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Ensuring the Validity of the Micro Foundation in DSGE Modelswith
Stochastic and Deterministic Trends
Martin Møller Andreasen�
School of Economics and ManagementUniversity of Aarhus,
Denmark
September 2, 2007
Abstract
The presence of stochastic and deterministic trends in DSGE
models may imply that thevalues of the agentsobjective functions
are innite. For the households, this might happenif the consumption
process has a su¢ ciently high growth rate and the subjective
discountfactor is very close to 1. The problem associated with
objective functions attaining innitevalues is that they do not have
an optimum. Hence, DSGE models with trends may haveinvalid micro
foundations because the optimal behavior of the agents is
indetermined. Ina rich DSGE model we derive su¢ cient conditions
which ensure that the householdsandthe rmsobjective functions do
not attain innite values. Based on these results we testthe
validity of the micro foundation in four calibrated or estimated
DSGE models from theliterature.Keywords: DSGE models, unit-roots,
moment generating functions, error distributions
�Email: [email protected]. Telephone number: +45 8942 2138.
I am grateful to David Skovmand andmy advisors Bent Jesper
Christensen, Henning Bunzel and Torben M. Andersen for useful
comments.
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1 Introduction
Dynamic Stochastic General Equilibrium (DSGE) models are often
specied with stochas-tic and deterministic trends. These features
are added to the models in order to explainthe non-stationary and
trending behavior in the time series for GDP, Consumption,
Invest-ments etc. Recent examples are the DSGE models in Ireland
(2004b), Ireland (2004a), Altig,Christiano, Eichenbaum & Linde
(2005), Negro, Schorfheide, Smets & Wouters (2005), An
&Schorfheide (2005), An (2005), Justiniano & Primiceri
(2005), Fernández-Villaverde & Rubio-Ramírez (2006),
Schmitt-Grohé & Uribe (2006) and Gorodnichenko & Ng (2007).
But, includingtrends in DSGE models is a non-standard extension of
the basic framework because these trendsmay imply that the
householdsor the rmsobjective functions attain innite values. For
thehouseholds optimization problem, this might happen if the
consumption process has a su¢ -ciently high growth rate and the
subjective discount factor (�) is very close 1. The
problemassociated with objective functions attaining innite values
is that they cannot be optimizedand hence we cannot determine the
optimal action of the economic agents. In other words,DSGE models
with trends may face the serious problem of not having a valid
micro foundationbecause the optimal behaviour is indetermined.
Although the problem seems obvious the issuehas not been addressed
in any of the papers listed above. Even in the two pioneer papers
byKing, Plosser & Rebelo (1988a) and King, Plosser & Rebelo
(1988b) is the problem only ad-dress in the case of a deterministic
trend in the model but not with a stochastic trend. Thepresent
paper addresses the problem and closes an important gab in the
literature by derivingsu¢ cient conditions which ensure a valid
micro foundation for DSGE models with deterministicand stochastic
trends.
We derive these su¢ cient conditions in a rich DSGE model which,
up to two minor excep-tions, nests the models considered in King et
al. (1988b), Ireland (2004a), Altig et al. (2005) andSchmitt-Grohé
& Uribe (2006). However, both exceptions are without loss of
generality for thepurpose of this paper. We return to this point in
the next section. In addition, our DSGE modelhas two new
interesting features. First, we include a non-stationary shock in
the householdsutility function as a way to specify long lasting
changes in the householdsintertemporal pref-erences. Second, our
model also introduces a new way of specifying public spendings in
DSGEmodels with trends. We nd this specication of public spendings
easier to motivate and hencemore realistic than the specication in
Schmitt-Grohé & Uribe (2006).
The rest of this paper is organized as follows. Section 2
presents our DSGE model. Section3 describes the su¢ cient
conditions which ensure that the householdsand the
rmsobjectivefunctions do not attain innite values. Based on these
conditions we examine the validity ofthe micro foundation for the
DSGE models in King et al. (1988b), Ireland (2004a), Altig et
al.(2005) and Schmitt-Grohé & Uribe (2006). Section 4
concludes.
2 The DSGE model
This section describes our DSGE model where we use the same
framework as in Schmitt-Grohé& Uribe (2006). The following two
reasons motivate our choice. First, Schmitt-Grohé & Uribe(2006)
show how to derive the exact nonlinear recursive representation of
the equilibrium condi-tions for DSGE models of this type. Thus,
when we in section 3 derive su¢ cient conditions which
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ensure the validity of the micro foundation then these
conditions will be independent of the ap-proximation method used to
solve the DSGE model. Second, Schmitt-Grohé & Uribe (2006) usea
exible specication of the labor markets which does not restriction
preferences to be separablein consumption and leisure. However,
such a restriction is required in the specication of thelabor
markets used in Altig et al. (2005).
The foundation of our DSGE model is the standard neoclassical
growth model with fourgroups of agents: i) households, ii) rms,
iii) a government and iv) a central bank. The economyis driven by
mutually independent structural shocks which we specify below. To
this basicstructure we add a number of extensions which may be
grouped as follows: First, nominalfrictions are introduced through:
i) sticky wages, ii) sticky prices, iii) a transactional demandfor
money by households and iv) a cash-in-advance constraint on a
fraction of the rmswage bill.Second, real frictions are added by
assuming: i) adjustment costs related to new investments,ii) a
variable capacity utilization rate of the capital stock, iii) habit
formation and money in thehouseholdsutility function, and iv)
imperfect competition in the goods and the labor markets.Finally,
stochastic and deterministic trends are added to the model.
Although our DSGE model is very rich it does not perfectly nest
the models by King et al.(1988b), Ireland (2004a), Altig et al.
