Limited Asset Markets Participation, Monetary Policy and ...

Limited Asset Markets Participation, Monetary Policy and(Inverted) Keynesian Logic.

Florin O. Bilbiie

University of Oxford, Nu¢ eld College andCEP, London School of Economics.

This Version: January 2006

Abstract. This paper incorporates limited asset markets participation in dy-namic general equilibrium and develops a simple analytical framework for mon-etary policy analysis. Aggregate dynamics and stability properties of an other-wise standard business cycle model depend nonlinearly on the degree of assetmarket participation. While �moderate�participation rates strengthen the roleof monetary policy, low enough participation causes an inversion of resultsdictated by (�Keynesian�) conventional wisdom. The slope of the �IS� curvechanges sign, the �Taylor principle� is inverted, optimal welfare-maximizingmonetary policy requires a passive policy rule and the e¤ects and propagationof shocks are changed. The conditions for these results to hold are relativelymild compared to some existing empirical evidence. Our results may justifyFed�s behavior during the �Great In�ation�period.

Keywords: limited asset markets participation; dynamic general equi-librium; aggregate demand; Taylor Principle; optimal monetary policy; real(in)determinacy.

JEL codes: E32; G11; E44; E31; E52; E58.

Address: Nu¢ eld College, University of Oxford, New Road, OX1 1NF, Oxford, UK; email:�orin.bilbiie@nu¢ eld.ox.ac.uk.I am particularly indebted to Roberto Perotti and Giancarlo Corsetti for numerous discussionsand advice. I am grateful to Jordi Galí, Andrew Scott, Mike Woodford and Roger Farmer forhelpful, detailed comments. I also bene�ted from comments at various stages from: Kosuke Aoki,Mike Artis, Gianluca Benigno, Paul Bergin, Giuseppe Bertola, Fabrice Collard, Mike Haliassos,Stephanie Schmitt-Grohe, Fabio Ghironi, Nobu Kiyotaki, David Lopez-Salido, Albert Marcet, AlexMichaelides, Tommaso Monacelli, John Muellbauer, Gernot Mueller, Kris Nimark, Fabrizio Perri,Jon Skinner and Jaume Ventura. Participants at the CEPR�s 2004 European Summer Symposiumin Macroeconomics in Tarragona, and seminars at London School of Economics, CREI-UniversitatPompeu Fabra, Nu¢ eld College, Oxford, European University Institute, Birkbeck, IGIER-Bocconiand DOFIN provided valuable comments. Special thanks to Roland Straub, my coauthor in relatedwork. All errors are mine.

1

2 FLORIN O. BILBIIE

1. Introduction

At the heart of modern macroeconomic literature dealing with monetary pol-icy issues lies some form of �aggregate Euler equation�, or �IS� curve: an inverserelationship between aggregate consumption today and the expected real interestrate. This relationship is derived from the households�individual Euler equationsassuming that all households substitute consumption intertemporally - for exampleusing assets. Normative prescriptions are then derived by using this equation asa building block, together with an in�ation dynamics equation (�Phillips curve�)derived under the assumption of imperfect price adjustment1.

This paper introduces limited asset markets participation (LAMP) into an oth-erwise standard dynamic general equilibrium model and studies the implications ofthis for monetary policy. We model LAMP in a way that has become standard inthe macroeconomic literature reviewed below. Namely, we assume that a fraction ofagents have zero asset holdings, and hence do not smooth consumption but merelyconsume their current disposable income, while the rest of the agents hold all assetsand smooth consumption2. This modelling choice is motivated both by direct dataon asset holdings and by an extensive empirical literature studying consumption be-havior. The latter seems to suggest that, regardless of whether aggregate time seriesor micro data are used, consumption tracks current income for a large fraction ofthe US population. To give just some prominent examples, Campbell and Mankiw(1989) used aggregate time series data to �nd that a fraction of 0.4 to 0.5 of theUS population merely consumed their current income. More recent studies usingmicro data also �nd that a signi�cant fraction of the US population fails to behaveas prescribed by the permanent income hypothesis (e.g. Hurst (2004), Johnson,Parker and Souleles (2004))3. Finally, direct data on asset holdings shows that alow fraction of US population holds assets in various forms4. Models incorporatingthis insight have been recently used in the macroeconomic literature. First, someversion of this assumption -whereby a fraction of agents does not hold physicalcapital- has been proposed by Mankiw (2000) and extended by Galí, Lopez-Salidoand Valles (2003) for �scal policy issues5. Second, it is the norm in the monetarypolicy literature trying to capture the �liquidity e¤ect�, where it is assumed thatasset markets are �segmented�(e.g. Alvarez, Lucas and Weber (2001)). This mod-elling choice has only recently been incorporated into the sticky-price monetarypolicy research in a paper that we review in detail below.

1See Woodford (2003) for a state-of-the art review of this literature. Earlier overviews com-prise, amongst others, Clarida, Gali and Gertler (1999) and Goodfriend and King (1997).

2In an appendix, we outline a simple model in which high enough proportional transactioncosts can rationalize limited participation. We also review some evidence concerning the magni-tude of these costs necessary to generate observed non-participation levels.

3Johnson et al. show that a large part of the US population consumed the unexpectedincrease in transitory income generated by the 2001 tax rebate and �nd that the response washigher for households with low wealth. Relatedly, Wol¤ and Caner (2004) use 1999 PSID data to�nd that 41.7 percent of the US population can be classi�ed as asset-poor when home equity isexcluded from net worth, whereas 25.9 percent are asset-poor based on net worth data.

