Model appendix
Frank Smets and Raf Wouters
November 20, 2006
1
Decision problems of …rms and households and equilibrium conditions
1.1
Final goods producers
The …nal good Yt is a composite made of a continuum of intermediate goods Yt (i) as in
Kimball (1995). The …nal good producers buy intermediate goods on the market, package Yt , and resell it to consumers, investors and the government in a perfectly competitive
market.
The …nal good producers maximize pro…ts. Their problem is:
maxYt ;Yt (i) Pt Yt
hR
1
;
s:t: 0 G YYt (i)
t
R1
0
p;t
Pt (i)Yt (i)di
i
di = 1 (
f;t )
where Pt and Pt (i) are the price of the …nal and intermediate goods respectively,
and G is a strictly concave and increasing function characterised by G(1) = 1. pt is an
exogenous process that re‡ects shocks to the aggregator function that result in changes in
the elasticity of demand and therefore in the markup. We will constrain pt 2 (0; 1). pt
follows the exogenous ARMA process:
ln
p
t
= (1
p) ln
p
+
p
p ln t 1
p
p t 1
+
p
t;
p;t
N (0;
p)
(1)
To simplify notiation, in what follows we leave out this argument.
The FOCs are:
Z 1
Yt (i) Yt (i)
f;t
(@Yt )
Pt =
G0
di
Yt 0
Yt
Yt
Yt (i) 1
(@Yt (i))
Pt (i)= f;t G0
Yt
Yt
resulting in
Z
Pt (i) 1 0 Yt (i) Yt (i)
Yt (i) = Yt G
G
di
Pt 0
Yt
Yt
As in Kimball (1995), the assumptions on G() imply that the demand for input Yt (i)
is decreasing in its relative price, while the elasticity of demand is a positive function of
the relative price (or a negative function of the relative output).
0 1
1
1.2
Intermediate goods producers
Intermediate good producer i uses the following technology:
Yt (i) = "at Kts (i)
t
1
Lt (i)
t
(2)
where Kts (i) is capital services used in production, Lt (i) is aggregate labour input and
is a …xed cost. t respresents the labour-augmenting deterministic growth rate in the
economy and "at is total factor productivity and follows the process:
ln"at = (1
z )ln"
a
a
z ln"t 1
+
+
a
t;
a
t
N (0;
a)
(3)
The …rm’s pro…t is given by:
Pt (i)Yt (i)
Wt Lt (i)
Rtk Kt (i):
where Wt is the aggregate nominal wage rate and Rtk is the rental rate on capital.
Cost minimization yields the conditions:
(@Lt (i)) :
(1
t (i)
(@Kts (i)) :
(1
t (i)
)"at Kts (i) Lt (i)
)t (1
"at Kts (i)
)t
1 L (i)1
t
= Wt
= Rtk
(4)
(5)
where t (i) is the Lagrange multiplier associated with the production function and equals
marginal cost M Ct .
Combining these FOCs and noting that the capital-labour ratio is equal across …rms
implies:
Wt
Lt
(6)
Kts =
1
Rtk
The marginal cost M Ct is the same for all …rms and equal to:
M Ct =
(1
)
(1
)
Wt1
Rtk
(1
)t
("at )
1
Under Calvo pricing with partial indexation, the optimal price set by the …rm that is
allowed to re-optimise results from the following optimisation problem:
max Et
Pet (i)
1
X
s=0
s
p
s
t+s Pt
t Pt+s
h
Pet (i)(
1
p
s
l=1 t+l 1
s:t: Yt+s (i) = Yt+s G0
where Pet (i) is the newly set price,
p
1
p
)
Pt (i)Xt;s
Pt+s
i
M Ct+s Yt+s (i)
t+s
is the Calvo probability of being allowed to optimise
2
s
Pt
] is the nominal discount factor
one’s price, t is in‡ation de…ned as t = Pt =Pt 1 , [ t Pt+s
t+s
for …rms (which equals the discount factor for the households that are the …nal owners of
R1
Yt (i)
the …rms), t = 0 G0 YYt (i)
Yt di and
t
Xt;s =
(
(
1 f or s = 0
1 p
) f or s = 1; :::; 1
1
p
s
l=1 t+l
)
The …rst-order condition is given by:
Et
1
X
s
p
s
t+s Pt
t Pt+s
s=0
Yt+s (i) Xt;s Pet (i) + Pet (i)Xt;s
1
G0 (xt+s )
=0
G0 1 (zt+s ) G00 (xt+s )
(7)
M Ct+s
where xt = G0 1 (zt ) and zt = PPt (i)
t.
t
The aggregate price index is in this case given by:
Pt = (1
1.3
p )Pt (i)G
0 1
Pt (i)
Pt
t
+
1
p
p
p t 1
Pt
1G
0 1
"
p
1
p
t 1
Pt
1 t
Pt
#
(8)
Households
Household j chooses consumption Ct (j), hours worked Lt (j), bonds Bt (j), investment
It (j) and capital utilisation Zt (j), so as to maximise the following objective function:
Et
1
X
s=0
1
s
1
(Ct+s (j)
Ct+s
1
1)
c
exp
c
c
1+
1
l
Lt+s (j)1+
l
subject to the budget constraint:
Ct+s (j) + It+s (j) +
Bt+s (j)
b
"t Rt+s Pt+s
Tt+s
(9)
h (j)L
Rk Zt+s (j)Kt+s
Bt+s 1 (j) Wt+s
t+s (j)
+
+ t+s
Pt+s
Pt+s
Pt+s
1 (j)
a(Zt+s (j))Kt+s
1 (j)
+
Divt+s
Pt+s
and the capital accumulation equation:
Kt (j) = (1
)t
1 (j)
+ "it 1
S
It (j)
It 1 (j)
It (j)
There is external habit formation captured by the parameter . The one-period bond
is expressed on a discount basis. "bt is an exogenous premium in the return to bonds, which
might re‡ect ine¢ ciencies in the …nancial sector leading to some premium on the deposit
rate versus the risk free rate set by the central bank, or a risk premium that households
require to hold the one period bond. "bt follows the stochastic process:
ln"bt =
b
b ln "t 1
+
3
b b
t; t
N (0;
b)
(10)
is the depreciation rate, S( ) is the adjustment cost function, with S( ) = 0;
S 0 ( ) = 0, S 00 ( ) > 0, and "it is a stochastic shock to the price of investment relative to
consumption goods and follows an exogenous process:
ln"it =
i
i ln "t 1
+
i i
t; t
N (0;
i)
(11)
Tt+s are lump sum taxes or subsidies and Divt are the dividends distributed by the labour
unions.
