Exercise 5.23.
Verify the Leontief utility function on
R
2
+
is not differentiable, but the induced
Walrasian demand function is differentiable.
5.2
Comparative statics
Suppose
φ
:
R
n
×
R
m
→
R
, which we will write as
φ
(
x
;
q
). The decision maker gets to choose
x
to
maximize
φ
, while
q
are given parameters over which the decision maker has no control. Sometimes
x
is referred to as the endogenous variable and
q
as the exogenous variable. The basic comparative
statics question is how the decision maker will adjust her optimal choice
x
*
(
q
) in response to changes
in the parameters
q
. In general, there can be multiple maximizers
x
*
(
q
), but we will make life easier
here and assume that there exists a unique solution.
30
5.2.1
Comparative statics without constraints using the Implicit Function Theorem
Suppose
φ
(
x
;
q
) is a smooth objective function, where
x
refers to the
n
dimensional vector of choice
(endogenous) variables and
q
refers to the
m
dimensional vector of exogenous parameters. Consider
the simple maximization problem:
x
*
(
q
) = arg max
φ
(
x
;
q
)
.
Suppose
φ
is strictly concave and differentiable, so the first order condition defines the maximizer:
f
(
x
;
q
) =
D
x
φ
(
x
;
q
) =
0
>
n
.
The Jacobian of
f
is
D
x
f
=
D
xx
φ
(
x
;
q
)
and is nonsingular since
φ
is strictly concave. The first part of the Implicit Function Theorem says
we can treat
x
*
(
q
) as continuously differentiable function, at least locally. Moreover, the second
part of the Implicit Function Theorem tells us how to compute the local response of the
x
*
(
q
) to
small changes in
q
:
D
q
x
*
(¯
q
)
=

[
D
x
f
(
x
*
(¯
q
); ¯
q
)]

1
D
q
f
(
x
*
(¯
q
); ¯
q
)
=

[
D
xx
φ
(
x
*
(¯
q
); ¯
q
)]

1

{z
}
n
×
n
D
qx
φ
(
x
*
(¯
q
); ¯
q
)

{z
}
n
×
m
By the Chain Rule, the change of the value function,
φ
*
(
q
) =
φ
(
x
*
(
q
);
q
) in response to changes
in
q
is:
D
q
φ
(
x
*
(¯
q
); ¯
q
)
=
D
q
φ
(
x, q
)

x
=
x
*
(¯
q
)
,q
=¯
q
+
=0
z
}
{
D
x
φ
(
x, q
)

x
=
x
*
(¯
q
)
,q
=¯
q
D
q
x
*
(
q
)
=
D
q
φ
(
x, q
)

x
=
x
*
(
q
)
In other words, for small changes, the “second order effect” of how the maximizer
x
*
responds to
q
is
irrelevant; simply compute the “first order effect” of how
q
changes the objective function evaluated
at the fixed maximizer
x
*
(
q
). This observation is sometimes called the Envelope (Pseudo)Theorem.
If
x
and
q
are scalars,
D
q
x
*
(
q
) becomes
∂x
*
∂q
=
∂
2
φ
∂q∂x
∂
2
φ
∂x
2
5.2.2
Comparative statics with constraints using the Implicit Function Theorem
Now suppose there are
k
equality constraints,
F
i
(
x
;
q
) = 0, with each
F
i
smooth, which we can
stack to form the
k
dimensional constraint
F
(
x
;
q
) =
0
k
. We are therefore considering the following
31
maximization problem:
x
*
(
q
) = arg max
φ
(
x
;
q
) subject to
F
(
x
;
q
) =
0
k
.
We assume all constraint qualifications are met. Form the Lagrangian:
L
(
λ, x
;
q
) =
φ
(
x
;
q
)

λ
>
F
(
x
;
q
)
.
The derivative of the Lagrangian is:
f
(
λ, x
;
q
)

{z
}
1
×
(
k
+
n
)
=
D
(
λ,x
)
L
(
λ, x
;
q
) =
1
×
k
z
}
{

F
(
x
;
q
)
;
D
x
φ
(
x
;
q
)

{z
}
1
×
n

λ
>
{z}
1
×
k
D
x
F
(
x
;
q
)

{z
}
k
×
n
.
At the maximum (
λ
*
, x
*
;
q
), we know
f
(
λ
*
, x
*
;
q
) =
0
>
k
+
n
. The Jacobian of
f
is:
D
(
λ,x
)
f

{z
}
(
k
+
n
)
×
(
k
+
n
)
=
k
×
k
z}{
0
k
×
n
z
}
{

D
x
F
(
x
;
q
)

D
x
F
(
x
;
q
)
>

{z
}
n
×
k
D
2
xx
φ
(
x
;
q
)

D
x
[
λ
>
D
x
F
(
x
;
q
)]

{z
}
n
×
n
.