CHAPTER 3 Quantitative Demand Analysis Copyright 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.
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 3-2
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. Chapter Overview 3-3
Elasticity The Elasticity Concept Measures the responsiveness of a percentage change in one variable resulting from a percentage change in another variable. The Elasticity Concept 3-4
The Elasticity Formula The Elasticity Concept The elasticity between two variables, GG and SS, is mathematically expressed as: EE GG,SS = %ΔGG %ΔSS When a functional relationship exists, like GG = ff SS, the elasticity is: EE GG,SS = ddgg SS ddss GG 3-5
Measurement Aspects of Elasticity Important aspects of the elasticity: Sign of the relationship: Positive. Negative. The Elasticity Concept Absolute value of elasticity magnitude relative to unity: EE GG,SS EE GG,SS > 1 GG is highly responsive to changes in SS. < 1 GG is slightly responsive to changes in SS. 3-6
Own Price Elasticity of Demand 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. EE QQXX dd,pp XX = %ΔQQ XX dd %ΔPP XX Sign: negative by law of demand. Magnitude of absolute value relative to unity: EE QQXX dd,pp XX > 1: Elastic. EE QQXX dd,pp XX < 1: Inelastic. EE QQXX dd,pp XX = 1: Unitary elastic. 3-7
Own Price Elasticity of Demand Linear Demand, Elasticity, and Revenue Price $40 $35 $30 $25 $20 $15 Linear Inverse Demand: PP = 40 0.5QQ Demand: QQ = 80 2PP Revenue = $30 20 = $600 Elasticity: 2 $30 20 = 3 Conclusion: Demand is elastic. Observation: Elasticity varies along a linear (inverse) demand curve $10 $5 Demand 0 10 20 30 40 50 60 70 80 Quantity 3-8
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. Own Price Elasticity of Demand 3-9
Own Price Elasticity of Demand Extreme Elasticities Price Demand EE QQXX dd,pp XX = 0 Perfectly elastic Demand EE QQXX dd,pp XX = Perfectly Inelastic Quantity 3-10
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. Own Price Elasticity of Demand 3-11
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: 1 + EE MMMM = PP EE When < EE < 1 then, MMMM > 0. When EE = 1 then, MMMM = 0. When 1 < EE < 0 then, MMMM < 0. Own Price Elasticity of Demand 3-12
Own Price Elasticity of Demand Demand and Marginal Revenue Price 6 PP Unitary MR Demand 0 1 3 6 Quantity Marginal Revenue (MR) 3-13
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. EE dd QQXX,PP = %ΔQQ XX dd YY %ΔPP YY If EE QQXX dd,pp YY If EE QQXX dd,pp YY > 0, then XX and YY are substitutes. < 0, then XX and YY are complements. 3-14
Cross-Price Elasticity in Action Cross-Price Elasticity 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 = % QQ CCCCCCCCCCCCCCC dd % QQ dd 10 CCCCCCCCCCCCCCC = 1.8 That is, demand for clothing is expected to decline by 1.8 percent when the price of food increases 10 percent. 3-15
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: RR = RR XX 1 + EE QQXX dd,pp XX + RR YY EE QQYY dd,pp XX % PP XX 3-16
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 EE QQXX,PP XX = 1.5 and the cross-price elasticity of demand between sodas and hamburgers is EE QQYY,PP XX = 4.0, what would happen to the firm s total revenues if it reduced the price of hamburgers by 1 percent? RR = $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. 3-17
Income Elasticity Income Elasticity Income elasticity Measures responsiveness of a percent change in demand for good X due to a percent change in income. EE dd QQXX,MM = %ΔQQ XX dd %ΔMM If EE QQXX dd,mm If EE QQXX dd,mm > 0, then XX is a normal good. < 0, then XX is an inferior good. 3-18
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 = %ΔQQ XX dd 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. 3-19
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. 3-20
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: QQ dd XX = αα 0 + αα XX PP XX + αα YY PP YY + αα MM MM + αα HH PP HH Own price elasticity: αα XX PP XX QQ XX dd. Cross price elasticity: αα YY PP YY QQ XX dd. Income elasticity: αα MM MM QQ XX dd. 3-21
Elasticities for Linear Demand Functions In Action The daily demand for Invigorated PED shoes is estimated to be QQ XX dd = 100 3PP XX + 4PP YY 0.01MM + 2AA XX 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. QQ XX dd = 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. Obtaining Elasticities From Demand Functions 3-22
