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NB
ER
WO
RK
[NG
PA
PE}
SER
IES
RA
TIO
NA
L
SPE
aJL
TIV
E 3
JBB
LE
S IN
AN
E0(
CH
NG
E R
AT
E T
A3E
T Z
ON
E
Wif
lem
H.
Bei
ter
Paol
o A. Pesenti
Wor
kim
Paper No. 3467
NA
TIO
NA
L
JRE
PU O
F E
OJN
CM
IC R
ESE
AR
UI
1050
Mas
sach
uset
ts A
venu
e C
ambr
idge
, NA
02
138
Oct
ober
199
0
We
wou
ld l
Bce
to
than
k T
orst
en P
erss
on,
thri
stoç
tier
Sim
s,
the
part
icip
ants
at
the
War
wic
k Su
mse
r Res
earc
h W
orks
hop
on 'E
xcha
re R
ates
ax
Fina
ncia
l M
arke
ts",
Jul
y 9-
27,
1990
, in
par
ticul
ar P
enzo
Ave
sani
, G
iuse
ppe
Ber
tola
, A
rdy
Cap
lin,
Ber
nard
tmas
, M
arcu
s M
iller
, R
ick
van
der
Ploe
g,
Paul
Wel
ler
ani t
he p
artic
ipan
ts a
t the
Inte
rnat
iona
l Fi
nanc
e W
orks
hop
at Y
ale,
in
pa
rtic
ular
Gia
ncar
lo C
orse
tti, V
ittor
io G
rilli
, E
cidi
i Ham
ada
and
Nou
rie]
. R
oubi
ni.
Paol
o Pe
sent
i w
ould
lik
e to
ack
elge
fina
ncia
l su
ppor
t fr
om t
he
Mat
e pe
r gl
i St
udi
Mon
etar
i, B
anca
ri e
Fin
anzi
ari
L.
Ein
audi
, Rom
e.
Thi
s pa
per
is p
art o
f N
BE
R' s
res
earc
h pr
cgra
m i
n In
tern
atio
nal
Stis
lies.
ny
op
inio
ns e
ress
i are
thos
e of
the
auth
ors
and
not
thos
e of
the
Nat
iona
l B
erea
u of
Eco
nom
ic R
esea
rch.
NB
ER
Wor
king
Pa
per
#346
7 O
ctob
er 1
990
RA
TIO
NA
L S
PEJI
AT
IVE
JB
BL
ES
IN A
N E
XQ
1PN
GE
R
AT
E T
AJE
I Z
ON
E
The
rec
ent
theo
ry o
f ex
chan
ge
rate
dyn
amic
s w
ithin
a t
arge
t zon
e ho
lds
that
exc
iiari
je r
ates
und
er a
cur
renc
y ba
rd a
re l
ess
resp
ansi
ve to
furd
asen
tal
shoc
ks t
han
exch
ange
rate
s un
der
a fr
ee f
loat
, pr
ovid
ed t
hat
the
inte
rven
tion
rule
s of
the
Cen
tral
Ban
k(s)
are
cat
uwn
kncM
ledg
e.
The
se r
esul
ts a
re d
eriv
ed
afte
r hav
ing
assu
med
a p
rior
i th
at e
xces
s vo
latil
ity du
e to
ratio
nal b
ubbl
es
does
not
occ
ur in
the
for
eign
exc
hang
e m
arke
t. In
this
pape
r we
cons
ider
inst
ead
a se
tup
in w
hich
the
exis
tenc
e of
spe
cula
tive
beha
vior
is a
dat
um t
he
Cen
tral
Ban
k ha
s to
dea
l w
ith.
We
show
tha
t th
e de
fens
e of
the
targ
et z
one
in t
he p
rese
nce
of b
ubbl
es i
s vi
able
if
the
Cen
tral
Ban
k ac
com
imxl
ates
spec
ulat
ive
atta
cks w
hen
the
latte
r ar
e co
nsis
tent
with
the
sur
viva
l of
the
targ
et z
one
itsel
f an
d ex
pect
atio
ns a
re s
elf-
fulf
illir
. T
hese
reB
ults
hol
d
for
a la
rge
clas
s of
exn
geno
us a
nd f
unda
men
tal-
depe
nden
t bub
ble
proc
esse
s.
We
show
tha
t th
e in
stan
tane
ous v
olat
ility
of
exch
ange
ra
tes
with
in a
bar
d is
no
t ne
cess
arily
les
s th
an th
e vo
latil
ity un
der f
ree
floa
t an
d an
alyz
e th
e
ixpl
icat
ions
for
inte
rest
rat
e di
ffer
entia
l dy
nam
ics.
Will
em F
l. B
aite
r Pa
olo
A.
Pese
nti
Ipar
toen
t of
Eco
nom
ics
Cep
arbn
ent o
f E
cono
mic
s Y
ale
Uni
vers
ity
Yal
e U
nive
rsity
19
72 Y
ale
Stat
ion
1972
Yal
e St
atio
n N
ew H
aven
, C
I 06
520—
1972
N
ew H
aven
, cr
065
20—
1972
I. I
ntro
duct
ion
The
rec
ent
liter
atur
e on
exc
hang
e ra
te t
arge
t zo
nes
insp
ired
by
Kru
gman
's
sem
inal
ar
ticle
on
the
subj
ect
(Kru
gman
[199
0])
repr
esen
ts
a su
cces
sful
mar
riag
e of
polic
y re
leva
nce a
nd t
echn
ical
inno
vatio
n. F
orm
al t
arge
t zon
es l
ike
the
exch
ange
rat
e
arra
ngem
ents
of th
e E
urop
ean M
onet
ary
Syst
em a
nd i
nfor
mal
tar
get z
ones
bet
wee
n th
e
U.S
. do
llar,
the
Japa
nese
yen
and
the
D—
Mar
k si
nce
the
mid
dle o
f th
e la
st d
ecad
e w
ere
a pr
omin
ent f
eatu
re o
f the
exc
hang
e ra
te s
yste
m o
f the
Eig
htie
s and
pro
mis
e to
be
so
for t
he N
inet
ies.
T
he t
echn
ical
mod
elin
g in
nova
tion
cons
ists
in
the
appb
catio
n of th
e
theo
ry o
f re
gula
ted
Bro
wni
an m
otio
n (s
ee e
.g.
Mal
liari
s an
d B
rock
[19
82],
Har
riso
n
[198
5], D
ixit
[198
8], K
arat
zas
and
Shre
ve [
1988
], D
umas
[1
989]
and
Flo
od a
nd G
arbe
r
[198
9]) to
the
stud
y of
the
beha
vior
of
a fl
oatin
g ex
chan
ge
rate
that
is c
onst
rain
ed
by
appr
opri
ate i
nter
vent
ions
not
to s
tray
out
side
som
e gi
ven
rang
e or t
arge
t zon
e.
The
lite
ratu
re o
n th
is s
ubje
ct i
s gr
owin
g ve
ry f
ast.
Am
ong
the
man
y re
cent
cont
ribu
tions
are
Kru
gman
[198
7], K
rugm
an
and
Rot
embe
rg [1
990]
, Mill
er a
nd W
elle
r
[198
8a,b
; 19
89;
1990
a,b,
c],
Mill
er a
nd S
uthe
rlan
d [1
990]
, Fro
ot a
nd O
bstf
eld
[198
9a,b
],
Ber
tola
an
d C
abal
lero
[1
989,
199
0],
Kle
in [
1989
], S
vens
son
[198
9; 1
990a
,b],
Pes
enti
[190
0], D
elga
do a
nd D
umas
[19
90],
Lew
is [1
990]
, A
vesa
ni [
1990
], Ic
hika
wa,
M
iller
and
Suth
erla
nd [1
990]
, and
Sm
ith a
nd S
penc
er [
1990
].
A c
omm
on
elem
ent
in a
ll of
the
se a
naly
ses
is t
hat
a "f
unda
men
tals
onl
y"
solu
tion
is c
hose
n fo
r th
e ex
chan
ge r
ate:
The
exc
hang
e ra
te, a
non—
pred
eter
min
ed s
tate
vari
able
, is e
xpre
ssed
as
a fu
nctio
n of
the
curr
ent a
nd e
xpec
ted
futu
re v
alue
s of
the
fund
amen
tals
onl
y.
The
lite
ratu
re i
n fa
ct
conc
entr
ates
on
mod
els
with
a
sing
le
fund
amen
tal.
Thi
s fu
ndam
enta
l is g
over
ned
by r
egul
ated
Bro
wni
an m
otio
n, th
at i
s by
unre
gula
ted
Bro
wni
an m
otio
n as
lon
g as
the
fund
amen
tal s
tays
with
in t
he lo
wer
and
uppe
r in
terv
entio
n th
resh
olds
and
by
(int
erm
itten
t) i
nter
vent
ions
tha
t ke
ep
the
fund
amen
tal
with
in t
hese
thr
esho
lds
whe
neve
r th
e un
regu
late
d B
row
nian
m
otio
n
thre
aten
s to
tak
e th
e fu
ndam
enta
l ou
tsid
e th
e in
terv
entio
n lim
its (
and
thus
the
—2—
exch
ange
ra
te o
utsi
de i
ts t
arge
t zo
ne).
T
his
perm
its t
he c
urre
nt
valu
e of
the
spot
exch
ange
ra
te to
be
expr
esse
d as
a n
on—
linea
r fun
ctio
n of
the
cur
rent
val
ue o
f th
e
fund
amen
tal o
nly.
The
firs
t ste
p in
this
arg
umen
t, th
e de
cisi
on to
sol
ve f
or th
e cu
rren
t val
ue o
f th
e
exch
ange
ra
te a
s a
func
tion
only
of
the
cur
rent
an
d ex
pect
ed fu
ture
val
ues
of t
he
fund
amen
tal,
mea
ns th
at s
pecu
lativ
e bu
bble
s ar
e ru
led o
ut.
In d
ynam
ic
linea
r ra
tiona
l exp
ecta
tions
mod
els
the
argu
men
ts
for
rulin
g ou
t
spec
ulat
ive
bubb
les
are
wel
l—kn
own,
if n
ot n
eces
sari
ly w
holly
con
vinc
ing.
In
man
y of
the
mos
t po
pula
r m
odel
s th
e bu
bble
pr
oces
ses
are
non—
stat
iona
ry.'
The
se
non—
stat
iona
ry
bubb
le p
roce
sses
th
en
lead
to
non
—st
atio
nary
be
havi
or o
f st
ate
vari
able
s su
ch a
s as
set
pric
es.
Unb
ound
ed g
row
th o
r de
clin
e in
the
se s
tate
vari
able
s for
cons
tant
val
ues o
f th
e fu
ndam
enta
ls is
arg
ued
to v
iola
te,
even
tual
ly, c
erta
in o
ften
onl
y
impl
icitl
y st
ated
feas
ibili
ty co
nstr
aint
s.2
Bla
ncha
rd [1
979]
ha
s de
velo
ped
an e
xam
ple
of a
non—
stat
iona
ry (
stoc
hast
ic o
r
dete
rmin
istic
) lin
ear b
ubbl
e th
at h
as a
fini
te ex
pect
ed l
ifet
ime.
T
he p
roba
bilit
y of
the
bubb
le su
rviv
ing
for
a gi
ven
peri
od o
f tim
e de
clin
es w
ith th
e le
ngth
of th
at p
erio
d at
a co
nsta
nt e
xpon
entia
l ra
te a
nd a
ppro
ache
s ze
ro a
sym
ptot
ical
ly
(see
als
o B
lanc
hard
and
Fisc
her
[198
9, c
hapt
er 5
]).
Alth
ough
th
e bu
bble
rem
ains
non
-sta
tiona
ry a
nd i
ts
expe
cted
val
ue g
row
s ex
pone
ntia
lly
with
out b
ound
, th
e fi
nite
exp
ecte
d lif
etim
e of
this
bubb
le m
ay m
ake
it le
ss v
ulne
rabl
e to
the
stan
dard
cri
tique
of n
on-s
tatio
nary
bub
bles
.
Mill
er a
nd W
elle
r [1
990a
[ an
d M
iller
, W
elle
r an
d W
illia
mso
n [1
989]
use
the
Bla
ncha
rd
bubb
le
to a
naly
ze e
xcha
nge
rate
beh
avio
r (b
oth
with
and
with
out a
tar
get
zone
) in
a
stoc
hast
ic ve
rsio
n of
the
Dor
nbus
ch o
vers
hoot
ing
mod
el (
Dor
nbus
ch
[197
6]).
Cer
tain
no
n—lin
ear m
odel
s al
so p
osse
ss n
on—
stat
iona
ry b
ubbl
e so
lutio
ns.
For
inst
ance
, the
def
latio
nary
an
d in
flat
iona
ry b
ubbl
es
stud
ied
by H
ahn
[198
2] a
nd b
y
Obs
tfel
d an
d R
ogof
f [1
983]
, al
thou
gh o
btai
ned
in n
on—
linea
r m
odel
s,
are
also
non—
stat
iona
ry.
In m
ore
gene
ral
non—
linea
r mod
els
it it
how
ever
qui
te c
omm
on t
o
—3—
find
stat
iona
ry b
ubbl
es.3
The
exc
hang
e ra
te t
arge
t zo
ne m
odel
pot
entia
lly p
rovi
des
a pa
rtic
ular
ly h
appy
non—
linea
r br
eedi
ng g
roun
d fo
r sp
ecul
ativ
e bu
bble
s.
Prov
ided
the
targ
et z
one
is
cred
ible
, th
at i
s pr
ovid
ed
ther
e is
cer
tain
ty th
at t
he i
nter
vent
ions
requ
ired
to
defe
nd
the
targ
et z
one
will
ta
ke
plac
e,
the
exch
ange
ra
te c
an n
eith
er ri
se n
or f
all
with
out
boun
d.
The
re a
re i
nfin
itely
man
y in
terv
entio
n ru
les
that
are
com
patib
le w
ith t
he
defe
nse
of a
giv
en t
arge
t zo
ne.
For
a ra
tiona
l sp
ecul
ativ
e bu
bble
to
exis
t, th
e
inte
rven
tion
rule
mus
t of c
ours
e be
con
sist
ent,
both
with
in th
e ta
rget
zon
e an
d at
the
boun
dari
es, w
ith th
e be
havi
or o
f th
e (r
egul
ated
) fun
dam
enta
ls.
Thi
s pa
per d
evel
ops
a
sim
ple
and
intu
itive
int
erve
ntio
n ru
le th
at e
ncom
pass
es a
s a
spec
ial
case
the
sol
utio
n
with
out
a bu
bble
, ye
t is
ge
nera
l en
ough
to
perm
it a
wid
e ra
nge
of e
xoge
nous
and
fund
amen
tal—
depe
nden
t de
term
inis
tic or
sto
chas
tic b
ubbl
es.
With
thi
s in
terv
entio
n
rule
the
argu
men
ts a
gain
st b
ubbl
es l
ose
thei
r bite
. A
non
—st
atio
nary
bub
ble
does
not
now
im
ply
non—
stat
iona
ry b
ehav
ior o
f th
e ex
chan
ge r
ate.
T
his
argu
men
t is
not
new
.
As
earl
y as
198
7 W
illia
m H
. B
rans
on,
in c
onve
rsat
ion
and
disc
ussi
on,
repe
ated
ly ra
ised
the
poss
ibili
ty th
at ta
rget
zon
es m
ight
lead
to "
inde
term
inac
y" of
the
exc
hang
e ra
te.
Thi
s pa
per c
an b
e vi
ewed
as
an a
naly
tical
conf
irm
atio
n of h
is c
onje
ctur
e.
In t
he n
ext
Sect
ions
we
expo
sit
the
sim
ples
t ve
rsio
n of
the
cred
ible
tar
get
zone
mod
el a
nd a
naly
ze its
beh
avio
r with
or
with
out s
pecu
lativ
e bu
bble
s.
II.
A T
arge
t Zon
e M
odel
With
or W
ithou
t Spe
cula
tive
Bub
bles
.
(a) T
he M
odel
.
Our
ana
lysi
s of
the
pla
ce a
nd r
ole
of s
pecu
lativ
e bu
bble
s in
an
exch
ange
rat
e
targ
et z
one
requ
ires
the
exch
ange
ra
te t
o be
a no
n—pr
edet
erm
ined
(fo
rwar
d—lo
okin
g)
—4—
,
stat
e va
riab
le,
but
does
not
m
ake
furt
her
dem
ands
on
de
scri
ptiv
e re
alis
m.
We
ther
efor
e fe
el
com
fort
able
in
usi
ng
Occ
am's
ra
zor
and
follo
win
g th
e bu
lk
of t
he
liter
atur
e on
tar
get
zone
s by
cas
ting
the
anal
ysis
in
ter
ms
of th
e si
mpl
est
poss
ible
linea
r in
tert
empo
ral s
ubst
itutio
n eq
uatio
n fo
r th
e ex
chan
ge
rate
and
a (
cont
inuo
us
time)
rand
om w
alk
with
dri
ft f
or th
e un
regu
late
d fun
dam
enta
l. (N
otab
le d
epar
ture
s
from
the
cult
of e
xtre
me
sim
plic
ity
are
Mill
er a
nd W
elle
r [1
988a
,b;
1989
; 19
90a,
b,c]
who
ana
lyze
stoc
hast
ic ve
rsio
ns o
f the
rich
er D
ornb
usch
ove
rsho
otin
g m
odel
.)
The
exc
hang
e ra
te p
roce
ss i
s gi
ven
in e
quat
ion
(1).
(1)
s(t)
dt =
f(t)
dt +
a'E
tds(
t)
a >
0.
Her
e s(
t)
is th
e na
tura
l lo
gari
thm
of t
he s
pot
nom
inal
exc
hang
e ra
te a
nd f(
t)
the
fund
amen
tal
dete
rmin
ant
of
the
exch
ange
ra
te.
Et
is
the
mat
hem
atic
al
expe
ctat
ion o
pera
tor c
ondi
tiona
l on
the
info
rmat
ion
avai
labl
e at t
ime
t to
the
priv
ate
sect
or a
nd th
e re
gula
tor.
Bot
h sO
) an
d f(
t)
are
assu
med
to b
e ob
serv
able
at ti
me
t,
i.e.
Ets
(t) =
sO)
and
Etf
(t) =
f(t)
, an
d th
e st
ruct
ure
of th
e m
odel
is
know
n.
