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Business Economics
Elasticityand its applications
Thomas & Maurice, Chapter 6
Herbert Stocker
Institute of International StudiesUniversity of Ramkhamhaeng
&Department of EconomicsUniversity of Innsbruck
Elasticities
Question: How ‘strongly’ reacts one variable inresponse of a change in another variable?
→ slope of a curve.
e.g. Demand for wheat in the USA:
Qd = 3550− 266P
where Qd is measured in million ‘bushels’ per yearand P is measured in US$.
Slope: dQd
dP= −266
i.e. if the price increases by one dollar the quantitydemanded decreases by 266 mio ‘bushels’!???
Elasticities
Problem: For the interpretation of the slope wehave to know the dimensions in which the unitsare measured!Alfred Marshall (1842-1924):proposed the use of relative changes!
Price-elasticity =Percentage change in quantity
Percentage change in price
=%∆Qd
%∆P
e.g. what is the percentage decrease in thequantity demanded of wheat if the price of wheatraises by one percent?
Elasticities
Elasticity: Percentage change in the dependentvariable resulting from a one percent increase inthe independent variable.
Elasticities are a very general concept to expressthe ‘strength of reaction’ of one variable inresponse to the change of another variable.
Main advantage of elsticities: free of
dimensions!
Two concepts:Arc-Elasticity: discrete change between two observedpoints.Point-Elasticity: infinitesimal change (function mustbe known).
Arc-Elasticity
When only two price-quantity pairs are known:(for discrete changes)
EB =%-Change of Y
%-Change of X
=
Y2−Y1
Y1
X2−X1
X1
≡∆YY1
∆XX1
≡∆Y∆XY1
X1
=∆Y
∆X
X1
Y1
Y
X
b before
X1
Y1
bc after
X2
Y2
∆X
∆Y
Arc-elasticity
For discrete changes:
e.g.: price raises from 2$ to 4$ ⇒ quantity decreasesfrom 20 kg to 15 kg.
P
Qd
bc
b∆P
P2 = 4
P1 = 2
bc
bc
∆Qd
15 = Qd2 Qd1 = 20
∆Qd = Qd1 − Qd2
= 20− 15 = 5
∆P = P1 − P2
= 2− 4 = −2
EBQd ,P
=
∆Qd
Qd1
∆PP1
=520−22
= −1
4
Arc-Elasticity
Problem: the values of the arc-elasticity dependon whether the price increases or decreases!
E 1Qd ,P
=
Qd1−Qd2
Qd1× 100
P1−P2
P1× 100
↔ E 2Qd ,P
=
Qd2−Qd1
Qd2× 100
P2−P1
P2× 100
Solution: Mid-Point Method
EQd ,P =
Qd1−Qd2
(Qd1+Qd2)/2× 100
P1−P2
(P1+P2)/2× 100
Point-Elasticity
For curvilinear demand curves: Slope is calculatedby using the derivative (∆ → d).
Point Elasticity:
EQd ,P =dQd
dPQd
P
=dQd
dP
P
Qd
P
Qd
bc
Qd1
P1b
Point-Elasticity
Example:
What is the price elasticity of demand for the followingdemand function:
Qd = 25− 2.5P
at the price P1 = 2, (⇒ Qd1 = 25− 2.5× 2 = 20):
EQd ,P =dQd
dP
P1
Qd1
= − 2.52
20= − 0.25
Point-Elasticity
Example Qd = 25− 2.5P
at a different price P2 = 4 demand is Qd2 = 15, andthe elasticity is therefore
EQd ,P =dQd
dP
P2
Qd2
= −2.54
15
= −2
3≈ −0.666̇
For linear functions, the value of the elasticity isdifferent at each point!
Point-Elasticity
Problem: For linear functions the elasticity is notconstant along the line!
EQd ,P =dQd
dP
Pi
Qdi
Solution: the elasticity is often calculated at themean value of the variable.
EQd ,P =dQd
dP
P
Qd
mit: P ≡ 1N
∑Ni=1 Pi und Qd ≡ 1
N
∑Ni=1Qdi
Elasticities
An elasticity shows the percentage change in thedependent variable (Y ) when the independent variable(X ) increases by one percent.
Elasticities are positive if the derivative is positive,i.e. if the variables move in the same direction.
Elasticities are negative if the variables move inopposite directions.
An elasticity measures how ‘strongly’ one variablereacts in response of a change in another variable.
