ELASTICITY





Principles of
Microeconomic Theory,
ECO 284
John Eastwood
CBA 247
523-7353
e-mail address:
John.Eastwood@nau.edu
1
Learning Objectives

Define and calculate the price elasticity of
demand

Explain what determines the price elasticity
of demand

Use the price elasticity to determine
whether a price change will increase or
decrease total revenue
2
Learning Objectives (cont.)

Define, calculate and interpret the income
elasticity of demand

Define, calculate and interpret the crossprice elasticity of demand

Define and calculate the elasticity of supply

Use elasticities to analyze tax incidence.
3
Learning Objectives

Define and calculate the price elasticity of
demand

Explain what determines the price elasticity
of demand

Use the price elasticity to determine
whether a price change will increase or
decrease total revenue
4
Elasticity
Elasticity measures the response of one
variable to changes in some other variable.
 Civil Engineers need to know the elasticity
of construction materials.
 Economists need to know the elasticity of
quantities demanded (and supplied).

5
Elasticity of Demand

How does a firm go about determining the
price at which they should sell their product
in order to maximize profit?
–
–

Profit = total revenue – total cost = TR - TC
Total Revenue = Price  Quantity =PQ
How does the government determine the tax
rate that will maximize tax revenue?
6
Price Elasticity of Demand, ed
ed measures the responsiveness of quantity
demanded of a product to a change in its
own price, ceteris paribus.
ed = (percentage change in Q ) divided by
d
(the percentage change in the Px)
7
Example

Assume that the price of crude oil has
increased by 100%, and that the quantity
demanded has fallen by 10%
ed = -10% / 100% = -0.1

For every 1% increase in price, the quantity
demanded fell by 0.1%
8
Computing Elasticity Using the
“Arc Formula”
e
d


Q2  Q1
P2  P1


(Q1  Q2 ) / 2 ( P1  P2 ) / 2
where P1 represents the first price, P2 the
second price, and Q1 and Q2 are the
respective quantities demanded.
Elasticity is dimensionless
(units divide out).
9
Arc Formula Notation

Some people prefer to write delta for
change, and “overbar” for average.
e
Q2  Q1
P2  P1


(Q1  Q2 ) / 2 ( P1  P2 ) / 2
e
Q P


Q
P
d
d
10
Calculating Elasticity

The changes in price and quantity are
expressed as percentages of the average
price and average quantity.
–
 ed
Avoids having two values for the price
elasticity of demand
is negative; its sign is ignored
11
Price (dollars per chip)
Calculating the
Elasticity of Demand
Original
point
410
400
390
Da
36
40
44
Quantity (millions of chips per year) 12
Price (dollars per chip)
Calculating the
Elasticity of Demand
Original
point
410
400
New
point
390
Da
36
40
44
Quantity (millions of chips per year) 13
Price (dollars per chip)
Calculating the
Elasticity of Demand
Original
point
410
P=
$20
400
New
point
390
Da
36
Q = 8
40
44
Quantity (millions of chips per year) 14
Price (dollars per chip)
Original
point
410
P=
$20
400
Pave =
$400
New
point
390
Da
36
Q = 8
40
44
Quantity (millions of chips per year) 15
Price (dollars per chip)
Calculating the
Elasticity of Demand
Original
point
410
ed = ?
P=
$20
400
Pave =
$400
New
point
Qave = 40
390
Da
36
Q = 8
40
44
Quantity (millions of chips per year) 16
Price (dollars per chip)
Calculating the
Elasticity of Demand
Original
point
410
ed = 20/5 = 4
P=
$20
400
Pave =
$400
New
point
Qave = 40
390
Da
36
Q = 8
40
44
Quantity (millions of chips per year) 17
Example -- Crude Oil


Assume P1 = $15/bbl, Q1 = 105 bbl/day, and
that P2 = $25/bbl, Q2 = 95 bbl/day
Calculate ed using this formula:
e
d
 Q2  Q1   P2  P1 



 (Q1  Q2) / 2   ( P1  P2) / 2 
18
Answer:
e
 (95  105)   (25  15) 





 (105  95) / 2   (15  25) / 2 
e
1
 10   10 

    10%  50%    0.2

5
 100   20 
d
d

For every 1% increase in price, Qd fell
0.2%.
19
Elasticity and Slope
ed and slope are inversely related.
e
Q P Q P
Q P






Q
P
Q P P Q
e
1
P
1
P

 

P
Q Slope Q
Q
d
d
20
Discussing ed


Note that ed is always negative (or zero)
because of the law of demand.
However, when discussing the value of ed ,
economists almost always use the absolute
value. Using | ed |, a larger value means
greater elasticity.
21
Elastic Demand, | ed |>1
If the percentage change in quantity
demanded is greater than the percentage
change in price, demand is said to be price
elastic.
 The demand for luxury goods tends to be
price elastic.
 Examples – see page 99 of McEachern.