For
the
purp
ose
of th
is p
aper
, the
econ
omic
int
erpr
etat
ion
of
f(t)
is
irr
elev
ant
exce
pt i
nsof
ar a
s it
affe
cts
the
cred
ibili
ty of
our
ass
umpt
ion t
hat,
with
eert
alnt
y, th
e
targ
et z
one
will
be
succ
essf
ully
def
ende
d.
A c
omm
on i
nter
pret
atio
n of
f is
in t
erm
s of
rela
tive
nom
inal
mon
ey s
tock
s m
inus
rela
tive
real
—in
com
e—re
late
d de
man
ds
for
real
mon
ey b
alan
ces,
i.e
.
* *
f=m
—m
—
k(y—
y )
whe
re
m i
s th
e lo
gari
thm
of th
e ho
me
coun
try n
omin
al m
oney
sto
ck,
y th
e lo
gari
thm
of h
ome
coun
try
real
out
put
and
k is
the
(co
mm
on)
inco
me
elas
ticity
of
mon
ey
dem
and.
St
arre
d va
riab
les
deno
te fo
reig
n qu
antit
ies,
In
thi
s ca
se
a de
note
s th
e
—5—
inve
rse o
f the
(com
mon
) in
tere
st s
emi—
elas
ticity
of m
oney
dem
and.
4
If th
e tw
o na
tiona
l rea
l out
puts
are
gov
erne
d by
exo
geno
us B
row
nian
mot
ion,
inte
rven
tion
mea
ns ch
ange
s in
the
rela
tive
mon
ey s
uppl
ies b
roug
ht a
bout
by
mon
etar
y
fina
ncin
g of
pub
lic
sect
or f
inan
cial
def
icits
, th
roug
h op
en m
arke
t op
erat
ions
or
by
unst
erili
zed
fore
ign
exch
ange
m
arke
t in
terv
entio
n.
As
show
n in
Bui
ter
[198
9] s
uch
inte
rven
tion
polic
ies
will
in
gene
ral
requ
ire
adju
stm
ents
to p
rim
ary
(non
—in
tere
st)
fisc
al
surp
luse
s to
be
sust
aina
ble
inde
fini
tely
fr
om a
tec
hnic
al p
oint
of
view
(i.e
. in
orde
r to
prev
ent e
ither
exh
aust
ion
of f
orei
gn e
xcha
nge
rese
rves
or u
nbou
nded
grow
th of
the
publ
ic d
ebt)
. A
pro
of th
at t
hey
are
polit
ical
ly f
easi
ble
and
cred
ible
is
beyo
nd t
he
scop
e of
this
pap
er.
In th
e ab
senc
e of
inte
rven
tion,
the
(unr
egul
ated
) fun
dam
enta
l f w
ould
fol
low
a
Bro
wni
an m
otio
n pr
oces
s w
ith d
rift
p an
d in
stan
tane
ous v
aria
nce c
.
di(t
) =
pdt
+ u1
dW1(
t)
clf>
0.
W1(
t)
is a
sta
ndar
d W
iene
r pr
oces
s (d
Wf
e 1 im
w
here
is
at
-to
inde
pend
ently
and
norm
ally
dist
ribu
ted
with
zer
o m
ean
and
unit
vari
ance
).
We
now
al
low
fo
r in
terv
entio
n at
the
upp
er a
nd l
ower
bo
unda
ries
of
the
exch
ange
rat
e ta
rget
zon
e. T
he re
gula
ted f
unda
men
tal
is g
over
ned
by
(2)
df(t
) =
pdt
+ yI
Wf(
t) —
dlu
+ d
lt
and I a
re t
he c
umul
ativ
e int
erve
ntio
ns
at th
e up
per
and
low
er b
ound
s of
the
exch
ange
rat
e ta
rget
zon
e, g
iven
by
5u an
d s1
resp
ectiv
ely
with
—
<
<
5u
< .
A
deta
iled s
peci
fica
tion
of th
ese
inte
rven
tions
is g
iven
bel
ow.
—6-
—
(b)
The
Sol
utio
n
The
gen
eral
sol
utio
n of
(1)
is
give
n in
equ
atio
n (3
).
B(t
) is
the
val
ue o
f th
e
spec
ulat
ive
bubb
le a
t tim
e t.
(3)
5(t)
=
i: E
tf(v)
e(v_
t)dv
+ B
(t).
The
bub
ble
B(t
) ca
n be
an
y st
ocha
stic
or
st
rict
ly d
eter
min
istic
pro
cess
satis
fyin
g
(4)
B(t
)dt
= a'
Etd
B(t
).
Not
e tha
t si
nce
s(t)
is
part
of t
he in
form
atio
n se
t at t
, so
is B
(t).
W
e re
stri
ct
B, w
here
it i
s st
ocha
stic
, to
follo
w a
dif
fusi
on p
roce
ss.
Exc
ept f
or t
he c
ase
of t
he
"Bla
ncha
rd b
ubbl
e",
all
exam
ples
w
ill
in f
act
rest
rict
it
furt
her
to b
e a
twic
e
cont
inuo
usly
di
ffer
entia
ble
func
tion o
f Bro
wni
an m
otio
ns.
Fina
lly,
it is
impo
rtan
t to
note
that
the
aut
hori
ties a
re a
ssum
ed t
o be
una
ble t
o
inte
rven
e di
rect
ly in
the
bub
ble
proc
ess
(and
in
deed
in t
he e
xcha
nge
rate
pro
cess
).
The
y ca
n on
ly r
egul
ate t
he fu
ndam
enta
l £
In t
he i
nter
ior
of th
e ta
rget
zon
e, t
he g
ener
al s
olut
ion
for
s as
a f
unct
ion
of
curr
ent f a
nd c
urre
nt B
is g
iven
by5
.A2(
f(t)
—C
(t))
(5
) s(
t) =
f(t)
+ B
(t) +
+
A1(
t)e
+ A
2(t)
w
ith
—
/7 +
2nc
2 crf
—7—
A1,
A2
and
C a
re c
onst
ant
as l
ong
as s
rem
ains
in t
he i
nter
ior
of t
he t
arge
t
zone
, hut
can
cha
nge
whe
n s
is a
t its
upp
er o
r lo
wer
bou
ndar
y.
In fa
ct w
e ha
ve
(6)
A1(
t) =
A1(
r)
A2(
t) =
A2(
r)
C(t
) =
—B
(r)
B(r
) is
the
val
ue o
f th
e bu
bble
at t
ime
r, w
hich
is th
e da
te o
f the
mos
t rec
ent
visi
t pri
or to
t of
the
exch
ange
rat
e s
to o
ne o
f th
e bo
unda
ries
, th
at i
s
(7)
r a S
upre
mum
{t' �
t, su
ch th
at s
(t')
= 5u
or S
O')
=
We
also
def
ine s to
be
a va
lue
off
both
str
ictly
gre
ater
than
s an
d ar
bitr
arily
clos
e to
. S
imila
rly,
s
is a
valu
e of
f st
rict
ly le
ss t
han
5 an
d ar
bitr
arily
clo
se to
s.
We
next
def
ine
two
criti
cal v
alue
s of
the
bubb
le,
Bu
and
Bt
Bu
is d
eter
min
ed,
toge
ther
with
A1,
A2
and
by e
quat
ions
(8a
—d)
.
A(s
-l-B
) A
(s-4
-B)
(8a)
5u
5u+
+B
uie
1 u
u+A
e 2
u u
(8b)
=
(1
+ 5u
X +
i A1
eA1(
5u+
+ A
2 eA
2(5u
)
A(s
+B
)
A(s
1-i-B
)
(Sc)
si
=ft
++
Bu+
Aie
1 u+
A2e
2 u
(Sd)
=
(1
+
+ A
1A1
eA1(
51+
+ A
2 eA
2(5)
.
—8—
Ana
logo
usly
, B
1 is
det
erm
ined
, to
geth
er w
ith A
1,
A2
and
f fr
om e
quat
ions
U
(9a-
d).
(9a)
s
= f
+
+ B
1 +
A1
eA1
1)
A2 e2
1)
U
U
a
(9b)
0
= (1
+
)(1
+ A
1 A
1 eA
1(5u
1) +
A2 e2
1))
U
A
(s1-
l-B
1)
(9c)
s1
= 8
1+ +
B
+ A
1 e1
(11)
+ A2
e 2
A (s
1+B
1) +
A2 e2
1)).
(9
d)
0 =
(1 +
f=
s1 +
A1A
1 e
1
If B
> B
(r)
> B
1, th
en
A1(
r) an
d A
2(i-
) ar
e de
fine
d, t
oget
her w
ith f
(r)
and
f1(r
) by
equ
atio
ns (l
Oa—
d).
A [f
(r)+
B(r
)]
(ba)
s
=f (
r)+
B(r
)++
A1(
r)e
l +
A2(
r)e
u u
a
(lob
) =
f1(r
) + B
() +
+
A1(
r)e
+ A
2(r)
e A
(bc)
0 =
(1 +
)(
1 +
A1
A1(
r)e
+ A
2A2(
r)e 2(
TT
) U
(lO
d) 0
= (1
+ q
T)-
)(l +
A1 A
1(r)
e'' + A
2 A
2(r)
e )
—9.
-.
Whe
n B
(r) � B
u A
1(r)
and
A2(
r) a
re d
eter
min
ed by
equ
atio
ns (l
la,b
).
(ha)
s =
s +
+ B
(r)
Ai(
r) e
iu +
A2(
r) eA
250
(ub)
0
= (1
+
+ A
1 A
1 ei
u +
A2
A2
eA25
).
Whe
n B
(r)
B1
, A
1(r)
and A
2(r)
are
dete
rmin
ed b
y eq
uatio
ns (l
ic,d
).
- .\i
A
s1
(hic
) 5r
s1 +
+
B(r
) + A
1(r)
e +
A2(
r) e
(lid
) 0
= (
1 +
+
A
1 el
l + A
2 A
2 e2
1).
Equ
atio
ns (i
oa,b
) de
fine
u
and
f1 fo
r giv
en
8u' s
1 an
d B
. E
quat
ions
(ioc
,d)
are
the
fam
iliar
tang
ency
or n
o—ar
bitr
age
cond
ition
s of
ten
refe
rred
to, i
nacc
urat
ely,
as
'sm
ooth
pa
stin
g' c
ondi
tions
, th
at e
xpec
ted
inte
rven
tions
in f
shou
ld
not
chan
ge t
he
valu
e of
the
exc
hang
e ra
te.
As
we
shal
l se
e in
Sec
tion
IV b
elow
, th
e va
lue
of t
he
bubb
le
may
dep
end
oo
the
curr
ent
valu
e of
th
e un
regu
late
d fu
ndam
enta
l. T
he
cond
ition
s (1
0c—
d), (
iib) a
nd (l
id) a
llow
for
thi
s.
Not
e th
at fo
r B0
< B
C B
t A1
and
A2
are
inde
pend
ent o
f the
val
ue o
f the
bub
ble,
and
the
refo
re c
onst
ant o
ver t
ime.
It
is
also
read
ily se
en t
hat
in th
is c
ase
df
d11
—10
—
Equ
atio
ns (l
la,b
) ch
arac
teri
ze a
one
—si
ded
'sm
ooth
pas
ting'
con
ditio
n at
the
uppe
r bo
unda
ry (
s =
on
ly.
Equ
atio
ns (l
lc,d
) ch
arac
teri
ze a
one
—si
ded
'sm
ooth
past
ing'
con
ditio
n at
the
low
er b
ound
ary
(s =
on
ly.
The
se n
on—
stan
dard
bou
ndar
y
cond
ition
s w
ill b
e in
terp
rete
d fu
rthe
r be
low
.
We
now
det
ail
the
inte
rven
tions
at
the
uppe
r an
d lo
wer
bo
unda
ries
of
the
exch
ange
rat
e ta
rget
zon
e.
From
equ
atio
n (5
), t
he e
xcha
nge
rate
in th
e ta
rget
zon
e ca
n
be w
ritte
n as
5(
t) =
g(f
(t),
B(r
)) +
B(t
).
The
bub
ble
at ti
me
t can
be
a fu
nctio
n of
the
curr
ent u
nreg
ulat
ed fu
ndam
enta
l f(
t) a
nd/o
r of
oth
er v
aria
bles
z(t
).
Not
e th
at
z(t)
co
uld
be a
fun
ctio
n of
nM
or
exoe
cted
fqe
valu
es o
f th
e un
regu
late
d fu
ndam
enta
l. T
here
fore
B(t
) =
b(f
(t),
z(t)
).
The
exc
hang
e ra
te i
n th
e ta
rget
zon
e
can
ther
efor
e be
wri
tten
as
5 =
h(f,
•), w
here
a
+ . Fo
r sim
plic
ity
we
Of
rest
rict
the
ana
lysi
s to
tho
se b
ubbl
es f
or w
hich
, ev
en i
f B(t
) de
pend
s on
1(
t),
the
func
tion
h(f,
.),
reta
ins
its f
amili
ar s
sha
pe w
ith a
uni
que
inte
rior
min
imum
an
d a
uniq
ue i
nter
ior
max
imum
. In
the
cas
e w
here
Bu
< B
(r) c
B,
u is
the
refo
re th
e
valu
e of
f fo
r w
hich
s =
5u an
d th
e h
func
tion
has
an i
nter
ior m
axim
um, t
hat
is u
is
defi
ned
by h
(fu;
)
= 5u
' =
0
and
•) <
0.
Not
e th
at b
ecau
se of
the
bubb
le,
s m
ay r
each
5u
at v
alue
s of
fun
dam
enta
l dif
fere
nt f
rom
u•
The
se w
ill
be
deno
ted f
. Sim
ilarl
y, s
may
reac
h s
at v
alue
s of
f d
iffe
rent
fro
m f
The
se w
ill b
e
deno
ted
f1.
We
cons
ider
th
e si
mpl
est
poss
ible
exa
mpl
e of
a t
arge
t zo
ne.
A p
laus
ible
inte
rpre
tatio
u of
the
form
al m
odel
is th
at t
here
is a
fully
cre
dibl
e co
mm
itmen
t tha
t the
valu
e of
the
exch
ange
ra
te w
ill b
e co
ntai
ned
with
in a
n ex
ogen
ousl
y gi
ven
boun
ded
rang
e, t
hat i
s —
z C
s
s 5u
< z
. T
he in
terv
entio
ns in
the
fun
dam
enta
l pro
cess
that
kee
p th
e ex
chan
ge r
ate
with
in th
is ra
nge o
ccur
onl
y w
hen
the e
xcha
nge
rate
is at
the
boun
dari
es.
Tw
o di
stin
ct k
inds
of
inte
rven
tions
are
requ
ired
to d
efen
d th
e ta
rget
zo
ne in
the
pres
ence
of a
bubb
le.
11 —
The
fir
st i
s a
pair
of
infi
nite
sim
al re
flec
ting
inte
rven
tions
, _dl
u <
0 a
t th
e
uppe
r bou
ndar
y if
f =
u an
d dI
1> 0
at t
he l
ower
bou
ndar
y of
the
targ
et z
one
if
=
The
se n
egat
ive i
nfin
itesi
mal
inte
rven
tions
at th
e up
per b
ound
ary
s =
s a
nd
posi
tive
infi
nite
sim
al in
terv
entio
ns a
t the
low
er b
ound
ary
s =
5, c
orre
spon
d to
the
orig
inal
sch
eme
anal
yzed
by
Kru
gman
[1
990]
. II
inte
rven
tions
in t
he i
nter
ior
of th
e
targ
et
zone
("
intr
amar
gina
l" i
nter
vent
ions
) w
ere
perm
itted
, a
pair
of
refl
ectin
g
inte
rven
tions
of
exog
enou
sly
give
n fi
nite
mag
nitu
des,
_A
Iu <
0
and
Al1
> 0
co
uld
be a
llow
ed (
see
Floo
d an
d G
arbe
r [19
89])
.
The
sec
ond
clas
s of
inte
rven
tions
com
pris
es i
nter
vent
ions
of f
inite
mag
nitu
de,
that
are
fun
ctio
ns o
f the
cha
nge
in th
e m
agni
tude
of th
e bu
bble
sin
ce t
he p
revi
ous
visi
t
to th
e bo
unda
ries
. T
hey
too
occu
r onl
y w
hen
the
exch
ange
rat
e is
at t
he b
ound
arie
s of
the
targ
et z
one.
T
hey
alw
ays
are
in t
he o
ppos
ite
dire
ctio
n fr
om t
he in
fini
tesi
mal
(typ
e I)
int
erve
ntio
ns at
the
sam
e bo
unda
ry.
We
inte
rpre
t th
ese
(typ
e II
) di
scre
te
inte
rven
tions
as
th
e pa
ssiv
e
acco
mm
odat
ion
of r
atio
nal
spec
ulat
ive
atta
cks
at t
he b
ound
arie
s of
the
tar
get
zone
,
that
are
con
sist
ent
with
the
sur
viva
l of
the
targ
et z
one.
W
e re
fer t
o su
ch s
tock
—sh
ift
port
folio
res
huff
ies
as "
sust
aini
ng"
or "
frie
ndly
" sp
ecul
ativ
e at
tack
s.
The
beh
avio
r of
the
regu
late
d fun
dam
enta
l is
give
n in
equ
atio
ns (1
2a) t
o (1
2c).
The
co
nditi
ons
in (
12a)
, (1
2b),
(12
c(a)
) an
d (1
2c(y
))
are
the
stan
dard
one
s fo
r
regu
late
d Bro
wni
an m
otio
n (s
ee e
.g.
Har
riso
n ]1
985,
p.
80])
. If
>
u an
d if
f =
whe
n =
5u' t
hey
char
acte
rize
a r
efle
ctin
g ba
rrie
r at
f =
u (
give
n by
(12
a),
(12b
)
and
(12c
(a))
. If
f1 c
s an
d if
f =
f1
whe
n s
= s,
th
ey c
hara
cter
ize
a re
flec
ting
barr
ier a
t f =
f1 g
iven
by
(12a
), (
12b)
and
(12
c('y
)).
The
oth
er c
ondi
tions
ar
e no
n-st
anda
rd a
nd a
re r
equi
red
for
our
anal
ysis
of
ratio
nal
bubb
les
in a
targ
et
zone
. If
>
5u'
they
cha
ract
eriz
e (d
iscr
ete)
acco
mm
odat
ed
spec
ulat
sve
atta
cks
from
f =
f to
I =
u' fo
llow
ed b
y (i
nfin
itesi
mal
)
refl
ectio
n at
f =
u (
give
n by
(12
a),
(12b
) an
d (1
2c(f
i)).
