· evidences results eci ca- costs. thatl and w del ion l t (1 1) l + 1 l t 1 w t 1 b w 2 ˆ w...
TRANSCRIPT
Macr
oeco
nom
ics
M1
2015–2016
Pat
rick
FE
VE
Ad
dre
ss:
Tou
lou
seS
choo
lof
Eco
nom
ics,
Un
iver
site
de
Tou
lou
se1,
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-Jac
ques
Laff
ont,
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508,
21A
llee
de
Bri
enn
e,31
000,
Tou
lou
se,
Fra
nce
.
emai
l:p
atri
ck.f
eve@
tse-
fr.e
u
Affi
liati
on:
Tou
lou
seS
choo
lof
Eco
nom
ics
(Un
iver
sity
ofT
oulo
use
I–C
apit
ole,
GR
EM
AQ
,ID
EI
and
IUF
)
Ver
yP
relim
inar
y(a
nd
Inco
mp
lete
)N
otes
Man
yT
ypos
.
Slid
eson
my
web
pag
eat
TS
E
http://www.tse-fr.eu/fr/people/patrick-feve
Intr
oduct
ion
Ob
ject
ives
of
this
cours
e:
•N
ota
cou
rse
inD
ynam
icP
rogr
amm
ing;
Not
aco
urs
ein
Tim
eS
erie
sE
con
omet
-
rics
;N
ota
cou
rse
inT
heo
reti
cal
Mac
roec
onom
ics;
Not
aco
urs
ein
Com
pu
ta-
tion
alE
con
omic
s;Ju
sta
Sim
ple
Mix
ofb
oth
!!!
•O
rgan
izat
ion
ofth
eM
acro
cou
rse
inM
1:on
ecl
ass
”Ad
van
ced
Mac
rofo
rth
e
doc
tora
ltra
ck”
(Rob
ert
Ulb
rich
),on
ecl
ass
of”A
dva
nce
dM
acro
”(P
atri
ckF
eve)
.
•10
tuto
rial
s.
•T
his
cou
rse
isn
ot(t
oom
uch
!!)te
chn
ical
(mor
ead
van
ced
cou
rses
inM
1M
acro
-
Doc
tora
lan
dM
2M
acro
1an
d2
cou
rses
inth
ed
octo
ral
trac
kof
TS
E).
•T
he
dyn
amic
opti
miz
atio
np
rob
lem
san
dth
ed
ynam
icge
ner
aleq
uili
bri
um
mod
-
els
are
con
sid
ered
inth
eir
sim
ple
stfo
rms.
•T
his
cou
rse
trie
sto
cove
rth
em
ajo
rfie
lds
ofm
oder
nm
acro
econ
omic
s(D
ynam
ic
Sto
chas
tic
Gen
eral
Equ
ilib
riu
mm
odel
s,la
bel
edn
owD
SG
Em
odel
s).
Org
aniz
ati
on
of
this
cours
e:
Th
ree
mai
nch
apte
rs(i
fw
eh
ave
enou
ghti
me!
!)
Chap
1:
Tw
oP
eri
od
Equilib
rium
Models
1.1
:A
nexch
an
ge
Eco
nom
y
•T
he
mod
el
•E
quili
bri
um
•C
entr
alp
lan
ner
pro
ble
m
•D
iscu
ssio
n
1.2
:A
Pro
du
ctio
nE
con
om
y
•T
he
mod
el
•E
quili
bri
um
•C
entr
alp
lan
ner
pro
ble
m
•D
iscu
ssio
n
Chap
2:
Models
and
Meth
ods
inM
acr
oeco
nom
icD
y-
nam
ics
2.1
:T
he
Perm
an
ent
Inco
me
Mod
el
•T
he
det
erm
inis
tic
case
•T
he
stoc
has
tic
case
2.2
:T
he
Dyn
am
icL
ab
or
Dem
an
dM
od
el
•T
he
det
erm
inis
tic
case
•T
he
stoc
has
tic
case
Chap
3:
Busi
ness
Cycl
eM
odels
3.1
:T
wo
Sim
ple
DS
GE
Mod
els
I-
Hab
itForm
ati
on
ina
Sim
ple
Equ
ilib
riu
mM
od
el
•T
he
mod
el
•S
olu
tion
•Q
uan
tita
tive
imp
licat
ion
s
II-
AS
toch
ast
icG
row
thM
od
el
•T
he
mod
el
•S
olu
tion
•Q
uan
tita
tive
imp
licat
ion
s
3.2
:R
BC
Mod
els
I-
AR
BC
Mod
el
wit
hC
om
ple
teD
ep
reci
ati
on
•T
he
mod
el
•S
olu
tion
•Q
uan
tita
tive
imp
licat
ion
s
II-
Ap
roto
typ
ical
RB
CM
od
el
(Op
tion
al)
•T
he
mod
el
•S
olu
tion
•Q
uan
tita
tive
imp
licat
ion
s
3.3
:B
eyon
dth
eR
BC
Mod
el
(Op
tion
al)
inP
rogre
ss
Rule
sof
the
gam
eof
this
cours
e:
•T
his
cou
rse
doe
sn
otre
quir
edad
dit
ion
alm
ater
ials
,i.
e.th
isco
urs
eis
alm
ost
self
cont
ain
ed.
•B
IBL
IOG
RA
PH
Y:
Dav
idR
omer
Ad
van
ced
Mac
roec
onom
ics
4th
Ed
itio
n,
Mc-
Gra
wH
ill,
2012
(see
chap
ters
5,7,
8,9,
12)
•A
pp
licat
ion
san
dex
erci
ses
are
cont
ain
edin
the
hom
ewor
ks:
slig
htm
odifi
cati
ons
(sim
ple
ran
d/o
rex
ten
sion
s)of
the
mod
els
exp
oun
ded
inth
eco
urs
e.
•A
mid
–ter
mex
amin
atio
n(T
wo
exer
cise
son
mac
roec
onom
icd
ynam
ics)
:40
%
•A
final
exam
inat
ion
:T
wo
exer
cise
son
mac
roec
onom
icd
ynam
ics:
60%
.
Th
equ
anti
tati
veap
pro
ach
toE
con
omic
Dyn
amic
s
1.F
orm
ula
tea
dyn
amic
(det
erm
inis
tic
orst
och
asti
c)m
odel
(par
tial
equ
ilib
riu
m–
hou
seh
old
s,fir
ms
–or
gen
eral
equ
ilib
riu
m):
obje
ctiv
es,
con
stra
ints
,ex
ogen
ous
dri
vin
gfo
rces
.
2.D
eter
min
eth
eop
tim
ald
ecis
ion
sto
the
asso
ciat
edd
ynam
icop
tim
izat
ion
pro
b-
lem
.
3.C
olle
ctal
lth
ere
leva
nteq
uat
ion
sth
atfu
llych
arac
teri
zeth
ein
tert
emp
oral
equ
i-
libri
um
.
4.S
olve
the
mod
elan
alyt
ical
ly(i
nth
isco
urs
e)or
num
eric
ally
ifto
oco
mp
lica
ted
(not
inth
isco
urs
e,se
em
yre
mar
kb
elow
).
5.E
stim
ate
and
test
the
mod
elfr
omac
tual
dat
a.If
the
mod
elfa
ils
(poo
rfit
of
the
dat
a),
goto
step
1an
dre
do
the
exer
cise
.If
not
,go
toth
en
ext
step
.
6.S
imu
late
the
mod
elto
eval
uat
eit
sd
ynam
ics
pro
per
ties
7.C
ond
uct
cou
nter
fact
ual
and
pol
icy
exp
erim
ents
(fisc
alan
d/o
rm
onet
ary
pol
icy
shoc
ks;
chan
gein
inst
itu
tion
san
dre
gula
tion
,...
)
Th
isco
urs
eon
lyd
eliv
ers
anal
ytic
alre
sult
sfr
om(v
ery
orm
ayb
eto
o)si
mp
led
ynam
ic
(det
erm
inis
tic
and
stoc
has
tic)
mod
els.
Mor
ere
alis
tic
case
sca
nb
eso
lved
inth
esa
me
way
bu
tth
eyn
eed
com
pu
ter
inte
nsi
ve
tech
niq
ues
inor
der
toge
tso
luti
ons,
esti
mat
ion
san
dsi
mu
lati
ons.
Th
ere
exis
tsn
owa
bu
nch
ofso
ftw
ares
than
can
be
use
dto
solv
ean
dsi
mu
late
dyn
amic
stoc
has
tic
gen
eral
(or
par
tial
)eq
uili
bri
um
mod
els.
See
DY
NA
RE
(fre
e
dow
nlo
adu
nd
erM
AT
LA
Bp
rogr
amm
ing)
amon
gm
any
oth
ers
(to
obta
init
,ju
st
wri
teth
en
ame
”DY
NA
RE
”inGoogle
.It
wil
lap
pea
rfir
st.)
,b
ut
we
pre
fer
tow
ork
wit
hsi
mp
ler
mod
els
bef
ore
any
imp
lem
enta
tion
sw
ith
this
use
ful
free
soft
war
e).
Ifyo
uar
eve
ryin
tere
stin
gin
this
soft
war
ean
dth
ew
ays
toim
ple
men
tou
rm
odel
econ
omie
sin
this
setu
p,
do
not
hes
itat
eto
ask
me
the
cod
es!
Warn
ing!!
!T
his
not
esm
ayco
ntai
nm
any
(Mat
hem
atic
alan
dE
ngl
ish
)ty
pos
!!!
Chap
2:
AT
wo
Peri
od
Equilib
rium
Model
2.1
:A
nexch
ange
Eco
nom
y
•T
he
mod
el
•E
quili
bri
um
•C
entr
alp
lan
ner
pro
ble
m
•D
iscu
ssio
n
2.2
:A
Pro
duct
ion
Eco
nom
y
•T
he
mod
el
•E
quili
bri
um
•C
entr
alp
lan
ner
pro
ble
m
•D
iscu
ssio
n
Chap
2:
AT
wo
Peri
od
Equilib
rium
Model
2.1
:A
nexch
ange
Eco
nom
y
The
Model
Ave
rysi
mp
lese
tup
two
per
iod
s
asi
ngl
e(r
epre
sent
ativ
e)co
nsu
mer
An
exch
ange
(en
dow
men
t)ec
onom
y
Old
idea
(Irv
ing
Fis
her
):th
eal
loca
tion
ofgo
ods
over
tim
e⇔
allo
ca-
tion
ofre
sou
rces
tod
iffer
ent
good
sat
agi
ven
tim
e
So
good
sat
diff
eren
td
ates
are
con
sid
ered
asd
iffer
ent
com
mod
itie
s.
Ele
men
tsof
the
Exc
han
geE
con
omy:
1)T
wo
com
mod
itie
s,d
enot
edC
1an
dC
2.
2)E
nd
owm
ents
ofth
ese
com
mod
itie
s:Y
1an
dY
2
An
imp
orta
ntp
oint
:Y
isnon
stora
ble
3)P
refe
ren
ces
ofth
ere
pre
sent
ativ
eco
nsu
mer
rep
rese
nted
byth
efo
l-
low
ing
uti
lity
fun
ctio
n
U(C
1,C
2)
We
will
spec
ify
this
uti
lity
fun
ctio
nla
tter
Th
eel
emen
ts1)
and
2)ch
arac
teri
zeth
efe
asib
leal
loca
tion
s
C1≤Y
1an
dC
2≤Y
2
Inad
dit
ion
,w
ear
ein
ad
eter
min
isti
cca
se.
Th
eag
ent
per
fect
lykn
ows
the
quan
titi
esin
seco
nd
per
iod
.
Tw
oim
por
tant
ques
tion
sof
inte
rest
:
i)H
owth
eco
nsu
mer
allo
cate
sco
nsu
mp
tion
over
tim
e?
ii)W
hat
are
the
det
erm
inan
tsof
pri
ces
and
quan
titi
es?
The
com
peti
tive
Equilib
rium
We
will
char
acte
rize
the
equ
ilib
riu
mal
loca
tion
sin
aco
mp
etit
ive
setu
p
(pri
ces
are
give
n)
We
den
oteP
1an
dP
2th
ep
rice
tod
ayan
dto
mor
row
,re
spec
tive
ly.
Th
ein
tert
emp
oral
bu
dge
tco
nst
rain
tis
give
nby
P1C
1+P
2C2≤P
1Y1
+P
2Y2
Inth
isec
onom
y,th
ep
rice
sP
1an
dP
2ad
just
such
that
Ct
=Yt∀t
=1,
2
Definit
ion
of
the
com
peti
tive
equilib
rium
Aco
mp
etit
ive
equ
ilib
riu
mis
asy
stem
ofp
rice
s{P
1,P
2}an
dal
loca
-
tion
s{C
1,C
2}su
chth
at:
1.T
he
(rep
rese
ntat
ive)
con
sum
erm
axim
izes
the
uti
lity
U(C
1,C
2)
sub
ject
toth
eb
ud
get
con
stra
int
P1C
1+P
2C2≤P
1Y1
+P
2Y2
forP
1an
dP
2gi
ven
.
2.S
up
ply
ofgo
ods
equ
als
dem
and
ofgo
ods
each
per
iod
:
C1
=Y
1,
C2
=Y
2
Applica
tion
Let
us
assu
me
the
follo
win
gu
tilit
yfu
nct
ion
U(C
1,C
2)=
log(C
1)+β
log(C
2)
Th
islo
g–u
tilit
yis
sep
arab
le(m
ore
gen
eral
,n
otse
par
able
,H
abit
for-
mat
ion
inco
nsu
mp
tion
,se
eb
elow
)
Th
ep
aram
eterβ>
0is
asu
bje
ctiv
ed
isco
unt
fact
or.
β=
1
1+θ
wh
ereθ>
0is
the
sub
ject
ive
ofp
refe
ren
cefo
rth
ep
rese
nt.
Th
eh
ouse
hol
dp
rob
lem m
axC
1,C
2
log(C
1)+β
log(C
2)
sub
ject
toth
ein
tert
emp
oral
bu
dge
tco
nst
rain
t
P1C
1+P
2C2
=P
1Y1
+P
2Y2
Solu
tion:
first
we
rep
lace
the
inte
rtem
por
alb
ud
get
con
stra
int
C2
=P
1
P2Y
1+Y
2−P
1
P2C
1
into
the
obje
ctiv
e
log(C
1)+β
log
( P 1 P2Y
1+Y
2−P
1
P2C
1)an
dth
enm
axim
ize
wit
hre
spec
ttoC
1.
FO
C
1 C1−βP
1
P2
1 C2
=0
oreq
uiv
alen
tly
β/C
2
1/C
1=P
2
P1
i.e.
the
mar
gin
alra
teof
sub
stit
uti
onof
con
sum
pti
onb
etw
een
two
per
iod
sis
equ
alto
the
rela
tive
pri
ce.
Now
,u
seth
isop
tim
alit
yco
nd
itio
nto
exp
ressC
2:
C2
=βP
1
P2C
1
(rm
k:if
the
equ
ilib
riu
mis
such
thatβP
1P
2=
1,p
erfe
ctco
nsu
mp
tion
smoo
thin
g)
Now
,re
pla
ceC
2in
toth
eb
ud
get
con
stra
int
P1C
1+P
2C2
=P
1Y1
+P
2Y2
P1C
1(1
+β
)=P
1Y1
+P
2Y2
Thu
s,w
eob
tain
C1
=1
1+β
( Y1
+P
2
P1Y
2)N
ow,
usi
ng
the
opti
mal
ity
con
dit
ion
,w
ed
edu
ceC
2
C2
=β
1+β
( P 1 P2Y
1+Y
2)T
he
exp
ress
ion
sfo
rC
1an
dC
2d
eter
min
eth
etw
od
eman
dfu
nct
ion
s.
Now
usi
ng
thes
etw
od
eman
dfu
nct
ion
san
dm
arke
tcl
eari
ng
con
di-
tion
ofgo
odm
arke
tin
each
per
iodY
1=C
1an
dY
2=C
2(S
up
-
ply
=D
eman
d),
we
can
ded
uce
the
(rel
ativ
e)eq
uili
bri
um
pri
ces
Y1
=1
1+β
( Y1
+P
2
P1Y
2)⇔
P2
P1
=βY
1
Y2
Not
eth
atw
eob
tain
exac
tly
the
sam
eeq
uili
bri
um
pri
ceif
we
use
the
con
dit
ion
Y2
=β
1+β
( P 1 P2Y
1+Y
2)
Rm
k1:
Th
ele
vel
ofco
nsu
mp
tion
each
per
iod
isd
irec
tly
ded
uce
d
from
the
reso
urc
eco
nst
rain
t(Ct
=Yt,t
=1,
2).
Th
isis
bec
ause
the
good
isp
eris
hab
le(i
tca
nn
otb
est
ored
orin
vest
ed,
we
will
inve
stig
ate
this
case
bel
ow).
So,
why
con
sid
erin
gan
inte
rtem
por
alm
odel
rath
erth
ana
rep
eate
d
stat
icm
odel
?(i
.e.
max
uti
lity
inp
erio
d1,
max
uti
lity
inp
erio
d2)
.
Th
isis
bec
ause
the
mod
elal
low
sto
det
erm
ine
first
the
inte
rtem
por
al
allo
cati
ons
forP
1an
dP
2gi
ven
atth
eh
ouse
hol
dle
vel
and
seco
nd
,
give
nth
eeq
uili
bri
um
con
dit
ion
son
good
mar
ket,
ital
low
sto
det
er-
min
eth
ep
rice
s.
Infa
ct,
for
agi
ven
spec
ifica
tion
ofu
tilit
y,th
isis
am
odel
ofin
tert
em-
por
alp
rice
sd
eter
min
atio
n(r
eal
inte
rest
rate
oras
set
pri
ces)
.
Am
ore
gen
eral
mod
elw
ith
ast
orag
ete
chn
olog
y(p
hysi
cal
cap
ital
)
will
pro
du
cesi
mila
rre
sult
s(t
he
com
pu
tati
onof
quan
titi
esis
mor
e
com
plic
ated
,se
eb
elow
).
Rm
k2:
Th
em
odel
can
not
det
erm
ineP
1an
dP
2se
par
atel
y.(t
he
two
dem
and
fun
ctio
ns
give
the
sam
ere
lati
vep
rice
s).
Th
em
odel
only
det
erm
ines
the
rela
tive
pri
cesP
1/P
2(o
rP
2/P
1).
To
see
this
,m
ult
iply
bot
hP
1an
dP
2by
asc
ale
fact
orλ>
0:
P1
=λP
1P
2=λP
2
You
will
obta
inth
esa
me
allo
cati
onfo
rC
1an
dC
2an
dth
esa
me
equ
ilib
riu
m.
So,
we
can
free
lych
oose
one
pri
cear
bit
rari
ly,
sayP
1=
1(t
he
nu-
mer
aire
)
Th
ism
ean
sth
atth
ese
con
dp
rice
ism
easu
red
inte
rms
ofu
nit
ofth
e
first
good
.
SoP
2/P
1=P
2is
the
inte
rtem
por
alp
urc
has
ing
pow
erof
the
con
-
sum
er.
On
eu
nit
ofco
nsu
mp
tion
tom
orro
wco
stsP
2u
nit
ofco
nsu
mp
-
tion
tod
ay.
Fin
ally
,th
eal
loca
tion{C
1=Y
1;C
2=Y
2}an
dth
ep
rice
s{P
1=
1;P
2=β
(Y1/Y
2)}
con
stit
ute
aco
mpe
titi
veeq
uil
ibri
um
ofth
eec
on-
omy.
The
Centr
al
Pla
nner
Pro
ble
m
Inth
isec
onom
y(w
ith
out
dis
tort
ion
s),t
he
two
wel
fare
theo
rem
sap
ply
.
1)th
eco
mp
etit
ive
allo
cati
ons
(C1,C
2)ar
eth
ose
obta
ined
bya
cent
ral
pla
nn
er
2)T
he
opti
mal
allo
cati
ons
can
be
sup
por
ted
asa
com
pet
itiv
eeq
ui-
libri
um
(wit
ha
pro
per
syst
emof
pri
ces)
Th
ece
ntra
lp
lan
ner
mu
stso
lve
max
C1,C
2
log(C
1)+β
log(C
2)
sub
ject
toth
etw
ob
ud
get
con
stra
ints
C1≤Y
1C
2≤Y
2
Not
eth
atth
ere
ish
ere
no
pri
ce,
bec
ause
the
cent
ral
pla
nn
erfu
lly
inte
rnal
izes
all
the
mar
ket
con
dit
ion
s.
Th
eso
luti
on
C1
=Y
1an
dC
2=Y
2
So
we
obta
inth
esa
me
allo
cati
ons.
Now
defi
ne
the
mar
gin
alu
tili
ties
∂U
∂C
1=
1 C1
,∂U
∂C
2=
β C2
Nex
tco
mp
ute
the
mar
gin
alra
teof
sub
stit
uti
on
∂U
∂C
2∂U
∂C
1
=βY
1
Y2
Ina
com
pet
itiv
eeq
uili
bri
um
,w
eh
ave
∂U
∂C
2∂U
∂C
1
=βY
1
Y2≡P
2
P1
So
the
two
wel
fare
theo
rem
sap
ply
.
Dis
cuss
ion
1-
The
real
inte
rest
rate
For
the
mom
ent,
we
hav
eco
nsi
der
edtw
op
rice
sP
1an
dP
2,b
ut
the
mod
elca
nb
eu
sed
tod
eter
min
eot
her
form
ofin
tert
emp
oral
pri
ces,
i.e.
the
real
inte
rest
rate
.
Wit
ha
slig
htm
odifi
cati
on,
we
can
det
erm
ine
the
equ
ilib
riu
mre
al
inte
rest
rate
.
Fir
stP
erio
dL
etu
sd
efin
eh
owth
eh
ouse
hol
dca
nsa
ve
S=Y
1−C
1
wh
ereS
den
otes
savi
ng
(S>
0)or
bor
row
ing
(S<
0).
