UCSC
AMS/ECON 11A
Supplemental Notes # 6
Elasticity
c 2006 Yonatan Katznelson
1.
Relative and percentage rates of change
The derivative of a differentiable function y = f (x) describes how the function changes.
The value of the derivative at a point, f 0 (x0 ), gives the instantaneous rate of change of the
function at that point, which we can understand in more practical terms via the approximation
formula
∆y = f (x0 + ∆x) − f (x0 ) ≈ f 0 (x0 ) · ∆x,
which is accurate when ∆x is sufficiently small, (see SN5 for more details). This formula
describes the change ∆y in terms of the change ∆x, which is very useful in a variety of contexts.
In other contexts however, it is sometimes more appropriate to describe the change in y in relative
terms. In other words, to compare the change in y to its previous value.
Suppose, for example that you learn that a firm’s revenue increased by $2.5 million over
the past year. An increase in revenue is generally a good thing, but how good is a $2.5 million
increase? The answer depends on the size of the firm, and more specifically, on the amount of
last year’s revenue. If in the previous year the firm’s revenue was $2.1 million, then an increase
of $2.5 million means that revenue more than doubled, which means that the firm’s business
is growing very robustly. On the other hand, if last year’s revenue was $350 million, then $2.5
million is an increase of less than 1%, which is far less impressive.
The relative change in the value of a variable or function is simply the ratio of the change
in value to the starting value. I.e., the relative change in y = f (x) is given by
∆y
f (x0 + ∆x) − f (x0 )
=
.
y
f (x0 )
Likewise, the relative rate of change (rroc) of a function y = f (x) is obtained by dividing the
derivative of the function by the function itself,
rroc =
f 0 (x)
1 dy
= ·
.
f (x)
y dx
(1)
Example 1
dy
Suppose that y = x2 + 1, then
= 2x, and the relative rate of change of y with respect to x
dx
is
2x
1 dy
·
= 2
.
y dx
x +1
When x = 4, the relative rate of change is 2 · 4/(42 + 1) = 8/17 ≈ 0.47, and when x = 10, the
relative rate of change is 2 · 10/(102 + 1) = 20/101 ≈ 0.198.
♣
The relative change, and the relative rate of change are often expressed in the percentage
terms, in which case they are usually called the percentage change and the percentage rate of
change, respectively. Thus, the percentage change in y is given by
%∆y =
∆y
f (x0 + ∆x) − f (x0 )
· 100% =
· 100%,
y
f (x0 )
1
and the percentage rate of change (%-roc) of the function y = f (x) is simply the relative rate
of change multiplied by 100%,
%-roc = rroc · 100% =
f 0 (x)
1 dy
· 100% = ·
· 100%.
f (x)
y dx
Example 1 (continued)
The percentage rate of change of the function y = x2 + 1 is
1 dy
2x
·
· 100% = 2
· 100%.
y dx
x +1
When x = 4, the percentage rate of change of the function is
8
· 100% ≈ 47%,
17
and when x = 10 the percentage rate of change is
20
· 100% ≈ 19.8%.
101
♣
Since the approximation formula provides an estimate for ∆y, it can also be used to estimate the percentage change, (∆y/y) · 100%. Specifically, if ∆x is sufficiently small, then the
approximation formula says that ∆y ≈ f 0 (x0 ) · ∆x, and therefore
%∆y =
∆y
f 0 (x0 ) · ∆x
· 100% ≈
· 100%.
y
f (x0 )
If we rearrange the factors on the right-hand side we obtain
Fact 1 (Approximation formula for percentage rate of change) If y = f (x) is differentiable at the point x = x0 and ∆x is sufficiently small, then
0
f (x0 )
f (x0 + ∆x) − f (x0 )
· 100% ≈
· 100% · ∆x.
(2)
%∆y =
f (x0 )
f (x0 )
In words, if ∆x is sufficiently small, then the percentage change in y is approximately equal to
the percentage rate of change of y multiplied by the change in x.
Example 1 (conclusion)
We have already seen that for y = x2 + 1, the percentage rate of change when x = 10 is
20/101 ≈ 19.8%. Now, suppose that x increases from x0 = 10 to x1 = 10.4, so that ∆x = 0.4.
Then, according to equation (2), the percentage change in y will be approximately
%∆y ≈ (19.8%) · (0.4) = 7.92%.
♣
2.
Elasticity
Economists frequently describe changes in percentage terms, and as described above, the
derivative can be used to compute the percentage rate of change of a function. Another measure
of change that we commonly see in economics is elasticity. Average elasticity is defined as a
ratio of percentage changes.
2
Definition 1 For the function y = f (x), the average elasticity, Ey/x , of the variable y
with respect to the variable x is given by the ratio
Ey/x =
(∆y/y) · 100%
%∆y
=
.
%∆x
(∆x/x) · 100%
(3)
Note that elasticity is a units-free measure of change, since the percentage units in the numerator and denominator cancel each other.
Example 2
A firm’s short-term production function is given Q = 20L1/4 , where Q is weekly output, measured in $1000’s and L is labor input, measured in 100’s of weekly worker-hours. In preparation
for the holiday season, the firm increases labor input from 4000 hours per week to 4300 hours
per week, in other words, L increases from L0 = 40 to L1 = 43. Correspondingly, the firm’s
output increases from Q0 = 20 · 401/4 ≈ 50.297, to Q1 = 20 · 431/4 ≈ 51.215.
