Elasticity as a measure of responsiveness

Y = Effect variable

X = Cause variable

Y = ƒ ( X )

Y = α – β X

Where α & β are the coefficients

Summing UP

Introductory

Economic

Lecture 6

Elasticity

Definitions

Computations

Recap

Y = α – βX in Y-X space

E (elastic)Y

X

β = slope = ∆Y / ∆X

C

A B

P

QR

IE (inelastic)

CA / AB > PQ / QR

O

Real world example

E ( elastic )Qd

P

C

A B

P

QR

IE ( inelastic )

Conventional representation

IE ( inelastic )P

R

Q P

B

AC

E ( elastic )

Qd

Slope of a demand curve

Slope of a demand curve = β

Higher slope = Inelastic demand curve (Steep)

Lower slope = Elastic demand curve (Flat)

Price elasticity of other variables

Y = ƒ ( X )

1. Y = Qd & X = Price Price elasticity of demand.

2. Y = Qs & X = Price Price elasticity of supply.

3. Y = Qd & X = Income Income elasticity of demand.

4. Y = Qda & X = Priceb Cross price elasticity of demand.

Formal definition of the four combinations

1. Price elasticity of demand

can be defined as

PЄd = Percentage change in Quantity Demanded Percentage change in Price

Where Є = Epsilon; universal notation for elasticity.

PЄd = Percentage change in Quantity DemandedPercentage change in Price

Example

If, for example, a 20% increase in the price of a product causes a 10% fall in the Quantity demanded , the price elasticity of demand will be:

PЄd = - 10% = - 0.5

20%

2. Price elasticity of supply

can be defined as

PЄs = Percentage change in Quantity Supplied

Percentage change in Price

Formal definition of the four combinations

PЄs = Percentage change in Quantity Supplied Percentage change in Price

Example

If a 15% rise in the price of a product causes a 15% rise in the quantity supplied, the price elasticity of supply will be:

PЄs = 15 % = 1

15 %

3. Income elasticity of demand

can be defined as

YЄd = Percentage change in Quantity Demanded

Percentage change in Income

Formal definition of the four combinations

YЄd = Percentage change in Quantity Demanded Percentage change in Income

ExampleIf a 2% rise in the consumer’s incomes causes an 8% rise in product’s demand, then the income elasticity of demand for the product will be :

YЄd = 8% = 4

2%

4. Cross price elasticity of demand

can be defined as

PbЄda = Percentage change in Demand for good a

Percentage change in Price of good b

Formal definition of the four combinations

PbЄda = Percentage change in Demand for good a

Percentage change in Price of good b

ExampleIf, for example, the demand for butter rose by 2% when the price of margarine rose by 8%, then the cross price elasticity of demand of butter with respect to the price of margarine will be.

PbЄda = 2% = 0.25

8%

PbЄda = Percentage change in Demand for good a Percentage change in Price of good b

Example

If, on the other hand, the price of bread (a compliment) rose, the demand for butter would fall. If a 4% rise in the price of bread led to a 3% fall in the demand for butter, the cross-price elasticity of demand for butter with respect to bread would be :

PbЄda = - 3% = - 0.75

4%

0 < |Є|< (for absolute values of elasticity)

Unit ElasticUnit ElasticPerfectly InelasticPerfectly Inelastic Perfectly ElasticPerfectly Elastic

ElasticElasticInelasticInelastic

ЄЄ = 1 = 1

ЄЄ = 0 = 0

ЄЄ < 1 < 1

ЄЄ >1 >1

ЄЄ = = αα

P

Qd

Qd

Qd

Qd

Qd

P

P

P

P

8

Total revenue and elasticity

Firm A Firm B

O O

* Not perfect competition

Firm A

O

6

100

10

90B

F

TC

D

AQd

P OAFD > OBTC

TR as P

Inelastic demand Curve

Firm B

P

QdO

R

U

Y V

U

Z6

40 100

7

OVZU > OYUR TR as P Elastic demand curve

O

6

10090B

F

TC

D

A

P

Numerical calculation of elasticity for firm A

Є = percentage change in Qd percentage change in P

= 90 – 100 10 – 6 100 6

= - 0.15

Qd

10

Numerical calculation of elasticity for Firm BЄ = percentage change in Qd percentage change in P

= 40 – 100 7 – 6 100 6

= - 3 . 6

Qd

P

O

R

U

Y V

U

Z6

40 100

7

Elastic demand between 2 points

O

6

168

K

P

8

TR as the P

L

ЄKL = percentage change in Qd percentage change in P

= 16– 8 ÷ 6 – 8 8 8 = - 4

Qd

Inelastic demand between 2 points

O

1

3628

G

P

3

TR as the P

H

ЄGH = percentage change in Qd percentage change in P = 36– 28 ÷ 1 – 3 28 3 = - 3

7

Qd

ЄLK = percentage change in Qd percentage change in P = 8 – 16 ÷ 8 – 6 16 6 = - 3

2

ЄKL = percentage change in Qd percentage change in P = 16– 8 ÷ 6 – 8 8 8 = - 4

Overview of previous example

Concept of arc elasticityAs Є = ∆ Q ÷ ∆ P

Q P To measure arc elasticity we take average values for Q and P respectively.

ЄKL = 16– 8 ÷ 6 – 8 = - 7 12 7 3

ЄLK = 8 – 16 ÷ 8 – 6 = - 7 12 7 3 average elasticity along arc KL or LK is - 7/ 3

Є = ∆ Q ÷ ∆ P Q P

Є = ∆ Q x P ∆ P Q

d = infinitely small change in price

Є = d Q x P d P Q

A straight line demand curve will have a different Є at each point on it except Є = 0 or Є = α .

Point elasticity

O

6

168

K

P

8

L

dP = -1

dQ 4

P at K = 8 = 1

Q 8

Є = - 4 x 1 = -4

P at L = 6 = -3Q 16 8 Є = - 4 x 3 = - 3

8 2

Previous example

Qd

P 60 - 15 P P2 Qd (000s)

0 60 0 0 60

1 60 -15 1 46

2 60 -30 4 34

3 60 -45 9 24

4 60 -60 16 16

5 60 -75 25 10

6 60 -90 36 6

Qd = 60 – 15P + P2

01234567

0 20 40 60 80

Qd (000s)

Quantity demanded

Pri

ce

PЄd = d Q x P d P Q

Differentiating the demand Equation

Given Qd = 60 – 15P + P2

then dQ/dP = -15 + 2P

Thus at a price of 3 for example, dQ/dP = -15 + ( 2 x 3 ) = -9 Thus price Elasticity of demand at Price 3 is - 9 x P/Q

= - 9 x 3 / 24 = - 9 / 8