Transcript Cross-elasticity of demand

Learning Objectives
• Define and measure elasticity
• Apply concepts of price elasticity,
cross-elasticity, and income elasticity
• Understand determinants of
elasticity
• Show how elasticity affects revenue
Price Elasticity of Demand
(E)
• Measures responsiveness or sensitivity of
consumers to changes in the price of a good
•
%Q
E
%P
• P & Q are inversely related by the law of
demand so E is always negative
– The larger the absolute value of E, the more
sensitive buyers are to a change in price
Calculating Price Elasticity of
Demand
• Price elasticity can be measured at
an interval (or arc) along demand, or
at a specific point on the demand
curve
– If the price change is relatively small, a
point calculation is suitable
– If the price change spans a sizable arc
along the demand curve, the interval
calculation provides a better measure
Computation of Elasticity Over
an Interval
• When calculating price elasticity of
demand over an interval of demand,
use the interval or arc elasticity
formula
Q Average P
E

P Average Q
So, arc price elasticity of demand =
Q2  Q1
P2  P1
Ep 

(Q1  Q2 ) / 2 ( P1  P2 ) / 2
•
•
•
•
•
Ep = Coefficient of arc price elasticity
Q1 = Original quantity demanded
Q2 = New quantity demanded
P1 = Original price
P2 = New price
Computation of Elasticity at a
Point
• When calculating price elasticity at a point
on demand, multiply the slope of demand
(Q/P), computed at the point of
measure, times the ratio P/Q, using the
values of P and Q at the point of measure
• Method of measuring point elasticity
depends on whether demand is linear or
curvilinear
The Price Elasticity of
Demand
• Point elasticity: measured at a given
point of a demand (or a supply) curve.
dQ P1
εP =
x
dP Q1
The Price Elasticity of
Demand
The point elasticity of a linear
demand function can be expressed
as:
Q P1
p 

P Q1
The Price Elasticity of
Demand
• Some demand curves have constant
elasticity over the relevant range
• Such a curve would look like:
Q = aP-b
where –b is the elasticity coefficient
• This equation can be converted to linear
by expressing it in logarithms:
log Q = log a – b(log P)
The Price Elasticity of
Demand
• Elasticity differs
along a linear
demand curve.
Price Elasticity of
Demand (E)
Elasticity
Responsiveness
E
Elastic
%Q%P E 1
Unitary Elastic
%Q%P E 1
Inelastic
%Q%P
Perfect elasticity: E = ∞
Perfect inelasticity: E = 0
E 1
Factors Affecting Price
Elasticity of Demand
• Availability of substitutes
– The better & more numerous the substitutes
for a good, the more elastic is demand
• Percentage of consumer’s budget
– The greater the percentage of the consumer’s
budget spent on the good, the more elastic is
demand
• Time period of adjustment
– The longer the time period consumers have to
adjust to price changes, the more elastic is
demand
The Price Elasticity of
Demand
• A long-run demand
curve will generally be
more elastic than a
short-run curve.
• As the time period
lengthens consumers
find way to adjust to
the price change, via
substitution or
shifting consumption
The Price Elasticity of
Demand
• There is a relationship between the price
elasticity of demand and revenue received.
– Because a demand curve is downward sloping, a
decrease in price will increase the quantity
demanded
– If elasticity is greater than 1, the quantity
effect is stronger than the price effect, and
total revenue will increase
Price Elasticity & Total
Revenue
Elastic
Unitary elastic
Inelastic
Quantity-effect
dominates
No dominant
effect
Price-effect
dominates
Price
rises
TR falls
No change in TR
TR rises
Price
falls
TR rises
No change in TR
TR falls
• As price decreases
– Revenue rises when
demand is elastic.
– Revenue falls when
it is inelastic.
– Revenue reaches its
peak when
elasticity of
demand equals 1.
• Marginal Revenue: The change in
total revenue resulting from changing
quantity by one unit.
Total Revenue
MR 
Quantity
• Since MR measures the rate of change
in total revenue as quantity changes, MR
is the slope of the total revenue (TR)
curve
Demand & Marginal Revenue
Unit sales (Q)
Price
0
$4.50
1
4.00
2
3.50
3
3.10
4
2.80
5
2.40
6
2.00
7
1.50
TR = P  Q
$
MR = TR/Q
--
0
$4.00
$4.00
$7.00
$3.00
$9.30
$2.30
$11.20
$1.90
$12.00
$0.80
$12.00
$0
$10.50
$-1.50
Demand, MR, & TR
Panel A
Panel B
• For a straight-line demand curve the
marginal revenue curve is twice as steep
as the demand.
• At the point where marginal revenue
crosses the X-axis, the demand curve is
unitary elastic and total revenue reaches
a maximum.
Linear Demand, MR, &
Elasticity
• Some sample elasticities
–
–
–
–
–
Coffee: short run -0.2, long run -0.33
Kitchen and household appliances: -0.63
Meals at restaurants: -2.27
Airline travel in U.S.: -1.98
Beer: -0.84, Wine: -0.55
MR, TR, & Price
Elasticity
Marginal
Total revenue
revenue
MR > 0 TR increases as
Q increases
MR = 0
MR < 0
(P decreases)
Price elasticity
of demand
Elastic
Elastic
(E> 1)
(E> 1)
Unit
Unitelastic
elastic
TR is maximized (E= 1)
(E= 1)
TR decreases as Inelastic
Inelastic
(E<
1)
Q increases
(P decreases)
(E< 1)
Marginal Revenue & Price
Elasticity
• For all demand & marginal revenue
curves, the relation between marginal
revenue, price, & elasticity can be
expressed as
1