Sec
ond
Per
iod
Con
sum
eth
ed
isp
osab
lein
com
e(1
+r)S
+Y
2,
C2
=(1
+r)S
+Y
2
wh
erer
den
otes
the
real
inte
rest
rate
.
Now
,m
axim
ize
the
inte
rtem
por
alu
tilit
y
max
C1,C
2
log(C
1)+β
log(C
2)
wit
hre
spec
tto
the
con
stra
intsC
1=Y
1−S
andC
2=
(1+r)S
+Y
2.
Not
eth
atw
eca
nge
tth
ein
tert
emp
oral
bu
dge
tco
nst
rain
tfr
omtw
o
abov
eco
nst
rain
ts
C2
=(1
+r)
(Y1−C
1)+Y
2
Now
,su
bst
itu
teth
isco
nst
rain
tin
toth
eob
ject
ive
max C1
log(C
1)+β
log(
(1+r)
(Y1−C
1)+Y
2)
and
max
imiz
ew
ith
resp
ect
toC
1.
FO
C
1 C1−β
(1+r)
1 C2
=0
oreq
uiv
alen
tly
1 C1
=β
(1+r)
1 C2
Th
isis
the
Eule
requati
on
onco
nsu
mp
tion
.
Now
,u
seth
eeq
uili
bri
um
con
dit
ion
s(d
eman
dof
good
s=su
pp
lyof
good
s)
C1
=Y
1,
C2
=Y
2
Not
eth
atat
equ
ilib
riu
m,
savi
ng
iseq
ual
toze
ro(S
=0)
.
Th
isal
low
sto
det
erm
ine
the
equ
ilib
riu
mre
alin
tere
stra
te
1
1+r
=βY
1
Y2
Inth
ep
revi
ous
mod
elve
rsio
n,
we
obta
ined
P2
P1
=βY
1
Y2
So,
we
ded
uce
1+r
=P
1
P2
Wit
hth
en
orm
aliz
atio
nP
1=
1,
1+r
=1 P2
Rm
k:
IfY
1=Y
2,w
eh
ave 1
1+r
=β≡
1
1+θ
so
r=θ
Tw
om
ain
det
erm
inan
tsfo
rr:
1)p
refe
ren
ces
(βorθ)
2)en
dow
men
ts
(Y1
andY
2)
2-S
toch
ast
icendow
ments
We
hav
eas
sum
edp
erfe
ctfo
resi
ght
ina
det
erm
inis
tic
setu
p.
Wh
ath
app
ens
ifY
isst
och
asti
c?
Inth
isca
se,
the
con
sum
erm
ust
exp
ect
futu
reY
and
thu
sfu
ture
con
sum
pti
on.
Am
odifi
cati
onof
our
setu
p:
Per
iod
1
Sav
ing
A=Y
1−C
1
Per
iod
2
C2
=Y
2+
(1+r)A
To
sim
plif
y,su
pp
ose
that
ther
eis
no
un
cert
aint
yon
the
retu
rnr
(a
finan
cial
cont
ract
sth
atd
eliv
erth
esa
me
retu
rnw
hat
ever
the
stat
eof
the
nat
ure
).S
o,r
isgi
ven
.
Y1
andY
2ar
en
owst
och
asti
cla
bor
inco
mes
inp
erio
d1
and
2.
Th
eE
ule
req
uat
ion
onco
nsu
mp
tion
bec
omes
1 C1
=βE
1
( (1+r)
1 C2
)w
her
eE
1is
the
exp
ecta
tion
con
dit
ion
alon
the
info
rmat
ion
set
in
per
iod
1,i.e
.w
hen
agen
tsm
ust
take
thei
rd
ecis
ion
s.
Ass
um
eth
atβ
(1+r)
=1.
Th
eE
ule
req
uat
ion
rew
rite
s
1 C1
=E
1
( 1 C2
)B
yth
eJe
nse
nin
equ
alit
y(L
etf
(x)
aco
nvex
fun
ctio
nan
dx
ara
nd
om
vari
able
,th
enE
(f(x
))≥f
(E(x
))or
sim
ply
show
itfr
oma
figu
re),
we
hav
e
E1
( 1 C2
) ≥(
1
E1(C
2)
)T
his
imp
lies
that
the
con
sum
pti
onw
illlo
wer
wit
hst
och
asti
cin
com
e,
i.e.
con
sum
ers
try
tosa
vem
ore
(cal
led
as“p
reca
uti
onar
y”sa
vin
g).
3-
Lab
or
supply
We
omit
anim
por
tant
vari
able
for
the
hou
seh
old
:la
bor
sup
ply
We
hav
eas
sum
edth
atla
bor
isin
elas
tic.
Her
e,w
ere
lax
this
assu
mp
tion
(all
owin
gfo
rel
asti
cla
bor
sup
ply
)in
atw
o–p
erio
dm
odel
(par
tial
equ
ilib
riu
m).
Tw
oop
pos
ite
forc
esth
atd
ive
lab
orsu
pp
ly:
sub
stit
uti
oneff
ect
and
wea
lth
effec
t.
Th
esu
bst
itu
tion
effec
t:L
abor
sup
ply
incr
ease
saf
ter
anin
crea
sein
real
wag
e.
Wea
lth
effec
t:L
abor
sup
ply
dec
reas
esaf
ter
anin
crea
sein
real
wag
e.
Uti
lity
fun
ctio
n
U(C
1,C
2,N
1,N
2)=
log(C
1)+β
log(C
2)+
log(
1−N
1)+β
log(
1−N
2)
wh
ereN
1an
dN
2re
pre
sent
lab
orsu
pp
lyin
per
iod
1an
d2,
resp
ec-
tive
ly.
Tim
een
dow
men
tis
nor
mal
ized
toon
e.
Th
ein
tert
emp
oral
bu
dge
tco
nst
rain
t
C1
+C
2
R≤W
1N1
+W
2N2
R
wh
ereR
=1
+r>
1an
dr
isth
ere
alin
tere
stra
te.
Now
,fo
rmth
eL
agra
ngi
an
L=U
(C1,C
2,N
1,N
2)−λ
( C1
+C
2
R−W
1N1−W
2N2
R
)w
her
eλ≥
0is
the
Lag
ran
gem
ult
ipli
er.
Max
imiz
eth
isL
agra
ngi
anw
ith
resp
ect
toC
1,C
2,N
1an
dN
2
FO
Cs
1/C
1−λ
=0
β/C
2−λ/R
=0
−1/
(1−N
1)+λW
1=
0
−β/(
1−N
2)+λW
2/R
=0
and
the
excl
usi
onre
stri
ctio
n
λ
( C1
+C
2
R−W
1N1−W
2N2
R
) =0
Not
eth
atλ>
0(b
ecau
seλ
=1/C
1)an
dth
enth
ein
tert
emp
oral
bu
dge
tco
nst
rain
tb
ind
s
C1
+C
2
R=W
1N1
+W
2N2
R
We
imp
oseβR
=1.
We
then
ded
uce
C2
=C
1
N1
=1−C
1/W
1
N2
=1−C
1/W
2
Nex
tsu
bst
itu
teC
2,N
1an
dN
2in
toth
eb
ud
get
con
stra
int.
We
obta
in
C1
=R
2(1
+R
)(W1
+W
2/R
)(=C
2)
N1
=1−
R
2(1
+R
)(1+
(1/R
)(W
2/W
1))
N2
=1−
R
2(1
+R
)((W
1/W
2)+
(1/R
))
IfW
1/W
2,th
enN
1in
crea
ses
andN
2d
ecre
ases
(th
esu
bst
itu
tion
effec
t
dom
inat
esth
ew
ealt
heff
ect)
.
Perm
anent
vers
us
transi
tory
changes
inre
al
wage
Per
man
ent
chan
ge
LetW
1=µW
1an
dW
2=µW
2w
ithµ>
1(a
per
man
ent
incr
ease
).
We
see
thatN
1an
dN
2ar
eu
naff
ecte
d.
At
the
sam
eti
me
the
con
sum
pti
onin
crea
ses
alo
t(s
eeth
ep
erm
anen
t
inco
me
mod
el)
Th
isri
sein
con
sum
pti
onca
nb
esu
stai
ned
byth
ein
crea
ses
inth
ere
al
wag
e(t
he
wea
lth
effec
tp
erfe
ctly
can
cels
out
the
sub
stit
uti
oneff
ect)
Tra
nsi
tory
chan
ge
LetW
1=µW
1an
dW
2=W
2w
ithµ>
1(a
tran
sito
ryin
crea
se).
So,N
1in
crea
ses
andN
2d
ecre
ases
.A
tth
esa
me
tim
e,co
nsu
mp
tion
incr
ease
sve
rylit
tle
bec
ause
the
incr
ease
inre
alw
age
isp
erce
ived
as
tran
sito
ry.
4-
Ric
ard
ian
equiv
ale
nce
For
the
mom
ent,
we
hav
eig
nor
edth
ego
vern
men
t.
We
use
asi
mp
leex
ten
sion
ofou
rex
chan
geec
onom
y.
Ou
rob
ject
ive
are
i)w
ew
ant
tod
escr
ibe
how
the
defi
nit
ion
ofth
eec
onom
yis
affec
ted
byth
ein
trod
uct
ion
ofgo
vern
men
tsp
end
ing
and
taxe
s.
ii)w
ew
ant
toex
plo
reth
eeff
ect
ofgo
vern
men
tp
olic
yon
pri
ces
and
quan
titi
es.
The
gove
rnm
ent
Th
ego
vern
men
td
oes
two
thin
gs:
1)th
ego
vern
men
tco
nsu
mes
quan
titi
es{G
1,G
2}of
good
sin
each
per
iod
;
2)th
ego
vern
men
tco
llect
slu
mp
-su
mta
xes{T
1,T
2}fr
omco
nsu
mer
tofin
ance
its
own
con
sum
pti
ons.
Gov
ern
men
tco
nsu
mp
tion
and
taxe
sin
per
iodt
(t=
1,2)
are
mea
-
sure
din
un
itof
the
dat
et
good
.
Tax
esan
dsp
end
ing
dec
isio
ns
are
rela
ted
thro
ugh
the
gove
rnm
ent’
s
(int
erte
mp
oral
)b
ud
get
con
stra
int.
P1G
1+P
2G2
=P
1T1
+P
2T2
Th
ep
rese
ntva
lue
ofta
xes
equ
als
the
pre
sent
valu
eof
gove
rnm
ent
spen
din
g(r
emin
dth
atP
2/P
1=
1/(1
+r)
)
An
imp
orta
ntas
sum
pti
on:
the
gove
rnm
ent
spen
din
gh
asn
oeff
ect
on
uti
lity
(an
din
ap
rod
uct
ion
econ
omy,
no
effec
ton
pro
du
ctio
n,
i.e.
un
pro
du
ctiv
ego
vern
men
tsp
end
ing,
pu
rely
was
tefu
l)
The
con
sum
er
Sam
eas
bef
ore,
bu
tth
ein
tert
emp
oral
bu
dge
tco
nst
rain
tb
ecom
es
P1C
1+P
2C2
=P
1(Y
1−T
1)+P
2(Y
2−T
2)
bec
ause
the
con
sum
erm
ust
now
pay
taxe
sin
each
per
iod
.
Defi
nit
ion
ofth
eco
mpe
titi
veeq
uil
ibri
um
Aco
mp
etit
ive
equ
ilib
riu
mco
nsi
sts
ofp
rice
s{P
1,P
2},
pri
vate
dec
i-
sion
s{C
1,C
2}an
dgo
vern
men
tp
olic
ies{G
1,G
2,T
1,T
2}su
chth
at:
•T
he
con
sum
erm
axim
izes
uti
lity
,gi
ven
pri
ces
and
taxe
s,su
bje
ctto
the
bu
dge
tco
nst
rain
t;
•T
he
gove
rnm
ent’
sd
ecis
ion
ssa
tisf
yit
sb
ud
get
con
stra
int;
•S
up
ply
=D
eman
dfo
rea
chgo
od:
C1
+G
1=Y
1C
2+G
2=Y
2
An
illu
stra
tion
En
dow
men
ts:{Y
1,Y
2}
Uti
lity:
maxC
1,C
2lo
g(C
1)+β
log(C
2)
Gov
ern
men
tsp
end
ing:{G
1,G
2}ar
egi
ven
Con
sum
er
Rep
lace
the
new
bu
dge
tco
nst
rain
t
C2
=P
1
P2(Y
1−T
1)+
(Y2−T
2)−P
1
P2C
1
into
the
obje
ctiv
e.N
ext,
max
imiz
eth
eu
tilit
yw
ith
resp
ect
toC
1(f
or
P1,P
2,T
1an
dT−
2gi
ven
).T
his
yiel
ds
1 C1−βP
1
P2
1 C2
=0
Sam
eeq
uat
ion
asb
efor
e.
Equ
ilib
riu
m
C1
=Y
1−G
1C
2=Y
2−G
2
Not
ice
that
dC
1
dG
1=dC
2
dG
2=−
1
Per
fect
crow
din
gou
tof
pri
vate
con
sum
pti
onby
pu
bli
csp
end
ing
,
bec
ause
the
end
owm
ent
are
give
nan
din
vari
ant
toG
.
Rm
kp
rod
uct
ive
gove
rnm
ent
spen
din
g
Exa
mp
le:
pu
blic
exp
end
itu
res
onin
fras
tru
ctu
re(r
oad
,p
ort,
com
mu
-
nic
atio
nsy
stem
s),
rese
arch
,b
asic
pro
visi
onof
edu
cati
onan
dh
ealt
h.
Asi
mp
lere
pre
sent
atio
n
Y1
=Y
1+ηG
1Y
2=Y
2+ηG
2
wh
ereη≥
0.
Wit
hth
issl
ight
mod
ifica
tion
,w
eob
tain
C1
=Y
1+
(η−
1)G
1≥Y
1−G
1C
2=Y
2+
(η−
1)G
2≥Y
2−G
2
and
dC
1
dG
1=dC
2
dG
2=η−
1
Wh
enη>
0n
op
erfe
ctcr
owd
ing–
out
effec
t,ifη
issu
ffici
entl
yla
rge,
crow
din
g–in
.E
nd
ofth
ere
mar
k.
Now
,fr
omC
1an
dC
2,w
eca
nd
eter
min
eth
ein
tert
emp
oral
pri
ces
P2
P1
=βC
1
C2≡βY
1−G
1
Y2−G
2
Not
eth
atn
eith
erp
rice
s,n
orqu
anti
ties
dep
end
son
taxe
san
dth
eir
tim
ing.
Tax
esar
ead
just
edin
ord
erto
sati
sfy
the
gove
rnm
ent’
s
bu
dge
tco
nst
rain
t.W
eh
ave
the
follo
win
gm
ain
resu
lt.
Resu
lt(R
icard
ian
Equiv
ale
nce
)If
pri
ces{P
1,P
2},
pri
vate
de-
cisi
ons{C
1,C
2}an
dgo
vern
men
tp
olic
ies{G
1,G
2,T
1,T
2}co
nst
itu
te
aco
mp
etit
ive
equ
ilib
riu
m,
then
an
ewta
xp
olic
y{T
1,T
2}sa
tisf
yin
g
P1G
1+P
2G2
=P
1T1
+P
2T2
will
con
stit
ute
aco
mp
etit
ive
equ
ilib
riu
mw
ith
the
sam
epri
ces,
pri
vate
deci
sions
and
govern
ment
spendin
g.
Pro
of:
Fir
stn
ote
thatP
1G1
+P
2G2
=G
isco
nst
ant
and
doe
s
not
vary
acro
ssth
etw
oeq
uili
bri
asi
nce
the
pri
ces
and
gove
rnm
ent
spen
din
gar
eth
esa
me
inea
cheq
uili
bri
um
.T
he
con
sum
er’s
inte
rtem
-
por
alb
ud
get
con
stra
int
only
dep
end
onth
ep
rese
ntva
lue
ofta
xes
P1T
1+P
2T2,
notT
1an
dT
2se
par
atel
y.F
rom
the
gove
rnm
ent
bu
d-
get
con
stra
int,
the
pre
sent
valu
esof
taxe
sar
eeq
ual
toG
and
isth
us
con
stan
t.S
o,th
eco
nsu
mer
’sb
ud
get
con
stra
int
rem
ain
su
naff
ecte
d,
and
soth
eop
tim
ald
ecis
ion
onC
1an
dC
2.T
hen
the
inte
rtem
por
al
pri
ces
are
the
sam
e.T
his
com
ple
tes
the
pro
of.�
Bec
ause
the
tim
ing
ofta
xes
hav
en
oeff
ect
oneq
uili
bri
um
pri
ces
and
quan
titi
es,
deb
tfin
anci
ng
orta
xfin
anci
ng
are
equ
ival
ent
(Ric
ard
ian
equ
ival
ence
)in
this
econ
omy.
Th
ed
ebt
ofth
ego
vern
men
tin
per
iod
1
B=G
1−T
1
Th
ep
revi
ous
resu
ltsa
ysth
atfo
ran
yle
vel
ofd
ebtB
will
pro
du
ceth
e
sam
eeq
uili
bri
um
.
IfB
incr
ease
,th
enw
en
eed
toin
crea
seta
xes
tom
orro
wto
sati
sfy
the
inte
rtem
por
algo
vern
men
tb
ud
get
con
stra
int
T2
=G
2+P
1
P2B≡
(1+r)B
Aim
port
ant
poin
tT
his
resu
ltd
oes
not
mea
nth
eth
ego
vern
men
t’
dec
isio
ns
hav
en
oeff
ect
onth
eec
onom
y(s
eeth
ed
iscu
ssio
nab
ove)
.It
just
stat
esth
atth
eti
min
gof
taxa
tion
doe
sn
otm
atte
r.
Depart
ure
sfr
om
Ric
ard
ian
Equiv
ale
nce
•S
hor
t–liv
edco
nsu
mer
.
•D
isto
rtio
nar
yta
xati
on.
•Im
per
fect
finan
cial
mar
ket.
2.2
:A
Pro
duct
ion
Eco
nom
y
•T
he
mod
el
•E
quili
bri
um
•C
entr
alp
lan
ner
pro
ble
m
•D
iscu
ssio
n
Inth
ep
revi
ous
setu
pof
anen
dow
men
tec
onom
y,th
ere
alin
tere
stra
te
only
dep
end
son
pre
fere
nce
san
den
dow
men
ts.
Th
isis
bec
ause
the
agen
tsh
ave
no
opp
ortu
nit
yto
pro
du
cego
ods.
We
con
sid
ern
owth
atag
ents
can
hav
eac
cess
toa
tech
nol
ogy
that
allo
ws
totr
ansf
orm
curr
ent
con
sum
pti
onin
tofu
ture
con
sum
pti
on.
The
Model
Th
eag
ent
can
now
inve
stin
phy
sica
lca
pit
al.
Per
iod
1T
he
agen
tin
vest
sK
un
its
ofp
hysi
cal
cap
ital
.
Per
iod
2A
tth
eb
egin
nin
gof
the
per
iod
,F
(K)
un
its
ofgo
ods
are
pro
du
ced
from
theK
un
its.
Nex
t,th
eK
isfu
llyd
estr
oyed
(fu
lld
epre
ciat
ion
).
So,
agen
tsre
ceiv
eF
(K)
inp
erio
d2.
Ass
um
pti
ons
onth
ep
rod
uct
ionF
(K):
i)F
(0)
=0
ii)F′ >
0an
dF′′<
0(i
ncr
easi
ng
and
con
cave
)
iii)
Inad
aco
nd
itio
ns:F′ (
0)=∞
andF′ (∞
)=
0.
Wit
hth
ein
vest
men
tin
phy
sica
lca
pit
alan
dth
ep
rod
uct
ion
tech
nol
-
ogy,
we
can
now
con
sid
era
new
acto
rin
the
econ
omy:
AF
irm
.
Th
efir
mca
nb
uy
(or
rent
)th
eca
pit
alfr
omth
eco
nsu
mer
ata
pri
ce,
den
otedQ
,p
eru
nit
ofca
pit
alan
dth
enca
nu
seit
top
rod
uce
extr
a
outp
utF
(K)
inp
erio
d2.
The
Fir
m’s
deci
sion
Th
efir
mch
oose
sop
tim
ally
the
leve
lofK
inp
erio
d2
such
that
she
max
imiz
esth
ep
rofit
:
Π=P
2F(K
)−QK
The
Consu
mer’
sdeci
sion
Th
ere
pre
sent
ativ
eco
nsu
mer
own
sth
eca
pit
alan
dth
efir
m.
Th
en
ewb
ud
get
con
stra
int
LetX
isth
eam
ount
ofca
pit
alre
nted
toth
efir
man
dΠ
the
pro
fit
that
she
rece
ives
from
own
ing
the
firm
.
Th
eco
nsu
mer
’sd
ecis
ion
isto
choo
seC
1,C
2an
dX
tom
axim
ize
the
inte
rtem
por
alu
tilit
y
U(C
1,C
2)
un
der
the
(new
)in
tert
emp
oral
bu
dge
tco
nst
rain
t
P1C
1+P
2C2≤P
1(Y
1−X
)+P
2Y2
+QX
+Π
The
com
peti
tive
equilib
rium
Defi
nit
ion
ofth
eeq
uil
ibri
um
Aco
mp
etit
ive
equ
ilib
riu
mis
ap
rice
syst
em{P
1,P
2,Q}
and
allo
ca-
tion
s{C
1,C
2,X,K}
such
that
i)T
he
con
sum
mer
max
imiz
esth
ein
tert
emp
oral
uti
lity
give
np
rice
s
and
sub
ject
toth
ein
tert
emp
oral
bu
dge
tco
nst
rain
t.
ii)T
he
firm
max
imiz
esp
rofit
give
np
rice
san
dth
ete
chn
olog
y.
iii)
Su
pp
ly=
Dem
and
for
each
good
s
C1
+K
=Y
1(g
ood
mar
ket
inp
erio
d1)
C2
=Y
2+F
(K)
(goo
dm
arke
tin
per
iod
2)
X=K
(cap
ital
mar
ket)
Applica
tion
Uti
lity
fun
ctio
n
U(C
1,C
2)=
log(C
1)+β
log(C
2)
En
dow
men
ts
Y1
=Y
Y2
=0
(th
ep
rod
uct
ion
allo
ws
toco
nsu
me
tom
orro
ww
ith
out
end
owm
ent)
F(K
)=Kα
(+fu
lld
epre
ciat
ion
ofp
hysi
cal
cap
ital
)
The
con
sum
er
max
C1,C
2,X
log(C
1)+β
log(C
2)
un
der
the
inte
rtem
por
alb
ud
get
con
stra
int.
