Quantitative Demand Analysis

CHAPTER 3 Quantitative Demand Analysis

© 2017 by McGraw-Hill Education. All Rights Reserved. Authorized only for instructor use in the classroom. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Learning Objectives 1. Apply various elasticities of demand as a quantitative tool to forecast changes in revenues, prices, and/or units sold. 2. Illustrate the relationship between the elasticity of demand and total revenues. 3. Discuss three factors that influence whether the demand for a given product is relatively elastic or inelastic. 4. Explain the relationship between marginal revenue and the own price elasticity of demand. 5. Show how to determine elasticities from linear and loglinear demand functions. 6. Explain how regression analysis may be used to estimate demand functions, and how to interpret and use the output of a regression. © 2017 by McGraw-Hill Education. All Rights Reserved.

2

The Elasticity Concept

The Elasticity Concept

• Demand - Supply = Big picture, directional analysis. • Magnitude of effects? • e.g. How much will our sales change if rivals cut their prices by 2 percent or a recession hits and household incomes decline by 2.5 percent? • Elasticities of demand as a quantitative forecasting tool • Elasticity – A measure of the responsiveness of one variable to changes in another variable; the percentage change in one variable that arises due to a given percentage change in another variable. © 2017 by McGraw-Hill Education. All Rights Reserved.

3-3

The Elasticity Concept

The Elasticity Concept • The elasticity between two variables, 𝐺 and 𝑆, is mathematically expressed as: %Δ𝐺 𝐸𝐺,𝑆 = %Δ𝑆 • When a functional relationship exists, like 𝐺 = 𝑓 𝑆 , the elasticity is: 𝑑𝐺 𝑆 𝐸𝐺,𝑆 = 𝑑𝑆 𝐺 G = Qd (Quantity demanded) and S = P (Price) © 2017 by McGraw-Hill Education. All Rights Reserved.

3-4

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.

G = Qd (Quantity demanded) and S = P (Price) © 2017 by McGraw-Hill Education. All Rights Reserved.

3-5

Own Price Elasticity of Demand

Own Price Elasticity of Demand • 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. 𝑑

𝐸𝑄

𝑋

𝑑

,𝑃𝑋

%Δ𝑄𝑋 = %Δ𝑃𝑋

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

𝑋

• 𝐸𝑄

𝑋

• 𝐸𝑄

𝑋

𝑑

,𝑃𝑋

𝑑

,𝑃𝑋

𝑑

,𝑃𝑋

> 1: Elastic.

< 1: Inelastic. = 1: Unitary elastic.

© 2017 by McGraw-Hill Education. All Rights Reserved.

3-6

Own Price Elasticity of Demand

Linear Demand, Elasticity, and Revenue Linear Inverse Demand: 𝑃 = 40 − 0.5𝑄 Demand: 𝑄 = 80 − 2𝑄 $30 $20 × 60 20 40 = $600 $800 • Revenue = $10

Price $40

$35

$30 $20 $10

−3 −1 • Elasticity: −2 × 60 = −0.333 20 40 elastic. unitary elastic. • Conclusion: Demand is inelastic.

$30 $25

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

$20 $15 $10

$5 Demand 0

10 20 30

40

50

60

70

80

© 2017 by McGraw-Hill Education. All Rights Reserved.

Quantity

3-7

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.

© 2017 by McGraw-Hill Education. All Rights Reserved.

3-8

Own Price Elasticity of Demand

Perfectly Elastic and Inelastic Demand Price Demand

𝐸𝑄𝑋 𝑑 ,𝑃𝑋 = 0 Perfectly elastic

Demand 𝐸𝑄𝑋𝑑 ,𝑃𝑋 = −∞

Perfectly Inelastic

Quantity

Aspirin vs. life saving drugs © 2017 by McGraw-Hill Education. All Rights Reserved.

3-9

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 • The more substitutes available for the good, the more elastic the demand for it • Demand for broadly defined commodities tends to be more inelastic than the demand for specific commodities (e.g., food vs. beef)

– Time/duration of purchase horizon • more inelastic in the short term than in the long term (e.g., demand for a taxi of a consumer who want to catch a flight)

– Expenditure share of consumers’ budgets • Goods that comprise a relatively small share of consumers’ budgets tend to be more inelastic than goods for which consumers spend a sizable portion of their incomes (e.g., salt). © 2017 by McGraw-Hill Education. All Rights Reserved.

3-10

Cross-Price Elasticity

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. 𝑑

𝐸𝑄

𝑋

– If 𝐸𝑄

𝑋 ,𝑃𝑌

– If 𝐸𝑄

𝑋

𝑑 𝑑

,𝑃𝑌

𝑑

,𝑃𝑌

%Δ𝑄𝑋 = %Δ𝑃𝑌

> 0, then 𝑋 and 𝑌 are substitutes. < 0, then 𝑋 and 𝑌 are complements.

