Section4: ComparativeStatics,Duality,andAfriat’sTheorem
ECON 201A, Fall 2010
GSIs: Omar Nayeem and Aniko Oery
These notes were originally prepared by Juan Sebasti´an Lleras during the Fall 2007 semester and have since been
revised by Juan Sebasti´an and us. We are grateful for his permission to continue using these resources this semester.
Summary
In the last couple of lectures, we wrapped up the material on comparative statics
using the Implicit Function Theorem and monotone comparative statics and then
studied the concept of Hicksian demand, in which the consumer’s problem is framed
as an expenditure minimization problem.
It turns out that there is a neat dual-
ity between this problem and the utility maximization problem that characterizes
Walrasian demand.
We then talked briefly about the types of inferences regarding rationality that can
be drawn from finite sets of consumption, wage, and price data. The main result of
this part is due to Afriat. This result shows (among other things) that rationality,
in the case of such a data set, is equivalent to the Generalized Axiom of Revealed
Preference (GARP).
4.1
Implicit Function Theorem and Comparative
Statics
The Implicit Function Theorem is a useful tool for analyzing the responses of opti-
mization problem solutions to small changes in the values of exogenous parameters.
Specifically, we are interested in observing how the solutions to the UMP change as
we vary prices and the wage.
Last time we stated the Implicit Function Theorem
and discussed some mathematical examples illustrating it. Today, we will apply it to
some economic settings in order to do comparative statics.
Theorem 4.1.
(Implicit Function Theorem)
Suppose
A
is an open set in
R
n
+
m
and
f
:
A
→
R
n
is continuously differentiable.
Let
D
x
f
refer to the
n
×
n
derivative
matrix of f with respect to its first arguments, i.e.
(
D
x
f
)
ij
=
∂f
i
∂x
j
. If
f
(¯
x,
¯
q
) =
0
n
and
4-1

Section 4: Comparative Statics, Duality, and Afriat’s Theorem
4-2
D
x
f
(¯
x,
¯
q
)
is nonsingular, then there exist open sets
A
¯
x
∋
¯
x
in
R
n
and
B
∋
¯
q
in
R
m
and a unique continuously differentiable function
g
:
B
→
A
¯
x
such that
{
(
g
(
q
)
, q
) :
q
∈
B
}
=
{
(
x, q
)
∈
A
¯
x
×
B
:
f
(
x, q
) =
0
n
}
.
Moreover,
D
q
g
(¯
q
) =
−
[
D
x
f
(¯
x,
¯
q
)]
−
1
D
q
f
(¯
x,
¯
q
)
.
Exercise 4.2.
(From 2005 exam). Suppose a firm produces two goods and its profit
function is
π
(
x
1
, x
2
) =
x
1
p
1
(
x
1
) +
x
2
p
2
(
x
2
)
−
c
(
x
1
, x
2
)
−
t
1
x
1
−
t
2
x
2
,
where
p
i
(
x
i
) is a concave function which denotes the market price if the firm produces
quantity
x
i
of good
i
;
c
(
x
1
, x
2
) is a convex function which denotes the cost of producing
(
x
1
, x
2
); and
t
i
denotes a tax on each unit of
i
that is sold. Use the Implicit Function
Theorem to find how changes in
t
1
,
t
2
affect the firm’s optimal production choice.