Finally, households choose the utilitsation rate of capital. The amount of e¤ective
capital that households can rent to the …rms is:
Kts (j) = Zt (j)Kt
1 (j)
(12)
The income from renting capital services is Rtk Zt (j)Kt 1 (j) , while the cost of changing
capital utilisation is Pt a(Zt (j))Kt 1 (j):
In equilibrium households will make the same choices for consumption, hours worked,
bonds, investment and capital utilization. The …rst-order conditions for consumption,
hours worked, bond holdings, investment, capital and capital utilisation can be written as
(dropping the j index):
(@Ct )
t
1
(@Lt )
(Ct
1
= exp
hCt
1
1)
(@ Kt )
(@ut )
c
1
l
Lt (j)1+
exp
c
(@Bt )
(@It )
c
1+
t
k
t
Rtk
Pt
t
c
1+
l
1
l
= "bt Rt Et
It
)
It 1
(Ct
L1+
t
Ct
l
(
c
k i
t "t
1
S(
= a0 (Zt )
c
1)Lt l =
t
Wth
Pt
t+1
t+1
It
It
)
It 1 It 1
It+1 It+1 2
+ Et kt+1 "it+1 S 0 (
)(
)
It
It
"
!
k
Rt+1
=
Et t+1
Zt+1 a(Zt+1 ) +
Pt+1
=
1)
S0(
(13)
k
t+1 (1
#
)
(14)
(15)
where t and kt are the Lagrange multipliers associated with the budget and capital
accumulation constraint respectively. Tobin’s is Qt = kt = t and equals one in the absence
of adjustment costs.
4
1.4
Intermediate labour union sector
Households supply their homogenous labour to an intermediate labour union which differentiates the labour services and sets wages subject to a Calvo schemeiates about the
wage with the intermediate labour packers.
Before going into the household’s and the union’s decision on labour supply and wage
setting, more details on the labor market are needed. Labor used by the intermediate
goods producers Lt is a composite:
Z
Lt =
1
Lt (l) 1+
1
w;t
1+
w;t
dl
:
(16)
0
There are labor packers who buy the labor from the unions, package Lt , and resell
it to the intermediate goods producers. Labor packers maximize pro…ts in a perfectly
competitive environment. From the FOCs of the labor packers one obtains:
1+ w;t
w;t
Wt (l)
Wt
Lt (l) =
Lt
(17)
Combining this condition with the zero pro…t condition one obtains an expression for the
wage cost for the intermediate goods producers:
Z
Wt =
1
1
w;t
Wt (l)
w;t
dl
(18)
0
We assume that
ln
w;t
w;t
follows the exogenous ARMA process:
= (1
w) ln w
+
w ln w;t 1
w w;t 1
+
w;t ; w;t
::
(19)
Labor packers buy the labour from the unions. The unions are an intermediate between
the households and the labor packers. The unions allocate and di¤erentiate the labour
services from the households and have market power: they can choose the wage subject
to the labour demand equation 17. The household’s budget constraint now also contains
the dividends of the union distributed to the households:
Ct+s (j) + It+s (j) +
Bt+s (j)
2
bt Rt+s Pt+s
+ At+s (j)
Tt+s
h (j)L
k u
Bt+s 1 (j) Wt+s
Divt+s Rt+s
t+s (j)
t+s (j)K t+s
+
+
+
Pt+s
Pt+s
Pt+s
Pt+s
1 (j)
a(ut+s (j))Kt+s
The household labour supply decision is the same for all households and is given by the
following FOC:
Wth
Pt
=
h
1
1
(Ct
c
hCt
1
1)
c
5
i
exp
t
c 1
1+ l
L1+
t
l
(
c
1)Lt l
1 (j)
The real wage desired by the households here re‡ects the marginal rate of substitution between leisure and consumption. The marginal disutility of labour is equal accross
households and equal to:
0
=
Ul;t
Labour unions take this marginal rate of substitution as the cost of the labour services
in their negotiations with the labour packers. The markup above the marginal disutility
is distributed to the households. However, the union is also subject to nominal rigidities
á la Calvo. Speci…cally, unions can readjust wages with probability 1
w in each period.
For those that cannot adjust wages, Wt (l) will increase at the determistic growth rate
and weighted average of the steady state in‡ation
and of last period’s in‡ation ( t 1 ).
~ t (l) that maximizes the wage
For those that can adjust, the problem is to choose a wage W
income in all states of nature where the union is stuck with that wage in the future:
max Et
ft (l)
W
1
X
s
s
w[
t+s Pt
t Pt+s
s=0
h
] Wt+s (l)
Wt+s (l)
and Lt+s (l) =
Wt+s
ft (l)( s
with Wt+s (l) = W
l=1
i
h
Wt+s
Lt+s (l)
1+ w;t+s
w;t+s
Lt+s
1
w
w
t+l 1
) for s = 1; :::; 1
The …rst order condition becomes:
(@Wt )
0 = Et
1
X
s
w
s
t Pt+s
s=0
(
1+
t+s Pt
w;t+s
)(
w;t+s
where
Xt;s =
(
(
s
l=1
2
6
4(Wt+s (l)
Xt;s
)Lt+s
Wt+s
w
t+l
ft (l)
Xt;s W
Wt+s
h
Wt+s
)
!