Rule of Thumb
Managers can get a rough estimate of price elasticityby asking two questions:
What price P customers you currently pay for theproduct?At what price A would customers stop buying myproduct altogether?The answers to this questions can be used tocalculate a rough estimate of the demandelasticity, since
EQd ,P =P
(P − A)
where P is the price and A is the vertical interceptof the plotted demand curve (the P-axis).
Rule of Thumb
Why?
P
Q
A
bP
P = A− s Q
Q =1
s[A− P]
dQ
dP=
−1
s
E =dQ
dP
P
Q
=−1
s
P1s[A− P]
=P
P − A
Graphical Derivation
Elasticity and Angle
Slope and Angle
Remember:
Hypotenuse
Adjacent
Oppositeleg
α
tanα =Opposite leg
Adjacent
= Slope
Elasticity and Angle
Y
X
Abc
α
XA
XA
YA YA
β
Y = f (X )dY
dX= tanα
YA
XA
= tan β
EA =dYdXYA
XA
= −tanα
tan β
Elasticity and Angle
Y
X
αβ
Abc
EA =dYdXYAXA
= −tanαtanβ
Y = f ( X )
XA
YA
Elasticity and Angle
P
Qd
α
β
Bbc
EA =dQddPQdBPB
= −tanαtan β
Qd = f ( P )
QdB
PB
Elasticity and Angle
0
1
2
3
4
0 1 2 3 4
P
Qd
α
bc
β
E1 = − tanαtan β = −
4431
= − 13
bc
γ
E2 = − tanαtan γ = −
4422
= −1
bc
δ
E3 = − tanαtan δ = −
4413
= −3
bc E4 = − tanαtan(0) = −
4404
= − 10 = −∞
bcE0 = − tanα
tan(90) = −4440
= − 1∞
= 0
Elasticity and Angle
P
Qd
α
β
A
bc
Positive Intercept:
tanα > tan β
EA =dQd
dPQdA
PA
=tanα
tan β> 1
Question: which value has the elasticity of a linearfunction that goes through the origin?
Functions with constant elasticity
Y = f (X ) = AX b
dY
dX= bAX b−1
Y
X=
AX b
X= AX b−1
EY ,X =dYdXYX
=bAX b−1
AX b−1= b
⇒ Elasticities of power-functions are always constant!
Functions with constant elasticity
Example:
Y = f (X ) = 3X 0.5
dY
dX= 0.5× 3X 0.5−1 = 0.5× 3X−0.5
Y
X=
3X 0.5
X= 3X 0.5−1 = 3X−0.5
EY ,X =dYdXYX
=0.5× 3X−0.5
3X−0.5= 0.5
Elasticities of log-linear Functions
lnY = f (lnX )
EY ,X =d lnY
d lnX=
dYYdXX
=dY
dX
X
Y
Example:
Y = f (X ) = 3X 0.5
lnY = ln 3 + 0.5 lnX
EY ,X =d lnY
d lnX= 0.5
Elasticities of log-linear Functions
EY ,X =d lnY
d lnX=
dYYdXX
Intuition:
d lnY
dY=
1
Y⇒ d lnY =
dY
Yd lnX
dX=
1
X⇒ d lnX =
dX
X
EY ,X =dYYdXX
=d lnY
d lnX
Elasticities of log-linear Functions
Example:
Y = AX b
lnY = lnA+ b lnX
EY ,X =d lnY
d lnX= b
Example:
Y = f (X ) = 0.25X−2
lnY = ln 0.25− 2 lnX
EY ,X =d lnY
d lnX= −2
Demand with constant elasticity
P
Qd
Qd = 2P−0.5
Qd = 0.25P−2
Elastic:
Qd = 0.25P−2
lnQd = ln 0.25− 2 lnP
EQd ,P = −2
Inelastic:
Qd = 2P−0.5
lnQd = ln 2− 0.5 lnP
EQd ,P = −0.5
Applications of Elasticities
Theory of Demand
Price Elasticity of Demand
Demand Function:
Qd = Qd(P ,M ,PS ,PC , . . .)
Price Elasticity of Demand:(sometimes called price or demand elasticity)
EQd ,P =dQd
dPQd
P
=dQd
dP
P
Qd
≈%∆Qd
%∆P
The Price Elasticity of Demand shows the
percentage decrease of the quantity demanded if price
ceteris paribus increases by one percent.
Price Elasticity of Demand
Demand is elastic if |EQd ,P | > 1, or
%∆Q > %∆P
(The quantity demanded responds more than proportionally to a a
change in price.