© 2017 by McGraw-Hill Education. All Rights Reserved.

3-11

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? %∆𝑄𝐶𝑙𝑜𝑡ℎ𝑖𝑛𝑔 𝑑 𝑑 −0.18 = ⇒ %∆𝑄𝐶𝑙𝑜𝑡ℎ𝑖𝑛𝑔 = −1.8 10 – That is, demand for clothing is expected to decline by 1.8 percent when the price of food increases 10 percent.

© 2017 by McGraw-Hill Education. All Rights Reserved.

3-12

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 + 𝐸𝑄

𝑋

𝑑

,𝑃𝑋

+ 𝑅𝑌 𝐸𝑄

𝑌

𝑑

,𝑃𝑋

© 2017 by McGraw-Hill Education. All Rights Reserved.

× %∆𝑃𝑋

3-13

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. © 2017 by McGraw-Hill Education. All Rights Reserved.

3-14

Income Elasticity

Income Elasticity • Income elasticity – Measures responsiveness of a percent change in demand for good X due to a percent change in income. 𝑑

𝐸𝑄

𝑋

– If 𝐸𝑄

– If 𝐸𝑄

𝑑

,𝑀

%Δ𝑄𝑋 = %Δ𝑀

𝑑

> 0, then 𝑋 is a normal good.

𝑑

< 0, then 𝑋 is an inferior good.

𝑋 ,𝑀 𝑋 ,𝑀

© 2017 by McGraw-Hill Education. All Rights Reserved.

3-15

Obtaining Elasticities From Demand Functions

Elasticities for Linear Demand Functions • From a linear demand function, we can easily compute various elasticities. • Given a linear demand function: 𝑑 𝑄𝑋 = 𝛼0 + 𝛼𝑋 𝑃𝑋 + 𝛼𝑌 𝑃𝑌 + 𝛼𝑀 𝑀 + 𝛼𝐻 𝑃𝐻 – Own price elasticity: 𝛼𝑋

𝑃𝑋 𝑄𝑋

– Cross price elasticity: 𝛼𝑌 – Income elasticity: 𝛼𝑀

𝑑

.

𝑃𝑌

𝑄𝑋

𝑀 𝑄𝑋 𝑑

𝑑

.

.

© 2017 by McGraw-Hill Education. All Rights Reserved.

3-16

Obtaining Elasticities From Demand Functions

Elasticities for Linear Demand Functions In Action The daily demand for Invigorated PED shoes is estimated to be: 𝑄𝑋 𝑑 = 100 − 3𝑃𝑋 + 4𝑃𝑌 − 0.01𝑀 + 2𝑃𝐴𝑋 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. 25 65 35 Cross-price elasticity: 4( ) = 2.15. 65 20,000 Income elasticity: −0.01( ) = −3.08. 65

– Own price elasticity: −3( ) = −1.15. – –

© 2017 by McGraw-Hill Education. All Rights Reserved.

3-17

Obtaining Elasticities From Demand Functions

Elasticities for Nonlinear Demand Functions • One non-linear demand function is the loglinear demand function: ln 𝑄𝑋 𝑑 = 𝛽0 + 𝛽𝑋 ln 𝑃𝑋 + 𝛽𝑌 ln 𝑃𝑌 + 𝛽𝑀 ln 𝑀 + 𝛽𝐻 ln 𝐻 – Own price elasticity: 𝛽𝑋 . – Cross price elasticity: 𝛽𝑌 . – Income elasticity: 𝛽𝑀 . © 2017 by McGraw-Hill Education. All Rights Reserved.

3-18

Obtaining Elasticities From Demand Functions

Elasticities for Nonlinear Demand Functions In Action An analyst for a major apparel company estimates that the demand for its raincoats is given by 𝑙𝑛 𝑄𝑋 𝑑 = 10 − 1.2 ln 𝑃𝑋 + 3 ln 𝑅 − 2 ln 𝐴𝑌 where 𝑅 denotes the daily amount of rainfall and 𝐴𝑌 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? 𝐸𝑄

𝑋

𝑑

,𝑅

= 𝛽𝑅 = 3. So, 𝐸𝑄

𝑋

𝑑

,𝑅

=

%∆𝑄𝑋 𝑑 %∆𝑅

⇒3=

%∆𝑄𝑋 𝑑 . 10

A 10 percent increase in rainfall will lead to a 30 percent increase in the demand for raincoats. © 2017 by McGraw-Hill Education. All Rights Reserved.

3-19

Regression Analysis

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.

© 2017 by McGraw-Hill Education. All Rights Reserved.

