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.. ECON 563Managerial Economics

Module 3: Elasticity, and Demand Estimation

Copyright 2017 Montclair State University

.. ECON 563 Managerial Economics

Module 3a: Brief Overview

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 de- mand and total revenues.

(3) Discuss three factors that influencewhether 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.

Learning Objectives (5) Show how to determine elasticities from linear and log-

linear demand functions. (6) Explain how regression analysis may be used to esti-

mate demand functions. (7) How to interpret and use the output of a regression.

.. ECON 563 Managerial Economics

Module 3b: Elasticity Concept

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.

• In case of demand, percentage change in quantity de- manded due to a given percentage change in price.

Definition of Elasticity • The elasticity between two variables, say P (price) and Q (quantity) is mathematically expressed as :

εQ,P = %∆Q %∆P .

• Here %∆Q denotes percentage change in quantity, Q and %∆P denotes percentage change in price, P .

Measurement of Elasticity Two important aspects of the elasticity are

• Sign of the relationship : • Positive, • Negative.

• Absolute value of elasticity magnitude relative to unity :

• |εQ,P | > 1 implies Q is highly responsive to changes in P . • |εQ,P | < 1 implies Q is slightly responsive to changes in P .

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. εQdX ,PX =

%∆QdX %∆PX .

• Sign : negative by law of demand. • Absolute value of elasticity magnitude relative to unity :

• |εQdX ,PX | > 1 : Elastic. • |εQdX ,PX | < 1 : Inelastic. • |εQdX ,PX | = 1 : Unitary elastic.

.. ECON 563 Managerial Economics

Module 3c: Linear Demand and Elasticity

Linear Demand, Elasticity, and Revenue • Consider a linear demand :

Inverse Demand : P = 40−2Q Demand : Q = 20−0.5P.

• P = 10, Q = 15, Revenue = P ×Q = 10× 15 = 150.

Linear Demand, Elasticity, and Revenue • There is a convenient expression for the elasticity of de- mand for linear demand functions.

εQ,P = 1

slope of inverse demand × P

Q .

• εQ,P = 1−2 × 10 15 = −0.333.

• Demand is therefore inelastic.

Linear Demand and Elasticity • It is easy to infer that elasticity varies along a linear in- verse demand curve.

• This is becasue the slope remains constant and the ra- tio of P and Q changes along the demand curve.

Total Revenue and Elasticity • When demand is elastic :

• A price increase (decrease) leads to a decrease (increase) in total revenue.

• When demand is inelastic : • A price decrease (increase) leads to a decrease (increase) in total revenue.

• When demand is unitary elastic : • Total revenue is maximized.

Two Extremes of Elasticity • Perfectly elastic (inverse) demand curve :

• is horizontal with εQ,P = −∞. • Perfectly inelastic (inverse) demand curve :

• is verticall with εQ,P = 0.

Price

Quantity

εQ,P = −∞

0

Perfectly elastic (inverse) demand

Price

Quantity

εQ,P = 0

0

Perfectly inelastic (inverse) demand

.. ECON 563 Managerial Economics

Module 3d: Applications

Marginal Revenue and εQ,P • The marginal revenue (MR) can be derived from a mar- ket demand curve.

• It measures the additional revenue due to a change in output.

MR = P

( 1 + εQ,P εQ,P

) .

• If −∞ < εQ,P < −1, MR > 0. • If εQ,P = −1, MR = 0. • If −1 < εQ,P , MR < 0.

Price 8

4

Quantity1890

Unit elastic demand, MR = 0.

Inelastic demand, MR < 0.

Elastic demand, MR > 0.

Linear Demand and Marginal Revenue

Cross Price Elasticity • Measures responsiveness of a percent change in de- mand for good X due to a percent change in the price of good Y .

εQdX ,PY = %∆QdX %∆PY

.

• If εQdX ,PY > 0, then X and Y are substitutes. • If εQdX ,PY < 0, then X and Y are complements.

Example • 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%, by how much will demand for clothing change ?

−0.18 = εQdX ,PY = %∆QdX

10 → %∆QdX = −1.8.

• So the demand for clothing is expected to decline by 1.8% when the price of food increases by 10%.

Cross Price Elasticity • Cross-price elasticity is important for firms selling mul- tiple 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 + εQdX ,PX

) +RY εQdY ,PX

] × %∆PX .

.. ECON 563 Managerial Economics

Module 3e: Income Elasticity

Income Elasticity • Measures responsiveness of a percent change in de- mand for good X due to a percent change in income.

εQdX ,M = %∆QdX %∆M .

• If εQdX ,M > 0, then X is a normal good. • If εQdX ,M < 0, then X is an inferior good.

Example • Suppose that the income elasticity of demand for trans- portation is estimated to be 1.80.

• If income is projected to decrease by 15%, what is the impact on the demand for transportation ?

1.8 = εQdX ,M = %∆QdX −15

→ %∆QdX = −27.

• Demand for transportation will decline by 27%. • Since demand decreases as income declines, transpor- tation is a normal good.

Elasticities for Linear Demand Functions • Given a linear demand function :

QdX = α0 + αXPX + αY PY + αMM + αHPH

• Own price elasticity : αX · PXQdX .

• Cross price elasticity : αY · PYQdX .

• Income elasticity : αM · MQdX .

Example Linear Demand QdX = 100− 3PX + 4PY − 0.01M + 2PH .

• where PX = $25 per unit, PY = $35, the company uti- lizes 50 units of advertising (PH), and average consu- mer income is $20, 000.

QdX = 100− 3(25) + 4(35)− 0.01(20000) + 2(50) = 65.

• Own price elasticity : −3 · 2565 = −1.15 • Cross price elasticity : 4 · 3565 = 2.15 • Income elasticity : −0.01 · 2000065 = −3.08

Elasticities for Non-linear Demand Functions • Given a non-linear demand function :

lnQdX = β0 + βX lnPX + βY lnPY + βM lnM + βH lnPH

• Own price elasticity : βX . • Cross price elasticity : βY . • Income elasticity : βM .

Example Demand for raincoats lnQdX = 10 − 1.25 lnPX + 3 lnR − 2 lnAY .

• whereR denotes the daily amount of rainfall andAY 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 ?

εQdX ,R = 3 = %∆QdX %∆R → 3 =

%∆QdX 10

.

• A 10% increase in rainfall will lead to a 30% increase in the demand for raincoats.

.. ECON 563 Managerial Economics

Module 3f: Demand Estimation

Regression Analysis How do we obtain information on the demand function ?

• Published studies • Hire consultant • Statistical technique called regression analysis using data on quantity, price, income and other important va- riables.

Ordinary 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 de- viation σ.

OLS Regression

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.

OLS Regression on Excel • To use the linear regression tool in Excel, the Data Ana- lysis Toolpak must be installed.

• To verify if it is installed, click Data from the Excel main menu.

• If you see the Data Analysis command in the Analysis group (far right), the Data Analysis Toolpak is already installed.

• If it is not installed, it is easy to install it from theOptions menu.

• Once installed, the linear regression command can be used directly.