Lecture Notes on Elasticity of Substitution
Ted Bergstrom, UCSB Economics 210A
March 3, 2011
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
differentiable 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
.
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 exam
ple, 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
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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 rea
sons, engineers measure the stretchability of a material by an “elasticity” of
the length of the material with respect to the force exerted on it.
The elasticity of the function
f
at a point of
x
can also be thought of as
the slope of a graph that plots ln
x
on the horizontal axis and ln
f
(
x
) on the
vertical axis. That is, suppose that we make the change of variables
u
= ln
x
and
v
= ln
y
and we rewrite the equation
y
=
f
(
x
) as
e
v
=
f
(
e
u
). Taking
derivatives of both sides of this equation with respect to
u
and applying the
chain rule, we have
e
v
dv
du
=
e
u
f
0
(
e
u
)
(3)
and hence
dv
du
=
e
u
f
0
(
e
u
)
e
v
=
xf
0
(
x
)
f
(
x
)
=
η
(
x
)
,
(4)
where the second equality in Expression 4 is true because
e
u
=
x
and
e
v
=
f
(
x
). Thus
dv
du
is the derivative of ln
f
(
x
) with respect to ln
x
. We sometimes
express this by saying that
η
(
x
) =
d
ln
f
(
x
)
d
ln
x
.
(5)
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 5, we have
ln
f
(
x
) =
η
ln
x
+
a
(6)
for some constant
a
. Exponentiating both sides of Equation 6, we have
f
(
x
) =
cx
η
(7)
where
c
=
e
a
. Thus we see that
f
has constant elasticity
η
if and only if
f
is
a “power function” of the form 7.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Bergstrom
 Economics, Derivative, Monotonic function, Convex function

Click to edit the document details