elasticity handout economics

7/24/2019 Elasticity Handout economics

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CHAPTER FOUR

ELASTICITY

We have seen in chapter three how a change in the price of the good results in

change in quantity demanded of that good in the opposite direction (movement

along the same demand curve); and how a change in income results in a

change in quantity demanded at every price. The same thing is said about the

changes in the price of related goods and other non-price determinants.

The question is now how to measure the magnitude of each change in quantity

demanded or supplied as a response to a change in one of the independentvariables. The same argument can be applied to the quantity supplied.

In order to have a better picture of the degree of responsiveness of quantity to a

change in one of the independent variable we have to understand the concept of

elasticity.

The Economic Concept of Elasticity

Elasticity is a measurement of the degree of responsiveness of the dependent

variable to changes in any of the independent variables.

In general elasticity is the percentage change in one variable in response to a

percentage change in another variable.

tCoefficienElasticityX%

Y%

VariablendependentI%

ariableVdependent%Elasticity =

=

=

Elasticity coefficientincludes a sign and a size. We need to interpret the sign

and the size of the coefficient.

Sign shows the direction of the relationship between the two variables. A

positive sign shows a direct relationship while a negative sign shows an inverse

relationship between the two variables.


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Size illustrates the magnitude of this relationship. In other words, it shows how

large the response of the dependent variable to the change in the independent

variable.

Large elasticity coefficient means that a small change in the independent

variable will result in a large change in the dependent variable (the opposite is

true).

Elasticity coefficient is a unit-free measure because in calculating the elasticity

we use the percentage change rather than the change to avoid the difficulty of

comparing different measurement units, and the percentages cancel out.

Changing the units of measurement of price or quantity leave the elasticity value

the same

Elasticity is an important concept in economic theory. It is used to measure the

response of different variables to changes in prices, incomes, costs, etc.

In addition to price and income elasticities of demand, you may estimate the

elasticity with respect to any of the other variables like advertisement and

weather conditions. You may even measure the elasticity of production to

various inputs or the elasticity of your grades in managerial economics to hours

of study.

This chapter covers some of the important types of elasticities.


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THE PRICE ELASTICITY OF DEMAND (Ed):

In the previous chapter we have discussed the movement of the quantity

demanded along a given demand curve as a result of change in the price of the

good. The direction of the movements reflects the law of demand that shows an

inverse (negative) relationship between P and Qd; the lower the price the greater

the quantity demanded.

When supply increases while demand stays

constant, the equilibrium price falls and the

equilibrium quantity increases. But does

the price fall by a large amount or a little?

And does the quantity increase by large

amount or a little? The answer depends on

the responsiveness of quantity demanded to a change in price.

We are now going to discuss the question of how sensitive the change in

quantity demanded is to a change in price. The response of a change in quantity

demanded to a change in price is measured by the price elasticity of demand.

Price elasticity of demand (Ed)is an economic measure that is used to

measures the degree of responsiveness of the quantity demanded of a good to

a change in its price, when all other influences on buyers plans remain the

same.

The price elasticity of demand is calculated by dividing the percentage change

in quantity demanded by the percentage change in price.

P/P

Q/Q

P%

Q%E dddd

=

=

Example:

Suppose P1= 7, P2= 8, Q1= 11, Q2= 10, then

If Pfrom 8 to7, Ed= -0.8

If Pfrom 7 to 8, Ed= -0.64

You can see that the value of Edis different depending on direction of change in

P even with the same magnitude.

Q0 QQ2

S0

P0

P1

D1

D2

Q1

P

S1

P2


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To solve this problem we use Arc elasticity

The arc elasticity of demandis measured over a discrete interval of a demand

(or a supply) curve.

To calculate the price elasticity of demand (Ed): We express the change in price

as a percentage of the average pricethe average of the initial and new price,

and we express the change in the quantity demanded as a percentage of the

average quantitydemandedthe average of the initial and new quantity. By using the average priceand average quantity, we get the same elasticity

value regardless of whether the price rises or falls.