1+ w;t+s
w;t+s
1
Xt;s Lt+s (l)
1 f or s = 0
1 w ) f or s = 1; :::; 1
1
)
Simplifying by substituting for the individual labour and multiplying with the optimal
wage
Et
1
X
s
w
s
s=0
or
Et
t+s Pt
ft (l)
(Xt;s W
t Pt+s
1
X
s=0
s
w
s
t+s Pt
t Pt+s
1+
h
Wt+s
)(
1
Lt+s (l)
w;t+s
w;t+s
)Lt+s (l)
w;t+s
h
h
w;t+s )Wt+s
(1 +
ft (l)Lt+s (l) = 0
Xt;s W
i
ft (l) = 0
Xt;s W
(20)
The aggregate wage expression is
Wt = [(1
f
w )Wt
1
w;t
+
w(
6
w
t 1
1
w
Wt
1)
1
w;t
]
w;t
:
(21)
1.5
Government Policies
The central bank follows a nominal interest rate rule by adjusting its instrument in response to deviations of in‡ation and output from their respective target levels:
Rt
=
R
Rt
R
1
R
"
Yt
Yt
1
t
2
#1
R
Yt =Yt
Yt =Yt
3
1
rt
(22)
1
where R is the steady state nominal rate (gross rate) and Yt is the natural output.
The parameter R determines the degree of interest rate smoothing. The monetary policy
shock rt is determined as
lnrt =
r lnrt 1
+
(23)
r;t
The central bank supplies the money demanded by the household to support the desired nominal interest rate.
The government budget constraint is of the form
Pt Gt + Bt
1
= Tt +
Bt
Rt
(24)
where Tt are nominal lump-sum taxes (or subsidies) that also appear in household’s
budget constraint. Government spending expressed relative to the steady state output
path gt = Gt =(Y t ) follows the process:
lngt = (1
g )lng
+
g lngt 1
+
ga lnZt
ga lnZt 1
+
g;t ; g;t
:::
(25)
where we allow for a reaction of government spending to respond on the productivity
process. Or a slightly di¤erent speci…cation:
lngt
1.6
ga lnZt
= (1
g )lng
+
g (lngt 1
ga lnZt 1 )
+
g;t ; g;t
:::
The natural output level
The natural output level is de…ned as the output in the ‡exible price and wage economy.
The question is which shocks need to be taken into account. More in particular the
question is whether the markup shocks in prices and wages must be taken into account
in the calculation of the natural output level. If the markup shock are not taken into
account this will lead to a trade-o¤ problem between output-gap stabilisation and in‡ation
stabilisation. Persistent markup shocks might result in persistent con‡icts between the
two objectives and therefore in persistent deviations of in‡ation from the in‡ation target.
7
1.7
Resource constraints
To obtain the market clearing condition for the …nal goods market …rst integrate the
HH budget constraint across households, and combine it with the government budget
constraint:
Z
Z
Z
k
Pt Ct + Pt It + Pt Gt
+
W
(j)L
(j)dj
+
R
K
(j)dj
P
a(u
)
Kt 1 (j)dj:
t
t
t
t
t
t
t
or in case of the labour unions:
Pt Ct + Pt It + Pt Gt
t+
Z
Wth (j)Lt (j)dj + Divt + Rtk
Z
Kt (j)dj
Pt a(ut )
Z
Kt
1 (j)dj:
Next, realize that
t
=
Z
t (i)di
=
Z
Pt (i)Yt (i)di
Rtk Kt ;
Wt Lt
R
where Lt = L(i)t di is total labor supplied by the labor packers (and demanded by
R
R
the …rms), and Kt = Kt (i)di = Kt (j)dj. Now replace the de…ntion of t into the
HH budget constraint, realize that by the labor and goods’packers’zero pro…t condition
R
R
R
Wt Lt = Wt (j)Lt (j)dj, or = Wth (j)Lt (j)dj + Divt and Pt Yt = P (i)t Y (i)t di and
obtain:
Pt Ct + Pt It + Pt Gt + Pt a(ut )Kt 1 = Pt Yt ;
or
Ct + It + Gt + a(ut )Kt
1
= Yt
(26)
where Yt is de…ned by eq(1) .
In the data we do observe Y t and Lt instead of Yt and Lt where
Z
Y t = Yt (i)di
Starting from eq (), the relationship between output and the aggregate inputs, labor
and capital, is:
Yt =
Z
1+ p;t
p;t
Pt (i)
Pt
= Yt (Pt )
= Yt (Pt )
= Yt (Pt )
1+ p;t
p;t
1+ p;t
p;t
1+ p;t
p;t
8
Z
Z
Pt
Yt di
Pt (i)
Pt (i)
1+ p;t
p;t
1+ p;t
p;t
1+ p;t
p;t
di
di
where P t =
and
R
Pt (i)
1+ p;t
p;t
p;t
1+ p;t
di
Lt =
=
Z
Z
Lt (j)dj
Wt (j)
Wt
= Lt (Wt )
where W t =
R
Wt (j)
1+ w;t
w;t
dj
1+ w;t
w;t
1+ w;t
w;t
Wt
Lt dj
1+ w;t
w;t
w;t
1+ w;t
In the …rst order approximation the barred concepts will equal the unbarred.