If
f1 <
s,
they
cha
ract
eriz
e
—12
—
acco
mm
odat
ed sp
ecul
ativ
e atta
cks
from
f
= f
to
I =
f, fo
llow
ed b
y re
flec
tion a
t
=1
(giv
en
by (
12a)
, (1
2b)
and
(12c
(6))
. If
f0
�
s,
they
cha
ract
eriz
e
acco
mm
odat
ed
spec
ulat
ive a
ttack
s fr
om I =
I (to
I =
s (g
iven
by
(12a
), (
12b)
and
(12c
(c))
. If
I � s
, th
ey c
hara
cter
ize
acco
mm
odat
ed
spec
ulat
ive a
ttack
s fr
om I
=
to I
= s
(giv
en b
y (1
2a),
(12
b) a
nd l
2c(s
)).
The
rea
sons
for
thes
e no
n—st
anda
rd
feat
ores
will
bec
ome
clea
r be
low
.
We
rest
rict
atte
ntio
n to
sta
rtin
g st
ates
s
and
star
ting
date
t
= 0
and
defi
ne t
as th
e in
stan
t im
med
iate
ly be
fore
t, th
at is
t_e
lim (
t—ht
) t-
4O
at>
o
The
regu
late
d fun
dam
enta
l is fo
rmal
ly d
efin
ed a
s fo
llow
s:
(12a
) 1(
t) e
pt +
dW
1—10
(t)
+ I/
t) fo
r all
t � 0
(12b
) 1n
and
I ar
e co
nsta
nt if
s1 <
s <
(12c
) jO
an
d I v
ary
whe
n =
or
s
= s
(I
) 1f
f0 >
5 an
d 1
< 5
then
:
(a)
II s
= s
and
I = 1
then
1U
is
cont
inuo
us
and
incr
easi
ng.
(jl)
If
s(t
) =
s an
d 1(
t) =
10,
then
1(
t) =
f0(t
) an
d (a
) ap
plie
s he
ncef
orth
.
('y)
II s
= s1
an
d I =
1 th
en I
is co
ntin
uous
and
incr
easi
ng.
( II
s(t_
) =
s1 an
d 1(
t) =
1,
then
1(t
) =
f/t)
and
()
appl
ies
henc
efor
th.
— 1
3—
(ha)
1f
f � s
then
(c)
If s
(t_)
= s
then
1(t
) =
s
If s
= s
then
(y)
or (
8)
appl
y.
(JIb
) If
1� s
then
(77)
If
s(t)
= s
then
1(t
) =
s1
If s
= s
then
(a)
or
(fi)
app
ly.
It w
ill h
e ap
pare
nt t
hat,
for
any
size
bub
ble
we
can
calc
ulat
e exa
ctly
bot
h th
e
mag
nitu
des
of th
e pa
ssiv
e or
acc
omm
odat
ing
disc
rete
int
erve
ntio
ns re
quir
ed to
allo
w
the
targ
et z
one
to s
urvi
ve a
nd th
e va
lues
of
the
fund
amen
tals
at w
hich
they
will
occ
ur.
In o
ur
rudi
men
tary
mod
el,
ther
e is
no
pos
sibi
lity
of t
he s
pecu
lativ
e bu
bble
infl
uenc
ing
the
beha
vior
of th
e fu
ndam
enta
l. A
bub
ble
in th
is m
odel
can
onl
y in
flue
nce
the
exch
ange
ra
te d
irec
tly.
If th
e pr
oces
s go
vern
ing
the
unre
gula
ted
fund
amen
tal
incl
uded
th
e ex
chan
ge
rate
as
an a
rgum
ent
then
, be
caus
e th
e bu
bble
af
fect
s th
e
exch
ange
ra
te,
it w
ill
indi
rect
ly i
nflu
ence
th
e be
havi
or o
f th
e fu
ndam
enta
l. A
n
inte
rest
ing
anal
ysis
of
a m
odel
in
whi
ch t
he b
ubbl
e af
fect
s th
e ex
chan
ge
rate
bot
h
dire
ctly
and
ind
irec
tly th
roug
h th
e fu
ndam
enta
l is
ana
lyze
d by
Mill
er—
Wel
ler [
1990
a]
and
by M
iller
, Wel
ler
and
Will
iam
son
[198
9].
Fina
lly,
the
beha
vior
of
the
auth
oriti
es at
the
edge
s of
the
tar
get
zone
can
be
sum
mar
ized
as
fol
low
s:
"Ref
lect
w
hen
this
is
suff
icie
nt;
acco
mm
odat
e whe
n th
is i
s
nece
ssar
y.
Ill.
Smoo
th P
astin
g a
Bub
bly E
xcha
nge R
ate
Con
side
r the
cas
e of
an e
xoge
nous
bub
ble,
tha
t is
one
tha
t do
es n
ot d
epen
d on
1.
The
bub
ble
can
be ei
ther
det
erm
inis
tic
or s
toch
astic
. T
he g
raph
ic re
pres
enta
tion o
f
this
cas
e is
sho
wn
in F
igur
es la
and
lb.
We
now
def
ine
the
bubb
le e
ater
of
the
exch
ange
rat
e fo
r a
give
n va
lue
of th
e
— 1
4—
bubb
le,
B(t
). T
he b
ubbl
e ea
ter
of s
fo
r B
(t)
satis
fies
equ
atio
n (5
) w
ith
A1,
A2
and
C
dete
rmin
ed by
equ
atio
ns (6
) an
d (l
Oa—
d) f
or th
e ac
tual
ly p
reva
iling
va
lue
of
the
bubb
le, t
hat
is f
or
C(t
) =
—B
(t).
T
hat
is t
he b
ubbl
e ea
ter
of
s fo
r B
=
B is
give
n by
—
A1(
f + B
) A
2(f +
B)
(13a
) s=
f++
B+
A1e
+
A2e
—
A1(
f +
B)
A2(
f +
B)
(13b
) 5u
=fu
+ +
B + A
1 e
u +
A2e
u
A (f
+
B)
A2(
f +
B)
(13c
) O
=l+
A1A
1e 1
u +
A2A
2e
u
—
A1(
f1 +
B)
A2(
f1 +
B)
(13d
) s1
=f,
+-j
-B +
A1e
+
A2e
A1(
f1+
B)
A2(
f1+
B)
(13e
) O
=1+
A1A
1e
+A
2A2e
Thi
s bu
bble
ea
ter
shif
ts
cont
inuo
usly
w
ith
B;
an in
crea
se (
decr
ease
) in
B
corr
espo
nds
to a
n eq
ual h
oriz
onta
l di
spla
cem
ent
to t
he le
ft (
righ
t) o
f th
e gr
aph.
N
ote
that
the
bubb
le e
ater
giv
es a
ref
eren
ce t
raje
ctor
y on
ly
and
does
not
sho
w t
he a
ctua
l
mov
emen
t of
f an
d s
whe
n B
ch
ange
s un
less
Bu
> B
(r)
> B
1 an
d ei
ther
s =
an
d f=
f or
s=
s1 a
nd f
=f1
. W
hen
s is
in t
he in
teri
or o
f th
e ta
rget
zon
e an
d B
u >
B(r
) > B
1, its
pos
ition
ca
n be
def
ined
with
ref
eren
ce t
o th
e pe
riod
r bu
bble
eat
er,
that
is
the
bubb
le
eate
r
corr
espo
ndin
g to
the
valu
e of
B t
hat p
reva
iled a
t the
pre
viou
s (m
ost
rece
nt) v
isit
of
ui oq 03 sAnqe lnq) 'j jo q2u aqj o; aq osjt w
oo j 'ojqqnq gems A
puapgjns t °a
au!! .ç aq jo 24211 a; o3 si 434JM 'J =
; axoq'T v A
npunoq zaddn 042 03
Aoua2utl jo lulod e oe 'T
a + (1a ';)2 =
g £q uaA
I2 '1 ioj ;aoa aqquq 0112 '(ui2uo alp q2rIoflp 01511 ,ç alp uo si qotqA
t) "iJ Jo 3J0 01J3 02 51
J O
jII{M 3043 03°N
;o; 10300 oqqnq 0112 0AO
O 0a —
1a 100!310A
0 ' + (a ')2 =
qdei2aipuoarT
put u put
20813043'11J=J put
°J=j
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oo siq; °s
puxq joddu 531 o 0201 a2uoqoxo 012 s2uuq
1H
02 ojqqnq
042 JO otis alp us aseanui w
e 1104M suodctsq T
eqht .ioprsuoo M
ON
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repunoq IO
MO
j 0112
30 9jj
3ulod Aoua2w
e3 t put Arepunoq ;oddn 0l
0u 3uiod A
ouo2ut t swq 4314M
+ (0a 'J)2 =
s A
q UO
A!2 '0
IOJ
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uoireiooidop
popadxa ss 3043 '5 JO
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042 03 3uiOd L
ull
'3p — s)o =
spa aousg
J = s
UIPIIO
013 4200143 0114 .91' 013 JO 212!; 012 03 504 (0
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us aqqnq aAs3ssO
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juo OM
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a 5 a) s 02
renba 10 11043 1030012 J stq ojqqnq oA
i3o8ou o2itj o put (1q < a <
o) s 11042 °1
504 ajqqnq O
A!3020u
'Asno2otuy °H
<
H ! 3043
115 03 rnba 10 1101{3
8101
oeq ajqqnq aAs3s od o2rei y
°a > a >
o ! 3043 11043 1030012 J
JO O
fijEA
0 0A2
(p—oi)
suolsonbo 4O0M
loJ 0110 51 ajqqnq oAslisod item
s y oqquq O
A!3!SO
d 0a2rej o Jo 3013 q o;n2sj 'ajqqnq aA
r3ssod j[tlS1 0 JO 0503 042
S3uOSO
ldOl
02 0102!J
•1a + (0a 'flP =
Lq U
OA
! s s
JO uo131sod 100300 012 '1H
5! a JO
Ofl[O
A [n30t 01J2
2 011112
30 J! U041
=
0 L3it1auo2 Jo 5501 315042!M
) 0a +
(0a 'j)2 = s
pa3ouap aq
UR
O
iOJ
10200 aqqnq 041 0H
aq
H Jo O
O10A
301J3 301 50fl0U
flO4 01j3 JO
ouo 02
— PT
—
—16
—
of
hi).
*2
w
ill
alw
ays
be to
the
lef
t of
ll•
From
11
*1 (o
r 11
2)
ther
e is
a
disc
rete
inte
rven
tion
(a s
tock
—sh
ift i
ncre
ase
in f
) w
hich
, at
a c
onst
ant
exch
ange
rat
e
= s,
mov
es t
he s
yste
m t
o Q
, the
poin
t of
tang
eocy
of
the
bobb
le e
ater
of
B1
and
the
uppe
r lim
it of
the
targ
et z
one.
T
he h
oriz
onta
l dis
tanc
e be
twee
n an
d hi
is
jus
t eq
ual
to t
he d
iffe
renc
e be
twee
n B
1 an
d B
°, i
.e.
— fj
= B
1 —
B°.
Mor
eove
r,
the
hori
zont
al d
ista
nce
betw
een
(1 an
d f
is a
lway
s gr
eate
r th
an t
he
dist
ance
bet
wee
n 11
0 an
d T
hese
tw
o di
stan
ces
wou
ld b
e eq
ual
only
if th
e sl
ope
of
the
bubb
le e
ater
wer
e 45
de
gree
s, w
hich
is n
ot p
ossi
ble.
*1
N
ote
that
at
Il
the
smoo
th p
astin
g (o
r no
—ar
bitr
age)
con
ditio
ns
(lO
a—d)
are
not s
atis
fied
. A
t Il
l in
Fig
ure
la th
e sm
ooth
pas
ting
cond
ition
s ar
e sa
tisfi
ed a
nd t
he
expe
cted
ra
te o
f cha
nge
of th
e ex
chan
ge r
ate
is n
egat
ive
(Ili
is to
the
righ
t of
the
45
line
thro
ugh
the
orig
in).
Il
l is
the
refo
re
cons
iste
nt w
ith c
redi
ble
inte
rven
tion
to
prev
ent
s fr
om ri
sing
abo
ve
s.d.
T
his
is n
ot th
e ca
se w
ith th
e la
rge b
ubbl
e sh
own
in
Figu
re l
b.
Ill
is to
the
left
of t
he 4
5* l
ine
thro
ugh
the
orig
in a
nd t
here
fore
rep
rese
nts
a po
sitiv
e ex
pect
ed
rate
of c
hang
e of
the
exc
hang
e ra
te,
inco
nsis
tent
w
ith
a cr
edib
le
defe
nse
of th
e up
per
limit
of th
e ta
rget
zon
e. In
this
cas
e th
e in
terv
entio
n rul
e le
ads
to
the
follo
win
g sc
enar
io.
From
Il
l a
one—
off
disc
rete
(st
ock-
shif
t) in
crea
se i
n f
brin
gs
the
syst
em to
11u
' tha
t is
the
poi
nt [
st,
su],
an a
rbitr
arily
sm
all
dist
ance
to th
e ri
ght
of t
he in
ters
ectio
n of
s =
an
d th
e 45
0 lin
e th
roug
h th
e or
igin
. Si
nce
s is
kep
t
cons
tant
at
s =
8n'
ther
e ar
e no
arb
itrag
e pro
fits
. Fr
om 1
l the
syst
em, w
ith A
1 an
d
A2
now
def
ined
by
the
one—
side
d 's
moo
th p
astin
g' c
ondi
tions
giv
en in
(lla
,b),
can
now
agai
n m
ove i
nto
the
inte
rior
of
the
targ
et z
one.
In F
igui
e lb
, the
gra
ph o
f s w
ith A
1 an
d A
2 de
term
ined
by
(ha)
and
(lib
) an
d B
= B
u de
note
d s'
, coi
ncid
es w
ith t
he b
ubbl
e ea
ter
of B
u Fo
r a
slig
htly
larg
er
valu
e of
B
, the
gra
ph o
f s
with
A
1 an
d A
2 de
term
ined
by
tbe
one
-sid
ed s
moo
th
past
ing
cond
ition
s (ha
) and
(lib
) is
deno
ted
s' '.
A la
rger
valu
e of
s
yiel
ds s
'' A
s B
go
es to
infi
nity
the
gra
ph o
f s
appr
oach
es a
vert
ical
line
thro
ugh
t1u
—17
—
Not
e th
at,
by c
onst
ruct
ion,
we
alw
ays
have
a l
ocal
max
imum
at t
he ta
ngen
cy
poin
t f2
(fo
r fin
ite v
alue
s of
B).
For
pos
itive
unr
egul
ated
real
izat
ions
of
f, s
tand
ard
infi
nite
sim
al re
flec
tion
stop
s th
e re
gula
ted
valu
e of
f f
rom
ris
ing
beyo
nd s.
For
nega
tive r
ealiz
atio
ns of
the
unre
gula
ted f
, the
sys
tem
mov
es (
give
n B
) in
to th
e in
teri
or
of th
e ta
rget
zon
e.
As
in t
he c
ase
of a
sm
all
bubb
le,
a va
riat
ion
in t
he v
alue
of th
e
bubb
le i
n th
e in
teri
or o
f the
targ
et zo
ne re
pres
ents
a ve
rtic
al d
ispl
acem
ent o
f the
gra
ph
of th
e ex
chan
ge r
ate
(with
con
stan
t val
ues
of A
1 an
d A
2).
With
a l
arge
bub
ble,
the
shap
e of
the
grap
h ch
ange
s ev
ery
time
the
exch
ange
rat
e vi
sits
the
bou
ndar
ies a
t a n
ew
valu
e of
B.
Sinc
e th
e bu
bble
gro
ws
expo
nent
ially
(i
n ex
pect
atio
n)
we
expe
ct
mor
e fr
eque
nt
inte
rven
tions
at
the
uppe
r (l
ower
) bo
unda
ry if
the
cur
rent
va
lue
of t
be b
ubbl
e is
posi
tive
(neg
ativ
e) th
an w
ith
a ze
ro b
ubbl
e.
We
cons
ider
the
se r
esul
ts
to b
e qu
ite
intu
itive
. O
ne
wou
ld e
xpec
t an
exc
hang
e ra
te s
ubje
ct t
o a
posi
tive
and
grow
ing
(in
expe
ctat
ion)
bub
ble
to s
pend
a lo
t of
time
at th
e up
per
boun
dary
of
the
targ
et z
one,
with
the
aut
hori
ties
inte
rven
ing
to c
ount
erac
t not
onl
y th
e un
regu
late
d fu
ndam
enta
l
but a
lso
the
bubb
le.
It i
s ap
pare
nt t
hat
whe
n s
reac
hes
5u w
ith
a la
rge
bubb
le,
the
stoc
k—sh
ift
incr
ease
in
f m
ust t
ake
the
syst
em to
a p
ositi
on s
tric
tly t
o th
e ri
ght
of t
he 4
50 l
ine
thro
ugh
the
orig
in.
On
the
450
line
the
expe
cted
rat
e of
dep
reci
atio
n is
zero
. If
we
impo
se t
he 's
moo
th p
astin
g' c
ondi
tions
w
hen
f =
= 5u
' (sm
all)
posi
tive r
ealiz
atio
ns o
f
incr
emen
ts
in t
he u
nreg
ulat
ed f
unda
men
tal
will
be
offs
et w
ith i
nfin
itesi
mal
refl
ectin
g
inte
rven
tions
. Neg
ativ
e re
aliz
atio
ns o
f df
wou
ld b
ring
the
sys
tem
to th
e le
ft o
f the
450
line
thro
ugh
the
orig
in,
that
is
to a
poi
nt w
ith a
pos
itive
exp
ecte
d ra
te o
f ch
ange
of s
.
Thi
s is
of
cour
se i
ncon
sist
ent w
ith
a cr
edib
le d
efen
se o
f the
tar
get
zone
. If
pri
vate
agen
ts
belie
ved
that
the
aut
hori
ties
wer
e st
abili
zing
th
e ex
chan
ge
rate
at 'e
u' u
b a
frie
ndly
spec
ulat
ive
atta
ck o
f in
fini
tesi
mal
size
wou
ld o
ccur
, br
ingi
ng th
e sy
stem
back
to
1u'
Thi
s am
ount
s to
sta
biliz
ing
the
fund
amen
tal
at f
= 5u
Si
nce
—18
—
Etd
s =
a(s
— f)
th
e ex
chan
ge
rate
wou
ld e
ffec
tivel
y be
exp
ecte
d to
be
fixe
d.