=+

+

=

+

+

=

+

+

=

+

+

=

=

=

12

12

12

12

12

12

12

12

12

12

12

12

12

12

12

12

avg

avg

d

dd

QQ

PP

PP

QQ

PP

PP

QQ

QQ

PPPP

QQ

QQ

2/)PP(PP

2/)QQ(

QQ

PP

Q

Q

P%

Q%E

Where,

Q1= the original (the old) quantity demanded, Q2= the new quantity demanded

P1= the original (the old) price, P2= the new price

Qavg= the average quantity, Pavg= the average price The formula yields a negative value, because price and quantity move in

opposite directions (law of demand). But it is the magnitude, or absolute value,

of the measure that reveals how responsive the quantity change has been to a

price change. Thus, we ignore the minus (negative) sign and use the absolute

value because it simply represents the negative relationship between P and Qd

Example:

Suppose P1= 7, P2= 8, Q1= 11, Q2= 10, then

71.0

2

78

78

2

1110

1110Ed =+

+

=

Now how to interpret the elasticity coefficient? What Ed= - 0.71 means?


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It means that if the price of the good increases (decreases) by 1% the quantity

demanded of the good decreases (increases) by 0.71%

Example:

Price($) Qd(bushels of Wheat)

8 207 406 605 80

What is the Ed if P increases from 6 to 7?

6.267

67

4060

6040Ed =

+

+

=

A 1% increase in P would result in a 2.6% decrease in Qd

Example:If a rightward shift in the supply curve leads to an increase in Qdby 10 % as a

result of a decrease in P by 5%.

a. Calculate Ed.

25

10

P%

Q%E dd =

=

=

b. Interpret Ed

Ed= 2 means that a decrease in P by 1% results in an increase in Qdby 2%c. What would be the increase in Qdif P decreases by 4%?

SinceP%

Q%E dd

= , then dQ% = ( dE ) ( P% ) = (-2) (-4%) = + 8 %,

Thus, a decrease in P by 4% results in an increase in Qdby 8%

d. What would be the decrease in P if Qdincreases by 6%

SinceP%

Q%E dd

= , then P% = %3

2

%6

E

Q%

d

d =

=

,

Thus, if Qdincreases by 6%, P decreases by 2%

However, if we want to measure Edat a single point rather than between two

points we should use pointelasticity of demand


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The point elasticity of demandmeasured at a given point of a demand (or a

supply) curve. It is the price elasticity for small changes in the price or for

changes around a point on the demand curve.

QP

dPdQ

P

dPQ

dQ

P%Q%

d

dd ==

=

The point elasticity of a linear demand function can be expressed as:

Q

P

P

Qd

=

Notice that the first term of the last formula is nothing but the slope of the

demand function with respect to the price.

Having this fact in mind you will easily remember that:

1. The value of the elasticity; varies along a linear demand curve, as P/Q

change even though, the slope is constant.

2. The value of the elasticity varies along a nonlinear demand curve as both

terms in the above equation varies from as we move along a nonlinear

demand curve.

3. The value of the elasticity is constant along the demand curve only in the

case of an exponential function in the form:

Qd= aP-b,

where the price elasticity of demand equals b, which can be proved as

follows:

baP

PbaP

Q

P

dP

dQb

1bd ===

This type of nonlinear equations can be expressed in linear form using logarithm

log Q = log a b (log P)

Example:

Calculate the elasticity of demand using the following equation:

Qd= 50P-3,


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3P

P

50

150

50

PP

P

150

P50

P)P(150

Q

P

dP

dQ4

43

43

4d =

=

===

Example:

Given Qd = 2000 - 20P, Find dwhen P=70

At P=70, Qd = 2000 20(70) = 2000 1400 = 600

33.2600

70)20(

Q

P

dP

dQd ===

Example:

If Qd=200 - 300P + 120I + 65T 250Pc + 400Ps,

and if I=10, T=60, Pc=15, Ps j=10, Find:

a. Edfor the price range $10 and $11

b. dat P =$10Qd= 200 - 300P + 120(10) + 65(60) 250(15) + 400(10)

= 200 - 300P + 1200 + 3900 3750 + 4000

Qd= 5550 300P

a. Edfor the price range $10 and $11 (Arc Elasticity)

At P=10, Qd= 5550 300(10) = 2550

At P=11, Qd= 5550 300(11) = 2250

31.125502250

1011

1011

25502250Ed =++

=

b. dat P =$10 (Point Elasticity)

2.12550

10)300(

Q

P

dP

dQd ===

Example:

Assume a company sells 10,000 units of its output at price of $100.

Suppose competitors decrease their price and as a result the companys sale

decrease to 8,000 units. Edin this price-quantity range is -2. What must be the

price if the company wants to sell the same number of units before its

competitors decrease their price?

Using 2QQ

PP

PP

QQE

12

12

12

12d =

+

+

=


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Q1=8000, Q2= 10000, P1= 100, P2=?

000,800,1P000,18

000,200P000,2

)000,18)(100P(

)100P(000,2

)000,8000,10)(100P(

)100P)(000,8000,10(

000,8000,10

100P

100P

000,8000,102

2

2

2

2

2

22

2

+=

+=

+

+=

+

+

=

For simplification, divide numerator and denominator of left-hand side by 1000

800,1P18

200P22

2

2

+=

200P2)1800P18(2 22 +=

-36P2+ 3600 = 2P2+ 200

-38P2= -3400

P2= -3400/-38 = 89.5TR1 = 100 X 8,000 = 800,000

TR2= 89.5 X 10,000 = 895,000

Since TR2 > TR1TR it is good to cut price but is not known since we do

not the TC

Example:

A 50% decrease in the price ofsalt caused the quantity demanded to increase

by10%. Calculate the price elasticity of demand for salt, explain the meaning of

your result and tell if the demand for salt is elastic or inelastic?

Ed= 10/50 = 0.20 which means a10% change in price results in a 2% Change in

the quantity demanded in the opposite direction.

Example:

Qd= 50 P3, is the demand curve equation for apple, calculate the price

elasticity of demand when P =3 and Q = 9.

93

127

9

3)3(3

Q

P

dP

dQ 2d ====


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Categories of Demand Elasticity

Using absolute value of Ed, we differentiate between five categories of elasticity

that range between zero and infinity.

1. Relatively Elastic Demand (Ed> 1)

IfP%

Q%E dd= > 1 % Qd> % P demand is elastic.

Consumers are very responsive to changes in P. Demand curve is flatter 1%

change in P results in a more than 1% change in Qd(in the opposite direction).

(if Ed= 2 that means if P by 1% Qd by 2%.)

Examples of elastic goods: cars, furniture, vacations, etc.

2. Relatively Inelastic Demand (Ed< 1)

IfP%

Q%E dd

= < 1 % Qd< % P demand

is inelastic.

Consumers are not very responsive to changes

in P. Demand Curve is steeper 1% (or) in P results in a less than 1%

(or) in Qd (if Ed= 0.70 that means if P by 1% Qdby 0.7%.) or (if P by

10% Qdby 7%.)

Examples of inelastic goods: medicine, food, etc.

If the price elasticity is between 0 and 1, demand is inelastic.

More Elastic

Qd

P

More Inelastic


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P

3. Unitary Elastic Demand (Ed= 1)

IfP%

Q%E dd

= = 1 % Qd= % P demand is

unit-elastic1% in P results in a 1% in Qd

4. Perfectly Elastic Demand (Ed= )

IfP%

Q%E dd

= =demand is perfectly elastic

horizontal demand curve the same price is

charged regardless of Qd(perfect competition).