Demand is unit elastic if |EQd ,P | = 1, or
%∆Q = %∆P
Demand is inelastic if 0 < |EQd ,P | < 1, or
%∆Q < %∆P
(The quantity demanded responds less than proportionally to a a
change in price.
Since the price elasticity is usually negative it is common to use the absolutevalue |EQd ,P |.
Linear Demand Function and Elasticity
In which point of a linear demand function theelasticity has the value −1?
elastic
inelastic
bc
bc
bc
EQd ,P = −1
EQd ,P = −∞
EQd ,P = 0
a2b
a2
a
P
Qd
Qd = a − bP
EQd ,P =dQd
dP
P
Qd
=−bP
a − bP= −1
bP = a − bP
P =a
2b
Qd = a − b( a
2b
)
=a
2
Special cases . . .
Perfectly inelastic de-mand: EQd ,P = 0
P
Qd
A change in price has no effect on
the quantity demanded!
Perfectly elastic demand:EQd ,P = −∞
P
Qd
A change in price has an infinitely
large effect on the quantity de-
manded!!
Determinants of price elasticity
Determinants of price elasticity: ceteris paribus
demand tends to be more elastic, . . .
the more and closer substitutes are available.
when the good is rather a luxury than a necessity.
the higher the proportion of income spent on thegood.
the longer the time period under consideration.
demand for durable goods tends to be more elasticthan demand for non-durables (consumers chooseto hold on to the good instead of replacing it).
Price elasticities for cars
Model Price Estimated EQd ,P
Mazda 323 $ 5,039 −6.358Nissan Sentra $ 5,661 −6.528Ford Escort $ 5,663 −6.031Honda Accord $ 9,292 −4.798Ford Taurus $ 9,671 −4.220Nissan Maxima $13,695 −4.845Cadillac Sevifle $24,544 −3.973Lexus LS400 $27,544 −3.085BMW 735i $37,490 −3.515
Source: Table V in S. Berry, Levinsohn, and A. Pakes, “Automobile Prices in Market Equilibrium”, Econometrica 63 (July1995): 841-890. [aus: D. Besanko & D. Braeutigam, Microeconomics (Wiley)]
⇒ probably cheaper cars are perceived more as substitutes than luxury cars.
Price Elasticity
Attention:
Even if the demand for the entire product israther inelastic the elasticity for the individualproducer might be quite large.
For example, the demand for eggs or potatoes israther inelastic, but the elasticity for the eggs orpotatoes of an individual farmer might be close toinfinity!
Price Elasticity of Demand
and Total Revenue
Elasticity & Total Revenue
Elastic Demand:P
Qd
bc
bc
Total revenue decreases
when price increases!
Inelastic Demand:P
Qd
bc
bc
Total revenue increases
when price increases!Total revenue is P × Qd (i.e. the hatched area)
Elasticity & Total Revenue
elastic
inelastic
E = −1
E = −∞
E = 0
a2
P
Qd
bc
bc
If demand is ineleas-tic price and totalrevenue move in thesame direction!
Elasticity & Total Revenue
elastic
inelastic
E = −1
E = −∞
E = 0
a2
P
Qd
bc
bc
If demand is elasticprice and total rev-enue move in oppositedirection!
Elasticity & Total Revenue
With Calculus:
TR = P × Q(P)
d(TR)
dP=
d [P × Q(P)]
dP
= Q + PdQ
dP
= Q
(
1 +dQ
dP
P
Q
)
= Q(1 + EQd ,P)
= Q(1− |EQd ,P |)
Elasticity & Total Revenue
d(TR)
dP= Q(1− |EQd ,P |)
d(TR)
dP= 0 for |EQd ,P | = 1 ⇒ Max.!
d(TR)
dP< 0 for |EQd ,P | > 1 ⇒ elastic
d(TR)
dP> 0 for |EQd ,P | < 1 ⇒ inelastic
Other Elasticities
Income Elasticity
Demand function: Qd = Qd(P ,M ,PS ,PC , . . .)
Income Elasticity of Demand:
EQd ,M =Percentage change in quantity demanded
Percentage change in income
=dQd
dMQd
M
=dQd
dM
M
Qd
≈%∆Qd
%∆M
The Income Elasticity of Demand shows the
percentage change in quantity demanded if income
ceteris paribus increases by one precent.
Income Elasticity of Demand
Normal goods (necessities): 0 < EQd ,M < 1:income elasticity is between 0 and 1.
