Chapter 4
Additional
Derivative Topics
Section 7
Elasticity of Demand
Introduction
 How does a change in price of an item affect the demand
for that item?
 When will a price increase lead to an increase in revenue?
 The answer to these types of questions will be answered in
this lesson.
 In this lesson we will learn the concepts:
• Relative rate of change
• Percentage rate of change
• Elasticity of demand
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Relative Rates vs Absolute Rates
 Suppose:
• 𝑓(𝑡) = the cost of a pair of shoes at time t (in years)
• 𝑔(𝑡) = the cost of a new car at time t (in years)
 Then:
• 𝑓′(𝑡) = 3 means the price of shoes is increasing at a rate
of $3 per year
• 𝑔′(𝑡) = 300 means that the price of a new car is
increasing at a rate of $300 dollars per year
 Does that mean that car prices are rising 100 times as fast
as shoe prices?
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Relative Rates vs Absolute Rates
 In absolute terms, the answer is yes. But this doesn’t take into
account the large price difference between cars and shoes.
 If shoe prices are increasing at a rate of $3 per year and the
average cost of a pair of shoes is $60, then the relative rate of
increase is:
•
3
60
•
300
15000
= .05 which means that shoe prices are increasing at a
relative rate of 5% per year.
 If the average cost of a car is $15,000 then the relative rate of
increase is:
= .02 for a relative rate of 2% per year
 Therefore, in a relative sense, car costs are increasing less
quickly than shoe prices.
4
Relative Rate of Change
 In general, if 𝑓(𝑡) is the price of an item at time t:
 The “absolute” rate of change is 𝑓′(𝑡)
𝑓′(𝑡)
 The relative rate of change is
𝑓(𝑡)
 In the world of economics and business, relative rates of
change are often more meaningful than absolute rates of
change.
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Relative and Percentage
Rates of Change
The relative rate of change is defined as 𝑓′(𝑥)
𝑓(𝑥)
They sometimes call this the logarithmic derivative of f (x)
because:
f (x) d
f (x)

dx
ln f (x)
The percentage rate of change of a function f (x) is
𝑓′(𝑥)
∙ 100
𝑓(𝑥)
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Example 1
Find the relative rate of change of f (x) = 50x – 0.01x2
𝑓′(𝑥)
50 − .02𝑥
=
𝑓(𝑥) 50𝑥 − 0.01𝑥 2
Find the relative rate of change when x = 10
𝑓′(𝑥)
50 − .02(10)
49.8
≈ 0.1
=
=
2
𝑓(𝑥) 50(10) − 0.01(10)
499
Find the percentage rate of change when x = 10
0.1 = 10%
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Example 2
 An investor estimates that if a piece of land is owned for t
years, then it will be worth 𝑓 𝑡 = 300 + 𝑡 2 (in thousands
of dollars). Find the relative rate of change (as a percent)
after 10 years.
𝑓′(𝑡)
2𝑡
=
𝑓(𝑡)
300 + 𝑡 2
20
2(10)
= .05 = 5%
=
=
2
400
300 + (10)
The land’s value is increasing at a relative
rate of 5% per year after 10 years.
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Example 3
A model for the real GDP (gross domestic product expressed in
billions of 1996 dollars) from 1995 to 2002 is given by
f (t) = 300t + 6,000
where t is years since 1990. Find the percentage rate of change
of f (t) for 1995.
For 1995, t = 5:
𝑓′(𝑡)
300
=
1
𝑓(𝑡)
300𝑡 + 6000
= .04
5 + 20
300
=
300(𝑡 + 20)
The GDP is increasing at a relative
1
rate of 4% per year in 1995.
=
𝑡 + 20
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Price vs Demand
 In general:
• If the price of an item increases, the demand decreases.
• If the price of an item decreases, the demand increases.
 Price and demand behave in “opposite” directions.
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Elasticity of Demand
Elasticity of demand describes how a change in the price of a
product affects the demand.
Assume that f (p) is the demand of a product at price p. Then
we define
relative rate of change in demand
Elasticity of Demand =
relative rate of change in price
p  f  ( p)
E ( p)  
f ( p)
(Derivation of this formula is given
in your textbook.)
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Elasticity of Demand
Interpretation
E(p)
Demand
Interpretation
E(p) < 1
Inelastic
E(p) > 1
Elastic
E(p) = 1
Unit
Demand is insensitive to changes in
price. (A change in price produces a
small change in demand.)
Demand is sensitive to changes in
price. (A change in price produces a
large change in demand.)
A change in price produces the same
change in demand.
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Example 4
For the price-demand equation
x = f (p) = 1875 - p2,
A) Determine whether demand is elastic, inelastic, or unit
for p = 15, 25, and 40.
B) Interpret the results if the price increases by 10% in
each case.
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Example 4
For the price-demand equation
x = f (p) = 1875 - p2,
A) Determine whether demand is elastic, inelastic, or unit
for p = 15, 25, and 40.
2
2𝑝
𝑝(−2𝑝)
𝑝 ∙ 𝑓′(𝑝)
=
=−
𝐸 𝑝 =−
2
1875 − 𝑝2
1875 − 𝑝
𝑓(𝑝)
E(15) = 0.27 < 1; demand is inelastic
E(25) = 1; demand has unit elasticity
E(40) = 11.6 > 1; demand is elastic
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Example 4
B) Interpret the results if the price increases by 10% in each
case.
When p=15, E(p)=.27 (inelastic)
So if the $15 price increases by 10%, then the demand
decreases by (.27)(10%) = .027%
When p=25, E(p)=1 (unit)
So if the $25 price increases by 10%, then the demand
decreases by (1)(10%) = 10%
When p=40, E(p)=11.6 (elastic)
So if the $40 price increases by 10%, then the demand
decreases by (11.6)(10%) = 1.16%
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Elasticity of Demand
for Different Products
Different products have different elasticities. If there
are close substitutes for a product, or if the product is
a luxury rather than a necessity, the demand tends to
be elastic. Examples of products with high
elasticities are jewelry, furs, or furniture.
On the other hand, if there are no close substitutes or
the product is a necessity, the demand tends to be
inelastic. Examples of products with low elasticities
are milk, sugar, and light bulbs.
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Homework
#4-7
Pg 260
(1-13 odd, 19,
21, 33, 35)
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