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Derivation and Log-linearization of Otsu(2007)s Small Open Economy Model
Chen-chiao Kuo
Otsu, Keisuke (2007), "A Neoclassical Analysis of the Asian Crisis: Business Cycle Accounting ofa Small Open Economy," IMES Discussion Paper Series No. 07-E-16, Bank of Japan.
1 The benchmark prototype economy
The representative consumer depends on an expected intertemporal discountedutility function that includes utility from consumption and disutility from labor:
maxU = E0
1Xt=0
tu(ct; lt)
The utility is:
u(ct; lt) = log(ct lt )
where parameters (> 0) and (> 1) represent the level and curvature of theutility cost of labor respectively.
subject to the budget constraint:
wt
ltlt+rtkt+tt+
dt+1Rdt
= ct+xtxt+dt+(dt+1)
The law of motion for capital:
kt+1 = xt + (1 )kt
where dene = (1 + )(1 + n), is the growth rate of labor augmentingtechnical progress and n is the population growth rate.
The debt adjustment cost function, (dt+1), as (dt+1d)2
2 , where d is thesteady state foreign debt.
Solution of consumersmaximization problem:
maxfct;lt;kt+1;dt+1g
U = E0
1Xt=0
t log (ct lt )
wt
ltlt + rtkt + tt +
dt+1Rdt
= ct + xt [kt+1 (1 )kt] + dt +(dt+1)
The Lagrangian
L =1Xt=0
t log(ct lt ) +1Xt=0
tt
8
@L@ct
= tuct tt = 0
@L@lt
= tult + ttwt
lt= 0
@L@kt+1
= ttxt + t+1t+1[t+1 + xt+1(1 )] = 0
@L@dt+1
= tt
"
Rdt 0(dt+1)
# t+1t+1 = 0
1
ct lt= t (A1)
l1t
ct lt= twt
lt
Use Equation (A1) substitute t, and we obtain:
l1t =wt
lt(A2)
txt = t+1[t+1 + xt+1(1 )]1
ct ltxt = Etuct+1[t+1 + xt+1(1 )] (A3)
t
"
Rdt 0(dt+1)
#= t+1
1
ct lt
"
Rdt 0(dt+1)
#= Etuct+1 (A4)
The rm produces a nal good (output y with a constant returns to scale) fromcapital and labor using a Cobb-Douglas production function.
yt = ztkt l1t
The rms prot maximization problem is:
= yt wtlt rtkt
Solution of rmsmaximization problem:
maxfkt;ltg
= ztkt l1t wtlt rtkt
@
@kt= ztk
1t l
1t rt = 0
@
@lt= (1 )ztkt lt wt = 0
ztk1t l
1t = rt
yt
kt= rt (A5)
(1 )ztkt lt = wt
(1 )ytlt= wt (A6)
Substitute (A6) into (A2), and we get the labor wedge.
l1t =wt
lt= (1 )yt
lt
1
lt
l1t = (1 )yt
lt
1
lt
Substitute (A5) into (A3), and we get the investment wedge.
xt1
ct lt= Et
"uct+1
yt+1kt+1
+ xt+1(1 )!#
Substitute 0(dt+1) = (dt+1 d) into (A4), and we get the foreign debtwedge.
1ct lt
"
R
1
dt (dt+1 d)
#= Et[uct+1]
The government collects tax revenues at date t, and the government budgetconstraint:
tt = (1 1 lt)wtlt + (xt 1)xt
Substitute the government budget constraint into the consumer budget con-straint and we can get the economys resource constraint at last:
wt
ltlt+ rtkt+ (1 1
lt)wtlt+ (xt 1)xt+ dt+1
Rdt= ct+ xtxt+ dt+(dt+1)
rtkt + wtlt xt + dt+1Rdt
= ct + dt +(dt+1)
rtkt + wtlt = xt + ct + dt dt+1Rdt
+(dt+1)
yt = ct + xt + dt dt+1Rdt
+(dt+1 d)2
2
where the trade balance is:
tbt = dt dt+1Rdt
+(dt+1 d)2
2
The equilibrium conditions in the detrended per-capita form:
yt = ct + xt + dtdt+1Rdt
+(dt+1 d)2
2(1)
yt = ztkt l1t (2)
l1t = (1 )yt
lt
1
lt(3)
xt1
ct lt= Et
"uct+1
yt+1kt+1
+ xt+1(1 )!#
(4)
1
ct lt
"
R
1
dt (dt+1 d)
#= Et[uct+1] (5)
Endogenous variables are : ct ,xt ,yt ,kt ,dt ,lt.
