Solutions to Problem Set 2
EC720.01  Math for Economists
Peter Ireland
Boston College, Department of Economics
Fall 2009
Due Thursday, September 24
1. Utility Maximization  SecondOrder Conditions
The following result specializes Theorem 19.8 from Simon and Blume’s book to provide ﬁrst
and secondorder conditions for a constrained optimization problem with two choice variables
and a single constraint that is assumed to bind at the optimum.
Theorem
Let
F
:
R
2
→
R
and
G
:
R
2
→
R
be twice continuously diﬀerentiable functions,
and consider the constrained optimization problem
max
x
1
,x
2
F
(
x
1
,x
2
) subject to
c
≥
G
(
x
1
,x
2
)
,
with parameter
c
∈
R
. Associated with this problem, deﬁne the Lagrangian
L
(
x
1
,x
2
,λ
) =
F
(
x
1
,x
2
) +
λ
[
c

G
(
x
1
,x
2
)]
.
Suppose there exist values
x
*
1
,
x
*
2
, and
λ
*
of
x
1
,
x
2
, and
λ
that satisfy the ﬁrstorder conditions
L
1
(
x
*
1
,x
*
2
,λ
*
) =
F
1
(
x
*
1
,x
*
2
)

λ
*
G
1
(
x
*
1
,x
*
2
) = 0
,
L
2
(
x
*
1
,x
*
2
,λ
*
) =
F
2
(
x
*
1
,x
*
2
)

λ
*
G
2
(
x
*
1
,x
*
2
) = 0
,
L
3
(
x
*
1
,x
*
2
,λ
*
) =
c

G
[(
x
*
1
,x
*
2
)
≥
0
,
λ
*
≥
0
,
and
λ
*
[
c

G
(
x
*
1
,x
*
2
)] = 0
.
Suppose also that
c

G
(
x
*
1
,x
*
2
), so that the constraint binds at the optimum, and that the
“bordered Hessian” matrix
H
=
0
G
1
(
x
*
1
,x
*
2
)
G
2
(
x
*
1
,x
*
2
)
G
1
(
x
*
1
,x
*
2
)
L
11
(
x
*
1
,x
*
2
,λ
*
)
L
21
(
x
*
1
,x
*
2
,λ
*
)
G
2
(
x
*
1
,x
*
2
)
L
12
(
x
*
1
,x
*
2
,λ
*
)
L
22
(
x
*
1
,x
*
2
,λ
*
)
satisﬁes the secondorder condition that

H

>
0, so that the determinant of
H
is strictly
positive. Then
x
*
1
and
x
*
2
are local maximizers of
F
(
x
1
,x
2
) subject to
c
≥
G
(
x
1
,x
2
).
Note that this result provides suﬃcient conditions for a solution to the problem: it says that
if the ﬁrst and secondorder conditions are satisﬁed, then the values of
x
*
1
and
x
*
2
constitute
at least a local maximum.
1