It is
easi
ly se
en th
at a
n ex
chan
ge r
ate
cann
ot re
mai
n fi
xed
for
any
fini
te in
terv
al o
f tim
e if
the
fund
amen
tal
is c
onst
ant
and
ther
e is
a n
on—
zero
spe
cula
tive
bubb
le.
Con
side
r
equa
tion
(3),
for
the
spec
ial
case
in w
hich
f is
kep
t con
stan
t at
s fo
reve
r.
The
sol
utio
n
for
the
exch
ange
rat
e is
giv
en b
y s
= 5u
+ B
whi
ch i
mpl
ies
that
, un
less
B =
0,
the
exch
ange
rat
e ca
nnot
be c
onst
ant.
The
re is
a s
impl
e, un
ifyi
ng c
hara
cter
izat
ion
of th
e in
terv
entio
ns th
at su
stai
n th
e
targ
et z
one.
It
is th
at th
e go
vern
men
t is
cre
dibl
y co
mm
itted
to in
fini
tesi
mal
refl
ectin
g
inte
rven
tions
at th
e bo
unda
ries
s1
an
d T
he v
alue
of
the
fund
amen
tal a
t w
hich
thes
e in
fini
tesi
mal
refl
ectin
g int
erve
ntio
ns at
the
mar
gins
occ
ur a
re d
eter
min
ed by
the
appr
opri
ate
two—
side
d or
one
—si
ded
"no—
arbi
trag
e"
(or
smoo
th
past
ing)
bou
ndar
y
cond
ition
s.
Con
side
r fi
rst t
he c
ase
of a
sm
all p
ositi
ve b
obbl
e sh
own
in F
igur
e la
. T
he v
alue
of th
e fu
ndam
enta
l at w
hich
the
exch
ange
rat
e re
ache
s, s
ay, t
he u
pper
bou
ndar
y u'
in Fi
gure
la
) is
dif
fere
nt fr
om th
e va
lue
of
f th
at c
hara
cter
izes
th
e ta
ngen
cy o
f th
e
bubb
le e
ater
and
the
upp
er b
ound
ary
(4 in
Fig
ure
la)
exce
pt i
n th
e ca
se o
f a
zero
bubb
le.
Sinc
e u
is le
ss t
han
u' th
e di
scre
te (
stoc
k-sh
ift)
jum
p in
the
fund
amen
tal
that
tak
es th
e sy
stem
to t
he re
flec
tion
poin
t Q
ca
n be
int
erpr
eted
as
a fr
iend
ly o
r
sust
aini
ng sp
ecul
ativ
e at
tack
whi
ch is
acc
omm
odat
ed
by th
e au
thor
ities
. *1
11
is
not
a c
redi
ble
eqm
hbrs
um i
f th
e pr
ivat
e se
ctor
kno
ws
the
gove
rnm
ent's
inte
rven
tion
rule
, as
the
re c
an b
e no
traj
ecto
ry s
= g(
f, B
0) +
B1
that
th
e up
per
boun
dary
if th
ere
is in
fini
tesi
mal
refl
ectio
n at
the
poi
nt w
here
s
= g
(f,
B0)
+ B
1 =
The
re is
a c
redi
ble
refl
ectin
g equ
ilibr
ium
at
Ili.
Not
e th
at
Etd
s =
n(su
_41)
dt>
Etd
s =
n(s0
_4)d
t
—19
—
Sinc
e th
e ex
pect
ed
rate
of
chan
ge o
f s
at
in F
igur
e la
is (
posi
tive
and)
high
er th
an t
he (n
egat
ive)
exp
ecte
d ra
te o
f cha
nge
of s
at
Il ,
the
shift
from
1u
to
can
he i
nter
pret
ed a
s an
inc
reas
e in
the
rela
tive
dem
and f
or h
ome
coun
try
mon
ey
due
to a
dec
reas
e in
the
dom
estic
—fo
reig
n in
tere
st d
iffe
rent
ial.6
T
he a
utho
ritie
s
oblig
ingl
y ac
com
mod
ate
this
frie
ndly
or s
usta
inin
g sp
ecul
ativ
e at
tack
.7 N
ote
that
if at
time
t th
e ex
chan
ge r
ate
reac
hes
s at
f =
fu(B
(rfl
that
is w
hen
the
curr
ent
valu
e
of th
e bu
bble
is t
he s
ame
as t
he v
alue
of
B
at th
e m
ost
rece
nt p
revi
ous
boun
dary
visi
t, th
e si
ze o
f th
e fr
iend
ily s
pecu
lativ
e at
tack
is o
bvio
usly
zer
o, a
nd o
nly
a re
flec
ting
inte
rven
tion o
ccur
s.
With
a la
rge b
ubbl
e (s
ee F
igur
e ib
) th
e st
ock-
-shi
ft i
ncre
ase i
n f
from
f =
f*1
to f
= s
can
also
be
inte
rpre
ted
as t
he p
assi
ve a
ccom
mod
atio
n of
a di
scre
te f
rien
dly
spec
ulat
ive
atta
ck.
The
bub
ble
eate
r of
the
larg
e bu
bble
B
1 in
Fig
ure
lb p
rodu
ces
a
tang
ency
po
int
with
the
upp
er b
ound
ary
at
whi
ch i
s to
the
lef
t of
the
45
line
thro
ugh
the
orig
in.
It th
eref
ore
has
a po
sitiv
e exp
ecte
d ra
te o
f cha
nge
of s,
whi
ch i
s
inco
nsis
tent
with
a c
redi
ble d
efen
se o
f th
e ta
rget
zon
e.
At
11u'
th
e ex
chan
ge r
ate
is
expe
cted
to a
ppre
ciat
e, w
hich
is c
onsi
sten
t with
a cr
edib
le ta
rget
zon
e.
The
pol
icy
rule
w
e us
e to
def
end
the
targ
et z
one
in t
he p
rese
nce
of r
atio
nal
spec
ulat
ive
bubb
les i
s no
t the
onl
y on
e ca
pabl
e of
del
iver
ing
the
surv
ival
of th
e ta
rget
zo
ne.
We
coul
d, f
or in
stan
ce, a
pply
the
one—
side
d sm
ooth
pas
ting
cond
ition
s [(
lla,b
) w
hen
s =
an
d (l
lc,d
) whe
n s
=
even
whe
n B
u >
B(r
) >
B1,
at a
ny e
xoge
nous
ly
give
n va
lue
of
f = u
> 5u
w
hen
s =
s an
d at
any
exo
geno
usly
gi
ven
valu
e of
=
<
whe
n s
= s
. W
e ch
ose
the
part
icul
ar r
ule
give
n in
(lO
a—d)
, (lla
,b)
and
(llc
,d)
both
bec
ause
it i
s qu
ite i
ntui
tive
and
beca
use
it is
as
clos
e as
pos
sibl
e to
the
orig
inal
mod
el.
In p
artic
ular
it i
nclu
des
the
trad
ition
al 's
moo
th p
astin
g" s
olut
ion
as
the
spec
ial
case
whe
n th
ere
is n
o bu
bble
.
—20
—
W. A
Sur
feit
of B
ubbl
es
Con
side
r ag
ain
the
gene
ral
solu
tion
to e
qnat
ion
(1)
give
n in
(3)
. T
he fi
rst t
erm
on th
e R.H
.S. o
f (3
) is
com
mon
ly c
alle
d th
e fu
ndam
enta
l so
lutio
n, s
1
(14)
5f
0) =
a
Etf(
v) e
(\T
_t)d
v
Con
side
r eq
uatio
n (1
5) b
elow
, w
hich
is a
spec
ial
case
of e
quat
ion
(5).
.\f
If
(15)
s=
f++
A1e
1 +
A2e
2
In th
e te
rmin
olog
y of
McC
allu
m
[198
3] th
is i
s th
e m
inim
al
stat
e so
lutio
n: i
t in
volv
es o
nly
the
stat
e va
riab
le(s
) an
d th
e st
ate
vari
able
s ent
er i
n a
"min
imal
" w
ay.
The
fir
st tw
o te
rms
on t
he R
.II.
S of
(15
) ar
e th
e fu
ndam
enta
l sol
utio
n fo
r th
e
unre
gula
ted f
unda
men
tal.
All
of th
e R
.H.S
. of
(15
), f
or s
ome
give
n (n
on—
zero
) val
ues
of A
1 an
d A
2, is
the
fund
amen
tal s
olut
ion
for t
he r
egul
ated
fund
amen
tal.
For
the
unre
gula
ted
fund
amen
tal,
non—
zero
va
lues
of
A1
and/
or A
2 w
ould
perm
it w
hat
Froo
t an
d O
bstf
eld
[198
9c]
call
intr
insi
c bu
bble
s, w
ithin
the
cla
ss o
f fu
nctio
ns ex
pres
sing
s(
t)
as a
func
tion
of c
urre
nt f
unda
men
tals
(f(
t))
ppjy
.
For
the
regu
late
d fu
ndam
enta
l, th
e ex
iste
nce
of in
trin
sic
bubb
les
wou
ld r
equi
re
that
f
ente
rs t
he s
olut
ion
for
the
exch
ange
rat
e in
the
int
erio
r of
the
targ
et z
one
thro
ugh
term
s oth
er th
an th
e on
es a
ppea
ring
(w
ith A
1 an
d A
2 de
term
ined
by
(lO
a—d)
,
(lla
,b)
or
(llc
,d))
on
the
11.1
1.5.
of
equ
atio
n (1
4)
and
with
no
othe
r va
riab
le(s
) in
clud
ed.
Such
intr
insi
c bu
bble
s ar
e cl
earl
y im
poss
ible
.
The
lite
ratu
re h
as g
ener
ally
cas
t its
dis
cuss
ion
of bu
bble
s fo
r our
cla
ss o
f mod
els
in t
erm
s of
the
beha
vior
of
s as
a f
unct
ion
only
of
curr
ent
f for
dif
fere
nt v
alue
s of A
1 an
d A
2.
For
exam
ple,
in
the
case
of
a fr
eely
flo
atin
g ex
chan
ge r
ate,
unl
ess
A1
= 0,
the
—21
—
devi
atio
n of
s fr
om f
will
bec
ome
infi
nite
as
I gro
ws
with
out b
ound
(si
nce
A1
> )
and
unle
ss A
2 =
0,
the
devi
atio
n of s
from
f w
ill b
ecom
e in
fini
te a
s f f
alls
with
out b
ound
(sin
ce
A2
< 0
).
A1
= A
2 =
0
rule
s ou
t on
ly
intr
insi
c bu
bble
s.
Man
y ki
nds
of
exch
ange
rat
e bu
bble
s ar
e po
ssib
le u
nder
a f
loat
ing
rate
eve
n if
A1
= A
2 =
0.
With
such
bub
bles
, s c
an d
evia
te fr
om f
(for
any
giv
en va
lue
of f
) by
even
tual
ly un
boun
ded
amou
nts.
It is
rec
ogni
zed
in t
he l
itera
ture
(Fr
oot
and
Obs
tfel
d [1
989b
] are
an
exam
ple)
,
that
the
choi
ce o
f A
1 an
d A
2 on
ly r
estr
icts
the
rela
tions
hip b
etw
een
s an
d f,
but
that
stat
e va
riab
les o
ther
than
I m
ay e
nter
the
solu
tion
for
s th
roug
h bu
bble
s.
In g
ener
al,
bubb
les
can
intr
oduc
e on
e or
mor
e ad
ditio
nal s
tate
var
iabl
es i
nto
the
solu
tion
for
s.
Subj
ect
to th
e co
nstr
alnt
that
the
se b
ubbl
es o
bey
(4),
th
ey c
an c
ause
s t
o de
viat
e in
alm
ost a
ny c
once
ivab
le m
anne
r fro
m g
(f,
.),
for
any
valu
es o
f A1
and
A2.
Wha
t is
not,
we
belie
ve,
reco
gniz
ed a
s cl
earl
y is
tha
t s
can
depe
nd o
n fi
n w
ays
othe
r th
an g
iven
by
g(f,
•)
(or
the
R.H
.S.
of (1
5)),
pro
vide
d s
also
dep
ends
(th
roug
h
B)
on a
t lea
st o
ne o
ther
sta
te v
aria
ble.
We
show
bel
ow th
at t
he b
ubbl
e at
tim
e t,
B(t
)
can
be a
fun
ctio
n of
the
unre
gula
ted
fund
amen
tal
f(t)
, pr
ovid
ed
it al
so d
epen
ds o
n
som
e ot
her
stat
e va
riab
le (
or st
ate
vari
able
s) z
(t),
with
P1.
1 =
0, w
hich
ens
ures
that
01
(t)
Etd
B =
csB
dt.
Thi
s im
plie
s th
at w
ithin
the
targ
et z
one,
dB
ca
n be
a fu
nctio
n of
df.
z(t)
ca
n (b
ut n
eed
not)
be
a fu
nctio
n ex
clus
ivel
y of
pas
t an
d/or
ant
icip
ated
fut
ure
valu
es
of t
he u
nreg
ulat
ed f
unda
men
tal.
We
call
this
cla
ss o
f ge
nera
lized
in
trin
sic
bubb
les "
fund
amen
tal-
.dep
ende
nt" b
ubbl
es.
(a) B
ubbl
es th
at d
on't
burs
t
We
begi
n by
con
side
ring
spe
cula
tive
bubb
les,
bot
h de
term
inis
tic an
d st
ocha
stic
,
that
don
't co
llaps
e.
Exa
mpl
es o
f non
—co
llaps
ing b
ubbl
e pr
oces
ses
satis
fyin
g (4
) in
clud
e
the
sim
ple
exog
enou
s de
term
inis
tic p
roce
ss
give
n in
(16
),
whe
re B
(0)
= B
0 is
an
— 2
2 —
arbi
trar
y in
itial
val
ue fo
r the
B p
roce
ss.
(16)
B
(t) =
B0e
4.
An
exam
ple
of a
si
mpl
e ex
ogen
ous
(bac
kwar
d—lo
okio
g) s
toch
astic
bub
ble
satis
fyin
g (4
) is
give
n in
equ
atio
ns (1
7a,b
), w
ith B
0 ar
bitr
ary.
(17a
) dB
(t)
= n
B(t
)dt +
abd
Wb
(llb
) B
(t) =
j.t e
abdW
b(v)
+
e'B
0.
The
par
amet
er
cb is
the
ins
tant
aoeo
us va
rian
ce o
f th
e bu
bble
pro
cess
. N
ote
that
5b
co
uld
be a
fuo
ctio
n of
s,
f,
t or
any
oth
er s
et o
f va
riab
les
with
out t
his
affe
ctin
g th
e fa
ct t
hat
equa
tion
(17a
) sat
isfi
es
(4).
Fo
r in
stan
ce, i
f U
B
is a
pos
itive
cons
tant
, th
e bu
bble
can
"ch
ange
sig
n".
With
geo
met
ric
Bro
wni
an
mot
ion
prop
ortio
nal t
o B
), s
ign
reve
rsal
s of
the
bub
ble
are
rule
d ou
t, al
thou
gh B
ca
n ch
ange
dIre
ctio
n.
Impo
rtan
t fo
r ou
r an
alys
is,
the
bubb
le p
roce
ss
can
itsel
f be
a
func
tion
of
curr
ent,
past
or
antic
ipat
ed f
utur
e va
lues
of
the
unre
gula
ted
fund
amen
tal
f.
One
exam
ple
is g
iven
in e
quat
ions
(18
a,b)
bel
ow.
In w
hat f
ollo
ws
the
vari
able
z(
t)
is
eith
er s
tric
tly d
eter
min
istic
or f
ollo
ws
a di
ffus
ion
proc
ess.
T
he z
(t)
proc
ess
is a
twic
e
cont
inuo
usly
dif
fere
ntia
ble
func
tion
of it
s ar
gum
ents
. T
he k
ey d
efin
ing
prop
erty
of
z(t)
is
tha
t it d
oes
not v
ary
with
f(t
), i
.e.
= 0.
Fo
r the
exa
mpl
e giv
en be
low
, U
f(t)
th
e z
proc
ess
give
n in
equ
atio
n (l
Sb)
(a b
ackw
ard—
look
ing
dete
rmin
istic
pro
cess
)
gene
rate
s, to
geth
er w
ith e
quat
ion
(18a
), a
bubb
le p
roce
ss th
at sa
tisfi
es e
quat
ion
(4).
—23
-—
(isa)
B
(t) =
kf(
t) +
z(t
) k
# —
1
(18b
) z(
t) =
et
_[k(
of(v
) —
#)]d
v +
ez0
is a
n ar
bitr
ary
initi
al v
alue
for
the
z pr
oces
s.
Not
e th
at t
he b
ound
ary
cond
ition
s
(iO
a—d)
ca
nnot
be
satis
fied
if
k =
—1
and
that
, gi
ven
the
boun
dary
cond
ition
s (i
ia,b
)
or (
lied)
, A1
and
A2
are
inde
term
inat
e if k
= —
1.
Mor
e ge
nera
lly, e
quat
ion
(18a
) will
hol
d pr
ovid
ed z
sat
isfi
es
(19)
E
tdz(
t) =
[k(
of(t
) - j)
+ o
z(t)
]dt.
The
z p
roce
ss
give
n in
(1
9)
(and
th
eref
ore
the
B
proc
ess
itsel
f) c
an b
e
forw
ard—
look
ing,
bac
kwar
d—lo
okin
g (o
r a
linea
r co
mbi
natio
n of
the
for
war
d—lo
okin
g
and
back
war
d—lo
okin
g so
lutio
ns),
it
can
be s
tric
tly d
eter
min
istic
or st
ocha
stic
.
An
exam
ple
of
a fo
rwar
d—lo
okin
g so
lutio
n fo
r z
(now
tr
eate
d as
a
non—
pred
eter
min
ed v
aria
ble)
whi
ch s
atis
fies
equ
atio
ns
(isa
) an
d (1
9)
(and
of
cou
rse
equa
tion
(4))
is g
iven
in e
quat
ion
(20)
.
(20)
z(
t) =
- J e(
t)E
tk(a
i(v)
- ji)
dv +
R(t
).
Her
e R
(t)
is a
ny
stri
ctly
det
erm
inis
tic
or s
toch
astic
pro
cess
w
hich
sat
isfi
es
Btd
R(t
)dt
= c
sR(t
)dt.
