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Lecture Notes on the Elasticity of Substitution
Ted Bergstrom UCSB Econ 210A
Today’s featured guest is “the elasticity of substitution.”
Elasticity of a function of a single variable
Before we meet this guest, let us spend a bit of time with a slightly simpler
notion, the elasticity of a a function of a single variable. Where
f
is a diﬀeren
tiable realvalued function of a single variable, we deﬁne the elasticity of
f
(
x
)
with respect to
x
(at the point
x
) to be
η
(
x
) =
xf
0
(
x
)
f
(
x
)
.
(1)
Another way of writing the same expression 1 is
η
(
x
) =
x
df
(
x
)
dx
f
(
x
)
=
df
(
x
)
f
(
x
)
dx
x
.
(2)
From Expression 2, we see that the elasticity of of
f
(
x
) with respect to
x
is the
ratio of the percent change in
f
(
x
) to the corresponding percent change in
x
.
There is yet another way to think of elasticities.
Measuring the responsiveness of a dependent variable to an independent
variable in percentage terms rather than simply as the derivative of the func
tion has the attractive feature that this measure is invariant to the units in
which the independent and the dependent variable are measured. For example,
economists typically express responsiveness of demand for a good to its price
by an elasticity.
1
In this case, the percentage change in quantity is the same
whether quantity is measured in tons or in ounces and the percentage change
in price is the same whether price is measured in dollars, Euros, or farthings.
Thus the price elasticity is a “unitfree” measure. For similar reasons, engineers
measure the stretchability of a material by an “elasticity” of the length of the
material with respect to the force exerted on it.
Suppose that we were to draw the graph of the function
f
on a loglog paper.
That is, let us deﬁne
z
= ln
x
, so that
x
=
e
z
. Let us also deﬁne
g
(
z
) = ln
f
(
e
z
).
Then applying the chain rule,
g
0
(
z
) =
e
z
f
0
(
e
z
)
f
(
e
z
)
=
xf
0
(
x
)
f
(
x
)
=
η
(
x
)
.
(3)
Since
g
(
z
) = ln
f
(
z
) and
z
= ln
x
, we can also express the result of equation 2
as
η
(
x
) =
d
ln
f
(
x
)
d
ln
x
.
(4)
1
Some economists ﬁnd it tiresome to talk about negative elasticities and choose to deﬁne
the priceelasticity as the absolute value of the percentage responsiveness of quantity to price.
1
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View Full DocumentIn general, the elasticity of
f
with respect to
x
depends on the value of
x
. For
example if
f
(
x
) =
a

bx
, then
η
(
x
) =

bx
a

bx
. In this case, as
x
ranges from
0 to
a/b
,
η
(
x
) ranges from 0 to
∞
. It is interesting to consider the special
case where the elasticity of
f
(
x
) with respect to
x
is a constant,
η
that does not
dependent on
x
. In this case, integrating both sides of Equation 3 with respect
to
z
, we have
g
(
z
) =
ηz
+
b
for some constants
a
and
b
. Since
g
(
z
) = ln
f
(
z
) and
z
= ln
x
, this equation is
equivalent to
ln
f
(
x
) =
η
ln
x
+
b.
(5)
Exponentiating both sides of Equation 5, we have
f
(
x
) =
cx
η
(6)
where
c
=
e
b
. Thus we see that
f
has constant elasticity
η
if and only if
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