Nontradable Goods and the Real Exchange Rate
Pau Rabanal
International Monetary Fund
Vicente Tuesta
CENTRUM Católica
This version: August 30, 2012
Abstract
This online appendix provides the equilibrium conditions of the model and the loglinear
approximation.
1
A
Appendix: The Baseline Model
In this appendix, we present the full version of a model with tradable and nontradable …nal
consumption goods, in the spirit of Stockman and Tesar (1995) and Dotsey and Duarte (2008).
We introduce sticky prices in both sectors to be able to study in‡ation dynamics and their role
in a¤ecting the real exchange rate.
A.1
Households
A.1.1
Preferences
Representative households in the home country are assumed to maximize the following utility
function:
Ut = E0
(1
X
t
t
t=0
"
log Ct
bCt
L1+'
t
1+'
1
#)
;
(1)
subject to the following budget constraint:
BtH 1 St BtF
+
Pt
Pt
St BtF
BtH
+
St BtF
Pt R t P R
t t
Pt Yt
1
Wt
Lt
Pt
t
(2)
E0 denotes the conditional expectation on information available at date t = 0;
is the intertem-
poral discount factor, with 0 <
+
Ct +
< 1: Ct denotes the level of consumption in period t, Lt denotes
labor supply. The utility function displays external habit formation with respect to the habit
stock, which is last period’s aggregate consumption of the economy Ct
1.
b 2 [0; 1] denotes the
importance of the habit stock. ' > 0 is inverse elasticity of labor supply with respect to the real
wage.
t
is a preference shock that follows an AR(1) process in logs
log
t
=
log
t 1
+ "t
(3)
We de…ne the consumption index as
1="
c
Ct
CtT
" 1
"
+ (1
1="
c)
CtN
" 1
"
"
" 1
;
where " is elasticity of substitution between the …nal tradable (CtT ) and …nal nontradable (CtN )
goods, and
c
is the share of …nal tradable goods in the consumption basket at home.
In this context, the consumer price index that corresponds to the previous speci…cation is
given by
Pt
h
c
PtT
1 "
+ (1
2
N
c ) Pt
1 "
i
1
1 "
;
where all prices are for goods sold in the home country, in home currency and at consumer level,
for both tradable and nontradable goods.
Demands for the …nal tradable and nontradable goods are given by:
CtT
CtN
A.1.2
=
c
= (1
PtT
Pt
c)
"
Ct ;
"
PtN
Pt
Ct :
Incomplete Asset Markets
For modelling simplicity, we choose to model incomplete markets with two risk-free one-period
nominal bonds denominated in domestic and foreign currency, and a cost of bond holdings is
introduced to achieve stationarity. Then, the budget constraint of the domestic households in
real units of home currency is given by:
BtH 1 St BtF
+
Pt
Pt
St BtF
BtH
+
St BtF
Pt R t P R
t t
Pt Yt
where Wt is the nominal wage, and
t
1
+
Wt
Lt
Pt
Ct +
t
(4)
are real pro…ts for the home consumer. BtH is the
holding of the risk free domestic nominal bond and BtF is the holding of the foreign risk-free
nominal bond expressed in foreign country currency. St is the nominal exchange rate, expressed
in units of home country currency per unit of foreign country. The function
(:) depends on the
net liability position (i.e. the negative net foreign asset position) of the home country, BtF , in
percent of GDP in the entire economy, and is taken as given by the domestic household.1
(:)
introduces a convex cost that allows to obtain a well-de…ned steady state, and captures the costs
of undertaking positions in the international asset market.2
A.2
Production Sector
The production of this economy is undertaken by three sectors. First, there is a …nal goods
sector, that uses intermediate tradable inputs from both countries and operates under perfect
competition, to produce the …nal tradable goods. This same sector also aggregates varieties of
the nontradable goods to produce a …nal nontradable good that is sold to households. The second
1
As Benigno, P.(2001) points it out, some restrictions on (:) are necessary: (0) = 1; assumes the value 1
only if BF;t = 0; di¤erentiable; and decreasing in the neighborhood of zero.
2
Another way to describe this cost is to assume the existence of intermediaries in the foreign asset market (which
are owned by the foreign households) who can borrow and lend to households of country F at a rate (1 + r ), but
can borrow from and lend to households of country H at a rate (1 + r ) (:) :
3
sector produces intermediate tradable goods, which are used as an input for the production of
…nal goods both in the home and in the foreign country. The third sector produces nontradable
goods, that are used as inputs in the production of the …nal nontradable good.