4Vissing-Jorgensen (2002) reports based on the PSID data that of US population 21.75 per-cent hold stock and 31.40 percent hold bonds. Data from the 1989 Survey of Consumer Finances(see e.g. Mulligan and Sala-i-Martin (2000)) shows that 59 percent of US population had nointerest-bearing �nancial assets, while 25 percent had no checking account either.

5The latter paper argues that this modelling assumption can help explaining the e¤ects ofgovernment spending shocks. See also Bilbiie and Straub (2003b).

LIMITED ASSET MARKETS PARTICIPATION AND MONETARY POLICY. 3

We show how the general equilibrium model with LAMP can be reduced toa familiar 2-equations system, consisting of a Phillips- and an IS- curve, whichnests the standard New Keynesian model; since the resulting system is very sim-ple, it might be of independent interest to some researchers. Notably, we capturethe in�uence of LAMP on aggregate dynamics through an unique parameter, theelasticity of aggregate demand to real interest rates, which depends non-linearly onthe degree of asset market participation and is at the core of the intuition for allour results. In a nutshell, we show that limited asset market participation has anon-linear e¤ect on most predictions of the standard full-participation model.

Interest rate changes modify the intertemporal consumption and labor supplypro�le of asset holders, agents who smooth consumption by trading in asset markets.This a¤ects the real wage, and the demand thereby of agents who have no assetholdings, are oversensitive to real wage changes, and insensitive directly to interestrate changes. Variations in the real wage (marginal cost) lead to variations inpro�ts and hence in the dividend income of asset holders. These variations caneither reinforce (if participation is not �too�limited) or overturn the initial impactof interest rates on aggregate demand. The latter case occurs if the share of non-asset holders is high enough and/or and the elasticity of labor supply is low enough,for the potential variations in pro�t income o¤set the interest rate e¤ects on thedemand of asset holders. This is the main mechanism identi�ed by this paper tochange dramatically the e¤ects of monetary policy as compared to a standard full-participation case whereby aggregate demand is completely driven by asset holders.

If participation is restricted below a certain threshold, the predictions arestrengthened: as more (but not �too many�) people do not hold assets, the link be-tween interest rates and aggregate demand becomes stronger, and monetary policyis more e¤ective. However, when participation is restricted beyond a given thresh-old (i.e. enough agents do not participate in asset markets), standard theoreticalprescriptions or predictions are reversed. First, the �IS curve�has a positive slope:current aggregate output is positively related to real interest rates. Secondly, the�Taylor Principle�(Woodford (2001)) is inverted: the central bank needs to adoptan passive policy rule whereby it increases the nominal interest rate by less thanin�ation (i.e. decreases the real interest rate), for policy to be consistent with aunique rational expectations equilibrium6. Relatedly, an interest rate peg can alsolead to a determinate equilibrium. Thirdly, the welfare-maximizing optimal policyproblem can still be cast in a linear-quadratic framework, but also requires a passiverule. And fourthly, the e¤ects of some shocks are overturned (for example, unan-ticipated positive shocks to interest rates are expansionary). In the limit (whennobody holds assets), aggregate demand ceases to be linked to interest rates andmonetary policy becomes ine¤ective. The required share of non-asset holders forthese results to hold can be compared to empirical estimates or to direct data onasset holding.

The paper closest related to ours is Galí, Lopez-Salido and Valles (2004, hence-forth GLV); that paper studies determinacy properties of interest rate rules in asticky-price model in which a fraction of agents does not hold physical capital and

6An inversion of the Taylor principle occurs independently under some other modellingchoices. For example, in continuous time, Dupor (2000) shows that merely introducing capi-tal invalidates the Taylor principle. A non-Ricardian �scal policy in the sense of Woodford (1996)can also require a passive policy rule for equilibrium determinacy, as noted also by Leeper (1991).

4 FLORIN O. BILBIIE

follows a �rule-of-thumb�. The general message of that paper is that the Taylor prin-ciple is not a good guide for policy under some parameterizations. Namely, GLVargue that if the central bank responds to current in�ation via a simple Taylorrule, when the share of �rule-of-thumb�agents is high enough the Taylor principleis strengthened : the response to in�ation needs to be higher than in the benchmarkmodel. On the contrary, for a rule responding either to past or future expectedin�ation, GLV suggest, based on numerical simulations, that for a high share ofnon-asset holders the policy rule needs to violate the Taylor principle to ensureequilibrium uniqueness.

The aspects that di¤erentiate our paper from GLV pertain to three issues:assumptions, girth and, where the focus of the paper does overlap, message. Con-cerning assumptions, we model the asset market explicitly and emphasize its in-teraction with the labor market, which is at the core of the intuition for all ourresults; any discussion of this is absent from GLV. We also abstract from physi-cal capital accumulation and non-separabilities in the utility function, since thesefeatures can by themselves dramatically change determinacy properties of interestrate rules7. This simpli�cation allows us to: focus on the role of LAMP exclusively,derive all results analitically, and hence provide clear economic intuition for them.Concerning girth, determinacy properties of interest rate rules are only a subsetof our paper�s focus, which regards these issues together with others (e.g. welfare-based optimal monetary policy) as part of a more general theme having to do withLAMP�s in�uence on the aggregate demand side. Finally, within the issue of deter-minacy properties, our conclusions are di¤erent from GLV�s as follows. We showanalitically that an �Inverted Taylor principle�holds in general when asset marketparticipation is restricted enough. This result depends only to a small extent onwhether the rule is speci�ed in terms of current or expected future in�ation. Asdiscussed in text in more detail, this is in contrast to GLV who, while having notedthe possibility to violate the Taylor principle for a forward-looking rule, also arguethat a strengthening of the Taylor principle is required for a contemporaneous ruleto result in equilibrium uniqueness. A very strong response to current in�ationwould also insure determinacy in our model, but we �nd the implied coe¢ cient ishigher than any plausible estimates, makes policy non-credible and would lead toviolation of the zero lower bound in case of small de�ations. We also show how theTaylor principle can be restored by either an appropriate response to output or viadistortionary redistributive taxation of dividend income.