22
Inelastic Demand, | ed |< 1
If the percentage change in quantity
demanded is smaller than the percentage
change in price, demand is said to be price
inelastic.
 The demand for necessities tends to be price
inelastic.

23
Perfectly Elastic D, ed = infinity
If quantity demanded drops to zero in
response to any price increase, demand is
said to be perfectly elastic.
 This corresponds to a horizontal demand
curve.
 Sounds unlikely, doesn’t it?
 Example: Demand for a small country’s
exports

24
Price
Inelastic and Elastic Demand
12
6
Elasticity = 
D3
Perfectly Elastic
Quantity
25
Perfectly Inelastic D, ed =0
If quantity demanded is completely
unresponsive to a change in price, demand
is said to be perfectly inelastic.
 This corresponds to a vertical demand
curve.
 Can you think of a vertical demand curve?

26
Price
Inelastic and Elastic Demand
D1
Elasticity = 0
12
Perfectly Inelastic
6
Quantity
27
Unit Elastic D, | ed |= 1
If the percentage change in quantity just
equals the percentage change in price,
demand is said to be unit elastic.
 While there are many goods that could be
unit elastic, there aren’t any we can identify
without statistical evidence.
 Example:

28
Price
Inelastic and Elastic Demand
Elasticity = 1
12
Unit Elasticity
6
D2
1
2
3
Quantity
29
ed and Total Revenue (TR)
Note that TR = P times Q = PQ.
 Will a change in price raise or lower total
revenue?
 It all depends on the price elasticity of
demand!

30
When Demand is Elastic, P and
TR vary inversely.
Since | ed | > 1, the percentage change in Qd
is greater than the percentage change in P.
 If P rises by, say, 1%, Qd will fall by more
than 1%.
 Therefore, if price is increased, total
revenue will decrease.
 If price is reduced, then TR will rise.

31
When Demand is Inelastic, P and
TR vary directly.
Since | ed | < 1, the percentage change in Qd
is smaller than the percentage change in P.
 If P rises by, say, 1%, Qd will fall by less
than 1%.
 Therefore, if price is increased, total
revenue will increase.
 If price is reduced, then TR will fall.

32
When Demand is Unit Elastic,
TR does not change.
Since | ed | = 1, the percentage change in Qd
equals the percentage change in P.
 If P rises by, say, 1%, Qd will fall by
exactly 1%.
 Therefore, if price is increased, total
revenue will stay the same.
 If price is reduced, TR will not change.

33
Some Real-World Price
Elasticities of Demand
Good or Service
Elastic Demand
Metals
Electrical engineering products
Mechanical engineering products
Furniture
Motor vehicles
Instrument engineering products
Professional services
Transportation services
Inelastic Demand
Gas, electricity, and water
Oil
Chemicals
Beverages (all types)
Clothing
Tobacco
Banking and insurance services
Housing services
Agricultural and fish products
Books, magazines, and newspapers
Food
Elasticity
1.52
1.30
1.30
1.26
1.14
1.10
1.09
1.03
0.92
0.91
0.89
0.78
0.64
0.61
0.56
0.55
0.42
0.34
0.12
34
Example: Demand for Oil and
Total Revenue
Assume demand is p = 60 - q
 TR = price x quantity =PQ
 Substituting 60-q for p gives, TR=(60-q)q
 Multiply through by q to get an equation for
TR, TR = 60q - q2
 TR will graph as a parabola.
 Let’s calculate TR and graph it with D.

35
Computing Total Revenue
Q
0
10
20
30
40
50
P
60
50
40
30
20
10
TR
0
500
800
900
800
500
Unit Analysis:
Q (bbl/day)
P ($/bbl)
TR = P ($/bbl.) times Q (bbl. /day) = TR ($/day)
60
0
0
36
60
900
800
700
600
500
400
300
200
100
0
50
40
30
20
10
0
0
5
10 15 20 25 30 35 40 45 50 55 60
Quantity (bbl./day)
Total Revenue ($/day)
Price ($/bbl.)
Demand (P), Total Revenue (TR),
and Marginal Revenue (MR)
P=AR
MR
TR
38
Price ($/bbl.)
Total Revenue as an Area
60
50
40
30
20
P=AR
10
0
0
5
10
15
20
25
30
35
40 45 50 55 60
Quantity (bbl./day)
39
Linear Demand and Point
Elasticity
 ed can be illustrated with geometry.
With a linear D, the slope is constant.
 We don’t need an arc to get the slope.
 Elasticity is inversely related to slope.