MR  P 1  
E

The Cross-Elasticity of
Demand
• Cross-elasticity of demand: The
percentage change in quantity
consumed of one product as a result
of a 1 percent change in the price of
a related product.
% Q A
EX 
% PB
The Cross-Elasticity of
Demand
• Arc Elasticity
Q2 A  Q1 A
P2 B  P1B
Ex 

(Q1 A  Q2 A ) / 2 ( P1B  P2 B ) / 2
The Cross-Elasticity of
Demand
• Point Elasticity
Q A PB
EX 

QA
PB
The Cross-Elasticity of
Demand
• The sign of cross-elasticity for
substitutes is positive.
• The sign of cross-elasticity for
complements is negative.
• Two products are considered good
substitutes or complements when the
coefficient is larger than 0.5.
Predicting Revenue Changes
from Two Products
Suppose that a firm sells to related goods. If the price of
X changes, then total revenue will change by:
 


R  RX 1  EQX , PX  RY EQY , PX  %PX
Income Elasticity
• Income Elasticity of Demand: The
percentage change in quantity
demanded caused by a 1 percent
change in income.
%Q
EY 
%Y
Income Elasticity
• Arc Elasticity
Q2  Q1
Y2  Y1
EY 

(Q1  Q2 ) / 2 (Y1  Y2 ) / 2
Income Elasticity
• Categories of income
elasticity
– Superior goods: EY > 1
– Normal goods: 0 >EY >1
– Inferior goods –
demand decreases as
income increases: EY < 0
Other Elasticity
Measures
• Elasticity is encountered every time a
change in some variable affects
quantities.
– Advertising expenditure
– Interest rates
– Population size
Uses of Elasticities
• Pricing.
• Managing cash flows.
• Impact of changes in competitors’
prices.
• Impact of economic booms and
recessions.
• Impact of advertising campaigns.
• And lots more!
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?
Answer: Lower price!
• 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.
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?
Answer
• Calls would increase by 25.92 percent!
EQX , PX
% QX
 8.64 
% PX
d
% QX
 8.64 
 3%
d
 3%   8.64   %QX
d
%QX  25.92%
d
Example 3: Impact 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?
Answer
• AT&T’s demand would fall by 36.24 percent!
EQX , PY
%QX
 9.06 
%PY
%QX
9.06 
 4%
d
 4%  9.06  %QX
d
%QX  36.24%
d
d
Elasticity of Supply
• Price Elasticity of Supply: The
percentage change in quantity supplied
as a result of a 1 percent change in
price.
% Quantity Supplied
ES 
% Price
• If the supply curve slopes upward and
to the right, the coefficient of supply
elasticity is a positive number.
Elasticity of Supply
• Arc elasticity
Q2  Q1
P2  P1
Es 

(Q1  Q2 ) / 2 ( P1  P2 ) / 2
Elasticity of Supply
• When the supply curve is more
elastic, the effect of a change in
demand will be greater on quantity
than on the price of the product.
• With a supply curve of low elasticity,
a change in demand will have a
greater effect on price than on
quantity.
Interpreting Demand Functions
• Mathematical representations of demand
curves.
• Example: d
QX  10  2 PX  3PY  2 M
• X and Y are substitutes (coefficient of PY
is positive).
• X is an inferior good (coefficient of M is
negative).
Linear Demand Functions
• General Linear Demand Function:
QX  0   X PX  Y PY   M M   H H
d
PX
EQX , PX   X
QX
Own Price
Elasticity
EQ X , PY
PY
 Y
QX
Cross Price
Elasticity
M
EQX , M   M
QX
Income
Elasticity
Example of Linear
Demand
•
•
•
•
Qd = 10 - 2P.
Own-Price Elasticity: (-2)P/Q.
If P=1, Q=8 (since 10 - 2 = 8).
Own price elasticity at P=1, Q=8:
(-2)(1)/8= - 0.25.
Log-Linear Demand
• General Log-Linear Demand Function:
ln QX d   0   X ln PX  Y ln PY   M ln M   H ln H
Own Price Elasticity :  X
Cross Price Elasticity :  Y
Income Elasticity :
M
Example of Log-Linear
Demand
• ln(Qd) = 10 - 2 ln(P).
• Own Price Elasticity: -2.
Graphical Representation of
Linear and Log-Linear Demand
P
P
D
Linear
D
Q
Log Linear
Q