READ BEFORE YOU APPLY - 3 Short Papers 3 pages each ----- DUE in 12hours

Instructor – Dr. Gabriel Axarlian

• Chapter 3 Quantitative Demand Analysis

o The Elasticity Concept

o Own Price Elasticity of Demand

o Cross Price Elasticity of Demand

o Income Elasticity of Demand

o Obtaining Elasticities from Demand Functions

The 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

• For example, the elasticity of your grade with respect to studying, denoted 𝐄𝐆,𝐒 is the percentage change in your grade (%ΔG) that will result from a given percentage change in the time you spend studying (%ΔS)

𝑬𝑮,𝑺 = %ΔG

%ΔS

%ΔG = ΔG

𝑮 𝒙𝟏𝟎𝟎 %ΔS =

ΔS

𝑺 𝒙𝟏𝟎𝟎

𝑬𝑮,𝑺 = %ΔG

%ΔS =

ΔG 𝑮 𝒙𝟏𝟎𝟎

ΔS 𝑺 𝒙𝟏𝟎𝟎

=

ΔG 𝑮 ΔS 𝑺

= 𝜟𝑮

𝑮

S

Δ𝑺 = 𝜟𝑮

Δ𝑺

S

𝑮

= 𝜟𝑮

Δ𝑺

S

𝑮

• Since in calculus, 𝜟𝑮

Δ𝑺 can be approximated by the derivative (slope of a curve) we can write

the elasticity equation as 𝑬𝑮,𝑺 =

dG

dS

S

𝑮

Own Price Elasticity of Demand

Own Price Elasticity of Demand A measure of the responsiveness of the quantity demanded of a good to a change in the price of that good; the percentage change in quantity demanded divided by the percentage change in the price of the good

• For a demand function 𝑸𝒙 𝒅 = 𝒇(𝑷𝒙,𝑷𝒚,𝑴,𝑯)

the own price elasticity of demand is denoted

𝑬𝑸𝒙,𝑷𝒙 = %Δ𝑸𝒙

𝒅

%Δ𝑷𝒙 =

Δ𝑸𝒙 𝒅

Δ𝑷𝒙

𝑷𝒙

𝑸𝒙 𝒅 = 𝒅𝑸𝒙

𝒅

𝒅𝑷𝒙

𝑷𝒙

𝑸𝒙 𝒅

• By the law of demand the own price elasticity of demand is always a negative number

• Demand is said to be elastic if the absolute value of the own price elasticity is greater than 1

𝐄𝐐𝐱,𝐏𝐱 > 𝟏

• Demand is said to be inelastic if the absolute value of the own price elasticity is less than 1

𝐄𝐐𝐱,𝐏𝐱 < 𝟏

• Demand is said to be unitary (or ‘unit’) elastic if the absolute value of the own price elasticity is equal to 1

𝐄𝐐𝐱,𝐏𝐱 = 𝟏

Elasticity of Total Revenue

• The absolute value of the own price elasticity gets larger as price increases

𝑸𝒙 𝒅 = 𝟖𝟎 − 𝟐𝑷𝒙

o The slope of this linear demand function is constant

Δ𝑸𝒙 𝒅

Δ𝑷𝒙 = −𝟐,

which implies 𝑬𝑸𝒙,𝑷𝒙 =

Δ𝑸𝒙 𝒅

Δ𝑷𝒙

𝑷𝒙 𝑸𝒙

increases in absolute value as 𝑷𝒙 increases

• This result tells us that the own price elasticity of demand varies along a linear demand curve

• When the absolute value of the own price elasticity is less than 1, an increase in price increases total revenue

• When the absolute value of the own price elasticity is greater than 1 an increase in price leads to a reduction in total revenue

• Total revenue is maximized at point E, where the own price elasticity equals −1