It c
an b
e pr
edet
erm
ined
or
non
—pr
edet
erm
ined
. O
ne p
ossi
ble
choi
ce f
or
R(t
) w
ould
be
the
proc
ess
give
n on
the
rig
ht—
hand
sid
e of
(iS
b).
Thi
s
wou
ld m
ake
the
bubb
le
B(t
) in
(is
a) a
func
tion
of c
urre
nt,
past
and
ant
icip
ated
futu
re v
alue
s of
the
unre
gula
ted f
unda
men
tal.
Ano
ther
inte
rest
ing
bubb
le p
roce
ss i
nvol
ving
f
is th
e fo
llow
ing:
—24
—
(21a
) B
(t) =
fO)z
(t)
(21b
) dz
(t) =
f(t)
[ai(t
) — p]
z(t)
dt +
adW
5 If
0 z(
t)=
0 1=
0
(dW
and
dW1
are
assu
med
to
be co
ntem
pora
neou
sly
inde
pend
ent)
.
An
exam
ple
of a
cla
ss o
f ra
tiona
l bub
bles
th
at h
as t
he p
rope
rty
that
-
01(t
) de
pend
s on
1(
t) is
giv
en in
(22a
—b)
.
(22a
) B
(t) =
f(t)
2 +
z(t)
(22b
) E
tdz(
t) =
[o(f
(t)2
+ z
(t))
— 2
f(t)
p — o]
dt.
A s
impl
e bac
kwar
d—lo
okin
g de
term
inis
tic
solu
tion
for
a th
at sa
tisfi
es (
22b)
is
(23)
z(
t) =
j eo(
t_v)
[a(v
)2 —
2f(
v) —
a]dt
+ e
ulz0
.
If B
ca
n de
pend
on
the
regu
late
d va
lue
of th
e fu
ndam
enta
l as
wel
l as
on
the
unre
gula
ted
fund
amen
tal,
we
can
gene
rate
the
very
unu
sual
bub
ble
give
n in
equ
atio
n
(24)
.
(24)
B
(t) =
- g(f
(t),
B(r
)) +
kf(
t) +
z(t
).
Thi
s bu
bble
is c
onsi
sten
t with
the
law
of
mot
ion
(4),
pro
vide
d th
at (2
5) h
olds
:
— 2
5 —
(25)
Etd
z(t)
= {—
a[g
(f(t
),.)
— kf
(t)
— z(
t)] +
[g'(f
(t),
.) —
k]p
+ g
' '(f(
t),.)
a}dt
Equ
atio
n (2
6) i
s an
exa
mpl
e of
a s
impl
e st
ocha
stic
proc
ess
for
z th
at
satis
fies
(25)
(26)
z(
t) =
eQ
(t) [{
-a[g
(f(v
),.)
- kf(
v)]
+ [g
' (f(
v),.)
—k]
p +
g
/ (fN
),.)
a}dv
+ cz
dWz(
v)] dv
+ e
z0.
Whi
le th
is b
ubbl
e is
con
sist
ent w
ith t
he l
aw o
f m
otio
n (4
) w
ithin
the
targ
et
zone
, it
does
not
per
mit
the
boun
dary
cond
ition
s to
be
satis
fied
. W
ith th
is p
roce
ss f
or
B(t
) th
e ex
chan
ge r
ate
is g
iven
by
(27)
s(
t) =
kf(
t) +
z(t)
.
Not
e th
at in
this
cas
e s
incr
ease
s or
dec
reas
es l
inea
rly
with
f
(if k
# 0
), t
hus
mak
ing
"sm
ooth
pas
ting"
bo
unda
ry c
ondi
tions
im
poss
ible
to
appl
y.
Whe
n k
= 0
, 5(
t)
is in
depe
nden
t of
f(t)
, al
thou
gh
z(t)
of
cou
rse
depe
nds
on p
ast v
alue
s of
f
(and
of f
l
(b)
Bub
bles
that
bur
st
Bla
ncha
rd [
1979
] pr
ovid
es
an i
nter
estin
g ex
ampl
e of
a r
atio
nal
spec
ulat
ive
bubb
le w
hose
exp
ecte
d du
ratio
n (t
ime
until
the
mom
ent a
col
laps
e oc
curs
) is
fini
te.
It
is
mos
t ea
sily
mot
ivat
ed i
n di
scre
te t
ime.
E
quat
ion
(28)
is
the
disc
rete
ana
log
of
equa
tion
(1).
(28)
5t
= 0
(1 +
u)f
t + (1
+ a
)'Ets
t+r
0> 0
— 2
6
The
sol
utio
n of
(28
) co
o±tio
n on
cur
rent
and
exp
ecte
d fu
ture
vue
s of
1t i
s
(29)
s
= o
(i +
0)_
i E0
(1 +
or'
Etf
t+ +
Bt
whe
re
Bt
satis
fies
(30)
E
tBt+
i = (1
+ o)
Bt.
Con
side
r th
e fo
llow
ing
proc
ess
for B
(31)
B
t+i =
(1 —
irr'(
i + o
)Bt
+ +
i w
ith p
roba
bilit
y 1
—
Bt+
i = 6t
+i
with
pro
babi
lity
r
whe
re
0< r
< 1
an
d
(32)
E
tct+
i = E
tct+
i = 0.
It is
eas
ily c
heck
ed t
hat
(31)
an
d (3
2) s
atis
fy (
30).
E
quat
ions
(31)
an
d (3
2)
defi
ne a
bub
ble
(sto
cbas
tic if
t
or
is a
ran
dom
var
iabl
e)
for
whi
ch th
ere
is a
cons
tant
pro
babi
lity
of co
llaps
e,
ir,
each
per
iod.
If
a c
olla
pse
occu
rs i
n pe
riod
t+
l, th
e ex
chan
ge r
ate
retu
rns
to it
s fu
ndam
enta
l val
ue if
=
0.
Mor
e ge
nera
lly, i
f
is r
ando
m,
the
expe
cted
po
st—
colla
pse
valu
e of
the
exc
hang
e ra
te i
s its
fund
amen
tal v
alue
, but
the
rea
lized
val
ue o
f th
e ex
chan
ge r
ate
can
diff
er f
rom
this
expe
ctat
ion b
y a
zero
mea
n fo
reca
st e
rror
. If
the
bubb
le c
olia
pses
in
peri
od
t+1
and
—27
—
is
non—
zero
, a
new
bub
ble
star
ts i
n pe
riod
t+
1 w
hicb
fol
low
s a
law
of
mot
ion
like
(31)
and
(32)
, bu
t pos
sibl
y w
itb a
dif
fere
nt v
alue
of
ir a
nd d
iffe
rent
(al
thou
gh st
ill
zero
mea
n) ra
ndom
dis
turb
ance
term
s c
and
c.
In co
ntin
uous
tim
e th
e sa
me
idea
is c
aptu
red
as f
ollo
ws.
ir
now
is t
he c
onst
ant
inst
anta
neou
s pr
obab
ility
of
colla
pse
of t
he b
ubbl
e.
Con
ditio
nal
on t
he b
ubbl
e no
t
havi
ng c
olla
psed
by
time
t, th
e pr
obab
ility
of
the
bubb
le l
astin
g til
l tim
e t +
At
is
e_iA
t (A
t 0)
. W
e co
nsid
er t
he f
ollo
win
g pr
oces
s fo
r B.
As
befo
re,
Wb
and
Wb
are
stan
dard
Wie
ner p
roce
sses
. C
ondi
tiona
l on
the
bubb
le h
avin
g la
sted
till
time
t, w
e
have
(33)
B
(t+
At)
= e
( r)
AtB
(t) +
Cb Jl
e(X
WbN
) w
ith p
roba
bilit
y
B(t
+A
t) =
b Je
b(v)
-i
rAt
with
pro
babi
lity
1 —
e
It fo
llow
s th
at t
he e
xpec
ted
rate
of
chan
ge o
f B
per
uni
t tim
e ov
er t
he in
terv
al
[t,
t+A
t] is
B B
(t+
At)
— B
(t)
oAt
(34
____
____
____
__ _
(e
—_1
)Bt
At
—
At
In t
he li
mit
as
At
0 th
is b
ecom
es
Etd
B(t
) =
oB
(t)d
t, as
req
uire
d. W
hile
the
bubb
le l
asts
, the
inst
anta
neou
s rat
e of
cha
nge
of t
he b
ubbl
e is
giv
en by
—28
—
dB(t
) (a
+ ir
)B(t
)dt +
cbdW
b(t)
.
Whi
le th
e ex
pect
ed r
ate
of c
hang
e of
the
bubb
le i
s E
tdB
= a
Bdt
, th
e ex
pect
ed
rate
of v
aria
tion
of
B
cond
ition
al
on th
e bu
bble
sur
vivi
ng t
he n
ext i
nsta
nt, i
s gi
ven
by (a
+ ir
)Bdt
.
Let
tc be
the
(ran
dom
) tim
e w
hen
the
bubb
le c
olla
pses
. If
the
valu
e of
B
at
tc
(a-i
-r)(
t —
v)
the
time
of c
olla
pse,
B
(tc)
= b
e
c dW
b(v)
is n
on—
zero
, a
new
bub
ble
will
sta
rt a
t tc
pos
sibl
y w
ith a
dif
fere
nt v
alue
of
ii,
diffe
rent
whi
te n
oise
pro
cess
es
dWb
and
dWb
and
diff
eren
t va
lues
for
ab
and
A B
lanc
hard
bubb
le i
s pe
rfec
tly
com
patib
le w
ith o
ur t
arge
t zon
e m
odel
. W
hen
it co
llaps
es t
here
will
be
a di
scre
te c
hang
e in
the
leve
l of
the
exch
ange
rat
e, w
hich
lie
s
on t
he t
raje
ctor
y ap
prop
riat
e to
the
new
(po
st—
colia
pse)
bub
ble.
Si
nce
this
cha
nge
is
unex
pect
ed,
no a
rbitr
age o
ppor
tuni
ties a
rise
.
zone
s w
ith fu
ndam
enta
l—de
pend
ent
bubb
les
If th
e va
lue
of th
e bu
bble
at t
ime
t is
a fu
nctio
n of
the
cont
empo
rane
ous
valu
e
of t
he u
nreg
ulat
ed f
unda
men
tal,
then
with
in t
he t
arge
t zo
ne
the
resp
onse
of
the
exch
ange
rat
e to
the
fund
amen
tal w
ill b
e af
fect
ed r
elat
ive t
o th
e ca
se o
f an
exo
geno
us
bubb
le (
and,
a—
fort
iori
, rel
ativ
e to
the
no—
bubb
le
case
). T
his
is s
trai
ghtf
orw
ard,
sinc
e
in th
is c
ase
= 1
. A
s lo
ng a
s j —
1,
the
tang
ency
cond
ition
s (b
c—cl
), (
llb)
and
(lld
) are
the
sam
e as
in th
e ca
se o
f an
exo
geno
us b
ubbl
e.
—29
—
V.
The
Dis
trib
utio
n Fun
ctio
n of
the
Exc
hang
e Rat
e in
a T
arge
t Zon
e in
the
Pres
ence
of a
Rat
iona
l Spe
cula
tive B
ubbl
e
By
Ito'
s lem
ma,
the
exch
ange
rat
e in
the
targ
et z
one
is a
dif
fusi
on p
roce
ss w
ith
stoc
hast
ic di
ffer
entia
l
(35)
ds
(t)
= [g
f(f,.
)p +
gff(
f,.)
4]dt
+ gf
(f,.)
cfdW
f + dB
.
Let
V
artx
(t')
deno
te t
he v
aria
nce
of
x(t')
con
ditio
nal
on t
he in
form
atio
n av
aila
ble
at t
ime
t, th
at i
s V
artx
(t')
z E
t[x(
t') —
E
tx(t
')]2.
T
he in
stan
tane
ous
varia
nce o
f the
bub
ble
proc
ess
is g
iven
by
o(f,
f, s
, B
, .)
S
ince
the
only
res
tric
tion
on t
he b
ubbl
e pr
oces
s is
tha
t E
tdB
= nB
dt,
it is
cer
tain
ly p
erm
issi
ble
for
the
inst
anta
neou
s va
rian
ce o
f th
e bu
bble
pr
oces
s to
dep
end
on
f, f
, s,
B
or
oth
cr
vari
able
s.
As
befo
re, r
ecog
nizi
ng t
hat
B m
ay d
epen
d on
f,
we
wri
te
- .
,9z
B=
b(f,
z) w
ith
Usi
ng s
elf—
expl
anat
ory
nota
tion w
e ob
tain
(36a
) E
ds(t
) =
(gf+
bf)p
+ b
Etd
z(t)
+ {(
gff+
bff)
o +
bzzU
+ 2b
zfPf
rfff
Cz]
dl
and
(36b
) V
artd
s(t)
= [(
gf+
bf)2
o + b
a + 2
(gf +
bf)b
zPfz
afcz
] dt
.
For
a bu
bble
pr
oces
s to
be
cons
iste
nt w
ith a
targ
et z
one
unde
r th
e sp
ecif
ied
inte
rven
tion
rule
, it
mus
t be
th
e ca
se t
hat,
for
any
size
bu
bble
, th
e bo
unda
ry
—30
—
cond
ition
s ch
arac
teri
ze
tang
ency
po
ints
tha
t ar
e lo
cal
max
ima
(min
ima)
if
the
tang
ency
is
at t
he u
pper
(lo
wer
) bo
unda
ry.
For
a bu
bble
tha
t is
con
sist
ent w
ith t
he
law
of m
otio
n in
(4)
but d
oes
not p
erm
it sm
ooth
pas
ting
acco
rdin
g to
(lo
ad),
we r
efer
to th
e bu
bble
in
(18a
,b)
with
k =
—1
or th
e bu
bble
giv
en in
(24)
and
(25
).
Not
e th
at t
he i
nsta
ntan
eous
cond
ition
al v
aria
nce
of th
e ch
ange
in
the
exch
ange
rate
no
long
er
in g
ener
al
goes
to
zero
as
the
boun
dari
es
of t
he t
arge
t zo
ne
are
appr
oach
ed.
As
the
exch
ange
ra
te
appr
oach
es
the
boun
dary
, the
var
ianc
e of
the
exog
enou
s co
mpo
nent
of
the
bubb
le n
eed
not g
o to
zer
o no
r nee
d gf
+ bf
tend
to z
ero.
If,
in t
he
spir
it of
O
bstf
eld—
Rog
off
and
Dib
a—G
rnss
man
, w
e ac
cept
th
e
prop
ositi
on th
at u
nder
fre
e fl
oatin
g bu
bble
s ca
nnot
occ
ur,
the
cnnd
ition
al
expe
ctat
ion
and
vari
ance
of th
e ch
ange
in
the
exch
ange
rat
e w
ith a
free
flo
at a
re (
letti
ng s
den
ote
the
exch
ange
rat
e un
der a
free
flo
at):
Ft
=
and
ds(t
2
Var
1 --
-- =
Triv
ially
, sin
ce b
ubbl
es c
an e
xist
in
a ta
rget
zon
e, t
he in
stan
tane
ous c
ondi
tiona
l
vari
ance
of
chan
ges
in t
he e
xcha
nge
rate
may
be
less
und
er fr
ee f
loat
ing
than
und
er a
targ
et z
one.
If
unco
vere
d in
tere
st p
arity
hol
ds, it
mus
t be
the
case
tha
t th
e in
tere
st
diff
eren
tial
S is
giv
en b
y
8(t)
e i(
t) —
i*(t
) = F
t ds
(t)/
dt.
It f
ollo
ws
that
the
inst
anta
neou
s con
ditio
nal
mea
n an
d va
rian
ce of
the
chan
ge in
the i
nter
est d
iffe
rent
ial in
the
targ
et z
one
are
give
n by
— 3
1 —
(37a
) E
t4 =
o{(g
f+bf
-1)P
+b5
Ftd
z +
(37b
) V
art4
.l = [o
(gf+
bç—
1)]2
a +
(Obz
)2O
+ 2o
2(gf
+bf
—1)
bpfc
fc5.
As noted in Svensson [1989], if
= 0
(a
nd therefore also fz = 0
) eq
uatio
ns
(36b
) and (37b) imply that (denoting the conditional standard deviation by SD)
ds
—1
dS
SDt() —
SD
t() =
Cf.
Thi
s fi
ndin
g of
a c
onst
ant t
rade
—of
f be
twee
n th
e in
stan
tane
ous v
aria
bilit
y of
the
chan
ge i
n th
e ex
chan
ge
rate
and
the
ins
tant
aneo
us
vari
abili
ty o
f the
cba
nge
in t
he
inte
rest
dif
fere
ntia
l is
no l
onge
r au
tom
atic
ally
valid
whe
n th
ere
is a
bub
ble,
as
can
be
veri
fied
by
insp
ectio
n of
(36b
) an
d (3
7b).
The
co
nditi
onal
in
stan
tane
ous
mea
n an
d va
rian
ce of
the
chan
ge i
n th
e in
tere
st
diff
eren
tial u
nder
a fr
ee f
loat
, dS
, ar
e: Etd
b(t)
= 0
and
Var
tds(
t) =
0.
With
or w
ithou
t a b
ubbl
e, t
he fr
ee f
loat
the
refo
re a
lway
s de
liver
s m
ore
stab
ility
in c
hang
es i
n th
e in
tere
st d
iffe
rent
ial
than
doe
s th
e ta
rget
zon
e.
The
rel
atio
nshi
p be
twee
n th
e le
vel
of t
he e
xcha
nge
rate
an
d th
e in
tere
st
diff
eren
tial in
the
targ
et z
one
is f
ound
by
notin
g th
at
95
lB
= o(
gf +
—
1)
—32
—
and
DB
=
f +
It fo
llow
s th
at
Os
gf+
-3-f
-
atgf
+
Whe
n the
re is
no
bubb
le a
nd t
he ra
nge o
f pe
rmis
sibl
e va
lues
of
f is
f1 �
f � 1
(t
hat i
s th
e va
lues
of
f co
rres
pond
ing
to t
he u
pwar
d—sl
opin
g par
t of
the
bubb
le e
ater
for
B =
0)
we
have
, sin
ce 0
� gf
< 1
fo
r f1
< f <
f0
, a
dow
nwar
d—sl
opin
g re
latio
n-
ship
bet
wee
n th
e le
vel of
the
exch
ange
rat
e an
d th
e in
tere
st d
iffe
rent
ial.
