Income Elasticity

(Normal Goods)

Elasticity

Income Elasticity

(Normal Goods)

Elasticity

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

Income Elasticity

(Normal Goods)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

Income Elasticity

(Normal Goods)

“Own” Price Elasticity

Income Elasticity

(Normal Goods) 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.

Income Elasticity

(Normal Goods)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.

Income Elasticity

(Normal Goods)

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

Income Elasticity

(Normal 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 softdrink 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!

Fall '97 Economics 205Principles of Microeconomics Slide 24

“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

Fall '97 Economics 205Principles of Microeconomics Slide 26

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

Fall '97 Economics 205Principles of Microeconomics Slide 27

Reference: Principles of Economics, 6/e by Karl Cas, Ray Fair

Slides prepared by: Fernando Quijano and Yvonn Quijano