Demand is elastic: a higher price reduces total revenue

Elastic

Inelastic

Unit-elastic

Demand is inelastic: a higher price increase total revenue

0 0 1 2 3 4 5 6 7 1098

$25 24 21

16

9

Quantity

Total revenue

0 1 2 3 4 5 6 7 1098

$10 9 8 7 6 5 4 3 2 1

Quantity

Price

$0 1 2 3 4 5 6 7 8 9

10

$0 9

16 21 24 25 24 21 16 9 0

10 9 8 7 6 5 4 3 2 1 0

Total

revenue

Quantity demanded

Demand Schedule and Total Revenue for a Linear Demand Curve

Price D

(Principle) Total Revenue Test – If demand is elastic an increase (decrease) in price will lead to a decrease (increase) in total revenue

If demand is inelastic, an increase (decrease) in price will lead to an increase (decrease) in total revenue

Total revenue is maximized at the point of unit elasticity

Example

Dell recently faced a dilemma regarding its pricing strategy for computers: Should it increase prices to boost cash flow or adopt a “cut price and make it up in volume” strategy?

Based on a careful analysis of its demand, the company decided to adopt the latter strategy and reduced prices in order to increase revenues

To understand why suppose the research department of a computer company estimates that the own price elasticity of demand for a particular desktop computer is −1.7 and it cuts price by 5%

𝑬𝑸𝒙,𝑷𝒙 = %Δ𝑸𝒙

𝒅

%Δ𝑷𝒙 = −𝟏.𝟕 %Δ𝑷𝒙 = −𝟓

%Δ𝑸𝒙 𝒅

−𝟓 = −𝟏.𝟕−𝟓 × × −𝟓

%Δ𝑸𝒙 𝒅 = 𝟖.𝟓

Since the percentage increase in quantity demanded is greater than the percentage decline in prices the price cut will raise the firm's sales revenues

• Producers of generic (unbranded) products, such as aspirin, may face a demand curve that is perfectly elastic

Quantity

Price

Demand

Perfectly Inelastic

𝐸 𝑄𝑋

𝑑,𝑃𝑋 = 0

Demand

𝐸 𝑄𝑋

𝑑,𝑃𝑋 = −∞

Perfectly elastic

• In contrast, when demand is perfectly inelastic, consumers do not respond at all to changes in price

Factors Affecting the Own Price Elasticity

• There are three factors that affect the magnitude of the own price elasticity of a good

1. Available Substitutes

• A key implication of the effect of the number of close substitutes on the elasticity of demand is that the demand for broadly defined commodities tends to be more inelastic than the demand for specific commodities

2. Time

3. Expenditure Share

• Goods that comprise a relatively small share of consumers' budgets tend to be more inelastic than goods for which consumers spend a sizable portion of their incomes

Marginal Revenue and the Own Price Elasticity of Demand

Quantity0

𝑃

MR

Price

6

Demand

1

6

Marginal Revenue (MR)

• The line labeled MR in the figure is the marginal revenue associated with each price– output pair on the demand curve

3

Quantity0

𝑃

MR

Price

6

Demand

1

6

Marginal Revenue (MR)

• Suppose consumers purchase 1 unit of output at a price of $5 per unit

• If the price falls to $4 con

Units Sold at $5 Units Sold at $4

Revenue$5 × 1 = $5

1 2

$4 × 2 = $8

Units Sold at $5 Units Sold at $4

Revenue

Quantity0

𝑃

MR

3

Price

6

Demand

1

6

Marginal Revenue (MR)

• By the total revenue test, this means that demand is elastic over this range

Unitary

• When elasticity is unitary, MR=0

$5 × 1 = $5

1 2

• Had the price reduction decreased total revenues, demand would be inelastic over the range and MR would be negative

$4 × 2 = $8

• The more inelastic the demand for a product, the greater the decline in revenue that results from a price cut

• This intuition leads to the following general relationship between marginal revenue and the elasticity of demand:

𝐌𝐑 = 𝑷 𝟏 + 𝑬

𝑬

where P is price and E is the own price elasticity of demand

• When −∞ < E < −1, demand is elastic, and the formula implies that MR is positive (ie E = -1.8)