Fir
st,
sub
stit
ute
the
con
-
stra
int
C2
=P
1
P2Y−( P 1 P
2−Q P
2
) X+
Π P2−P
1
P2C
1
FO
Cs
wit
hre
spec
ttoC
1
1 C1−βP
1
P2
1 C2
=0⇔
1 C1
=βP
1
P2
1 C2
(Eu
ler
equ
atio
n)
wit
hre
spec
ttoX −β
1 C2
P1
P2
+β
1 C2
Q P2
=0⇔
P1
=Q
The
firm
max K
ΠP
2Kα−QK
FO
C(M
argi
nal
pro
du
ctiv
ity
ofca
pit
alis
equ
alto
its
real
un
itco
st)
αKα−
1=Q P
2
Cap
ital
dem
and
K=
( Q αP
2
) 1/(α−
1)
Equ
ilib
riu
m
Goo
dm
arke
tin
per
iod
1
C1
=Y−K
Goo
dm
arke
tin
per
iod
2
C2
=Kα
Cap
ital
mar
ket
(+th
eco
nd
itio
nQ
=P
1)
X=K
=
( P 1 αP
2
) 1/(α−
1)
Fro
mth
eE
ule
req
uat
ion
onco
nsu
mp
tion
,w
ed
eter
min
eth
ere
lati
ve
pri
ce
P2
P1
=βC
1
C2≡βY−K
Kα
Fro
mth
efir
m’s
opti
mal
ity
con
dit
ion
,w
eh
ave
P2
P1
=K
αKα
We
ded
uce
βY−K
Kα
=K
αKα⇔
αβ
(Y−K
)=K
and
we
obta
inth
ele
vel
ofth
ep
hysi
cal
cap
ital
K=ωY
wh
ere
ω=
αβ
1+αβ∈
(0,1
)
We
then
ded
uce
the
con
sum
pti
ons
inp
erio
d1
and
2
C1
=(1−ω
)YC
2=
(ωY
)α
Not
eth
atth
eco
nsu
mer
doe
sn
otco
nsu
me
all
the
end
owm
ent
infir
st
per
iod
,b
ut
choo
seto
save
(in
the
form
ofp
hysi
cal
cap
ital
).
Th
esa
vin
gis
then
use
toco
nsu
me
tom
orro
w(w
ith
any
end
owm
ent
inp
erio
d2)
,ac
cord
ing
toth
ete
chn
olog
y(K
α)
Th
ep
aram
eterω
can
be
inte
rpre
ted
asth
esa
vin
gra
te(ω
=Y
1/K
).
Th
em
odel
pro
vid
esm
icro
-fou
nd
atio
ns
for
this
par
amet
er.
Itd
epen
ds
ontw
o“d
eep
”p
aram
eter
sre
flect
ing
the
pre
fere
nce
s(β
)
and
the
tech
nol
ogy
(α).
Th
esa
vin
gis
anin
crea
sin
gfu
nct
ion
ofβ
andα
:
∂ω
∂α
=β
(1+αβ
)2∂ω
∂β
=α
(1+αβ
)2
Mor
ep
atie
ntag
ents
(βla
rger
)sa
vem
ore.
Mor
eeffi
cien
tte
chn
olog
y
(αla
rger
)cr
eate
sin
cent
ives
toin
vest
.
Fin
ally
,fr
omC
1an
dC
2,w
eca
nd
eter
min
eth
ere
lati
vep
rice
s
P2
P1
=βC
1
C2≡β
(1−ω
)Y
(ωY
)α
Not
eth
atw
eca
nd
efin
eas
pre
viou
sly
the
real
inte
rest
rate
usi
ng
the
rela
tion
P2
P1
=β
1
1+r
The
centr
al
pla
nner
max
C1,C
2
U(C
1,C
2)
un
der
the
two
con
stra
ints
C1
+K≤Y
1C
2≤Y
2+F
(K)
oreq
uiv
alen
tly
C2≤Y
2+F
(Y1−C
1)
App
lica
tion
:U
sin
gth
eab
ove
assu
mp
tion
,th
isp
rob
lem
bec
omes
max
C1,C
2,K
log(C
1)+β
log(C
2)
Aft
erre
pla
cem
ent
ofth
eco
nst
rain
tin
toth
eob
ject
ive,
we
obta
in
max K
log(Y−K
)+β
log(Kα
)⇔
max K
log(Y−K
)+αβ
log(K
)
FO
C
−1
Y−K
+αβ
1 K=
0
Th
isyi
eld
s K=ωY
C1
=(1−ω
)YC
2=
(ωY
)α
Th
esa
me
allo
cati
onth
anth
eon
eof
the
com
pet
itiv
eeq
uili
bri
um
.
Chap
2:
Models
and
Meth
ods
inM
acr
oeco
nom
icD
y-
nam
ics
We
use
two
use
fulm
odel
sto
intr
odu
cem
eth
ods
and
con
cep
tsof
mod
-
ern
mac
rod
ynam
ics.
2.1
:T
he
Perm
anent
Inco
me
Model
•T
he
det
erm
inis
tic
case
•T
he
stoc
has
tic
case
2.2
:T
he
Dynam
icL
ab
or
Dem
and
Model
•T
he
det
erm
inis
tic
case
•T
he
stoc
has
tic
case
We
do
not
con
sid
erh
ere
two
oth
eru
sefu
lm
odel
s:
Am
odel
of(p
hysi
cal)
cap
ital
/inv
estm
ent
dec
isio
ns
wit
had
just
men
t
cost
s(s
eeth
eex
erci
se)
Am
odel
ofd
ynam
icla
bor
sup
ply
(see
the
nex
tch
apte
rin
asi
mp
le
two-
per
iod
mod
el)
Chap
2:
Models
and
Meth
ods
inM
acr
oeco
nom
icD
y-
nam
ics
2.1
:T
he
Perm
anent
Inco
me
Model
•T
he
det
erm
inis
tic
case
•T
he
stoc
has
tic
case
Th
eK
eyn
esia
nco
nsu
mp
tion
fun
ctio
n(k
eyfu
nct
ion
inth
eK
eyn
esia
n
mod
el,
i.e.
the
valu
eof
the
gove
rnm
ent
spen
din
gm
ult
iplie
r)
Ct
=Co
+cYt
Ct
isth
ere
alco
nsu
mp
tion
andYt
the
real
(aft
erta
x)in
com
e.
Co
den
otes
ap
osit
ive
con
stan
tan
dc∈
(0,1
)is
the
mar
gin
alp
rop
en-
sity
toco
nsu
me.
[In
sert
afig
ure
]
Em
pir
ical
stu
die
s(a
fter
the
WW
II)
hav
equ
esti
oned
the
emp
iric
al
rele
van
ceof
this
fun
ctio
n(f
orex
.,w
ith
alo
ng
dat
asa
mp
le,
esti
ma-
tion
resu
lts
sugg
estc'
1)+
Th
eore
tica
lex
ten
sion
s:re
lati
vein
com
e
app
roac
hes
,in
clu
din
gre
fere
nce
inco
me
and
ratc
het
effec
t(s
eeD
ues
en-
ber
ryin
50’)
,an
db
asic
form
ula
tion
sof
hab
itp
ersi
sten
ce(s
eeB
row
n
in19
54).
Mor
ere
cent
ly,
the
per
man
ent
inco
me
mod
elal
low
sto
dis
con
nec
tth
e
curr
ent
con
sum
pti
onfr
omth
ecu
rren
tla
bor
inco
me.
Her
e,th
ep
erm
anen
tin
com
em
odel
wit
han
infin
ite
hor
izon
,fo
llow
-
ing
Hal
l(1
978)
and
all
the
rece
ntd
evel
opm
ents
onth
eco
nsu
mp
tion
fun
ctio
ns.
Sim
ple
rfo
rms:
atw
op
erio
dm
odel
ofco
nsu
mp
tion
(alr
ead
yd
one
in
Mic
roco
urs
esan
din
par
tial
equ
ilib
riu
m;
we
will
inve
stig
ate
bel
owa
sim
ple
two–
per
iod
setu
p,
bu
tin
age
ner
aleq
uili
bri
um
setu
p)
Un
der
stan
din
gin
tert
emp
oral
con
sum
pti
ond
ecis
ion
sis
esse
ntia
lin
mod
ern
Mac
roec
onom
icD
ynam
ics.
Her
e,w
eu
sea
(too
!!an
dp
rob
ably
not
real
isti
c)si
mp
lean
dtr
acta
ble
setu
pfr
omw
hic
hw
eca
nob
tain
anal
ytic
also
luti
ons
(an
dth
enw
eca
n
anal
yze
them
exte
nsi
vely
)
The
Dete
rmin
isti
cC
ase
Th
ese
tup
A(r
epre
sent
ativ
e)h
ouse
hol
d(o
rco
nsu
mer
)w
illse
ekto
max
imiz
e
(eve
ryp
erio
d)
the
follo
win
gob
ject
ive
fun
ctio
n
∞ ∑ i=0
βi U
(Ct+i)
un
der
ase
quen
ceof
(in
stan
tan
eou
s)b
ud
get
con
stra
int
At+
1=
(1+r)At
+Yt−Ct∀t
wh
ereU
(Ct)
isth
ein
stan
tan
eou
su
tilit
yfu
nct
ion
(sp
ecifi
edb
elow
).
Rm
k:th
eu
tilit
yfu
nct
ion
inp
erio
dt
only
dep
end
son
the
curr
ent
con
sum
pti
on(e
xten
sion
:n
on-s
epar
able
uti
lity
fun
ctio
n,
ex.
hab
it
form
atio
nin
con
sum
pti
on)
β∈
(0,1
)is
the
sub
ject
ive
dis
cou
ntra
te(p
refe
ren
ces)
rth
ere
al(c
onst
ant)
inte
rest
rate
(we
assu
mer>
0),
give
n
At
isth
ele
vel
of(r
eal)
finan
cial
wea
lth
Yt
isth
ere
al(n
etof
taxa
tion
)la
bor
inco
me,
her
esu
pp
osed
tob
e
know
nin
each
per
iod
(we
will
con
sid
erla
tter
the
stoc
has
tic
case
)
Ct
isth
ere
alco
nsu
mp
tion
Sta
tus
ofth
eva
riab
les
At
isa
stat
e(o
rp
re-d
eter
min
ed)
vari
able
,w
hic
hsu
mm
ariz
esth
ep
ast
dec
isio
ns
onco
nsu
mp
tion
,gi
ven
the
(pre
sent
and
pas
t)re
aliz
atio
ns
ofth
ela
bor
inco
me.
Yt
isan
exog
enou
sva
riab
le
Rm
k1:
no
dec
isio
non
lab
orsu
pp
ly,
orla
bor
isco
nst
ant.
Ifn
ot,
solu
tion
ism
ore
com
plic
ated
.
Rm
k2:
the
elas
tici
tyof
lab
orsu
pp
lyis
sub
ject
toem
pir
ical
cont
ro-
vers
ies
(mac
rost
ud
ies,
larg
e;m
icro
stu
die
s,ve
rysm
all)
Ct
isth
eco
ntro
lva
riab
le
An
Imp
ort
ant
Pro
pert
yC
ontr
ollin
gCt
iseq
uiv
alen
tto
cont
rol-
lingAt+
1.
Pro
of
dir
ectl
yd
edu
ced
from
the
bu
dge
tco
nst
rain
t
At+
1+Ct
=(1
+r)At
+Yt
and
the
abov
em
enti
oned
stat
us
ofth
eva
riab
les.
Conse
quence
We
can
rep
lace
the
con
stra
int
into
the
obje
ctiv
ein
ord
erto
sim
ply
get
the
opti
mal
dec
isio
non
con
sum
pti
on.
Rm
kN
ote
that
we
can
form
ula
teth
ep
rob
lem
ina
sim
ple
recu
rsiv
e
way
.L
etV t
the
valu
eof
this
dyn
amic
pro
ble
min
per
iodt
V t=
∞ ∑ i=0
βi U
(Ct+i)≡U
(Ct)
+βU
(Ct+
1)+β
2 U(Ct+
2)+...
now
,rew
rite
this
valu
ein
per
iodt+
1(j
ust
man
ipu
late
the
tim
ein
dex
)
V t+
1=
∞ ∑ i=0
βi U
(Ct+
1+i)≡U
(Ct+
1)+βU
(Ct+
2)+β
2 U(Ct+
3)+...
Now
mu
ltip
lyV t
+1
byβ
βV t
+1
=β
∞ ∑ i=0
βi U
(Ct+
1+i)≡βU
(Ct+
1)+β
2 U(Ct+
2)+β
3 U(Ct+
3)+...
and
then
sub
trac
tβV t
+1
fromV t
V t−βV t
+1
=U
(Ct)⇔V t
=U
(Ct)
+βV t
+1
So,
the
valu
eof
this
inte
rtem
por
ald
ecis
ion
pro
ble
mto
day
iseq
ual
to
the
curr
ent
uti
lity
plu
sth
ed
isco
unt
edva
lue
tom
orro
w.
Giv
enth
eop
tim
ald
ecis
ion
onco
nsu
mp
tion
Ct,
we
can
obta
inea
s-
ilyco
mp
ute
the
wel
fare
fun
ctio
nof
the
con
sum
er(u
sefu
lto
eval
uat
e
the
wel
fare
imp
licat
ion
ofva
riou
sec
onom
icp
olic
y,fo
rex
amp
lefis
cal
pol
icy)
Tw
oeq
uiv
alen
tE
con
omie
s
i)A
Sm
all
Op
en
Eco
nom
y
Th
eeq
uili
bri
um
onth
ego
odm
arke
t
Yt
=Ct
+I t
+Gt
+Xt−IMt
wh
ereYt,Ct,I t
,Gt,Xt
andIMt
den
ote
outp
ut,
con
sum
pti
on,
in-
vest
men
t,go
vern
men
tsp
end
ing,
exp
orts
and
imp
orts
,re
spec
tive
ly
(all
vari
able
sar
eex
pre
ssed
inre
alte
rms)
.
Th
ecu
rren
tac
cou
nteq
uat
ion
Bt+
1=
(1+r)Bt
+TBt
wh
ereBt
isth
en
etw
ealt
hof
the
dom
esti
cec
onom
yw
ith
resp
ect
to
the
rest
ofth
ew
orld
.T
he
(con
stan
t)re
alin
tere
stra
te(r>
0)is
give
n(b
yth
ere
stof
the
wor
ld).
Let
us
defi
neTBt
the
trad
eb
alan
ce(o
rn
etex
por
ts)
TBt
=Xt−IMt
No
rela
tive
pri
ce(a
sin
gle
hom
ogen
ous
good
isp
rod
uce
db
oth
hom
e
and
abro
ad)
Com
bin
ing
the
pre
viou
seq
uat
ion
syi
eld
s
Bt+
1=
(1+r)Bt
+(Yt−I t−Gt)−Ct
wh
ereYt−I t−Gt
isas
sum
edto
be
exog
enou
s
Equ
ival
ence
:A=B
andY
isn
owd
efin
edas
net
ofp
riva
tein
vest
men
t
and
pu
blic
spen
din
g(Yt−I t−Gt)
ii)
AG
row
thM
odel
Equ
ilib
riu
mon
the
good
mar
ket
Yt
=Ct
+I t
wh
ereYt,Ct
andI t
den
ote
outp
ut,
con
sum
pti
onan
din
vest
men
t,
resp
ecti
vely
.
Law
ofm
otio
nof
phy
sica
lca
pit
alKt
Kt+
1=
(1−δ)Kt
+I t
wh
ereδ∈
[0,1
]is
aco
nst
ant
dep
reci
atio
nra
te.
Th
ete
chn
olog
y
Yt
=Zt
+aKt
wh
ereZt
rep
rese
ntch
ange
sin
the
pro
du
ctio
n,
ind
epen
den
tly
from
the
leve
lofKt.a>
0is
asc
ale
par
amet
er(s
eeth
eex
erci
sean
dch
ap.
2).
Not
eth
atth
eca
pit
alst
ock
tod
ayis
pre
-det
erm
ined
and
inve
stm
ent
dec
isio
nto
day
will
bec
ome
pro
du
ctiv
eto
mor
row
.
Ifw
eco
mb
ine
the
thre
eeq
uat
ion
sab
ove,
we
get:
Kt+
1=
(1−δ
+a
)Kt
+Zt−Ct
Equ
ival
ence
:A
=K
,r
=a−δ
(th
ere
alin
tere
stra
teis
equ
alth
e
mar
gin
alp
rod
uct
ivit
yof
cap
ital
min
us
the
dep
reci
atio
nra
te,
i.e.
the
real
retu
rnof
phy
sica
lca
pit
al)
andYt
=Zt.
Opti
mality
Condit
ion
We
assu
me
that
the
uti
lity
take
sth
efo
llow
ing
form
U(Ct)
=α
0Ct−α
1 2C
2 t
wh
ereα
0,α
1>
0.
Th
em
argi
nal
uti
lity
islin
ear
and
dec
reas
ing
inC
U′ (Ct)
=α
0−α
1Ct
(U′′
=−α
1<
0)
To
sim
ply
get
the
opti
mal
dec
isio
non
con
sum
pti
on,
rep
lace
the
con
-
stra
int
into
the
obje
ctiv
e
∞ ∑ i=0
βi U
(Ct+i)
=U
(Ct)
+βU
(Ct+
1)+β
2 U(Ct+
2)+...
wh
ere
Ct
=(1
+r)At
+Yt−At+
1
Ct+
1=
(1+r)At+
1+Yt+
1−At+
2
Ct+
2=
(1+r)At+
2+Yt+
2−At+
3
and
soon
.
Don
’tfo
rget
:co
ntro
llin
gCt⇔
cont
rolli
ngAt+
1ev
ery
per
iod
(in
-
crea
sin
gcu
rren
tco
nsu
mp
tion
iseq
uiv
alen
tto
red
uce
savi
ng!
)
U(Ct)
+βU
(Ct+
1)+β
2 U(Ct+
2)+...
=U
((1
+r)At+Yt−At+
1)+
βU
((1
+r)At+
1+Yt+
1−At+
2)
+β
2 U((
1+r)At+
2+Yt+
2−At+
3)+....
Th
enm
axim
ize
the
obje
ctiv
ew
ith
resp
ect
toAt+
1.T
his
will
give
you
the
opti
mal
con
sum
pti
on/s
avin
gd
ecis
ion
inp
erio
dt.
You
can
alw
ays
red
oth
eex
erci
sefo
rot
her
per
iod
s(s
eeth
ed
iscu
ssio
n
bel
ow).
Fir
stO
rder
Con
dit
ion
(den
oted
FO
C,
her
eaft
er)
inp
erio
dt.
−U′ (Ct)
+β
(1+r)U′ (Ct+
1)=
0
Th
iseq
uat
ion
isca
lled
the
Eule
requati
on
onco
nsu
mp
tion
.
Th
isd
ecis
ion
isin
tert
emp
oral
bec
ause
incr
easi
ng
con
sum
pti
onto
day
will
red
uce
savi
ng
and
then
con
sum
pti
onto
mor
row
.
Th
eag
ent
will
thu
sfa
cea
trad
e–off
(int
erte
mp
oral
sub
stit
uti
on)
be-
twee
nco
nsu
mp
tion
tod
ayan
dco
nsu
mp
tion
tom
orro
w.
Equ
ival
entl
y
U′ (Ct)
=β
(1+r)U′ (Ct+
1)⇔
U′ (Ct)
U′ (Ct+
1)=β
(1+r)
Th
em
argi
nal
rate
ofsu
bst
itu
tion
ofco
nsu
mp
tion
bet
wee
np
erio
dt
and
per
iodt
+1
iseq
ual
toth
ed
isco
unt
edre
alin
tere
stra
te(r
elat
ive
pri
ce).
Rm
k:th
isop
tim
alit
yco
nd
itio
nis
sati
sfied
ever
yp
erio
d(i
fw
ere
do
this
opti
miz
atio
np
rob
lem
atot
her
per
iod
st
+h
).
U′ (Ct+h)
=β
(1+r)U′ (Ct+h
+1)
forh
=0,
1,....
So,
up
the
tim
ein
dex
,th
eop
tim
alp
lan
for
con
sum
p-
tion
rem
ain
sth
esa
me.
Th
eco
nsu
mp
tion
dec
isio
nis
thu
sti
me
con
sist
ent.
Inot
her
wor
ds,
give
nth
est
ruct
ure
ofth
eth
eore
tica
lm
odel
,th
ere
isn
oin
cent
ive
for
the
con
sum
erto
mod
ify
thei
rop
tim
alp
lan
sov
er
tim
e.
Now
,w
eas
sum
e
β(1
+r)
=1
Su
bje
ctiv
ed
isco
unt
fact
or=
Ob
ject
ive
dis
cou
ntra
te
To
see
this
,le
tβ
=1/
(1+θ)
,w
her
eθ>
0is
the
sub
ject
ive
rate
of
tim
ep
refe
ren
ce;
θ(o
rβ
)is
ad
eep
par
amet
erre
pre
sent
ing
ake
yin
tert
emp
oral
beh
av-
ior:θ
larg
e(β
smal
l)m
ean
san
imp
atie
ntag
ent,
wh
erea
sθ
smal
l(β
clos
eto
one)
mea
ns
ap
atie
ntag
ent.
