Elasticity

Fall '97Economics 205Principles of

Microeconomics

This is a PowerPoint presentation on the fundamentalsof the concept of “elasticity” as used in principles ofeconomics.

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R. Larry Reynolds


MicroeconomicsSlide 2

Elasticity

· Elasticity is a concept borrowed from physics· Elasticity is a measure of how responsive a

dependent variable is to a small change in an independent variable(s)

· Elasticity is defined as a ratio of the percentage change in the dependent variable to the percentage change in the independent variable

· Elasticity can be computed for any two related variables



Elasticity [cont. . . ]

· Elasticity can be computed to show the effects of:· a change in price on the quantity demanded [ “a

change in quantity demanded” is a movement on a demand function]

· a change in income on the demand function for a good

· a change in the price of a related good on the demand function for a good

· a change in the price on the quantity supplied· a change of any independent variable on a

dependent variable



“Own” Price Elasticity

· Sometimes called “price elasticity”· can be computed at a point on a demand

function or as an average [arc] between two points on a demand function

· ep, are common symbols used to represent price elasticity

· Price elasticity [ep] is related to revenue· “How will a change in price effect the total

revenue?” is an important question.



Elasticity as a measure of responsiveness

· The “law of demand” tells us that as the price of a good increases the quantity that will be bought decreases but does not tell us by how much.

· ep [“own”price elasticity] is a measure of that information]

· “If you change price by 5%, by what percent will the quantity purchased change?



pechange in quantity demanded

change in price%

%

or, ep % Q

% P

At a point on a demand function this can be calculated by:

ep =

Q2 - Q1

Q1

P2 - P1

P1

Q2 - Q1 = Q

P2 - P1 = P=

QQ1

PP1



Q

Q1

P

P1

ep =

Price decreases from $7 to $5

3

Px

Qx/ut

D

$5B

5

$7A

P1 =

P2 =

P2- P1 = 5 - 7 = P = -2P = -2

Q1 = Q2 =

Q2 - Q1 = 5 - 3 = Q = +2

Q = +2

+2

7

3[2/3 = .66667]

[-2/7=-.28571]

= % Q = 67%

% P = -28.5%= -2.3 [rounded]

The “own” price elasticity of demandat a price of $7 is -2.3

This is “point” price elasticity. It is calculated at a pointon a demand function. It is not influenced by the directionor magnitude of the price change.

.

There is a problem! If theprice changes from $5 to$7 the coefficient of elasticity is different!

-2



3

Px

Qx/ut

D

$5B

5

$7A

Q

Q1

P

P1

ep =

When the price increases from $5 to $7,

P1 =

P2 =P = +2

+2

5

Q1=Q2=

Q = -2

-2

5

[-2/5 = -.4]

[+2/5 = .4]

= % Q = -40%

% P = 40%= -1 [this is called “unitary elasticity]

the ep = -1 [“unitary”]

ep = -1

In the previous slide, when the price decreased from $7 to $5, ep = -2.3

ep = -2.3The point price elasticity is different at every point!

There is an easier way!



An easier way!

Q1PP1

ep =

QQ1 =

Q

Q1

P1

P*

By rearranging terms

=P1

Q1*

Q

Pthis is the slope of thedemand function

this is a point onthe demandfunction

Q P1

Q1

= *Pep

Given that when:P1 = $7, Q1 = 3

P2 = $5, Q2= 5

P2- P1 = 5 - 7 = P = -2

Q2 - Q1 = 5 - 3 = Q = +2

Then,Q

P +2 -2

= = -1

This is the slope of the demand Q = f(P)

-1

P1 = $7, Q1 = 3

7

3= -2.33

On linear demand functions the slope remains constant so you just put in P and Q



P1 = $7, Q1 = 3

P2 = $5, Q2= 5

P2- P1 = 5 - 7 = P = -2Q2 - Q1 = 5 - 3 = Q = +2

3

Px

Qx/ut

D

$5B

5

$7A

The following information wasgiven

Q = f (P)

The slope of the demand function

[Q = f(P)] is Q

P =+2

-2= -1

The slope-intercept form Q = a + m P- 1

What is the Qintercept?

Px must decreaseby 5.

The slope [-1] indicates that for every1 unit increase in Q, Px will decrease by 1. Since Px must decrease by 5, Q must increase by 5

Q increases by 5

Q = 10

Q = 10 when Px = 0

10

The equation for the demandfunction we have been using isQ = 10 - 1P. A table can beconstructed.



