Foundations of International Macroeconomics
1
Workbook
2
Maurice Obstfeld, Kenneth Rogo
f
, and Gita Gopinath
Chapter 6 Solutions
1.
(a) We look at the symmetric e
ﬃ
cient incentivecompatible contract.
That contract maximizes an equallyweighted average of Home and Foreign
expected utility,
E
{
u
(
C
)
}
+E
{
u
(
C
∗
)
}
,
subject to the constraints
−
η
Y
∗
≤
P
(
²
)
≤
η
Y,
which must hold for all
N
possible realizations of the shock
²
. The Lagrangian
for the contracting problem is
L
=m
a
x
P
(
²
)
N
X
1=1
π
(
²
i
)
'
u
£
±
Y
+
²
i
−
P
(
²
i
)
/
+
u
£
±
Y
−
²
i
+
P
(
²
i
)
/“
−
N
X
1=1
λ
(
²
i
)
£
P
(
²
i
)
−
η
(
±
Y
+
²
i
)
/
−
N
X
1=1
(
²
i
)
£
−
P
(
²
i
)
−
η
(
±
Y
−
²
i
)
/
.
The
rstorder condition with respect to
P
(
²
i
)is
π
(
²
i
)
{
−
u
0
[
C
(
²
i
)] +
u
0
[
C
∗
(
²
i
)]
}
−
λ
(
²
i
)+
(
²
i
)=0
,
(1)
and the complementary slackness conditions are
λ
(
²
i
)
£
η
(
±
Y
+
²
i
)
−
P
(
²
i
)
/
=0
,
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
f
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(
²
i
)
£
η
(
f
Y
−
²
i
)+
P
(
²
i
)
/
=0
.
Despite the Forbidding Formalism oF the KuhnTucker conditions, the solu
tion to the problem can be characterized rather simply. Consider the so
lution in states where the incentive constraints do not bind, that is when
λ
(
²
i
)=
(
²
i
)=0
.
Across these states, the
rstorder condition (1) reduces
to
u
0
[
C
(
²
i
)] =
u
0
[
C
∗
(
²
i
)]
,
and so
C
(
²
i
)=
C
∗
(
²
i
)
.
There is thereFore a range
[
−
e, e
] such that inside this interval
C
=
C
∗
. The bound
e
is easily Found as
the largest
²
such that
²
≤
η
¡
f
Y
+
²
¢
, implying
e
=
η
1
−
η
f
Y.
(2)
(b
)
±or
²>e
, the incentive constraint
P
(
²
i
)
≤
η
(
f
Y
+
²
i
)p
r
ev
en
t
sFu
l
l
insurance, so
P
(
²
i
)=
e
+
η
(
²
i
−
e
)[whe
rewehavesub
s
t
i
tu
tedFo
r
f
Y
From
equation (2)]. Since the equilibrium is symmetric, For
²<
−
e
,
P
(
²
i
)=
−
e
+
η
(
²
i
+
e
).
To graph the payments schedule that the contract implies, put
P
(
²
)on
the vertical axis and
²
onthehor
izonta
lax
is
.Thepaymentsschedu
lepasses
through the origin, has slope 1 over [
−
e,e
], and has slope
η
outside that
interval. See
gure 6.1.
2.
This problem is completely parallel to the problem in the text. There are
only two di
f
erences: the zeropro
t condition For lenders is now
N
X
1=1
π
(
²
i
)
P
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