Appl
.Mat
h.J.Chi
ne
s
eUni
v.Se
r
.B
2005,20(4):384392
POLLUTI
ON,GOVERNMENTEXPENDI
TURE,
TAXESAND STOCHASTI
CGROWTH
Zha
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qi
ng HuShi
ge
ng Wa
ngHa
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un
.Thi
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udi
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sas
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l
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ygove
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.Mat
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h.J.Chi
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s
eUni
v.Se
r
Appl
386
.20,No.4
Vol
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mot
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s
umeduY andduZ ar
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r
r
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duY,
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ondi
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i
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ons
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ant
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nt
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ame s
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c
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at
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dk dK
=
,
k
K
(7)
andt
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nme
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dT= dG.
(8)
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U(C(t
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,
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0
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.t
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AKduY,K(0)= K0,
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Pr
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nt
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et
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uef
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t
i
on
∞
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-t
)
V(K(t
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)= maxEt e
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)- θ)ds
,
C,g
t
ac
c
or
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o (6).I
nt
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r
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at
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on me
t
hod,weobt
ai
nt
heBe
l
l
man
e
quat
i
on
ρV= max{U+ LV},
C,g
(10)
whe
r
e
U= U(CP- θ),
LV= Vt+ VK(g'
AK - C)+ VZα
Z+
1
1
VKK(g0'
AK)2σ2
VZZZ2σ2
Y+
Z.
2
2
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i
r
s
t
or
de
rc
ondi
t
i
onsc
anbewr
i
t
t
e
nas
U'
(CP- θ)P- θ = VK,
-θ
(11)
-θ
θ
δU'
(CP )P µ= AgVK,
-θ
-θ
ρVK =- θ
δ'
U'
(CP )P µ+ LVK,
(12)
(13)
whe
r
e
µ=
C
,
K
2 2
LVK =VtK + VKAg'+ VKK(Ag'- µ+ A2g'
K + VKZα
Z+
0σ
Y)
1
1
VKKK(g0'
VKZZZ2σ2
AK)2σ2
Y+
Z.
2
2
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ne
qui
l
i
br
i
um,wepos
t
ul
at
et
heval
uef
unc
t
i
onas
V(K,Z)= aU(K1- θδ'Zθδ'),
(14)
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r
et
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oe
f
f
i
c
i
e
ntai
st
obede
t
e
r
mi
ne
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f
f
e
r
e
nt
i
at
i
ngVwi
t
hr
e
s
pe
c
tt
oK andZyi
e
l
ds
KVKK =[(1- θ
δ'
)σ'- 1]VK,
K2VKKK =[(1- θ
δ'
)σ'- 1][(1- θ
δ'
)σ'- 2]VK,
(15)
2
ZVKZ =θ
δ'
σ'
VK,Z VKZZ = θ
δ'
σ'
(θ
δ'
σ'- 1)VK.
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δ
^ θ
g= µ.
A
(16)
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r
i
vi
ngf
r
om (11),weobt
ai
n
a=
^
gθδσ'
.
(1- θ
δ'
)µσAθδ'σ'
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ZhangXue
qi
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389
and gi
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t
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qui
l
i
br
i
um vl
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l
ye
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e
s
s
e
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e
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i
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nt
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oduc
et
heval
uef
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t
i
on
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-t
)
U(c
(s
)P(s
)- θ)ds
,
V(k(t
),K(t
),Z(t
),t
)= maxEt e
c
t
ac
c
or
di
ngt
o(22).TheBe
l
l
mane
quat
i
oni
s
ρV= max{U+ LV},
c
(23)
whe
r
e
U=U(c
P- θ),
LV=Vt+ Vk(τ
'
Ak- τ
)+ VK(g'
AK - C)+ VZα
Z+
cc
1
1
1
Vkk(τ
'
Ak)2σ2
'
g0'
A2kKσ2
VKK(g0'
AK)2σ2
VZZZ2σ2
0
0
Y + VkKτ
Y+
Y+
Z.
2
2
2
Thef
i
r
s
t
or
de
rc
ondi
t
i
onsc
anbewr
i
t
t
e
nas
U'
(c
P- θ)P- θ = Vk,
(24)
ρVk = LVk,
(25)
whe
r
e
2 2
LVk =Vtk + VkAτ
'+ Vkk(τ
'
Ak- τ
+τ
'
σ2
Z+
cc
0A k
Y)+ VkZα
VkK(g'
AK - C+ τ
'
g0'
A2Kσ2
0
Y)+
VkkKτ
'
g0'
A2kKσ2
0
Y+
1
Vkkk(τ
'
Ak)2σ2
0
Y+
2
1
1
VkKK(g0'
AK)2σ2
VkZZZ2σ2
Y+
Z.
