Quantitative Demand Analysis 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 Chapter Overview 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

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.

The Elasticity Concept Measures the responsiveness of a percentage change in one variable resulting from a percentage change in another variable.

The Elasticity Formula The Elasticity Concept The Elasticity Formula The elasticity between two variables, 𝐺 and 𝑆, is mathematically expressed as: 𝐸 𝐺,𝑆 = %Δ𝐺 %Δ𝑆 When a functional relationship exists, like 𝐺=𝑓 𝑆 , the elasticity is: 𝐸 𝐺,𝑆 = 𝑑𝐺 𝑑𝑆 𝑆 𝐺

Measurement Aspects of Elasticity 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: 𝐸 𝐺,𝑆 >1 𝐺 is highly responsive to changes in 𝑆. 𝐸 𝐺,𝑆 <1 𝐺 is slightly responsive to changes in 𝑆.

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. 𝐸 𝑄 𝑋 𝑑 , 𝑃 𝑋 = %Δ 𝑄 𝑋 𝑑 %Δ 𝑃 𝑋 Sign: negative by law of demand. Magnitude of absolute value relative to unity: 𝐸 𝑄 𝑋 𝑑 , 𝑃 𝑋 >1: Elastic. 𝐸 𝑄 𝑋 𝑑 , 𝑃 𝑋 <1: Inelastic. 𝐸 𝑄 𝑋 𝑑 , 𝑃 𝑋 =1: Unitary elastic.

Linear Demand, Elasticity, and Revenue Own Price Elasticity of Demand Linear Demand, Elasticity, and Revenue Price Linear Inverse Demand: 𝑃=40−0.5𝑄 Demand: 𝑄=80−2𝑄 $40 Revenue = $10×60=$600 Elasticity: −2× $10 60 =−0.333 Conclusion: Demand is inelastic. Revenue = $30×20=$600 Elasticity: −2× $30 20 =−3 Conclusion: Demand is elastic. Revenue = $20×40=$800 Elasticity: −2× $20 40 =−1 Conclusion: Demand is unitary elastic. $35 $30 $25 Observation: Elasticity varies along a linear (inverse) demand curve $20 $15 $10 $5 Demand 10 20 30 40 50 60 70 80 Quantity

Total Revenue Test When demand is elastic: When demand is inelastic: 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.

Extreme Elasticities Price Demand 𝐸 𝑄 𝑋 𝑑 , 𝑃 𝑋 =0 Perfectly Demand Own Price Elasticity of Demand Extreme Elasticities Price Demand 𝐸 𝑄 𝑋 𝑑 , 𝑃 𝑋 =0 Perfectly elastic Demand 𝐸 𝑄 𝑋 𝑑 , 𝑃 𝑋 =−∞ Perfectly Inelastic Quantity

Factors Affecting the Own Price Elasticity 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.

Elasticity and Marginal Revenue 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=𝑃 1+𝐸 𝐸 When −∞<𝐸<−1 then, 𝑀𝑅>0. When 𝐸=−1 then, 𝑀𝑅=0. When −1<𝐸<0 then, 𝑀𝑅<0.

Demand and Marginal Revenue Own Price Elasticity of Demand Demand and Marginal Revenue Price 6 Elastic Unitary 𝑃 MR Inelastic Demand 1 3 6 Quantity Marginal Revenue (MR)

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. 𝐸 𝑄 𝑋 𝑑 , 𝑃 𝑌 = %Δ 𝑄 𝑋 𝑑 %Δ 𝑃 𝑌 If 𝐸 𝑄 𝑋 𝑑 , 𝑃 𝑌 >0, then 𝑋 and 𝑌 are substitutes. If 𝐸 𝑄 𝑋 𝑑 , 𝑃 𝑌 <0, then 𝑋 and 𝑌 are complements.

