### A.3 Elasticity - Meet the Faculty

```Readings
Baye 6th edition or 7th edition, Chapter 3
BA 445 Lesson A.3 Elasticity
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Overview
Overview
BA 445 Lesson A.3 Elasticity
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Overview
Own Price Elasticity measures how much the demand for a good responds to a
change in its own price. Demand is more inelastic (less responsive) when
products are distinguished from competitors.
Elasticity and Revenue are inversely related: If demand is elastic, then
increasing price decreases revenue from that product; if inelastic, increasing
price increases revenue. — So, elasticity affects optimal prices and quantities.
Cross Elasticity measures how the demand for one good responds to a change
in the price of another good. Cross elasticity affects the optimal choice of prices
and quantities for firms supplying multiple products.
Demand Functions are typically linear or log-linear. Linear demand simplifies
computing equilibrium price, quantity and surplus. Log-linear demand simplifies
computing elasticity.
BA 445 Lesson A.3 Elasticity
3
Own Price Elasticity
Own Price Elasticity
BA 445 Lesson A.3 Elasticity
4
Own Price Elasticity
Overview
Own Price Elasticity of Demand measures how much the
demand for a good responds to a change in its own price.
 For example, elasticity measures how much demand for
cars responds to a change in the price of cars.
 Demand is more inelastic (less responsive) when
products are distinguished from competitors.
Elasticity can also measure how much demand for a good
responds to a change in the price of any other good.
 For example, how much the demand for cars responds
to a change in the price of motorcycles.
BA 445 Lesson A.3 Elasticity
5
Own Price Elasticity
Two extremes of own price elasticity are:
• Perfect inelasticity, demand is perfectly insensitive
• Demand for an artificial limb may be perfectly inelastic for some
range of prices.
• Perfect elasticity, demand is perfectly sensitive
• Demand for Farmer Smith’s apples.
Price
Perfect
Inelasticity
Perfect
Elasticity
Price
Demand
Demand
Quantity
BA 445 Lesson A.3 Elasticity
Quantity
6
Own Price Elasticity
The primary factor determining own-price elasticity
is the number of substitutes competing against that good.
 Fewer substitutes implies lower demand elasticity.
 Innovating or distinguishing a product (making it
different) from competing goods reduces substitutes.
 Distinguishing a product may raise demand and profit, or
lower demand and profit.
 Google and Apple Inc. are innovative firms, with many
distinguished products. All have inelastic demand, but
not all have been profitable.
• Profitable Apple products include the Apple II computer, iPod,
iPhone, and the iPad tablet computer.
• Unprofitable Apple products include the Apple III computer and
the Newton tablet computer.
BA 445 Lesson A.3 Elasticity
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Own Price Elasticity
Graphing distinguished successes and failures
• The new product is more innovative or distinguished
than the generic product, and so has lower elasticity.
• New product demand may be higher or lower than
generic demand. (Higher demand means higher profit.)
Price
Profitable
innovation
Price
iPod
Demand
Generic
MP3
Player
Demand
Quantity
Unprofitable
innovation
Generic
Tablet
Computer
Demand
Newton
Demand
BA 445 Lesson A.3 Elasticity
Quantity
8
Own Price Elasticity
The Simplest Definition of Elasticity
• The simplest precise elasticity measure EQ,P of how
much the demand Q for a good X responds to a change
in price P (either the price Px of the same good X or the
price Py of another good Y) is a ratio of changes
DQ / DP
• For one example, if demand Qx for cars decreases 10 in
response to a \$20 increase in the price Px of cars, then
elasticity DQx/DPx = -10/20 = -0.5
• That definition is unacceptable, however, because it
depends on units of measure.

