Chapter 3 Quantitative Demand Analysis

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2 Overview I. The Elasticity Concept Own Price Elasticity Elasticity and Total Revenue Cross-Price Elasticity Income Elasticity II. Demand Functions Linear Log-Linear III. Regression Analysis 3-2

3 The Elasticity Concept How responsive is variable G to a change in variable S E G, S % G = % S If E G,S > 0, then S and G are directly related. If E G,S < 0, then S and G are inversely related. If E G,S = 0, then S and G are unrelated. 3-3

4 The Elasticity Concept Using Calculus An alternative way to measure the elasticity of a function G = f(s) is E = G, S dg ds S G If E G,S > 0, then S and G are directly related. If E G,S < 0, then S and G are inversely related. If E G,S = 0, then S and G are unrelated. 3-4

5 Own Price Elasticity of Demand E, P % = % P d Negative according to the law of demand. Elastic: Inelastic: Unitary: E, P >1 E, P <1 E, P =1 3-5

6 Perfectly Elastic & Inelastic Demand Price Price D D Perfectly Elastic (, = ) E P uantity Perfectly Inelastic ( E, P = uantity 0) 3-6

7 Elastic Own-Price Elasticity and Total Revenue Increase (a decrease) in price leads to a decrease (an increase) in total revenue. Inelastic Increase (a decrease) in price leads to an increase (a decrease) in total revenue. Unitary Total revenue is maximized at the point where demand is unitary elastic. 3-7

8 Elasticity, Total Revenue and Linear Demand P 100 TR

9 Elasticity, Total Revenue and Linear Demand P 100 TR

10 Elasticity, Total Revenue and Linear Demand P 100 TR

11 Elasticity, Total Revenue and Linear Demand P 100 TR

12 Elasticity, Total Revenue and Linear Demand P 100 TR

13 Elasticity, Total Revenue and Linear Demand P Elastic TR Elastic 3-13

14 Elasticity, Total Revenue and Linear Demand P Elastic TR Inelastic Elastic Inelastic 3-14

15 Elasticity, Total Revenue and Linear Demand P Elastic Unit elastic TR Unit elastic Inelastic Elastic Inelastic 3-15

16 Demand, Marginal Revenue (MR) and Elasticity P Elastic Unit elastic Inelastic MR For a linear inverse demand function, MR() = a + 2b, where b < 0. When MR > 0, demand is elastic; MR = 0, demand is unit elastic; MR < 0, demand is inelastic. 3-16

17 Factors Affecting the Own-Price Elasticity Available Substitutes The more substitutes available for the good, the more elastic the demand. Time Demand tends to be more inelastic in the short term than in the long term. Time allows consumers to seek out available substitutes. Expenditure Share Goods that comprise a small share of consumer s budgets tend to be more inelastic than goods for which consumers spend a large portion of their incomes. 3-17

18 Cross-Price Elasticity of Demand E, P Y % = % P Y d If E,P Y > 0, then and Y are substitutes. If E,P Y < 0, then and Y are complements. 3-18

19 Predicting Revenue Changes from Two Products Suppose that a firm sells to related goods. If the price of changes, then total revenue will change by: R = ( R ( ) ) 1 + E, P + RY E, P % P Y 3-19

20 Income Elasticity E, M % = % M d If E,M > 0, then is a normal good. If E,M < 0, then is a inferior good. 3-20

21 Uses of Elasticities Pricing. Managing cash flows. Impact of changes in competitors prices. Impact of economic booms and recessions. Impact of advertising campaigns. And lots more! 3-21

22 Example 1: Pricing and Cash Flows According to an FTC Report by Michael Ward, AT&T s own price elasticity of demand for long distance services is AT&T needs to boost revenues in order to meet it s marketing goals. To accomplish this goal, should AT&T raise or lower it s price? 3-22

23 Answer: Lower price! Since demand is elastic, a reduction in price will increase quantity demanded by a greater percentage than the price decline, resulting in more revenues for AT&T. 3-23

24 Example 2: uantifying the Change If AT&T lowered price by 3 percent, what would happen to the volume of long distance telephone calls routed through AT&T? 3-24

25 Answer: Calls Increase! Calls would increase by percent! E, P % 8.64 = 3% 3% % = 8.64 = ( 8.64) d d = = 25.92% % % P % d d 3-25

26 Example 3: Impact of a Change in a Competitor s Price According to an FTC Report by Michael Ward, AT&T s cross price elasticity of demand for long distance services is If competitors reduced their prices by 4 percent, what would happen to the demand for AT&T services? 3-26

27 Answer: AT&T s Demand Falls! AT&T s demand would fall by percent! E, P Y % 9.06 = 4% 4% 9.06 = % = 9.06 = d d % % P % = 36.24% d d Y 3-27

28 Interpreting Demand Functions Mathematical representations of demand curves. Example: d = 10 2P + 3P 2M Y Law of demand holds (coefficient of P is negative). and Y are substitutes (coefficient of P Y is positive). is an inferior good (coefficient of M is negative). 3-28

29 Linear Demand Functions and Elasticities General Linear Demand Function and Elasticities: d = 0 α + α P + α P + α M + Y Y M α H H P E, P = α Own Price Elasticity E, P Y = α Y P Cross Price Elasticity Y E M, M = α M Income Elasticity 3-29

30 Example of Linear Demand d = 10-2P. Own-Price Elasticity: (-2)P/. If P=1, =8 (since 10-2 = 8). Own price elasticity at P=1, =8: (-2)(1)/8=

31 Log-Linear Demand General Log-Linear Demand Function: ln = β + β ln P + β ln P + β ln M + β ln H d 0 Y Y M H Own Price Elasticity : Cross Price Elasticity : Income Elasticity : β β β Y M 3-31

32 Example of Log-Linear Demand ln( d ) = 10-2 ln(p). Own Price Elasticity:

33 Graphical Representation of Linear and Log-Linear Demand P P D D Linear Log Linear 3-33

34 Regression Analysis One use is for estimating demand functions. Important terminology and concepts: Least Squares Regression model: Y = a + b + e. Least Squares Regression line: Confidence Intervals. t-statistic. Yˆ = aˆ + b ˆ R-square or Coefficient of Determination. F-statistic. 3-34

35 An Example Use a spreadsheet to estimate the following log-linear demand function. ln = β + β ln P + e x 0 x x 3-35

36 Summary Output Regression Statistics Multiple R 0.41 R Square 0.17 Adjusted R Square 0.15 Standard Error 0.68 Observations ANOVA df SS M S F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept ln(p)

37 Interpreting the Regression Output The estimated log-linear demand function is: ln( x ) = ln(p x ). Own price elasticity: (inelastic). How good is our estimate? t-statistics of 5.29 and indicate that the estimated coefficients are statistically different from zero. R-square of 0.17 indicates the ln(p ) variable explains only 17 percent of the variation in ln( x ). F-statistic significant at the 1 percent level. 3-37

38 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. 3-38

Managerial Economics

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