Our results can be perhaps most relevant for analyzing (i) developing economies,in which participation in asset markets is notoriously limited; (ii) historical episodesduring which even developed economies experienced exceptionally low asset marketparticipation. Regarding the latter, many authors have argued that policy beforeVolcker was �badly�conducted along one or several dimensions, which led to worsemacroeconomic performance as compared to the Volcker-Greenspan era. One suchargument relies upon the estimated pre-Volcker policy rule non ful�lling the �Tay-lor principle�, hence containing the seeds of macroeconomic instability driven bynon-fundamental uncertainty (CGG (2000), Lubik and Schorfheide (2004)). In acompanion paper (Bilbiie and Straub (2004)), we take a positive standpoint and

7E.g. capital accumulation in itself may overturn the Taylor principle, at least in continuoustime, as emphasized by Dupor (2000).


argue that Fed policy was better managed than conventional wisdom dictates, if �-nancial market imperfections were pervasive during the �Great In�ation�period. Weuse split the data in two conventional sub-samples (pre-1979 and post-1982, elim-inating the �Volcker disin�ation) and �rst estimate an aggregate IS curve �ndingthat its slope changed sign, consistent with our theory. We hence estimate a modelsimilar with this paper�s using Bayesian methods and �nd that this change indeedcame from a change in the degree of asset market participation. The tremendous�nancial innovation and deregulation process in the 1979-1982 period and the ab-normally high degree of regulation in the 1970�s provide some support to this view.Moreover, we con�rm previous literature�s �nding that monetary policy switchedfrom passive in the former sample to active in the latter. However, in contrast toprevious studies, we show that a passive policy rule implied a determinate equilib-rium, was close to optimal policy and allows for the e¤ects of fundamental shocks tobe studied. Other parameter estimates and the results of some stochastic simula-tion exercises using the estimated shock processes are also in line with stylized factsand some other empirical �ndings. Therefore, the change in �nancial imperfectionsmight help explain both the change in macroeconomic performance and the changein the policy response; the abrupt change in the policy rule might not be a merecoincidence, but an optimal response to the structural change.

The rest of the paper is organized as follows. Sections 2 and 3 introduce theLAMP general equilibrium model and its reduced log-linear form, and discuss ourcore results intuitively. A discussion of the labor market equilibrium useful forfurther intuition is also presented. Section 4 outlines the �Inverted Taylor Principle�and discusses ways to restore the Taylor principle by making interest rates respondto the output gap. Section 5 analyzes optimal monetary policy, Section 6 calculatesanalytically the responses of the economy to cost-push, technology and sunspotshocks under various scenarios and Section 7 concludes. Most technical details arecontained in the Appendices.

2. A General Equilibrium Model with LAMP

The model we use is a standard cashless dynamic general equilibrium model,augmented for limited asset markets participation. The latter feature is introducedby assuming that some of the households are excluded from asset markets, whileothers trade in complete markets for state-contingent securities (including a marketfor shares in �rms). The failure to trade in asset markets could come from avariety of sources, having to do with either preferences or market frictions. Weemphasize market frictions, and in Appendix A outline a simple asset pricing modelwith proportional transaction costs. We show how a distribution of proportionaltransaction costs can be found that rationalizes the exclusion of a given share ofhouseholds from asset markets. In light of this insight, in the remainder of thepaper we assume the fraction of non-asset holders to be exogenous, as in mostpapers on market segmentation and limited participation, e.g. Alvarez, Lucas andWeber (2001). Our baseline model is similar to GLV (2004), but there are importantdi¤erences in assumptions, focus and conclusions that have been emphasized in theIntroduction.

A role for monetary policy is introduced by assuming that prices are slow toadjust. There is a continuum of households, a single perfectly competitive �nal-good producer and a continuum of monopolistically competitive intermediate-goods

6 FLORIN O. BILBIIE

producers setting prices on a staggered basis. There is also a monetary authoritysetting its policy instrument, the nominal interest rate.

2.1. Households. There is a continuum of households [0; 1] ; all having thesame utility function U (:) : A 1 � � share is represented by households who areforward looking and smooth consumption, being able to trade in all markets forstate-contingent securities: �asset holders� or savers. Each asset holder (subscriptS denotes the representative asset holder) chooses consumption, asset holdings andleisure solving the standard intertemporal problem: maxEt

P1i=0 �

iU (CS;t+i; NS;t+i),subject to the sequence of constraints:

BS;t +S;t+1Vt � ZS;t +S;t (Vt + PtDt) +WtNS;t � PtCS;t:Asset holder�s momentary felicity function takes the additively separable log-

CRRA form U (CS;t; NS;t) = lnCS;t � !N1+'S;t = (1 + ') ; which has been used in

many DSGE studies8. � 2 (0; 1) is the discount factor, ! > 0 indicates how leisureis valued relative to consumption, and ' > 0 is the inverse of the labor supplyelasticity. CS;t; NS;t are consumption and hours worked by saver (time endowmentis normalized to unity), BS;t is the nominal value at end of period t of a portfolioof all state-contingent assets held, except for shares in �rms. We distinguish sharesfrom the other assets explicitly since their distribution plays a crucial role in therest of the analysis. ZS;t is beginning of period wealth, not including the payo¤ ofshares. Vt is average market value at time t shares in intermediate good �rms, Dt

are real dividend payo¤s of these shares and S;t are share holdings.Absence of arbitrage implies that there exists a stochastic discount factor �t;t+1

such that the price at t of a portfolio with uncertain payo¤ at t + 1 is (for state-contingent assets and shares respectively):