A.2.1
Final Goods Sector
The …nal tradable good is consumed by domestic households. This good is produced by a continuum of …rms, each producing the same variety, labelled by YtT , using intermediate home Xth
and foreign Xtf
goods with the following technology:
1
1
YtT =
where
Xth
1=
x
1=
x)
+ (1
1
Xtf
is the elasticity of substitution between home-produced and foreign-produced imported
intermediate goods, and
x
is the share of home goods in the production function. We further
assume symmetric home-bias in the composite of intermediate tradable goods. The corresponding
composite of home and foreign intermediate tradable goods abroad is given by
1
YtT =
Xth
1=
x)
(1
1
+
1=
x
Xtf
1
Xth and Xtf , that denote the amount of home and foreign intermediate tradable inputs to produce
the …nal tradable good at home, are also Dixit-Stiglitz aggregates of all types of home and foreign
…nal goods, with elasticity of substitution :
Z 1
h
Xth (h)
Xt
1
1
dh
0
and
Z
Xtf
where Xth (h) and
Xtf (f )
0
1
Xtf (f )
1
1
df
denote individual quantities from intermediate tradable goods producers
at home and foreign. The equivalent quantities for foreign …nal tradable goods producers are
Xth (h) and Xtf (f ). Optimizing conditions by …nal tradable goods producers deliver the following
demand functions:
Xth (h) =
Xtf (f )
x
= (1
Pth (h)
Pth
x)
Ptf (f )
Ptf
Pth
PtT
!
YtT ; Xth (h) = (1
Ptf
PtT
!
YtT ;
4
Xtf
(f ) =
x)
x
Pth (h)
Pth
Ptf (f )
Ptf
Pth
PtT
!
Ptf
PtT
YtT
!
YtT
where
Pth
and
Z
1
0
PtT
Z
1
1
Pth (h)1
=
dh
Pth
x
;
1
Ptf
+ (1
1
0
1
Ptf
x)
1
Ptf (f )1
1
df
:
1
1
We assume that the law of one price holds for intermediate inputs, such that Pth (h) = Pth (h)St ,
and Ptf (f ) = Ptf (f )St , where St is the nominal exchange rate.
The production of the …nal nontradable good is given by:
Z
YtN
1
0
where we assume the same elasticity
XtN (n)
1
1
dn
> 1 than in the case of …nal tradable goods produced
within country H. The price level for nontradables is
Z
PtN
A.2.2
0
1
1
1
pN
t (n)
1
dn
Intermediate Non-Tradable Goods Sector
The intermediate nontradable sector produces di¤erentiated goods that are aggregated by …nal good producing …rms, and ultimately used for …nal consumption by domestic households
only. Each …rm produces intermediate nontradable goods according to the following production
function
YtN (n) = At ZtN LN
t (n)
(5)
where At is a labor augmenting aggregate world technology shock which has a unit root with
drift, as in Galí and Rabanal (2005):
log At = g + log At
1
+ "at
(6)
This shock also a¤ects the intermediate tradable sector production function. Hence, real variables in both countries grow at a rate g. ZtN is the country-speci…c productivity shock to the
nontradable sector at time t which evolves according to an AR(1) process in logs
log ZtN = (1
N
) log(Z N ) +
Z;N
log ZtN 1 + "Z;N
t
(7)
Firms in the nontradable sector face a Calvo lottery when setting their prices. Each period, with
probability 1
N,
…rms receive a stochastic signal that allows them to reset prices optimally.
5
We assume that there is partial indexation with a coe¢ cient 'N to last period’s sectorial in‡ation
rate for those …rms that do not get to reset prices. As a result, …rms maximize the following
pro…ts function:
M axPtN (n) Et
1
X
82
N
Pt+k
>
1
N
>
<6 Pt (n) PtN 1
6
t;t+k
4
>
Pt+k
>
:
k
N
k=0
subject to
N;d
Yt+k
(n)
=
"
PtN (n)
N
Pt+k
!