9
1.8
Exogenous Processes
There are seven exogenoues processes in the model:
Technology process:
lnZt = (1
z )lnZ
+
z lnZt 1
+
z;t
+
;t
Investment relative price process:
ln
t
= (1
)ln +
ln
t 1
Intertemporal preference shifter (…nancial risk premium process):
lnbt = (1
b) ln b
+
b ln bt 1
+
b;t
Government spending process:
lngt = (1
lngt
g )lng
ga lnZt
+
= (1
g lngt 1
g )lng
+
+
ga lnZt
g (lngt 1
ga lnZt 1
+
ga lnZt 1 )
+
g;t
Monetary Policy Shock:
lnrt =
r lnrt 1
+
r;t
Price Mark-up shock:
ln
p;t
= (1
Wage Mark-up shock: ln
w;t
and where the innovations
p) ln p
= (1
+
w) ln w
+
p p;t 1
w ln w;t 1
+
p;t
w w;t 1
+
w;t
are distributed as i.i.d. Normal innovations:
N (0;
i;t
2
p ln p;t 1
i)
Detrending and steady state
2.1
Intermediate goods producers
The model can be detrended with the deterministic trend and nominal variables can be
replaced by their real counterparts. Lower case variables represent detrended real variables
which can be considered as stationary processes that have a well de…ned steady state: for
instance
kt = Kt = t ; wt = Wt =(Pt t ); rtk = Rtk =Pt ; t = t c t
The aggregate production function 2 becomes
10
yt (i) = Zt kt (i) (Lt (i))1
(27)
1
wt
Lt
rtk
(28)
1
w
L
rk
Equation 6 becomes:
kt =
and at st.st.:
k =
The maginal cost expression ?? becomes:
wt1
(1
M Ct
=
Pt
mct =
rtk
)(1
(29)
)Z
t
Option 1: Expression ?? becomes
1
Pet
p;t
1
X
Et
s=1
= 0
s
p
(1 +
s
p;t )M Ct
t+s Pt
1
t Pt+s
Yt (i) +
Pet (i)(
p;t+s
1
p
s
l=1 t+l 1
p
)
(1 +
p;t+s )M Ct+s
Yt+s (i)
becomes:
1
(e
pt
p;t
1
X
Et
(1 +
s s (1
p
p;t )mct ) yt (i)
c )s
1
pet (i)
p;t+s
t
s=1
= 0
t+s
+
1
p
s
l=1 t+l 1
( sl=1 t+l )
(
p
)
(1 +
p;t+s )mct+s
!
yt+s (i)
or
1
(e
pt
(1 +
p;t )mct ) yt (i)
p;t
1
X
s s s t+s
Et
p
t
s=1
= 0
1
p;t+s
+
pet (i)
(
1
p
s
l=1 t+l 1
( sl=1 t+l )
p
)
(1 +
p;t+s )mct+s
!
yt+s (i)
(30)
s
c s and
c (t+s) is the real discount factor:
where pet = Pet =Pt ;
= s
t+s = t+s
Note that in case c = 1 the expression simpli…es to the standard problem with as the
discount factor. This implies that in steady state:
p~ = (1 +
p)
(1
)
(1
)
w1
rk Z
1
1
p;t
p;t
Expression ?? becomes:
1
1 = (1
pt p;t +
p )~
p(
11
p
t 1
1
1
p
t
)
:
(31)
which means that:
p~ = 1
Option 2: Equation (7)
Et
1
X
s
p[
s=0
Et
s
t+s Pt
t Pt+s
]Yt+s (i)
1
X
s s s t+s
yt+s (i)
p
t
s=0
1+
1
G0 (xt+s )
G0 1 (zt+s ) G00 (xt+s )
1
1+
G0 1 (z
G0 (xt+s )
00
t+s ) G (xt+s )
1
G0 (xt+s )
G0 1 (zt+s ) G00 (xt+s )
Pet (i)Xt;s
pet (i)Xt;s
p
Xt+s
1
G0 1 (z
G0 (xt+s )
mct+s = 0
00
t+s ) G (xt+s )
(32)
where
p
Xt+s
= 0 for s = 0 or else (
In steady state this gives (using G0
1 + [1
mc]
1 (G0 (1))
s
l=0 t+l )
= x = 1)
1
G0 (1)
G0 1 (G0 (1)) G00 (1)
G00 (1)
1+ 0
G (1)
= 0
= mc
The implied markup is equal to
1
mu(1) =
1+
G00 (1)
G0 (1)
=
G0 (1)
G00 (1)
G0 (1)
G00 (1) +
=
1
"(1)
= (1 +
"(1) 1
p)
G0 (1)
G00 (1)
which corresponds with the elasticity of demand "(1) =
The aggregate price index in the case of the Calvo model becomes (8):
1 = (1
0
p )pt (i)G
1
[pt (i) ] +
1
p
p t 1
1
p
t
G0
1
h
p
1
1
p
t 1
t
i
In steady-state1
1 = (1
p )p
1 = p (i)G
0 1
(i)G0
1
[p (i) ] +
pG
0 1
[ ]
[z ]
1 = p (i)
1
Recall that zt = pt (i) t , hence z = p (i) and G0
12
1
(z ) = x = 1;
= G0 (1) , G0
M Ct+s = 0
1
( ) = 1:
(33)
Recall that aggregate pro…ts are equal to:
t
= Pt Yt
Rtk Kt :
Wt Lt