(2005) and Schmitt-Grohé & Uribe (2006) for the followingtwo
reasons. First, we do not include the same stationary shocks as in
these models becausethe conditions we derive for the validity of
the micro foundation are una¤ected by these shocks.Second, our
specication of the labor markets di¤ers from the specication in
Altig et al. (2005).The reason being, that they assume that each
household supplies labor to only one labor marketwhereas we assume
the existence of a representative family which supplies labor to
all labormarkets. Again, this di¤erence is without loss of
generality for the conditions ensuring thevalidity of the micro
foundation.
2.1 The households
For sake of clarity the presentation of the
householdsoptimization problem is split into threesubsections which
describe the households i) preferences, ii) constraints and iii)
rst-order-conditions.1
2.1.1 The householdspreferences
We start by assuming that the behavior of the households may be
described by a representativefamily with a continuum of members.
Each member of this family has the same amount ofconsumption, hours
of work and money holdings. The familys preferences are specied by
autility function dened over real per capita consumption (ct), per
capita labor supply (ht) andreal per capita money holdings
�mht�
Ut = Et
1Xl=0
�l"h;t+lu�ct+l � bct�1+l; ht+l;mht+l
�(1)
Here Et denotes the conditional expectation given information
available at time t and � 2 [0; 1[is the subjective discount
factor. The function u (�; �; �) is a period utility index which we
assume
1All the derivations can be found in a technical appendix
available on request.
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has the form
u�ct � bct�1; ht;mht
�=
�(ct � bct�1)1��5 (ct � etz�t )
�5�(1��4)(1��3)
1��3� (2)
�6 (1� ht)+�7 exp(��8
h1+�9t
1 + �9
)!�4 (1��3)� 11� �3
+�10
�mht =z
�t
�1��11 � 11� �11
(z�t )(1��4)(1��3)
where b 2 [0; 1], �3 2 ]0; 1[ [ ]1;1[, �4 2 ]0; 1[, �5 = f0; 1g,
�6 � 0, �7 � 0, �8 � 0, �9 � 0,�10 � 0 and �11 2 ]0; 1[[ ]1;1[. We
also require that: i) �6 6= 0 or �7 6= 0, ii) ct=ct�1 > b and
iii)ct > etz
�t for all t and all realizations to ensure that the utility
index in (2) is always well-dened.
We do emphasize that the utility index in (2) should not be
consider as an unrestricted functionwhich could be taken to the
data. Our intension is only to set up a utility index which
reducesto the utility indexes use in the related papers with
appropriate restrictions.
The condition ct=ct�1 > b may impose an upper bound less than
1 on the parameter b whichspecics the level of the internal habit
e¤ect in the consumption good. The name of this habite¤ect is due
to the fact that the present habit level is determined by the
familys own consumptionin the previous period (bct�1). On the other
hand, the variable et denotes an external habit e¤ectand di¤ers
from the rst habit e¤ect by being exogenous to the representative
family. Notice,that the external habit is scaled by z�t which is an
overall measure of technological progress inthe economy. Adopting
this scaling of et ensures that the external habit e¤ect does not
diminishalong the balanced-growth path. We leave the form of the
external habit e¤ect unspecied andonly require that et is a
function of stationary variables.2 Finally, the labor supply in (2)
isnormalized such that ht 2 [0; 1[.
In the table below we show how our utility function nests the
various specications in thefour related papers.
Table 1: Restrictions on the Utility ParametersThis table shows
what restrictions that are needed in our utility function in order
to get the utilityfunctions in the related papers.
b �3 �4 �5 �6 �7 �8 �9 �10 �11King et al. (1988b) 0 ! 1 none 0 1
0 none none 0 noneIreland (2004a) 0 ! 1 12 0 0 1 none 0 0 noneAltig
et a l. (2005) none ! 1 12 0 0 1 none 1 0 noneSchm itt-G rohé &
Urib e (2006) none none none 0 1 0 none none 0 none
The novel feature of our utility function in (1) is the
non-stationary exogenous shock, denoted"h;t, which introduces long
lasting preference shocks into the economy. The process for "h;t
isspecied based on the gross growth rate �"h;t+1 � "h;t+1="h;t
where we assume
ln��"h;t+1
�= �"h ln
��"h;t
�+ �"ht+1 (3)
2For instance, we could dene et � b ct�1z�t sincect�1z�t
is stationary. Then etz�t = bct�1.
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where �"h 2 ]�1; 1[ and "h;0 � 1. The error terms f�"ht g
1t=1 are assumed to be independent
and identical distributed according to a general probability
distribution. We denote this by�"ht+1 s iid. Notice, that �"h;ss =
1 in the steady state. Although, the idea of specifying shocksto
the householdsintertemporal preferences is widely used in
literature the specication in (1)and (3) is new. This follows from
the fact that the process for ln "h;t is typically assumed tobe
stationary in the literature whereas we assume that the process for
ln "h;t is integrated oforder one and thus non-stationary. This
implies that "h;t has a stochastic trend of the formexp
�Pti=1 ai
where ai is a stationary variable. In section 2.6 and section 3
of this paper we
show that the specication in (1) and (3) is a feasible extension
of the framework developed bySchmitt-Grohé & Uribe (2006).
Following the standard assumption in the literature the
consumption good is constructedfrom a continuum of di¤erentiated
goods (ci;t; i 2 [0; 1]) and the aggregation function
ct =
�Z 10c��1�
i;t di
� ���1
(4)
Here � > 1 is the intratemporal elasticity of substitution
across the di¤erentiated goods. Thedemand for ci;t with nominal
price Pi;t is found by solving the following problem for each
levelof ct
Minci;t�0
Cost =
Z 10Pi;tci;tdi St:
�Z 10c��1�
i;t di
� ���1
� ct (5)
This implies that the demand for good i is given by
ci;t =
�Pi;tPt
���ct (6)
where Pt �hR 1
0 P1��i;t di
i1=(1��)is the nominal price index in the economy. Hence, the
ination
rate is given by �t � Pt=Pt�1.