Itfo
llow
sth
at
β(1
+r)
=1⇔
r=θ
Now
,w
eu
seth
em
argi
nal
uti
lity
U′ (Ct)
=α
0−α
1Ct
and
we
sub
stit
ute
this
mar
gin
alu
tilit
yin
toth
eE
ule
req
uat
ion
α0−α
1Ct
=β
(1+r)
(α0−α
1Ct+
1)
Bec
auser
=θ,
this
Eu
ler
equ
atio
nre
du
ces
to
Ct
=Ct+
1
Th
eop
tim
ald
ecis
ion
for
the
(rep
rese
ntat
ive)
hou
seh
old
isto
mai
ntai
n
(per
fect
smoo
thin
g)co
nst
ant
its
con
sum
pti
onov
erti
me.
Aga
in,
this
opti
mal
dec
isio
nis
sati
sfied
ever
yp
erio
d
Ct+h
=Ct+h
+1
forh
=0,
1,....
Th
isco
nsu
mp
tion
dec
isio
nis
thu
sti
me
con
sist
ent.
Th
eE
ule
req
uat
ion
defi
nes
the
stru
ctu
ral
form
ofth
em
odel
,b
ut
this
isn
otth
eso
luti
on.
Definit
ion:
The
solu
tion
ofany
dynam
icm
odelexpre
sses
the
contr
ol
vari
able
(s)
as
a(l
inear)
funct
ion
of
the
state
vari
able
(s)
and
the
exogenous
vari
able
(s).
Inou
rca
se,
the
solu
tion
for
the
con
sum
pti
onw
illb
eof
the
follo
win
g
linea
rfo
rm
Ct
=φ
1At
+φ
2Yt
wh
ere
the
par
amet
ersφ
1an
dφ
2ar
ed
eter
min
edby
the
mod
el’s
pro
p-
erti
esan
da
fun
ctio
nof
the
stru
ctu
ral
par
amet
ers
(see
our
com
pu
ta-
tion
sb
elow
).
Com
puta
tion
of
the
solu
tion
(forw
ard
subst
ituti
ons)
⇒F
rom
the
inst
anta
neo
us
bu
dge
tco
nst
rain
tto
the
inte
rtem
por
al
bu
dge
tco
nst
rain
t
Let
us
star
tfr
omth
ein
stan
tan
eou
sb
ud
get
con
stra
int
inp
erio
dt:
At+
1=
(1+r)At
+Yt−Ct
Bec
auser>
0an
dth
us
1+r>
1,th
efir
stor
der
diff
eren
ceeq
uat
ion
isu
nst
able
bac
kwar
d(c
urr
ent
wea
lth
asa
fun
ctio
nof
pas
tw
ealt
h).
Bu
tit
isst
able
forw
ard
(cu
rren
tw
ealt
has
afu
nct
ion
offu
ture
wea
lth
)
[In
sert
figu
res
that
illu
stra
teth
isp
rop
erty
]
At
=1
1+rAt+
1+
1
1+rCt−
1
1+rYt
Now
,w
rite
this
equ
atio
nin
per
iodt
+1
At+
1=
1
1+rAt+
2+
1
1+rCt+
1−
1
1+rYt+
1
and
then
sub
stit
ute
this
bu
dge
tco
nst
rain
tin
per
iodt
+1
into
the
bu
dge
tco
nst
rain
tin
per
iodt
At
=1
(1+r)
2At+
2+1
(1+r)
2Ct+
1−1
(1+r)
2Ct+
1+1
1+rCt−
1
1+rYt
Aga
in,
wri
teth
eb
ud
get
con
stra
int
inp
erio
dt+
2an
dth
ensu
bst
itu
te
this
exp
ress
ion
intoAt,
....
and
cont
inu
e.
Th
isyi
eld
s
At
=limT→∞
1
(1+r)TAt+T
+1
1+r
limT→∞
T−
1 ∑ i=0
1
(1+r)i(Ct+i−Yt+i)
Now
,im
pos
eth
e“t
ran
sver
salit
y”co
nd
itio
n
limT→∞
1
(1+r)TAt+T
=0
mea
nin
gte
chn
ical
lyth
atth
ew
ealt
hca
nn
otex
plo
de
atso
me
rate
that
exce
edsr.
Wit
hth
isre
stri
ctio
non
At,
the
inte
rtem
por
alb
ud
get
con
stra
int
be-
com
es
At
=1
1+r
∞ ∑ i=0
1
(1+r)i(Ct+i−Yt+i)
oreq
uiv
alen
tly
At
+Ht
=1
1+r
∞ ∑ i=0
1
(1+r)iCt+i
wh
ereHt
defi
nes
the
non
–fin
anci
alw
ealt
h(o
r“h
um
an”
wea
lth
),i.e
.
all
that
you
can
det
erm
inis
tica
lly
earn
du
rin
gyo
ur
(in
finit
e)p
arti
ci-
pat
ion
onth
ela
bor
mar
ket.
Ht
=1
1+r
∞ ∑ i=0
1
(1+r)iYt+i
To
obta
inth
eco
nsu
mp
tion
fun
ctio
n,
we
first
nee
dto
com
pu
te
1
1+r
( Ct
+1
1+rCt+
1+
1
(1+r)
2Ct+
2+...)
Fro
mth
eE
ule
req
uat
ion
onco
nsu
mp
tion
,w
ekn
owth
atth
eop
tim
al
dec
isio
nsa
tisfi
es
Ct+h
=Ct+h
+1
forh
=0,
1,2,....
Itfo
llow
sth
at
Ct
=Ct+
1=Ct+
2=...
Ifw
ere
pla
ceth
isin
toth
eab
ove
equ
atio
n,
we
obta
in
1
1+r
( Ct
+1
1+rCt+
1+
1
(1+r)
2Ct+
2+...) =
1
1+rCt
( 1+
1
1+r
+1
(1+r)
2+...)
Bec
auser>
0,th
en1/
(1+r)<
0,so
the
sequ
ence
( 1+
1
1+r
+1
(1+r)
2+...) =
∞ ∑ i=0
1
(1+r)i
conv
erge
sto
afin
ite
num
ber
.
We
hav
e
∞ ∑ i=0
1
(1+r)i
=1
1−
11+r
≡1
+r
r
We
obta
in
1
1+r
∞ ∑ i=0
1
(1+r)iCt+i
=Ct r
Now
,if
we
rep
lace
this
exp
ress
ion
into
At
+Ht
=1
1+r
∞ ∑ i=0
1
(1+r)iCt+i
we
get
Ct
=r
(At
+Ht)
Th
eco
nsu
mp
tion
fun
ctio
nd
efin
esa
linea
rre
lati
onb
etw
een
the
cur-
rent
con
sum
pti
onan
dth
efin
anci
alan
dhu
man
wea
lth
.
Ver
yd
iffer
ent
from
the
Key
nes
ian
con
sum
pti
onfu
nct
ion
that
ex-
pre
sses
the
curr
ent
con
sum
pti
onas
alin
ear
fun
ctio
nof
the
curr
ent
inco
me)
.
Th
eva
riab
leHt
defi
nes
the
non
–fin
anci
al(o
rhu
man
)w
ealt
h.
Itre
pre
sent
sal
lth
ein
com
eth
atan
agen
tca
nob
tain
from
its
lab
or
inco
me
over
its
(in
finit
e)lif
etim
e.
Her
e,w
eas
sum
ep
erfe
ctfo
resi
ght
(th
ed
eter
min
isti
cca
se),
soth
e
agen
tp
erfe
ctly
know
sth
ese
quen
ce
Yt,
Yt+
1,Yt+
2,...
For
sim
plic
ity,
assu
me
that
the
lab
orin
com
eis
con
stan
t
Yt
=Y∀t
Pu
ttin
gth
isin
toth
en
on–fi
nan
cial
wea
lth
Ht
=1
1+r
∞ ∑ i=0
1
(1+r)iYt+i,
we
obta
in
Ht
=1
1+rYt
∞ ∑ i=0
1
(1+r)i
=1
1+rYt1
+r
r≡Yt r
Now
,u
sin
gth
isex
pre
ssio
n,
we
can
ded
uce
the
con
sum
pti
onfu
nct
ion
Ct
=r
(At
+Ht)≡rA
t+Yt
Th
ecu
rren
tco
nsu
mp
tionCt
isa
linea
rfu
nct
ion
ofcu
rren
tin
com
eYt
+fin
anci
alre
venu
esrA
t.
Now
,re
pla
ceth
isex
pre
ssio
nin
toth
ein
stan
tan
eou
sb
ud
get
con
stra
int
At+
1=
(1+r)At
+Yt−Ct⇔
At+
1=At
Th
eco
nsu
mp
tion
let
the
leve
lof
the
finan
cial
wea
lth
con
stan
t.
Inth
isec
onom
y,th
eop
tim
ald
ecis
ion
onco
nsu
mp
tion
allo
ws
tosm
ooth
the
con
sum
pti
onan
dto
let
con
stan
tth
efin
anci
alw
ealt
h.
Rm
k:
An
oth
erw
ayto
com
pu
teth
eso
luti
onis
the
met
hod
ofu
nde
-
term
ined
coeffi
cien
ts.
Th
eb
asic
idea
isto
iden
tify
the
form
ula
for
the
par
amet
ersφ
1an
d
φ2
inth
ep
ostu
late
deq
uat
ion
Ct
=φ
1At
+φ
2Yt
that
sati
sfied
bot
hth
eE
ule
req
uat
ion
onco
nsu
mp
tion
(op
tim
ald
e-
cisi
ons)
and
the
bu
dge
tco
nst
rain
t(r
estr
icti
ons
onth
ein
tert
emp
oral
allo
cati
onof
con
sum
pti
on).
Fir
stre
pla
ceth
ep
ostu
late
deq
uat
ion
into
the
Eu
ler
equ
atio
non
con
-
sum
pti
on
φ1A
t+1
+φ
2Yt+
1=φ
1At
+φ
2Yt
we
know
that
Yt
=Y
∀t
Th
en,
φ1A
t+1
+φ
2Y=φ
1At
+φ
2Y=⇔
At+
1=At
Now
use
the
bu
dge
tco
nst
rain
t
At+
1=
(1+r)At
+Yt−Ct
and
sub
stit
ute
the
pos
tula
ted
equ
atio
nfo
rCt
(1+r)At
+Yt−Ct
=At⇔
(1+r)At
+Yt−φ
1At−φ
2Yt
=At
Iden
tify
the
term
s (1+r)−φ
1=
1,
1−φ
2=
0
Th
isgi
ves
us
the
two
par
amet
ersφ
1=r
andφ
2=
1.
The
Sto
chast
icC
ase
Th
ese
tup
A(r
epre
sent
ativ
e)h
ouse
hol
dw
illse
ekto
max
imiz
e(e
very
per
iod
)
Et
∞ ∑ i=0
βi U
(Ct+i)
un
der
ase
quen
ceof
(in
stan
tan
eou
s)b
ud
get
con
stra
int
(as
bef
ore)
At+
1=
(1+r)At
+Yt−Ct
Th
est
atu
san
dth
ed
efin
itio
nof
the
vari
able
sar
eth
esa
me
asb
efor
e,
bu
tn
owth
eva
riab
leYt
isst
och
asti
c(f
orex
amp
le,
stoc
has
tic
afte
r
tax
real
wag
e).
Th
eag
ent
know
sth
est
och
asti
cp
roce
ssof
this
vari
able
(sh
ekn
ows
the
pro
bab
ility
dis
trib
uti
onofYt)
.
Th
eop
erat
orEt
den
otes
the
con
dit
ion
alex
pec
tati
onop
erat
or.
Opti
mality
condit
ion
Th
esa
me
asb
efor
e,u
pto
the
con
dit
ion
alex
pec
tati
onop
erat
orEt.
Rm
kR
atio
nal
Exp
ecta
tion
s:
Her
e,w
eas
sum
era
tion
alex
pec
tati
ons
Wh
atar
era
tion
alex
pec
tati
ons?
Exp
ecta
tion
slie
atth
eco
reof
econ
omic
dyn
amic
s(i
nd
ivid
ual
beh
av-
ior,
bu
tm
ost
imp
orta
ntly
equ
ilib
riu
mp
rop
erti
es).
Lon
gtr
adit
ion
inec
onom
ics,
sin
ceK
eyn
es,
bu
tp
ut
forw
ard
byth
e
new
clas
sica
lec
onom
y(L
uca
s,P
resc
ott,
Sar
gent
inth
e70
’an
d80
’).
Th
ete
rm“r
atio
nal
exp
ecta
tion
s”is
mos
tcl
osel
yas
soci
ated
toR
ober
t
Lu
cas
(Nob
elL
aure
ate,
Un
iver
sity
ofC
hic
ago)
,b
ut
the
rati
onal
ity
of
exp
ecta
tion
sd
eep
lyex
amin
edb
efor
e(s
eeM
uth
,19
60)
We
con
sid
erth
efo
llow
ing
defi
nit
ion
.
Def
Age
nts
form
ula
teex
pect
atio
ns
insu
cha
way
that
thei
rsu
b-
ject
ive
prob
abil
ity
dist
ribu
tion
ofec
onom
icva
riab
les
(con
diti
onal
onth
eav
aila
ble
info
rmat
ion
)co
inci
des
wit
hth
eob
ject
ive
prob
-
abil
ity
dist
ribu
tion
ofth
esa
me
vari
able
(acc
ordi
ng
toa
mea
sure
ofth
est
ate
ofn
atu
re)
inan
equ
ilib
riu
m.
Exp
ect
ati
ons
should
be
consi
stent
wit
hth
em
odelV
Solv
ing
the
modelis
findin
gan
exp
ect
ati
on
funct
ion
(see
belo
ww
hen
we
will
solv
eth
em
odel.)
Wit
hth
isd
efin
itio
n(a
nd
som
ere
stri
ctio
ns
onth
em
odel
,i.e
.re
cur-
sive
rep
rese
ntat
ion
),w
eas
sum
eth
atag
ents
know
the
mod
elan
dth
e
pro
bab
ility
dis
trib
uti
onof
exog
enou
sva
riab
les
(or
shoc
ks)
that
hit
the
econ
omy.
Et≡E
(./It)
wh
ereI t
den
otes
the
info
rmat
ion
set
inp
erio
dt,
i.e.
wh
enth
eco
nsu
mer
(or
firm
)ag
ent
mak
esit
sd
ecis
ion
onco
nsu
mp
tion
s
(or
lab
or/c
apit
ald
eman
d).
Her
e,I t
incl
ud
esal
lth
eh
isto
ries
ofC
andY
,i.e
.{C
t,Ct−
1,....
;Yt,Yt−
1,...}
.
EtCt+
1is
thu
sth
elin
ear
pro
ject
ion
of
Ct+
1on{C
t,Ct−
1,....
;Yt,Yt−
1,...}
Pro
pert
ies
of
rati
onal
exp
ect
ati
ons
Pro
pert
y1:
No
syst
emat
icb
ias.
Let
the
exp
ecta
tion
erro
rεc t
+1
=
Ct+
1−EtCt+
1.T
his
erro
rte
rmsa
tisfi
es:
Etεc t+
1=
0
Pro
of:
Str
aigh
tfor
war
d(u
sin
gEt(A
+B
)=EtA
+EtB
andEtEtA
=
EtA
):
Etεc t+
1=Et(Ct+
1−EtCt+
1)=EtCt+
1−EtEtCt+
1
=EtCt+
1−EtCt+
1=
0
Pro
pert
y2:
Exp
ecta
tion
erro
rsd
on
otex
hib
itan
yse
rial
corr
ela-
tion
.
Pro
of:
Str
aigh
tfor
war
du
sin
gth
eco
nd
itio
nal
auto
–cov
aria
nce
fun
c-
tion
(see
the
pre
viou
sch
apte
rfo
ra
form
ald
efin
itio
nof
this
fun
ctio
n):
Covt(εc t
+1,εc t
)=Et(εc t
+1εc t)−Et(εc t
+1)Et(εc t
)
=Et(εc t
+1)εc t−Et(εc t
+1)εc t
=0
An
exam
ple
:an
AR
(1)
pro
cess
forYt
(see
bel
owth
esp
ecifi
cati
on
ofla
bor
inco
me
wag
es)
Yt
=ρYt−
1+εy t
Th
ein
form
atio
nse
tin
per
iodt
isgi
ven
byal
lth
ere
aliz
atio
ns
ofth
e
ran
dom
vari
ableY
from
per
iodt
(all
the
his
tory
atp
erio
dt
ofth
e
vari
ableY
),i.e
.
I t={Y
t,Yt−
1,...}
Fro
mth
isd
efin
itio
n,
we
get
EtYt+
1=E
(ρYt
+εy t
)=ρEtYt
+Etεy t+
1=ρYt
+Etεy t+
1
Sin
ceεy t+
1is
anin
nov
atio
n,
itis
orth
ogon
alto
the
info
rmat
ion
set
and
thu
sEtεy t+
1=
0.
Itfo
llow
sth
at
EtYt+
1=ρYt
Th
eex
pec
tati
oner
rors
εy t+1
=Yt+
1−EtYt+
1=Yt+
1−ρYt
thu
ssa
tisfi
es
Etεy t+
1=
0
Th
isp
rop
erty
ofra
tion
alex
pec
tati
ons
coin
cid
esw
ith
the
opti
mal
fore
-
cast
ofan
econ
omet
rici
anw
ho
use
the
obse
rvat
ion
san
dth
eA
R(1
)
pro
cess
tofo
rmu
late
anop
tim
alfo
reca
stofY
(on
est
ep–a
hea
d)
We
now
gob
ack
toou
rst
och
asti
cse
tup
Et
∞ ∑ i=0
βi U
(Ct+i)
=Et
( U(Ct)
+βU
(Ct+
1)+β
2 U(Ct+
2)+...)
wh
ere
Ct
=(1
+r)At
+Yt−At+
1
Ct+
1=
(1+r)At+
1+Yt+
1−At+
2
Ct+
2=
(1+r)At+
2+Yt+
2−At+
3
Aga
in,
don
’tfo
rget
:co
ntro
llin
gCt⇔
cont
rolli
ngAt+
1ev
ery
per
iod
(in
crea
sin
gcu
rren
tco
nsu
mp
tion
iseq
uiv
alen
tto
red
uce
savi
ng!
)
Et
( U(Ct)
+βU
(Ct+
1)+β
2 U(Ct+
2)+...) =
Et(U
((1+r)At+Yt−At+
1)
+βU
((1
+r)At+
1+Yt+
1−At+
2)
+β
2 U((
1+r)At+
2+Yt+
2−At+
3)+....
)
Th
enm
axim
ize
the
obje
ctiv
ew
ith
resp
ect
toAt+
1.
Rm
k:b
ecau
seAt
isp
re-d
eter
min
edan
dYt
iskn
own
inp
erio
dt
(an
d
r>
0is
con
stan
t),
the
leve
lof
finan
cial
wea
lth
inp
erio
dt
+1
is
know
n!
Et( −U
′ (Ct)
+β
(1+r)U′ (Ct+
1)) =
0
Th
isis
the
Eule
requati
on
onco
nsu
mp
tion
,b
ut
now
inex
pec
ta-
tion
s.W
eas
sum
eth
esa
me
uti
lity
fun
ctio
nas
bef
ore
and
we
imp
ose
the
sam
ere
stri
ctio
nβ
(1+r)
=1.
Th
isyi
eld
s
Et(−Ct
+Ct+
1)=
0
By
the
linea
rity
(in
con
sum
pti
on)
ofth
eE
ule
req
uat
ion
Et(−Ct
+Ct+
1)=−EtCt
+EtCt+
1
and
usi
ng
the
fact
that
the
curr
ent
con
sum
pti
onis
know
nin
per
iodt
(EtCt
=Ct)
,w
ed
edu
ceCt
=EtCt+
1
oreq
uiv
alen
tly
Et∆Ct+
1=
0
wh
ere
∆Ct+
1=Ct+
1−Ct.
We
can
also
wri
te
Ct+
1=Ct
+εc t
+1
wh
ere
the
ran
dom
vari
ableεc t
+1
sati
sfies
Etεc t+
1=
0
Imp
ort
ant
Quanti
tati
ve
Implica
tions
1)T
he
curr
ent
con
sum
pti
onis
anop
tim
alp
red
icto
rof
futu
reco
n-
sum
pti
on.
2)T
he
chan
gein
con
sum
pti
onis
un
pre
dic
tab
lean
dd
oes
not
dis
pla
y
seri
alco
rrel
atio
n.
3)T
he
con
sum
pti
onfo
llow
sa
ran
dom
wal
k.
Th
ese
quan
tita
tive
imp
licat
ion
sca
nb
ete
sted
easi
lyfr
omac
tual
dat
a
(see
the
dis
cuss
ion
bel
ow)
Aga
in,
we
mu
stso
lve
the
mod
el,
i.e.
we
mu
std
eriv
eth
eco
nsu
mp
tion
fun
ctio
n.
We
star
tfr
omth
ein
stan
tan
eou
sb
ud
get
con
stra
int
inp
erio
dt
At+
1=
(1+r)At
+Yt−Ct
All
the
vari
able
sar
ekn
own
inp
erio
dt.
Th
ism
ean
sth
atif
we
app
lyth
eco
nd
itio
nal
exp
ecta
tion
oper
ator
,w
e
get
the
sam
efo
rmu
lati
on.
Aga
in,
we
form
ula
tea
forw
ard
rep
rese
ntat
ion
ofth
isco
nst
rain
t
At
=1
1+rAt+
1+
1
1+rCt−
1
1+rYt
Now
,w
rite
this
equ
atio
nin
per
iodt
+1
At+
1=
1
1+rAt+
2+
1
1+rCt+
1−
1
1+rYt+
1
Th
ele
vel
ofw
ealt
hin
per
iodt
+1
iskn
own
inp
erio
dt,
bu
tth
e
vari
able
sCt+
1,Yt+
1an
dAt+
2ar
eu
nkn
own
.
Th
isis
bec
ause
the
inco
me
isa
stoc
has
tic
vari
able
and
thu
sag
ent
mu
stfo
rmu
late
(rat
ion
al)
exp
ecta
tion
sab
out
the
futu
rere
aliz
atio
ns
ofth
isva
riab
le.