(2.1) BS;t = Et [�t;t+1ZS;t+1] and Vt = Et [�t;t+1 (Vt+1 + Pt+1Dt+1)]

Note that the Euler equation for shares iterated forward gives the fundamental pric-

ing equation: Vt = Et

1Xi=t+1

�t;iPiDi:The riskless gross short-term nominal interest

rate Rt is a solution to:

(2.2)1

Rt= Et�t;t+1

Substituting the no-arbitrage conditions (2.1) into the wealth dynamics equationgives the �ow budget constraint. Together with the usual �natural�no-borrowinglimit for each state, this will then imply the usual intertemporal budget constraint:

(2.3) Et

1Xi=t

�t;iPiCS;i � ZS;t + Vt + Et

1Xi=t

�t;iWiNS;i

Maximizing utility subject to this constraint gives the �rst-order necessary andsu¢ cient conditions at each date and in each state:

�UC (CS;t+1)

UC (CS;t)= �t;t+1

Pt+1Pt

!N'S;t =

1

CS;t

Wt

Pt

8This function is in the King-Plosser-Rebelo class and leads to constant steady-state hours.In Section 3.3 we conduct some robustness checks for an utility function that does not.


along with (2.3) holding with equality (or alternatively �ow budget constraint holdwith equality and transversality conditions ruling out overacummulation of assetsand Ponzi games be satis�ed: lim

i!1Et [�t;t+iZS;t+i] = lim

i!1Et [�t;t+iVt+i] = 0):

Using (2.3) and the functional form of the utility function, the short-term nominalinterest rate must obey:

1

Rt= �Et

�CS;tCS;t+1

PtPt+1

�:

The rest of the households on the [0; �] interval have no assets9: �non-assetholders� (for instance because, as in Appendix A, they face a common, large enoughproportional cost to trade in asset markets). The problem of the representativenon-asset holder indexed by H is then equivalent to:

(2.4) maxCH;t;NH;t

lnCH;t � !N1+'H;t

1 + 's.t. CH;t =

Wt

PtNH;t:

The �rst order condition is:

(2.5) !N'H;t =

1

CH;t

Wt

Pt;

which further allows reduced-form solutions for CH;t and NH;t (functions only ofWt=Pt and exogenous processes). Due to the very form of the utility function,hours are constant for these agents: the utility function is chosen to obtain constanthours in steady state, and this agent is �as if�she were in the steady state always.In this case labour supply of non-asset holders is �xed, no matter '; as incomeand substitution e¤ects cancel out. While this facilitates algebra, it is in no waynecessary for our results (elastic labor supply will be discussed below). Hours aregiven by: NH;t = !�

11+' and consumption will track the real wage to exhaust

the budget constraint (see Section 3.3 for preferences that do not lead to constantsteady state hours).

2.2. Firms. The �rms�problem is completely standard and can be skippedby some readers without loss of continuity. The �nal good is produced by arepresentative �rm using a CES production function (with elasticity of substi-tution ") to aggregate a continuum of intermediate goods indexed by i: Yt =�R 1

0Yt (i)

("�1)="di�"=("�1)

: Final good producers behave competitively, maximizing

pro�t PtYt�R 10Pt (i)Yt(i)di each period, where Pt is the overall price index of the �-

nal good and Pt (i) is the price of intermediate good i. The demand for each interme-diate input is Yt(i) = (Pt (i) =Pt)

�"Yt and the price index is P

1�"t =

R 10Pt (i)

1�"di.

Each intermediate good is produced by a monopolist indexed by i using atechnology given by: Yt(i) = AtNt(i)�F; if Nt(i) > F and 0 otherwise:F is a �xedcost assumed to be common to all �rms: this can be treated as a free parametergoverning the degree of returns to scale. Cost minimization taking the wage asgiven, implies that nominal marginal cost is MCt =Wt=At: The pro�t function in

real terms is given by: Dt (i) = [Pt(i)=Pt]Yt(i) � (Wt=Pt)Nt(i); which aggregatedover �rms gives total pro�ts Dt = [1� (MCt=Pt)�t]Yt. Total pro�ts are rebated

9These households are labeled �non-traders�by Alvarez, Lucas and Weber, �rule-of-thumb�or�non-Ricardian�by GLV, and �spenders�by Mankiw 2000.

8 FLORIN O. BILBIIE

to the asset holders as dividends. The term �t is relative price dispersion de�nedfollowing Woodford (2003) as �t �

R 10(Pt (i) =Pt)

�"di and will play a major role

in the welfare analysis.To introduce a role for monetary policy in a¤ecting the real allocation in this

simple cashless model we follow Calvo (1983) and Yun (1996) and introduce stickyprices. Intermediate good �rms adjust their prices infrequently, � being both thehistory-independent probability of keeping the price constant and the fraction of�rms that keep their prices unchanged. Asset holders (who in equilibrium will holdall the shares in �rms) maximize the value of the �rm, i.e. the discounted sum offuture nominal pro�ts, using the relevant stochastic discount factor �t;t+i used asthe pricing kernel for nominal payo¤s:

maxPt(i)

Et

1Xs=0

(�s�t;t+s [Pt(i)Yt;t+s(i)�MCt+iYt;t+s(i)] ;

subject to the demand equation:The optimal price of the �rm obeys:

P ot (i) = Et

1Xs=0

�s�t;t+sP"�1t+s Yt+s

EtP1k=0 �

k�t;t+kP"�1t+k Yt+k

MCt+s

In equilibrium each producer that chooses a new price Pt(i) in period t willchoose the same price and the same level output, hence the price index is: P 1�"t =

(1 � �) (P ot )1�"

+ �P 1�"t�1 . The combination of these two conditions leads in thelog-linearized equilibrium to the well known New Keynesian Phillips curve givenbelow.

