CHAPTER 3. Quantitative Demand Analysis

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1 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.

2 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

3 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

4 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

5 The Elasticity Formula The Elasticity Concept 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 S ds G 3-5

6 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: E G,S E G,S > 1 G is highly responsive to changes in S. < 1 G is slightly responsive to changes in S. 3-6

7 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. E QX d,p X = %ΔQ X d %ΔP X Sign: negative by law of demand. Magnitude of absolute value relative to unity: E QX d,p X > 1: Elastic. E QX d,p X < 1: Inelastic. E QX d,p X = 1: Unitary elastic. 3-7

8 Own Price Elasticity of Demand Extreme Elasticities Price Demand E QX d,p X = 0 Perfectly elastic Demand = E QX d,p X Perfectly Inelastic Quantity 3-8

9 Own Price Elasticity of Demand Linear Demand, Elasticity, and Revenue Price $40 $35 $30 $25 $20 $15 Linear Inverse Demand: P = Q Demand: 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 $10 $5 Demand Quantity 3-9

10 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 3-10

11 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 Own Price Elasticity of Demand 3-11

12 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 + E MR = P E When < E < 1 then, MR > 0. When E = 1 then, MR = 0. When 1 < E < 0 then, MR < 0. Own Price Elasticity of Demand 3-12

13 Own Price Elasticity of Demand Demand and Marginal Revenue Price 6 P Unitary MR Demand Quantity Marginal Revenue (MR) 3-13

14 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. E QX d,p Y = %ΔQ X d %ΔP Y If E QX d,p Y > 0, then X and Y are substitutes. If E QX d,p Y < 0, then X and Y are complements. 3-14

15 Cross-Price Elasticity in Action Cross-Price Elasticity Suppose it is estimated that the cross-price elasticity of demand between clothing and food is 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 % Q d 10 Clothing = 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

16 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 = R X 1 + E QX d,p X + R Y E QY d,p X % P X 3-16

17 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 E QX,P X = 1.5 and the cross-price elasticity of demand between sodas and hamburgers is E QY,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, $2, % = $100 That is, lowering the price of hamburgers 1 percent increases total revenue by $

18 Income Elasticity Income Elasticity Income elasticity Measures responsiveness of a percent change in demand for good X due to a percent change in income. E d QX,M = %ΔQ X d %ΔM If E QX d,m If E QX d,m > 0, then X is a normal good. < 0, then X is an inferior good. 3-18

19 Income Elasticity Income Elasticity in Action Suppose that the income elasticity of demand for transportation is estimated to be If income is projected to decrease by 15 percent, what is the impact on the demand for transportation? 1.8 = %ΔQ 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. 3-19

20 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

21 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. 3-21

22 Elasticities for Linear Demand Functions In Action The daily demand for Invigorated PED shoes is estimated to be Q X d = 100 3P X + 4P Y 0.01M + 2A 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 = $ $ $20, = 65 units. Own price elasticity: = Cross-price elasticity: = Income elasticity: , = Obtaining Elasticities From Demand Functions 3-22

23 Obtaining Elasticities From Demand Functions Elasticities for Nonlinear Demand Functions One non-linear demand function is the loglinear 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. 3-23

24 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 = 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 QX d,r = β R = 3. So, E QX d,r = % Q X d % R 3 = % Q d X. 10 Obtaining Elasticities From Demand Functions A 10 percent increase in rainfall will lead to a 30 percent increase in the demand for raincoats. 3-24

25 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. 3-25