cross-price elasticity: .– Income elasticity: .

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Elasticities for Nonlinear Demand Functions• One non-linear demand function is the log-

linear demand function:

– Own price elasticity: .– Cross price elasticity: .– Income elasticity: .

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Elasticities for Nonlinear Demand FunctionsIn Action

• An analyst for a major apparel company estimates that the demand for its raincoats is given by

where denotes the daily amount of rainfall and 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?. So, .

A 10 percent increase in rainfall will lead to a 30 percent increase in the demand for raincoats.

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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.

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Regression Analysis

Regression Line and Least Squares Regression• True (or population) regression model

– unknown population intercept parameter.– unknown population slope parameter.– random error term with mean zero and standard deviation .

• Least squares regression line

– least squares estimate of the unknown parameter .– least squares estimate of the unknown parameter.

• The parameter estimates and , represent the values of and that result in the smallest sum of squared errors between a line and the actual data.

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Regression Analysis

Excel and Least Squares Estimates

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SUMMARY OUTPUT

Regression StatisticsMultiple R 0.87R Square 0.75Adjusted R Square 0.72Standard Error 112.22Observations 10.00

ANOVA Df SS MS F Significance F

Regression 1 301470.89 301470.89 23.94 0.0012Residual 8 100751.61 12593.95Total 9 402222.50

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 1631.47 243.97 6.69 0.0002 1068.87 2194.07Price -2.60 0.53 -4.89 0.0012 -3.82 -1.37

Estimated Demand:

�̂�=1631.47�̂�=−2.60

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

• t-statistics rule of thumb– When , we are 95 percent confident the true

parameter is in the regression is not zero.

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Regression Analysis


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SUMMARY OUTPUT





Regression Analysis

𝑠𝑒 (̂𝑎)=243.97𝑠𝑒 (̂𝑏)=0.53

, the intercept is different from zero. , the intercept is different from zero.

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.

– Ranges between 0 and 1.• Values closer to 1 indicate “better” fit.

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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.

– is total observations.– is the number of estimated coefficients.– is the degrees of freedom for the regression.

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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.

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Regression Analysis


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SUMMARY OUTPUT





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:

– Functional relationships with multiple variables:• Multiple regression example:

or

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Regression Analysis


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SUMMARY OUTPUT




Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 135.15 20.65 6.54 0.0006 84.61 185.68Price -0.14 0.06 -2.41 0.0500 -0.29 0.00Advertising 0.54 0.64 0.85 0.4296 -1.02 2.09Distance -5.78 1.26 -4.61 0.0037 -8.86 -2.71

Regression Analysis

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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Documents

CHAPTER 3 Quantitative Demand Analysis McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved