CHAPTER 3

Quantitative Demand Analysis

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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 basic analysis to measure different elasticities.

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Chapter Overview

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

Copyright © 2014 by the McGraw-Hill Companies, Inc. All rights reserved.

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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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Own Price Elasticity of Demand

Copyright © 2014 by the McGraw-Hill Companies, Inc. All rights reserved.

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Extreme Elasticities

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Quantity

DemandPrice

Perfectly Inelastic

𝐸𝑄𝑋𝑑 ,𝑃 𝑋

=0

Demand𝐸𝑄𝑋

𝑑 ,𝑃 𝑋=−∞

Perfectly elastic

Own Price Elasticity of Demand

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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Own Price Elasticity of Demand

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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Own Price Elasticity of Demand

Demand and Marginal Revenue

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Quantity0

𝑃

MR

3

Price

6

Elastic

Demand

Own Price Elasticity of Demand

1

6

Inelastic

Unitary

Marginal Revenue (MR)

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• 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

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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Cross-Price Elasticity

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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Obtaining Elasticities From Demand Functions

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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Obtaining Elasticities From Demand Functions

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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Obtaining Elasticities From Demand Functions

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

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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