If the \$20 increase in the price Px of cars were measured as a
2000 cent increase, then DQx/DPx changes from -0.5 to -10/2000
= -0.005
BA 445 Lesson A.3 Elasticity
9
Own Price Elasticity
The Percent Definition of Elasticity
• One acceptable precise elasticity measure EQ,P of how
much the demand Q for a good X responds to a change
in price P is a ratio of percentages
EQ,P = %DQ / %DP
• For one example, if demand Qx for cars decreases 10%
in response to a 20% increase in the price Px of cars,
then elasticity EQx,Px = %DQx/%DPx = -10/20 = -0.5
• For another example, if demand Qx for cars increases
6% in response to a 2% increase in the price Py of
motorcycles 5%, then EQx,Py = %DQx/%DPy = 6/2 = 3
BA 445 Lesson A.3 Elasticity
10
Own Price Elasticity
The Sign of Elasticity tells the direction of a
relationship:
• If EQ,P < 0, then Q and P are negatively (or inversely)
related.
 For example, the observed law of demand for all
goods considered in this class is that demand is
negatively related to its own price, so own-price
elasticity of demand is negative, EQx,Px < 0.
• If EQ,P > 0, then Q and P are positively (or directly)
related.
• If EQ,P = 0, then Q and P are unrelated.
BA 445 Lesson A.3 Elasticity
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Own Price Elasticity
Percentages in the Definition of Elasticity have both
• One good property of percentages: they do not depend
on units of measure.
 Doubling price from \$1 to \$2 in dollars, or from 100c
to 200c in cents, is the same percent increase.
• One bad property of percentages: they are inaccurate.
 100 to 200 is a 100% increase, but 200 to 100 is only
a 50% decrease. So, the percent definition of
elasticity depends on whether price increases or
decreases.
BA 445 Lesson A.3 Elasticity
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Own Price Elasticity
The Derivative Definition of Elasticity
• An alternative acceptable elasticity measure uses
derivatives
EQ,P = dQ/dP . P/Q
• The derivative definition shares the good property of the
percentage definition (elasticity does not depend on the
units of measure).
• The derivative definition avoids the inaccuracy of the
percentage definition. The derivative definition of
elasticity does not depend on whether price increases or
decreases
 Elasticity EQ,P is the same whether P increases (dP >
0), or P decreases (dP < 0)
BA 445 Lesson A.3 Elasticity
13
Own Price Elasticity
Own Price Elasticity of Demand
EQx,Px = %DQx / %DPx or EQx,Px = dQx/dPx . Px/Qx
is classified into three categories, according to its
magnitude:
• Elastic (sensitive demand): | EQx,Px | > 1
• Inelastic (insensitive demand): | EQx,Px | < 1
• Unit elastic: | EQx,Px | = 1
BA 445 Lesson A.3 Elasticity
14
Elasticity and Revenue
Elasticity and Revenue
BA 445 Lesson A.3 Elasticity
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Elasticity and Revenue
Overview
Elasticity and Revenue are related when a supplier
changes either price or quantity. When price increases,
revenue increases if demand is inelastic but revenue
decreases if demand is elastic. Likewise, when quantity
increases, revenue decreases if demand is inelastic but
revenue increases if demand is elastic. — So, elasticity
affects whether to increase or decrease price or
quantity.
BA 445 Lesson A.3 Elasticity
16
Elasticity and Revenue
Choosing price or output quantity is the simplest
management decision. Each firm chooses one variable,
with the other variable defined by demand.


Examples of choosing price include choosing the advertised price
first, then producing just enough to satisfy demand at that price.
Examples of choosing output quantity include choosing the output
first, then adjusting price just enough to sell all output.
Price
Price
Setting Quantity
Setting Price
Demand
P
P
Q
Quantity
Q
BA 445 Lesson A.3 Elasticity
Demand
Quantity
17
Elasticity and Revenue
First, consider firms that choose price.
Predicting revenue change from a price change
follows the formula
DR = Rx(1+EQx,Px) . %DPx
where
 DR is the change in revenue
 Rx is the initial revenue from good X
 EQx,Px is the own price elasticity of demand for good X
 %DPx is the change in the price Px of good X expressed
as a fraction.