Any price increase would cause demand

to fall to zero. Shifts in supply curve results in no change in price. Examples:

identical products sold side by side, agricultural products.

5. Perfectly Inelastic Demand (Ed= 0)

IfP%

Q%E dd

= = 0 demand is perfectly inelastic

a vertical demand curve demand is

completely inelastic. Qdremains the same

regardless of any change in price. Shifts in supply

curve results in no change in Qd. Examples: medicine of heart diseases or

diabetes such as insulin A good with a vertical demand curve has a demand

with zero elasticity.

We conclude from the five categories above that the more flatter is the demand

curve the more elastic is the demand and the more steeper is the demand curve

the more inelastic is the demand

QdQd

DPS1

S2

0D

P

Qd

D

Qd

PS1

S2


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Elasticity along straight line demand curve

Elasticity of demand (Ed) is not the slope of the demand curve.

dQ

PSlope

= ,

Elasticity:P%

Q%E dd=

For a straight-line (linear) demand curve the slope is constant (i.e., the slope is

the same at every point along the curve). It is equal to the change in price over

the change in quantity demanded.

Although the slope is constant, price elasticity varies along a linear demand

curve.

The following equation shows the relationship between the elasticity and the

slope of a straight line demand curve

dd

d

d

ddddd

Q

P

slope

1

Q

P

P

Q

P

P

Q

Q

P/P

Q/Q

P%

Q%E =

=

=

=

=

Since the slope of straight-line demand curve is constant, slope

1

is also

constantelasticity varies as a result of variation ofdQ

P; i.e. straight-line

demand curve elasticity depends on the values of Qdand P

Ed> 1

Ed=1

Ed< 1

Ed =

Ed = 0Q

P


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1. When P = 0, Ed= 0 (perfectly inelastic)

2. When Q = 0, Ed= (perfectly elastic)

3. Edincreases as we move upward along a straight-line demand curve (from

the inelastic range to the elastic one) (as P and Q)

4. Eddecreases as we move downward along the straight-line demand curve

(as Pand Q).

Thus, along downward sloping demand curve, demand is elastic when price is

high, inelastic when price is low and unit-elastic at the midpoint of the demand

curve.

Pricing Strategy: The Relationship between P, Ed, and TR

Managers of profit maximizing firms are usually concern with the best pricing

strategy.

There is a relationship between the price elasticity of demand and revenue

received.

Total revenue (TR)equals the total amount of money a firm receives from the

sales of its product

TR = P X Q.

TR is affected by changes in both P and Qd. But as we know by now the law of

demand implies that an increase in P will result in a decrease in Qd.

Thus, an increase in P may or may not lead to greater TR. This depends on

which effect is the largest, price effect or the effect of quantity demanded.

The size of the price elasticity of demand coefficient, tells us which of these two

effects is largest.o If demand is elastic (Ed>1) % Qd> % P

10 %in P results in more than 10 %in sales TR

10 %in P results in more than 10 %in sales TR

o If demand is inelastic (Ed


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10 %in P results in less than 10 %in sales TR

o If demand is unit elastic (Ed=1) % Qd= % P

10 %in P results in 10 %in sales TR does not change

10 %in P results in 10 %in sales TR does not change

TR

E>1 E


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Marginal revenueis the revenue generated by selling one additional unit of the

product

Itis the change in total revenue resulting from changing quantity by one unit.

Q

TRMR

=

For a straight-line demand curve the marginal revenue curve is twice as steep

as the demand.

To sell more, often price must decline, so MR is often less than the price.

When EP= -. MR = P.

At the point where marginal revenue crosses the X-axis, the demand curve is

unitary elastic and total revenue reaches a maximum.

A product maximizing manager will expand product as long as the additional unit

produced adds more to TR than adding to TC; i.e., expand production as long

as MR > MC and Mis positive.