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

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= 𝑅 𝑋 1+ 𝐸 𝑄 𝑋 𝑑 , 𝑃 𝑋 + 𝑅 𝑌 𝐸 𝑄 𝑌 𝑑 , 𝑃 𝑋 ×%∆ 𝑃 𝑋

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 𝐸 𝑄 𝑋 , 𝑃 𝑋 =−1.5 and the cross-price elasticity of demand between sodas and hamburgers is 𝐸 𝑄 𝑌 , 𝑃 𝑋 =−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.

Income Elasticity Income elasticity 𝐸 𝑄 𝑋 𝑑 ,𝑀 = %Δ 𝑄 𝑋 𝑑 %Δ𝑀 Measures responsiveness of a percent change in demand for good X due to a percent change in income. 𝐸 𝑄 𝑋 𝑑 ,𝑀 = %Δ 𝑄 𝑋 𝑑 %Δ𝑀 If 𝐸 𝑄 𝑋 𝑑 ,𝑀 >0, then 𝑋 is a normal good. If 𝐸 𝑄 𝑋 𝑑 ,𝑀 <0, then 𝑋 is an inferior good.

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= %Δ 𝑄 𝑋 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.

Other Elasticities 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.

Elasticities for Linear Demand Functions 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: 𝑄 𝑋 𝑑 = 𝛼 0 + 𝛼 𝑋 𝑃 𝑋 + 𝛼 𝑌 𝑃 𝑌 + 𝛼 𝑀 𝑀+ 𝛼 𝐻 𝑃 𝐻 Own price elasticity: 𝛼 𝑋 𝑃 𝑋 𝑄 𝑋 𝑑 . Cross price elasticity: 𝛼 𝑌 𝑃 𝑌 𝑄 𝑋 𝑑 . Income elasticity: 𝛼 𝑀 𝑀 𝑄 𝑋 𝑑 .

Elasticities for Linear Demand Functions In Action Obtaining Elasticities From Demand Functions Elasticities for Linear Demand Functions In Action The daily demand for Invigorated PED shoes is estimated to be 𝑄 𝑋 𝑑 =100−3 𝑃 𝑋 +4 𝑃 𝑌 −0.01𝑀+2 𝑃 𝐴 𝑋 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. 𝑄 𝑋 𝑑 =100−3 $25 +4 $35 −0.01 $20,000 +2 50 =65 units. Own price elasticity: −3 25 65 =−1.15. Cross-price elasticity: 4 35 65 =2.15. Income elasticity: −0.01 20,000 65 =−3.08.

Elasticities for Nonlinear Demand Functions Obtaining Elasticities From Demand Functions Elasticities for Nonlinear Demand Functions One non-linear demand function is the log-linear demand function: ln 𝑄 𝑋 𝑑 = 𝛽 0 + 𝛽 𝑋 ln 𝑃 𝑋 + 𝛽 𝑌 ln 𝑃 𝑌 + 𝛽 𝑀 ln 𝑀 + 𝛽 𝐻 ln 𝐻 Own price elasticity: 𝛽 𝑋 . Cross price elasticity: 𝛽 𝑌 . Income elasticity: 𝛽 𝑀 .

Elasticities for Nonlinear Demand Functions In Action 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 𝑄 𝑋 𝑑 =10−1.2 ln 𝑃 𝑋 +3 ln 𝑅 −2 ln 𝐴 𝑌 where 𝑅 denotes the daily amount of rainfall and 𝐴 𝑌 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? 𝐸 𝑄 𝑋 𝑑 ,𝑅 = 𝛽 𝑅 =3. So, 𝐸 𝑄 𝑋 𝑑 ,𝑅 = %∆ 𝑄 𝑋 𝑑 %∆𝑅 ⇒3= %∆ 𝑄 𝑋 𝑑 10 . A 10 percent increase in rainfall will lead to a 30 percent increase in the demand for raincoats.

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.

Regression Line and Least Squares Regression 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= 𝑎 + 𝑏 𝑋 𝑎 least squares estimate of the unknown parameter 𝑎. 𝑏 least squares estimate of the unknown parameter 𝑏. The parameter estimates 𝑎 and 𝑏 , represent the values of 𝑎 and 𝑏 that result in the smallest sum of squared errors between a line and the actual data.