Luxury or superior goods: EQd ,M > 1:if income ceteris paribus increases by one percentthe quantity demanded will increase by more thanone percent! Example: lobster, . . .
Inferior goods: EQd ,M < 0:the quantity demanded decreases if incomeincreases! Example: second-hand clothes, rice, . . .
Income Elasticity of Demand
Income elasticity of demand can be important forfirms:
Demand for luxuries increases more thanproportional with income, markets for luxuriestend to grow more rapidly than markets fornormal and inferior goods.
Firms can try to target marketing campaigns toconsumer groups with higher income elasticity.
Developing countries are often specialized inprimary production with low income elasticities.
Income Elasticity of Demand
Estimates of the Income Elasticity of Demand forSelected Food ProductsProduct Estimated EQd ,M Product Estimated EQd ,M
Cream 1.72 Milk 0.50Peaches 1.43 Butter 0.37Apples 1.32 Potatoes 0.15Fresh peas 1.05 Margarine −0.20Oranges 0.83 Flour −0.36Eggs 0.44
Source: Daniel B. Suits, “Agriculture”, in: The Structure of American Industry, W. Adams and J. Brock, eds.(Englewood, Nj: Prentice Hall), 1995;H. S. Houthhakker and Lester D. Taylor, “Consumer Demand in the United States, 1929-1970” (Cambridge, MA:Harvard University Press), 1966.taken from: D. Besanko & D. Braeutigam, Microeconomics (Wiley)
Cross Price Elasticity
Cross-price elasticity of demand: measureshow demand for Good X varies with changes inthe price of another Good Y .
Substitute goods have positive cross elasticity.Complementary goods have negative cross elasticity.
Defines relevant market in which differentproducts compete.
Cross Price Elasticity
Demand function: Qd = Qd(P ,M ,PS ,PC , . . .)
Cross Price Elasticity of Demand:
EQd ,PS=
Percentage change in demand for good A
Percentage change in price for good B
=
dQdA
dPB
QdA
PB
=dQdA
dPB
PB
QdA
≈%∆QdA
%∆PB
The Cross Price Elasticity of Demand shows the
percentage change in the demand for a good, if the
price of another good changes by one percent.
Cross Price Elasticity
Substitutes: (e.g. Cafe and Tea)⇒ cross price elasticity is positiveif tea becomes more expensive the demand for cafe
increases.
EQd ,PS=
dQd
dPS
PS
Qd
> 0
Complementary Goods: (e.g. cafe and sugar)⇒ cross price elasticity is negativeif cafe becomes more expensive the demand for sugar
decreases.
EQd ,PC=
dQd
dPC
PC
Qd
< 0
Price and Cross Price Elasticities
Demand for Price of Beef Price of Pork Price of Chicken
Beef −0.65 0.01 0.20Pork 0.25 −0.45 0.16Chicken 0.12 0.20 −0.65
Source: Daniel B. Suits, “Agriculture”, in: The Structure of American Industry, W. Adams and J. Brock, eds.(Englewood, Nj: Prentice Hall), 1995entnommen aus: D. Besanko & D. Braeutigam, Microeconomics (Wiley)
Price elasticities are on the main diagonale, off the main diagonal are thecross price elasticities.e.g.: −0.65 is the price elasticity of beef,0, 01 is the cross price elasticity of the demand for beef with respect to theprice of pork.
Price and Cross Price Elasticities
Sometimes useful to judge whether markets are‘related’.
Price of Price of Price of Price ofDemand for Sentra Escort LS400 735iSentra −6.528 0.078 0.000 0.000Escort 0.454 −6.031 0.001 0.000LS400 0.000 0.001 −3.085 0.093735i 0.000 0.001 0.032 −3.515
Source: S. Berty Levinsohn, and A. Pakes, “Automobile Prices in Market Equilibrium”, Econometrica 63 (July 1995):841-890.entnommen aus: D. Besanko & D. Braeutigam, Microeconomics (Wiley)
Diagonal elements: the price elasticity of demandOff-diagonal elements: the cross-price elasticity of demand.
Demand for Coca- and Pepsi Cola
Econometrically estimated demand functions:
Qdc = 26.17− 3.98Pc + 2.25Pp + 2.60Ac − 0.62Ap + 0.99M + . . .
Qdp = 17.48− 5.48Pp + 1.40Pc + 2.83Ap − 4.81Ac + 1.92M + . . .