2.3. Monetary policy. We consider two policy frameworks prominent in theliterature. First, we study instrument rules in the sense of a feedback rule forthe instrument (short-term nominal interest rate) as a function of macro variables,mainly in�ation. We focus on rules within the family (where variables with a stardenote variables calculated under �exible prices, de�ned below):

(2.6) Rt = (R�t )��R

�EtPt+1Pt

�� YtY �t

��x

e"t :

We shall also consider targeting rules under discretionary policymaking, wherebythe path of the nominal rates is found by optimization by the central bank - this isdescribed in detail in Section 6 below. Such a framework will also imply a behavioralrelationship for the instrument rule, but this is only an implicit instrument rule.

2.4. Market clearing, aggregation and accounting. All agents take asgiven prices (with the exception of monopolists who reset their good�s price in agiven period), as well as the evolution of exogenous processes. A rational expec-tations equilibrium is then as usually a sequence of processes for all prices andquantities introduced above such that the optimality conditions hold for all agentsand all markets clear at any given time t. Speci�cally, labor market clearing re-quires that labor demand equal total labor supply, Nt = �NH;t + (1� �)NS;t:State-contingent assets are in zero net supply (markets are complete and agents


trading in them are identical), whereas equity market clearing implies that shareholdings of each asset holder are:

S;t+1 = S;t = =1

1� �:

Finally, by Walras�Law the goods market also clears (this is also the social resourceconstraint) Yt = Ct; where Ct � �CH;t + (1� �)CS;t is aggregate consumption.

2.5. Steady state and linearized equilibrium. We study the dynamics ofthe above model by taking a (log-)linear approximation of it around the uniquenon-stochastic steady state. The latter is found by evaluating the optimality con-ditions in the absence of shocks and assuming that all variables are constant.From the Euler equation of asset holders, the steady-state riskless interest rate isR = ��1; where R � 1+r: De�ning the steady-state net mark-up as � � ("� 1)�1,the share of real wage in total output can easily be shown to be: WN=PY =(1 + FY ) = (1 + �) ; while pro�ts�share in total output is: DY = D=Y = (�� FY ) = (1 + �) :We assume that hours are the same for the two groups in steady state, NH = NS =N . Then, using the budget constraint for each group, consumption shares in totaloutput are:

CSY=1 + FY1 + �

+1

1� �� FY1 + �

;CHY

=1 + FY1 + �

Since preferences are homogenous, steady-state consumption shares are also equalacross groups, since intratemporal optimality conditions evaluated at steady-stateimply:

CH = CS =1

!N'

W

P:

This instead requires either restrictions on technology making the share of assetincome zero in steady state, or an appropriate taxation scheme (something we haveabstracted from until now, but shall return to when analyzing optimal policy). Forexample, if the share of the �xed cost is equal to net markup � = FY the share ofpro�ts in steady-state DY is zero, consistent with evidence and arguments in i.a.Rotemberg and Woodford (1995), and with the very idea that the number of �rmsis �xed in the long run10. Consumption shares are then:

CHY

=CSY= CY = 1:

Alternatively, we could assume that either (i) hours worked are not equal acrossgroups or (ii) preferences are not homogenous across groups. Any of these assump-tions would allow for consumption shares to be di¤erent across groups in steady-state, complicating the subsequent analysis. We studied such generalizations in anearlier working paper version (Bilbiie (2004)) and have found that all results of thesimplest case carry through largely unaltered. Therefore, we have decided to adoptthe simplest setup in order to facilitate exposition.

We proceed by taking a (log-)linear approximation of the equilibrium condi-tions around this steady state. We let a small-case letter denote the log-deviation ofa variable from its steady state value, e.g. yt = log (Yt=Y ) ' (Yt � Y ) =Y; with theexception of pro�ts, which are de�ned as a fraction of steady-state output (since

10However, pro�ts will vary across �rms and over time around this steady-state. Indeed, asit shall soon become clear, pro�t -and asset income- variations are at the heart of this paper�sintuition.

10 FLORIN O. BILBIIE

their steady-state value D is zero), i.e. dt ' Dt=Y . The linearized equilibriumconditions are conveniently summarized in Table 1, where we have already imposedasset market clearing, including = (1� �)�1, and substituted the steady-stateratios found above. We use these equations to express dynamics in terms of ag-gregate variables only; this makes our model readily comparable with the standardfull-participation framework (see CGG (1999), Woodford (2003)) and amenable topolicy exercises.

Table 1 here.