2.1.2 The constraints on the households
The rst constraint on the households originates from basic
assumptions in the labor markets.In the framework developed by
Schmitt-Grohé & Uribe (2006) labor decisions in the
householdare assumed to be made by a central authority within the
household, which we think of as aunion. This union supplies labor
monopolistically to a continuum of labor markets, indexed byj 2 [0;
1], and faces a labor demand given by (Wj;t=Wt)�~� hdt in each
market. A derivation ofthis equation is postponed to the
presentation of the rmsoptimization problem. At this pointit is su¢
cient to know that: i) Wj;t is the nominal wage charged by the
union in the jth labormarket, ii) Wt is a nominal wage index and
iii) hdt is a measure of the total labor demand inthe economy. Both
Wt and hdt are considered exogenous by the union. Furthermore, we
assumethat the union determines the wages in each labor market and
supplies enough labor to meetlabor demand in all markets. This
implies that the total labor supply to market j at time t isgiven
by
hjt =
�wj;twt
��~�hdt (7)
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where wj;t � Wj;t=Pt and wt � Wt=Pt. Hence, the total labor
supply (ht) across all marketsmust satisfy the resource
constraint
ht =R 10 h
jtdj (8)
The second constraint is also related to the labor markets and
describes how the unioncan change wages. We follow Schmitt-Grohé
& Uribe (2006) and assume that in each pe-riod the union cannot
set the nominal wages optimally in a fraction ~� 2 [0; 1[ of
randomlychosen labor markets. In these labor markets the wages are
set according to the rule Wj;t =
Wj;t�1��hz�;t�1�
ht�1�~�. The parameter ~� 2 [0; 1]measures the degree of
indexation to �hz�;t�1�ht�1.
Here, �hz�;t�1 denotes the householdsgross growth rate target in
real wages and �ht�1 denotes
the householdstarget for the gross ination rate. Hence, for ~� =
0 there is no wage stickinesswhich is the case consider by King et
al. (1988b) and Ireland (2004a). If we on the other handlet �ht�1 =
�t�1 and �
hz�;t = �z�;ss then we get the same specication as in
Schmitt-Grohé &
Uribe (2006) and if we further let ~� = 1 then we get the
specication in Altig et al. (2005).
The third constraint is the law of motion for the physical
capital stock (kt) which is assumedto be owned by the households.
We adopt the standard assumption in the literature by letting
kt+1 = (1� �) kt + it�1� S
�itit�1
��(9)
The parameter � 2 [0; 1] is the depreciation rate for the
capital stock and it is gross investments.The function S (�) = �2
(
itit�1
� �i;ss)2 with � � 0 adds investment adjustment costs to
theeconomy based on changes in the growth rate of investments. The
value for the growth rate ininvestments in steady state
��i;ss
�is determined such that there are no adjustments costs
along
the balanced-growth path.
The fourth constraint is the householdsreal period by period
budget constraint
Etrt;t+1xht+1 + ct (1 + l (vt)) + �
�1t (it + a (ut) kt) +m
ht + nt
=xht +m
ht�1
�t+ rkt utkt +
R 10wj;thj;tdj + �t (10)
The function l (�) determines the transactional costs imposed on
the households based on thevelocity vt � ct=mht . Equation (10)
also introduces capital adjustment costs through the functiona (ut)
where ut is the capacity utilization rate of the capital stock. We
assume standard functionalforms for both functions, i.e.
l (vt) = �1vt + �2=vt � 2 (�1�2)0;5 (11)
a (ut) = 1 (ut � 1) +
22(ut � 1)2 (12)
where �1 and �2 are subject to the constraint that l (�) � 0 and
ut is normalized to 1 in thesteady state. Furthermore, we require
that 1 � 0 and 2 � 0. The left hand side of (10) is
thehouseholdstotal expenditures in period t which are used to: i)
purchase state-contingent claims�Etrt;t+1x
ht+1
�, ii) consumption including transaction costs (ct [1 + l
(vt)]), iii) investments and
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costs of providing capital services to the rms���1t (it + a (ut)
kt)
�, iv) the real money holdings�
mht�and paying transfers (nt) to the government. Notice, that
��1t is the real price in terms
of consumption goods for investing and selling capital services
to the rms. Changes in �t areoften referred to as embodied
technology changes because this type of technological progress
isembodied in the economys capital stock. The right hand side of
(10) is the householdstotalwealth in period t which consists of: i)
pay-o¤ from state-contingent assets purchased in periodt� 1
�xht =�t
�, ii) the real money holdings from the previous period
�mht�1=�t
�, iii) income from
selling capital services to the rms�rkt utkt
�, iv) labor income
�R 10wj;thj;tdj
�and v) dividends
received from the rms (�t). Since all these assumptions and
frictions are standard in theliterature (see Christiano, Eichenbaum
& Evans (2005), Altig et al. (2005) and Schmitt-Grohé&
Uribe (2004)) we keep the presentation short and only introduce
notation.
The nal constraints are a no-Ponzi-game condition and a
no-arbitrage restriction on thegross one-period nominal interest
rate, Rt;1 � 1:
2.1.3 The rst-order-conditions for the households
The householdsobjective is to maximize the utility function in
(1) with respect to the processesfor ct, xht+1, ht, kt+1, it, ut,
m
ht and wj;t given the constraints listed in the previous
subsection.
