Foundations of International Macroeconomics
1
Workbook
2
Maurice Obstfeld, Kenneth Rogo
f
, and Gita Gopinath
Chapter 8 Solutions
1
. First think about the
continuoustime
case. At time
t
=0thema
rke
t
t
=
T
level from time
T
onward. In that case, we can show that the exchange
rate is
indeterminate
,sotha
tth
egov
e
rnm
en
t
spo
l
icyi
sno
tcoh
e
r
en
t(
in
that it does not tie down a unique market equilibrium). Let perfectforesight
equilibrium be described by the continuoustime Cagan model [eq. (70) on
p. 559 of the book],
m
t
=
e
t
−
η
o
e
t
.
Let
e
a
T
be an arbitrary time
T
exchange rate and suppose the market
rmly
expects that rate to prevail. Then the preceding Cagan equation, coupled
with the terminal condition
e
T
=
e
a
T
, shows that the exchange rate path
e
t
=
1
η
Z
T
t
exp[(
t
−
s
)
/
η
]
m
s
d
s
+exp[(
t
−
T
)
/
η
]
e
a
T
will equilibrate markets for
t
∈
[0
,T
].
T
when con
fronted with this exchange rate path. The authority has no choice, in view
of its vow of a constant exchange rate from
T
on, but to set the fundamental
m
T
=
e
a
T
and to hold
m
t
=
e
a
T
for all
t>T
.W
h
y
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T,
∞
authority must validate (ratify) any market expectation whatsoever. Any
initial exchange rate can be an equilibrium because each is conditioned on a
di
f
erent expectation of what the (constant) money supply path will be from
date
T
onward.
The situation is subtly di
f
erent in
discrete time
,inwh
ichcasethemode
l
is
m
t
=
e
t
−
η
(
e
t
+1
−
e
t
)
.
The basic reason discrete time makes a di
f
erence is that now, if the market
rmly foresees a date
T
rate of
e
a
T
,thereare
two
distinct ways the author
ity can ful
ll its promise of a constant exchange rate from date
T
on (two
alternatives which collapse to one in continuous time). First, the authority
could still set
m
t
=
e
a
T
for
t
=
T
and for
t>T
, as in the continuoustime
setting±in which case the exchange rate again is not uniquely determined.
Alternatively, if the authority can commit not to adjust
m
T
fully to validate
e
a
T
, but instead only to set
m
t
=
e
a
T
for
t
strictly greater than
T
only, we
again get an exchange rate path constant at
e
a
T
starting on date
T
.Inth
i
s
second case, however, the exchange rate is uniquely determined. To
xideas,
suppose the authority can commit to a
xed value
m
T
for date
T
.I
nt
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 Winter '08
 YuChinChe
 Exchange Rate, Foreign exchange market

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