3. Aggregate dynamics

First, as noted above hours of non-asset holders are constant nH;t = 0; there-fore, their consumption tracks real wage, cH;t = wt and total labor supply (fromthe labor market clearing condition) is nt = [1� �]nS;t:

Using these two expressions and asset holders�labor supply equation into thede�nition of total consumption we �nd: ct = �

1��'nt+ cS;t. Finally, using the pro-duction function, we obtain the equivalent of the �planned expenditure line�familiarfrom standard Keynesian models (see for example David Romer�s textbook):

(3.1) ct = c

�yt+; rrt�; at

�=

�

1� �'

1 + �yt + cS;t �

�

1� �'at

This equation links aggregate expenditure to current total income (output),consumption of asset holders and exogenous technology. Note that (3:1) is not areduced-form relationship since c; y; cS are all endogenous variables, which will bedetermined in general equilibrium. However, we can think of (3:1) as a schedule inthe (y; c) space, for a given level of cS;t: In that sense, we can say that aggregatedemand (expenditure) depends positively on current income and negatively on theex-ante real interest rate rrt � rt�Et�t+1 (the latter coming from the Euler equa-tion of asset-holders in Table 1). We can de�ne the (partial) �marginal propensityto consume�out of current income11 as @ct=@yt = �

1��'1+� > 0:

The marginal propensity to consume is increasing in (i) the share of non-assetholders �, for this means that a higher fraction of total population simply consumesthe real wage and is insensitive to interest rate movements and (ii) the extent towhich labor supply is inelastic '; for this implies that small variations in hours (andoutput) are associated to large variations in real wage and hence in the consumptionof non-asset holders. Therefore, the aggregate propensity to consume depends�nally on income distribution, which changes as aggregate income and the wagerate change; this gives the model a distinctly Keynesian �avor. In fact, togetherwith the condition that consumption equal output ct = yt; equation (3:1) leads to a�Keynesian cross�in case @c=@y < 1: this is pictured in Figure 1 by the black thickline labelled �K�: A positive but low enough � makes the economy �more Keynesian�since the propensity to consume out of current income becomes larger than zero,its value under full participation (whereby the PEL line is the horizontal axis):

However, note that the marginal propensity to consume out of current income(output) @c=@y can become greater than one. This case, which we label �non-Keynesian�, occurs when enough agents consume their wage income wt (� high)

11This is in fact a partial marginal propensity, i.e. keeping �xed consumption of asset holderscS ; since in equilibrium all output is consumed. We will loosely refer to @ct=@yt as �the marginalpropensity to consume�in the remainder.


and/or wage is sensitive enough to real income yt (' high), more precisely when:

(3.2) � > �� =1

1 + '= (1 + �):

This is illustrated in Figure 1 by the red thin line labelled �NK�plotting (3:1) in thiscase along with the ct = yt schedule, resulting in a �Non-Keynesian cross�. An in-crease in the real interest rate moves the (3:1) schedule rightward (by intertemporalsubstitution) leading to higher consumption and output. An increase in technologyalso moves the (3:1) schedule rightward and boosts consumption and output, sincethe resulting increase in real wage boosts consumption of non-asset holders.

Fig.1: The Keynesian and Non-Keynesian Crosses.

An immediate implication of the above is that the aggregate IS curve, relatingtotal output growth with the ex-ante real interest rate, swivels (its slope changessign). Consumption of asset holders is related to total output, combining (3:1) withct = yt; by:

(3.3) cS;t = �yt + (1 + �) (1� �) at ; where � � 1� '�

1� �1

1 + �= 1� @c

@y;

Note that � becomes negative when @c=@y > 1, i.e. precisely when (3.2) holds.Consumption of asset holders can be negatively related to total output since anincrease in demand can only be satis�ed by movements of (as opposed to move-ments along) the labor supply schedule when enough people hold no assets andlabor supply is inelastic enough. But the necessary rightward shift of labor sup-ply can only come from a negative income e¤ect on consumption of asset holders.This negative income e¤ect is ensured in general equilibrium by a potential fall individend income. Note that asset holders have in their portfolio (1� �)�1 shares:if total pro�ts fell by one unit, dividend income of one asset holder would fall by(1� �)�1 > 1 units12. The potential decrease in pro�ts is a natural result of inelas-tic labor supply, since the increase in marginal cost (real wage) would more than

12In the standard model all agents hold assets, so this mechanism is completely irrelevant.Any increase in wage exactly compensates the decrease in dividends, since all output is consumedby asset holders.


outweigh the increase in sales (hours). Therefore, this mechanism relies on the con-sumption of the wealth-owning agents being sensitive to changes in the money-valueof their wealth.

Importantly, note that despite the potential decrease, in general equilibriumactual pro�ts may not fall, precisely due to the negative income e¤ect making assetholders willing to work more; for as a result of this e¤ect hours will increase bymore and marginal cost by less, preventing actual pro�ts from falling. In fact, forcertain combinations of parameters, shocks or policies our model would not implycountercyclical pro�ts in equilibrium (or at least implies more procyclical pro�tsthan a standard full-participation model with countercyclical markups). This is animportant point, since it is widely believed that pro�ts are procyclical (see Section7 for further discussion). It is also important to note that the negative income e¤ectdoes not mean that for a given increase in output, the consumption of asset holderswill necessarily decrease. In fact, if the increase in output is due to technology,cS;t will increase in most cases (i.e. when the equilibrium elasticity of output totechnology is less than (1 + �)

�1� ��1

�):

Having expressed consumption of asset holders as a function of aggregate out-put, we can now substitute it back into the Euler equation and �nd an aggregateEuler equation, or �IS curve�:

(3.4) yt = Etyt+1 � ��1 [rt � Et�t+1] + (1 + �)�1� ��1

�[at � Etat+1]

Direct inspection of (3:4) suggests the impact that LAMP has on the dynamicsof a standard business cycle model through modifying the elasticity of aggregatedemand to real interest rates ��1 in a non-linear way. For high enough partic-ipation rates � < �� (where the latter is given by (3:2)) we are in a �Keynesian�region, whereby real interest rates restrain aggregate demand. As � increases to-wards ��, the sensitivity to interest rates increases in absolute value, making policymore e¤ective in containing demand. However, once � is above the threshold ��

we move to the �non-Keynesian� region where increases in real interest rates be-come expansionary (see also Figure 1). As � tends to its upper bound of 1, ��1decreases towards zero - policy is ine¤ective when nobody holds assets. We willcall �non-Keynesian� an economy in which participation in asset markets is limitedenough such that � < 0. Finally, note that the only way for � to be independent of� is for ' to be zero, i.e. labor supply of asset holders be in�nitely elastic. In thiscase, consumption of all agents is independent of wealth, making the heterogeneityintroduced in this paper irrelevant.

