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: mandreasen@econ.au.dk. 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.

1

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

2

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.

3

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.

4

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)

5

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

6

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.

7

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)

8

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

9

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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