In doing so, the households take the processes for wt, rkt , hdt
, rt;t+1, �t, �
ht , �t, �t, �
hz�;t and nt
as given. This is also the case for the initial conditions for
c0, xh0 , k0, i�1, mh�1, and wj;0. We
let the lagrange multipliers for constraints (8),(9) and (10) be
�l�t+lwt+l~�t+l
, �lqt+l�t+l and �l�t+l,
respectively, which leads to the following
rst-order-conditions:3
ct : �t ="h;tuc
�ct � bct�1; ht;mht
�� b�Et"h;t+1uc
�ct+1 � bct; ht+1;mht+1
�1 + l (vt) + vtl0 (vt)
(13)
xht+1 : rt;t+1 = ��t+1�t
1
�t+1for all states (14)
ht : �"h;tuh�ct � bct�1; ht;mht
�=�twt~�t
(15)
kt+1 : �t = Et��t+1
"rkt+1ut+1 ���1t+la (ut+1) + qt+1 (1� �)
qt
#(16)
it : �t = �tqt�t
�1� S
�itit�1
�� itit�1
S0�
itit�1
��(17)
+�Et�tqt+1�t+1
�it+1it
�2S0�it+1it
�ut : r
kt = �
�1t a
0 (ut) (18)
mht : v2t l0 (vt) = 1� �Et
��t+1�t�t+1
��"h;tum
�ct � bct�1; ht;mht
��t
(19)
3The variable to the left of each equation denotes the variable
for which the equation is a rst-order-condition.
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wj;t : wj;t =
(~wt if market j is optimizing
wj;t�1��hz�;t�1�
ht�1�~�
else(20)
~wt : Et
1Xl=0
(�~�)l �t+lhdt+l
�wt+l~wt
�~�(Xtl)
�~��(~� � 1)~�
~wtXtl +MRSh;ct+l
�= 0 (21)
In (21) we use the notation Xtl �Qli=1
��hz�;t�1�i�
ht+i�1
�~��t+i
and MRSh;ct ��"h;tuh(ct�bct�1;ht;mht )
�tto
simplify the expression. Equation (13) shows that changes in the
householdstime preferencesthrough "h;t a¤ect the value of �t which
may be interpreted as the expected marginal utility ofincome. The
standard expression for the nominal stochastic discount factor
appears in (14) andpricing a one-period zero-coupon bond gives the
familiar Euler-equation
�t = �Rt;1Et
��t+1�t+1
�(22)
For an interpretation of the other rst-order-conditions we refer
to Schmitt-Grohé & Uribe(2006) and Christiano, Eichenbaum &
Evans (2001). Following the procedure described inSchmitt-Grohé
& Uribe (2006), the exact recursive representation of (21) is
given by f1t �f2t = 0where
f1t = �thdt
�wt~wt
�~� �~� � 1~�
�~wt + Et�~�
�~wt+1~wt
�~��1 �~��1t+1 f1t+1��hz�;t�
ht
�~�(~��1) (23)f2t = �"h;tuh
�ct � bct�1; ht;mht
�hdt
�wt~wt
�~�+ Et�~�
�~wt+1~wt
�~� �~�t+1f2t+1��hz�;t�
ht
�~�~� (24)
2.2 The rms
The production in the economy is assumed to be undertaken by a
continuum of rms, indexed byi 2 [0; 1]. Here, we adopt the standard
assumptions saying that each rm supplies a di¤erentiablegood
�ysi;t
�to the goods market which is characterized by monopolistic
competition with no
exit or entry. Furthermore, all rms have access to the same
technology given as follows
ysi;t =
�F (ki;t; zthi;t)� z�t if F (ki;t; zthi;t)� z�t > 0
0 else(25)
where F (�) � k�i;t (zthi;t)1�� with � 2 ]0; 1[ and � 0. Here,
ki;t and hi;t denote physical capital
and labor services used by the ith rm. As in the case of the
di¤erentiated consumption goods,
rm is demand in the jth labor market�hji;t
�is given by the solution to the standard problem
Minhji;t�0
Cost =
Z 10Wj;th
ji;tdj St:
�Z 10
�hji;t
� ~��1~�dj
� ~�~��1
� hi;t (26)
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Here Wj;t is the nominal wage paid to labor services in labor
market j. The solution to theproblem is
hji;t =
�Wj;tWt
��~�hi;t (27)
where Wt �hR 1
0W1�~�t dj
i1=(1�~�)is the nominal wage index. Aggregating equation (27)
over all
producers and deningR 10 h
ji;tdi � h
jt and
R 10 hi;tdi � h
dt gives (7), the labor demand faced by
the union.
The variable zt in (25) denotes an aggregate neutral technology
shock. Now, let us dene thevariable z�t by the relation z
�t � �
�=(1��)t zt which implies that we may interpret z
�t as an overall
measure of technological progress in the economy. Hence, scaling
by z�t in (25) ensures thatthe rmsxed costs do not diminish along
the balanced-growth path. Letting �z;t � zt=zt�1and ��;t � �t=�t�1
we assume that
ln
��z;t+1�z;ss
�= ��z ln
��z;t�z;ss
�+ �
�zt+1 (28)
ln
���;t+1��;ss
�= ��� ln
���;t��;ss
�+ �
��t+1 (29)
and let z0 � 1 and �0 � 1. Here, ��zt+1 s iid: and ���t+1 s iid.
We also require that ��z 2 ]�1; 1[
and ��� 2 ]�1; 1[. Equations (28) and (29) imply that the
processes for ln zt and ln�t havea stochastic trend and the
deterministic trends are ln�z and ln��, respectively. These
resultsfollows directly from the MA-representations for the
processes in (28) and (29). In the case of
neutral technology shocks we have that ln��z;t�z;ss
�= at where at is a stationary variable. Hence,
ln zt = ln zt�1 + ln�z;ss + at.