In terms of detrended variables we then have, using 2 :
t
Pt
= yt
t
rtk kt
wt Lt
= Zt kt L1t
wt Lt
1
=
Zt kt Lt
=
(1
)
(1
)1
=
h
=
1
mct
wt Lt
1
wt
Zt wt (rtk )
1
Zt wt
1
(rtk )
1
wt Lt
1
1
At steady state we can use 1 = p (i) = (1 +
= (1 +
=
2.2
wt Lt
1
p )mc
1)
p
Lt
i wL
t t
1
to get st. st. pro…ts:
w L
1
w L
1
p
Households
Expression ?? and ?? become:
t
t
respectively with where
t
=
t
= b1 exp
1
c
1+
= b1t exp
ct
l
L1+
t
l
(ct
= b2t Rt Et [
and
1
c
1+
l
1+
1)
c
1
t+1 t+1 ]
=( =
L
(h= )ct
c
c
l
(34)
(35)
) . At steady state:
c
(1
(h= ))
c
1
R =
Equation 12 and ?? become:
kt = ut kt
kt = [(1
1=
;
)= ] kt
(36)
1
+
13
t
1
S(
it
it
) it
1
(37)
which deliver the steady state relationships:
k
= u k = =k =
i
= (1=
) (1
= (1
(1
(1
)= ) k
)= ) k
under the assumption that S( ) = 0; u = 1; and
= 1:
The FOC with respect to investment, capital, and capital utilization are:
1 = Qkt
(@It )
1
t
c
+( =
(@ Kt )
c
Qt = ( =
S
)Et
(
it
it
1
ct
t
c t+1
ct
t
rtk = a0 (ut )
(@ut )
k=
t
t
with
c t+1
t+1
t+1
)Et
it
it
S0
h
"
it
it
1
1
it+1
it
Qkt+1 t+1 S 0
k
(rt+1
ut+1
it+1
it
2
a(ut+1 )) + Qt+1 (1
#)
i
)
= Qt so that
1 = Qkt
(@It )
it
1 S
it
(
"
t
t+1
+( )Et
1
Qkt+1
t+1 S
t
(@ Kt )
t+1
Qt = ( )Et
it
it
S0
k
[(rt+1
ut+1
0
it
it
1
1
it+1
it
it+1
it
a(ut+1 )) + Qt+1 (1
#)
(38)
)]
(39)
2
t
rtk = a0 (ut )
(@ut )
(40)
so that in steady state where Q = 1, u = 1; and a(u ) = 0:
1 = [(rk + (1
)]
In the variant for the utilisation cost, the accumulated utilisation costs becomes
akt =
a(ut+s (j))
1 s
k t+s
1 (j)
+
(1
)
akt
1
Option 1: Expressed in terms of detrended variables, equations ?? and 21 become:
Et
1
X
s s
t+s Lt+s (j)
w
s=0
where
0
Ul;t
=
h
b1t
t
=
=
h
1
1
w;t+s
hCt
b1t exp
1
1
(Ct
c
1
c
[(Ct
(Ct
hCt
"
0
Ul;t+s
(j)
t+s
1
1)
c 1
1+ l
1
1)
(Ct hCt
hCt 1 )] Lt l
c
#
Xt;s f
(1 + w;t+s ) +
Wt (j) = 0
Pt+s
i
+ `t exp
c 1
1+ l
L1+
t
Lt (j)1+ l (Ct hCt
i
l
c + `
t ( c 1)Lt
1)
14
c
1)
l
(
c
c
1)Lt l
for `t = 0:
Et
1
X
s s t+s
( t+s )Lt+s (j)
w
w;t+s
s=0
Et
1
X
s
s
w( =
ss
) s(
s=0
1
w
( s
+ l=0s t+l 1
( l=0 t+l 1 )
Et
1
X
w
)
"
1
s t+s
[(ct
1 )] Lt
l
[ [(ct
(1 +
w;t+s )
(h= )ct
"
1
w;t+s
[(ct
(h= )ct
1 )] Lt
l
(1 +
+
1 )] Lt
w;t+s
w
et (j) = 0
s=0
1
t+s )Lt+s (j)
#
s s
s
w ( ) ( t+s )Lt+s (j)
(h= )ct
w;t+s )
+
s
1
w
l=0 t+l 1
( sl=0 t+l 1 )
(
l
(1 +
(
s
1
w
l=0 t+l 1
( sl=0 t+l 1 )
or option 2: from (20)
1
X
s
s
w[
t+s Pt
t Pt+s
s=0
1
]Lt+s (l)
w;t+s
h
(1 +
i
ft (l) = 0
Xt;s W
h
w;t+s )Wt+s
which becomes
Et
1
X
s s s(1
w
c)
Et
Lt+s (l)
w;t+s
t
s=0
1
X
t+s
1
s
w
s s t+s
s=0
"
1
Lt+s (l)
w;t+s
t
With
h
wt+s
"
(1 +
0
Ul;t
=
t
h
w;t+s )wt+s
(1 +
= [(ct
s
1
w
l=0 t+l 1
( sl=0 t+l 1 )
s
1
w
l=0 t+l 1
( sl=0 t+l 1 )
(
h
w;t+s )wt+s
(
(h= )ct
t
1 )] Lt
w
)
w
)
#
w
et (l) = 0
#
w
et (l) = 0 (42)
l
in steady state
)wh = w
f
Ul;0
h
with w =
= [(c
(1 +
w
= (1
(h= )c )] L
h= )c L
l
l
and where
f
w )Wt
Wt = (1
becomes
1
w;t
+
w(
w(
w
1
w
w
t 1
Wt
1)
1
wt
1
w;t
1
wt = (1
et w;t +
w )w
1
t 1
w
1
t
which imply at steady state:
w =w
~ = (1 +
15
w )w
h
1)
w;t
1
w;t
)
#
w
et (j) = 0
w;t+s )
(41)
Et
w
w;t
(43)
w
)
#
w
et (j) = 0
2.3
Resource constraints
The resource constraint(s) become:2
Ct + It + Gt + a(ut )Kt
ct + it + y gt + a(ut )kt
1
= Yt
1=
= yt :
(44)
and
y_ t = Zt kt L1t
Yt =
P_t
Pt
:
! 1+
becomes
yt = (p_t )
p;t
p;t
1+ p;t
p;t
Y_ t
y_ t