2
2
(26)
I
ne
qui
l
i
br
i
um,s
i
mi
l
ar
l
yt
oThe
or
e
m 1,wepos
t
ul
at
et
heval
uef
unc
t
i
oni
nt
hef
or
m
V= b
U(kK- θδ'Zθδ'),
whe
r
ebi
st
obede
t
e
r
mi
ne
d.Thi
shast
hepr
ope
r
t
y
kVkk =- σVk,KVkK =- θ
δ'
σ'
Vk,ZVkZ = θ
δ'
σ'
Vk,k2Vkkk = σσVk,
kKVkkK = σθ
δ'
σ'
Vk,K2VkKK = θ
δ'
σ'
(1+ θ
δ'
σ'
)Vk,Z2VkZZ = θ
δ'
σ'
(θ
δ'
σ'- 1)Vk.
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s
t
i
ngui
s
hf
r
om t
hec
e
nt
r
al
l
ypl
anne
de
c
onomy,weus
e* t
ode
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et
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qui
l
i
br
i
um i
n
dk dK
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t
hede
c
e
nt
r
al
i
z
e
de
c
onomy.I
nt
hemac
r
oe
c
onomi
ce
qui
l
i
br
i
um, =
k
K
c
*
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andµ =
i
sac
ons
t
ant
ni
nge
quat
i
ons(6)wi
t
he
qui
l
i
br
i
um c
ondi
t
i
ons(7)and
k
(8),weobt
ai
n
ψ* = τ
'
A- τµ* = g'
A- µ* ,
(27)
c
τ
0= g
0.
(28)
Subs
t
i
t
ut
i
ngt
he
s
ee
qui
l
i
br
i
um c
ondi
t
i
onsi
nt
oe
quat
i
ons(24),(25)and(26),yi
e
l
ds
µ* =
whe
r
e
M + (1- F)Aτ
,
τ
cF
(29)
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c=
Ag- Aτ
g- τ
= ^
.
(Aτ+ M)/F- Ag g(1+ θ
δ)/(θ
δ)- g
391
(37)
Subs
t
i
t
ut
i
ng(36)i
nt
o(37),yi
e
l
ds
^
g- g/δ
^
τ
.
(38)
c=
^
g(1+ θ
δ)/(θ
δ)- g
^
^
^= 0,
:I
,τ
Fr
om (38),wehavet
hef
ol
l
owi
ngc
onc
l
us
i
ons
fg= g/δ= τ
andt
hegove
r
nme
nt
c
^
^
.Name
l
y,
e
xpe
ndi
t
ur
ei
sf
ul
l
yf
i
nanc
e
dbyat
ax on c
api
t
ali
nc
ome.I
fg< g/δ,τ
c< 0
^
^
^
,andgove
c
ons
umpt
i
ons
houl
dbes
ubs
i
di
z
e
d.I
fg/δ< g< g(1+ θ
δ)/(θ
δ),τ
r
nme
nt
c> 0
e
xpe
ndi
t
ur
es
houl
dbef
i
nanc
e
dbyt
axe
sbot
honc
api
t
ali
nc
omeandc
ons
umpt
i
on.I
fg>
^
^
.Name
g(1+θ
δ)/(θ
δ),τ
l
y,t
hes
ubs
i
dy t
ot
hec
ons
umpt
i
on by t
hegove
r
nme
nt
c< - 1
e
xc
e
e
dst
hec
ons
umpt
i
on.Thi
sc
as
ei
si
mpos
s
i
bl
e.Thepr
oofofThe
or
e
m 3i
sc
ompl
e
t
e
d.
Se
c
ond,we dt
e
r
mi
ne t
he opt
i
malpol
l
ut
i
on i
nt
he f
i
r
s
t
be
s
topt
i
mum e
qui
l
i
br
i
um
. Gi
c
ondi
t
i
ons
ve
nt
he i
ni
t
i
alphys
i
c
alc
api
t
als
t
oc
k val
ue K0,by (6),we de
r
i
ve t
he
aggr
e
gat
ec
api
t
alac
c
umul
at
i
one
quat
i
on
K(t
)= K0e
xp{(ψ-
1 22
A σY)t+ AuY(t
)}.
2
Si
mi
l
ar
l
y,by(4),t
hepol
l
ut
i
onabat
e
me
ntt
e
c
hnol
ogyac
c
umul
at
i
one
quat
i
onc
anbede
r
i
ve
d
as
xp{(αZ(t
)= Z0e
12
σZ)t+ uZ(t
)}.
2
I
ne
qui
l
i
br
i
um,wehave
K(t
) E[K(t
)] K0
=
=
e
xp{(ψ- α)t
}.