The optimal production reached when MR = MC and M= 0

Units produced over and above the optimal level will have negative Mbecause

for these units MR < MC

When TR is maximized, MR = 0 and MC (positive value) is definitely greater

than MR.

The conclusion here is that if the manager maximizes TR the firm will make less

than max profit.

Ed Demand MR P TR

Ed>1 Elastic MR >0

Ed< 1 Inelastic MR < 0

Ed= 1 Unit elastic MR = 0 - Max.


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The above graph shows that:

o Ed >1 Demand elastic MR>0 P and TR move in the opposite

direction (negative relationship)

o Ed< 1 Demand inelastic MR < 0 P and TR move in the same

direction (positive relationship)o Ed= 1 Demand unit elastic MR = 0 TR is maximum

Example:

If a company wants to its TR when Ed= 0.75, it should P

Example:

If a company wants to its TR when Ed= 1.5, it should P

Ed< 1

*

MR

P

E > 1;

MR > 0

E = 1;

MR=0

E < 1;

MR< 0

0

00

TR

TR

Ed> 1

Ed= 1

Q

Q*Q


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Example:

If Ed= 1, an in P by 15%, Qdby 15%, TR will not change

Example:

Given Qd= 20 2P,

Find the price range for which

a. D is elastic

b. D is inelastic

c. D is unit elastic

d. If the firm increases P to $7, is TR increasing or decreasing?

Answer:

Qd= 20 2P P = 10 0.5Q

TR = 10Q 0.5Q2

MR = 10 Q

When MR = 0, 10 Q =0 Q = 10 and P = 10 0.5(10) = 5

At this P and Q, Ed= 1

a. D is elastic for price range above 5 (or Q less than 10)

b. D is inelastic for price range below 5 (or Q above 10)

c. D is unit elastic at P = 5 and Q = 10

d. If the firm chooses to increase the price to $7 and 7 is in the elastic part,

TR will be decreasing

Example:

Given Qd= 150 10P, find Q and P at which d= -1

Since Qd= 150 10P P = (150/10) (1/10) Q = 15 0.1Q

TR = 15Q 0.1Q2

MR = dTR/dQ = 15 0.2Q(Note that the slope of MR equation is twice the slope of the inverse demand

equation).

TR reaches maximum when MR = 0 (Q that max TR is the same as Q that

makes MR= 0

Set MR =0 15 0.2Q = 0 Q =15/0.2= 75 and P = 15 0.1(75) = 7.5


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So, at Q=75 and P=7.5

MR = 0 and 175

75

75

5.7)10(

Q

P

dP

dQd =

=

== TR is maximized

Find P and Q at which Ed>1 and Ed1 at P > 7.5, and Q < 75

Ed 75

Example:

P Q TR MR Ed

10 1 10 --- --

9 2 18 8 6.33

8 3 24 6 3.40

7 4 28 4 2.14

6 5 30 2 1.44

5 6 30 0 1.00

4 7 28 -2 0.69

3 8 24 -4 0.47

2 9 18 -6 0.29

1 10 10 -8 0.16

Ed> 1 (elastic demand

Ed= 1 (unitary elastic), TR is

max and MR is zero

Ed< 1 (inelastic demand

Exercise:

From the graph to the right

a. calculate Ed

b. When P increases what would happen to TR?

Ed= 1 and TR remains the same.

The area (0-5-a-20) = the area (0-4-b-25)

a

b

5

4

2 20

D

P


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MR and Elasticity

The relationship between Marginal Revenue (MR), price (P), and price the

elasticity of demand (Ed),can be stated using the formula:

+=d

11PMR

Clearly the equation shows that if Ed-1, MR

must be negative; and if Ed = -1, MR must be zero.

