Economics, 3rd Edition

%

ΔP

%

ΔQ

Substituting our known values 

into the formula we get:

1

0.8

=

%

ΔP

10

1.25

=

Δ%P

10

%

ΔP = 12.5

To bring about a reduction in 

demand of 10 per cent, the price of 

motor vehicles would have to rise by 

12.5 per cent.

Other Elasticities

Income and cross elasticity of demand 

are all treated in exactly the same way 

as the analysis of price elasticity of 

demand above. 

Point income elasticity would be:

Income

 

 

elasticity

 

 

of

 

 

demand

 

 

=

 

 

ΔQ

ΔY

 

 

·

 

 

Y

Q

Using calculus:

Income

 

 

elasticity

 

 

of

 

 

demand

 

 

=

 

 

dQ

dY

 

 

·

 

 

Y

Q

For cross elasticity the formulas 

would be:

Cross

 

 

elasticity

 

 

of

 

 

demand

 

 

=

 

 

ΔQa

ΔPb

 

·

 

Pb

Qa

Where Qa is the quantity demanded 

of one good, a, and Pb is the price of 

a related good, b (either a substitute 

or a complement). For a substitute

dQa

dPb

 

> 0 and for a complement,

dQa

dPb

 

< 0.

Cross

 

 

elasticity

 

 

of

 

 

demand

 

 

=

 

 

dQa

dPb

 

·

 

Pb

Qa

In Chapter 3 we saw that demand 

can be expressed as a multivariate 

function where demand is dependent 

on a range of variables which include 

price, incomes, tastes and so on. It is 

possible to calculate the elasticities 

of all these other factors using the 

same principles as those outlined 

above. In each case it is usual to cal-

culate the elasticity with respect to a 

change in one of the variables whilst 

holding the others constant.

For example, take the demand 

 

equation 

Q

= 1,400 − 4P + 0.04Y.  

This equation tells us that demand is 

dependent on the price and also the 

level of income.

From this equation we can calcu-

late the price elasticity of demand and 

the income elasticity of demand. In 

this example we will use calculus to 

find both elasticities assuming P 

= 50 

and Y 

= 8,000.

Given these values:

Q

= 1,400 − 4(50) + 0.04(8,000)

Q

= 1,400 − 200 + 320

Q

= 1,520

With:

dQ

dP

= −4

ped

= –

 

4a

50

1,520

b

ped

= –

 

0.132

Given:

dQ

dY

= 0.04

Income elasticity of demand

 

 

(yed)

dQ

dY

 

·

 

Y

Q

 

= 0.04a

8,000

1,520

b

yed

= 0.21

Now look at this demand equation:

Qa

  

=

 

100

  

−

  

8Pa

  

−

  

6Pb

  

+

  

4Pc

  

+

  

0.015Y

This equation gives the relation-

ship between demand and the prices 

of other goods labelled a,  b and c 

respectively. We can use this to find 

the respective cross elasticities.

Assume that the price of good a is 

20, the price of good b, 40, the price 

of good c, 80 and Y 

= 20,000.

Substituting these into our func-

tion gives:

Qa

= 100 − 8Pa − 6Pb + 4Pc

   

    

+ 0.015Y

Qa

= 100 − 8(20) − 6(40) + 4(80)

   

    

+ 0.015(20,000)

Qa

= 100 − 160 − 240 + 320 + 300

Qa

= 320

The change in demand of good a 

with respect to changes in the price 

of good b is given by:

dQa

dPb

= −6

The

cross

 

elasticity

 

of

 

demand

= −6a

40

320

b

= −6(0.125)

= –0.75

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CHAPTER 4  ELASTICITY AND ITS APPLICATIONS  93

The relationship between goods 

a and b is that they are complements –  

a rise in the price of good b will lead 

to a fall in the quantity demanded of 

good a.

The change in demand of good a 

with respect to changes in the price 

of good c is given by:

dQa

dPc

= 4

Cross elasticity of demand

= 4a

80

320

b

= 4(0.25)

= 1

In this case the relationship 

between the two goods is that they 

are substitutes – a rise in the price 

of good c would lead to a rise in the 

quantity demanded of good a.

Price Elasticity of Supply

Many of the principles outlined 

above apply also to the price elasti-

city of supply. The formula for the 

price elasticity of supply using the 

point method is:

Price elasticity of supply

=

ΔQs

ΔP

·

P

Qs

Using calculus:

Price elasticity of supply

=

dQs

dP

·

P

Qs

However, we need to note a 

particular issue with price elasti-

city of supply which relates to the 

graphical representation of supply 

curves.

This is summarized in the following:

t
 A straight line supply curve inter-

secting the 

y-axis at a positive 

value has a price elasticity of 

supply

 > 1

t
 A straight line supply curve 

passing through the origin has a 

price elasticity of supply 

= 1

t
 A straight line supply curve 

Intersecting the 

x-axis at a posit-

ive value has a price elasticity of 

supply 

< 1

To see why any straight line 

 

supply  curve passing through the 

origin has a price elasticity of sup-

ply of 1 we can use some basic 

knowledge of geometry and similar 

triangles.