3-20

Regression Analysis

Regression Line and Least Squares Regression • True (or population) regression model 𝑌 = 𝑎 + 𝑏𝑋 + 𝑒 – 𝑎 unknown population intercept parameter. – 𝑏 unknown population slope parameter. – 𝑒 random error term with mean zero and standard deviation 𝜎.

• Least squares regression line ෠ 𝑌 = 𝑎ො + 𝑏𝑋 – 𝑎ො least squares estimate of the unknown parameter 𝑎. – 𝑏෠ least squares estimate of the unknown parameter 𝑏.

෠ represent the values • The parameter estimates 𝑎ො and 𝑏, of 𝑎 and 𝑏 that result in the smallest sum of squared errors between a line and the actual data. © 2017 by McGraw-Hill Education. All Rights Reserved.

3-21

Regression Analysis

Excel and Least Squares Estimates SUMMARY OUTPUT Regression Statistics Multiple R 0.87 R Square 0.75 Adjusted R Square 0.72 Standard Error 112.22 Observations 10.00

Estimated Demand: 𝑄 = 1631.47 − 2.60𝑃𝑅𝐼𝐶𝐸 𝑎ො = 1631.47 𝑏෠ = −2.60

ANOVA Df Regression Residual Total

Intercept Price

1 8 9

SS 301470.89 100751.61 402222.50

Coefficients Standard Error 1631.47 243.97 -2.60 0.53

MS 301470.89 12593.95

F Significance F 23.94 0.0012

t Stat P-value Lower 95% Upper 95% 6.69 0.0002 1068.87 2194.07 -4.89 0.0012 -3.82 -1.37

© 2017 by McGraw-Hill Education. All Rights Reserved.

3-22

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.

• 95 Percent Confidence interval rule of thumb – 𝑎ො ± 2𝜎𝑎ො – 𝑏෠ ± 2𝜎𝑏෠

• t-statistics rule of thumb – When 𝑡 > 2, we are 95 percent confident the true parameter is in the regression is not zero. © 2017 by McGraw-Hill Education. All Rights Reserved.

3-23

Regression Analysis

Excel and Least Squares Estimates SUMMARY OUTPUT Regression Statistics Multiple R 0.87 R Square 0.75 Adjusted R Square 0.72 Standard Error 112.22 Observations 10.00

෢ = 243.97 𝑠𝑒 (𝑎) ෢ = 0.53 𝑠𝑒(𝑏) 𝑡𝑎ො = 6.69 > 2, the intercept is different from zero. 𝑡𝑏෠ = −4.89 < 2, the intercept is different from zero.

ANOVA Df Regression Residual Total

Intercept Price

1 8 9

SS 301470.89 100751.61 402222.50

Coefficients Standard Error 1631.47 243.97 -2.60 0.53

MS 301470.89 12593.95

F Significance F 23.94 0.0012

t Stat P-value Lower 95% Upper 95% 6.69 0.0002 1068.87 2194.07 -4.89 0.0012 -3.82 -1.37

© 2017 by McGraw-Hill Education. All Rights Reserved.

3-24

Regression Analysis

Evaluating the Overall Fit of the Regression Line

• R-Square

– Also called the coefficient of determination. – Fraction of the total variation in the dependent variable that is explained by the regression. 𝐸𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑆𝑆𝑅𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑅 = = 𝑇𝑜𝑡𝑎𝑙 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑆𝑆𝑇𝑜𝑡𝑎𝑙 2

– Ranges between 0 and 1. • Values closer to 1 indicate “better” fit. © 2017 by McGraw-Hill Education. All Rights Reserved.

3-25

Regression Analysis

Evaluating the Overall Fit of the Regression Line • Adjusted R-Square – A version of the R-square that penalize researchers for having few degrees of freedom. 𝑛−1 2 2 𝑅 =1− 1−𝑅 𝑛−𝑘 – 𝑛 is total observations. – 𝑘 is the number of estimated coefficients. – 𝑛 − 𝑘 is the degrees of freedom for the regression. © 2017 by McGraw-Hill Education. All Rights Reserved.

3-26

Regression Analysis

Evaluating the Overall Fit of the Regression Line • The 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. © 2017 by McGraw-Hill Education. All Rights Reserved.

3-27

Regression Analysis

Excel and Least Squares Estimates SUMMARY OUTPUT Regression Statistics Multiple R 0.87 R Square 0.75 Adjusted R Square 0.72 Standard Error 112.22 Observations 10.00

ANOVA Df Regression Residual Total

Intercept Price

1 8 9

SS 301470.89 100751.61 402222.50

Coefficients Standard Error 1631.47 243.97 -2.60 0.53

MS 301470.89 12593.95

F Significance F 23.94 0.0012

t Stat P-value Lower 95% Upper 95% 6.69 0.0002 1068.87 2194.07 -4.89 0.0012 -3.82 -1.37

© 2017 by McGraw-Hill Education. All Rights Reserved.

3-28