Foundations of International Macroeconomics
1
Workbook
2
Maurice Obstfeld, Kenneth Rogo
f
, and Gita Gopinath
Chapter 2 Solutions
1.
(a) The current account identity can be written as
B
s
+1
=(1+
r
)
B
s
+
TB
s
.
Now just plug in the assumed trade balance rule.
(b) Using the answer to part a, for any
ξ
>
0
,
−
∞
X
s
=
t
1
1+
r
¶
s
−
t
ξ
rB
s
=
−
∞
X
s
=
t
1
1+
r
¶
s
−
t
ξ
r
[1 + (1
−
ξ
)
r
]
s
−
t
B
t
=
−
ξ
rB
t
1
−
1+(1
−
ξ
)
r
1+
r
=
−
(1 +
r
)
B
t
.
(c) Under the rule above, debt grows without bound if
ξ
<
1. But once the
debt is as big as
Y/r
, the country can honor its foreign commitments only
if debt stops growing and consumption is zero forever. Thus, the suggested
rule must entail
negative
consumption levels at some point, which are inad
missible. To see directly why, consider the constantoutput case, in which
TB
s
=
Y
−
C
s
=
−
ξ
rB
s
so that the payback rule implies
C
s
=
Y
+
ξ
rB
s
.
Notice that since
B
s
→−∞
,C
s
must at some point become negative. The
rule therefore is consistent with intertemporal solvency only if we counterfac
tually allow for negative consumption levels: the price of high consumption
today would be infeasibly high trade surpluses later on. In general, suppose
output grows at the gross rate 1 +
g
,sothat
Y
s
=(1+
g
)
s
−
t
Y
t
.Un
le
s
s1+
g
1
By Maurice Obstfeld (University of California, Berkeley) and Kenneth Rogo
f
(Prince
ton University).
c
MIT Press, 1996.
2
c
MIT Press, 1998. Version 1.1, February 27, 1998. For online updates and correc
tions, see http://www.princeton.edu/ObstfeldRogo
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View Full Documentis at least as great as the gross growth rate of debt, which was shown to be
1+
r
(1
−
ξ
) in part a, the external debtoutput ratio is unbounded. Thus
the minimal payback fraction
ξ
consistent with intertemporal solvency and
positive consumption is
ξ
=1
−
(
g/r
) (which is positive if we assume that
g<r
).
2.
(a) The expected utility E
t
U
t
is a weighted average over di
f
erent life
spans, with weights equal to the survival probabilities:
E
t
U
t
=(
1
−
ϕ
)[
u
(
C
t
)] +
ϕ
(1
−
ϕ
)[
u
(
C
t
)+
β
u
(
C
t
+1
)] +
+
ϕ
2
(1
−
ϕ
)
h
u
(
C
t
)+
β
u
(
C
t
+1
)+
β
2
u
(
C
t
+2
)
i
+
....
(b) The result follows simply by expanding the expression in part a and
grouping terms together.
3.
Recall that with isoelastic utility,
u
(
C
)=
C
1
−
1
σ
1
−
1
σ
,
σ
>
0
,
log
C,
σ
=1
.
Using the intertemporal Euler equation, we thus obtain,
(1 +
r
)
β
=1=
1
E
t
(
C
t
+1
C
t
¶
−
1
/
σ
)
.
(1)
Since consumption has a conditional lognormal distribution, the natural log
of the gross consumption growth rate is conditionally normally distributed:
log
C
t
+1
C
t
∼
N
E
t
‰
log
C
t
+1
C
t
±
,
Var
t
‰
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 Winter '08
 YuChinChe
 Normal Distribution, current account, Þrm

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