Thi
s re
latio
n-
ship
is
qu
alita
tivsl
y un
affe
cted
by
the
bubb
le i
f th
e cu
rren
t va
lue
of th
e bu
bble
is
inde
pend
ent o
f th
e cu
rren
t val
ue o
f th
e fu
ndam
enta
l, if
we
agai
n re
stri
ct th
e an
alys
is t
o
the
upw
ard—
slop
ing
part
of
the
h(f,
B(r
), B
(t))
= g
(f)
+ B
(t)
func
tion.
T
here
are
,
how
ever
, no
goo
d re
ason
s fo
r th
is
rest
rict
ion
whe
n th
ere
are
bubb
les,
so
the
rela
tions
hip
betw
een
the
leve
l of
the
exch
ange
rat
e an
d th
e in
tere
st d
iffe
rent
ial c
an
chan
ge s
ign.
T
his
is s
how
n in
Fig
ure
2 fo
r an e
xoge
nous
bub
ble.
Har
riso
n [1
985,
pp.
89—
92]
show
s th
at t
he s
tead
y st
ate
or u
ncon
ditio
nal d
ensi
ty
func
tion
ir
of B
row
nian
mot
ion
f w
ith i
nsta
ntan
eous
var
ianc
e cr
an
d dr
ift
p w
hich
is re
gula
ted
on th
e cl
ose
inte
rval
[f1,
is
giv
en by
:
71ff
) =
[I0 —
f1]
if
p =
0 (t
he u
nifo
rm d
ensi
ty)
= ee
if
p
0 (t
runc
ated
exp
onen
tial d
ensi
ty)
— 3
3 —
-2
0 =
2/w
f
With
out a
bub
ble
it fo
llow
s th
at, s
ince
s
= g
(f)
is a
str
ictly
incr
easi
ng fu
nctio
n
of
f w
ith c
ontin
uous
fi
rst
deri
vativ
e on
the
open
int
erva
l f0
) an
d th
e im
plic
it
(inv
erse
) fu
nctio
n f =
g(s
) th
eref
ore
exis
ts o
n th
e ta
rget
zon
e (e
xcep
t at
the
end
poin
ts),
the
stea
dy s
tate
den
sity
func
tion
of s,
cc(
s)
say,
is g
iven
by:
(5)7
r( j
s))
Sj<
5<5
g'(f
(s))
u
Sven
sson
[1
989]
sho
ws
that
th
e re
sulti
ng d
istr
ibut
ion
of e
xcha
nge
rate
s is
U—
shap
ed.
With
a b
ubbl
e, h
owev
er,
the
long
—ru
n be
havi
or o
f the
exc
hang
e ra
te i
s qu
ite
diff
eren
t. Fi
rst
of a
ll, o
ne o
f the
sta
te v
aria
bles
, th
e bu
bble
, is
non—
stat
iona
ry a
nd
unre
gula
ted.
It
ther
efor
e do
es n
ot p
osse
ss a
ste
ady
stat
e di
stri
butio
n. T
he o
ther
sta
te
vari
able
, f,
is
of c
ours
e al
so n
on—
stat
iona
ry w
hen
unre
gula
ted,
bu
t re
gula
tion
may
st
ill e
nsur
e th
at it
has
a s
tead
y st
ate
dist
ribu
tion.
Not
e, h
owev
er t
hat
boun
ds s
et b
y
the
cont
rolle
r are
bou
nds
on s
, no
t on
f.
It is
cle
arly
pos
sibl
e fo
r f
to b
ecom
e a
nega
tive
num
ber s
mal
ler
than
the
val
ue of
f1 as
soci
ated
with
B =
U
(for
inst
ance
if a
posi
tive
bubb
le g
row
s st
eadi
ly a
nd t
here
is a
long
seq
uenc
e of
, on
ave
rage
, ne
gativ
e re
aliz
atio
ns o
f df
with
in th
e ta
rget
zon
e).
Sim
ilarl
y, f
can
bec
ome
a po
sitiv
e num
ber
larg
er th
an t
he v
alue
of f0
asso
ciat
ed w
ith
B =
0
if a
neg
ativ
e bu
bble
ste
adily
gro
ws
in s
ize
and
ther
e is
a lo
ng r
un o
f, o
n av
erag
e, p
ositi
ve re
aliz
atio
ns o
f df
.8
Whi
le th
e de
riva
tion
of th
e st
eady
-sta
te d
istr
ibut
ion o
f s
for m
ost i
nter
estin
g ca
ses
(inc
ludi
ng
the
exog
enou
s st
ocha
stic
bub
ble
and
the
stoc
hast
ic b
ubbl
e th
at
depe
nds
on
f) r
emai
ns to
be
done
, a
few
gen
eral
rem
arks
can
be
mad
e.
Con
side
r fi
rst t
he ca
se o
f an
exog
enou
s, n
on-s
toch
astic
bubb
le.
The
asy
mpt
otic
—34
—
)
dist
ribu
tion
of th
e ex
chan
ge r
ate
will
, if
the
bubb
le i
s po
sitiv
e,
put
all
its p
roba
bilit
y
mas
s ne
ar i
ts
uppe
r bo
unda
ry
valu
e s.
In t
his
very
sim
ple
case
, th
e ta
rget
zon
e
does
n't
kill
the
bubb
le, b
ut t
he b
ubbl
e ef
fect
ivel
y ki
lls t
he t
arge
t zo
ne a
nd t
urns
it,
asym
ptot
ical
ly, in
to a
fixe
d ex
chan
ge r
ate
regi
me.
Now
con
side
r the
cas
e w
here
the
bubb
le i
s st
ocha
stic
. If
we
have
the
exo
geno
us
geom
etri
c bub
ble
dB =
caB
dt +
cTB
BdW
B,
tbe
bubb
le c
anno
t cha
nge
sign
, al
thou
gh it
can
chan
ge d
irec
tion.
T
ake
the
case
with
B
(O)
> 0
. If
the
bub
ble
ever
rea
ches
B =
0, it
dies
and
we
are
back
hen
cefo
rth
in th
e ca
se s
tudi
ed b
y Sv
enss
oo.
Cle
arly
we
expe
ct t
o ha
ve a
non—
dege
nera
te d
istr
ibut
ion
of
s in
thi
s ca
se,
but
still
with
mor
e
mas
s at
s th
an w
ithou
t a b
ubbl
e.
u
If t
he e
xoge
nous
bu
bble
pr
oces
s is
giv
en in
stea
d by
dB
= o
Bdt
+ C
BdW
B
with
>
0
but c
onst
ant,
B
will
, in
a re
pres
enta
tive h
isto
ry, a
ssum
e bo
th p
ositi
ve
and
nega
tive
valu
es.
The
bub
ble
neve
r die
s in
this
cas
e.
Rel
ativ
e to
the
case
stu
died
by S
vens
son
we
wou
ld e
xpec
t the
lon
g—ru
n de
nsity
fun
ctio
n to
hav
e m
ore
mas
s at
the
two
end
poin
ts b
ut w
ith n
o pa
rtic
ular
bia
s to
war
ds ei
ther
s0
or s
. A
sym
ptot
ical
ly,
the
infl
uenc
e of
the
initi
al c
ondi
tions
fo
r B
on
the
den
sity
func
tion
for
s sh
ould
vani
sh.
We
hope
to b
ack
up th
ese
intu
ition
s with
sol
id a
naly
sis
in th
e ne
ar fu
ture
.
VT
. Bub
bles
, tar
get z
ones
an
d in
terv
entio
n rul
es: w
hat g
ives
whe
n th
ey a
re in
cons
iste
nt?
Our
vie
w o
f wha
t hap
pens
to
spec
ulat
ive
bubb
les
unde
r a c
redi
ble
targ
et z
one
is
rath
er d
iffe
rent
fro
m th
at o
f M
iller
, W
elle
r an
d W
illia
mso
n [198
91 (
henc
efor
th M
WW
).
The
for
mal
ana
lysi
s in
the
ir p
aper
is c
ast
in t
erm
s of
Bla
ncha
rd b
ubbl
es,
but
wou
ld
appl
y a—
fort
iori
to
the
non
—co
llaps
ing
bubb
les
we
cons
ider
ed.
MW
W p
ut
thei
r
argu
men
t su
ccin
ctly
:
If al
l mar
ket p
artic
ipan
ts be
lieve
that
, whe
n th
e ex
chan
ge ra
te
hits
the
edg
e of
the
ban
d, t
he a
utho
ritie
s will
def
end
the
rate
bp
sud
den
inte
rven
tion
(des
igne
d to
pro
duce
a
jum
p in
the
ra
te),
th
is m
ust
caus
e th
e bu
bble
to
bur
st.
But
thi
s is
—35
—
inco
nsis
tent
with
the
exi
sten
ce o
f a b
ubbl
e in
the
fir
st p
lace
: If
all k
now
this
col
laps
e is
cer
tain
to o
ccur
at t
ime
t, al
l w
ish
to s
ell t
he c
urre
ncy a
t t—
e. B
ut th
en c
olla
pse
will
occ
ur at
t—e.
R
ep ea
ting
this
arg
umen
t we
find
that
col
laps
e m
ust
occu
r at
time
zero
that
is, a
ll su
ch bu
bble
s ar
e "s
tran
gled
at b
irth
".
(Mill
er,
Wel
ler
and
Will
iam
son
1g8g
]).
It sh
ould
be
reco
gniz
ed
that
in
this
pap
er w
e ha
ve
chan
ged
the
inte
rven
tion
rule
s fo
llow
ed b
y th
e au
thor
ities
from
wha
t was
ass
umed
by
MW
W a
nd b
y th
e ot
her
cont
ribu
tors
to th
is li
tera
ture
. In
add
ition
to th
e st
anda
rd re
flec
tion
polic
ies,
we
allo
w
the
acco
mm
odat
ion
by
the
auth
oriti
es o
f su
stai
ning
sp
ecul
ativ
e at
tack
s.
The
inte
rven
tion
polic
ies
cons
ider
ed by
MW
W an
d in
the
rest
of t
his l
itera
ture
, do
not h
ave
this
fea
ture
. It
sho
uld
ther
efor
e co
me
as n
o su
rpri
se t
hat
from
dif
fere
nt a
ssum
ptio
ns
we
reac
h di
ffer
ent
conc
lusi
ons.
Giv
en th
e as
sum
ptio
n th
at t
he o
nly
form
of
inte
rven
tion
is r
egul
atio
n at
the
boun
dari
es
thro
ugh
infi
nite
sim
al r
efle
ctin
g op
erat
ions
, th
e co
nclu
sion
th
at
this
conf
igur
atio
n is
inc
onsi
sten
t w
ith r
atio
nal b
ubbl
es f
ollo
ws.
O
ne c
omm
on re
spon
se t
o
this
inc
onsi
sten
cy i
s to
trea
t the
bub
ble
as a
resi
duaL
T
he re
ason
ing
goes
as
follo
ws:
at
*1
a po
int l
ike
LI
in F
igur
e la
ther
e w
ould
be
a di
scon
tinuo
us
antic
ipat
ed fa
ll in
the
va
lue
of s
fr
om
5u
to
;, th
e va
lue
of t
he e
xcha
nge
rate
at
f = u
' on t
he b
ubbl
e
eate
r fo
r B
= 0,
gi
ven
by
s =
g(f
, B
0) +
B0,
int
erpr
etin
g B
° to
equ
al z
ero.
T
his
viol
ates
the
no—
arbi
trag
e co
nditi
on.
Wha
t is
pres
ente
d in
the
pre
viou
s pa
ragr
aph
as a
"re
al—
time"
eve
nt, t
he fa
ll of
th
e ex
chan
ge r
ate
from
5u to
s,
can
in fa
ct b
e no
mor
e th
an th
e de
sign
atio
n of
s as
the
only
val
ue o
f th
e ex
chan
ge
rate
rea
ched
whe
n =
tu ,
tha
t is
con
sist
ent w
ith t
he
inte
rven
tion
rule
. E
quiv
alen
tly, B
a 0
is t
he o
nly
valu
e of
the
bubb
le c
onsi
sten
t with
*1
th
e in
terv
entio
n ru
le.
If th
e sy
stem
real
ly w
ere
(som
ehow
) at
12
, th
ere
wou
ld b
e an
inco
nsis
tenc
y.
We
cann
ot d
educ
e ho
w t
he e
xcha
nge
rate
wou
ld b
ehav
e st
artin
g fr
om
an in
cons
iste
nt po
sitio
n.
Not
e th
at i
t is
not j
ust
the
assu
mpt
ion
of a
cre
dibl
e ex
chan
ge r
ate
targ
et z
one
—36
—
that
pr
oduc
es th
e in
cons
iste
ncy,
bu
t th
e co
mbi
natio
n of
a c
redi
ble
targ
et z
one
and
a
part
icul
ar in
terv
entio
n ru
le t
hat
can
only
sup
port
the
targ
et z
one
if n
o bu
bble
exi
sts.
Our
alte
rnat
ive
inte
rven
tion
rule
, w
hich
has
a d
iscr
ete
(sto
ck—
shif
t) in
crea
se in
f
at
*1
1 *1
Il
fr
om f1
1 to
f11
and
a co
rres
pond
ing
mov
emen
t of
the
equi
libri
um fr
om
11
to
fli,
cons
iste
ntly
com
bine
s a
cred
ible
tar
get z
one
and
ratio
nal b
ubbl
es.
An
obvi
ous
ques
tion
wou
ld s
eem
to
be:
Why
wou
ld t
he a
utho
ritie
s ad
opt a
rule
that
per
mits
the
pres
ence
of
bubb
les
rath
er th
an a
rule
that
pre
clud
es t
heir
exi
sten
ce?
Putti
ng t
he q
uest
ion
this
way
pr
ejud
ges
the
ausw
er,
beca
use
it as
sum
es t
hat,
give
n an
inc
onsi
sten
cy b
etw
een
havi
ng a
cred
ible
tar
get z
one,
a ra
tiona
l bub
ble
and
a
part
icul
ar i
nter
vent
ion
polic
y,
the
inco
nsis
tenc
y m
ust
be re
solv
ed
by d
ropp
ing
the
bubb
le,
rath
er th
an b
y dr
oppi
ng th
e ta
rget
zon
e or
the
inte
rven
tion
polic
y.
We
pref
er
to v
iew
the
exis
tenc
e of
a b
ubbl
e as
a
: A
bub
ble
eith
er e
xist
s or
it
does
n't,
and
the
auth
oriti
es m
ust
desi
gn
thei
r in
terv
entio
n po
licy
acco
rdin
gly,
if
the
y w
ish
to
supp
ort t
he ta
rget
zon
e.
A p
artic
ular
inte
rven
tion
rule
(e.
g. i
nfin
itesi
mal
refl
ectin
g in
terv
entio
ns
at th
e
boun
dari
es
with
out
acco
mm
odat
ion
of f
rien
dly
spec
ulat
ive
atta
cks)
may
of
cour
se b
e
inco
nsis
tent
with
hav
ing
a cr
edib
le
targ
et z
one
in t
he p
rese
nce
of b
ubbl
es.
Thi
s
impl
ies,
in
our
view
, a
colla
pse
eith
er o
f th
e ta
rget
zon
e or
of
the
inte
rven
tion r
ule,
but
not
of t
he b
ubbl
e,
whi
ch i
s no
t an
obj
ect
of c
hoic
e at
som
e in
itial
dat
e, n
ot e
ven
a
colle
ctiv
e on
e.
Giv
en th
e ex
iste
nce
of a
bub
ble,
a cr
edib
le t
arge
t zon
e re
quir
es a
polic
y
rule
spe
cifi
catio
n th
at p
erm
its th
e bu
bble
and
the
targ
et z
one
to c
oexi
st.
Of c
ours
e it
mus
t als
o be
cap
able
of h
andl
ing t
he n
o—bu
bble
case
.
Our
rul
e is
an
exam
ple
of s
uch
a co
nsis
tent
polic
y.9
VII
. Con
clus
ions
The
the
ory
of e
xcha
nge
rate
beh
avio
r w
ithin
a t
arge
t zon
e as
dev
elop
ed i
n th
e
—37
—
rece
nt
liter
atur
e ho
lds
that
exc
hang
e ra
tes
unde
r a
curr
ency
ba
nd r
egim
e ar
e le
ss
resp
onsi
ve t
o fu
ndam
enta
l sho
cks
than
exc
hang
e ra
tes
unde
r fre
e fl
oat,
prov
ided
that
the
inte
rven
tion
rule
s of
the
Cen
tral
Ban
k(s)
are
com
mon
kno
wle
dge.
M
oreo
ver,
the
re
alw
ays
exis
ts a
tra
de—
off
betw
een
the
inst
anta
neou
s vo
latil
ity o
f ch
ange
s in
the
exch
ange
rat
es a
nd c
hang
es i
n th
e in
tere
st r
ate
diff
eren
tial,
inde
pend
ently
of th
e si
ze o
f
the
band
and
the d
egre
e of
cred
ibili
ty of
the
targ
et z
one.
The
se r
esul
ts a
re d
eriv
ed a
fter
havi
ng
assu
med
a
prio
ri th
at
"rat
iona
l ex
cess
vol
atili
ty"
(due
to
so
calle
d ra
tiona
l
bubb
les)
doe
s no
t occ
ur in
the
for
eign
exc
hang
e m
arke
t. Im
plic
itly
or e
xplic
itly,
it
is
assu
med
th
at s
pecu
lativ
e bu
bble
s ar
e in
com
patib
le
with
the
exi
sten
ce a
nd t
he
pers
iste
nce o
f a c
redi
ble t
arge
t zon
e, s
o th
at th
ey n
ever
mat
eria
lize.
We
cons
ider
inst
ead
a se
tup
in w
hich
the
exis
tenc
e of
spe
cula
tive
beha
vior
is
a
datu
m t
he C
entr
al B
ank
has
to d
eal
with
. W
e sh
ow t
hat
ther
e is
no
inco
mpa
tibili
ty
betw
een
the
exis
tenc
e of a
tar
get z
one
and
the
pres
ence
of
ratio
nal b
ubbl
es.
Rat
her,
th
ere
are
inte
rven
tion
rule
s th
at
shou
ld
be
follo
wed
by
th
e C
entr
al B
ank
whe
n
spec
ulat
ive
bubb
les
aris
e, a
nd th
ese
sam
e ru
les
incl
ude
as a
spec
ial
case
the
trad
ition
al
polic
ies
for d
efen
ding
an
exch
ange
rat
e ba
nd w
hen
spec
ulat
ive
bubb
les d
o no
t occ
ur.
