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 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, 𝐺 and 𝑆, is mathematically expressed as:

𝐸𝐺,𝑆 =%Δ𝐺

%Δ𝑆• When a functional relationship exists, like 𝐺 =𝑓 𝑆 , the elasticity is:

𝐸𝐺,𝑆 =𝑑𝐺

𝑑𝑆

𝑆

𝐺

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The Elasticity Concept

Measurement Aspects of Elasticity

• Important aspects of the elasticity:

– Sign of the relationship:

• Positive.

• Negative.

– Absolute value of elasticity magnitude relative to unity:

• 𝐸𝐺,𝑆 > 1 𝐺 is highly responsive to changes in 𝑆.

• 𝐸𝐺,𝑆 < 1 𝐺 is slightly responsive to changes in 𝑆.

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The Elasticity Concept

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.

𝐸𝑄𝑋𝑑,𝑃𝑋 =%Δ𝑄𝑋

𝑑

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

• 𝐸𝑄𝑋𝑑,𝑃𝑋> 1: Elastic.

• 𝐸𝑄𝑋𝑑,𝑃𝑋 < 1: Inelastic.

• 𝐸𝑄𝑋𝑑,𝑃𝑋 = 1: Unitary elastic.

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Own Price Elasticity of Demand

Extreme Elasticities

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Quantity

Demand

Price

Perfectly Inelastic

𝐸𝑄𝑋

𝑑,𝑃𝑋= 0

Demand

𝐸𝑄𝑋𝑑,𝑃𝑋= −∞

Perfectly elastic

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: 𝑃 = 40 − 0.5𝑄Demand: 𝑄 = 80 − 2𝑃

• Revenue = $30 × 20 = $600

• Elasticity: −2 ×$30

20= −3

• Conclusion: Demand is elastic.

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

Own Price Elasticity of Demand

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

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.

– Product durability

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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:

𝑀𝑅 = 𝑃1 + 𝐸

𝐸– When −∞ < 𝐸 < −1 then, 𝑀𝑅 > 0.– When 𝐸 = −1 then, 𝑀𝑅 = 0.– When −1 < 𝐸 < 0 then, 𝑀𝑅 < 0.

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Own Price Elasticity of Demand

Demand and Marginal Revenue

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Quantity0

𝑃

MR

3

Price

6

Demand

Own Price Elasticity of Demand

1

6

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.

𝐸𝑄𝑋𝑑,𝑃𝑌=%Δ𝑄𝑋

𝑑

%Δ𝑃𝑌– If 𝐸𝑄𝑋𝑑,𝑃𝑌

> 0, then 𝑋 and 𝑌 are substitutes.

– If 𝐸𝑄𝑋𝑑,𝑃𝑌< 0, then 𝑋 and 𝑌 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?

−0.18 =%∆𝑄𝐶𝑙𝑜𝑡ℎ𝑖𝑛𝑔

𝑑

10⇒ %∆𝑄𝐶𝑙𝑜𝑡ℎ𝑖𝑛𝑔

𝑑 = −1.8

– 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:

∆𝑅 = 𝑅𝑋 1 + 𝐸𝑄𝑋𝑑,𝑃𝑋+ 𝑅𝑌𝐸𝑄𝑌𝑑,𝑃𝑋

×%∆𝑃𝑋

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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 𝐸𝑄𝑋,𝑃𝑋 = −1.5 and the cross-price elasticity of demand between sodas and hamburgers is 𝐸𝑄𝑌,𝑃𝑋 = −4.0, what would happen to the firm’s total revenues if it reduced the price of hamburgers by 1 percent?∆𝑅 = $4,000 1 − 1.5 + $2,000 −4.0 −1%= $100– 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.

𝐸𝑄𝑋𝑑,𝑀=%Δ𝑄𝑋

𝑑

%Δ𝑀– If 𝐸𝑄𝑋𝑑,𝑀

> 0, then 𝑋 is a normal good.

– If 𝐸𝑄𝑋𝑑,𝑀< 0, then 𝑋 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?

1.8 =%Δ𝑄𝑋

𝑑

−15– 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:

𝑄𝑋𝑑 = 𝛼0 + 𝛼𝑋𝑃𝑋 + 𝛼𝑌𝑃𝑌 + 𝛼𝑀𝑀 + 𝛼𝐻𝑃𝐻

– Own price elasticity: 𝛼𝑋𝑃𝑋

𝑄𝑋𝑑.

– Cross price elasticity: 𝛼𝑌𝑃𝑌

𝑄𝑋𝑑.

– Income elasticity: 𝛼𝑀𝑀

𝑄𝑋𝑑.

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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

𝑄𝑋𝑑 = 100 − 3𝑃𝑋 + 4𝑃𝑌 − 0.01𝑀 + 2𝐴𝑋

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.– 𝑄𝑋

𝑑 = 100 − 3 $25 + 4 $35 − 0.01 $20,000 + 2 50 =65 units.

– Own price elasticity: −325

65= −1.15.

– Cross-price elasticity: 435

65= 2.15.

– Income elasticity: −0.0120,000

65= −3.08.

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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:

ln𝑄𝑋𝑑

= 𝛽0 + 𝛽𝑋 ln 𝑃𝑋 + 𝛽𝑌 ln 𝑃𝑌 + 𝛽𝑀 ln𝑀 + 𝛽𝐻 ln𝐻

– Own price elasticity: 𝛽𝑋.

– Cross price elasticity: 𝛽𝑌.

– Income elasticity: 𝛽𝑀.

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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

𝑙𝑛 𝑄𝑋𝑑 = 10 − 1.2 ln 𝑃𝑋 + 3 ln𝑅 − 2 ln𝐴𝑌

where 𝑅 denotes the daily amount of rainfall and 𝐴𝑌the level of advertising on good Y. What would be the impact on demand of a 10 percent increase in the daily amount of rainfall?

𝐸𝑄𝑋𝑑,𝑅= 𝛽𝑅 = 3.

So, 𝐸𝑄𝑋𝑑,𝑅=

%∆𝑄𝑋𝑑

%∆𝑅⇒ 3 =

%∆𝑄𝑋𝑑

10.

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

Conclusion• Elasticities are tools you can use to quantify

the impact of changes in prices, income, and advertising on sales and revenues.

• Managers can quantify the impact of changes in prices, income, advertising, etc.

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