BA 445 Lesson A.3 Elasticity
18
Elasticity and Revenue
For example, suppose
 You only sell burgers.
 Your current revenue from burgers is \$100.
 The own price elasticity is -0.5
 You increase price 10%
Then, the formula
DR = Rx(1+EQx,Px) . %DPx
implies revenue changes
DR = (\$100(1-0.5)) . (0.10) = \$5
That is, revenue increases \$5.
BA 445 Lesson A.3 Elasticity
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Elasticity and Revenue
When demand is inelastic, revenue moves in the
same direction of a change in price.
DR = Rx(1+EQx,Px) . %DPx
• When demand is inelastic, |EQx,Px| < 1
 EQ ,P > -1, so 1+ EQ ,P > 0, so if price increases,
x x
x x
%DP > 0 and DR > 0.
 Increased price implies increased total revenue.
 It is definitely profitable to raise price since raising
price increases revenue and decreases as higher
price leads to lower supply quantity.)
BA 445 Lesson A.3 Elasticity
20
Elasticity and Revenue
When demand is elastic, revenue moves in the
opposite direction of a change in price.
DR = Rx(1+EQx,Px) . %DPx
• When demand is elastic, |EQx,Px| > 1
 EQ ,P < -1, so 1+ EQ ,P < 0, so if price increases,
x x
x x
%DP > 0 and DR < 0.
 Increased price implies decreased total revenue.
 It may be profitable to lower price since lowering
price increases revenue. (Profit also depends on
how much cost increases as lower price leads to
higher supply quantity.)
BA 445 Lesson A.3 Elasticity
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Elasticity and Revenue
Now consider firms that choose quantity.
Revenue change from a quantity change is the
opposite of the revenue change from a price change.
• When demand is inelastic, as quantity increases

Price decreases because of the law of demand.

So, revenue increases since revenue and price move in
opposite directions.

Marginal revenue MR is the change in revenue as quantity
increases, so MR < 0
• When demand is elastic, as quantity increases

Price decreases.

Revenue decreases.

Marginal revenue is positive.
• When demand is unit elastic, marginal revenue is 0
BA 445 Lesson A.3 Elasticity
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Elasticity and Revenue
Graphing elasticity and revenue when demand is
linear
P
100
TR
0
10
20
30
40
50 Q
0
BA 445 Lesson A.3 Elasticity
Q
23
Elasticity and Revenue
Graphing elasticity and revenue when demand is
linear
P
100
TR
80
800
0
10 20
30
40
50
Q
0
10
BA 445 Lesson A.3 Elasticity
20
30
40
50 Q
24
Elasticity and Revenue
Graphing elasticity and revenue when demand is
linear
P
100
TR
80
1200
60
800
0
10
20 30
40
50
Q
0
10
BA 445 Lesson A.3 Elasticity
20
30
40
50 Q
25
Elasticity and Revenue
Graphing elasticity and revenue when demand is
linear
P
100
TR
80
1200
60
40
800
0
10
20
30 40
50
Q
0
10
BA 445 Lesson A.3 Elasticity
20
30
40
50 Q
26
Elasticity and Revenue
Graphing elasticity and revenue when demand is
linear
P
100
TR
80
1200
60
40
800
20
0
10
20
30
40
50
Q
0
10
BA 445 Lesson A.3 Elasticity
20
30
40
50 Q
27
Elasticity and Revenue
Where quantity is less than 25, a price decrease causes a quantity
increase and an increase in revenue. So, demand is elastic since
price and revenue are negatively related (and quantity and revenue
are positively related).
P
100
TR
Elastic
80
1200
60
40
800
20
0
10
20
30
40
50
Q
0
10
20
30
40
50 Q
Elastic
BA 445 Lesson A.3 Elasticity
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Elasticity and Revenue
Where quantity is greater than 25, a price decrease causes a
quantity increase and a decrease in revenue. So, demand is
inelastic since price and revenue are positively related (and
quantity and revenue are negatively related).
