Solutions to Problem Set 7
EC720.01  Math for Economists
Peter Ireland
Boston College, Department of Economics
Fall 2009
Due Thursday, October 29
Consider an economy populated by a large number of identical consumers, each of whom
takes
s
0
as given, and chooses sequences
{
c
t
}
∞
t
=0
and
{
s
t
}
∞
t
=1
to maximize the utility function
∞
X
t
=0
β
t
u
(
c
t
)
subject to the budget constraint
d
t
s
t

c
t
p
t
≥
s
t
+1

s
t
for all
t
= 0
,
1
,
2
, ...
.
1. The KuhnTucker Formulation
With the Lagrangian for this problem written as
L
=
∞
X
t
=0
β
t
u
(
c
t
) +
∞
X
t
=0
π
t
+1
s
t
+
d
t
s
t

c
t
p
t

s
t
+1
the firstorder condition for
c
t
is
β
t
u
0
(
c
t
)

π
t
+1
p
t
= 0
and the firstorder condition for
s
t
is
π
t
+1
+
π
t
+1
d
t
p
t

π
t
= 0
.
The first of these two conditions must hold for all
t
= 0
,
1
,
2
, ...
and the second must hold
for all
t
= 1
,
2
,
3
, ...
. Together with the binding constraint
d
t
s
t

c
t
p
t
=
s
t
+1

s
t
for all
t
= 0
,
1
,
2
, ...
, these conditions form a system of three equations in the three unknowns
c
t
,
s
t
, and
π
t
+1
.
1
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2. The Maximum Principle
The Hamiltonian for the consumer’s problem can be defined as
H
(
s
t
, π
t
+1
;
t
) = max
c
t
β
t
u
(
c
t
) +
π
t
+1
d
t
s
t

c
t
p
t
.
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 Fall '09
 IRELAND
 Economics, Expression, Trigraph, firstorder condition, dt pt

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