e
d
Q P
1
P
1
P

 
 

P Q P Q Slope Q
Q
40
ed and Linear Demand
60
50
40
P
P
30
D
20
10
0
M
O
0
M
5
10
15
20
25
30
T
35
40
45 50
Quan
55
60
41
ed Using Line Segments
The formula for ed may be rewritten in
terms of the length of line segments.
 O is the origin, T is the x-intercept, and M is
a point between O and T.

e
d
Q P MT MP MT

 


P Q MP OM OM
42
Elasticity at the Midpoint




| ed | =MT/OM 50
40
for any linear
demand curve. 30
20
If M is the middle,10
O
0
then MT=OM.
0
60
ed = | -1| = 1
P
P
D
M
M
5
10
15
20
25
30
T
35
40
45 50
Quan
55
60
Unit Elastic at the
midpoint.
43
Elasticity at Higher Prices
If M is left of the
middle, then
MT>OM.
60
40
| ed | =MT/OM.
30
| ed | > 1
10
Demand is elastic
at higher
prices.
P
50
P
D
20
M
O
M
T
0
0
5
10
15
20
25
30
35
40
45 50
Quan
55
60
44
Elasticity at Lower Prices
If M is right of
the middle,
then MT<OM.
60
50
40
| ed | =MT/OM.
30
| ed | < 1
10
Demand is
inelastic at
lower prices.
P
P
20
M
M
O
D
T
0
0
5
10
15
20
25
30
35
40
45 50
Quan
55
60
45
Two Extremes
At the point where the demand curve
intercepts the vertical axis, ed is infinite or
perfectly elastic.
 At the point where the demand curve
intercepts the horizontal axis, ed = 0, that is,
demand is perfectly inelastic.

46
Determinants of ed

Number of substitutes
–
–
quality
availability
Budget proportion
 Time

–
–
to respond
to consume
47
Other Elasticity Concepts

Income Elasticity of Demand, ey

Cross Price Elasticity of Demand, ex,z

Price Elasticity of Supply, es
48
Income Elasticity of Demand, ey
 ey measures the change in demand for a
good (X) in response to a change in income
(Y), ceteris paribus.

If ey > 0, X is a normal good.

If ey < 0, X is an inferior good.
49
Computing Income Elasticity

Q
e
y

Q
Y
Y

With Q1 and Q2,
find the change in
quantity and the
average quantity .
Given Y1 and Y2,
find the change in
income and the
average income.
51
Example Computations

Q
e
y

Q

Y
Y

Median annual
family income rose
from $39,000 to
$41,000 per year.
The demand for
electricity rose from
79,000 GWh to
81,000 GWh.
Normal or inferior?
52
Cross Price Elasticity of D, ex,z
 ex,z measures the responsiveness of the
demand for one good to a change in the
price of another good, ceteris paribus.
 ex,z = (% change in demand for X ) divided
by (% change in PZ)
54
Using Cross Price Elasticity
 ex,z > 0 tells us the goods X and Z are
substitutes.
 ex,z < 0 tells us the goods X and Z are
complements.
 ex,z = 0 tells us the goods X and Z are
unrelated.
55
Computing Cross-Price Elasticity

 QX
e
x ,z

 PZ
QX
PZ

With QX1 and QX2,
find the change in
quantity and the
average quantity .
Given PZ1 and PZ2,
find the change in
price and the
average price.
56
Example Computations

 QX
e
x ,z

 PZ
QX
PZ


The price of
gasoline rose from
$.75 to $1.25/gal.
The demand for
Subarus rose from
9/day to 11/day.
The demand for
Cadillacs fell by
10%.
57
Subaru Example

Let X = Subarus, and Z = gasoline.

Find ex,z .
Are Subarus and gasoline related goods?
 If so, are they complements or substitutes?

58
Cadillac Example

Let X = Cadillacs, and Z = gasoline.

Find ex,z .
Are Cadillacs and gasoline related goods?
 If so, are they complements or substitutes?

60
Price Elasticity of Supply, es
 es measures the
 Qs

es
P
Qs
responsiveness of
quantity supplied
to a change in the
good’s price.
P
62
Example Computations

 Qs

es
P
Qs

P

The price of corn
fell from $3/bu. to
$1/bu.
The quantity
supplied of corn fell
from 101,000 bu to
99,000 bu.
Compute the price
elasticity of supply.
63
es Along a Supply Curve
70
60
50
e
u
P
40
30
20
y
10
0
Su
Si
Se
i
x
O
-10 0
5
10
Q
15
20
25
30
35
40
45 50
Quan
55
60
66
es Using Line Segments

Rewrite the formula for es in terms of point
elasticity. Note the relationship with the
slope. Use length of line segments to get es
Q P
1
P




es P Q Slope Q
67
A Supply Curve with es = 1

Find es at point u:

Note that Su is unit elastic at any point.
Q P 0Q uQ





1
es P Q uQ 0Q
68
A Supply Curve with es <1

Find es at point i.