where
_ )t
(P
Pt
p_t =
P~t
)
Pt
= [(1
p )(
= [(1
pt
p )~
1+ p;t
p;t
1+ p;t
p;t
+
Lt =
_ t
(p)
+
p(
_t
W
Wt
! 1+
While
_ )t
(P
Pt
p(
w;t
w;t
becomes
1+ w;t
w;t
_ t
Lt = (w)
1 t
1
1
)
)
1+ p;t
p;t
1+ p;t
p;t
]
p;t
1+ p;t
p;t
1+ p;t
]
_ t
(L)
_ t
(L)
where
At steady state we have:
c +i +g y =y :
and
y = Z k L1
:
and
y_ = y ; L_ = L :
2.4
Government Policies
The Taylor rule becomes:
Rt
=
R
2
Using gt =
Gt
Y t
,
Rt
R
Gt
t
1
R
"
t
1
yt
yt
= Y gt :
16
2
#1
R
yt =yt
yt =yt
1
1
3
rt
(45)
3
Steady state
Combining the steady state expressions:
From (28)
k =
1
w
L :
rk
(1
)
From (30)
p~ = 1 = (1 +
p)
(1
)
w1
(rk ) Z
1
From (35)
1
R =
From (36) and (37):
k
=
k ;
i
= (1
(1
)= ) k :
From (42):
w =w
~ = (1 +
with
wh =
Ul;0
= [(c
w )w
(h= )c )] L
h
= (1
l
h= )c L
From (34):
= b1 exp
c
1+
1
l
L
1+
l
c
c
(1
(h= ))
From (39) and (40)
1 = ( )[(rk ) + (1
and
rk =
1
=
1
(1
c
)
(1
rk = a0 (u )
From (27):
y = Z k L1
From (44):
i
c
+
+g =1
y
y
17
)]
:
)
c
l
The ratio
wh L
c
1
1+
1
1+
1
1+
=
=
=
with
k
y +
=
y
y
w L
w c
1
rk k
c
w
1
k y
rk
y c
w
(
L
)
k
1
and
c
y
4
= (1
g
= (1
g
i
)
y
i k
)
k y
Log-linearized model
Eq. (29) becomes:
mc
b t = (1
Eq. (31) becomes:
b
p~t =
Option1:
)w
bt +
p
1
p
( bt
bt
Z
rbtk
(46)
p bt 1 )
(47)
Eq. (30) becomes:
e )
d(p
t
Et
(1 +
1
X
p )d(mct )
s s s
p
mcd(
d(e
pt ) + d(
(
s=1
= 0
p;t ) +
1
p
s
l=1 t+l 1
( sl=1 t+l )
p
)
)
(1 +
minus the same expression for time t+1 multiplied with
1
(1
+
)
p
p
(1
e )
d(p
t
p
p )d(mct )
1 1
p
d( t )
p
)
p
(1 +
mcd(
(
p
p )d(mct+s )
p
p
p;t )
1
(
p
)
)2
mcd(
(1
d(
)
p
!
t+1 )
e ))
(d(p
t+1
= 0
in terms of deviations from steady state this becomes:
1
(1
+
= 0
)
p
p
(1
p
b
p~t
)
(1 +
( p bt
c t)
p )mc(mc
bt+1 )
18
mc
b
p p;t
p
(1
p
)
b
p~t+1
!
p;t+s )
using 47 to substitute the optimal price expression
1
(1
+
p
)1
p
(1
) (1
p
(1
p
p
( bt
p)
p)
= 0
p bt 1 )
( p bt
(mc
c t)
bt+1 )
mc
p
p
b
p p;t
(1
)1
p
p
( bt+1
regrouping
( bt
(1
=
p
p bt 1 )
( bt+1
p
)(1
p)
p
(mc
c t + mc
(1 +
=
p)
(1
b
p
bt
p bt 1
p)
Finally we obtain:
bt =
(1
)(1
p
(1 +
p
)
p)
p
mc
b t+
if c = 1; the expression
also DSSW).
p
1+
=
p
p ) ( p bt
+(
p p;t )
)(1
p
p bt )
bt+1
(mc
c t + mc
bp;t +
bt+1 )
p
1+
p
bt
b
p p;t )
1
+
1+
Et [bt+1 ]
(48)
p
and the traditional expression is obtained again (see
Option 2:
For the ‡exible price case the derivation is relative simple. Remember the presence
of the markup shock in the aggregtor function results in:3
d(
G00 (xt )
d(G0
1
(zt )) +
G0
G0
1 (z
1 (z
t)
t )G
00 (x )
t
)
d(
p;t )
+ d(pt (i))
d(mct ) = 0
d(G00 (xt ))
d(
p;t )
+ d(pt (i))
d(mct ) = 0
G00 (xt )
3
1
G0 1 (zt ) 000
d(z
)
+
G d(xt) d( p;t ) + d(pt (i))
t
G00 (xt )
G0 1 (z) 000
G00 (x) 1
d(p
)
+
G d(pt ) d( p;t ) + d(pt (i))
t
G00 (x)
G00 (x)
G0 1 (z) 000
d(pt ) + 00
G d(pt ) d( p;t ) + d(pt (i))
G (x)
1
(2 + 00
G000 )pbt (i) mc d
p;t
G (x)
The shock has been scaled with
G000 (x)G0 1 (z)
=
19
G000 (1)
:
G0 (1)
d(mct ) = 0
d(mct ) = 0
d(mct ) = 0
mcmc
ct = 0
p bt )
where the deviations in the markup are exppressed in percentage of the marginal cost
h
pbt (i) = h
1+
G00 (1)
G0 (1)
2+
G000
G00 (1)
i
i (mc
ct + d
p;t )
Linearization the aggregate price expression (33) (in stst: G0
G0 (1) and = 0)
0 = (1
+
= x = 1 and
=
p)
p
0 = (1
b
pet (i) =
G0 (1)
b
pe (i)
pet (i) + 00 b
G (1) t
G0 (1)
bt + p 00 bt