Th
en,
we
app
lyth
eco
nd
itio
nal
exp
ecta
tion
Et
onth
isb
ud
get
con
-
stra
int
At+
1=
1
1+rEtAt+
2+
1
1+rEtCt+
1−
1
1+rEtYt+
1
and
then
sub
stit
ute
this
exp
ecte
db
ud
get
con
stra
int
inp
erio
dt
+1
into
the
bu
dge
tco
nst
rain
tin
per
iodt
At
=1
(1+r)
2EtAt+
2+1
(1+r)
2EtCt+
1−1
(1+r)
2EtCt+
1+1
1+rCt−
1
1+rYt
Aga
in,
wri
teth
eb
ud
get
con
stra
int
inp
erio
dt
+2,
take
exp
ecta
tion
s
con
dit
ion
alon
the
info
rmat
ion
set
inp
erio
dt
and
then
sub
stit
ute
this
exp
ress
ion
intoAt,
....
and
cont
inu
e.
Th
isyi
eld
s
At
=limT→∞Et
1
(1+r)TAt+T
+1
1+r
limT→∞Et
T−
1 ∑ i=0
1
(1+r)i(Ct+i−Yt+i)
Now
,im
pos
eth
e“t
ran
sver
salit
y”co
nd
itio
n(i
nex
pec
tati
ons)
limT→∞Et
1
(1+r)TAt+T
=0
mea
nin
gte
chn
ical
lyth
atth
ew
ealt
hca
nn
otex
plo
de
inex
pec
tati
ons
atso
me
rate
that
exce
edsr.
Th
ein
tert
emp
oral
bu
dge
tco
nst
rain
tis
give
nby
At
=1
1+rEt
∞ ∑ i=0
1
(1+r)i(Ct+i−Yt+i)
oreq
uiv
alen
tly
At
+Ht
=1
1+rEt
∞ ∑ i=0
1
(1+r)iCt+i
wh
ereHt
defi
nes
the
non
–fin
anci
alw
ealt
h(o
r“h
um
an”
wea
lth
),i.e
.
all
that
you
can
exp
ect
toea
rnd
uri
ng
you
r(i
nfin
ite)
par
tici
pat
ion
on
the
lab
orm
arke
t.
Ht
=1
1+rEt
∞ ∑ i=0
1
(1+r)iYt+i
To
obta
inth
eco
nsu
mp
tion
fun
ctio
n,
we
first
nee
dto
com
pu
te
1
1+rEt
( Ct
+1
1+rCt+
1+
1
(1+r)
2Ct+
2+...)
Fro
mth
eE
ule
req
uat
ion
onco
nsu
mp
tion
,w
ekn
owth
atth
eop
tim
al
dec
isio
nsa
tisfi
es
Ct+h
=Et+hCt+h
+1
forh
=0,
1,2,....
Mor
ep
reci
sely
,w
eh
ave
EtCt+
1=Ct
Et+
1Ct+
2=Ct+
1
Et+
2Ct+
3=Ct+
2
and
soon
.N
owu
se,
the
seco
nd
Eu
ler
equ
atio
n(E
ule
req
uat
ion
in
per
iodt
+1)
Et+
1Ct+
2=Ct+
1
and
app
lyth
eco
nd
itio
nal
exp
ecta
tion
sop
erat
orEt
EtEt+
1Ct+
2=EtCt+
1
By
defi
nit
ion
ofra
tion
alex
pec
tati
ons,
the
diff
eren
ceb
etw
een
the
in-
form
atio
nse
tin
per
iodt
andt
+1
isu
np
red
icta
ble
,so
EtEt+
1Ct+
2=EtCt+
2
We
then
ded
uce
EtCt+
2=EtCt+
1=Ct
Ifw
ep
erfo
rmth
esa
me
exer
cise
onth
eth
ird
Eu
ler
equ
atio
n,
we
get
EtCt+
3=EtCt+
2=EtCt+
1=Ct
Mor
ege
ner
ally
,w
eh
ave
EtCt+h
=EtCt+h−
1=...
=EtCt+
1=Ct
for
som
ep
osit
ive
inte
gerh
.
Fro
mth
eab
ove
com
pu
tati
ons,
we
can
easi
lyd
eter
min
e
1
1+rEt
( Ct
+1
1+rCt+
1+
1
(1+r)
2Ct+
2+...) =
1
1+rCt
( 1+
1
1+r
+1
(1+r)
2+...)
Bec
auser>
0,th
en1/
(1+r)<
0,so
the
sequ
ence
( 1+
1
1+r
+1
(1+r)
2+...) =
∞ ∑ i=0
1
(1+r)i
conv
erge
sto
afin
ite
num
ber
.
We
hav
e
∞ ∑ i=0
1
(1+r)i
=1
1−
11+r
≡1
+r
r
We
obta
in
1
1+rEt
∞ ∑ i=0
1
(1+r)iCt+i
=Ct r
Now
,if
we
rep
lace
this
exp
ress
ion
into
At
+Ht
=1
1+rEt
∞ ∑ i=0
1
(1+r)iCt+i
we
get
Ct
=r
(At
+Ht)
We
mu
stn
owd
eter
min
eth
en
on–fi
nan
cial
wea
lthHt.
To
do
that
,w
en
eed
tosp
ecif
ya
stoc
has
tic
pro
cess
forYt.
Th
isst
och
asti
cp
roce
ssis
per
fect
lykn
own
byth
eco
nsu
mer
.
We
assu
me
thatYt
isd
eter
min
edby
the
follo
win
gp
roce
ss
Yt
=Y
+Yt
wh
ereY
isa
know
np
osit
ive
con
stan
t(t
he
det
erm
inis
tic
per
man
ent
com
pon
ent)
and
the
stoc
has
tic
(pos
sib
lyh
igh
lyp
ersi
sten
t)co
mp
onen
t
isgi
ven
by
Yt
=ρYt−
1+εy t
wh
ereρ∈
[0,1
].T
he
ran
dom
term
sati
sfies
Etεy t+
1=
0
Th
en
on–fi
nan
cial
wea
lth
isgi
ven
by
Ht
=1
1+rEt
∞ ∑ i=0
1
(1+r)iYt+i,
So,
we
nee
dto
com
pu
te EtYt,EtYt+
1,EtYt+
2,...
Fro
mth
eab
ove
pro
cess
forYt,
we
ded
uce
EtYt
=Yt≡Y
+Yt
EtYt+
1=Et( Y+
Yt+
1) =Y
+EtYt+
1
=Y
+Et(ρYt
+εy t+
1)=Y
+ρYt
+Etεy t+
1
=Y
+ρYt
EtYt+
2=Et( Y+
Yt+
2) =Y
+EtYt+
2
=Y
+Et(ρYt+
1+εy t+
2)=Y
+ρEtYt+
1+Etεy t+
2
=Y
+ρ
2 Yt
and
we
cont
inu
e...
Th
en
on–fi
nan
cial
wea
lth
isth
end
edu
ced
Ht
=1
1+rY
∞ ∑ i=0
1
(1+r)i
+1
1+rYt
∞ ∑ i=0
ρi
(1+r)i
Sin
ceρ∈
[0,1
]an
dr>
0,it
follo
ws
that
ρ
(1+r)<
1
and
the
sequ
ence
ρi
(1+r)i
isco
nver
gent
.
We
obta
in
Ht
=Y r
+1
1+r−ρYt
Fro
mth
isex
pre
ssio
n,
we
see
that
the
pro
cess
forYt
mat
ters
alo
tfo
r
the
non
–fin
anci
alw
ealt
h.
Th
isis
bec
ause
agen
tsu
seth
est
och
asti
cp
roce
ssto
mak
eth
eir
fore
cast
abou
tth
eir
futu
rein
com
e.
Th
ism
ean
sth
atn
on–fi
nan
cial
wea
lth
isn
otin
vari
ant
toth
ep
aram
-
eter
sof
the
pro
cess
.
Usi
ngHt,
we
can
ded
uce
the
con
sum
pti
onfu
nct
ion
Ct
=rA
t+Y
+r
1+r−ρYt
Rm
k:
Th
eu
nde
term
ined
coeffi
cien
tsin
ast
och
asti
cse
tup
.
Th
eb
asic
idea
isto
iden
tify
the
form
ula
for
the
par
amet
ersφ
0,φ
1
andφ
2in
the
pos
tula
ted
(lin
ear)
equ
atio
n
Ct
=φ
0+φ
1At
+φ
2Yt
that
sati
sfied
bot
hth
eE
ule
req
uat
ion
onco
nsu
mp
tion
(op
tim
ald
e-
cisi
ons)
and
the
bu
dge
tco
nst
rain
t(r
estr
icti
ons
onth
ein
tert
emp
oral
allo
cati
onof
con
sum
pti
on).
Fir
stre
pla
ceth
ep
ostu
late
deq
uat
ion
into
the
Eu
ler
equ
atio
non
con
-
sum
pti
on
φ0
+φ
1EtAt+
1+φ
2EtYt+
1=φ
0+φ
1At
+φ
2Yt
We
know
that
EtAt+
1=At+
1b
ecau
seA
isp
re-d
eter
min
ed
and
Yt+
1=ρYt
Th
en,
φ1A
t+1
+φ
2ρYt
=φ
1At
+φ
2ρYt
Now
use
the
bu
dge
tco
nst
rain
t
At+
1=
(1+r)At
+Y
+Yt−Ct
and
sub
stit
ute
the
pos
tula
ted
equ
atio
nfo
rCt
φ1(
1+r)At
+φ
1Y+φ
1Yt−φ
1Ct
=φ
1At
+φ
2Yt
φ1(
1+r)At
+φ
1Y+φ
1Yt−φ
1φ0−φ
2 1At−φ
1φ2Yt
=φ
1At
+φ
2Yt
Iden
tify
the
term
s
φ1(
1+r)−φ
2 1=φ
1φ
1Y−φ
1φ0
=0
φ1−φ
1φ2
+φ
2ρ=φ
2
Th
isgi
ves
us
the
thre
ep
aram
eter
sφ
0=Y
,φ
1=r
andφ
2=r/
(1+
r−ρ
).
The
Lu
cas
Cri
tiqu
e
Lu
cas
(197
6)cl
aim
sth
atst
and
ard
mac
roec
onom
etri
cm
odel
sca
nn
ot
be
use
dfo
rp
olic
yev
alu
atio
n,
bec
ause
that
the
con
sum
pti
onfu
nct
ion
(am
ong
oth
erp
riva
teb
ehav
iors
)ar
en
otin
vari
ant
top
olic
yp
aram
e-
ters
.
Inot
her
wor
ds,
thes
em
acro
econ
omet
ric
mod
els
wro
ngl
yas
sum
eth
at
the
par
amet
ers
ofth
eco
nsu
mp
tion
fun
ctio
nd
oes
not
chan
gew
hen
agen
tsm
odif
yth
eir
exp
ecta
tion
saf
ter
ap
olic
ych
ange
.
To
see
this
,su
pp
ose
that
the
econ
omet
rici
anw
ron
gly
assu
mes
that
the
con
sum
pti
onfu
nct
ion
ofth
efo
rm:
Ct
=c 0
+c 1Yt
+c 2At
Th
issp
ecifi
cati
onis
rich
erth
anth
est
and
ard
keyn
esia
nco
nsu
mp
tion
fun
ctio
n,
bec
ause
itin
clu
des
som
ew
ealt
heff
ects
.
Bu
t,th
issp
ecifi
cati
onas
sum
esth
atth
ep
aram
eterc 1
isp
olic
yin
vari
-
ant.
Fro
mth
em
odel
’sso
luti
on,
this
par
amet
eris
give
nby
∂Ct
∂Yt
=r
1+r−ρ
Th
isp
aram
eter
isn
otin
vari
ant
toρ
,th
ep
aram
eter
sth
atgo
vern
sth
e
exp
ecta
tion
sab
out
futu
rein
com
e.
To
illu
stra
teth
isla
ckof
inva
rian
ce,
assu
me
pu
retr
ansi
tory
inco
me
shoc
k(ρ
=0)
∂Ct
∂Yt
=r
1+r
Th
ese
nsi
tivi
tyof
con
sum
pti
onto
inco
me
isve
rysm
all
(r=
0.01
per
quar
ter,
sose
nsi
tivi
tyar
oun
d1%
).
Th
isis
bec
ause
agen
tsex
pec
tson
lytr
ansi
tory
chan
ges
inth
ela
bor
inco
me.
Bec
ause
they
wan
tto
smoo
thth
eir
con
sum
pti
onov
erti
me,
they
al-
mos
td
isco
nn
ect
thei
rco
nsu
mp
tion
from
tran
sito
ryin
com
es.
Ass
um
en
owp
erm
anen
tin
com
esh
ocks
(ρ=
1).
Th
ese
nsi
tivi
tyis
now
∂Ct
∂Yt
=1
So,
the
resp
onse
isn
owon
e-to
-on
e,b
ecau
seag
ents
exp
ect
now
per
-
man
ent
chan
ges.
Model’Solu
tion
Ifw
ere
pla
ceth
eco
nsu
mp
tion
fun
ctio
nin
toth
ein
stan
tan
eou
sb
ud
get
con
stra
int
inp
erio
dt,
we
obta
in
At+
1=At
+1−ρ
1+r−ρYt
Inp
erio
dt−
1,w
eal
soh
ave
At
=At−
1+
1−ρ
1+r−ρYt−
1
Now
,d
efin
e∆At
=At−A−
1an
dta
keth
efir
std
iffer
ence
ofth
e
con
sum
pti
onfu
nct
ion ∆Ct
=∆At
+r
1+r−ρ
∆Yt
=1−ρ
1+r−ρYt−
1+
r
1+r−ρ
∆Yt
=r
1+r−ρYt−
rρ
1+r−ρYt−
1
∆Ct
=r
1+r−ρεy t
So
the
con
sum
pti
onfu
nct
ion
rew
rite
s(a
ran
dom
wal
k)
Ct
=Ct−
1+
r
1+r−ρεy t
Not
eth
atw
hen
com
pu
tin
gth
eco
nsu
mp
tion
fun
ctio
n,
we
det
erm
ine
the
rela
tion
ship
bet
wee
nan
inn
ovat
ion
onco
nsu
mp
tion
εc tan
dan
inn
ovat
ion
onst
och
asti
cin
com
eεy t
εc t=
r
1+r−ρεy t
Fro
mth
eab
ove
equ
atio
n,
we
can
det
erm
ine
the
dyn
amic
resp
onse
s
ofco
nsu
mp
tion
toan
un
exp
ecte
dsh
ock
tost
och
asti
cla
bor
inco
me.
Th
isis
give
nby
the
follo
win
gex
pre
ssio
n(I
RF
for
Imp
uls
eR
esp
onse
Fu
nct
ion
):
IRFc(h
)=∂Ct+h
∂εy t
forh
=0,
1,2,...
[In
sert
afig
ure
]
We
ded
uce
IRFc(
0)=
r
1+r−ρ,IRFc(
1)=
r
1+r−ρ,IRFc(
2)=
r
1+r−ρ,
Th
eIR
Fis
anin
crea
sin
gfu
nct
ion
ofρ
,co
nsu
mp
tion
ism
ore
sen
siti
ve
toan
inn
ovat
ion
tola
bor
inco
me.
ASp
eci
al
Case
Un
itro
otin
the
inco
me
pro
cess
(ρ=
1),s
up
por
ted
byth
eac
tual
dat
a
inin
du
stri
aliz
edco
unt
ries
. Yt
=Yt−
1+εy t
So,
the
chan
gein
lab
orin
com
e∆Yt
isgi
ven
by
∆Yt
=εy t
We
ded
uce
from
the
abov
efo
rmu
la
Ct
=rA
t+Yt
At+
1=At
∆Ct
=∆Yt
Qu
anti
tati
veim
plic
atio
ns
1)V
olat
ility
σ(∆Ct)
=σ
(∆Yt)
Inth
eac
tual
dat
a(i
nin
du
stri
aliz
edco
unt
ries
,w
her
eC
ism
easu
red
asth
esu
mof
non
–du
rab
lego
ods
and
serv
ices
)
σ(∆Ct)
=0.
7σ(∆Yt)
“Exc
ess
Sm
ooth
nes
sP
uzz
le”
(Dea
ton
,19
87)
2)P
ersi
sten
ce
∆Ct
=εy t
Con
sum
pti
onch
ange
sd
on
otd
isp
lay
seri
alco
rrel
atio
n.
Inth
ed
ata,
Corr(
∆Ct,
∆Ct−
1)'
0.3
So
con
sum
pti
onch
ange
sar
ep
red
icta
ble
.O
ther
emp
iric
alst
ud
ies
hav
e
show
nth
ata
set
ofre
leva
ntva
riab
les
(pas
tco
nsu
mp
tion
,p
asin
com
e)
hel
pto
pre
dic
tfu
ture
con
sum
pti
on.
Ext
ensi
ons
-H
abit
per
sist
ence
inco
nsu
mp
tion
-B
orro
win
gco
nst
rain
t
2.2
:T
he
Dyn
am
icL
ab
or
Dem
an
dM
od
el
Sta
nd
ard
theo
ryon
lyco
nsi
der
shor
t–ru
nan
dlo
ng–
run
lab
ord
eman
d
wit
hou
tan
yex
plic
itad
just
men
tp
roce
ssb
etw
een
thes
etw
osi
tuat
ion
s.
Th
eth
eory
ofla
bor
dem
and
wit
had
just
men
tco
sts
offer
san
opp
or-
tun
ity
tofil
lth
ega
pb
etw
een
thes
estw
osi
tuat
ion
s.
Quest
ion:
Why
adju
stm
ent
cost
son
lab
or??
Poss
ible
Expla
nati
ons:
Cos
tof
adju
stin
gem
plo
ymen
t
∆Lt
=Ht−Ft
wh
ereHt
den
otes
hir
ings
andFt
den
otes
firin
gs.
•H
irin
gco
sts:
trai
nin
gco
sts,
re–o
rgan
izat
ion
cost
s
•F
irin
gco
sts:
legi
slat
ion
(em
plo
ymen
tp
rote
ctio
n)
Em
pir
ical
stu
die
ssu
gges
t
•In
US
,h
irin
gco
sts
are
hig
han
dou
tstr
ipth
eco
stof
sep
arat
ion
•In
cou
ntri
esw
her
est
ron
gle
gal
mea
sure
sar
ein
pla
ceto
ensu
re
job
secu
rity
(man
yco
unt
ries
ofco
ntin
enta
lE
uro
pe)
,th
eco
sts
of
sep
arat
ion
far
outs
trip
recr
uit
men
tco
sts.
The
theore
tica
lse
tup:
ad
ynam
icla
bor
dem
and
mod
el
Pos
sib
lefo
rms
ofad
just
men
tco
sts.
Tw
ove
rsio
ns:
•D
eter
min
isti
cse
tup
.
•S
toch
asti
cse
tup
.
The
Cost
sof
Lab
or
Adju
stm
ents
Qu
adra
tic
Cos
ts
Let
us
con
sid
erth
efir
stsp
ecifi
cati
on
C(Lt,Lt−
1)=b 2
(Lt−Lt−
1−a
)2
wh
erea,b>
0.
Ad
vant
age:
this
fun
ctio
nin
trod
uce
san
asym
met
ryb
etw
een
the
cost
ofp
osit
ive
and
neg
ativ
eva
riat
ion
sin
emp
loym
ents
,si
ncea>
0.
Lim
it:
wh
enL
isco
nst
ant,
ther
eex
ists
stri
ctly
pos
itiv
eco
sts
ofad
-
just
ing
emp
loym
ent
(!!!)
,si
ncea>
0.
To
elim
inat
eth
isp
rob
lem
,se
ta
=0
inth
ep
revi
ous
spec
ifica
tion
C(Lt,Lt−
1)=b 2
(Lt−Lt−
1−a
)2
wh
erea>
0.
Bu
tn
ow,
cost
sar
eco
nvex
and
sym
metr
ic.
Asy
mm
etri
cC
onve
xC
osts
:
C(Lt,Lt−
1)=−
1+
exp
(Lt−Lt−
1)−a
(Lt−Lt−
1)+b 2
(Lt−Lt−
1)2
Dep
end
ing
onth
eva
lue
ofa
,th
em
argi
nal
cost
ofan
incr
ease
in
emp
loym
ent
isgr
eate
r(o
rsm
alle
r)th
anth
atof
are
du
ctio
n.
Wh
ena
=0,
we
retr
ieve
the
quad
rati
cca
se.
Not
eth
atw
henL
isco
nst
ant,
the
cost
ofad
just
men
tis
zero
.
An
oth
erfo
rm:
C(Lt,Lt−
1)=b h 2
(Lt−Lt−
1)2
ifLt−Lt−
1≥
0
and
C(Lt,Lt−
1)=b f 2
(Lt−Lt−
1)2
ifLt−Lt−
1≤
0
wh
ereb h,bf>
0
Lin
ear
Cos
ts:
C(Lt,Lt−
1)=b h
(Lt−Lt−
1)ifLt−Lt−
1≥
0
and
C(Lt,Lt−
1)=−b f
(Lt−Lt−
1)ifLt−Lt−
1≤
0
wh
ereb h,bf>
0
Th
ep
aram
eterb h
rep
rese
nts
the
un
itco
stof
ah
irin
gan
db f
isth
e
un
itco
stof
firin
g.
Th
ead
just
men
tof
emp
loym
ent
isas
ymm
etri
csi
nceb h6=b f
Fix
edC
osts
:
Fir
ms
face
ast
rict
lyp
osit
ive
cost
wh
en
Lt−Lt−
16=
0,
bu
tth
eyar
en
otsu
bje
ctto
any
cost
if
Lt−Lt−
1=
0
The
Dete
rmin
isti
cSetu
p
Th
eva
lue
ofth
efir
mV t
isth
ed
isco
unt
edsu
mof
inst
anta
neo
us
pro
fits
V t=
∞ ∑ i=0
( 1
1+r
) i Πt+i
wh
erer>
0is
aco
nst
ant
real
inte
rest
rate
and
the
inst
anta
neo
us
pro
fitis
give
nby
Πt
=Yt−WtLt−C(Lt,Lt−
1)
wh
ereWt
isth
ere
alw
age
(th
ep
rice
ofYt
isu
sed
asa
nu
mer
aire
.