In
the
stan
dard
m
odel
, th
e de
fens
e of
a
targ
et
zone
is
gu
aran
teed
by
inte
rmitt
ent
vari
atio
ns
in d
omes
tic c
redi
t (s
ay
thro
ugh
open
m
arke
t op
erat
ions
)
and/
or n
on s
teri
lized
fore
ign
mar
ket
inte
rven
tions
th
at t
ake
plac
e w
hen
the
exch
ange
rate
hits
one
of
the
limits
of
the
band
. For
inst
ance
, whe
n th
e ex
chan
ge r
ate
reac
hes
the
uppe
r bo
unda
ry, t
he s
tock
of f
orei
gn r
eser
ves
(or
the
stoc
k of
dom
estic
cre
dit)
is
redu
ced
in o
rder
to p
reve
nt th
e ex
chan
ge r
ate
from
ral
sing
fur
ther
. H
euri
stic
ally
, th
ese
oper
atio
ns
can
be c
hara
cter
ized
as i
nfin
itesi
mal
re
flec
ting
inte
rven
tions
. A
lthou
gh th
e
size
of
th
e re
flec
ting
oper
atio
ns
may
be
qua
ntita
tivel
y lim
ited,
th
e in
duce
d
expe
ctat
ions
stab
ilize
the
exch
ange
rat
e ev
en b
efor
e th
e up
per
or lo
wer
lim
it is
reac
hed.
We
show
tha
t in
the
cum
bub
ble
setu
p re
flec
ting
inte
rven
tions
are
insu
ffic
ient
.
In
fact
, w
hen
spec
ulat
ive
bubb
les
aris
e,
the
com
pani
on
phen
omen
a of
spe
cula
tive
— 3
8 —
atta
cks
on t
he t
arge
t zo
ne
mus
t oc
cur
as
wel
l. T
hese
sp
ecul
ativ
e at
tack
s ar
e
stoc
k—sh
ift r
eshu
ffle
s of
pri
vate
age
nts'
fina
ncia
l po
rtfo
lios
led
by ra
tiona
l exp
ecta
tions
of ch
ange
s in
the
rat
e of
exc
hang
e ra
te d
epre
ciat
ion.
A
s an
exa
mpl
e, if
the
rate
of
appr
ecia
tion
of th
e ex
chan
ge r
ate
wer
e su
dden
ly e
xpec
ted
to i
ncre
ase,
rat
iona
l pri
vate
agen
ts w
ould
in
crea
se
thei
r de
man
d fo
r ho
me
coun
try
mon
ey
and/
or d
ecre
ase
thei
r
dem
and
for f
orei
gn c
ount
ry m
oney
due
to
a de
crea
se i
n th
e do
mes
tic—
fore
ign
inte
rest
rate
dif
fere
ntia
l.
We
show
tha
t th
e de
fens
e of
the
targ
et z
one
in t
he p
rese
nce
of b
ubbl
es is
guar
ante
ed if
the
Cen
tral
Ban
k ac
com
mod
ates
spe
cula
tive
atta
cks
wbe
n th
e la
tter
are
frie
ndly
, th
at is
whe
n th
ey a
re c
onsi
sten
t with
the
sur
viva
l of
the
targ
et z
one
itsel
f
(gtv
en
the
Cen
tral
Ban
k's
role
).
Thi
s in
tuiti
ve p
olic
y ru
le
is c
ompa
tible
with
self
—fu
lfill
ing e
xpec
tatio
ns: a
fri
endl
y at
tack
occ
urs
only
if a
gent
s kno
w th
at i
t will
be
acco
mm
odat
ed,
and
exac
tly b
ecau
se
of t
he p
assi
ve
acco
mm
odat
ion
the
spec
ulat
ive
atta
ck s
nsta
in8
the
targ
et z
one.
H
ence
, th
e po
licy
rule
is
sum
mar
ized
by
the
max
im
"Ref
lect
w
hen
this
is s
uffi
cien
t; ac
com
mod
ate
whe
n th
is is
nec
essa
ry".
We
show
tha
t w
hen
the
exch
ange
ra
te h
its o
ne o
f th
e bo
unda
ries
w
ith
a
non—
zero
bub
ble,
an
acco
mm
odat
ed
frie
ndly
spe
cula
tive
atta
ck o
ccur
s, f
ollo
wed
by
refl
ectin
g in
terv
entio
ns.
Man
y of
the
con
clus
ions
re
ache
d in
the
exi
stin
g lit
erat
ure
do
not
appe
ar
suff
icie
ntly
rob
ust w
hen
spec
ulat
ive
bubb
les
are
cons
ider
ed a
s w
ell.
Firs
t, it
is n
ot tr
ue
anym
ore
that
the
ins
tant
aneo
us v
olat
ility
of
exch
ange
ra
tes
with
in a
tar
get
zone
is
alw
ays
less
tha
n th
e in
stan
tane
ous v
olat
ility
of e
xcha
nge
rate
und
er fr
ee f
loat
, pro
vide
d
that
, for
fam
iliar
reas
ons,
in
free
flo
at n
o sp
ecul
ativ
e bu
bble
ari
ses.
Se
cond
, the
find
ing
of a
con
stan
t tr
ade—
off
betw
een
the
inst
anta
neou
s va
riab
ility
of
the
chan
ge i
n th
e
exch
ange
ra
te a
nd t
he i
nsta
ntan
eous
var
iabi
lity
of t
he c
hang
e in
the
inte
rest
rat
e p
diff
eren
tial
is n
o lo
nger
auto
mat
ical
ly va
lid w
hen
ther
e is
a b
ubbl
e.
Mor
eove
r, t
he s
tand
ard
theo
ry c
hara
cter
izes
a s
tabl
e rel
atio
n be
twee
n th
e le
vel
—39
—
of th
e ex
chan
ge r
ate
and
the
expe
cted
rate
of
depr
ecia
tion
(equ
al to
the
int
eres
t rat
e
diff
eren
tial
if u
ncov
ered
int
eres
t par
ity ho
lds)
. A
ccor
ding
to
this
mod
el,
the
high
er th
e
exch
ange
ra
te,
the
low
er
the
inte
rest
ra
te d
iffe
rent
ial.
Mor
eove
r th
e in
tere
st r
ate
diff
eren
tial
is a
lway
s ne
gativ
e in
the
nei
ghbo
rhoo
d of
the
uppe
r bo
unda
ry o
f th
e
exch
ange
ra
te
band
; th
e op
posi
te r
esul
t ho
lds
in t
he n
eigh
borh
ood
of t
he l
ower
boun
dary
. W
e sh
ow th
at t
his
rela
tion
is n
o lo
nger
sta
ble
whe
n bu
bble
s ari
se.
—40
—
I
NO
TE
S
'Con
side
r a
dyna
mic
lin
ear
ratio
nal
expe
ctat
ions
m
odel
w
ith
cons
tant
coef
fici
ents
th
at h
as
the
usua
l sa
ddle
poin
t co
nfig
urat
ion
(as
man
y pr
edet
erm
ined
vari
able
s, n
1 sa
y,
as
stab
le c
hara
cter
istic
root
s an
d as
m
any
non—
pred
eter
min
ed
vari
able
s, n
2 sa
y, a
s un
stab
le c
hara
cter
istic
ro
ots)
. T
rans
form
the
syst
em to
can
onic
al
vari
able
s, b
y di
agon
aliz
ing
it or
by
usin
g Jo
rdan
's c
anon
ical
for
m.
Gro
up t
oget
her t
he
n2
dyna
mic
equ
atio
ns
cont
aini
ng
the
cano
nica
l va
riab
les
who
se
hom
ogen
eous
equa
tions
are
gov
erne
d by
th
e un
stab
le r
oots
. T
he
gene
ral
solu
tion
for
thes
e
non—
pred
eter
min
ed
cano
nica
l va
riab
les
can
cont
ain
an n2
—di
men
sion
al bu
bble
pro
cess
that
mus
t sa
tisfy
the
hom
ogen
eous
eq
uatio
n sy
stem
of
the
n2 n
on—
pred
eter
min
ed
cano
nica
l va
riab
les.
B
ubbl
e pr
oces
ses
will
th
eref
ore
be n
on—
stat
iona
ry
(in
expe
ctat
ion)
. T
he s
tate
var
iabl
es o
f th
e m
odel
, w
hich
are
linea
r co
mbi
natio
ns
of t
he
cano
nica
l va
riab
les,
will
, if
ther
e is
a
bubb
le,
be n
on—
stat
iona
ry.
If th
ere
are
mor
e
non—
pred
eter
min
ed
stat
e va
riab
les
than
uns
tabl
e ro
ots,
st
atio
nary
bub
bles
w
ill
of
cour
se b
e po
ssib
le e
ven
in li
near
mod
els.
2Sis
nila
r ar
gum
ents
ca
n be
m
ade
for
non—
stat
iona
ry
bubb
les
(and
th
e
non—
stat
iona
ry
beha
vior
of
ke
y en
doge
nous
va
riab
les
freq
uent
ly a
ssoc
iate
d w
ith
non—
stat
iona
ry
bubb
les)
in
cert
ain
non—
linea
r mod
els.
Fo
r in
stan
ce, i
n ov
erla
ppin
g
gene
ratio
ns
(OL
G)
mod
els
of a
co
mpe
titiv
e ec
onom
y w
ith
a si
ngle
pe
rish
able
com
mod
ity
and
with
no
n—in
tere
st—
bear
ing o
utsi
de m
oney
as
the
only
sto
re o
f val
ue,
ratio
nal d
efla
tiona
ry b
ubbl
es (
i.e.
non—
stat
iona
ry b
ubbl
es w
ith th
e pr
ice l
evel
dec
linin
g
to z
ero)
hav
e be
en s
how
n to
be
infe
asib
le.
Tak
e for
inst
ance
the
case
of
a tw
o—pe
riod
OL
G m
odel
with
a c
onst
ant p
opul
atio
n in
whi
ch th
e no
min
al m
oney
sto
ck is
con
stan
t,
—41
—
all m
oney
is
held
by
the
old
and
only
the
you
ng r
ecei
ve a
con
stan
t bou
nded
end
owm
ent
stre
am o
f th
e si
ngle
com
mod
ity. I
f the
pri
ce le
vel
wer
e to
fall
with
out b
ound
, rea
l cas
h
bala
nces
wou
ld b
e in
crea
sing
with
out b
ound
and
the
dem
and f
or th
e co
mm
odity
by t
he
old
gene
ratio
n w
ould
incr
ease
with
out b
ound
. Si
nce
the
supp
ly of
the
good
eac
h pe
riod
is b
ound
ed a
bove
by
the
sum
of
the
end
owm
ents
of
the
you
ng,
soon
er
or l
ater
the
dem
and
for
the
com
mod
ity m
ust
outs
trip
the
sup
ply.
A
seq
uenc
e of
mon
ey
pric
es
falli
ng t
o ze
ro c
an th
eref
ore
not
be r
atio
nal e
xpec
tatio
ns
equi
libri
um (s
ee e
.g.
Hah
n
[198
2, p.
10]
).
Unl
ess
very
st
rong
re
stri
ctio
ns a
re p
lace
d on
tbe
pri
vate
util
ity f
unct
ions
,
ratio
nal i
nfla
tiona
ry b
ubbl
es (w
ith t
he p
rice
lev
el i
ncre
asin
g w
ithou
t bou
nd d
espi
te a
cons
tant
nom
inal
mon
ey s
tock
) can
exi
st in
suc
h an
eco
nom
y.
The
seq
uenc
e of
risi
ng
pric
es w
ould
, in
the
limit,
dri
ve t
he r
eal v
alue
of m
oney
to
zero
. T
he s
tead
y st
ate
to
whi
ch s
uch
a m
odel
co
nver
ges
is t
hat
of a
non
—m
onet
ary
econ
omy.
T
o ru
le
out
infl
atio
nary
bub
bles
as
w
ell,
Obs
tfel
d an
d R
ogof
f [1
983]
, in
an
inf
inite
—liv
ed
repr
esen
tativ
e ag
ent
mod
el,
impo
sed
a po
litic
al—
tech
nolo
gica
l re
stri
ctio
n on
th
e
term
inal
val
ue
of m
oney
: th
e go
vern
men
t fr
actio
nally
ba
cks
the
curr
en,c
y by
guar
ante
eing
a m
inim
al r
eal r
edem
ptio
n va
lue
for
mon
ey.
Eve
n if
pri
vate
age
nts
are
not
com
plet
ely
cert
ain
that
th
ey c
an r
edee
m t
heir
mon
ey i
n an
y gi
ven
peri
od, t
his
suff
ices
to
rule
out
spe
cula
tive h
yper
infl
atio
ns.
Whi
le th
is a
ssum
ptio
n on
gove
rnm
ent
beha
vior
seem
s qu
ite a
d—ho
c,
it is
oft
en c
ited
as t
he s
econ
d bl
ade
of th
e sc
isso
rs t
hat
cut
the
lifel
ine
of n
on—
stat
iona
ry r
atio
nal
spec
ulat
ive
bubb
les,
bo
th d
efla
tiona
ry
and
infl
atio
nary
. D
iba
and
Gro
ssm
an
[198
8] a
lso
deri
ve s
uffi
cien
t co
nditi
ons
for r
ulin
g ou
t
non—
stat
iona
ry i
nfla
tiona
ry bu
bble
s.
3Sar
gent
an
d W
alla
ce's
"U
nple
asan
t Mon
etar
ist A
rith
met
ic"
mod
el
(Sar
gent
and
Wal
lace
[19
81])
can
exh
ibit
stat
iona
ry r
atio
nal
bubb
les.
E
ven
a fi
rst
orde
r
dete
rmin
istic
non
—lin
ear
diff
eren
ce
equa
tion
may
ex
hibi
t va
riou
s ki
nds
of p
erio
dic
— 4
2 —
solu
tions
or c
haot
ic b
ehav
ior
(see
e.g
. B
enha
bib
and
Day
[19
81,
1982
] an
d G
rand
mon
t
[198
5]).
St
atio
nary
bub
bles
ar
e ea
sily
gen
erat
ed
by s
uch
mod
els
(Aza
riad
is
[198
1],
Aza
riad
is a
nd G
uesn
erie
[1
986]
, C
hiap
pori
an
d G
uesn
erie
[1
988]
, W
oodf
ord
[198
7],
Farm
er a
nd
Woo
dfor
d [1
986]
).
Seco
nd o
rder
det
erm
inis
tic n
on—
linea
r di
ffer
entia
l
equa
tions
can
gen
erat
e lim
it cy
cles
and
thi
rd o
rder
non
—lin
ear
diff
eren
tial
equa
tions
can
exhi
bit
chao
tic b
ehav
ior.
A
gain
, su
ch m
odel
s ca
n su
ppor
t st
atio
nary
rat
iona
l
spec
ulat
ive b
ubbl
es.
4Tiii
s in
terp
reta
tion
of f
com
es f
rom
the
tw
o—co
untr
y m
ini—
mod
el o
utlin
ed
belo
w.
All
vari
able
s exc
ept
inte
rest
rat
es a
re in
nat
ural
log
arith
ms.
m
is
the
hom
e
coun
try
nom
inal
mon
ey s
tock
, p
the
hom
e co
untr
y pr
ice
leve
l, y
hom
e co
untr
y re
al
GD
P, i
th
e ho
me c
ount
ry sh
ort
nom
inal
int
eres
t ra
te a
nd
s th
e sp
ot p
rice
of
fore
ign
exch
ange
. C
orre
spon
ding
for
eign
cou
ntry
var
iabl
es a
re s
tarr
ed.
— p
= k
y —
i, k
, .1
> 0
(H
ome
mon
etar
y eq
uilib
rium
)
* *
* *
— p
= k
y —
X
i (F
orei
gn m
onet
ary
equi
libri
um)
* p
= p
+
(P
urch
asin
g Pow
er P
arity
)
Etd
s(t)
= [i
(t) —
i*(t
)]dt
(U
ncov
ered
Int
eres
t Par
ity)
From
this
ver
y si
mpl
e m
odel
we
obta
in t
he fo
llow
ing
rela
tion:
s(t)
dt =
]m(t
) — m
*(t)
— k(
y(t)
— y*
(t))
]dt +
AE
tds(
t).
Thi
s co
rres
pond
s to
equ
atio
n (1
) w
ith .
1 =
o (the
int
eres
t sem
i—el
astic
ity o
f
—43
—
* *
mon
ey d
eman
d) a
ndf=
m—
m
—k(
y—y
).
5Equ
atio
n (5
) is
of c
ours
e eq
uiva
lent
to
the
perh
aps
mor
e fa
mili
ar fo
rm
—
Af(
t)
—
A 1(
t)
s(t)
=fO
)+B
(t)+
+A
1(t)
e 1
+A
2(t)
e 2
whe
re A
1 an
d A
2 ar
e co
nsta
nt a
s lo
ng a
s s
is in
the
inte
rior
of t
he ta
rget
zon
e, b
ut
can
chan
ge w
hen
s is
at o
ne o
f its
bou
ndar
ies.
Fo
r m
any
bubb
les,
incl
udin
g th
e
exog
enou
s on
es, t
he r
epre
sent
atio
n gi
ven
in (
5) i
s at
trac
tive
beca
use i
t can
bri
ng n
ut
clea
rly
the
hori
zont
al sh
ift
nf t
he b
ubbl
e ea
ter
(def
ined
bel
ow)
whe
n th
e m
agni
tude
of
the
bubb
le v
arie
s.
6Whi
le t
he e
xpec
ted
rate
of
chan
ge o
f s
at th
e re
flec
tion
poin
ts o
n th
e up
per
boun
dary
(po
ints
like
or
flu
) is
al
way
s le
ss t
han
that
at
poin
ts l
ike
111,
the
*1
expe
cted
rate
of c
hang
e at
II
can,
for
ver
y sm
all
bubb
les,
be n
egat
sve.
7The
no
tion
of a
frie
ndly
or
sust
aini
ng
spec
ulat
ive
atta
ck i
s re
late
d to
the
"sus
tain
ing"
mon
ey
dem
and
by a
rbitr
ageu
rs in
one
of
the
solu
tions
to
the
"go
ld
stan
dard
pa
rado
x" p
ropo
sed
by B
uite
r an
d G
rilli
[1
989]
. Se
e al
so
Kru
gman
an
d
Rot
embe
rg 11
9901
.
8The
inf
imum
of f
is
whe
re th
e do
wnw
ard—
slop
ing p
art
of th
e bu
bble
eat
er f
or
Bu
cuts
the
upp
er b
ound
ary.
