Download - Ch 6 Demand Elasticity
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Demand Elasticity
Measuring Buyer Responsiveness
to Changes in Demand
Determinants
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Objectives of Discussion
Review the concept ofdemand elasticity
Discuss the factors that affect a firms demand
elasticity
Illustrate calculation of demand elasticity
Illustrate the relationship between demand
elasticity & a firms Total Revenue & Marginal
Revenue
Review the other frequently used elasticity
indices--Income Elasticity & Cross Elasticity
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Measuring Responsiveness of Demand
Elasticity provides a means forcomparing responsiveness of demandof a given product:
To changes in one of its DemandDeterminants
To responsiveness of other products tochanges in theirDemand Determinants
Between different groups of consumers
At different points in time
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Price Elasticity of Demand (E)
Index measuring buyers responsiveness to changesin own-price of a good
Needs to be independent of the base volume andscale ofmeasurement of the good
Ratio of the %( Q to % (P provides such an index
Eis always negative because of Law of Demand
Eis calculated for price changes along a given demand
curve--all other demand determinants are constant
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Price Elasticity of Demand (E)
Elasticity Responsiveness E
Elastic
Unitary Elastic
Inelastic
% Q % P ( " (
% Q % P ( ! (
% Q % P ( (
E " 1
E 1
E 1
|E| ranges from 0 to g
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Price Elasticity Examples
Elastic Demand:
(P = -10% and %(Qresponds by +15%
Inelastic Demand:
(P = -10% and %(Qresponds by +5%
Unitary elastic
(P = -10% and %(Qresponds by +10%
Question: IfE= -1.6, what(P is needed to increasesales by 20%?
Question: IfE= -0.6 whateffect will a +10%(P have onQd?
5.1%10
%15 !
!E
5.010
5!
!E
0.1%10
%10 !
!E
%5.126.1
%20%% !
!(
!(E
QP d
%606.
%10%% !
!!E
PQd
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Factors Affecting Demand
Elasticity Availability of substitutes:
Number and closeness of substitutes--more and closer means greater
elasticity
A related factor is how widely, or narrowly, a market is defined:
Demand for travel is much more inelastic than demand for train travel
Demand for a product is more elastic at the firm level than the industry level
Percent of consumers budget spent on item
The smaller the percent, the more inelastic
Nature of the good
Demand for necessities tends to be more inelastic than non-necessities
Demand for durable goods tends to be more elastic than for non-
durable
Length of time period over which elasticity is measured
Demand is more inelastic in short run than long-run
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Calculating Price Elasticity of Demand
If the price change isrelatively small, apoint calculation issuitable
If the price changespans a sizable arcalong the demandcurve, the intervalcalculation provides abetter measure
Q PE
P Q
(! v
(
Average
Average
Q
P
P
QE y
(
(!
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Calculating Elasticity at a Point
First requirement is to determine the slope of the
demand curve at the point in question
For linear demand curve, slope is a negative constant
For non-linear demand curve, slope is found by finding slope
of tangent to demand curve at point in question
Point Elasticity Formula:
Where (Q/ (P is the slope of the demand function
With some algebraic manipulation can show that:
)/ ( aPPE !
Where a is intercept of inverse direct demand curve on price axis
Q
P
P
Q
PP
QQ
E y(
(!
(
(
!
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Elasticity (Generally) Varies Along a
Demand Curve
For linear demand, price and Evary directly The higher the price, the more elastic is demand
The lower the price, the less elastic is demand
For curvilinear demand,
slope changes at each point on demand curve
no general rule about the relation between price and
elasticity
Exception to rule:
Special case of which has a constantprice elasticity (equal to ) for all prices
bQ aP
b
!y Special case of which has a constantprice elasticity (equal to ) for all prices
bQ aP
b
!y
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Constant Elasticity of Demand(Figure 6.3)
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Price Elasticity & Total Revenue (TR)
TR = P x Q
Price and quantity move in opposite directions on a
demand curve
Move on demand curve has a price effect & an
opposing quantity effect
price effect is effect on TR from a price change holding Q
fixed
quantity effect is effect on TR from a quantity change
holding P fixed
Impact on TR depends on two things:
Which effect is stronger
The direction of the movement on the demand curve (i.e. P
increase vs P decrease)
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Price Elasticity & Total Revenue
Elastic
Quantity-effect
dominates
Unitary elastic
No dominant effect
Inelastic
Price-effect dominates
Pricerises
Pricefalls
TR
lls
TR ris s
N g i TR
N g i TR
TR ris s
TR lls
% Q % P ( " ( % Q % P ( ! ( % Q % P ( (
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TR, MR & the Demand Curve
TR = P x Q
Linear demand curve: P = a + bQ
Substituting in TR forP
TR = (a + bQ)Q = aQ + bQ2
MR is change in TR from selling one more unit ofQ
Using some simple calculus we can show that:
bQa
d
d2
Q
(TR)MR !!
From above we can see that MR has same intercept on price axis as the
demand curve and is twice as steep
@ we can show that when MR = 0, Q is at one-half the amount that it
attains when P = 0 on the demand curve
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Linear Demand, MR, & Elasticity(Figure 6.5)
Inverse D: P = 6 - .05Q
MR= 6 - .10Q
TR= PxQ= (6 - .05Q)xQ=6Q- .05Q2
Q=40: TR= 160
Q=60: TR= 180
Q=65: TR= 178.75
160
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MR, TR, & Price Elasticity
Marginal
revenue Total revenuePrice elasticity of
demand
MR > 0 Elastic (E>
1)
MR = 0 Unit elastic (E=1)
MR < 0 Inelastic (E 1)
TR decreases asQ increases(Pdecreases)
TR is maximized
TR increases
as Q increases(Pdecreases)
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MR & Price Elasticity
Saw earlier that the effect on TR of a price change depends on
size of price elasticity coefficient, E
Recalling that MR is change in TR associated with unit changes
in output, we can express this as:
dQ
dPQ
dQ
dQP yy!MR
dQQPd
dQTRdMR )()(
y
!!
Using the product rule for derivatives, we can show:
Manipulating some of the terms in the above expression we can show:
Definition of price
elasticity
1dQ
dP
P
QPMR y! )
11(
Q
P
dP
dQP
y
! )1
1(E
P !
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Price Elasticity & MR
Since E is always negative, we can express the
above as:
)1
1(MRE
P !
)
1
1(MR EP!
From the above we can see that:
when E> 1, the term in ..) will be positive so MR will bepositive for each unit increase in quantity
when E< 1, the term in ..) will be negative so MR will benegative for each unit increase in quantity
when E= 1, the term in ..) will be zero, so MR will also be 0
Since MR is the change in TR associated with a change in Q, the effects
of elasticity on TR follow from the above
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Income Elasticity
Income elasticity is measured as ratio of % change in Q to
% change in income, holding all other demand
determinants fixedparibusceteris
%
%
M
QE
M
(
(!
EM is positive for most commodities--i.e. normal goods
j EM is smaller for necessities than for non-necessities
y EM
is negative for goods that are considered relatively
inferior in consumption
y EM can be calculated using either an arc approach or a
point approach
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Cross Elasticity
Cross elasticity is measured as ratio of % change in Q to
% change in price of some other good, holding all other
demand determinants fixed
paribusceteris%
%
B
A
X
P
Q
E (
(
!
Primary use of EX is to measure closeness of substitutes
and complements
EX is positive for substitute commodities
EX is negative for complements