N
Pt+k
1
PtN 1
9
>
>
=
7
N 7 N;d
M Ct+k 5 Yt+k (n)
>
>
;
3
'N
!'N #
YtN
(8)
(9)
where YtN;d (n) is total individual demand for a given type of nontradable good n, and YtN is
aggregate demand for nontradable goods, as de…ned above.
discount factor, where
t
=
t
Ct bCt
1
t;t+k
=
k
t+k
t
is the stochastic
is the marginal utility of consumption. M CtN corresponds
to the real marginal cost in the nontradable sector. From cost minimization:
M CtN =
A.2.3
Wt
Pt ZtN At
Intermediate Tradable Goods Sector
The intermediate tradable sector produces di¤erentiated goods that are sold to the …nal sector
goods producers in the home and foreign countries. Most functional forms are similar to those
presented for the nontradable sector.
Each …rm produces tradable intermediate goods according to the following production function
Yth (h) = At Zth Lht (h)
(10)
where Zth is the country-speci…c productivity shock to the intermediate goods tradable sector at
time t which evolves according to an AR(1) process in logs
log Zth = (1
h
) log(Z h ) +
Z;h
log Zth
1
+ "Z;h
t
(11)
Firms in the intermediate tradable sector face the same Calvo lottery as …rms in the intermediate
nontradable sector, with relevant parameters h and 'h :
82
h
>
h (h) Pt+k 1
>
P
1
<
t
X
Pth 1
6
k
6
M axP h (h) Et
h t;t+k
4
t
>
Pt+k
>
k=0
:
6
'h
9
>
>
=
7
h 7 h;d
M Ct+k 5 Yt+k (h)
>
>
;
3
(12)
subject to
h;d
h
h
Yt+k
(h) = Xt+k
(h) + Xt+k
(h)
"
!
!'h #
h
Pt+k
Pth (h)
1
=
h
Pt+k
Pth 1
Xth
(13)
where Yth;d (h) is total individual demand for a given type of tradable intermediate good h,
and Xth is aggregate demand for intermediate good h, consisting of home demand, and foreign
demand:
Xth =
"
x
Pth
PtT
YtT + (1
x)
Pth
PtT
YtT
#
M Cth corresponds to the real marginal cost in the nontradable sector. From cost minimization:
Wt
M Cth =
Pt Zth At
A.2.4
Market Clearing
We assume that the demand shock is allocated between tradable and nontradable goods in the
same way that private consumption is. Hence the market clearing conditions for both types of
…nal goods, consisting of private consumption and the demand shock in the tradadable sector,
are:
YtT = CtT + GTt
YtN = CtN + GN
t
T
where GN
t ; Gt follow AR(1) processes in logs. The bond market clearing conditions are
BtH + BtH = 0
(14)
BtF + BtF = 0
(15)
For the nontradable intermediate goods, the market clearing condition is:
YtN (n) = XtN ; for all n 2 [0; 1]
(16)
while for the intermediate tradable goods sector it is:
Yth (h) = Xth (h) + Xth (h); for all h 2 [0; 1]
(17)
For the labor market:
Lt = Lht + LN
t =
Z 1
Z
h
=
Lt (h)dh +
0
0
7
(18)
1
LN
t (n)dn
A.3
Optimizing, Market Clearing Conditions, and Monetary Policy
In this subsection we present the full set of equations characterizing the symmetric equilibrium.
Since all agents in each economy are equal, then the per capita and aggregate consumption levels
are equal (Ct = Ct ), as well as the net foreign assets levels (BtF = BtF ).
A.3.1
Households
The Euler equations for home and foreign households, and the optimal condition of holdings by
home household of the foreign bond are:
where
t
Pt
Pt+1
P
Rt t
Pt+1
t
=
E t Rt
t+1
t
=
Et
t+1
t
=
St BtF
Pt Yt
E t Rt
Qt+1
Qt
t+1
is the marginal utility of consumption:
t
t
= UC (Ct ) =
= UC (Ct ) =
t
Ct
t
Wt
= L'
t
Pt
Wt
= (Lt )'
Pt
where:
Lt = Lht + LN
t
and
Lt = Lht + LN
t
8
1
t
Ct
The labor supply decisions in each country are:
t
bCt
b Ct
1
Household demand for …nal tradable and nontradable goods are given by:
CtT
=
c
CtN
= (1
CtT
=
CtN
= (1
"
PtT
Pt
Ct ;
c)
Ct :
"
PtT
Pt
c
"
PtN
Pt
Ct ;
"
PtN
Pt
c)
Ct
and the CPI’s in each country are given by:
h
Pt
Pt
T 1 "
c Pt
PtT
c
+ (1
1 "
c)
+ (1
1 "
PtN
1
1 "
;
1
1 "
1 "
PtN
c)
i
The real exchange rate is
Qt =
A.3.2
S t Pt
Pt
Final Goods Producers
The production of …nal tradable goods in both countries is given by:
1
1
YtT =
1=
x
Xth
+ (1
1=
x)
Xtf
1=
x
Xtf
1
and
1
1
YtT =
(1
1=
x)
Xth
+
1
Demand for intermediate tradable goods is:
Xth =
x
Xtf = (1
where
Pth
Pth
PtT
x)
Z
0
1
YtT ; Xth = (1
Ptf
PtT
!