Rm
k:
As
bef
ore,
we
can
defi
ne
the
valu
eof
the
firm
ina
recu
rsiv
e
way
.
V t=
Πt
+
( 1
1+r
) V t+1
Th
ead
just
men
tco
stfu
nct
ion
isgi
ven
by
C(Lt,Lt−
1)=b 2
(Lt−Lt−
1)2
wh
ereb≥
0is
the
adju
stm
ent
cost
par
amet
er.
Wt
isas
sum
edto
be
per
fect
lykn
own
(no
shoc
ks)
Th
ep
rod
uct
ion
fun
ctio
nis
give
nby
Yt
=α
0Lt−α
1 2L
2 t
wh
ereα
0,α
1>
0.T
he
pro
du
ctio
nfu
nct
ion
sati
sfies
∂Y
∂Y
=α
0−α
1Lt>
0
⇔Lt<
(α0/α
1)(w
eas
sum
eth
atth
isre
stri
ctio
nh
old
s)
∂2 Y
∂Y
2=−α
1<
0
Dec
reas
ing
retu
rnto
scal
e.
Ifα
1=
0,co
nst
ant
retu
rnto
scal
ean
dth
em
argi
nal
lab
orp
rod
uct
ivit
y
isin
dep
end
ent
from
the
leve
lof
outp
ut
(or
emp
loym
ent)
.
Inte
rtem
pora
ldeci
sions
Th
ed
ecis
ion
onLt
tod
ayaff
ect
pro
fits
tod
ay:
pro
du
ctio
n(F
(Lt)
),
lab
orco
st(W
t),
cost
ofad
just
men
t((b/
2)(Lt−Lt−
1)2 )
Th
ed
ecis
ion
onLt
tod
ayaff
ect
pro
fits
tom
orro
w:
cost
ofad
just
men
t
((b/
2)(Lt+
1−Lt)
2 )
So,
choi
ces
onLt
are
inte
rdep
end
ent.
Ifb
=0,
the
valu
eof
the
firm
isth
ed
isco
unt
edsu
mof
ind
epen
den
t
pro
fitfu
nct
ion
s(a
sequ
ence
ofst
atic
con
dit
ion
s)
Lt
=α
0−Wt
α1
Sta
tus
of
the
vari
able
s
•Wt
isan
exogenous
det
erm
inis
tic
vari
able
.
•Lt−
1is
ap
re-d
eter
min
edva
riab
le
•{W
t,Lt−
1}ar
eth
est
ate
vari
able
s(t
he
solu
tion
will
be
ali
nea
r
fun
ctio
nof
thes
estw
ova
riab
les)
•Lt
isa
the
choi
ceva
riab
le.
FO
C(w
rtLt)
∂V t∂Lt
=0
⇔
∂Π
∂Lt
+
( 1
1+r
) ∂Πt+
1
∂Lt
=0
⇔
α0−α
1Lt−Wt−b(Lt−Lt−
1)+βb(Lt+
1−Lt)
=0
wh
ereβ
=1/
(1+r)∈
(0,1
).
Rm
kIfα
1=
0(c
onst
ant
retu
rnto
scal
e,se
eth
eex
erci
sein
tuto
rial
clas
s),
the
FO
Cre
du
ces
to
α0−Wt−bN
Ht
+βbN
Ht+
1=
0
wh
ereNHt
=∆Lt
are
the
net
hir
ings
.∆Lt>
0(¡
0)→
NHt>
0
(¡0)
.
We
stu
dy
the
caseα
1>
0
LetL
the
stea
dy
stat
eva
lue
ofLt
andLt
=Lt−L
.
Th
est
ead
y–st
ate
ofL
solv
es:
α0−α
1L−W−b(L−L
)+βb(L−L
)=
0
wh
ich
red
uce
sto
α0−α
1L−W
=0
So,
L=
(α0−W
)/α
1
the
“lon
g–ru
n”
lab
ord
eman
d.
We
hav
e
∂L
∂W
=−
1 α1<
0
Ap
erm
anen
tin
crea
sein
the
real
wag
e(t
he
real
cost
ofla
bor
)re
du
ce
lon
g-ru
nem
plo
ymen
t.
Wit
hou
rd
efin
itio
nofLt,
the
FO
Cre
wri
tes
−α
1Lt−Wt−b(Lt−Lt−
1)+βb(Lt+
1−Lt)
=0
or
βbL
t+1−
(α1
+b
+βb)Lt
+bL
t−1
=Wt
Lt+
1−
(α1
+b
+βb)
βb
Lt
+1 βLt−
1=
1 βbW
t
Th
ese
con
dor
der
diff
eren
ceeq
uat
ion
adm
its
the
follo
win
gm
ult
ivar
i-
ate
auto
regr
essi
vere
pre
sent
atio
n[ L
t+1
Lt
] =
[ (α1+b+βb)
βb
−1 β
10
][Lt
Lt−
1
] +
[ 1 βb
0
00
] [ Wt
0
]N
ow,
con
cent
rate
only
onth
ed
ynam
icp
rop
erti
esofLt
and
ign
ore
Wt
(or
setWt
=0,
i.e.Wt
isal
way
sat
its
stea
dy–
stat
eva
lue)
[ Lt+
1
Lt
] =A
[ Lt
Lt−
1
]w
her
e
A=
[ (α1+b+βb)
βb
−1 β
10
]T
ost
ud
yth
ed
ynam
icp
rop
erti
esof
the
mod
el,
we
mu
stch
arac
teri
ze
the
eige
nval
ues
ofA
.
Th
ese
are
obta
ined
from
det
(A−λI 2
)=
0
Inou
rn
otat
ion
s∣ ∣ ∣ ∣ ∣(α
1+b+βb)
βb
−λ−
1 β1
−λ
∣ ∣ ∣ ∣ ∣=0
Th
isgi
ves
λ2−λT
+D
=0
wh
ereT
=trace
(A)
andD
=det
(A).
Giv
enA
,w
eob
tain
T=
(α1
+b
+βb)
βb
and
D=
1 β
We
can
now
defi
ne
the
char
acte
rist
iceq
uat
ion
P(λ
)=λ
2−λT
+D≡
(λ−λ
1)(λ−λ
2)
Letλ
=0.
We
hav
e
P(0
)=D≡
1 β>
1
We
also
know
that
P(0
)=λ
1λ2
At
leas
ton
ero
ot(s
ayλ
2)is
grea
ter
than
one.
Let
us
defi
neP
(1).
We
obta
in
P(1
)=
1−T
+D≡
1−
(α1
+b
+βb)
βb
+1 β
Itfo
llow
sth
at
P(1
)=−α
1
βb<
0
We
know
that
P(1
)=
(1−λ
1)(1−λ
2)
AsP
(1)<
0,w
ed
edu
ceλ
1<
1an
dλ
2>
1.
Pro
pert
yT
he
lab
ord
eman
dm
odel
wit
had
just
men
tco
sts
dis
pla
ys
asa
dd
lep
ath
pro
per
ty,
i.e.
itex
ists
au
niq
ue
pat
hto
war
dth
est
ead
y–
stat
e.
Com
puta
tion
of
the
solu
tion
We
know
thatλ
1<
1an
dλ
2>
1
and
her
ew
eas
sum
eWt
=0.
Now
,co
nsi
der
the
bac
ksh
ift
oper
atorB
,su
chth
at
BLt
=Lt−
1
Th
eF
OC
wri
tes Lt+
1−
(α1
+b
+βb)
βb
Lt
+1 βLt−
1=
0
oreq
uiv
alen
tly Lt+
1
( 1−
(α1
+b
+βb)
βb
B+
1 βB
2) =0
Usi
ng
the
char
acte
rist
icp
olyn
omia
l,w
eob
tain
Lt+
1(1−λ
1B)(
1−λ
2B)
=0
We
ded
uce
( ˜ Lt+
1(1−λ
2B)) =
0
wh
ere
˜ Lt+
1=
(1−λ
1B)Lt+
1=Lt+
1−λ
1Lt
Th
eab
ove
equ
atio
nre
wri
tes
˜ Lt+
1−λ
2˜ Lt
=0
˜ Lt+
1=λ
2˜ Lt
˜ Lt
=1 λ2Et˜ Lt+
1
Sin
ceλ
2>
1,w
eh
ave
1/λ
2<
1.S
oth
em
odel
mu
stb
eso
lved
forw
ard
,i.e
.by
succ
essi
veit
erat
ion
son
the
futu
re.
Aft
erim
pos
ing
the
“tra
nsv
ersa
lity”
con
dit
ion
limT→∞
( 1 λ2
) T ˜ Lt+T
=0
we
ded
uce
˜ Lt
=0
Now
,u
sin
gth
ed
efin
itio
nofLt,
we
get
Lt
=λ
1Lt−
1
Now
usi
ng
the
defi
nit
ion
ofLt,
we
ded
uce
Lt
=(1−λ
1)L
+λ
1Lt−
1
or
Lt
=(1−λ
1)(α
0−W
)
α1
+λ
1Lt−
1
So,
the
lab
ord
eman
dca
nd
epar
tfr
omit
slo
ng–
run
(”fr
icti
onle
ss”)
valu
e,d
ue
toth
eco
sts
ofad
just
ing
emp
loym
ent
bet
wee
ntw
op
erio
ds.
Th
esp
eed
ofad
just
men
tto
war
dL
will
dep
end
onth
ead
just
men
t
cost
sp
aram
eter
.
Forb
smal
l,th
eco
stis
smal
lto
o,an
dth
ed
iscr
epan
cyb
etw
eenL
and
Lis
not
per
sist
ent.
Con
vers
ely,
wh
enb
isla
rge,
the
adju
stm
ent
pro
cess
tow
ard
the
lon
g
run
valu
eof
emp
loym
ent
can
be
very
per
sist
ent.
The
Eff
ect
of
on
incr
ease
inW
Let
us
sup
pos
etw
ore
alw
agesW
1an
dW
2,
W1<W
2
Itfo
llow
sth
at
L1>L
2
Now
,w
ew
ant
toev
alu
ate
the
tran
siti
onp
ath
forLt
from
L1
toL
2.
We
star
tw
ithLt
=L
1.N
owu
sin
gth
eso
luti
onfo
rLt
wit
hth
en
ew
real
wag
e,w
ed
edu
ce
Lt+
1=
(1−λ
1)L
2+λ
1Lt≡
(1−λ
1)L
2+λ
1L1
On
ep
erio
daf
ter
Lt+
2=
(1−λ
1)L
2+λ
1Lt+
1≡
(1−λ
1)L
2+λ
1((1−λ
1)L
2+λ
1L1)
At
the
limit
(wh
enh→∞
),w
eob
tain
limh→∞Lt+h
=limh→∞λh 1L
1+
(1−λ
1)L
2limh→∞
h ∑ i=0
λh 1
Sin
ceλ
1<
1,w
eh
ave
limh→∞λh 1L
1=
0
and
limh→∞
h ∑ i=0
λh 1
=1
1−λ
1
We
ded
uce
limh→∞Lt+h
=L
2
Th
ela
bor
mon
oton
ical
lyco
nver
geto
war
dit
sn
ewst
ead
y–st
ate.
[IN
SE
RT
AF
IGU
RE
]
The
Sto
chast
icSetu
p
Th
eva
lue
ofth
efir
mV t
isth
eex
pec
ted
dis
cou
nted
sum
ofp
rofit
.
V t=Et
∞ ∑ i=1
( 1
1+r
) i Πt+i
wh
ereEt
isth
eco
nd
itio
nal
exp
ecta
tion
sop
erat
or.
r>
0(t
he
con
stan
tre
alin
tere
stra
te)
and
the
inst
anta
neo
us
pro
fitis
give
nby
Πt
=Yt−WtLt−b 2
(Lt−Lt−
1)2
wh
ereWt
isth
ere
alw
age
(th
ep
rice
ofYt
isu
sed
asa
nu
mer
aire
and
b≥
0is
the
adju
stm
ent
cost
par
amet
er.
Th
ep
rod
uct
ion
fun
ctio
nis
give
nby
Yt
=α
0Lt−α
1 2L
2 t
wh
ereα
0,α
1>
0.
Rm
k1
Pro
per
ties
ofth
ep
rod
uct
ion
fun
ctio
n
∂Y
∂Y
=α
0−α
1Lt>
0
⇔Lt<
(α0/α
1)(w
eas
sum
eth
atth
isre
stri
ctio
nh
old
sfo
ral
lth
e
stat
esof
the
nat
ure
)
∂2 Y
∂Y
2=−α
1<
0
Dec
reas
ing
retu
rnto
scal
e.Ifα
1=
0,co
nst
ant
retu
rnto
scal
ean
dth
e
mar
gin
alla
bor
pro
du
ctiv
ity
isin
dep
end
ent
from
the
leve
lof
outp
ut
(or
emp
loym
ent)
.
Rm
k2
Sh
ocks
Her
e,w
eas
sum
eth
aton
lyre
alw
age
isa
ran
dom
vari
able
.W
ith
-
out
diffi
cult
yw
eca
nal
soin
clu
de
aT
FP
shoc
kin
toth
ep
rod
uct
ion
fun
ctio
n
Yt
=α
0,tLt−α
1 2L
2 t
wh
ereα
0,t
isn
owa
ran
dom
vari
able
We
spec
ify
ast
och
asti
cp
roce
ssfo
rWt.
We
assu
me
thatWt
can
be
dec
omp
osed
into
two
elem
ents
:on
eis
det
erm
inis
tic,
one
isst
och
asti
c
Wt
=W
+Wt
Wd
eter
min
isti
c,Wt
stoc
has
tic.
Th
est
och
asti
cco
mp
onen
tWt
follo
ws
anA
uto
regr
essi
vep
roce
ssof
ord
eron
e(A
R(1
)):
Wt
=ρwWt−
1+σε,wε w
,t
wh
ereρw∈
[0,1
],σε,w>
0an
dε w
,tis
iidw
ith
zero
mea
nan
du
nit
vari
ance
.
Inad
dit
ion
,it
veri
fies:
Etεw,t
+1
=0
Rm
k3
Aga
in,
we
assu
me
her
era
tion
alex
pec
tati
ons
Rm
k4
Inte
rtem
por
ald
ecis
ion
s
Th
ed
ecis
ion
onLt
tod
ayaff
ect
pro
fits
tod
ay:
pro
du
ctio
n(F
(L))
,
lab
orco
st(W
),co
stof
adju
stm
ent
((b/
2)(Lt−Lt−
1)2 )
Th
ed
ecis
ion
onLt
tod
ayaff
ect
pro
fits
tom
orro
w:
cost
ofad
just
men
t
((b/
2)(Lt+
1−Lt)
2 )
So,
choi
ces
onLt
are
inte
rdep
end
ent.
Bu
t,b
ecau
seWt
isst
och
asti
c,th
iscr
eate
sa
dec
isio
nab
out
anex
-
pec
ted
vari
ableEtLt+
1
Ifb
=0,
the
valu
eof
the
firm
isth
ed
isco
unt
edsu
mof
ind
epen
den
t
pro
fitfu
nct
ion
s(a
sequ
ence
ofst
atic
con
dit
ion
s)
Rm
k5
Sta
tus
ofth
eva
riab
les
•Wt
isan
exogenous
stoc
has
tic
vari
able
.
•Lt−
1is
ap
re-d
eter
min
edva
riab
le
•{W
t,Lt−
1}ar
eth
est
ate
vari
able
s(t
he
solu
tion
will
be
ali
nea
r
fun
ctio
nof
thes
estw
ova
riab
les)
•Lt
isa
the
choi
ceva
riab
le.
FO
C(w
rtLt)
∂V t∂Lt
=0
⇔
∂Π
∂Lt
+
( 1
1+r
) Et∂
Πt+
1
∂Lt
=0
⇔
α0−α
1Lt−Wt−b(Lt−Lt−
1)+βbE
t(Lt+
1−Lt)
=0
wh
ereβ
=1/
(1+r)in
(0,1
)
Rm
kIfα
1=
0(c
rs),
the
FO
Cre
du
ces
to
α0−Wt−bN
Ht
+βbE
tNHt+
1=
0
wh
ereNHt
=∆Lt
are
the
net
hir
ings
.∆Lt>
0(¡
0)→
NHt>
0
(¡0)
.S
eeth
eH
omew
ork.
Her
e,w
eso
lve
the
mod
elin
the
mor
ege
ner
alca
sew
her
eα
1>
0
Sec
ond
ord
erd
iffer
ence
equ
atio
nw
ith
stoc
has
tic
real
wag
e
βbE
tLt+
1−
(βb
+b
+α
1)Lt
+bL
t−1
=(W
t−α
0)
Div
ide
byβb
EtLt+
1−
(βb
+b
+α
1)
βb
Lt
+1 βLt−
1=
(Wt−α
0)
βb
As
inth
ep
revi
ous
sect
ion
,d
efin
ela
bor
and
real
wag
eat
det
erm
inis
tic
stea
dy
stat
e(W
t=W
).
Fro
mth
eab
ove
equ
atio
n,
we
imm
edia
tely
obta
in
L=α
0−W
α1
Th
enea
chva
riab
lein
the
FO
Cca
nb
eex
pre
ssed
ind
evia
tion
from
thei
rst
ead
y–st
ate
valu
es:
EtLt+
1−
(βb
+b
+α
1)
βb
Lt
+1 βLt−
1=
1 βbW
t
Not
ice
that
Wt
=Wt−W≡Wt
Now
,co
nsi
der
agai
nth
eb
acks
hif
top
erat
orB
,su
chth
at
BLt
=Lt−
1
Th
isal
low
sto
rew
rite
Et
( Lt+
1
( 1−
(βb
+b
+α
1)
βb
B+
1 βB
2))=
1 βbW
t
Th
ed
ynam
ics
ofla
bor
adju
stm
ent
isex
actl
yth
esa
me
asin
the
de-
term
inis
tic
case
.
So,
we
hav
eon
ero
otla
rger
than
un
ity
(sayλ
2>
1)an
dth
ese
con
d
bet
wee
n0
and
1(s
ayλ
1∈
[0,1
)).
Up
toth
esh
ockWt
and
con
dit
ion
alex
pec
tati
ons,
the
mod
elis
very
sim
ilar
toth
eon
eob
tain
edin
the
det
erm
inis
tic
case
.
Now
rew
rite
the
exp
ecta
tion
s
Et
( ˜ Lt+
1(1−λ
2B)) =
1 βbW
t
wh
ere
˜ Lt+
1=
(1−λ
1B)Lt+
1=Lt+
1−λ
1Lt
Th
eab
ove
equ
atio
nre
wri
tes
Et
( ˜ Lt+
1−λ
2˜ Lt) =
1 βbW
t
Et˜ Lt+
1=λ
2˜ Lt
+1 βbW
t
˜ Lt
=1 λ2Et˜ Lt+
1−
1 βb
1 λ2Wt
Sin
ceλ
2>
1,w
eh
ave
1/λ
2<
1.S
oth
em
odel
mu
stb
eso
lved
forw
ard
,i.e
.by
succ
essi
veit
erat
ion
son
the
futu
re.
Wri
teth
iseq
uat
ion
int+
1(t
ake
care
,d
on’t
forg
etto
adju
stth
eti
me
ind
exon
the
avai
lab
lein
form
atio
nse
t!)
˜ Lt+
1=
1 λ2Et+
1˜ Lt+
2−
1 βb
1 λ2Wt+
1
and
then
sub
stit
ute
this
equ
atio
nin
toth
ep
revi
ous
one
˜ Lt
=
( 1 λ2
) 2 EtEt+
1˜ Lt+
2−
1 βb
1 λ2Wt−
1 βb
( 1 λ2
) 2 EtW
t+1
and
cont
inu
e....
.(r
emar
kth
atEtEt+
1=Et
orEtEt+
1Et+q
=Et)
At
the
limit
we
obta
in
˜ Lt
=limT→∞
( 1 λ2
) T Et˜ Lt+T−
1 βb
limT→∞Et
1 λ2
T ∑ i=0
( 1 λ2
) i Wt+i
or
˜ Lt
=−
1 βbE
t1 λ2
∞ ∑ i=0
( 1 λ2
) i Wt+i
asw
eim
pos
eth
e”t
ran
sver
salit
y”co
nd
itio
n
limT→∞
( 1 λ2
) T Et˜ Lt+T
=0
To
solv
eth
iseq
uat
ion
,w
em
ust
com
pu
teth
esu
cces
sive
exp
ecta
tion
s
EtW
t,EtW
t+1,EtW
t+2,EtW
t+3,....
Now
,w
eu
seth
est
och
asti
cp
roce
ssfo
rWt
EtW
t=Wt
EtW
t+1
=ρwWt
EtW
t+2
=ρ
2 wWt
EtW
t+3
=ρ
3 wWt
and
cont
inu
e...
Aft
er,
dis
cou
ntea
chex
pec
tati
on
i=
0:
1×EtW
t=Wt
i=
1:
( 1 λ2
) ×EtW
t+1
=
( ρ w λ2
) Wt
i=
2:
( 1 λ2
) 2 ×EtW
t+2
=
( ρ w λ2
) 2 Wt
....
i=q
:
( 1 λ2
) q ×EtW
t+q
=
( ρ w λ2
) q Wt
Th
enco
llect
thes
eco
mp
uta
tion
s
˜ Lt
=−
1 βb
1 λ2Wt
∞ ∑ i=0
( ρ w λ2
) i
Sin
ce
( ρ w λ2
) <1
Th
ese
ries
isco
nver
gent
and
its
sum
isgi
ven
by
∞ ∑ i=0
( ρ w λ2
) i =1
1−ρw λ2
≡λ
2
λ2−ρw>
0
Now
,re
pla
ceth
emin
˜ Lt
˜ Lt
=−
1 βb
1 λ2
λ2
λ2−ρwWt
Th
issi
mp
lifies
˜ Lt
=−
1 βb
Wt
λ2−ρw
Now
usi
ng
the
defi
nit
ion
of˜ Lt,
we
ded
uce
Lt
=λ
1Lt−
1−
1 βb
Wt
λ2−ρw
and
the
defi
nit
ion
ofLt
andWt
Lt
=(1−λ
1)L
+λ
1Lt−
1−
1 βb
Wt−W
λ2−ρw
wh
ere
Wt−W
=Wt
Wt
=ρwWt−
1+σε,wε w
,t
wh
ereρw∈
[0,1
],σε,w>
0an
dε w
,tis
iidw
ith
zero
mea
nan
du
nit
vari
ance
.