Exogenous variables are: zt, dt, xt, and lt.
We need the capital accumulation law to close the model.
kt+1 = xt + (1 )kt (6)
2 Steady state of the prototype model
From equation(5), the calculation process is below:
1
ct lt
"
R
1
dt (dt+1 d)
#= Et[uct+1]
1c l [
R
1
d (d d)] = uc
1
d= R (ss1)
From equation(4), the calculation process is below:
xt1
ct lt= Et
"uct+1
yt+1kt+1
+ xt+1(1 )!#
x =
"z
k
l
1+ x(1 )
#
x
= z
k
l
1+ x(1 )
k
l
1=1
z
"x
x(1 )
#
k
l=
(1
z
"x
x(1 )
#) 11
(ss2)
From equation(6), the calculation process is below:
kt+1 = xt + (1 )kt
k = x+ (1 )k
x = [ (1 )]k (ss3)
From equation(2), the calculation process is below:
yt = ztkt l1t
y = zkl1 = zk
l
1k = z
k
l
l
From equation(1), the calculation process is below:
yt = ct + xt + dtdt+1Rdt
+(dt+1 d)2
2
y = c+ x+ d dRd
+(d d)2
2
c = y x d+ dRd
(d d)2
2= z
k
l
1k [ (1 )]k d+ d
Rd
c = zk
l
l [ (1 )]
k
l
l d+ d
Rd
c = zk
l
l [ (1 )]
k
l
l d+ d
c = zk
l
l [ (1 )]
k
l
l + ( 1)d (ss4)
From equation(3), the calculation process is below:
l1t = (1 )yt
lt
1
lt
l1 = (1 )yl
1
l
l = (1 )zk
l
1 l
l = (1 ) z
k
l
1 l
l =
"(1 ) z
k
l
1 l
#1
(ss5)
We obtain kl rst, and in order obtain l, then k, then c, then y, then x.
k =k
ll
c = zk
l
l [ (1 )]
k
l
l + d( 1)
y = zk
l
l
x = [ (1 )]k
d = d
3 Log-linearization
c^t = log ct log c
x^t = log xt log x
y^t = log yt log y
k^t = log kt log k
l^t = log lt log l
z^t = log zt log z
d^t =dtd 1
^dt = log dt log d
^xt = log xt log x
^ lt = log lt log l
Equilibrium conditions in the detrended per-capita form:
yt = ct + xt + dtdt+1Rdt
+(dt+1 d)2
2(1)
yt = ztkt l1t (2)
l1t = (1 )yt
lt
1
lt(3)
xt1
ct lt= Et
"uct+1
yt+1kt+1
+ xt+1(1 )!#
(4)
1
ct lt
"
R
1
dt (dt+1 d)
#= Et[uct+1] (5)
kt+1 = xt + (1 )kt (6)
From equation (1)
yt = ct + xt + dtdt+1Rdt
+(dt+1 d)2
2
yey^t = cec^t + xex^t +d^t + 1
dhd^t+1 + 1
di
Rde^dt
+hd^t+1 + 1
d d
i22
y(1+y^t) = c (1 + c^t)+x(1+x^t)+d^t + 1
dd^t+1d+ d
Rd(1 + ^dt)
+d^t+1d+ d
d
22
y + yy^t = c+ cc^t + x+ xx^t + dd^t + d d^t+1d+ dRd(1 + ^dt)
+d^t+1d
22
where y = c+ x+ d dRd andd^t+1d
22 0
y + yy^t = c+ cc^t + x+ xx^t + dd^t + d d^t+1d+ dRd(1 + ^dt)
dRd
+d
Rd
Rd(1 + ^dt)yy^t = Rd(1 + ^dt)cc^t +Rd(1 + ^dt)xx^t +Rd(1 + ^dt)dd^td^t+1dd+Rd(1 + ^dt)
d
Rd
Rdyy^t = Rdcc^t +Rdxx^t +Rddd^tdd^t+1 d+ d+ d^dt
y^t =c
yc^t +
x
yx^t +
d
yd^t d
Rdyd^t+1 +
d
Rdy^dt (1.c)
equation (2)
yt = ztkt l1t
yey^t = zez^t(kek^t)(lel^t)1 = (zkl1)ez^t+k^t+(1)l^t
ey^t = ez^t+k^t+(1)l^t
(1 + y^t) = 1 + z^t + k^t + (1 )l^ty^t = z^t + k^t + (1 )l^t (2.c)