T
he s
upre
mum
of f
is
whe
re th
e do
wnw
ard—
slop
ing
part
of th
e bu
bble
eat
er fo
r B
1 cu
ts th
e lo
wer
bou
ndar
y.
sepa
rate
arg
umen
t th
at m
ight
be
mad
e ag
ains
t bu
bble
equ
ilibr
ia ta
kes
aim
—44
—
at it
s m
ost s
trik
ing
feat
ure:
th
e po
ssib
ility
of v
ery
freq
uent
int
erve
ntio
ns at
the
edge
s
of th
e zo
ne.
Con
side
r for
sim
plic
ity
a de
term
inis
tic bu
bble
tha
t st
arts
fro
m a
pos
itive
initi
al
valu
e.
It m
ight
be
argu
ed t
hat
the
inte
rven
tions
in
the
fund
amen
tal th
at a
re re
quir
ed
to o
ffse
t su
ch a
non
—st
atio
nary
bub
ble
are
not s
usta
inab
le be
caus
e, a
s th
e m
agni
tude
of
the
bubb
le g
row
s ov
er t
ime,
inte
rven
tions
at
the
uppe
r bou
ndar
y ca
n be
exp
ecte
d to
beco
me
mor
e an
d m
ore
freq
uent
. Fi
nite
inte
rnat
iona
l res
erve
s (a
ssum
ing
tbes
e ar
e th
e
inte
rven
tion m
ediu
m) a
re b
ound
to b
e ex
haus
ted i
n du
e co
urse
.
It s
houl
d be
not
ed th
at a
very
sim
ilar a
rgum
ent c
an b
e m
ade
even
if th
ere i
s no
bubb
le,
as l
ong
as th
ere
is p
ositi
ve dr
ift i
n th
e fu
ndam
enta
l. W
hile
the
expe
cted
va
lue
of t
he f
unda
men
tal g
row
s lin
earl
y (a
t a c
onst
ant r
ate
p) r
athe
r th
an e
xpon
entia
lly
as
the
bubb
le d
oes,
the
drif
t of
the
unre
gula
ted f
unda
men
tal w
ill a
lso
in fi
nite
tim
e ca
use
any
fini
te s
tock
of r
eser
ves
to b
e ex
haus
ted
with
pro
babi
lity
one.
Ind
eed,
eve
n w
ithou
t
drif
t in
the
fun
dam
enta
l (an
d w
ithou
t a
bubb
le),
a
stoc
hast
ic fu
ndam
enta
l pr
oces
s
driv
en b
y B
row
nian
mot
ion
will
bri
ng a
ny fi
nite
stoc
k of
res
erve
s do
wn
to a
ny p
ositi
ve
low
er b
ound
in fi
nite
tim
e w
ith p
roba
bilit
y one
(se
e B
uite
r [19
89])
.
The
pro
blem
s of
inte
rnat
iona
l res
erve
exh
aust
ion
crea
ted
by t
he b
ubbl
e ar
e of
cour
se
less
se
vere
if
the
bub
ble
is
a B
lanc
hard
bub
ble,
whi
ch h
as f
inite
exp
ecte
d
dura
tion.
Not
e al
so
that
in
the
mos
t co
mm
on
inte
rpre
tatio
n of
ou
r m
odel
, th
e
fund
amen
tal
f st
ands
for
rel
ativ
e ho
me
coun
try
mon
ey s
uppl
y m
inus
rel
ativ
e re
al
inco
me
— r
elat
ed m
oney
dem
and.
Tak
ing
rela
tive n
omin
al m
oney
sto
cks
as t
he o
bjec
t
of r
egul
atio
n, th
ere
is n
othi
ng
in t
he l
ogic
of
the
mod
el t
hat
requ
ires
in
tern
atio
nal
rese
rves
to
be u
sed
to re
gula
te th
e m
oney
sto
cks.
D
omes
tic c
redi
t ex
pans
ion
(whe
ther
refl
ectin
g op
en m
arke
t ope
ratio
ns o
r m
onet
ary
fina
ncin
g of
gov
ernm
ent
budg
et d
efic
its)
can
achi
eve t
he s
ame
mon
etar
y ob
ject
ives
.
—45
—
RE
FE
RE
NC
ES
Ave
sani
, R
enzo
C.
[199
0],
"End
ogen
ousl
y D
eter
min
ed
Tar
get
Zon
e an
d C
entr
al B
ank
Opt
imal
Pol
icy
With
Lim
ited R
eser
ves"
, m
imeo
, U
nive
rsity
of T
rent
o, J
une.
Aza
riad
is, G
. [1
9811
, "S
elf—
fulf
illin
g Pr
ophe
sies
",
Jour
nal
of E
cono
mic
T
heor
y, 2
5, p
p.
380—
396.
____
____
and
R. G
uesn
erie
[19
86],
"Su
nspo
ts a
nd C
ycle
s",
Rev
iew
of Ec
onom
ic S
tudi
es,
53,
pp. 7
87—
806.
Ben
habi
b, J
. an
d R
. D
ay
[198
1],
Rat
iona
l Cho
ice
and
Err
atic
Beh
avio
ur",
R
evie
w of
E
cono
mic
Stud
ies,
48,
pp.
459
—71
.
____
____
and
___
____
_ [1
982]
, A
Cha
ract
eriz
atio
n of
Err
atic
Dyn
amic
s in
the
Ove
rlap
ping
G
ener
atio
ns M
odel
", Jo
urna
l of E
cono
mic
Dyn
amic
s an
d C
ontr
ol,
, pp.
37—
55.
Ber
tola
, G
iuse
ppe
and
Ric
ardo
J.
Cab
alle
ro
[198
9],
"Tar
get
Zon
es a
nd
Rea
lignm
ents
",
Mim
eo,
Dec
embe
r.
____
____
and
___
____
[19
90],
"Res
erve
s an
d R
ealig
nmen
ts in
a T
arge
t Z
one"
, m
imeo
, Ju
ne.
Bla
ncha
rd,
Oliv
ier
J. [
1979
], "
Spec
ulat
ive
Bub
bles
, C
rash
es a
nd R
atio
nal E
xpec
tatio
ns",
E
cono
mic
Let
ters
, pp.
3 87
—9.
____
____
and
Sta
nley
Fis
cher
[19
89],
L
ectu
res
on M
acro
econ
omic
s, M
IT P
ress
, Cam
brid
ge,
Mas
sach
uset
ts.
Bui
ter,
Will
em II
. [19
89],
"A
Via
ble
Gol
d St
anda
rd R
equi
res
Flex
ible
Mon
etar
y an
d Fi
scal
Po
licy"
, Rev
iew
of E
cono
mic
Stu
dies
, 56
, Jan
uary
, pp.
101
—18
.
____
____
and
Vitt
orio
Gri
lli [
1989
], "
The
'Gol
d St
anda
rd P
arad
ox' a
nd I
ts R
esol
utio
n",
NB
ER
Wor
king
Pa
per N
o. 3
178,
Nov
embe
r.
Chi
appo
ri,
P.
and
R.
Gue
sner
ie
[198
6],
"End
ogen
ous
Fluc
tuat
ions
Und
er
Rat
iona
l E
xpec
tatio
ns",
Eur
opea
n Eco
nom
ic R
evie
w,
32, p
p. 3
89—
97.
Del
gado
, Fr
anci
sco
and
Ber
nard
Dum
as [
1990
], "
Mon
etar
y C
ontr
actin
g B
etw
een
Cen
tral
B
anks
And
The
Des
ign
of S
usta
inab
le
Exc
hang
e—R
ate
Zon
es",
mim
eo,
Mar
ch.
Dib
a, B
. T
. an
d H
. I.
Gro
ssm
an
[198
8],
"Rat
iona
l In
flat
iona
ry
Bub
bles
", J
ourn
al o
f M
onet
ary
Eco
nom
ics,
21,
pp.
35—
46.
Dix
it, A
vina
sh [
1988
], "
A S
impl
ifie
d E
xpos
ition
of
Som
e R
esul
ts
Con
cern
ing
Reg
ulat
ed
Bro
wni
an M
otio
n", W
orki
ng P
aper
, Pri
ncet
on U
nive
rsity
, A
ugus
t.
Dor
nbus
ch,
Rud
iger
[19
76],
"E
xpec
tatio
ns a
nd
Exc
hang
e R
ate
Dyn
amic
s",
Jour
nal
of
Polit
ical
Eco
nom
y, 8
, pp
. 11
61—
76.
46 —
Dum
as, B
erna
rd [1
989]
, Sup
er C
onta
ct a
nd R
elat
ed O
ptim
ality
Con
ditio
ns:
A S
uppl
emen
t to
Avi
nash
Dia
lt's
"A S
impl
ifie
d E
xpos
ition
of
Som
e R
esul
ts
Con
cern
ing
Reg
ulat
ed
Bro
wni
an M
otio
n", N
BE
R T
echn
ical
Wor
king
Pap
er N
o. 7
7, A
pril.
Farm
er, R
. and
M. W
oodf
ord
[198
6], "
Self
—fu
lfill
ing
Prop
hesi
es A
nd T
he B
usin
ess
Cyc
le",
C
AR
ESS
Dis
cuss
ion
Pape
r.
Floo
d, R
ober
t P. a
nd P
eter
M. G
arbe
r [19
89],
"T
he L
inka
ge B
etw
een
Spec
ulat
ive A
ttack
an
d T
arge
t Zon
e M
odel
s of
Exc
hang
e R
ates
", N
BE
R W
orki
ng P
aper
No.
291
8.
Froo
t, K
enne
th A
. an
d M
auri
ce
Obs
tfel
d [1
989a
],
"Sto
chas
tic P
roce
ss
Switc
hing
: So
me
Sim
ple
Solu
tions
", N
BE
R W
orki
ng P
aper
No.
299
8, J
une.
____
____
_ an
d __
____
___
[198
9b[,
"E
xcha
nge R
ate
Dyn
amic
s un
der
Stoc
hast
ic R
egim
e Sh
ifts
: A
Uni
fied
App
roac
h. m
imeo
, Ju
ne.
____
____
_ an
d __
____
__ [1
989c
], "I
ntri
nsic
Bub
bles
: T
he C
ase
of S
tock
Pr
ices
", N
BE
R
Wor
king
Pa
per N
o. 3
091,
Sep
tem
ber.
Gra
ndm
ont,
J. [
1985
], "
On
End
ogen
ous
Com
petit
ive
Bus
ines
s C
ycle
s",
Eco
nom
etri
ca,
53,
pp. 9
95—
105.
Hah
n, F
rank
[19
82],
Mon
ey a
nd I
nfla
tion,
Bas
il B
lack
wel
l Pu
blis
hers
, Oxf
ord.
Har
riso
n, J
. M
icha
el [
1985
], B
row
nian
Mot
ion
and
Stoc
hast
ic F
low
Sys
tem
s, J
ohn
Wile
y an
d So
ns,
New
Yor
k.
Ichi
kaw
a,
M. M
.H. M
iller
and
A.
Suth
erla
nd [1
990]
, "E
nter
ing
a Pr
eann
ounc
ed
Cur
renc
y B
and"
, War
wic
k E
cono
mic
Res
earc
h Pa
pers
, No.
347
, M
arch
.
Kar
atza
s I.
an
d S.
E
. Sh
reve
[1
988]
, B
row
nian
m
otio
n an
d st
ocha
stic
cal
culu
s,
Spri
nger
—V
erla
g, N
ew Y
ork.
Kle
in,
Mic
hael
W.
1989
[, "
Play
ing
With
The
Ban
d: D
ynam
ic E
ffec
ts O
f Tar
get Z
ones
In
An
Ope
n E
cono
my'
, mim
eo,
Aug
ust.
Kru
gman
, Pa
ul [
1987
], "
Tri
gger
Str
ateg
ies
and
Pric
e D
ynam
ics
in E
quity
and
For
eign
E
xcha
nge
Mar
kets
", N
BE
R W
orki
ng P
aper
No.
245
9, D
ecem
ber.
____
____
[198
9],
"Tar
get Z
ones
with
Lim
ited
Res
erve
s",
Dis
cuss
ion
Pape
r, M
IT.
[199
0],
"Tar
get
Zon
es a
nd
Exc
hang
e R
ate
Dyn
amic
s",
Qua
rter
ly J
ourn
al o
f E
cono
mic
s, f
orth
com
ing.
Kru
gman
, P.
and
J. R
otem
berg
[1
990]
, "T
arge
t Z
ones
With
Lim
ited
Res
erve
s",
mim
eo,
July
.
Lew
is, K
aren
K.
[199
0], O
ccas
iona
l In
terv
entio
ns to
Tar
get R
ates
with
a F
orei
gn E
xcha
nge
App
licat
ion"
, mim
eo,
N.Y
.U.
June
.
McC
aJlu
m,
Ben
nett
T.
ç198
3[,
"On
Non
—un
ique
ness
in R
atio
nal E
xpec
tatio
ns M
odel
s: A
n A
ttem
pt a
t Pe
rspe
ctiv
e,
' Jo
urna
l of M
onet
ary
Eco
nom
ics,
11, M
arch
, pp
. 139
—68
.
—47
—
Maf
fiar
is, A
. G
. an
d W
. A
. B
rock
[19
821,
St
ocha
stic
Met
hods
in E
cono
mic
s an
d Fi
nanc
e,
Nor
th—
Hol
land
, A
mst
erda
m.
Mill
er,
Mar
cus,
H.
and
Paul
Wel
ler
[198
8a],
"So
lvin
g St
ocha
stic
Sad
dlep
oint
Sys
tem
s:
A
Qua
litat
ive
Tre
atm
ent
with
Eco
nom
ic A
pplic
atio
ns",
W
arw
ick
Eco
nom
ic
Res
earc
h Pa
per
No.
309
(C
oven
try,
Eng
land
: Uni
vers
ity of
War
wic
k).
____
____
and
___
____
_ [1
988b
}, "T
arge
t Z
ones
, C
urre
ncy O
ptio
ns a
nd M
onet
ary
Polic
y",
snim
eo,
July
.
____
____
and
[1
989]
, "A
Qua
litat
ive
Ana
lysi
s of
Sto
chas
tic S
addl
epat
hs
and
its
App
licat
ion t
o E
xcha
nge R
ate R
egim
es",
snim
eo,
Uni
vers
ity of
War
wic
k.
____
____
an
d __
____
__
[199
0a],
"Cur
renc
y B
ubbl
es
Whi
ch A
ffec
t Fu
ndam
enta
ls:
A
Qua
litat
ive
Tre
atm
ent"
, The
Eco
nom
ic J
ourn
al, V
ol.
100,
No.
400
, Su
pple
men
t 19
90,
pp.
170—
179.
____
____
an
d __
____
__
[199
0b],
"Cur
renc
y B
ands
, T
arge
t Z
ones
an
d C
ash
Lim
its:
Thr
esho
lds f
or M
onet
ary
and
Fisc
al P
olic
y", C
EPR
Dis
cuss
ion
Pape
r N
o. 3
82, M
arch
.
____
____
an
d __
____
__
[199
0c],
Exc
hang
e R
ate
Ban
ds
with
Pri
ce I
nert
ia.
War
wic
k U
nive
rsity
, m
imeo
, Fe
brua
ry.
Mill
er,
M.H
. an
d A
. Su
ther
land
[199
0],
"Bri
tain
's R
etur
n to
Gol
d an
d Im
pend
ing
Ent
ry
into
the
EM
S: E
xpec
tatio
ns, J
oini
ng C
ondi
tions
and
Cre
dibi
lity"
, m
imeo
, Ju
ly.
____
____
__
____
__ a
nd J
ohn
Will
iam
son
[198
9],
"The
Sta
biliz
ing
Prop
ertie
s of
Tar
get
Zon
es",
in
Ral
ph C
. B
ryan
t, D
avid
A.
Cur
rie,
Ja
cob
A.
Fren
kel,
Paul
R.
Mas
son
and
Ric
hard
Po
rtes
ed
s.,
Mac
roec
onom
ic
Polic
ies
in
an I
nter
depe
nden
t W
orld
, 11
fF,
Was
hing
ton,
D.C
., pp
. 248
—27
1.
Obs
tfel
d, M
auri
ce a
nd K
enne
th R
ogof
f [1
983]
, "Sp
ecul
ativ
e H
yper
infl
atio
ns In
Max
imiz
ing
Mod
els:
Can
We
Rul
e T
hem
Out
?" J
ourn
al o
f Pol
itica
l Eco
nom
y, 9
1, p
p. 6
75—
687.
Pese
nti,
Paol
o A
. [1
990]
, "E
xcha
nge
Rat
e St
ocha
stic
D
ynam
ics
and
Tar
get
Zon
es:
An
Intr
oduc
tory
Sur
vey"
, m
imeo
, Ja
nuar
y.
Sarg
ent,
Tho
mas
J.
and
Nei
l W
alla
ce ]1
981]
, "S
ome
Unp
leas
ant
Mon
etar
ist A
rith
met
ic",
Fe
dera
l Res
erve
Ban
k of
Min
neap
olis
Qua
rter
ly R
evie
w,
5(3)
; pp
. 1—
17.
Smith
, Gre
gor
W.
and
Mic
hael
G.
Spen
cer
[199
0],
"Est
imat
ion
and
Tes
ting
in M
odel
s of
E
xcha
nge
Rat
e T
arge
t Zon
es a
nd P
roce
ss S
witc
hing
", m
imeo
, Ju
ly.
Sven
sson
, L
ars
E. 0
. [1
989]
, "T
arge
t Zon
es a
nd In
tere
st R
ate
Var
iabi
lity"
, NB
ER
Wor
king
Pa
per
No.
321
8, D
ecem
ber.
____
____
_ [1
990a
],
"The
Ter
m S
truc
ture
Of
Inte
rest
Rat
e D
iffe
rent
ials
In
A T
arge
t Zon
e:
The
ory
and S
wed
ish
Dat
a", m
imeo
, Feb
ruar
y.
____
____
[199
0b],
"The
Sim
ples
t Tes
ts o
f T
arge
t Zon
e C
redi
bilit
y", N
BE
R W
orki
ng P
aper
, N
o. 3
394,
Jun
e.
Woo
dfor
d,
M.
[198
7],
"Lea
rnin
g T
o B
elie
ve I
n Su
nspo
ts",
m
imeo
, U
nive
rsity
of C
hica
go
Gra
duat
e Sc
hool
of
Bus
ines
s.