p bt 1
G (1)
)b
pe (i) + ( p bt 1 bt )
1 [z]
p
p
1
t
p
G0 (1)
bt
G00 (1)
p bt 1 )
(bt
p
1
Linearize the FOC for the Calvo wase with indexation (32):
Et
Et
1
X
1
G0 (xt+s )
s s s t+s
y
(i)(
)
t+s
p
G0 1 (zt+s ) G00 (xt+s )
t
s=0
1
X
1
G0 (xt+s )
s s s t+s
y
(i)(
)
t+s
p
G0 1 (zt+s ) G00 (xt+s )
t
s=0
Et
(
1
X
s s s
p
s=0
G000
d(xt+s ) +
d(e
pt (i)) + d(
G00 (x)
(1 + G0
1
(zt+s )
pet (i)Xt;s G0
+
p
Xt+s
Xt;s
p )
Xt+s
G00 (x)
1
G00 (xt+s ) pet (i)Xt;s
)
p
G0 (xt+s ) Xt+s
1 (z
d(mct+s )
t+s )G
00 (x
t+s ;
mct+s = 0
(49)
p;t+s )
(50)
d(
p;t+s )+
(51)
p;t+s )
(52)
d(zt+s ))
= 0
Et
+
1
X
s s s
p
s=0
G000
G
d(e
pt (i)) + d(
d(pt+s ) +
00
Xt;s
p )
Xt+s
G00 (x)
G00
= 0
20
d(mct+s )
d(pt+s ))
d(
mct+s = 0
Et
1
X
s s s
p
d(e
pt (i)) + d(
s=0
Et
1
X
s s s
p
Xt;s
p )
Xt+s
d(
p;t+s )
d(mct+s ) +
Xt;s
G000
G000
)d(e
p
(i))
+
(2
+
)d( p )
t
00
00
G
G
Xt+s
(2 +
s=0
G000
d(pt+s ) + d(pt+s )) = 0
G00
d(mct+s )
d(
p;t+s )
=0
where
(
Xt;s
d( p ) = d(
Xt+s
=
1
p
s
l=1 t+l 1
( sl=1 t+l )
p 1 1
p
p
)
)
p
1
p
(
p
)
)2
(
d(
+
G000
)
G00 1
p
0 = (2 +
+
d(
G000
)
G00 1
1
p
)
p
(2 +
pebt (i)
G000
)
G00
G000
)
G00 1
1 1
p
1
1
(
p
(1 +
(2 +
p
1
p
G00
G0 )
(mc
ct
G000
G00 )
( bt
+d
p;t ) +
0 = (( bt
+
p
(1 +
(2 +
p
1
p
p
(1
p)
p
G00 (1
G0 )
G000
G00 )
1
( p bt
p
( bt+1
bt+1 )
)(1
p
21
1 1
)
:::
d(mct )
p
)
)2
( bt+1
p
p)
d(
p;t)
!
d(
t+1 )
(1 +
bt
p
p b t 1 ))
p
p
)
p
1
p
(
p
p
(1
ped
t+1 (i)
(
p
)2
(
p
1
1
p
(
bt
p b t 1 ))
t+2 )
d(e
pt+1 (i))
p
p
d(
+ :::
p
p
using the expression for b
pet (i)
0 =
p
d( t )
(2 +
)
t+1 )
p
p
p
p
)2
(
G000
)
G00 1
1 1
p
1
p
(
t+1 )
(2 +
p
p
(1
d(e
pt (i))
G000
(2 + 00 )
G
)
p
(1
1
p
p
minus the same equation for t+1 multiplied with
0 = (2 +
1 1
p
d( t ) +
G00
)(mc
ct + d
p;t )
G0
!
bt+1
(
p bt )
(mc
ct + d
p;t )
p bt )
p
(
1
)2
p
)
bt+1
!
bt
0 =
+
( p bt
(1 +
(2 +
bt =
p)
(1 +
p bt 1
G00
G0
G000
G00
p bt 1
(1 +
with A =
(2 +
bt+1 )
) (1
)
bt+1 +
p
)(1
p
p
bt+1 + A
+
G00
G0 )
G000
G00 )
p bt
p
p ( p bt
(1
p)
bt+1 )
(mc
ct + d
p;t )
)(1
p
p)
p
(mc
ct + d
p;t )
(53)
If we consider the special case in which the aggregator function G reduces to the DixitStiglitz aggregator, we can see that option 1 and option 2 results in the same NKPC: in that
case G(x) = (x)1=(1+ ) and A = 1. The following relation between A and the curvature
holds:
A =
=
(1 +
(2 +
G00
G0 )
G000
G00 )
1
=
+1
=
"
1
1 +1
G00
d( xG
0)
d(")
G000
=
= 1 + " + " 00
d(x)
d(x)
G
Eq. 28 becomes:
b
kt = w
bt
Eq. 34 becomes:
(1
h= )bt = (1
using wh =
(1
h
[ wc L =
Ul;0
h= )bb1 t
= [(ct
(h= )ct
h= )bt = (1
h= )bb1 t
1
1+
1
w
rk ky
y
c
]
ct
cb
bt :
rbtk + L
+
1 )] Lt
l
ct
cb
c (h=
)b
ct
1
(54)
+ (1
h= )(
c
bt
1)L1+ l L
:
+
c (h=
)b
ct
1
+(
c
bt
1)(wh L=c )L
(55)
Eq. 35 becomes:
Eq. 36 becomes:
b = bb2 + R
bt + Et [bt+1 ]
t
t
Eq. 37 becomes:
b
k t = (1
b
kt = u
bt + b
kt
i b
) kt
k
1
22
+
i
k
Et [bt+1 ]:
(56)
1:
t
+
i b
it :
k
(57)
Eq. 38 becomes:
Qkt
t
+ Et
it
it
1 S
(
t+1
it
it
S0
1
it
it
1
1
it+1
it
[Qkt+1 t+1 S 0
t
2
it+1
it
)
]
=1
becomes:
b k + bt
Q
t
bit =
Eq. 39 becomes:
bk =
Q
t
2 00b
S it
1
(1 +
t + Et [
+
)
(bit
2 00b
S it 1
1
+(
)bit+1 +
rk
rk + (1
t+1 ] +
)
+(
)
2 00b
S it+1
1
2 S 00
(
)
1
b kt +
Q
k
Et [rt+1
]+
rk
2 00b
S it
2 S 00
=0
bt
(58)
(1
)
Et [Qkt+1 ]:
+ (1
)
where the steady state expression (rk + (1
)) = 1 is used.