Inad
dit
ion
,L
isth
elo
ng–
run
(”fr
icti
onle
ss”)
lab
ord
eman
d:
L=α
0−W
α1
Sim
ilar
toth
ed
eter
min
isti
cca
se,
bu
tn
owit
incl
ud
esa
ran
dom
vari
-
able
,Wt
that
rep
rese
nts
the
exce
ssre
alw
age
(wit
hre
spec
tto
the
lon
g–ru
nre
alw
age)
.
Wh
enit
take
sla
rge
and
per
sist
ent
valu
es,
the
lab
ord
eman
dw
ill
dec
reas
ep
ersi
sten
tly.
Anoth
er
Imp
ort
ant
Asp
ect
Th
eL
UC
AS
CR
ITIQ
UE
(AG
AIN
!!)
Th
ela
bor
dem
and
equ
atio
n
Lt
=(1−λ
1)L
+λ
1Lt−
1−
1 βb
Wt−W
λ2−ρw
isn
otin
vari
ant
toth
efo
rmof
the
stoc
has
tic
pro
cessWt,
i.e.
itis
very
sen
siti
vetoρw
.
Ifch
ange
sin
the
real
wag
esar
etr
ansi
toryρw
=0,
the
red
uce
dfo
rm
for
lab
ord
eman
dis
Lt
=(1−λ
1)L
+λ
1Lt−
1−
1 βb
Wt−W
λ2
Wh
enth
ese
chan
ges
are
very
per
sist
ent
(ρw'
1),
the
lab
ord
eman
d
rew
rite
s
Lt
=(1−λ
1)L
+λ
1Lt−
1−
1 βb
Wt−W
λ2−
1
Itis
easy
tove
rify
1 λ2<
1
λ2−
1
So,
lab
ord
eman
dis
less
sen
siti
veto
real
wag
e,w
hen
thei
rch
ange
s
are
per
ceiv
edas
tran
sito
ry(c
omp
ared
tove
ryp
ersi
sten
t).
Com
puta
tion
of
Dynam
icR
esp
onse
s:
Th
eIR
Fs
(Im
pu
lse
Res
pon
seF
un
ctio
n)
are
give
nby
∂Lt+h
∂ε w
,t
forh≥
0is
the
hor
izon
ofth
ere
spon
se.
Let
us
con
sid
er
Lt
=(1−λ
1)L
+λ
1Lt−
1+µWt
wh
ereµ
isgi
ven
by
µ=−
1 βb
1
λ2−ρw
We
nor
mal
izeσε,w
toon
e.
h=
0
∂Lt
∂ε w
,t=µ∂Wt
∂ε w
,t=µ
h=
1
∂Lt+
1
∂ε w
,t=∂Lt+
1
∂Lt
∂Lt
∂ε w
,t+µ∂Wt+
1
∂Wt
∂Wt
∂ε w
,t
∂Lt+
1
∂ε w
,t=λ
1µ+ρwµ
=µ
(λ1
+ρw
)
Rm
k:
the
resp
onse
wh
enh
=1
can
be
grea
ter
than
onim
pac
t
(h=
0)w
hen
λ1
+ρw>
1
For
anyh
,th
ech
ain
rule
diff
eren
ciat
ion
∂Lt+h
∂ε w
,t=
∂Lt+h
∂Lt+h−
1···∂Lt
∂ε w
,t+µ∂Wt+h
∂Wt+h−
1···∂Wt
∂ε w
,t
[IN
SE
RT
AF
IGU
RE
WIT
HA
HU
MP
–SH
AP
ED
RE
-
SP
ON
SE
]
Em
pir
ical
Evid
en
ces
Sum
mary
of
the
resu
lts
For
conv
enie
nce
,m
ost
ofem
pir
ical
wor
ksu
ses
aqu
adra
tic
spec
ifica
-
tion
for
adju
stm
ent
cost
s.
Ass
um
ing
thatL
andW
are
obse
rvab
le,
the
par
amet
ers
ofth
em
odel
are
obta
ined
from
the
equ
atio
n
Lt
=(1−λ
1)L
+λ
1Lt−
1−Wt−
1 βbW
λ2−ρw
wh
ereλ
1an
dλ
2ar
efu
nct
ion
sof
the
”dee
p”
par
amet
ersβ
,b
andα
1
andL
isa
fun
ctio
nofL
,α
0an
dα
1.
At
the
sam
eti
me,
the
pro
cess
ofWt
isgi
ven
by
Wt−W
=Wt
Wt
=ρwWt−
1+σε,wε w
,t
Th
eec
onom
etri
cian
mu
stac
cou
ntfo
rth
ecr
oss
equ
atio
nre
stri
ctio
ns
bet
wee
nth
eeq
uat
ion
ofL
and
the
equ
atio
nofW
(see
the
par
amet
er
ρw
).
Th
iseq
uat
ion
ofL
isob
tain
edaf
ter
solv
ing
the
mod
el(c
omp
uta
tion
ofth
ere
du
ced
form
).
To
obta
inth
isre
du
ced
form
,it
isn
eces
sary
tosp
ecif
yth
ep
roce
ssof
W.
An
oth
erap
pro
ach
,u
sin
gth
eF
OC
ofth
efir
m
Et( βbL
t+1−
(βb
+b
+α
1)Lt
+bL
t−1−
(Wt−α
0))
=0
LetZt
ase
tof
inst
rum
enta
lva
riab
les
Zt
={L
t,Lt−
1,...,Lt−q,Wt,Wt−
1,...,Wt−q,Cste}
asu
bse
tof
the
info
rmat
ion
set
use
dby
the
firm
.T
he
FO
Cca
nb
e
rew
ritt
en
E[(βbL
t+1−
(βb
+b
+α
1)Lt
+bL
t−1−
(Wt−α
0))|Zt]
=0
Th
issu
gges
tsth
atth
em
odel
’sp
aram
eter
sca
nb
ees
tim
ated
usi
ng
an
IVes
tim
ator
(GM
M,
Han
sen
(198
2)).
Th
isap
pro
ach
can
be
gen
eral
ized
tom
ore
com
ple
xad
just
men
tco
sts
fun
ctio
ns.
•T
he
quad
rati
cad
just
edco
sts
fun
ctio
nis
less
and
less
sup
por
ted
by
the
dat
a(s
ymm
etry
and
conv
exit
y).
•D
ata
favo
ras
ymm
etri
c,p
iece
wis
elin
ear
and
fixed
cost
sap
pro
ach
es.
•T
he
spee
dof
adju
stm
ent
ofla
bor
dem
and
isre
lati
vely
hig
hin
US
.
•T
he
resu
ltis
not
affec
ted
ifw
eco
nsi
der
mu
ltip
lein
pu
ts.
•F
irm
sad
just
hou
rsof
wor
km
ore
rap
idly
than
the
num
ber
ofw
ork-
ers
(ad
just
men
tco
sts
are
hig
her
for
wor
kers
than
for
hou
rs)
•E
mp
loym
ent
mor
era
pid
lyin
US
than
anyw
her
eel
se.
•M
ore
rap
idad
just
men
tin
Eu
rop
eth
anin
Jap
an.
Adiffi
cult
yId
enti
ficat
ion
ofad
just
men
tco
sts
par
amet
er.
Rem
ind
that
the
solu
tion
take
sth
efo
rm
Lt
=λ
1Lt−
1+µWt
wh
ereµ
=−
1 βb
1λ
2−ρw
.
For
sim
plic
ity,
nor
mal
ize
the
par
amet
erµ
=1.
Th
isyi
eld
s Lt
=(λ
1+ρw
)Lt−
1−
(λ1ρw
)Lt−
2+σε,wε w
,t
Su
pp
ose
that
the
econ
omet
rici
and
oes
not
obse
rve
the
real
wag
eWt
bu
ton
lyth
ela
bor
inp
ut.
Th
e(u
nco
nst
rain
ed)
red
uce
dfo
rmis
give
nby
Lt
=β
1Lt−
1+β
2Lt−
2+ut
Letβ
1an
dβ
2th
ees
tim
ated
valu
esofβ
1an
dβ
2.
Fro
m
β1
=λ
1+ρw
β2
=−λ
1ρw
we
ded
uce
λ1
=β
1±√ β
2 1+
4β2
2ρw
=β
1−λ
1
Tw
oop
pos
ite
con
clu
sion
s:
•λ
1la
rge
andρw
smal
l:la
rge
adju
stm
ent
cost
san
dsm
allp
ersi
sten
ce
ofsh
ocks
.
•λ
1sm
alla
ndρw
larg
e:sm
alla
dju
stm
ent
cost
san
dla
rge
per
sist
ence
ofsh
ocks
.
Su
mm
ary
an
dC
on
clu
sion
•T
her
eex
ists
sub
stan
tial
cost
sof
adju
stin
gem
plo
ymen
t(h
irin
gve
r-
sus
firin
gco
sts,
US
vers
us
cont
inen
tal
Eu
rop
e)
•W
hen
adju
stm
ent
cost
sar
equ
adra
tic,
the
firm
grad
ual
lyad
just
s
the
lab
orin
pu
t.
•In
the
stoc
has
tic
case
,th
ere
du
ced
form
onem
plo
ymen
tis
not
inva
rian
tto
the
spec
ifica
tion
ofth
eex
ogen
ous
real
wag
es.
Chap
3:
Busi
ness
Cycl
eM
odels
3.1
:T
wo
Sim
ple
DS
GE
Mod
els
I-
Hab
itForm
ati
on
ina
Sim
ple
Equ
ilib
riu
mM
od
el
•T
he
mod
el
•S
olu
tion
•Q
uan
tita
tive
imp
licat
ion
s
•T
akin
gth
em
odel
toth
ed
ata
II-
AS
toch
ast
icG
row
thM
od
el
•T
he
mod
el
•S
olu
tion
•Q
uan
tita
tive
imp
licat
ion
s
•T
akin
gth
em
odel
toth
ed
ata
3.2
:R
BC
Mod
els
I-
AR
BC
Mod
el
wit
hC
om
ple
teD
ep
reci
ati
on
•T
he
mod
el
•S
olu
tion
•Q
uan
tita
tive
imp
licat
ion
s
•T
akin
gth
em
odel
toth
ed
ata
II-
Ap
roto
typ
ical
RB
CM
od
el
(Op
tion
al)
•T
he
mod
el
•S
olu
tion
•Q
uan
tita
tive
imp
licat
ion
s
•T
akin
gth
em
odel
toth
ed
ata
3.3
:B
eyon
dth
eR
BC
Mod
el
(Op
tion
al)
3.1
:T
wo
Sim
ple
DSG
EM
odels
I-
Habit
Form
ati
on
ina
Sim
ple
Equilib
rium
Model
Hab
itP
ersi
sten
cein
con
sum
pti
on
An
oth
erfo
rmof
hab
itp
ersi
sten
ce:
leis
ure
(Eic
hen
bau
m,
Han
se,
Sin
-
glet
on(1
988)
,W
enn
(199
8),
...)
Tw
om
odel
s:
A)
Hab
itp
ersi
sten
cein
the
per
man
ent
inco
me
mod
el:
Hom
ewor
k
B)
Hab
itp
ersi
sten
cein
asi
mp
leD
SG
Em
odel
B)
Hab
itp
ersi
sten
cein
asi
mp
leD
SG
Em
odel
Asi
mp
leD
SG
Em
odel
wit
hou
tca
pit
alac
cum
ula
tion
Equ
ilib
riu
mon
the
good
mar
ket Yt
=Ct
Pro
du
ctio
nec
onom
y.
Tec
hn
olog
yu
sed
byfir
ms
(CR
Sw
ith
lab
oras
the
sole
vari
able
inp
ut)
Yt
=Ztht
wh
ereZt
isa
tech
nol
ogy
(TF
P)
per
turb
atio
n(t
he
only
one
we
con
-
sid
erin
this
sim
ple
econ
omy)
Op
tim
alit
yco
nd
itio
nfo
rth
ere
pre
sent
ativ
eco
mp
etit
ive
firm
:
Zt
=wt
wh
erewt
isth
ere
alw
age.
Hou
seh
old
pro
ble
m
Et
∞ ∑ i=0
log(Ct−bC
t−1)−ht
Com
ments
:
-lo
gu
tilit
yin
con
sum
pti
on
-h
abit
per
sist
ence
inco
nsu
mp
tion
(b≥
0)
-lin
ear
uti
lity
inle
isu
re(o
rla
bor
sup
ply
)
Meanin
gof
habit
pers
iste
nce
Mar
gin
alu
tilit
yin
per
iodt
∂U
∂Ct
=1
Ct−bC
t−1
Th
em
argi
nal
uti
lity
isd
ecre
asin
ginCt
bu
tin
crea
sin
ginCt−
1p
ro-
vid
edb>
0.T
his
crea
tes
inte
rtem
por
alco
mp
lem
enta
rity
inco
n-
sum
pti
ond
ecis
ion
.
Inot
her
wor
ds,
ifth
eco
nsu
mer
wan
tsto
mai
ntai
nit
su
tilit
y,fo
ra
give
nva
lue
ofit
sp
ast
con
sum
pti
on,
he
mu
stch
oose
ah
igh
valu
eof
con
sum
pti
onto
day
.
So
hig
her
con
sum
pti
onye
ster
day
tran
slat
esin
toev
enm
ore
hig
her
con
sum
pti
onto
day
.
Bu
dge
tco
nst
rain
t
Ct
=wtht
+Πt
wh
ere
Πt
are
the
pro
fits
rece
ived
from
the
firm
s(r
mk:
ateq
uili
bri
um
thes
ep
rofit
sar
eze
ro)
Tw
oco
ntro
ls:C
andh
Now
,re
pla
ceth
eb
ud
get
con
stra
int
into
the
obje
ctiv
eVh
isth
eon
ly
one
cont
rol
Etlo
g(wtht−bw
t−1ht−
1)−ht+β
(log
(wt+
1ht+
1−bw
tht)−ht+
1)+....
FO
C
wt
Ct−bC
t−1−βbE
twt
Ct+
1−bC
t=
1
wt
(1
Ct−bC
t−1−βbE
t1
Ct+
1−bC
t) =1
1
Ct−bC
t−1−βbE
t1
Ct+
1−bC
t=
1 Zt
Non
–lin
ear
forw
ard
look
ing
equ
atio
nw
ith
rati
onal
exp
ecta
tion
s.
Diffi
cult
(not
imp
ossi
ble
,in
this
sim
ple
setu
p)
toco
mp
ute
the
solu
-
tion
.
An
sim
ple
appro
xim
ati
on:
Log
–lin
ear
app
roxi
mat
ion
(ext
ensi
vely
use
dfo
rm
ore
com
ple
xm
odel
s
inth
eD
SG
Elit
erat
ure
)
Let
the
follo
win
gfu
nct
ion y
=f
(x1,x
2,...,xn
)
Lin
ear
app
roxi
mat
ion
:
y=f
(x1,x
2,...,xn
)+
n ∑ i=1
∂y
∂xi∣ ∣ ∣ ∣ ∣ ∣ x=x
(xi−xi)
wh
ere
y=f
(x1,x
2,...,xn
)
Th
isre
wri
tes
y−y
=
n ∑ i=1
∂y
∂xi∣ ∣ ∣ ∣ ∣ ∣ x=x
(xi−xi)
Now
div
ide
bot
hsi
de
byy
:
y−y
y=
n ∑ i=1
∂y
∂xi∣ ∣ ∣ ∣ ∣ ∣ x=x
1 y(xi−xi)
oreq
uiv
alen
tly
y−y
y=
n ∑ i=1
∂y
∂xi∣ ∣ ∣ ∣ ∣ ∣ x=x
xi y
xi−xi
xi
Th
elo
g–lin
ear
app
roxi
mat
ion
ofy
isth
engi
ven
by
y=
n ∑ i=1
∂y
∂xi∣ ∣ ∣ ∣ ∣ ∣ x=x
xi yxi
wh
ere
y=
(y−y
)/y'
log(y
)−
log(y
)
and
xi
=(xi−xi)/xi'
log(xi)−
log(xi)
Exa
mp
les:
1)P
rod
uct
ion
fun
ctio
n
y t=z thα t
Yis
the
outp
ut,Zt
the
TF
Pan
dht
the
lab
orin
pu
t.
Th
elo
g–lin
ear
app
roxi
mat
ion
y t=hα
(z/y
)zt
+zαhα−
1 (h/y
)ht
Fro
my
=zh
α,
this
red
uce
sto
:
y t=z t
+αht
2)E
quili
bri
um
con
dit
ion
Clo
sed
econ
omy
y t=c t
+i t
+g t
Th
elo
g–lin
ear
app
roxi
mat
ion
y t=c yc t
+i yi t
+g yg t
c/y
,i/y
andg/y
rep
rese
ntth
eav
erag
esh
ares
(c/y
+i/y
+g/y
=1
byco
nst
ruct
ion
)or
stea
dy–
stat
eva
lues
inth
em
odel
’sla
ngu
age.
3)E
ule
req
uat
ion
onco
nsu
mp
tion
log
uti
lity
and
stoc
has
tic
retu
rns
(see
asse
tp
rici
ng
mod
els)
1 Ct
=βEtRt+
11
Ct+
1
wh
ereR
isth
egr
oss
retu
rnon
asse
tsb
etw
een
per
iod
st
andt
+1.
Ste
ady
stat
e
βR
=1
Log
–lin
eari
zati
on:
−Ct
=EtRt+
1−EtCt+
1
or
Ct
=−EtRt+
1+EtCt+
1
Applica
tion
toth
em
odel
wit
hhabit
:
1
Ct−bC
t−1−βbE
t1
Ct+
1−bC
t=
1 Zt
Det
erm
inis
tic
stea
dy–
stat
e
1−βb
C(1−b)
=1 Z
Var
iab
led
efin
itio
n
Xt
=1
Ct−bC
t−1
wit
hst
ead
yst
ate
X=
1
C(1−b)
Th
em
odel
bec
omes
Xt
=βbE
tXt+
1+
1 Zt
Log
–lin
eari
zati
on
Xt
=βbE
tXt+
1−
(1−βb)Zt
For
war
dsu
bst
itu
tion
s
Xt
=−
(1−βb)Et
∞ ∑ i=0
(βb)i Zt+i
Log
–nor
mal
tech
nol
ogy
shoc
ks
Zt
=ρzZ−
1+εz t
wh
ere
E(εz t)
=0
V(εz t)
=σ
2 z
Itfo
llow
sth
at
Xt
=−
1−βb
1−βbρzZt
Now
,w
eco
mp
ute
the
log–
linea
riza
tion
ofX
Xt
=−
1
1−bC
t+
b
1−bC
t−1
Aft
ersu
bst
itu
tion
,w
eob
tain
Ct
=bC
t−1
+(1−b)
(1−βb)
1−βbρz
Zt
So,
the
con
sum
pti
onfo
llow
san
AR
(2)
pro
cess
bec
auseZ
follo
ws
an
AR
(1)
pro
cess
(1−bB
)(1−ρzB
)Ct
=(1−b)
(1−βb)
1−βbρz
εz t
wh
ereB
isth
eb
acks
hif
top
erat
or,
i.e.BCt
=Ct−
1.
Fro
mth
isre
du
ced
form
,w
eca
nco
mp
ute
mom
ents
(vol
atil
ity,
AC
Fs)
and
IRF
s
The
Luca
sC
riti
que
Again
and
Again
:a
chan
geinρz
will
affec
tth
ere
du
ced
form
!
The
behavio
rof
hours
:W
eh
ave
Yt
=Ct
=Zt
+ht
Aft
era
sub
stit
uti
onin
toth
eco
nsu
mp
tion
fun
ctio
n,
one
gets
Zt
+ht
=b(Zt−
1+ht−
1)+
(1−b)
(1−βb)
1−βbρz
Zt
For
ave
ryp
ersi
sten
tte
chn
olog
ysh
ock
(ρz
1),
we
hav
e
ht
=bht−
1−bεz t
Th
ete
chn
olog
ysh
ocks
has
an
egat
ive
and
per
sist
ent
effec
ton
hou
rs
(b>
0).
Inth
isca
se,
the
IRF
sar
e:
IRF
(0)
=−b,IRF
(1)
=−b2,...,IRF
(h)
=−bh
+1
Why
?:C
onsu
mp
tion
is”s
tick
y”d
ue
toh
abit
per
sist
ence
.
Ap
erm
anen
tin
crea
sein
the
real
wag
ele
ads
top
ut
the
extr
are
sou
rces
onle
isu
re(t
he
inco
me
effec
tsd
omin
ate
the
sub
stit
uti
oneff
ect)
.
So,
ap
erfe
ctly
com
pet
itiv
em
odel
ofth
eb
usi
nes
scy
cle
can
pro
du
ce
ad
ecre
ase
inh
ours
(du
eto
stro
ng
lab
orsu
pp
lyeff
ects
)
Inco
ntra
dic
tion
wit
hG
ali
(199
)w
ho
clai
med
that
the
stan
dar
dn
ea-
clas
sica
lm
odel
can
not
exp
lain
edth
isfe
atu
re(f
oun
din
the
dat
a).
Con
vers
ely,
aK
eyn
esia
nm
odel
can
rep
licat
eth
is.
Rea
son
?S
imp
le.