We now have a �rst glance at the magnitude of � required for our results to holdquantitatively. To that end, in Figure 2 we plot the threshold �� as a function of ',assuming a conventional value for steady-state markup of � = 0:2 corresponding toan elasticity of substitution of intermediate goods � of 6. Values under the curve givethe Keynesian (� > 0) case, whereas above the curve we have the non-Keynesianeconomy with � < 0.


Fig. 2: Threshold share of non-asset holders as a function of inverse labor supplyelasticity. Above the threshold we have the �LAMP economy�where the Inverted Taylor

Principle applies.

For Keynesian logic to work, the Frisch elasticity of labor supply (and of intertem-poral substitution in labor supply), should be high, and the higher, the higher theshare of non-asset holders �. For a range of ' between 1 (unit elasticity) and 10(0.1 elasticity) the threshold share of non-asset holders should be lower than 0.5 toas low as around 0.1 respectively. Compared to some empirical evidence by Camp-bell and Mankiw (1989) that place this at around 0:4-0:5 for the US economy fordata running up to the mid eighties, or to data on asset holdings reviewed in theIntroduction, this shows that the required share of non-asset holders to end up inthe non-Keynesian case is not empirically implausible.

3.1. Aggregate supply and gap dynamics. In the foregoing we have fo-cused on the aggregate demand side and hence ignored aggregate supply, the dy-namics of in�ation and its interaction with the above insights, to which we now turn.By substituting (3:3) and the production function in the labor supply equation ofasset holders we obtain an expression relating real wage to total output:

wt = �yt � 'at; where � � 1 + '= (1 + �) � 1 � �:

Substituting this in the expression for real marginal cost in Table 1 and further inthe New Phillips curve we obtain:

(3.5) �t = �Et�t+1 + �yt � (1 + ') at; where � � �:

This is the Phillips curve written over the level of output, where is de�ned inTable 113.

We now seek to write both the IS and Phillips curve in terms of the output gapxt � yt � y�t ; de�ned as the di¤erence between actual output and natural outputy�t : Natural levels of all variables are de�ned as values occurring in the notionalequilibrium in which all prices are �exible (� = 0), and hence in�ation is nil andreal marginal cost (and markup) are constant. We see directly from (3:5) that

13Note that the Phillips curve is invariant to the share of non-asset holders � and henceidentical to that obtained in the standard framework; this is due to the assumption that steadstate consumption shares are equal across groups. In the more general case studied in the workingpaper (Bilbiie (2004)) the presence of non-asset holders modi�es � (the elasticity of marginalcost to movements in the output gap) and hence the response of in�ation to aggregate demandvariations. However, the induced modi�cations are quantitatively minor.


natural output is14: y�t = (1 + ')��1at; so marginal cost is related to the output

gap by: mct = �xt + �1ut where, following e.g. Clarida, Galí and Gertler (1999)

or Galí (2002) we also introduce cost-push shocks ~ut = �1ut, i.e. variations inmarginal cost not due to variations in excess demand. These could come fromthe existence of sticky wages creating a time-varying wage markup, time-varyingelasticity of substitution among intermediate goods or other sources creating thisine¢ ciency wedge although we do not model this explicitly here (see Woodford,2003). Therefore, the Phillips curve written in terms of the output gap is:

(3.6) �t = �Et�t+1 + �xt + ut:

Finally, returning to the aggregate demand side and evaluating the IS curve under�exible prices we can de�ne the natural interest rate15: r�t = [1 + � (1� �=�)] [Etat+1 � at] :Using this, the IS curve can be rewritten in terms of the output gap:

(3.7) xt = Etxt+1 � ��1 [rt � Et�t+1 � r�t ] ;The Phillips and IS curves (3:6) and (3:7) ; together with the feedback interest

rate rule fully determine dynamics of endogenous variables as a function of ex-ogenous shocks. Note that while we have not speci�ed a process for technology,the model can be solved even for non-stationary technology at since r�t (which isthe relevant shock for determining dynamics) is stationary. For instance, one canassume that technology growth (�at � at � at�1) is given by an AR(1) process�at = �a�at�1 + "at , which implies shocks to technology have permanent e¤ectson the level of output (see Galí (1999)).

3.2. Further intuition: the labor market. The key to understanding theresults obtained here is the labor market equilibrium - and its interaction with theasset market. In system (3.8) we outline the labor supply and the equilibrium wage-hours locus. The labor supply schedule LS represents the locus of wages and hoursfor a given level of consumption of asset holders (all the intertemporal substitutionin labor supply comes naturally from asset holders). The equilibrium wage-hourslocus labeled WN is derived taking into account the endogenous relationship be-tween asset holders�consumption and total output (3:3) and the production func-tion. This schedule is invariant to endogenous forces in equilibrium (in fact, it willbe shifted by technology shocks only).

LS : wt ='

1� �nt + cS;t(3.8)

WN : wt =

�(1 + �) � +

'

1� �

�nt + (1 + �) at

A �non-Keynesian economy�(� < 0) has an intuitive interpretation in labor marketterms, for it implies that the equilibrium wage-hours locus is less upward sloping

14Following the discussion above, note that under �exible prices consumption of asset holderswill always increase in response to technology shocks (despite its partial elasticity to total output� being negative) since c�S;t = [1 + � (1� �=�)] at and hence is procyclical. Real pro�ts under�exible prices are given by d�t = [�= (1 + �)] y

�t ; and are evidently procyclical.