To proof the relationship between MR and Ed, (for your information only)

We know that TR = P X Q

)1(1PMR,Thus

P

QX

dQ

dP1,So

Q

PX

dP

dQ

But

)dQ

dPX

P

Q1(P)

dQ

dPXQ

P

11(P

dQ

dPXQ

P

PpMR

P/PbytermsecondtheMultiply

dQ

dPQP

dQ

dPQ

dQ

dQPX)PXQ(

dQ

d

dQ

dTRMR

d

d

d

+=

=

=

+=+=+=

+=+===

If P = 20 and d= -4 find MR

+=

4

1120MR

MR = 20 (1 - 0.25)

= 20 (0.75) = 15


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Factors Affecting Demand Elasticity

Demand for some goods and services is elastic whereas for other goods and

services is inelastic.

Elasticity does not only differ from one good to another but also it may differ for

a particular product at different prices.

The elasticity of demand is computed between points on a given demand curve.

Hence, the price elasticity of demand is influenced by all determinants of

demand.

We can summarize the main factors that affect Edas:

1. Availability and closeness of Substitutes

When a large number of substitutes are available, consumers respond to a

higher price of a good by buying more of the substitute goods and less of the

relatively more expensive one. So, we would expect a relatively high price

elasticity of demand for goods or services with many close substitutes, but

would expect a relatively inelastic demand for goods with few close substitutes.

Example:

Dell computer, for example, has many substitutes. So its price elasticity ofdemand is highly elastic because the consumers can easily shift to the other

substitutes if the price of Dell computer increases

Example:

Pepsi and Coke are very close substitutes. So, the availability of Pepsi makes

the price elasticity of demand of Coke very high. Any increase in the price of

Coke will result in a huge shift of consumers to Pepsis purchase.

Furthermore, the broader the definition of the good, the lower the elasticity since

there is less opportunity for substitutes. The narrower the definition of the good

the higher the elasticity, since there are more substitutes.


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Example:

A buyer who likes Japanese cars and has relative preference for Toyota

products may have higher price elasticity of demand for Camry than the price

elasticity of demand for Toyota cars. His price elasticity of demand for Toyota

cars is higher than the price elasticity of demand for Japanese cars. And his

price elasticity of demand for Japanese cars is higher than the price elasticity of

demand for cars in general. Why?

Example:

Consider the relative price elasticity of demand for a good such as apples

compared to a good such as fruits. What is the difference between apples and

fruits? Apples are, of course, a fruit but so are lots of other goods as well.

Hence, more substitutes exist for apples than exist for the broader category of

fruits. We have already determined that as the number of substitutes increase

then so does that goods relative price elasticity of demand.

2. Proportion of total expenditures to Income

The higher the proportion of income spent on the good, the higher the elasticity

of demand. Expensive good take a greater proportion of an individuals incomeand expenditures than the inexpensive goods; so expensive good are more

elastic.

Example:

Consider the price elasticity of demand for a good such as a pen compared to

that for a good such as a car. One of the big differences between these two

types of goods is that the price of a pen is small as a proportion of the income

while the price of a car is typically a large percentage of income. Doubling theprice of pens will not, therefore, have a big impact on ones income. However,

doubling the price of cars will have a large impact on ones income.

Thus, the demand for high-priced goods such as cars tends to be more price

elastic than the demand for low-priced good such as bread or salt.


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3. The Time Elapsed Since Price Change (Length of Time)

Over time, demand tends to be more elastic because time is available to search

for substitutes for a good when a longer time period is considered.

Example:

Consider what happens as the price of a good such as gasoline doubles. People

respond to the higher price by decreasing their use of gas. However, in just a

short time period it is more difficult to do this than in a longer period. Essentially,

the longer the time period people have to adjust, the more alternatives they can

find to reduce their consumption of gas. For example, they might be able to

move closer to work, buy a more fuel-efficient car, use public transportation,

arrange with friends to go in on car, etc.

Thus, in short run, the response is very limited demand is less elastic; over

time, demand tends to be more elastic because time is available to search for

substitutes and adjust to the new situation

4. Necessary vs. Luxury goods

Demand for necessary goods, goods that are critical to our everyday life and

have no close substitute, is relatively inelastic (food, medicine).