All rms are assumed to maximize the present value of their
nominal dividend payments,denoted di;t. That is, each rm
maximizes
di;t � Et1Xl=0
rt;t+lPt+l�i;t+l (30)
where the expression for the real dividend payments from the ith
rm��i;t�are given below in
(32). When doing so, the rms face ve constraints. The rst is
related to the good produced bythe ith rm. The total amount of good
i is allocated to: i) consumption including transactioncosts, ii)
public spendings (�gi;t), iii) investments and iv) costs of
providing capital servicesto the rms. We make the standard
assumption that the aggregation function for the threelatter
components coincides with the aggregation function for consumption
in (4). Hence, therestriction on the aggregate demand can be
written as
ydt = ct (1 + l (vt)) + �gt +��1t (it + a (ut) kt) (31)
In addition, we assume that the rms satisfy demand, i.e. ysi;t �
ydi;t 8i 2 [0; 1].The second restriction is a cash-in-advance
constraint on a fraction � of the rmspayments
to workers. Thus, the money demanded by the ith rm is mfi;t =
�wthi;t. This assumption is
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also standard in the literature and serves the purpose of
motivating demand for money at therm level.
The third constraint is the budget restriction which gives rise
to the expression for realdividends from rm i in period t
�i;t = (Pi;t=Pt) ydi;t � rkt ki;t � wthi;t �m
fi;t
�1�R�1t;1
�(32)
�Etrt;t+1xfi;t+1 +mfi;t � �
�1t
�xfi;t +m
fi;t�1
�The rst term in (32) denotes the real revenue from sales of the
ith good. The rms ex-penditures are allocated to: i) purchase of
capital services
�rkt ki;t
�, ii) payments to the work-
ers (wthi;t) and iii) opportunity costs of holding money due to
the cash-in-advance constraint�mfi;t
�1�R�1t;1
��. The nal terms in (32) constitute the change in the rms real
nancial
wealth.The fourth constraint introduces staggered price
adjustments. We make the standard as-
sumption that in each period a fraction � 2 [0; 1[ of randomly
picked rms are not allowedto set the optimal nominal price of the
good they produce. Instead, these rms update their
prices according to the rule Pi;t = Pi;t�1��ft�1
��where � 2 [0; 1] and �ft�1 is the rmsination
rate target. Hence, for � = 0 there is no price stickiness which
is the case consider by Kinget al. (1988b) and Ireland (2004a). If
we let �ft�1 = �t�1 then we get the specication used
inSchmitt-Grohé & Uribe (2006) and if we further let � = 1 then
we get the setup in Altig et al.(2005).
The fth constraint is a no-Ponze-game condition.
Given these constraints rm i maximizes the present discounted
value of dividend paymentswith respect to xfi;t, m
fi;t, hi;t, ki;t and Pi;t given the processes for Rt;1, Pt, wt,
r
kt , zt, z
�t , y
dt , �
ht ,
�t and the nominal stochastic discount factor between period t
and period t+ l, denoted rt;t+l.As in Schmitt-Grohé & Uribe
(2006) we assume, without loss of generality, that xfi;t +m
fi;t = 0
in all periods and states. Dening rt;t+lPt+lmci;t+l as the
lagrange multiplier for the constraintysi;t � ydi;t, the
rst-order-conditions are
hi;t : mci;tztF2 (ki;t; zthi;t) = wt
�1 + �
�1� 1
Rt;1
��(33)
ki;t : mci;tF1 (ki;t; zthi;t) = rkt (34)
Pi;t : Pi;t =
�~Pt if rm i is optimizing
Pi;t�1��ht�1
��else
(35)
~Pt : Et
1Xl=0
rt;t+lPt+l�l
~PtPt
!��Y ��tl y
dt+l
"� � 1�
~PtPtYtl �mci;t+l
#= 0 (36)
where Ytl �Qli=1
��ft+i�1
���t+i
. Following the procedure described in Schmitt-Grohé & Uribe
(2006)
we derive the exact recursive representation of (36) as �x1t +
(1� �)x2t = 0 where
x1t = ydtmct~p
���1t + Et��
�t+1�t
�~pt~pt+1
����10@��ft
���t+1
1A�� x1t+1 (37)10
-
x2t = ydt ~p��t + Et��
�t+1�t
�~pt~pt+1
���0@��ft
���t+1
1A1�� x2t+1 (38)
2.3 The government
This section describes the role of the government in the
economy. The scal policy is speciedas the following process for the
aggregated public spendings
�gt = w�gt (z
�t )1��g gt (39)
where �g 2 [0; 1] and gt is some unspecied exogenous stationary
process. Equation (39) clearlynests the specication of public
spendings in King et al. (1988b), Ireland (2004a), Altig et
al.(2005) and Schmitt-Grohé & Uribe (2006). We prefer the
specication where �g = 1, implyingthat �gt = wtgt, because public
spendings are mostly used to provide labor intensive servicessuch
as law and order, education, administration etc. Thus, public
spendings are approximatelyproportional to wages. Changes in the
public spendings unrelated to the wage level are pickedup by the
exogenous process for gt. This could be expenditures related to
large scal reforms,wars etc. Notice, that equation (39) implies
that the ratio between public spendings and totaldemand is constant
along the balanced growth path.
Part of the public spendings is nanced by seigniorage. If we let
mt � mht +R 10m
fi;tdi be
the total amount of outstanding real money then seigniorage is
given by mt�mt�1=�t. To keepthings simple, we assume that there
exists lump-sum transfers (nt) which are set to ensure thatthe
governments intertemporal budget constraint always holds. Thus,
given the process for gtthis policy regime is Ricardian.
2.4 The central bank
The monetary policy is conducted by the central bank and
generally its behavior may eitherbe specied by a rule for the
interest rate or by a rule for the money stock. It turns out
thatthe specic nature of these policy rules is unimportant for the
validity of the micro foundation,provided that these rules are
based on stationary variables. Therefore, we choose not to
specifymonetary policy explicitly but simply require that the
policy rule should be based on stationaryvariables.