• When E = −1, demand is unitary elastic, and marginal revenue is zero

• When −1 < E < 0, demand is inelastic, and marginal revenue is negative ((ie E = -.8)

𝐌𝐑 = 𝑷 𝟏 + (−𝟏.𝟖)

−𝟏.𝟖 > 𝟎

𝐌𝐑 = 𝑷 𝟏 + (−𝟏)

−𝟏 = 𝟎

𝐌𝐑 = 𝑷 𝟏 + (−.𝟖)

−.𝟖 < 𝟎

Cross-Price Elasticity

Cross-Price Elasticity A measure of the responsiveness of the demand for a good to changes in the price of a related good; the percentage change in the quantity demanded of one good divided by the percentage change in the price of a related good

𝑬𝑸𝒙,𝑷𝒚 = %Δ𝑸𝒙

𝒅

%Δ𝑷𝒚

o 𝑬𝑸𝒙,𝑷𝒚 > 0 whenever goods X and Y are substitutes

o 𝑬𝑸𝒙,𝑷𝒚 < 0 whenever goods X and Y are compliments

• Clothing and food have a cross-price elasticity of −0.18

• Cross-price elasticities play an important role in the pricing decisions of firms that sell multiple products

• Generally, suppose a firm's revenues are derived from the sales of two products, X and Y

𝑹 = 𝑹𝒙 + 𝑹𝒚 • The impact of a small percentage change in the price of product X on the total revenues of

the firm is

∆𝑹 = 𝑹𝒙 𝟏 + 𝑬𝑸𝒙,𝑷𝒙 + 𝑹𝒚𝑬𝑸𝒚,𝑷𝒙 × %Δ𝑷𝒙

what would happen to the firm's total revenues if it reduced the price of hamburgers by 1 percent?

Suppose a restaurant earns $4,000 per week in revenues from hamburger sales (product X) and $2,000 per week from soda sales (product Y)

∆𝑹 = [

𝑹𝒙 = $𝟒𝟎𝟎𝟎 𝑹𝒚 = $𝟐𝟎𝟎𝟎

If the own price elasticity of demand for burgers is

𝑬𝑸𝒙,𝑷𝒙 = −𝟏.𝟓

and the cross-price elasticity of demand between sodas and hamburgers is

𝑬𝑸𝒚,𝑷𝒙 = −𝟒.𝟎

∆𝑹 = 𝑹𝒙 𝟏 + 𝑬𝑸𝒙,𝑷𝒙 + 𝑹𝒚𝑬𝑸𝒚,𝑷𝒙 × %Δ𝑷𝒙

%Δ𝑷𝒙 = −.𝟎𝟏

∆𝑹 = −𝟏𝟎𝟎𝟎𝟎 × −.𝟎𝟏

∆𝑹 = 𝟒𝟎𝟎𝟎 −.𝟓 − 𝟖𝟎𝟎𝟎 × −.𝟎𝟏

𝟒𝟎𝟎𝟎(𝟏 − 𝟏.𝟓)+2000(−𝟒)]× −.𝟎𝟏

∆𝑹 = −𝟐𝟎𝟎𝟎 − 𝟖𝟎𝟎𝟎 × −.𝟎𝟏 _ __ __ __ __ __ __

∆𝑹 = $𝟏𝟎𝟎

Example

Income Elasticity

Income Elasticity A measure of the responsiveness of the demand for a good to changes in the consumer income; the percent change in quantity demanded divided by the percentage change in income

𝑬𝑸𝒙𝒅,𝑴 =

%Δ𝑸𝒙 𝒅

%Δ𝑴

o 𝑬𝑸𝒙𝒅,𝑴 > 0 when X is a normal good

o 𝑬𝑸𝒙𝒅,𝑴 < 0 when X is an inferior good

• Since the income elasticity is positive, we transportation is a normal good

• Since the income elasticity for transportation is greater than 1, we know that expenditures on transportation grow more rapidly than income.