YtT ; Xtf (f ) =
1
Pth (h)1
1
dh
;
9
Ptf
Pth
PtT
x)
Z
0
1
x
Ptf
PtT
YtT
!
YtT
1
Ptf (f )1
1
df
:
The price of …nal tradable goods is:
PtT
=
x
Pth
x
Pth
1
+ (1
x)
Ptf
x)
Ptf
1
1
1
and
PtT
=
1
+ (1
1
1
1
Since we assumed that the law of one price holds for intermediate goods, it also holds in the
aggregate, such that Pth = Pth St , and Ptf = Ptf St , where St is the nominal exchange rate.
A.3.3
Nontradable Goods Producers
The price setting equations are given by the following optimal expressions:
9
8
!
1
k
'N
Y
X
>
>
N
>
(
)
k k
t+s 1
N YN >
>
>
>
M
C
t+k
N
>
>
N
t+k t+k >
=
<
t+s
N
p^t
s=1
k=0
=
E
t
!1
>
k
1
(
1) >
PtN
'N
>
>
Y
X
N
N
>
>
Pt+k
)
(
k k
t+s 1
>
>
N
>
>
Y
t+k
N
:
N
Pt+k t+k ;
t+s
s=1
k=0
where
M CtN =
Wt
;
Pt ZtN At
YtN = CtN + GN
t
The evolution of the price level of nontradables is
PtN
where
N
t 1
=
N
PtN 1
'N
N
t 1
1
1
+ (1
N)
1
p^N
t
1
PtN 1
.
PtN 2
The production function is:
YtN = At ZtN LN
t :
In the foreign country these expressions are:
8
9
!
'N
1
k
N
X
Y
>
>
>
t+s
1
k k
N YN >
>
>
>
M
C
t+k
N
>
>
N
t+k t+k >
<
=
N
t+s
p^t
s=1
k=0
=
Et
!1
>
1
k
(
1) >
PtN
'N
>
>
X
Y
N
N
>
>
Pt+k
(
)
k
t+s 1
>
>
k
N
>
>
Y
t+k
:
N
N
t+k ;
P
t+k
t+s
s=1
k=0
10
where
M CtN =
Wt
;
Pt ZtN At
YtN = CtN + GN
t
The evolution of the price level of nontradables is
PtN
where
N
t 1
=
PtN 1
N
N
t 1
1
'N
+ (1
N
)
p^N
t
1
1
1
PtN 1
.
PtN 2
The production function is:
YtN = At ZtN LN
t :
A.3.4
Intermediate Traded Goods Producers
The price setting equations are given by the following optimal expressions:
9
8
!
1
k
'h
X
Y
>
>
h
>
(
)
k k
t+s 1
h Yh >
>
>
>
M
C
t+k
h
>
>
t+k t+k >
h
=
<
t+s
h
p^t
s=1
k=0
Et
=
!1
>
k
1
(
1) >
Pth
'h
>
>
Y
X
h
h
>
>
Pt+k
)
(
k
t+s 1
>
>
h
k
>
>
Y
t+k
h
;
:
t+k
h
P
t+k
t+s
s=1
k=0
where
M Cth =
Wt
;
Pt Zth At
Yth = Xth + Xth :
The evolution of the price level of …nal tradables is
Pth
where
h
t 1
=
Pth
Pth
1
h
Pth 1
h
t 1
'h 1
+ (1
h)
p^ht
1
1
1
.
2
The production function is:
Yth = At Zth Lht
In the foreign country, this expressions are:
8
1
X
>
>
k k
>
>
>
f
f
<
p^t
k=0
=
Et
1
(
1) >
Ptf
>
> X k k
>
>
:
f
t+k
k
Y
s=1
t+k
k
Y
s=1
k=0
11
'f
f
t+s 1
f
t+s
'f
f
t+s 1
f
t+s
!