Demand for luxury goods, goods with many substitutes and we would like to

have but are not likely to buy unless our income jumps or the price declines

sharply, is relatively elastic (cars, traveling to foreign countries for vacation).

Nevertheless, what is one person's luxury is another person's necessity

5. Durability of the product:

The demand for durable goods (such as cars) tends to be more price elastic

than the demand for non-durable goods, such as foods.

This is because durable goods have the possibility of postponing purchase,

have the possibility of repairing the existing ones, and the possibility of buying

used ones.


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As a result a small percentage change in the prices of durable goods cause

larger percentage change in the quantity demanded.

The Elasticity of Derived Demand:

The demand for intermediate goods (goods used in producing the final good) is

called a derived demand, since the demand for these goods is directly

associated with the demand for the final good. The derived demand for a

specific intermediate good will be more inelastic:

1. The more essential is that good to the production of the final good.

2. The more inelastic the demand for the final good.

3. The smaller the share of that good in the cost of producing the final good.

4. The shorter the time passes after the price changes.


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Example:

The manager of Global Food Inc heard the news that government plans to give

a 15% raise to all its employees who represent 70% of the labor force of the

country. If the estimated income elasticity of demand for global food products is

0.85, find the expected change in the demand for the firm products.

%Y = 15% X 70% =10.5%

5.10

Q%85.0

Y%

Q%E

d

dY

=

=

=

%Qd= 10.5 X 0.85 = 8.9%

Examples:1. If peoples average income increased from BD300 to BD350 per month and

as a result their purchase of orange juice increased from 5000 liters to 5800

liters per month, Calculate EY

EY= 0.96.

The increase in income by 10% results in an increase in the Qdof orange

juice by 9.6% .Orange juice is a normal, necessary good. People buy more of

it when their income increases.2. If peoples average income increased from BD300 to BD350 per month and

as a result their purchase of used mobiles decreased from 400 units to 300

units per month, Calculate EY

EY= - 1.86.

The increase in income by 10% results in a decrease in the Qdof used

mobiles by 18.6%. Since the sign is negative this means the mobile is an

inferior good. People buy less of it when their income increases.

3. If income by 5% and Qdby 10% EY= +2 normal, luxury good

4. If income by 5% and Qdby 10% EY= -2 inferior good


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Page 25 of 34

CROSS ELASTICITY OF DEMAND

The decision to buy a good depends not only on its price but also on the price

and availability of other goods (substitutes or complements).

We know that as the price of related good changes, the demand for the goodwill also change.

What we want to know here is how much will quantity demanded rise or fall as

the price of the related good changes. That is, how elastic is the demand curve

in response to changes in prices of related goods.

Cross elasticity measures the responsiveness of Qdof a particular good to

changes in the prices of its substitutes and its complements.

If X and Y are two goods, the cross elasticity of demand is the percentagechange in Qdof good X to the percentage change in price of good Y

The arc elasticity formula:

y1y2

y1y2

x1x2

x1x2

y1y2

y1y2

x1x2

x1x2

y

xR

PP

PP

QQ

QQ

2

PP

PP

2

QQ

QQ

P%

Q%E

+

+

=

+

+

=

=

For small price changes, the cross elasticity may be calculated as a point

elasticity using the following formula:

x

y

y

x

y

y

x

x

y

xR

Q

PX

dP

dQ

P

dP

Q

dQ

P%

Q%E =

=

=

When the cross elasticity of demand has a positive sign, the two goods are

substitute goods.

When the cross elasticity of demand has a negative sign, the two goods are

complementary goods

When ER=0 no relation between PX and DY

The size of cross elasticity of demand coefficient is primarily used to indicate the

strength of the relationship between the two goods in question.


