Suggested Solutions to EC2102 Macroeconomic Analysis I
Tutorial 1, Week 3 (January 2428, 2011)
Question 1
(
i
)
Let us set up the maximization problem for Mr.
j
, dropping the superscript
j
for now. Mr.
j
wants to
max
c
1
;c
2
u
(
c
1
) +
°u
(
c
2
)
s:t: c
1
+
c
2
1 +
r
=
y
1
+
y
2
1 +
r
°
!:
Rewriting the lifetime budget constraint, we have that
c
2
(
c
1
) = (
!
±
c
1
) (1 +
r
)
:
Using the above and the fact that
u
(
c
t
) =
p
c
t
for
t
= 1
;
2
, Mr
j
°s maximization
problem is just
max
c
1
p
c
1
+
°
p
(
!
±
c
1
) (1 +
r
)
:
(
ii
)
The F.O.C. to Mr
j
°s maximization problem in
(
i
)
, dropping the superscript
j
, is
1
2
p
c
°
1
+
±
°
(1 +
r
)
2
p
(
!
±
c
°
1
) (1 +
r
)
= 0
;
(1)
which yields
c
°
1
=
!
°
2
(1 +
r
) + 1
=
y
1
(1 +
r
) +
y
2
°
°
2
(1 +
r
) + 1
±
(1 +
r
)
:
(2)
From the ±rst period budget constraint, in equilibrium
s
°
1
=
y
1
±
c
°
1
=
y
1
±
y
1
(1 +
r
) +
y
2
°
°
2
(1 +
r
) + 1
±
(1 +
r
)
:
From the lifetime budget constraint, in equilibrium,
c
°
2
= (
!
±
c
°
1
) (1 +
r
) =
²
°
2
(1 +
r
)
°
2
(1 +
r
) + 1
³
[
y
1
(1 +
r
) +
y
2
]
:
1
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Putting the superscripts back and substituting
y
A
1
= 10
,
y
A
2
= 50
, and
y
B
1
= 50
and
y
B
2
= 10
;
we have that
c
A
°
1
=
10 (1 +
r
) + 50
°
°
2
(1 +
r
) + 1
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 Spring '11
 Tan
 Debt, Subscript and superscript, lifetime budget constraint, Mr J

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