2.5 Aggregation
An explicit aggregation is necessary in the goods and labor
markets. This is due to the di¤er-entiated consumption goods and
the large number of labor markets. The aggregation in ourDSGE model
is identical to the aggregation described by Schmitt-Grohé &
Uribe (2006). Belowwe briey summarize the aggregated relations for
sake of completeness.
We start by considering the aggregate goods market where the
resource constraint reads
F�utkt; zth
dt
�� z�t =
�ct (1 + l (vt)) + �gt +�
�1t (it + a (ut) kt)
�st (40)
11
-
st = (1� �) ~p��t + �
0B@ �t��ft�1
�~�1CA�
st�1 (41)
for utkt �R 10 ki;tdi. and h
dt �
R 10 hi;tdi. These equations are derived by summing over all
goods
while taking into account that i) rms have access to the same
technology which is homogenousof degree one and ii) the ratio
ki;t=hi;t is constant across all rms. The state variable st is
equalto or greater than one and in case of fully exible prices (� =
0) we have st = 1. So, st measuresthe resource costs due to the
presence of sticky prices.
The aggregated relations for the rmsrst-order-conditions for
labor and capital and totaldividend payments, �t �
R 10 �i;tdi, are
mctztF2
�utkt; zth
dt
�= wt
�1 + �
�1� 1
Rt;1
��(42)
mctF1
�utkt; zth
dt
�= rkt (43)
�t = ydt � rkt utkt � wthdt
h1 + �
�1�R�1t;1
�i(44)
The resource constraint in the aggregate labor market resembles
the constraint in the goodsmarket and is given by
ht = hdt ~st (45)
~st = (1� ~�)�~wtwt
��~�+ ~�
�wt�1wt
��~� ��hz�;t�1�ht�1�~��t
!�~�~st�1 (46)
Recall that ht is the total labor supply and hdt is the total
labor demand. The state variable ~st isequal to or greater than one
and in case of fully exible wages (~� = 0) we have ~st = 1.
Equation(45) therefore implies an unemployment level of hdt (1�
~st) � 0 which is a cost of having stickywages.
Aggregating the real money holdings gives
mt = mht + �wth
dt (47)
Finally, we derive the relationship between the real optimal
price ~pt �~PtPtand the ination
rate (�t) and the relationship between the real wage index (wt)
and the optimal real wage ( ~wt)
1 = (1� �) ~p1��t + �
0@��ft�1
���t
1A1�� (48)
w1�~�t = (1� ~�) ~w1�~�t + ~�w
1�~�t�1
��hz�;t�1�
ht�1�~�
�t
!1�~�(49)
12
-
2.6 Solving the DSGE model
The three non-stationary processes for the shocks in our DSGE
model imply that some of thevariables in the model are
non-stationary. An easy way to get around this problem when
solvingthe model is to transform the economy such that we only have
equilibrium conditions withstationary variables. The solution to
the transformed economy is then easy to approximateby standard
solution methods for DSGE models. We get the desired solution of
our DSGEmodel by transforming the solution back into the original
setting. Thus, we only need to showhow to construct the transformed
economy. We proceed as follows: First, observe that ct, wt,~wt,
ydi;t, y
dt , �i;t, �t, x
2t , �gt, nt, m
ht and mt all are cointegrated with 1=z
�t in such a way that
ct=z�t ; wt=z
�t and so on are stationary. Likewise, r
kt and qt cointegrate with �t and it; kt+1 and
ki;t+1 cointegrate with 1=(�tz�t ). Finally, �t and f2t
cointegrate with 1=(z
� (1��3)(1��4)�1t "h;t) and
1=(z� (1��3)(1��4)t "h;t), respectively. All the remaining
variables in the model are stationary - in
particular, the labor supply, the interest rate and the ination
rate. If "h;t = 1 for all t we obtainthe same cointegrating results
as in Schmitt-Grohé & Uribe (2006). Next, we construct
thetransformed variables by multiplying the variables with their
corresponding cointegrating factor.These transformed variables are
denoted by the corresponding capital letters. I.e. Ct � ct=z�t ,Rkt
� rkt�t and so on. The equilibrium conditions are then rewritten in
terms of the transformedvariables in order to get the transformed
economy.4
It is interesting to note that a long lasting preference shock
("h;t) does not a¤ect variables likereal consumption and real
production in the long run. Only the householdsexpected
marginalvalue of income (�t) and the control variable f2t related
to the labor markets are a¤ected by along lasting preference shock
in the long run.
3 Su¢ cient conditions for a valid micro foundation
This section derives su¢ cient conditions which ensure the
validity of the micro foundation inour DSGE model. We introduce the
following notation to ease the presentation below
F� � (1� �4) (1� �3)�
1� � (50)
Fz � (1� �4) (1� �3) (51)
f (ht) � �6 (1� ht)+�7 exp
(��8
h1+�9t
1 + �9
)!�4(52)
We rst consider the conditions which ensure that the
householdsutility functions have anite value in the case of power
preferences for the habit adjusted consumption good.
4A list of the equilibrium conditions for the untransformed and
the transformed economy is given in the paperstechnical appendix.
This appendix is available on request.