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Page 26 of 34

Two products are considered good substitutes or complements when the

coefficient is larger than 0.5 (in absolute terms)

Example:

If P1x= 20, P2x= 30

Q1y= 200 Q2y= 250

Q1z= 150 Q2z= 140

Determine the relationship between X and Y, and the relationship between X

and Z

ER(xy)= 0.556 X and Y are strong substitutes

ER(xz)= - 0.172 X and Z are mild complements

Example:

Given QA= 3 2PA+1.5Y + 0.8PB 3PC

If PA= 2, Y=4,PB=2.5,PC=1

Calculate

a. ER between A and B

b. ER between A and C

Solution,

dQA/dPB= 0.8, dQA/dPC= -3,

QA= 3 2(2) +1.5(4) + 0.8(2.5) 3(1) = 4

a. 5.04

5.28.0 === X

Q

PX

dP

dQE

A

B

B

A

R Strong Substitutes

b. 75.04

13 === X

Q

PX

dP

dQE

A

C

C

AR Strong Complements

Example:

Nissan Maxima and Toyota Camry are competing substitutes in the market for

small passenger cars. The Nissan Manager would like to predict the negative

effect of Toyotas 15% discount on Camry during Ramadhan. From previous

years, Nissan manager has an estimate of the cross elasticity of 2.0 between

these two brands.


27/28

Page 27 of 34

Given this information, calculate the expected effect on Nissan sales of Maxima

cars.

Solution

%15

%2

%

%

=

=

=

Maxima

Camry

Mamima

R

QP

QE

%QMaxima= 2 X (-15%) = -30%

Maxima sales are expected to drop by 30% as a result of Toyota discounts

during Ramadhan.

Exercise

Find the point price elasticity, the point income elasticity, and the point crosselasticity at P=10, Y=20, and PR=9, if the demand function were estimated to be:

Qd= 90 - 8P + 2Y + 2PR

Is the demand for this product elastic or inelastic? Is it a luxury or a necessity?

Does this product have a close substitute or complement? Find the point

elasticities of demand.

Solution

First find the quantity at these prices and income:Qd= 90 - 8P + 2Y + 2PR= 90 -8(10) + 2(20) + 2(9) =90 -80 +40 +18 = 68

Ed= (Q/P)(P/Q) = (-8)(10/68)= -1.17 which is elastic

EY= (Q/ Y)(Y/Q) = (2)(20/68) = +.59 which is a normal good, but a necessity

ER= (Qx/ PR)(PR/Qx) = (2)(9/68) = +.26 which is a mild substitute


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NET OR COMBINED EFFECT OF ELASTICITY

To find the total effect of change in more than one variable on the quantity

demanded, we may combine the effect of price elasticity of demand (Ed),

income elasticity of demand (EY), and cross elasticity of demand (ER), and or

any other elasticity, thus calculating the net effect of theses changes.

Most managers find that prices and income change every year.

By definition we know that:

o Ed= %Q/ %P %Q = Ed(%P)

o EY= %Q/ %Y %Q = EY(%Y)

o ER= %Q/ %PR%Q = ER(%PR)

If you knew the price, income, and cross price elasticities, then you can forecast

the percentage changes in quantity.

Combining these effects (assuming independent and additive functions) we

have:

%Q = Ed(%P) + EY(%Y) + ER(%PR)

Where, P is price, Y is income, and PRis the price of a related good.

Example:

LTC has a price elasticity of -2, and an income elasticity of 1.5 for its laptops.

The cross elasticity with another brand is +.50

a. What will happen to the quantity sold if LTC raises price 3%, income rises

2%, and the other brand companies raises its price 1%?

b. Will Total Revenue for this product rise or fall?

Solution

a. %Q = Ed(%P) + EY(%Y) + ER(%PR)

= -2 (3%) + 1.5 (2%) +0.50 (1%) = -6% + 3% + 0.5% = -2.5%.We expect sales to decline.

b. Total revenue will rise slightly (about + 0.5%), as the price went up 3%

and the quantity of laptops sold falls 2.5%.

elasticity handout economics

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