Qdc quantity demanded of Coca-Cola (ten million cases)Qdp quantity demanded of Pepsi (ten million cases)Pc price of Coca-Cola (dollars per ten cases)Pp price of Pepsi (dollars per ten cases)Ac advertising expenditures on behalf of Coca-ColaAp advertising expenditures on behalf of PepsiM disposable income in the United States
All prices expressed in 1986 U.S. dollars!
Elasticities: Coca Cola und Pepsi ColaBy inserting the means (e.g. Pc = 12, 96, Pp = 8, 16; Ac = 5, 89; . . . ) one cancalculate the elasticities in the mean:
Price, Cross-Price, and Income Elasticities of Demandfor Coca-Cola and Pepsi
Elasticity Coca-Cola PepsiPrice elasticity of demand −1.47 −1.55Cross-price elasticity of demand 0.52 0.64Income elasticity of demand 0.58 1.38
Source: Gasmi, F., J.J. Laffont and Q. Vuong (1992): “Econometric Analysis of Collusive Behaviour in the Soft DrinkMarket”, Journal of Economics and Marketing Strategy, Vol. 1entnommen aus: D. Besanko & D. Braeutigam, Microeconomics (Wiley)
Advertising Elasticity of Demand
Advertising elasticity of demand: the percentagechange in quantity demanded of a good relative tothe percentage change in advertising dollars spenton that good.
Marketing studies: e.g. Tellis, 1988; Sethuramanand Tellis, 1991; Hoch, et al, 1995
Advertising elasticities of demand tend to bemuch smaller than price elasticities of demand (bya factor 10-15).
Supply Elasticity
Price elasticity of supply is the percentage change inquantity supplied resulting from a percent change inprice.
ES ,P =dSdPSP
=dS
dP
P
S≥ 0
Supply is elastic when ES ,P > 1
Supply is inelastic when ES ,P < 1
Supply Elasticity
Determinants of Elasticity of Supply:
Time period: Supply is more elastic in the longrun!
Ability of sellers to change the amount of thegood they produce.(Beach-front land is probably inelastic, whilebooks, cars, or manufactured goods are ratherelastic)
Special cases . . .
Perfectly Inelastic Supply:
ES ,P = 0
P
S
An increase in price leaves the
quantity supplied unchanged!
Perfectly Elastic Supply:
ES ,P = ∞
P
Qs
Below the price the quantity sup-plied is zero, above it is infinite!
Price Elasticity in Marketing
Managerial Price Sensitivity Analysis
The price elasticity of demand (in marketingliterature often called price sensitivity) is one ofthe most important variables for managers.
A managerial analysis of price elasticity should bea written document that can be criticized andimproved over time.
It should include some of the following questions:
Reference Price
Substitutes and Reference Price:
Are there close substitutes to the product, and if so,are the buyers (or a segment of buyers) usually awarethereof when making a purchase? Can they compareprices?Can buyers speed up or delay purchases based onexpectations of future prices?
How difficult is it for buyers to compare offers ofdifferent suppliers?
Switching Cost & Expenditure Share
To what extent have buyers already madeinvestments (monetary and/or psychological) thatthey would need to incur again if they switchedsuppliers?
For how long are buyers presumably ‘locked’ bythose expenditures?How significant are buyers expenditures for theproduct?
For end consumers mainly the portion of income isimportant.For business customers also the absolute price mightbe important.
Fairness
Buyers are more sensitive to a product’s pricewhen it is outside the range that they perceive as‘fair’ or ‘reasonable’.
How does the current price compare with pricespeople have paid in the past?
What do buyers expect to pay for similarproducts?
Do customers perceive the product as ‘necessity’or as a discretionary purchase?
Framing Effect
Prospect Theory: (D. Kahneman & A. Tversky)Essential idea: people ‘frame’ purchasing decisionsin their minds as a bundle of gains and losses.
Consumers tend to be more price sensitive whenthey perceive the price as a ‘loss’ rather than aforegone ‘gain’.
Additionally, they are more price sensitive whenthe price is paid separately rather than as part ofa bundle.
Framing Effect
Example: (Prospect Theory)Gas station A sells gasoline for $1.20 and gives a$0.10 per liter discount if the buyer pays with cash.Gas station B sells gasoline for $1.10 and charges a$0.10 surcharge if the buyer pays with credit card.
Most people choose station A.
We’ll have to say a lot more about priceelasticities when we study optimal pricing onimperfect markets.