!1
9
>
f
f >
>
>
M Ct+k
Yt+k
>
=
f
Pt+k
f
Pt+k Yt+k
>
>
>
>
>
;
where
Wt
M Ctf =
Pt Ztf At
;
Ytf = Xtf + Xtf :
The evolution of the price level of …nal tradables is
Ptf
where
f
t 1
=
Ptf
1
Ptf
2
Ptf 1
f
1
'f
f
t 1
+ (1
f
)
1
1
p^ft
1
.
The production function is:
Ytf = At Ztf Lft
A.3.5
Monetary Policy
Monetary policy in both countries is conducted with a Taylor rule that targets CPI in‡ation and
output growth deviation from steady-state values:
Rt = R(1
Rt = R
A.3.6
(1
r)
r)
Pt =Pt
Rt r 1
Rt
1
r
r)
(1
Pt =Pt
1
Yt =Yt 1
1+g
r)
Yt =Yt
1+g
(1
r) y
(1
1
exp("rt )
r) y
exp("rt )
Demand Shocks
GTt = (GT )(1
N (1
GN
t = (G )
A.3.7
(1
1
GT
)
(GTt 1 )
GT
(GN
t 1)
GN
)
GN
GTt = (GT )(1
GT
GN
= (GN )(1
t
GN
)
(GTt 1 )
)
(GN
t 1)
T
exp("G
t )
N
exp("G
t )
T
)
exp("G
t
GT
GN
N
exp("G
t
)
Trade Balance and Net Foreign Asset Dynamics
We present the evolution of the trade balance and net foreign assets of the home country, since
the de…nition those in the foreign country will mirror those in the home country. Holdings of
foreign bonds depend on the trade balance (N Xt ) as follows
St BtF
Pt R t
St BtF
Pt Yt
=
12
St BtF
Pt
1
+ N Xt
Since international trade only occurs at the intermediate goods level, net exports equal exports
minus imports of intermediate goods:
N Xt =
Ptf Xtf
Pth Xth
Pt
Finally, we de…ne nominal GDP to be equal to aggregate nominal private and public consumption,
hence Pt Yt = PtT (CtT + GTt ) + PtN (CtN + GN
t ).
13
B
Appendix: Log-Linear version of the Model
Euler equations
b ct =
b
ct =
(1 + g
(1 + g
b) (rt
b ) rt
Et 4pt+1 ) + (1 + g) Et ct+1 + (1 + g
Et 4pt+1 + (1 + g) Et ct+1 + (1 + g
b
b) 1
b
b ) 1
Risk sharing
Et (qt+1
where
(1 + g) Et ct+1 b
(1 + g) Et ct+1 b ct
(1 + g b)
(1 + g b )
b
b
+ 1
1
t
t + bt
qt ) =
0 (0) Y;
bt =
St BtF
Pt
ct
t
(19)
t
(20)
(21)
1:
Y
The labor supply schedules are given by:
b
b
1+g
e
ct
e
ct 1 +
"a
1+g b
(1 + g b)
(1 + g b) t
1+g
b
b
= 'lt +
e
ct
e
ct 1 +
"a
1+g b
(1 + g b )
(1 + g b ) t
!
~ t = 'lt +
(22)
!
~t
(23)
Technology
yeth = lth + zth
"at
yeth = lth + zth
yetN = ltN + ztN
"at
(25)
"at
(26)
yetN = ltN + ztN
Consumer price in‡ation
4pt =
T
c 4pt
4pt =
c
+ (1
4pTt + (1
(24)
"at
(27)
N
c ) 4pt
c
(28)
) 4pN
t
(29)
+
st )
(30)
) 4pft
(31)
Tradable in‡ation
4pTt =
4pTt =
h
x 4pt
x
f
x ) (4pt
+ (1
(4pht
st ) + (1
x
Price setting in the nontradable sector
4pN
t
N
'N 4pN
t 1 = Et 4pt+1
14
'N 4pN
+
t
N
w
et
ztN
tN
t
(32)
4pN
t
where
tN
=
N
pN
t
N
'N 4pN
t 1 = Et 4pt+1
= (1
N ) (1
N) = N;
N
'N 4pN
t
= (1
+
N
) (1
N
N
w
et
)=
ztN
N
tN
t
; tN = pN
t
(33)
pt ; and
pt .