Z(t
) E[Z(t
)] Z0
The
n,s
ubs
t
i
t
ut
i
nge
quat
i
ons(16),(19)andt
heabovee
quat
i
oni
nt
ot
hee
xpr
e
s
s
i
onofP,
weobt
ai
nt
heopt
i
malpol
l
ut
i
onas
^
P=
M
Y
)] δ'
^ AE[K(t
= g- δ
=A θ
δ
E[Z(t
)]
N
GZ
δ δ'
(
)
-δ
K0
Z0
δ'
( ) ( ) exp{δ'(ψ- α)t}.
(39)
Fr
om (39),wec
ande
r
i
vet
hef
ol
l
owi
ngpr
opos
i
t
i
on:
^
^
;i
→∞;
Pr
opos
i
t
i
on1.i
fψ=α,P i
sac
ons
t
antwhi
c
hi
si
r
r
e
l
e
vantt
ot
i
met
fψ<α,P→0ast
^
→ ∞ andt
whi
l
ei
fψ> α,P→ ∞ ast
hi
sc
as
ei
si
nf
e
as
i
bl
e.Name
l
y,i
nor
de
rt
oe
l
i
mi
nat
e
.
pol
l
ut
i
one
f
f
i
c
ac
i
ous
l
y,t
hee
c
onomi
cgr
owt
hr
at
ec
an'
tbet
oof
as
t
~6 Conc
l
us
i
ons
Weanal
yz
eas
t
oc
has
t
i
cgr
owt
hmode
lwi
t
hpol
l
ut
i
on.Byus
i
ngs
t
oc
has
t
i
copt
i
mi
z
at
i
on
me
t
hod we de
r
i
ve t
he opt
i
malgr
owt
hr
at
e,t
he opt
i
malgove
r
nme
nte
xpe
ndi
t
ur
e,t
he
.I
,we
opt
i
malt
axpol
i
c
yandt
heopt
i
malpol
l
ut
i
onunde
re
qui
l
i
br
i
um c
ondi
t
i
ons
nt
hi
spape
r
,s
c
ons
i
de
rt
hegove
r
nme
nthasnode
bt
oas
t
r
ai
ght
f
or
war
de
xt
e
ns
i
oni
st
oi
nt
r
oduc
ebond
i
nt
oc
api
t
al ac
c
umul
at
i
on e
quat
i
on.Anot
he
re
xt
e
ns
i
on i
st
oc
ons
i
de
rt
he pol
l
ut
i
on
392
.Mat
.B
Appl
h.J.Chi
ne
s
eUni
v.Se
r
.20,No.4
Vol
.
abat
e
me
ntt
e
c
hnol
ogyase
ndoge
nous
Re
f
e
r
e
nc
e
s
～
1 Ana Ba
. Endoge
l
c
a
o Re
i
s
nous gr
owt
ha
nd t
he pos
s
i
bi
l
i
t
y ofe
l
i
mi
na
t
i
ng pol
l
ut
i
on,J
our
na
lof
,2001,42:360373.
Envi
r
onme
nt
a
lEc
onomi
c
sa
ndMa
na
ge
me
nt
2 Che
nJ
hyHwa,La
iChi
ngChong,Shi
e
hJ
hyYua
n.Ant
i
c
i
pa
t
e
de
nvi
r
onme
nt
a
lpol
i
c
ya
ndt
r
a
ns
i
t
i
ona
l
,Envi
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dyna
mi
c
si
na
ne
ndoge
nousgr
owt
hmode
l
r
onme
nt
a
la
ndRe
s
our
c
eEc
onomi
c
s
^
3 Mor
,
ga
neChe
ve.I
r
r
e
ve
r
s
i
bi
l
i
t
yofpol
l
ut
i
ona
c
c
umul
a
t
i
on,Envi
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nt
a
la
ndRe
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our
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eEc
onomi
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4 Zha
ngXue
qi
ng,HuShi
ge
ng,Wa
ngHa
i
j
un.A s
t
oc
ha
s
t
i
ce
ndoge
nousgr
owt
hmode
lwi
t
hpol
l
ut
i
on(i
n
Chi
ne
s
e),J
our
na
lofHua
z
hongUni
ve
r
s
i
t
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i
e
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ur
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ls
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i
e
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5 Tur
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c
r
oe
c
onomi
cpol
i
c
i
e
s
owt
ha
nd we
l
f
a
r
ei
n as
t
oc
ha
s
t
i
ce
c
onomy,I
nt
e
r
na
t
i
ona
l
981.
Ec
onomi
cRe
vi
e
w,1993,35:953-
6 As
e
aP,Tur
novs
kyS.Ca
pi
t
a
li
nc
omet
a
xa
t
i
ona
ndr
i
s
kt
a
ki
ngi
nas
ma
l
lope
ne
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s
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novs
ky S. On t
he r
ol
e ofgove
r
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nti
n as
t
oc
hs
t
i
c
a
l
l
y gr
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ng,WuFuke.Ma
t
he
ma
t
i
c
a
lAna
l
ys
i
sofMa
c
r
oe
c
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e),Be
i
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i
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