Inth
esh
ort–
run
,th
ele
vel
ofou
tpu
tis
det
erm
ined
byth
ele
vel
ofd
eman
d.
Su
pp
ose
ap
osit
ive
tech
nol
ogy
shoc
kZt
Yt
=Ztht
Th
eou
tpu
tYt
doe
sn
otch
ange
inth
esh
ort
run
.
So,
anin
crea
seinZ
red
uce
sth
ele
vel
ofh
,b
ecau
seth
esa
me
quan
tity
Yca
nb
ep
rod
uce
dw
ith
less
lab
orin
pu
t.(S
eeth
efig
ure
)
The
case
of
random
walk
tech
nolo
gy
shock
sw
ith
dri
ft:
∆Zt
=γz
+εz t
wh
ereγz>
1is
the
gros
sgr
owth
rate
ofte
chn
ical
pro
gres
s(a
nd
thu
s
grow
thra
teof
the
econ
omy
alon
gth
eb
alan
ced
grow
thp
ath
γz
=γy
=γc
Not
ice
that
hou
rsd
on
otd
isp
lay
this
tren
dp
atte
rn(h
ours
are
first
–
ord
eran
dse
con
d–o
rder
stat
ion
ary)
.
Thu
s,a
stat
ion
ary
vers
ion
ofth
isec
onom
yw
ould
be
obta
ined
by
defl
atin
gth
eeq
uili
bri
um
con
dit
ion
sbyZ
.
All
vari
able
sar
eth
end
efin
edin
dev
iati
onfr
omth
est
och
asti
ctr
end
Z.
See
the
com
puta
tion
of
the
solu
tion
ina
more
genera
l
case
wit
hca
pit
al
acc
um
ula
tion.
II-
Sto
chast
icA
KM
odel
Are
pre
sent
ativ
eh
ouse
hol
dw
illse
ekto
max
imiz
e
Et
∞ ∑ i=0
log(C
+i)
(no
leis
ure
choi
ces)
un
der
the
con
stra
ints
Cap
ital
accu
mu
lati
on
Kt+
1=
(1−δ)Kt
+I t
Agg
rega
tere
sou
rces
con
stra
int
Yt
=Ct
+I t
Tec
hn
olog
y
Yt
=ZtK
t
Kis
the
sole
inp
ut;
con
stan
tre
turn
tosc
ale
inth
ere
pro
du
cib
lefa
ctor
.
Zt
isa
TF
Psh
ock.
Pu
ttin
gth
ese
thre
eeq
uat
ion
sto
geth
eryi
eld
s
Kt+
1=
(1−δ)Kt
+ZtK
t−Ct
or
Kt+
1=
(1−δ
+Zt)Kt−Ct
Her
ew
ed
on
otim
pos
efu
lld
epre
ciat
ion
,i.e
.δ∈
[0,1
).
Aft
ersu
bst
itu
tion
ofth
isco
nst
rain
tin
toth
eob
ject
ive,
the
FO
Cis
give
nby
1 Ct
=βEt(
1−δ
+Zt+
1)1
Ct+
1
Her
e(1−δ+Zt)
rep
rese
ntth
ere
alin
tere
stra
te(m
argi
nal
pro
du
ctiv
ity
ofca
pit
aln
etof
dep
reci
atio
n)
Solv
ing
the
model:
Kt
isp
red
eter
min
edan
dth
usEtK
t+1
=Et(
(1−δ)Kt+ZtK
t−Ct)
=
Kt+
1.
Now
div
ide
and
mu
ltip
lyby
Kt+
1th
eri
ght–
han
dsi
de
ofth
eE
ule
r
equ
atio
n.
Th
isim
plie
s
1 Ct
=βEt(
1−δ
+Zt+
1)Kt+
1
Kt+
1
1
Ct+
1
Th
enu
mer
ator
is
(1−δ
+Zt+
1)Kt+
1=
(1−δ)Kt+
1+Zt+
1Kt+
1=
(1−δ)Kt+
1+Yt+
1
Now
,use
the
equ
ilib
riu
mco
nd
itio
non
the
good
mar
ket
inp
erio
dt+
1
Yt+
1=Ct+
1+I t
+1
and
the
law
ofm
otio
non
the
cap
ital
stoc
k
Kt+
2=
(1−δ)Kt+
1+I t
+1
Kt+
2=
(1−δ)Kt+
1+Yt+
1−Ct+
1⇔
Kt+
2+Ct+
1=
(1−δ)Kt+
1+Yt+
1
Th
isyi
eld
s
Kt+
1
Ct
=βEtC
t+1
+Kt+
2
Ct+
1
or
Kt+
1
Ct
=βEt
( 1+Kt+
2
Ct+
1
)
LetXt
=Kt+
1Ct
,th
eE
ule
req
uat
ion
rew
rite
s
Xt
=βEt(
1+Xt+
1)
Sin
ceβ∈
(0,1
),th
eso
luti
onis
obta
ined
byfo
rwar
dsu
bst
itu
tion
s:
Xt
=β
1−β
Th
isyi
eld
s
Ct
=1−β
βKt+
1
Now
,su
bst
itu
teth
iseq
uat
ion
into
the
law
ofm
otio
nof
cap
ital
Kt+
1=
(1−δ
+Zt)Kt−Ct
Th
isyi
eld
s
Kt+
1=β
(1−δ
+Zt)Kt
All
the
oth
erag
greg
ate
vari
able
sca
nb
eth
end
edu
ced
.
Outp
ut
Dynam
ics
We
hav
e
Yt
=ZtK
t
So
Yt
Yt−
1=
Zt
Zt−
1
Kt
Kt−
1
Now
,u
sin
gth
ep
roce
ssofK
,it
com
es
Yt
Yt−
1=
Zt
Zt−
1β
(1−δ
+Zt−
1)
Tak
ing
logs
(low
erle
tter
s),
we
hav
e
∆y t
=∆z t
+lo
g(β
)+
log(
1−δ
+Zt−
1)
Non
–lin
ear
inZ
(exc
ept
wh
enδ
=1)
,so
no
exp
licit
solu
tion
for
the
mom
ents
.Com
plic
ated
likel
ihoo
dfu
nct
ion
.
Con
vers
ely,
the
mod
elis
easy
tosi
mu
late
and
thu
sm
omen
tsca
nb
e
obta
ined
bysi
mu
lati
on.
An
oth
erap
pro
ach
:co
mp
ute
anap
pro
xim
ate
solu
tion
byli
nea
riza
tion
ofth
em
odel
(see
the
pre
viou
sm
odel
).
3.2
:R
BC
Models
I-
RB
CM
odel
wit
hco
mple
tedepre
ciati
on
We
con
sid
eran
econ
omy
wit
ha
rep
rese
ntat
ive
hou
seh
old
and
are
p-
rese
ntat
ive
firm
.T
he
firm
pro
du
ces
anh
omog
enou
sgo
odw
ith
the
follo
win
gp
rod
uct
ion
fun
ctio
n
Yt
=ZtK
1−α
thα t
wit
hα∈
(0,1
).Kt
isth
eca
pit
alst
ock
andht
den
otes
the
lab
or
inp
ut.Zt
isa
ran
dom
vari
able
that
rep
rese
nts
chan
ges
inT
otal
Fac
tor
Pro
du
ctiv
ity
(TF
P).
Th
eca
pit
alst
ock
evol
ves
acco
rdin
gto
Kt+
1=
(1−δ)Kt
+I t
wh
ereδ
isth
ed
epre
ciat
ion
rate
.I t
den
otes
inve
stm
ent.
Her
e,w
e
assu
me
com
ple
ted
epre
ciat
ion
(δ=
1)
Kt+
1=I t
Th
ere
pre
sent
ativ
eh
ouse
hol
dco
nsu
mesCt
and
wor
ksht
ever
yp
erio
d.
Th
eh
ouse
hol
dse
eks
tom
axim
ize
Et
∞ ∑ i=0
βi( l
og(Ct+i)−V
(ht+i)
)
V(.
)is
aco
nvex
fun
ctio
n.β∈
(0,1
)is
the
dis
cou
ntfa
ctor
and
Et
den
otes
the
con
dit
ion
alex
pec
tati
ons
oper
ator
.T
he
equ
ilib
riu
m
con
dit
ion
onth
ego
odm
arke
tis
give
nby
Yt
=Ct
+I t
Th
ed
ynam
icop
tim
izat
ion
pro
ble
m
Pre
-det
erm
ined
vari
able
:Kt
Ch
oice
vari
able
:Ct
andht
Rm
k:C
ontr
ollin
gCt
iseq
uiv
alen
tto
cont
rolKt+
1
maxEt
∞ ∑ i=0
βi( l
og(Ct+i)−V
(ht+i)
)
sub
ject
to
Kt+
1=I t
Yt
=Ct
+I t
Yt
=ZtK
1−α
thα t
Com
bin
ing
the
thre
eeq
uat
ion
syi
eld
s
Kt+
1=ZtK
1−α
thα t−Ct
Now
,re
pla
ceth
eco
nst
rain
t
Ct
=ZtK
1−α
thα t−Kt+
1
into
the
obje
ctiv
e
Et
∞ ∑ i=0
βi( lo
g(Zt+iK
1−α
t+ihα t+i−Kt+i+
1)−V
(ht+i))
and
max
imiz
eth
isfu
nct
ion
wit
hre
spec
ttoKt+
1an
dht.
On
eob
tain
s
−1 Ct
+αβEt
[( Yt+
1
Kt+
1
)(1
Ct+
1
)] =0
oreq
uiv
alen
tly
1 Ct
=αβEt
[( Yt+
1
Kt+
1
)(1
Ct+
1
)]T
hus
isth
eE
ule
req
uat
ion
onco
nsu
mp
tion
(in
aD
SG
Em
odel
)
V′ (ht)
=1 Ct(1−α
)Yt
ht
Th
isre
pre
sent
the
mar
gin
alra
teof
sub
stit
uti
onb
etw
een
lab
orsu
pp
ly
and
con
sum
pti
on.
Reso
luti
on
For
war
dsu
bst
itu
tion
s
Wit
hco
mp
lete
dep
reci
atio
n,
the
phy
sica
lca
pit
alev
olve
sac
cord
ing
to
Kt+
1=I t
Su
bst
itu
tein
toth
eth
eE
ule
req
uat
ion
onco
nsu
mp
tion
1 Ct
=αβEt
[( Yt+
1
I t
)(1
Ct+
1
)]
I tis
know
nin
per
iodt.
So,
we
can
mov
eou
tth
ein
vest
men
tfr
omth
e
righ
th
and
sid
eof
the
Eu
ler
equ
atio
n
I t Ct
=αβEt
[ Y t+1
Ct+
1
]F
rom
the
equ
ilib
riu
mco
nd
itio
non
the
good
mar
ket
inp
erio
dt
+1
Yt+
1=Ct+
1+I t
+1
we
obta
in
I t Ct
=αβEt
[ C t+1
+I t
+1
Ct+
1
]or
I t Ct
=αβEt
[ 1+I t
+1
Ct+
1
]
For
war
dsu
bst
itu
tion
s(αβ<
1)
Sol
uti
on
I t Ct
=αβ
1−αβ
I t=
αβ
1−αβCt
Usi
ng
the
equ
ilib
riu
mco
nd
itio
non
the
good
mar
ket
Yt
=Ct
+I t⇒
Yt
=Ct
1−αβ⇔
Ct
Yt
=1−αβ
(con
stan
t)
I t Yt
=αβ
(con
stan
tsa
vin
gra
te)
Lab
or
V′ (ht)
=1 Ct(1−α
)Yt
ht⇔
htV′ (ht)
=(1−α
)Yt
Ct
So,
lab
oris
con
stan
t(w
ed
enot
eh
).
Fro
mth
ela
wof
mot
ion
onp
hysi
cal
cap
ital
Kt+
1=Yt−Ct⇔
αβYt
and
the
pro
du
ctio
nfu
nct
ion Yt
=ZtK
1−α
thα
we
obta
in
Kt+
1=αβZtK
1−α
thα
Analy
zing
the
dynam
icpro
pert
ies
Now
,ta
keth
iseq
uat
ion
inlo
gs(a
llva
riab
les
are
pos
itiv
e)
kt+
1=c k
+z t
+αkt
wh
erec k
isa
con
stan
tte
rmth
atd
epen
ds
onth
ed
eep
par
amet
ers
of
the
mod
el.
Th
eeq
uat
ion
for
outp
ut
(in
logs
)is
y t=c y
+z t
+αkt
wh
erec y
isa
con
stan
t.
For
sim
plic
ity
(wit
hou
tlo
ssof
gen
eral
ity)
,se
tth
istw
oco
nst
ant
term
s
toze
ro.
Now
,in
trod
uce
the
bac
ksh
ift
oper
atorB
,i.e
.
Bkt+
1=kt
orBkt
=kt−
1orBy t
=y t−
1
kt+
1=z t
+αkt⇔
kt+
1=z t
+αBkt+
1
⇔(1−αB
)kt+
1=z t⇔
kt+
1=
z t1−αB
⇔kt
=z t−
1
1−αB
Now
,in
sert
this
equ
atio
nin
toth
eou
tpu
t(i
nlo
g)eq
uat
ion
y t=z t
+α
z t−
1
1−αB⇔
(1−αB
)yt
=(1−αB
)zt
+αz t−
1
y t=αy t−
1+z t
Now
assu
me
thatz t
isan
iidsh
ock
(pu
rely
tran
sito
ry)
z t=εz t
Imp
uls
ere
spon
seof
outp
ut
(in
log,
soth
isis
anel
asti
city
)to
ate
ch-
nol
ogy
inn
ovat
ionεz t
inp
erio
dt.
IRFy(h
)=∂y t
+h
∂εz t
wit
hh
=0,
1,2,....
Fro
mth
eab
ove
equ
atio
n,
we
obta
in
IRFy(h
)=αh
[IN
SE
RT
FIG
UR
E]
Rm
k1:
the
resp
onse
ofco
nsu
mp
tion
and
inve
stm
ent
(in
logs
)ar
e
exac
tly
the
sam
e.
So,
atr
ansi
tory
tech
nol
ogy
imp
rove
men
th
asa
lon
gla
stin
gp
osit
ive
effec
ton
real
vari
able
s.
Rm
k2:
no
effec
ton
lab
or.
Rm
k3:
Th
eL
uca
scr
itiq
ue
doe
sn
otap
ply
her
e.T
he
solu
tion
(Im
ean
the
red
uce
dfo
rmp
aram
eter
s)is
not
affec
ted
byth
ep
aram
eter
sth
at
sum
mar
ize
the
stoc
has
tic
pro
cess
ofz t
.
II-
RB
CM
odel
wit
hL
inear
Uti
lity
inL
eis
ure
For
sim
plic
ity,
the
mod
elin
clu
des
only
ara
nd
omw
alk
inp
rod
uct
ivit
y
(Zt)
.
Sim
ilar
resu
lts
are
obta
ined
wit
ha
stat
ion
ary
lab
orw
edge
shoc
k.
Th
ein
tert
emp
oral
exp
ecte
du
tilit
yfu
nct
ion
ofth
ere
pre
sent
ativ
eh
ouse
-
hol
dis
give
nby
Et
∞ ∑ i=0
log(C
+i)−χV
(Ht)
wh
ereχ>
0,β∈
(0,1
)d
enot
esth
ed
isco
unt
fact
oran
dEt
isth
e
exp
ecta
tion
oper
ator
con
dit
ion
alon
the
info
rmat
ion
set
avai
lab
leas
ofti
met.
Ct
isth
eco
nsu
mp
tion
att
andHt
rep
rese
nts
the
hou
seh
old
’sla
bor
sup
ply
.
Th
ere
pre
sent
ativ
efir
mu
ses
cap
italKt
and
lab
orHt
top
rod
uce
the
hom
ogen
eou
sfin
algo
odYt.
Th
ete
chn
olog
yis
rep
rese
nted
byth
efo
llow
ing
con
stan
tre
turn
s–to
–
scal
eC
obb
–Dou
glas
pro
du
ctio
nfu
nct
ion
Yt
=Kα t
( ZtH
t)1−α,
wh
ereα∈
(0,1
).Zt
isas
sum
edto
follo
wan
exog
enou
sp
roce
ssof
the
form
∆lo
g(Zt)
=σzε z,t,
wh
ereε z,t
isii
dw
ith
zero
mea
nan
du
nit
vari
ance
.
Th
eca
pit
alst
ock
evol
ves
acco
rdin
gto
the
law
ofm
otio
n
Kt+
1=
(1−δ)Kt
+I t,
wh
ereδ∈
(0,1
)is
the
con
stan
td
epre
ciat
ion
rate
.
Fin
ally
,th
efin
algo
odca
nb
eei
ther
con
sum
edor
inve
sted
Yt
=Ct
+I t.
FO
Cs
+E
quili
bri
um
con
dit
ion
s+
Exo
gen
ous
vari
able
1 Ct
=βEt
( 1−δ
+αYt+
1
Kt+
1
) 1
Ct+
1
V′ (Ht)
1=
1 Ct(1−α
)Yt
Ht
Kt+
1=
(1−δ)Kt
+Kα t
( ZtH
t)1−α−Ct
∆lo
g(Zt)
=σzε z,t,
Inth
ism
odel
,th
ete
chn
olog
ysh
ockZt
ind
uce
sa
stoc
has
tic
tren
din
to
outp
ut,
con
sum
pti
on,
inve
stm
ent
and
cap
ital
.
Acc
ord
ingl
y,to
obta
ina
stat
ion
ary
equ
ilib
riu
m,
thes
eva
riab
les
mu
st
be
det
ren
ded
asfo
llow
s
y t=Yt
Zt,
c t=Ct
Zt,
i t=I t Zt,
kt+
1=Kt+
1
Zt.
Wit
hth
ese
tran
sfor
mat
ion
s,th
eap
pro
xim
ate
solu
tion
ofth
em
odel
isco
mp
ute
dfr
oma
log–
linea
riza
tion
ofth
est
atio
nar
yeq
uil
ibri
um
con
dit
ion
sar
oun
dth
isd
eter
min
isti
cst
ead
yst
ate.
Her
ew
eas
sum
e
V′ (Ht)
=1
i.e.
the
uti
lity
islin
ear.
⇒M
ore
deta
ils
(FO
C,
stati
onary
vers
ion
of
the
model,
Ste
ady–st
ate
,lo
g–lineari
zati
on)
Th
elo
g–lin
eari
zati
onof
equ
ilib
riu
mco
nd
itio
ns
arou
nd
the
det
erm
in-
isti
cst
ead
yst
ate
yiel
ds
k t+1=
(1−δ)
( k t−σzε z,t
)+y k y t−
c k c t
(1)
ht
= y t−
c t(2
)
y t=α
( k t−σzε z,t
)+
(1−α
) h t(3
)
Etct+
1= c t+
αβy kEt( y t+1
− k t+1
−σzε z,t
+1)
(4)
wh
erey/k
=(1−β
(1−δ)
)/(αβ
)an
dc/k
=y/k−δ.
Aft
ersu
bst
i-
tuti
onof
(2)
into
(3),
one
gets
y t− k t=
−σzε z,t−
1−α
α c t
Now
,u
sin
gth
eab
ove
exp
ress
ion
,(1
)an
d(4
)re
wri
te
Etct+
1=ϕ c t
wit
hϕ
=α
1−β
(1−α
)(1−δ)∈
(0,1
)(5
)
k t+1=ν 1 k t−
ν 1σzε z,t−ν 2 c t
wit
hν 1
=1 βϕ>
1an
dν 2
=1−β
(1−δ(
1−α
2 ))
α2 β
(6)
Asν 1>
1,(6
)m
ust
be
solv
edfo
rwar
d
k t=σzε z,t
+
( ν 2 ν 1
) limT→∞Et
T ∑ i=0
( 1 ν 1
) i c t+i+
lim
T→∞Et
( 1 ν 1
) T kt+T
Exc
lud
ing
exp
losi
vep
ath
es,
i.e.
limT→∞Et(1/ν
1)T k t+T
=0,
and
usi
ng
(5),
one
gets
the
dec
isio
nru
leon
con
sum
pti
on:
c t=( ν 1−ϕ
ν 2
)( kt−σzε z,t
)(7
)
Aft
ersu
bst
itu
tin
g(7
)in
to(6
),th
ed
ynam
ics
ofca
pit
alis
give
nby
:
k t+1=ϕ
( k t−σzε z,t
)(8
)
Th
ep
ersi
sten
cep
rop
erti
esof
the
mod
elis
thu
sgo
vern
edby
the
pa-
ram
eterϕ∈
(0,1
).
Th
ed
ecis
ion
rule
sof
the
oth
er(d
eflat
ed)
vari
able
sar
esi
mil
arto
equ
a-
tion
(7).
Th
eco
nsu
mp
tion
toou
tpu
tra
tio
isgi
ven
by
c t− y t=
ν cy
( k t−σzε z,t
)=ν cy
( −ϕ
1−ϕLσzε z,t−
1−σzε z,t
)=−ν cy
( σ zε z,t
1−ϕL
)w
her
eν cy
=α
(ν1−ϕ−ν 2
)/ν 2
isp
osit
ive.
Th
ela
tter
exp
ress
ion
show
sth
atth
eco
nsu
mp
tion
toou
tpu
tra
tio
follo
ws
exac
tly
the
sam
est
och
asti
cp
roce
ss(a
nau
tore
gres
sive
pro
cess
ofor
der
one)
asth
ed
eflat
edca
pit
allo
g(Kt/Zt−
1)in
equ
atio
n(8
).
Fro
mth
eab
ove
exp
ress
ion
,w
eca
nd
irec
tly
ded
uce
the
beh
avio
rof
hou
rs:
(1−ϕL
)ht
=ν cyσzε z,t
(9)
Hou
rsw
orke
din
crea
seaf
ter
ate
chn
olog
yim
pro
vem
ent
and
dis
pla
ys
am
onot
onic
dyn
amic
resp
onse
.
3.3
:B
eyond
the
RB
CM
odel
(Opti
onal)
Inpro
gre
ss