Price setting in the intermediate tradable good sector
4pht
4pft
where
tT = pTt
h
= (1
pt ; and
'h 4pht
'f 4pft
h ) (1
tT =
= Et 4pht+1
1
1
= Et 4pft+1
h) = h;
pTt
'h 4pht +
'f 4pft
= (1
f
+
) (1
f
w
et
h
f
f
w
et
)=
f
zth
ztf
tht
tft
tTt
(34)
tTt
; tht = pht pTt ; tft = pft
(35)
pTt ;
pt .
Final consumption demand
e
cTt =
"tTt + e
ct
(36)
"tN
ct
t +e
(38)
e
cTt =
" tTt + e
ct
(37)
e
cN
=
t
" tN
ct
t +e
(39)
tht + yetT
(40)
tft + yetT
(42)
e
cN
t =
Intermediate tradable and nontradable demand
x
eht =
x
eht =
tht + yetT
x
eft =
tft + yetT
x
eft =
Relative Price Index
Let´s de…ne tt =
pft
,
ph
t
(41)
(43)
since the law of one price holds tt =
following relative prices as a function of tt .
tht =
tht
(1
=
tft
=
ttf
x ) tt
ph
t
pft
; then we can write the
(44)
x tt
(45)
x tt
= (1
tt =
(46)
x ) tt
(47)
Relative prices
t t = tt
1
+ 4st + 4pft
15
4pht
(48)
tTt = tTt
1
+ 4pTt
c
tN
t =
tTt = tTt
1
1
(49)
tTt
1
(50)
c
+ 4pTt
4pt
c
tN
=
t
qt = qt
4pt
(51)
tTt
1
(52)
c
+ 4st + 4pt
4pt
(53)
Taylor rules
rt =
rt =
r rt 1
r rt 1
+ (1
+ (1
r
r)
4pt +
y 4yt
+ "rt
(54)
)
4pt +
y 4yt
+ "rt
(55)
Net foreign assets and net exports
ebt
where ebt =
BtF St
Pt Y
n
fxt =
1 e
bt
1+g
Xf
Y
x
eht
1
is the debt to GDP ratio, and n
fxt =
=n
fxt
x
eft
(56)
tt
N Xt
Y ,
(57)
and where we have assumed balanced
trade in the steady state. To solve for the steady-state ratios, we have that:
YT
GT
CT
=
=
=
Y
G
C
c
YN
GN
CN
=
=
=1
Y
G
C
Xh
=
YT
x;
Xf
=1
YT
c
x
Therefore tradable GDP over total GDP is
Xh
Xh Y T
= T
=
Y
Y Y
Hence
x c:
Xf
Xf XT Y T
= T T
= (1
Y
X Y Y
x) c:
Market clearing:
y~tT = (1
)~
cTt + g~tT
16
(58)
where
equals the fraction of government spending over total output. For the nontradable good,
the market clearing condition is:
y~tN = (1
)~
cN
~tN
t + g
(59)
Finally, for the intermediate tradable goods sector:
yeth =
Total real GDP
total labor
T
c (tt
yet =
lt =
eht
xx
+ (1
eht
x) x
+ yetT ) + (1
N
c ) (tt
(60)
+ yetN )
Xh
YN
h
l
+
lN
Xh + Y N t
Xh + Y N t
(61)
(62)
For the foreign country:
yet =
y~tT = (1
)~
cTt +
g~tT
(63)
y~tN = (1
)~
cN
t +
g~tN
(64)
eft
x) x
(65)
yetf =
T
c (tt
lt =
eft
xx
+ (1
+ yetT ) + (1
N
c ) (tt
+ yetN )
Yf
YN
f
l
+
lN
t
Yh+YN
Yf +YN t
(66)
(67)
Mapping variables in the model with observable variables.
e
ct
e
ct
e
ct
e
ct
yet
Shocks
yet
yet
yet
1
= 4ct
1
= 4ct
1
= 4yt
1
= 4yt
t 1
+ "t
=
t 1
+ "t
zth =
Z;h h
zt 1
t
t
=
ztf =
Z;f
ztf
17
1
"at
(68)
"at
(69)
"at
(70)
"at
(71)
+ "Z;h
t
+ "Z;f
t
ztN =
ztN =
Z;N N
zt 1
Z;N
+ "Z;N
t
ztN 1 + "Z;N
t
gtT =
G;T T
gt 1
+ "G;T
t
gtN =
G;N N
gt 1
+ "G;N
t
gtT =
G;T
gtT
+ "G;T
t
gtN =
G;N
gtN 1 + "G;N
t
and "at ; "rt ; "tr are iid shocks.
18
1