DEMAND ELASTICITY

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Overview

• Context: Product manager wants to estimate impact of pricechange on sales (quantity and revenue). How sensitive is demandto price? How important is the pricing of competing products?

• Concepts: demand elasticity, cross-elasticity

• Economic principle: sometimes reducing price attracts many morecustomers, sometimes very few

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How sensitive is demand to price changes?

• Example 1: world oil demand decreases by 1.3 million barrels aday when price increases from $50 to $60 dollars per barrel.Would you consider the demand for oil very sensitive or not verysensitive to price?

• Example 2: demand for sugar in Europe decreases by 1 milliontones per day when average retail price increases from e.80 toe.90 per kilo. Can you compare the demand for sugar in Europeto the worldwide demand for oil?

• Problem: by measuring the slope of the demand curve, we arestuck with units: barrels, dollars, kilos, euros, and so on.

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Demand elasticity: definition

✏ ⌘d q

q

d p

p

=d q

d p

p

q

=d log q

d log p

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Demand elasticity: definition

✏ ⌘d q

q

d p

p

=d q

d p

p

q

=d log q

d log p

⇡� q

q

� p

p

=% � quantity

% � price

3

Demand elasticity: definition

✏ ⌘d q

q

d p

p

=d q

d p

p

q

=d log q

d log p

⇡� q

q

� p

p

=% � quantity

% � price

⇡ � log quantity

� log price

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Demand elasticity: notes

• Elasticity and slope are not the same

• Elasticity is independent of units

• Knowing price change, quantity change may be estimated basedon elasticity:

� q

q

⇡ ✏� p

p

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Examples

Product Elasticity

Milk -0.5

Cigarettes -0.5

Beer -0.8

Apples -1.3

US luxury cars in US -1.9

Foreign luxury cars in US -2.8

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Elasticity illustrated: linear demand

p

q

✏ = 0

|✏| < 1

|✏| = 1

|✏| > 1

✏ = �1

•

•

•

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Elasticity illustrated: constant elasticity

p

q

✏ = �1

|✏| > 1

|✏| < 1✏ = 0

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Elasticity, price, and revenue

Revenue ⌘ R = p ⇥ q. Therefore:

�R

R

=� (p ⇥ q)

(p ⇥ q)⇡ q� p + p� q

(p ⇥ q)

=� p

p

+� q

q

=� p

p

+ ✏� p

p

=� p

p

(1 + ✏)

If price falls, then:

•Revenue rises if ✏ < �1 (that is, |✏ | > 1)

•Revenue falls if ✏ > �1 (that is, |✏ | < 1)

•Revenue is unchanged if ✏ = �1

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Elasticity, price, and revenue

Examples: for a 1% decrease in price,

•Cigarettes: revenue falls approx 0.5% = �.1%⇥(1+(�.5))

•U.S. luxury cars: revenue rises approx 0.9%

•Foreign luxury cars: revenue rises approx 1.8%

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Revenue change from price decrease

�p

�q

p

q

E

1

E

2

q

p

Loss L = q (�� p), Gain G = p � q

G > L i↵ p � q > q (�� p)

i↵

� q

� p

p

q

< �1

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Cross-price elasticity

• Idea: How sensitive is demand for your product to prices ofcompeting products? Answer: Cross-price elasticity.

✏ij

=

d q

i

q

i

d p

j

p

j

• Jargon:

� If ✏ij

> 0, we say i and j are substitutes

� If ✏ij

< 0, we say i and j are complements

� If ✏ij

= 0, we say i and j are independent

• Examples?

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Income elasticity

• Idea: How sensitive is demand for your product to consumerincome? Answer: income elasticity.

✏y

=

d q

q

d y

y

• Jargon:

� Inferior good: ✏y

< 0

� Normal good: ✏y

> 0

� Necessity: 0 < ✏y

< 1

� Luxury: ✏y

> 1

• Examples?

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Example: U.S. gasoline demand (cont)

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Example: gasoline demand

• Based on U.S. data from 1953–2004,

ln q = �16.1� 0.03 ln p + 1.17 ln y � 0.33 ln c + 0.85 ln n

where

q: gasoline consumption

p: gasoline price

y : per capita income

c : price of cars

n: population

• What is the gasoline demand elasticity? Income elasticity? Crossprice elasticity w.r.t. cars? How do you classify the good“gasoline”?

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Example: gasoline demand

• From 1953 to 2004, p, y , c and n increased at the following

annual rates: 3.9, 2.2, 2.0, 1.2%. How much do you expect

demand to have grown?

• Recall that d z

z

= ✏zx

d x

x

, for any z and x . Hence,

� q

q

= �.03⇥ 3.9% + 1.17⇥ 2.2%� 0.33⇥ 2 + .85⇥ 1.2%

= �.117% + 2.574%� 0.66% + 1.02%

= 2.817%

• Note: actual growth rate was 2.7%

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Takeaways

• The elasticity (of demand) measures sensitivity of buyers tochanges in price

• It’s useful for computing impact of price changes on quantity andrevenue

• Cross-price and income elasticities measure sensitivity to otherfactors

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