Eq. 40 becomes:
rk rbtk = a00 u
bt :
(60)
or in case of accumulated costs (assuming s = 0) expressed relative to output:
a0 (u )k
akt =
= rk k ubt +
ubt +
1
1
akt
akt
1
1
Option 2 for wage setting:
Eq. 42 becomes:
(1 +
w )w
w
(1
w
= 0
h
w
e
w
bth +
)
ww
w bt
+
c
w;
h
t
(1
w
(1
w
e
w
w
e
)
)
w
bt+1
bet +
w
w
(1
w
w
e
)
bet+1
w
Using Eq. 43:
1
1
and
results in
w
w
bt
w
1
w
bth =
w
1
w
bt
1
1
b
ct
h=
+
w
w
1
h=
b
ct
h=
23
w w
bt
1
1
1
+
w
b
l Lt
(59)
bt
1
b~ t :
=w
(1 +
w )w
(1
+
w
e
h
(1
w
w
(1
w
= 0
w
e
w
e
(1
w)
w)
(1
) (1
w)
w)
w
bt
[w
bt+1
w
w (1
= 0
(1 +
(1
w
)w
bt
)(1
w
(1 +
[w
bt+1
w bt
1
+
w
bt
w
+
y +
y
Eq. (45) becomes:
bt
R
=
b
R Rt 1
+
yt
3 (b
w
(1
) (1
w)
bt
w )w
w w
w
bt+1
(1
bt+1
w)
t
+ (1
1
)
y +
y
bt+1
1]
bt ]
w
bt
1]
w w
w )b t+1
w
w)
bt+1
c
w;
t
bt + y +
L
y
yt
R )( 1 b t +
2 (b
f lex
f
lex
b
(y
ybt 1 )) + rt
t
24
bt
w w
w)
c
w;
bt
w w
bt ]
(61)
(1 +
+
bt
w
bt ]
i b
rk k
c
b
ct +
it +
u
bt :
y
y
y
b
kt + (1
ybt
w
w
w
bt+1
w bt 1
w
e
(1 +
w
[w
bth +
Eq. (27) becomes (remember ybt = b
y_ t ):
ybt =
bt
w
bt +
w)
+
w
w
+
1
w
bt +
(1
1
Eq. (44) becomes:
ybt = gbt +
w
w
bth +
+
w )b t
w
+
w
w
w )b t
w
bt
w
w)
w
w
w
ew
bt + w
ew
bt
t
[w
bt
)(1
w
w
=
1
) (1
[w
bt
c
w;
h
1
+(1
+
ww
) (1
w
w
w
bth +
(62)
bt
Z
ybtf lex ))
(63)
(64)
5
Estimated equations and rescaling residuals
5.1
For the sticky economy
Eq. (46):
mc
b t = (1
Eq (53)
(1 +
p)
Eq (61)
bt =
(1 +
=
(1
w
p bt 1
)w
bt
)(1
w
(1 +
+
Et [bt+1 ] + A
Eq (54)
(1
)(1
p
p)
p
w
bt
w )b t
+
Eq (56)
=
bt
Z
rbtk
)w
bt +
Et [w
bt+1 ]
1
w)
[
1
w bt 1
1
b
ct
h=
h=
b
ct 1 +
h=
Et [bt+1 ] + c
w; t
+
1
(h= )
Et [b
ct+1 ] +
b
ct
(1 + (h= ))
(1 + (h= ))
(1 h= ) b2 b
(bt + Rt Et [bt+1 ])
c (1 + (h= ))
h
i
( c 1)(wh L=c )
b t+1
b t ):
(Et L
L
c (1 + (h= ))
b
kt = w
bt
Eq (60)
b
l Lt
1
b
ct
(mc
c t) + d
p;t
1
bt :
rbtk + L
rk rbtk = a00 u
bt :
Eq (57)
b
k t = (1
i b
) kt
k
Eq (58)
bit =
1
(1 +
)
(bit
1+(
1
+
i
k
t
h
i
)Et bit+1 +
25
+
i b
it :
k
1
2 S 00
b kt +
Q
1
2 S 00
bt
w
bt ]
Eq (59)
b k = bb2
Q
t
t
Eq. (64):
bt
(R
bt
R
Et [bt+1 ]) +
=
b
R Rt 1
+
yt
3 (b
Eq. (63):
Eq. (62):
5.2
ybt =
y +
y
rk
rk
+ (1
+ (1
ybt
1
For the ‡exible economy
k
Et [rt+1
]+
(1
)
Et [Qkt+1 ]:
rk + (1
)
yt
R )( 1 b t +
2 (b
f lex
b
(y
ybtf lex
1 )) + rt
t
b
kt + (1
ybt = gbt +
)
)
y +
y
bt + y +
L
y
i b
rk k
c
b
ct +
it +
u
bt :
y
y
y
26
ybtf lex ))
bt
Z