For a simple demand function: Q = 10 - 1P

price quantity ep TotalRevenue

$0 10

$1 9

$2 8

$3 7

$4 6

$5 5

$6 4

$7 3

$8 2

$9 1

$10 0

The slope is -1 The intercept is 10using our formula,

ep=Q P1

Q1*P

ep =Q P1

Q1P *

the slope is -1,

(-1)

price is 7

7

at a price of $7, Q = 3

3= -2.3

-2.3

Calculate ep at P = $9Q = 1

ep = (-1) 91

= -9

Calculate ep for all other price and quantity combinations. -9

0-.11

-.25-.43-.67

-1.

-1.5

-4.

undefined



For a simple demand function: Q = 10 - 1P

price quantity ep TotalRevenue

$0 10

$1 9

$2 8

$3 7

$4 6

$5 5

$6 4

$7 3

$8 2

$9 1

$10 0

-2.3

-9

0-.11

-.25-.43-.67

-1.-1.5

-4.

undefined

Notice that at higher prices the absolute value of the priceelasticity of demand, ep is greater.

Total revenue is price times quantity; TR = PQ.0

9162124

2524

211690

Where the total revenue [TR]is a maximum, epis equalto 1

In the range where ep< 1, [less than 1 or “inelastic”], TR increases asprice increases, TR decreases as Pdecreases.

In the range where ep> 1, [greater than 1 or “elastic”], TR decreases as price increases, TR increases as P decreases.



3

Px

Qx/ut

D

$5B

5

$7A

To solve the problem of a point elasticity that is different for every price quantitycombination on a demand function, an arc price elasticity can be used. This arc priceelasticity is an average or midpoint elasticity between any two prices. Typically,the two points selected would be representative of the usual range of prices in the time frame under consideration.

The formula to calculate the average or arc price elasticity is:

ep=Q P1 + P2

Q1 + Q2*PThe average or arc ep between$5 and $7 is calculated,

ep=Q P1 + P2

Q1 + Q2*P

Slope of demand

QP = - 1

-1

P1 = $7, Q1 = 3

P2 = $5, Q2= 5

P2- P1 = 5 - 7 = P = -2Q2 - Q1 = 5 - 3 = Q = +2

P1 + P2 = 12

12

Q1 + Q2 = 8

8= - 1.5

The average ep between $5 and $7 is -1.5



Given: Q = 120 - 4 P

Price Quantity e p T R

$ 10

$ 20

$ 25

$ 28

Calculate the point ep at eachprice on the table.

Calculate the TR at each priceon the table.

Calculate arc ep at between$10 and $20.

Calculate arc ep at between$25 and $28.

Calculate arc ep at between $20 and $28.

Graph the demand function [labeling all axis and functions], identify which ranges on the demand function are price elastic and which areprice inelastic.



Given: Q = 120 - 4 P

Price Quantity e p T R

$ 10

$ 20

$ 25

$ 28

Calculate the point ep at eachprice on the table.

80

40

20

8

- . 5

-2

-5

-14

Calculate the TR at each priceon the table. TR = PQ

$800

$800

$500

$224

Calculate arc ep at between$10 and $20. ep = -1

Calculate arc ep at between$25 and $28. ep = -7.6

Calculate arc ep at between $20 and $28. ep = -4

Graph the demand function [labeling all axis and functions], identify which ranges on the demand function are price elastic and which areprice inelastic. At what price will TR by maximized? P = $15



Q/ut

Price

120

30

ep = -115

60

|ep | > 1 [elastic]

The top “half” of the demand function is elastic.

|ep | < 1inelastic

The bottom “half” of the demand function is inelastic.

Graphing Q = 120 - 4 P,

TR

TR is a maximumwhere ep is -1 or TR’s slope = 0

When ep is -1 TR is a maximum.When |ep | > 1 [elastic], TR and P move in opposite directions. (P hasa negative slope, TR a positive slope.)

When |ep | < 1 [inelastic], TR and P move in the same direction. (P and TR both have a negative slope.)

Arc or average ep is the average elasticity between two point [or prices]

pointep is the elasticity at a point or price.

Price elasticity of demand describeshow responsive buyers are to change in the price of the good. The more “elastic,” the more responsive to P.