13
-
Proposition 1 Let �3 2 ]0; 1[[ ]1;1[. The following conditions
are su¢ cient for a nite valuefor the representative familys
utility function, that is for Ut 2 R
1. a)
�������
Ct�bCt�1�z�;t�1��5
(Ct�et)�5�(1��4)(1��3)
f (ht)1��3
�����
-
Table 2: Error distributions
Logistic Laplace Normal Subbotin
Density, f (x) e� x�
��1+e�
x�
�2 12�e�jxj=� e�0:5( x� )2�p2� v
expf�0:5jz=��jvg��2(1+1=v)�(1=v)Domain x 2 R x 2 R x 2 R x 2
RSymmetric yes yes yes yes
Restrictions � > 0 � > 0 � > 0 � > 0; v > 1; �
��2�2=v�(1=v)�(3=v)
� 12
Mean 0 0 0 0Variance �2�2=3 2�2 �2 �2
M.g.f, MX (t) ��tsin(��t)1
1��2t2 e0:5�2t2
P1k=0
�21v ��
�k[1+(�1)k]�((k+1)=v)2�(1=v)�(k+1)
The third condition in proposition 1 states that E�exp
���"ht
1���"h
��� < 1 and if there is
no long lasting preference shocks in the model (i.e. ��"ht = 0
for t and all realizations) then
we get the standard condition � < 1. For the distributions
mentioned in table 2 it holds that
E
�exp
���"ht
1���"h
��> 1. Thus, for these distributions the presence of long
lasting preferences
shocks imposes a stronger restriction on the value of �. The
intuition behind this result is thatthe householdssubjective
discounting factor (�) now must o¤ set two e¤ects in order to geta
nite value for the utility function: i) the innite utility stream
and ii) the stochastic trendgenerated by "h;t.
We interprete the fourth condition in proposition 1 by rst
considering the case with onlydeterministic trends in the processes
for technology. In this case the condition reduces to��F��;ss�
Fzz;ss < 1. In general, ��;ss > 1 and �z;ss > 1 but the
sign of F� and Fz depends
on the value of �3. If �3 > 1, which is probably the most
realistic case, then F�; Fz < 0 and thedeterministic trends
operate as additional discounting factors in the householdsutility
function.If �3 < 1 then F�; Fz > 0 and the opposite situation
is the case. Notice, that if our DSGE mod-els does not have
deterministic growth in embodied technology (��;ss = 1) then the
conditionis ��Fzz;ss < 1 which is exactly the same condition as
in King et al. (1988a) where �
� � ��Fzz;ssis called the e¤ective time rate of preference.6 On
the other hand, if we only have stochastictrends and no
deterministic trends in our DSGE model
�i.e. ��;ss = �z;ss = 1
�then condition
4 is
Et
"exp
(��"ht+1
1� ��"h
)#Et
"exp
(���t+1F�
1� ���
)#Et
"exp
(��zt+1Fz
1� ��z
)#� < 1
which is an even more restrictive condition on � than condition
3. Hence, adding deterministictrends to a model with stochastic
trends may be recommend based on proposition 1 because
thedeterministic trends are most likely to operate as additional
discounting factors.
6However, in King et al. (1988b) where the neoclassical growth
model is extended with stochastic trends, theauthors do not state
the conditions for niteness of the households lifetime utility.
15
-
The next proposition considers the limiting case where �3 ! 1
implying log preference forthe habit adjusted consumption good and
separability between this good and the labor supply.
Proposition 2 Let �3 ! 1. The following conditions are su¢ cient
for a nite value for therepresentative familys utility function,
that is for Ut 2 R
1. a)���ln�Ct�bCt�1�z�;t����
-
Proof. See the technical appendix.
In relation to proposition 3, recall that �t is the households
expected marginal value ofincome and �i;t is real dividend payments
from rm i. Both variables are expressed in deviationfrom the
stochastic and deterministic growth path in the economy. The
important thing tonotice is that conditions 2 and 3 in proposition
3 are both satised if either the conditions inproposition 1 or the
conditions in proposition 2 hold. In the latter case, this follows
from thefact that �3 ! 1 implies that F� ! 0 and Fz ! 0.
The proofs of proposition 1 and proposition 2 show that we only
use the following fourproperties from our DSGE model in the
derivations: i) the specication of the householdsutility functions,
ii) the fact that ct = �z�;tCt and m
ht = �z�;tM
ht , iii) stationarity of the labor
supply (ht) and iv) the law of motion for �z�;t and "h;t. Hence,
Proposition 1 and 2 may beapplied in relation to all DSGE models
with these four properties even though these DSGEmodels di¤er from
our DSGE model along other dimensions. We highlight this important
resultin the following lemma:
Lemma 1 Proposition 1 and 2 are valid for all DSGE models which
have the following charac-teristics:
1. The utility function in (1)
2. Consumption and money holdings are given by ct = z�tCt and
mht = z
�tM
ht
3. The labor supply (ht) is stationary
4. z�t � ��=(1��)t zt, ln�z;t s AR(1), ln��;t s AR(1) and
ln�"h;t s AR(1)
A similar lemma also holds for proposition 3
Lemma 2 Proposition 3 are valid for all DSGE models which have
the following characteristics:
1. The prot function in (32)
2. A complete market of state contigent claims
3. rkt = Rkt =�t, ki;t+1 = �tz
�tKi;t+1, wt = z
�tWt, y
dt = z
�t Y
dt and m
ft = z
�tM
ft
4. The ination rate (�t) is a stationary processes
5. Conditions 1 to 4 in Lemma 1 are satised
17
-
3.1 Evaluating the boundness conditions
This section examines the boundness conditions in the three
propositions from above in greaterdetail. For this purpose we
impose the assumption:
Assumption 1 All variables in the economy, execpt exogenous
state variables, must bewithin a nite distance from the economys
stochastic and deterministic growth path.
Assumption 1 means that all variables in the transformed
economy, execpt exogenous statevariables, are bounded from above
and from below. We exclude exogenous state variables inAssumption 1
because assuming that these state variables should be bounded would
contradictwith the specied law of motions for the variables.