Elasticities for Nonlinear Demand Functions One non-linear demand function is the loglinear demand function: ln QQ XX dd Obtaining Elasticities From Demand Functions = ββ 0 + ββ XX ln PP XX + ββ YY ln PP YY + ββ MM ln MM + ββ HH ln HH Own price elasticity: ββ XX. Cross price elasticity: ββ YY. Income elasticity: ββ MM. 3-23
Elasticities for Nonlinear Demand Functions In Action An analyst for a major apparel company estimates that the demand for its raincoats is given by llll QQ XX dd = 10 1.2 ln PP XX + 3 ln RR 2 ln AA YY where RR denotes the daily amount of rainfall and AA YY 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? EE QQXX dd,rr = ββ RR = 3. So, EE QQXX dd,rr = % QQ XX dd Obtaining Elasticities From Demand Functions % RR 3 = % QQ dd XX. 10 A 10 percent increase in rainfall will lead to a 30 percent increase in the demand for raincoats. 3-24
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. 3-25
Regression Analysis Regression Line and Least Squares Regression True (or population) regression model YY = aa + bbbb + ee aa unknown population intercept parameter. bb unknown population slope parameter. ee random error term with mean zero and standard deviation σσ. Least squares regression line YY = aa + bb XX aa least squares estimate of the unknown parameter aa. bb least squares estimate of the unknown parameter bb. The parameter estimates aa and bb, represent the values of aa and bb that result in the smallest sum of squared errors between a line and the actual data. 3-26
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: QQ = 1631.47 2.60PPPPPPPPPP aa = 1631.47 bb = 2.60 ANOVA Df SS MS F Significance F Regression 1 301470.89 301470.89 23.94 0.0012 Residual 8 100751.61 12593.95 Total 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 2194.07 Price -2.60 0.53-4.89 0.0012-3.82-1.37 3-27
Regression Analysis Evaluating Statistical Significance Standard error Measure of how much each estimated coefficient varies in regressions based on the same true demand model using different data. Confidence interval rule of thumb aa ± 2σσ aa bb ± 2σσ bb t-statistics rule of thumb When tt > 2, we are 95 percent confident the true parameter is in the regression is not zero. 3-28
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 ssss (aa) = 243.97 ssss(bb) = 0.53 tt aa = 6.69 > 2, the intercept is different from zero. tt bb = 4.89 < 2, the intercept is different from zero. ANOVA Df SS MS F Significance F Regression 1 301470.89 301470.89 23.94 0.0012 Residual 8 100751.61 12593.95 Total 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 2194.07 Price -2.60 0.53-4.89 0.0012-3.82-1.37 3-29
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. Regression Analysis RR 2 = EEEEEEEEEEEEEEEEEE VVVVVVVVVVVVVVVVVV TTTTTTTTTT VVVVVVVVVVVVVVVVVV = SSSS RRRRRRRRRRRRRRRRRRRR SSSS TTTTTTTTTT Ranges between 0 and 1. Values closer to 1 indicate better fit. 3-30
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. RR 2 = 1 1 RR 2 nn 1 nn kk nn is total observations. kk is the number of estimated coefficients. nn kk is the degrees of freedom for the regression. Regression Analysis 3-31
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. Regression Analysis 3-32
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 301470.89 23.94 0.0012 Residual 8 100751.61 12593.95 Total 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 2194.07 Price -2.60 0.53-4.89 0.0012-3.82-1.37 3-33
Regression for Nonlinear Functions and Multiple Regression Regression techniques can also be applied to the following settings: Nonlinear functional relationships: Nonlinear regression example: ln QQ = ββ 0 + ββ pp ln PP + ee Functional relationships with multiple variables: Multiple regression example: QQ dd XX = αα 0 + αα XX PP XX + αα MM MM + αα HH PP HH + ee or ln QQ dd XX = ββ 0 + ββ XX ln PP XX + ββ MM ln MM + ββ HH ln PP HH + ee Regression Analysis 3-34
SUMMARY OUTPUT Excel and Least Squares Estimates 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 Regression Analysis Coefficients Standard Error 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 3-35
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. 3-36