Use of Price Elasticity

· Ruffin and Gregory [Principles of Economics, Addison-Wesley, 1997, p 101] report that:

· short run epof gasoline is = .15 (inelastic)

· long run epof gasoline is = .78 (inelastic)

· short run epof electricity is = . 13 (inelastic)

· long run epof electricity is = 1.89 (elastic)

· Why is the long run elasticity greater than short run?

· What are the determinants of elasticity?



Determinants of Price Elasticity

· Availability of substitutes [greater availability of substitutes makes a good relatively more elastic]

· Portion of the expenditures on the good to the total budget [lower portion tends to increase relative elasticity]

· Time to adjust to the price changes [longer time period means there are more adjustment possible and increases relative elasticity

· Price elasticity for “brands” is tends to be more elastic than for the category of goods



An application of price elasticity.The price elasticity of demand for milk is estimated between -.35 and -.5. Using -.5 as a reasonable figure, there are several important observations that can be made.

What effect does a10% increase in the Pmilk

have on the quantity thatindividuals are willing to buy?

ep % Q

% P

ep % Q

% P

Since ep = -.5

-.5 =+10%

To solve for % Q

Multiply both sides by +10%(+10%)x ( ) x (+10%)-5% = % QA 10% increase in the price of milk would reduce the quantity demanded by about5%.

Pmilk

Qmilk

DmilkP1

Q1

P2

A 10% increasein P

Q2

reduces Qby 5%

+10%

-5%

If price were decreased by 5%, whatwould be the effect on quantity demanded?



ep % Q

% PThe price elasticity of demand is a measure ofthe % Q that will be “caused” by a % P.

If the price elasticity of demand for air travel was estimated at -2.5, whateffect would a 5% decrease in price have on quantity demanded ?

-2.5 = % Q

% P- 5%= +12.5% change in quantity demanded

If the price elasticity of demand for wine was estimated at -.8, whateffect would a 6% increase in price have on quantity demanded ?

-.8 = % Q

% P+6%= -4.8% decrease in quantity demanded



If the price elasticity of demand for milk were -.5, the effects of a price change on total revenue [TR] can also be estimated.

Since ,

ep % Q

% P

When ep < 1, demand is “inelastic. “ This means that the % Q< % P. Since the % price decrease is greater than the % increase in Q,TR [TR = PQ] will decrease.

When ep < 1, a price decrease will decrease TR; a price increase willincrease TR, Price and TR “move in the same direction.” [inelastic demand with respect to price]

When ep > 1, demand is “elastic.” This means that the % Q> % P. When the % price decrease is less than the % increase in Q, TR [TR = PQ] will increase.

When ep > 1, a price decrease will increase TR; a price increase will decrease TR, price and TR “move in opposite directions.” [elastic demand wrt price]



Graphically this can be shown

P

Q/ut

D

at the midpoint, ep = -1

P1

Q1

TR

TR is a maximum

TR

TR = PQ, so the maximum TR is therectangle 0Q1 EP1

0

E

elastic

price risesP2

Q2

(P2

Q2)

is less

than (P1 Q1) Loss inTR whenP

+TR

As price rises into the elastic rangethe TR will decrease. Notice that in this range the slope of demandis negative, the slope of TR ispositive

Price andTR move in opposite directions



P

Q/ut

D

inelastic

TR

at the midpoint, ep = -1

0

EP1

Q1

TR is a maximum

TR = P1 Q1

[Maximum]

TRWhen price elasticity of demand isinelastic

A price decrease

P0

Q0

results in a smaller PQ [TR]

will result ina decrease in TR [PQ]. notice thatboth TR and Demand have a negative slope in the inelastic range of the demand function.Price and TR “move in the samedirection.”

A price decrease will reduceTR; a price increase will increase TR. Note that this information is usefulbut does not provide information about profits!



“Own” Price Elasticity of Demand

· ep is a measure of the responsiveness of buyers to changes in the price of the good.

· ep will be negative because the demand function is

negatively sloped.· A linear demand function will have unitary elasticity at

its “midpoint.” AT THIS POINT TR IS A MAXIMUM!· A linear demand function will be more “elastic” at

higher prices and tends to be more “inelastic” in the lower price ranges



Inelastic ep

· When ep< 1 [less than 1] the demand is “inelastic”· The %Q< %Pbuyers are not

very responsive to changes in price.· An increase in the price of the good

results in an increase in total revenue [TR], a decrease in the price decreases TR. Price and TR move in the same direction



P

Q/ut

D1

D1 is a “perfectly elastic”demand function.

ep % Q

% P

For an infinitesimally smallchange in price, Q changes by infinity.