Excel and Least Squares Estimates 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 SS MS F Significance F Regression 1 301470.89 23.94 0.0012 Residual 8 100751.61 12593.95 Total 9 402222.50 Coefficients t Stat P-value Lower 95% Upper 95% Intercept 1631.47 243.97 6.69 0.0002 1068.87 2194.07 Price -2.60 0.53 -4.89 -3.82 -1.37 Estimated Demand: 𝑄=1631.47−2.60𝑃𝑅𝐼𝐶𝐸 𝑎 =1631.47 𝑏 =−2.60

Evaluating Statistical Significance 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 𝑎 ±2 𝜎 𝑎 𝑏 ±2 𝜎 𝑏 t-statistics rule of thumb When 𝑡 >2, we are 95 percent confident the true parameter is in the regression is not zero.

Excel and Least Squares Estimates 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 SS MS F Significance F Regression 1 301470.89 23.94 0.0012 Residual 8 100751.61 12593.95 Total 9 402222.50 Coefficients t Stat P-value Lower 95% Upper 95% Intercept 1631.47 243.97 6.69 0.0002 1068.87 2194.07 Price -2.60 0.53 -4.89 -3.82 -1.37 𝑠𝑒 (𝑎) =243.97 𝑠𝑒 (𝑏) =0.53 𝑡 𝑎 = 6.69 >2, the intercept is different from zero. 𝑡 𝑏 = −4.89 <2, the intercept is different from zero.

Evaluating Overall Regression Line Fit: R- Square Regression Analysis 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. 𝑅 2 = 𝐸𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑇𝑜𝑡𝑎𝑙 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 = 𝑆𝑆 𝑅𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑆𝑆 𝑇𝑜𝑡𝑎𝑙 Ranges between 0 and 1. Values closer to 1 indicate “better” fit.

Evaluating Overall Regression Line Fit: Adjusted R-Square 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. 𝑅 2 =1− 1− 𝑅 2 𝑛−1 𝑛−𝑘 𝑛 is total observations. 𝑘 is the number of estimated coefficients. 𝑛−𝑘 is the degrees of freedom for the regression.

Evaluating Overall Regression Line Fit: F-Statistic 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.

Excel and Least Squares Estimates 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 SS MS F Significance F Regression 1 301470.89 23.94 0.0012 Residual 8 100751.61 12593.95 Total 9 402222.50 Coefficients t Stat P-value Lower 95% Upper 95% Intercept 1631.47 243.97 6.69 0.0002 1068.87 2194.07 Price -2.60 0.53 -4.89 -3.82 -1.37

Regression for Nonlinear Functions and Multiple Regression 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: ln 𝑄= 𝛽 0 + 𝛽 𝑝 ln 𝑃 +𝑒 Functional relationships with multiple variables: Multiple regression example: 𝑄 𝑋 𝑑 = 𝛼 0 + 𝛼 𝑋 𝑃 𝑋 + 𝛼 𝑀 𝑀+ 𝛼 𝐻 𝑃 𝐻 +𝑒 or ln 𝑄 𝑋 𝑑 = 𝛽 0 + 𝛽 𝑋 ln 𝑃 𝑋 + 𝛽 𝑀 ln 𝑀 + 𝛽 𝐻 ln 𝑃 𝐻 +𝑒

Excel and Least Squares Estimates Regression Analysis Excel and Least Squares Estimates SUMMARY OUTPUT Regression Statistics Multiple R 0.89 R Square 0.79 Adjusted R Square 0.69 Standard Error 9.18 Observations 10.00 ANOVA   Df SS MS F Significance F Regression 3 1920.99 640.33 7.59 0.182 Residual 6 505.91 84.32 Total 9 2426.90 Coefficients t Stat P-value Lower 95% Upper 95% Intercept 135.15 20.65 6.54 0.0006 84.61 185.68 Price -0.14 0.06 -2.41 0.0500 -0.29 0.00 Advertising 0.54 0.64 0.85 0.4296 -1.02 2.09 Distance -5.78 1.26 -4.61 0.0037 -8.86 -2.71

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.