P
100
TR
Elastic
80
1200
60
Inelastic
40
800
20
0
10
20
30
40
50
Q
0
10
20
Elastic
BA 445 Lesson A.3 Elasticity
30
40
50 Q
Inelastic
29
Elasticity and Revenue
Unit elasticity divides elasticity from inelasticity.
P
100
TR
Unit elastic
Elastic
Unit elastic
80
1200
60
Inelastic
40
800
20
0
10
20
30
40
50
Q
0
10
20
Elastic
BA 445 Lesson A.3 Elasticity
30
40
50 Q
Inelastic
30
Elasticity and Revenue
Marginal Revenue is the extra revenue from increasing output. It is
positive when output is less than 25 and demand is elastic, and is
negative when output is greater than 25 and demand is inelastic.
P
100
TR
Unit elastic
Elastic
Unit elastic
80
1200
60
Inelastic
40
800
20
0
10
20
30
40
50
Q
0
10
20
Elastic
BA 445 Lesson A.3 Elasticity
30
40
50 Q
Inelastic
31
Elasticity and Revenue
P
100
Elastic
80
60
40
20
0
10
20
For any linear inverse demand
function, P(Q) = a - bQ, then
MR(Q) = a - 2bQ. So,
Unit elastic
• MR > 0, where demand is
elastic
Inelastic
• MR = 0, where demand is
unit elastic
• MR < 0, where demand is
inelastic
40
50
Q
MR
BA 445 Lesson A.3 Elasticity
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Elasticity and Revenue
Example: To maximize revenue
when demand is
Q = 12 – 0.2 P,
first invert demand, to
TR
0.2 P = 12 – Q and
P = 60 – 5Q.
Then, compute marginal
revenue
MR = 60 – 10Q.
Then, set MR = 0, to get Q = 6.
Then, set P = 60 – 5(6) = 30.
Demand is unit elastic when
revenue is maximized.
Unit elastic
Q
0
Elastic
BA 445 Lesson A.3 Elasticity
Inelastic
33
Cross Elasticity
Cross Elasticity
BA 445 Lesson A.3 Elasticity
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Cross Elasticity
Overview
Cross Price Elasticity measures how the demand for one
good responds to a change in the price of another good.
Cross elasticity affects the optimal choice of prices and
quantities for firms supplying multiple products.
BA 445 Lesson A.3 Elasticity
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Cross Elasticity
Cross Price Elasticity of Demand is defined like own
price elasticity
EQx,Py = %DQx / %DPy or EQx,Py = dQx/dPy . Px/Qy

Unlike the negative own price elasticity EQx,Px < 0, cross
price elasticity can be positive or negative, depending
on how good relate.

If EQx,Py > 0, then X and Y are (gross) substitutes.

If EQx,Py < 0, then X and Y are (gross) complements.
BA 445 Lesson A.3 Elasticity
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Cross Elasticity
Predicting revenue change from a price change
follows the formula for multiple products
DR = (Rx(1+EQx,Px) + RyEQy,Px) . %DPx
where
 DR is the change in revenue from the two products
 Rx is the initial revenue from good X
 EQx,Px is the own price elasticity of demand for good X
 Ry is the initial revenue from good Y
 EQy,Px is the cross elasticity of demand for good Y
 %DPx is the change in the price Px of good X expressed
as a fraction.
BA 445 Lesson A.3 Elasticity
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Cross Elasticity
For example, suppose
 You sell only burgers and fries.
 Current revenue is \$100 from burgers, \$50 from fries.
 The own price elasticity of burgers is -0.5 (inelastic).
 The cross price elasticity of fries when the price of
burgers changes is -2 (gross complements).
 You increase burger price 20%
Then, the formula
DR = (Rx(1+EQx,Px) + RyEQy,Px) . %DPx
implies revenue changes
DR = (\$100(1-.5) + \$50(-2)) . (0.20) = -\$10
That is, revenue decreases \$10.