= undefined

perfectly elasticep = undefined

.

Buyers are very responsive to price changes. An infinitely small change in pricechanges Q by infinity.

D2

perfectlyinelastic

ep = 0

D2 is a “perfectly inelastic” demand function, no matter howmuch the price changes the same amount is bought. Buyersare not responsive to price changes! ep = 0, perfectly inelastic.

0

P 0= 0

.

As the demand function becomes more horizontal, [buyers are more responsiveto price changes],ep approaches infinity.

De



Examples

· Goods that are relatively price elastic· lamb, restaurant meals, china/glassware,

jewelry, air travel [LR], new cars, Fords· in the long run, eptends to be greater

· Goods that are relatively price inelastic· electricity, gasoline, eggs, medical care,

shoes, milk· in the short run, eptends to be less



Income Elasticity[normal goods]

ey % Qx

% Y[Where Y = income]

Income elasticity is a measure of the change in demand [a “shift” of the demand function] that is “caused” by a change in income.

.

Q/ut

P

D

At a price of P1 , the quantity demandedgiven the demand D is Q1 .

P1

Q1

D is the demand function when the income is Y1 .

For a “normal good” an increasein income to Y2 will “shift” thedemand to the right. This is anincrease in demand to D2.

Due to increase in income,

demandincreases

D2

The increase in income, Y, increases demand to D2. The increase in demand results in a larger quantity being purchased at the

same Price [P1]..

Q2

% Y > 0; % Q> 0; therefore,

ey >0 [it is positive]



Income Elasticity [continued. . .][normal goods]

ey % Qx

% Y

Q/ut

P

D1

A decrease in income is associated with a decrease inthe demand for a normal good.

At income Y1, the demand D1 representsthe relationship between P and Q. At a price [P1] the quantity [Q1] is demanded.

P1

Q1

For a decrease in income [-Y],the demand decreases; i.e. shiftsto the left,

A decrease in income, decreasesdemand

D2

at the price [P1 ], a smaller Q2 will be purchased.

Q2

% Y < 0 [negative]; % Q < 0 [negative];

so, ep > 0 [ positive]

For either an increase or decrease in income

the ep is positive. A positive relationship [positive correlation] between Y and Qis evidence of a normal good.



When income elasticity is positive, the good is considered a “normalgood.” An increase in income is correlated with an increase in the demand function.

ey % Qx

% Y

ey % Qx

% Y + % Y

+ % Qx+ ey

A decrease in income is associated with a decrease in the demand function.

- % Y

- % Qx

+ ey

For both increases and decreases in income, ey is positive

.

The greater the value of ey, the more responsive buyersare to a change in their incomes.

When the value of ey is greater than 1, it is called a “superior good.”

.

The |% Qx| is greater than the |% Y|.Buyers are very responsive to changes inincome. Sometimes “superior goods” arecalled “luxury goods.”



ey % Qx

% Y

D1

Income Elasticity[inferior goods]

There is another classification of goods where changes in income shift the demand function in the “opposite” direction.

An increase in income [+Y] reduces demand.

Q/ut

P

P1

Q1

decreasesdemand

D2

Q2

+Y

- %Qx

-%Qx

-ey =

.

An increase in income reduces the amount that individualsare willing to buy at each priceof the good. Income elasticity

is negative: - ey

The greater the absolute valueof - ey, the more responsive buyersare to changes in income

.



D1

Income Elasticity[inferior goods]

A decrease in income [-Y] increases demand.

Q/ut

P

P1

Q1

D2

Q2

ey % Qx

% Y-Y

+%Qx

Decreases in income increase the demand for inferior goods.

+%Qx - ey

.

A decrease in income [-Y] results in an increase in demand,the income elasticity of demand is negative

.

For both increases and decreases in income the income elasticity is negativefor inferior goods. The greater the

absolute value of ey, the more responsivebuyers are to changes in income



Income Elasticity

· Income elasticity [ey] is a measure of the effect of an income change on demand. [Can be calculated as point or arc.]

· ey > 0, [positive] is a normal or superior good an increase in income increases demand, a decrease in income decreases demand.· 0 < ey < 1 is a normal good

· 1 < ey is a superior good

· ey < 0, [negative] is an inferior good



Examples of ey

· normal goods, [0 < ey < 1 ], (between 0 and 1) · coffee, beef, Coca-Cola, food, Physicians’

services, hamburgers, . . .