Equation (3)
l1t = (1 )zt kt
lt
!1
lt
lel^t
1= (1 )zez^t
kek^t
lel^t
1 le
^ lt
l1e(1)l^t = (1 )zez^tkek^tlel^t 1 le
^ lt
ll1e(1)l^t+^ lt = (1 )zez^tkek^tlel^t
ll1 h1 + ( 1) l^t + ^ lti = (1 )zkl 1 + z^t + k^t l^t
llh1 + ( 1) l^t + ^ lt
i= (1 )zkl1
1 + z^t + k^t l^t
where l l = (1 )y and y = zkl1
(1 )yh1 + ( 1) l^t + ^ lt
i= (1 )y
1 + z^t + k^t l^t
1 + ( 1) l^t + ^ lt =
1 + z^t + k^t l^t
^ lt = z^t + k^t l^t ( 1) l^t
^ lt = z^t + k^t + (1 ) l^t (3.c)
equation (4)
xt1
ct lt= Et
"uct+1
yt+1kt+1
+ xt+1(1 )!#
xe^xt
1
cec^t lel^t= Et
24 1cec^t+1 lel^t+1
yey^t+1k1ek^t+1+xe^xt+1(1 )
!35
Ignore Et for a moment
xe^xt
cec^t+1 lel^t+1
=
yey^t+1k1ek^t+1 + xe^xt+1(1 )
cec^t lel^t
LHS
xe^xtcec^t+1 xe^xtlel^t+1
= xce^xt+c^t+1 xle^xt+l^t+1
= xc (1 + ^xt + c^t+1) xl1 + ^xt + l^t+1
= xc+ xc^xt + xcc^t+1 xl xl ^xt xll^t+1
= (c l) x^xt + (c l) x + xcc^t+1 xll^t+1
RHS
yey^t+1k1ek^t+1 + xe^xt+1(1 )
cec^t lel^t
=
2664yey^t+1k1ek^t+1cec^t yey^t+1k1ek^t+1lel^t
+xe^xt+1(1 )cec^t xe^xt+1(1 )lel^t
3775
=
2664cyke
y^t+1k^t+1+c^t lykey^t+1k^t+1+l^t
+(1 )cxe^xt+1+c^t (1 )lxe^xt+1+l^t
3775
=
24 cyk 1 + y^t+1 k^t+1 + c^t lyk 1 + y^t+1 k^t+1 + l^t+(1 )cx (1 + ^xt+1 + c^t) (1 )lx
1 + ^xt+1 + l^t
35
=
2664cyk
1 + y^t+1 k^t+1
+ cyk c^t lyk
1 + y^t+1 k^t+1
lykl^t + (1 )cx (1 + ^xt+1) + (1 )cxc^t(1 )lx (1 + ^xt+1) (1 )xll^t
3775
=
24 (c l) yk 1 + y^t+1 k^t+1+ hyk + (1 )xi cc^t(1 ) (c l) x (1 + ^xt+1)
hyk + (1 )x
ill^t
35
=
24 (c l) yk 1 + y^t+1 k^t+1+ xcc^t(1 ) (c l) x (1 + ^xt+1) xll^t
35= (c l)
hyk
1 + y^t+1 k^t+1
+ (1 )x (1 + ^xt+1)
i+xcc^txll^t
= (c l)"
yk + yk y^t+1 ykk^t+1
+(1 )x + (1 )x^xt+1
#+ xcc^t xll^t
= (c l) x + (c l)"yk y^t+1 ykk^t+1+(1 )x^xt+1
#+ xcc^t xll^t
= (c l) x + 0@ (c l) yk y^t+1 (c l) ykk^t+1+(c l) (1 )x^xt+1 + x cc^t x ll^t
1A
where x = hyk + x(1 )
iand x =
yk + x(1 )
LHS=RHS(c l) x^xt + (c l) x + xcc^t+1 xll^t+1
= (c l) x+ 0@ (c l) yk y^t+1 (c l) ykk^t+1+(c l) (1 )x^xt+1 + x cc^t x ll^t
1A
(c l) x^xt = 0@ (c l) yk y^t+1 (c l) ykk^t+1+(c l) (1 )x^xt+1 + x cc^t x ll^t
1Axcc^t+1 + xll^t+1
x^xt =
0@ yk y^t+1 ykk^t+1 + (1 )x^xt+1+ x(cl)cc^t x(cl)ll^t
1A xc(c l)c^t+1 +
xl
(c l) l^t+1
^xt =
x
y
ky^t+1
x
y
kk^t+1 +
(1 )
^xt+1 +c
(c l)c^t l
(c l) l^t
c(c l)c^t+1 +
l
(c l) l^t+1
Add back expectation
^xt =
x
y
kEty^t+1 k^t+1
+
c
(c l) (c^t Etc^t+1) (4.c)
l
(c l)l^t Etl^t+1
+
(1 )Et^xt+1
equation (5)
1
ct lt
"
R
1
dt (dt+1 d)
#= Et[uct+1]
1
cec^t lel^t
"
R
1
de^dt
d^t+1 + 1
d d
#= Et
1
cec^t+1 lel^t+1
Ignore Et for the moment.