In percentage terms, the change in labor input is %∆L = (3/40) · 100% = 7.5%, and
the change in output is %∆Q ≈ (0.918/50.297) · 100% ≈ 1.825%. The firm’s average labor
elasticity of output is therefore
EQ/L =
%∆Q
1.825%
≈
≈ 0.243,
%∆L
7.5%
♣
in this case.
The instantaneous rate of change, a.k.a. the derivative, was defined by considering the limit
of average rates of change over smaller and smaller intervals. In the same way, we define the
point elasticity by considering the limit of average elasticity, as the length of interval of change
goes to 0.
Definition 2 For the function y = f (x), the point elasticity of y with respect to x is
denoted η y/x , and is defined by
η y/x = lim Ey/x = lim
∆x→0
∆x→0
%∆y
.
%∆x
(4)
The point elasticity is often just called the elasticity.
The limit in equation (4) is actually easy to compute, but to do so, we need to express the
ratio (%∆y)/(%∆x) in more familiar terms. Namely, we have
∆y
∆y
· 100%
∆y x
%∆y
y
y
=
=
=
· .
∆x
∆x
%∆x
∆x y
· 100%
x
x
Now, if we substitute the expression on the right-hand side of the equation above into the
definition of point elasticity in equation (4), then we find that
%∆y
∆y x
∆y
x
dy x
η y/x = lim
= lim
·
= lim
· =
· ,
∆x→0 %∆x
∆x→0 ∆x y
∆x→0 ∆x
y
dx y
or, simply put,
η y/x =
dy x
· .
dx y
3
(5)
Comment: Given the function y = f (x), the elasticity η y/x is also a function of the variable x,
η y/x =
f 0 (x)
· x,
f (x)
and we see that the point elasticity is simply the relative rate of change of y with respect to x,
multiplied by x.
Example 3
Returning to the production function, Q = 20L1/4 , from Example 2, let’s compute the elasticity
of Q with respect to L at the point L0 = 40. First we compute η Q/L ,
η Q/L =
dQ L
1
L
1 20
1
·
= 20 · L−3/4 ·
= ·
· L−3/4 · L3/4 = ,
dL Q
4
4 20
4
20L1/4
i.e., the labor elasticity of output is constant in this case! In particular, the (point) elasticity of
output with respect to labor input at the point L0 = 40 is equal to 0.25.
♣
Example 4
Household demand for a certain good is given by the function q = 20 ln(0.001y 3 + 1), where q
is the number of units of the good consumed per year, and y is the annual disposable household
income, measured in $1000’s. The income elasticity of demand is
η q/y =
dq y
0.003y 2
y
0.003y 3
· = 20 ·
·
=
.
dy q
0.001y 3 + 1 20 ln(0.001y 3 + 1)
(0.001y 3 + 1) ln(0.001y 3 + 1)
If a household’s disposable income is $20000, then their income elasticity of demand for this
good is
24
η q/y ≈ 1.2136.
=
9 · ln 9
y=20
If a household’s disposable income is $30000, then their income elasticity of demand is
81
η q/y =
≈ 0.8681,
28 · ln 28
y=30
and if their income is $40000 then the income elasticity is
192
η q/y =
≈ 0.7076.
65 · ln 65
y=40
Can you show that η q/y is a decreasing function of y, for this demand function?
♣
What do these numbers mean? To understand how the (point) elasticity of y with respect
to x may be interpreted, we return to the definition,
%∆y
.
∆x→0 %∆x
η y/x = lim
This means that when ∆x is sufficiently small, the ratio on the right is very close to the elasticity,
i.e., when ∆x is sufficiently small
%∆y
≈ η y/x .
%∆x
If we multiply both sides of this approximate equality by %∆x, we obtain
4
Fact 2 (The approximation formula for elasticity) If y = f (x) and ∆x is sufficiently small,
then
(6)
%∆y ≈ η y/x · (%∆x) .
Example 4 (continued)
How will a household’s consumption of the good change, if their annual disposable income
increases from $20000 to $21000? The $1000 increase represents a 5% increase, in other words
%∆y = 5%, so according to the approximation formula above,
%∆q ≈ η q/y · (%∆y) ≈ 1.2136 · 5% ≈ 6.068%.
y=20
In other words, consumption will increase by a little more than 6%. If a household’s income
increases from $40000 to $42000, again a 5% increase, then according to the approximation
formula
%∆q ≈ η q/y · (%∆y) ≈ 0.7076 · 5% ≈ 3.538%.
y=40
♣
Example 5
The supply curve1 for the widget market is given by qs = 300(0.5p − 10)4/3 , where qs is the
quantity of widgets supplied weekly, and p is the price of a widget. The price-elasticity of supply
is
4
p
dqs p
2p
1/3
·
= 300 · (0.5p − 10) · (0.5) ·
.
η qs /p =
=
4/3
dp qs
3
1.5p − 30)
300(0.5p − 10)
When p = 30, the price-elasticity of supply is η qs /p = 60/15 = 4, and when p = 60 the
elasticity is η qs /p = 120/60 = 2. In general, as the price rises, the price elasticity of supply in
this example decreases, but it never goes below η 0 = 1.5.
Can you prove the last statement above? You should begin your attempt by phrasing the
statement in terms of a limit.
♣
To be continued...
1
This model for supply only makes sense once the price of widgets is high enough so that the supply is
positive, i.e., when p > 20.
5