BA 445 Lesson A.3 Elasticity
38
Demand Functions
Demand Functions
BA 445 Lesson A.3 Elasticity
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Demand Functions
Overview
Demand Functions are typically linear or log-linear.
Linear demand simplifies computing equilibrium price,
quantity and surplus. Log-linear demand simplifies
computing elasticity.
BA 445 Lesson A.3 Elasticity
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Demand Functions
Interpreting Linear Demand
Example:
QX = 10 – 2PX + 3PY + 5M
• Law of demand holds (coefficient of PX is negative).
• X and Y are gross substitutes (coefficient of PY is
positive).
• X is a normal good (coefficient of income M is positive).
BA 445 Lesson A.3 Elasticity
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Demand Functions
Computing Elasticity from Linear Demand
Use the derivative definition of elasticity:
QX = 10 – 2PX + 3PY + 5M
• Own price elasticity (depends on price and quantity):
EQx,Px = dQx/dPx . Px/Qx = - 2 Px/Qx
• Cross price elasticity (depends on price and quantity):
EQx,Py = dQx/dPy . Py/Qx = 3 Py/Qx
BA 445 Lesson A.3 Elasticity
42
Demand Functions
Computing Elasticity from Log-Linear Demand
Use the derivative definition of elasticity:
ln(QX) = 10 – 2ln(PX) + 3ln(PY) + 5ln(M)
• Own price elasticity (not depend on price and quantity):
EQx,Px = dQx/dPx . Px/Qx = - 2
• Cross price elasticity (not depend on price and quantity):
EQx,Py = dQx/dPy . Py/Qx = 3
BA 445 Lesson A.3 Elasticity
43
Demand Functions
Graphs of Linear and Log-Linear Demand
Price
Price
Linear
Demand
Log Linear
Demand
Quantity
BA 445 Lesson A.3 Elasticity
Quantity
44
Summary
Summary
BA 445 Lesson A.3 Elasticity
45
Summary
Applications of Elasticity
• Pricing and managing cash flows.
• Effect of changes in competitors’ prices.
BA 445 Lesson A.3 Elasticity
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Summary
Example 1: Pricing and Cash Flows
• According to an FTC Report by Michael Ward, AT&T’s
own price elasticity of demand for long distance services
is -8.64.
• AT&T needs to boost revenues in order to meet it’s
marketing goals.
• To accomplish this goal, should AT&T raise or lower it’s
price?
BA 445 Lesson A.3 Elasticity
47
Summary
• Since demand is elastic, a reduction in price will increase
quantity demanded by a greater percentage than the
price decline, resulting in more revenues for AT&T.
BA 445 Lesson A.3 Elasticity
48
Summary
Example 2: Quantifying the Change
• If AT&T lowered price by 3 percent, what would happen
to the volume of long distance telephone calls routed
through AT&T?
BA 445 Lesson A.3 Elasticity
49
Summary
• Calls would increase by 25.92 percent.
EQX , PX
%DQX
 8.64 
%DPX
d
%DQX
 8.64 
 3%
d
 3%   8.64  %DQX
d
%DQX  25.92%
d
BA 445 Lesson A.3 Elasticity
50
Summary
Example 3: Effect of a change in a competitor’s price
• According to an FTC Report by Michael Ward, AT&T’s
cross price elasticity of demand for long distance
services is 9.06.
• If competitors reduced their prices by 4 percent, what
would happen to the demand for AT&T services?
BA 445 Lesson A.3 Elasticity
51
Summary
• AT&T’s demand would fall by 36.24 percent.
EQX , PY
%DQX
 9.06 
%DPY
d
%DQX
9.06 
 4%
d
 4%  9.06  %DQX
d
%DQX  36.24%
d
BA 445 Lesson A.3 Elasticity
52
Review Questions
Review Questions
 You should try to answer some of the review
questions (see the online syllabus) before the next
class.
request to discuss their answers to begin the next class.
 Your upcoming Exam 1 and cumulative Final Exam
will contain some similar questions, so you should
eventually consider every review question before taking