· Superior goods, [ ey > 1], (greater than 1)

· movie tickets, foreign travel, wine, new cars, . . .

· Inferior goods, [ey < 0], (negative)

· flour, lard, beans, rolled oats, . . .



Cross-Price Elasticity

· Cross-price elasticity [exy] is a measure of how responsive the demand for a good is to changes in the prices of related goods.

· Given a change in the price of good Y [Py ], what is the effect on the demand for good X [Qy ]?

· exy is defined as:

xyx

y

eQP

%

%



Cross-price elasticity of demand , [exy][substitutes]

.

When the price of pork increases, it will tend to increase the demand for beef. People will substitute beef, which is relatively cheaper, forpork, which is relatively more expensive.

pork/ut

[pri

ce o

f pork

]

Pp

Dp

When pork is $1.50, Qp porkis purchased.

1.50

Db

beef/ut

[pri

ce o

f beef]

Pb

Qp

When beef is $2, Qb beefis purchased.

2

Qb

price of pork increases

2

The quantity demanded of pork decreases.

Qp’

-Qp

at Pb = $2 more beef will be bought to substitute for the smaller

quantity of pork.

increase demand

Db’

Qb’

for an increase

in Ppork, demand forbeef increases



Cross-price elasticity

· In the case of beef and pork· the ebp is not the same as epb

· ebp is the % change in the demand for beef with respect to a % change in the price of pork

· epb is the % change in the demand for pork with respect to a % change in the price of beef

· beef may not be a good substitute for pork · pork may not be a good substitute for beef



Cross-price elasticity of demand , [exy][substitutes]

The cross elasticity of the demand for beef with respect to the

price of pork, ebeef-pork or ebp can be calculated:

ebp =% Q of beef

%P of pork

An increase in the price of pork,

+ Pp

“causes” an increase in the demand for beef.

+ Qb+ebppositive

cross elasticity is positive

ebp =% Q of beef

%P of pork

A decrease in the price of pork,

- Pp

“causes” a decrease in the demand for beef.

- Qb+ebppositive

If goods are substitutes, exy will be positive. The greater the coefficient, the more likely they are good substitutes.



Cross-price elasticity of demand , [exy][compliments]

colour books

Pc

crayons

Pc

Dp

a decrease in the price of crayons,

P1

Q1

$3

2000

Po

Q2

increases the quantity demanded of crayons

as more crayons arepurchased, the demand for colour books increases.

DcDc’

increasedemand

2500

At the sameprice a larger quantity will be bought

ebc =% Q of b

%P of c

-Pc

- Pc

+ Qb

+ Qb- ebcnegative

for compliments, the cross elasticity is negative for priceincrease or decrease.



Cross-Price Elasticity

· exy > 0 [positive], suggests substitutes, the higher the coefficient the better the substitute

· exy < 0 [negative], suggests the goods are compliments, the greater the absolute value the more complimentary the goods are

· exy = 0, suggests the goods are not related

· exy can be used to define markets in legal proceedings



Elasticity of Supply

· Elasticity of supply is a measure of how responsive sellers are to changes in the price of the good.

· Elasticity of supply [ep] is defined:

seQuantity Supplied

price%

%



Q /ut

P

Given a supply function,

supply

at a price [P1], Q1 is produced and offeredfor sale.

P1

Q1

At a higher price [P2],

P2

a largerquantity, Q2, will be producedand offered for sale.

Q2

+P

+Q

The increase in price [ P ], inducesa larger quantity goods [ Q]for sale.

The more responsive sellers are to

P, the greater the absolute value of es.

[The supply function is “flatter”ormore elastic]

Elasticity of supply

es = %Qsupplied

%P



Q /ut

PThe supply function is amodel of sellers behavior.

Sellers behavior is influenced by:1. technology

2. prices of inputs3. time for adjustment

market periodshort runlong runvery long run

4. expectations 5. anything that influences costs of production

taxesregulations, . . .

Se

a perfectly elastic supply [es is undefined.]

Sia perfectly inelasticsupply, es = 0

as supply approaches horizontal es

approaches infinity



Elasticity

· Price elasticity of demand [measures a move on a demand function caused by change in price/arc or point]· elastic, inelastic or unitary elasticity

· income elasticity [measures a shift of a demand function associated with a change in income]· superior, normal, and inferior

· cross elasticity· measure the shift of a demand function for a good

associated with the change in the price of a related good · [compliment/substitute]

· price elasticity of supply [measures move on a supply curve]

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