Si is inelastic at any point.
Q P xQ iQ xQ






1
es P Q iQ 0Q 0Q
69
Supply Curves May Not Touch
the x-axis or y-axis.
Si is unrealistic.
 It implies that the firm would supply
positive quantities of its product at a price
of zero (or at a negative price)!
 As we will learn later, a firm will shut down
if the price of its product falls too low.
Thus, we should draw supply curves that
begin at a positive (Q, P).

70
A Supply Curve with es >1

Find es at point e. Se is elastic at any point.

As y gets larger, es gets larger.

As P gets larger, es approaches 1.
Q P 0Q 0 P 0 P
es  P  Q  yP  0Q  yP  1
71
Perfectly Elastic Supply


the slope is zero.
Cost per unit is
constant.
Example: One
consumer may buy
as many apples as
s/he wishes at the
going price.
Price ($/unit)
 es is infinite when
70
60
50
40
S
D
30
20
10
0
0
5
10 15 20 25 30 35 40 45 50 55 60
Quantity (units/time)
72
Perfectly Inelastic Supply
 es is zero when the slope


is infinite.
Price has no effect on the
quantity supplied.
e.g.: Once the crop is
ready to harvest, the
farmer will do so as long
as s/he can earn at least
the cost of harvesting it.
P
Smarket period
D
Pmin
0
Qs
Q
73
Determinants of es :
The degree of substitutability of resources
among different productive activities.
 Time -- Given more time, producers are
able to make more adjustments to their
production processes in response to a given
change in price.

74
Elasticity and the Burden of a Tax
The economic incidence of taxation falls on
the persons who suffer reduced purchasing
power because of the tax.
 The legal incidence falls on the persons who
are required by law to pay the tax to the
government.

75
Tax Burden
Demand for Tonic: P = $42 - 3Q
 Let Supply be: P = -3 + 2Q. (es <1.)
 Solve for equilibrium quantity:

–
–
–

-3 + 2Qe = 42 - 3Qe
5Qe = 45
Qe = 9 pints per day (|ed|<1 if Q>7.)
Solve for equilibrium price:
–
Pe = 42 - 3Qe = 42 - 27 = $15 per pint.
76
Legal incidence on seller:
Add the tax to Supply: P= -3+2Q+10=7+2Q
 Solve for new quantity:

–
–
–

7 + 2Qn = 42 - 3Qn
es>1
5Qn = 35
Qn = 7 pints per day (|ed|=1 if Q=7.)
Solve for gross & net price:
–
–
Pgross = 42 - 3Qn = 42 - 21 = $21 per pint.
Pnet = - 3 + 2Qn = -3 + 14 = $11 per pint.
77
Price ($/pint)
Specific Tax on the Seller
45
40
35
30
25
20
15
10
5
0
Demand
Supply
S + Tax
0
2
4
6
8
Quantity (pints/day)
10
12
14
78
Legal incidence on buyer:
Subtract tax from Demand: P= 42-3Q-10
 Solve for new quantity:

–
–
–

-3 + 2Qn = 32 - 3Qn
5Qn = 35
Qn = 7 pints per week (|ed|=1 if Q=7.)
Solve for gross & net price:
–
–
Pgross = 42 - 3Qn = 42 - 21 = $21 per pint.
Pnet = 32 - 3Qn = 32 - 21 = $11 per pint.
79
Price ($/pint)
Specific Tax on the Buyer
45
40
35
30
25
20
15
10
5
0
Demand
Supply
D - Tax
0
2
4
6
8
Quantity (pints/day)
10
12
14
80
Compute|ed| and es

Before the tax Pe = $15/pint and Qe = 9 pints/week
The slope of D = -3, while the slope of S = 2.
e
d
1
P 1 15

    0.56
Slope Q 3 9
1
P 1 15
es  Slope  Q  2  9  0.83
81
Now Who Pays the Tax?

Consumers now pay $21 per pint
–

Vendors now receive $21 per pint,
–
–
–

$6 / pint more than before the tax
but must pay the $10 per pint tax.
Sellers keep only $11 per pint.
$4 / pint less than before
Buyers respond less to a change in price, so
they pay more of the tax.
82