Figure 4.13 shows a straight line 

supply curve S

1

 passing through the 

origin. The slope of the supply curve 

is given by 

ΔP

ΔQ

s

. We have high-

lighted a triangle, shaded green, with 

the ratio 

ΔP

ΔQ

s

 relating to a change in 

price of 7.5 and a change in quantity 

of  1. The larger triangle formed by 

taking a price of 22.5 and a quantity 

of 3 shows the ratio of the price and 

quantity at this point (P/Q). The two 

triangles formed by these are both 

classed as similar triangles – they 

have different lengths to their three 

sides but the internal angles are 

all the same. The ratio of the sides 

must therefore be equal as shown by 

equation (1) below:

 

ΔP

ΔQs

=

P

Qs

 

(1) 

Given our definition of point 

elasticity of supply, if we substi-

tute equation 1 into the formula and 

rearrange we get:

Price elasticity of supply

=

ΔQs

ΔP

 

P

Qs

Therefore:

Price elasticity of supply

= 1

Elasticity and Total  

Expenditure/Revenue

We have used the term ‘total 

expenditure’ in relation to the 

demand curve to accurately reflect 

the fact that demand is related to 

buyers and when buyers pay for 

products this represents expendit-

ure. Many books use the term 

expenditure and revenue inter-

changeably and in this short section 

we are going to refer to revenue.

FIGURE 4.13

Price (

€)

Quantity

1

2

3

4

5

6

5

0

10

25

30

20

15

S

1

Q

P

∆P

∆Q

s

---

94  PART 2  SUPPLY AND DEMAND: HOW MARKETS WORK

Total revenue is found by mul-

tiplying the quantity purchased 

by the average price paid. This is 

shown by the formula:

TR

= P × Q

Total revenue can change if either 

price or quantity, or both, change. 

This can be seen in Figure 4.14 

where a rise in the price of a good 

from P

o

 to P

1

 has resulted in a fall in 

quantity demanded from Q

o

 to Q

1

.

We can represent the change in 

price as 

ΔP so that the new price is 

(P

+ ΔP) and the change in quant-

ity as 

ΔQ so that the new quantity is 

(Q

+ ΔQ) so TR can be represented 

thus:

TR

= (P + ΔP)(Q + ΔQ)

If we multiply out this expression 

as shown then we get:

TR

= (P + ΔP)(Q + ΔQ)

TR

= PQ + PΔQ + ΔPQ + ΔPΔQ

In Figure 4.14, this can be seen 

graphically.

As a result of the change in price 

there is an additional amount of 

revenue shown by the blue rectangle 

(Q

 

ΔP). However, this is offset by the 

reduction in revenue caused by the fall 

in quantity demanded as a result of the 

change in price shown by the green 

rectangle  (P

 

ΔQ). There is also an 

area indicated by the yellow rectangle 

which is equal to 

ΔPΔQ. This leaves us 

with a formula for the change in TR as:

ΔTR = QΔP + PΔQ + ΔPΔQ

Let us substitute some figures into 

our formula to see how this works in 

practice. Assume the original price 

of a product is 15 and the quantity 

demanded at this price is 750. When 

the price rises to 20 the quantity 

demanded falls to 500.

Using the equation:

TR

= PQ + PΔQ + ΔPQ + ΔPΔQ

TR is now:

TR

= 15(750) + 15(–250)

+ 5(750) + 5(–250)

TR

= 10,000

The change in TR is:

ΔTR = QΔP + PΔQ + ΔPΔQ

ΔTR = 750(5) + 15(–250) + 5(–250)

ΔTR = 3,750 − 3,750 − 1,250

ΔTR = –1,250

In this example the effect of the 

change in price has been negative 

on  TR. We know from our analysis 

of price elasticity of demand that 

this means the percentage change 

in quantity demand was greater than 

the percentage change in price  – 

in other words, price elasticity of 

demand must be elastic at this point 

(

>1). For the change in TR to be pos-

itive, therefore, the price elasticity of 

demand must be 

<1.

We can express the relationship 

between the change in TR and price 

elasticity of demand as an inequality 

as follows:

Price elasticity of demand

 

=

ΔQ

ΔP

·

P

Q

> 1

The original TR is found by multiplying the original price ( P

o

) by the original quantity ( Q

o

) and is shown by the red 

+ green 

rectangles.

Price

Quantity

Q

1

Q

0

P

1

P

0

D

1

∆ PQ

∆P∆Q

Q∆P

P∆Q

Th

i i l TR i f

d

FIGURE 4.14

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CHAPTER 4  ELASTICITY AND ITS APPLICATIONS  95

When price increases, revenue 

decreases if price elasticity of 

demand meets this inequality. 

Equally, for a price increase to result 

in a rise in revenue ped must meet 

the inequality below:

Price elasticity of demand

=

ΔQ

ΔP

·

P

Q

< 1