cec^t+1 lel^t+1
R
1de
^dt
d^t+1d+ d d
=
cec^t lel^t
cec^t+1 R1
de^dtcec^t+1dd^t+1+lel^t+1 R 1de^dt+l
el^t+1dd^t+1 = cec^t lel^t
LHS
cec^t+1 R1
de^dt cec^t+1dd^t+1 + lel^t+1 R 1de^dt + l
el^t+1dd^t+1
= cdec^t+1^dt R cec^t+1dd^t+1 + l
del^t+1^dt R + l
el^t+1dd^t+1
=
24 cd (1 + c^t+1 ^dt) R c (1 + c^t+1)dd^t+1ld
1 + l^t+1 ^dt
R + l
1 + l^t+1
dd^t+1
35
=24 cd (1 + c^t+1) R cd ^dt R c (1 + c^t+1)d^t+1
ld
1 + l^t+1
R +
l
dR^dt + l
1 + l^t+1
dd^t+1
35
where R1d= or R d =
1
= c (1 + c^t+1) c^dt cdd^t+1 l1 + l^t+1
+ l ^dt+ l
dd^t+1
= hc+ cc^t+1 c^dt 1cdd^t+1 l ll^t+1 + l ^dt + 1ldd^t+1
i
= h(c l) + cc^t+1 (c l) ^dt 1 (c l)dd^t+1 ll^t+1
i
RHS
cec^t lel^t
= hc (1 + c^t) l
1 + l^t
i
= c+ cc^t l ll^t
LHS=RHS
"(c l) + cc^t+1 (c l) ^dt1 (c l)dd^t+1 ll^t+1
#=
c+ cc^t l ll^t
cc^t+1 (c l) ^dt 1 (c l)dd^t+1 ll^t+1 = cc^t ll^t
(c l) ^dt = cc^t+1 cc^t ll^t+1 + ll^t 1 (c l)dd^t+1
^dt =c
(cl) (c^t+1 c^t)l
(cl)l^t+1 l^t
1dd^t+1
Add back expectation.
^dt =c
(c l) (Etc^t+1 c^t)l
(c l)Etl^t+1 l^t
1dEtd^t+1 (5.c)
equation (6)
kt+1 = xt + (1 )ktkek^t+1 = xex^t + (1 )kek^t
k(1 + k^t+1) = x(1 + x^t) + (1 )k(1 + k^t)kk^t+1 = xx^t + (1 )kk^t
k^t+1 =x
kx^t + (1 )k^t (6.c)
We will solve the model by the linearizing the equations that characterize thecompetitive equilibrium around the steady state.
Here again the log-linearized model :
y^t =c
yc^t +
x
yx^t +
d
yd^t d
Rdyd^t+1 +
d
Rdy^dt (1.c)
y^t = z^t + k^t + (1 )l^t (2.c)
^ lt = z^t + k^t + (1 ) l^t (3.c)
^xt =
x
y
kEty^t+1 k^t+1
+
c
(c l) (c^t Etc^t+1) (4.c)
l
(c l)l^t Etl^t+1
+
(1 )Et^xt+1
^dt =c
(c l) (Etc^t+1 c^t)l
(c l)Etl^t+1 l^t
1dEtd^t+1 (5.c)
k^t+1 =x
kx^t + (1 )k^t (6.c)