Notice, that Assumption 1 does not rule outan innite consumption
level, for instance, since the actual consumption level (ct) is
given by therelation ct = Ctz�t and the aggregated measure of
technology z
�t may tend to innity for t!1.
It is important to realize that our Assumption 1 of not being
too fare away from the stochasticand deterministic growth path is
the same assumption which implicitly is used when solvingDSGE
models by local procedures like the log-linearization approach.
Hence, Assumption 1 isactually a standard assumption in the
literature.
We proceed with the following three Lemmas.
Lemma 3 Assumption 1 implies that for all t and all
realizations:
1. j�tj
-
Proof. See the technical appendix.
The implication of Lemma 3 to Lemma 5 is that all of the
boundness conditions fromproposition 1 to proposition 3 are satised
except one. The exception is the case with log
preference for the consumption good and b > 0. Here, we
cannot show that���ln�Ct�bCt�1�z�;t���� 0. But, the conditionsin
proposition 1 are clearly more restrictive than the requirements in
proposition 2 and thisfact should also be taken into account. A nal
alternative is to have log preferences for theconsumption good and
an external habit e¤ect (�5 = 1). This combination has the
advantagethat proposition 2 still applies and Assumption 1 is su¢
cient to garantee that all the boundnessconditions in the
proposition are satised in this particular case.
In the beginning of the paper we stated that the stationary
shocks in the papers by Kinget al. (1988b), Ireland (2004a), Altig
et al. (2005) and Schmitt-Grohé & Uribe (2006) do nota¤ect the
conditions for the validity of the micro foundation. The reason
being that none ofthese papers have stationary shocks which enter
directly into the utility function. If this is thecase then
proposition 1 to 3 would still hold, but we might no longer be able
to apply the samemethod to show that many of the boundness
conditions are in fact satised.
3.2 Testing the validity of the micro foundation in four DSGE
models
This section examines the validity of the micro foundations for
the DSGE models in the followingfour papers: i) King et al.
(1988b), ii) Ireland (2004a), iii) Altig et al. (2005) and iv)
Schmitt-Grohé & Uribe (2006). To handle the boundness
conditions in proposition 1 to 3 we imposeAssumption 1 in the
following evaluation of the models.
First, consider the models by King et al. (1988b) and Ireland
(2004a). These models have logpreferences for the consumption good
and no internal habit e¤ect (b = 0). Thus, the problemwith log
preferences and internal habit formation is not present in these
models. Moreover,the two models do not have long lasting preference
shocks and no embodied technology shocks.Hence, only Et
�����zt+1���
-
Second, the model by Altig et al. (2005) also use log
preferences for the consumption goodand an internal habit e¤ect. As
mentioned above this combination may be a problem for theboundness
conditions in proposition 2 if the internal habit e¤ect is very
strong. The highestestimate of b in Altig et al. (2005) is 0:73.
The paper uses quarterly data and it thereforeseems reasonable to
assume that the boundness conditions in proposition 2 holds.
Moreover,the model by Altig et al. (2005) has long lasting neutral
and embodied technology shocks so alsoEt������t+1��� < 1 and Et
�����zt+1��� < 1 are required according to proposition 2. The
probability
distributions for the shocks are not specied but the conditions
clearly hold if we in additionassume symmetric error
distributions.
Finally, the DSGE model by Schmitt-Grohé & Uribe (2006) is
calibrated to the case of logpreferences even though the model is
set up also to encompass power preferences. Furthermore,the model
by Schmitt-Grohé & Uribe (2006) has an internal habit e¤ect of
a moderate degree(b = 0:69) and long lasting neutral and embodied
technology shocks. Hence, the requirementsare Et
������t+1��� < 1 and Et �����zt+1��� < 1 in addition to
the boundness conditions. Schmitt-Grohé & Uribe (2006) do not
specify probability distributions for the two shocks but again
theconditions hold if the error distributions are assumed to be
symmetric.
Summing up, we nd that all the four DSGE models have a valid
micro foundation providedwe add an assumption of symmetric error
distributions in the cases where the functional formfor the
distributions are not specied.
4 Conclusion
This paper closes an important gab in the literature by deriving
su¢ cient conditions whichensure the validity of the micro
foundation for DSGE models with stochastic and deterministictrends.
In addition, we show how to introduce long lasting preference
shocks in the householdsutility function. This feature is new
compared to the existing DSGE models since these modelsonly specify
long lasting shocks to the economys production technology. Finally,
this papersuggests a new way of specifying public spendings by
making these spendings proportional tothe real wage.
An important implication of this paper is that every DSGE model
with stochastic and/ordeterministic trends should be specied such
that they fulll the conditions in proposition 1to proposition 3. In
the case of log preferences for the consumption good the conditions
arerelatively weak, in particular if the long lasting preference
shocks are left out. In this case,assuming symmetric error
distributions is su¢ cient as shown in the previous section. On
theother hand, in the case of power preferences for the consumption
good the conditions are morerestrictive and greater care most be
taken when setting up the DSGE model. For instance, aDSGE model
with power preferences for the consumption good should never be
specied witha stochastic trend where the innovation in this trend
follows a Student-t distribution. But, forstationary shocks Student
-t distributed shocks may still be used without any problems.
On an empirical level, future research could be devoted to
estimate or calibrate DSGEmodels with power preferences since DSGE
models with trends have mostly been estimated orcalibrated based on
log preferences for the consumption good. Furthermore, it would
also be ofgreat interest to examine whether the restrictions on �
in the three propositions are binding, inparticular when long
lasting shocks to the householdsutility function are added to the
model.
20
-
On a theoretical level, future research could derive su¢ cient
conditions ensuring the validityof the micro foundation in DSGE
models where the growth rates for the long lasting shocksevolve
according to ARMA(p,q) processes and/or have stochastic
volatility.
21
-
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