15Note that r�t is stationary even when at is not. Moreover, the sign of the response of r�t to

at is the same as under full participation: permanent technology shocks have permanent e¤ectson natural output and positive temporary e¤ects on the r�t (since � < �), whereas temporarytechnology shocks cause a fall in r�t .


than (and hence cuts from above) the labor supply curve. Intuitively, the presenceof non-asset holders generates overall a �negative income e¤ect�, which cannot beobtained ceteris paribus when � = 0 and � = 1. In the latter, standard case, thewage-hours locus is more upward sloping than LS. The di¤erence between the twois the intertemporal elasticity of substitution in consumption, normalized to 1 inour case (multiplied by returns to scale 1+� ). Ceteris paribus, if the labor demandshifts out, labor supply shifts leftward due to the usual income e¤ect, since agentsanticipate higher income and higher consumption. If labor supply shifts up dueto a positive income e¤ect, same e¤ect makes labor demand shift out. This givesa WN locus more upward sloping than the labor supply curve LS. The thresholdvalue for � for this insight to change is the same as that making � < 0 and given in(3.2) above. When the share of non-asset holders is higher than this threshold (orequivalently for a given share, labor supply of asset holders is inelastic enough), thewage-hours locus becomes less upward sloping than the labor supply. An intuitionfor that follows, as illustrated in Figure 3 where we assume that the real interestrate is kept constant for simplicity16.

Take �rst an exogenous outward shift in labor demand. Keeping supply �xed,there would be an increase in real wage and an increase in hours. The increase inthe real wage would boost consumption of non-asset holders, amplifying the initialdemand e¤ect. When labor supply is relatively inelastic, this increase in wage islarge and the increase in hours is small compared to that necessary to generatethe extra output demanded; note that the e¤ect induced on demand is larger, thehigher the share of non-asset holders. The only way for supply to meet demand isfor labor supply to shift right. This is insured in equilibrium by the potential fallin pro�ts resulting from: (i) increasing marginal cost (since wage increases) and (ii)the weak increase in hours and hence in output and sales. This is like and indirectnegative income e¤ect induced on asset holders by the presence of non-asset holders.Next consider a shift in labor supply, for example leftward as would be the case ifconsumption of asset holders increased. Keeping demand �xed, wage increases andhours fall. The increase in wage (and the increase in consumption of asset holdersitself) has a demand e¤ect due to sticky prices. As labor demand shifts right, thereal wage would increase by even more; hours would increase, but by little dueto the relatively inelastic labor supply (the overall e¤ect would again depend onthe relative slopes of the two curves). The increase in the real wage means extrademand through non-asset holders�consumption17. To meet this demand, only wayfor increasing output is an increase in labor supply, which instead obtains only iflabor supply shifts right, which is insured as before by the potential fall in pro�ts.This explains why in a �non-Keynesian economy� the wage-hours locus cuts thelabor supply curve from above. This instead will help our intuition in explainingthe further results18. Note that such a wage-hours locus implies that the model

16How the nominal interest rate reacts to in�ation, generated here by variations in demand,will be crucial in the further analysis.

17The assumptions on preferences ensuring constant steady-state hours are less crucial thanit might seem. Below we consider alternative preference speci�cations.

18Note that the intuition for real indeterminacy to obtain in standard models (see e.g. Ben-habib and Farmer 1994) requires the wage hours locus be upward sloping but cut the labor supplycurve from below. This is also the case in standard sticky-price models, and gives rise to a certainrequirement for the monetary policy rule to result into real determinacy - see below. Our intuition


generates a higher partial elasticity of hours to the real wage, and more so morenegative � is.

Fig. 3: The equilibrium wage-hours locus and labor supply curve with LAMP.

Having derived the equilibrium wage-hours locus gives us a simple way of think-ing intuitively about the e¤ects of shocks and of monetary policy in general; mon-etary policy, by changing nominal interest rates, modi�es real interest rates andhence shifts the labor supply curve (by changing the intertemporal consumptionpro�le of asset holders). But this has no e¤ect on the wage-hours locus by con-struction, since this describes a relationship that holds in equilibrium always andis shifted only by technology shocks.

3.3. Robustness. One might rightly wonder whether the mere theoreticalpossibility of a change in the sign of � is entirely dependent upon the speci�cationof preferences. It turns out this possibility is robust to two obvious candidates:an elastic labor supply of non-asset holders, and a non-unitary elasticity of in-tertemporal substitution in consumption. For completion, we brie�y study theseextensions jointly. Consider preferences given by a general CRRA utility functionfor both agents j ( is relative risk aversion for both agents and also inverse ofintertemporal elasticity of substitution in consumption for asset holders):

(3.9) Uj (:; :) =C1� j;t

1� � !N1+'j;t

1 + '

Following the same method as before one can show that the solution to non-assetholders�problem will be (in log-linearized terms, where elasticity of hours to wage� � (1� ) = ( + ') is positive i¤ < 1):

nH;t = �wt; cH;t = (1 + �)wt

will be that having the wage-hours locus cut the labor suply from above, changes determinacyproperties in a certain way.


For asset holders, the new Euler equation and intratemporal optimality in log-linearized form are:

EtcS;t+1 � cS;t = �1 (rt � Et�t+1) ; 'nS;t = wt � cS;tUsing the same method as previously, one �nds that the new condition to be ful�lledin order for � to become negative and hence end up in a �non-Keynesian economy�:

(3.10) � >1

1 + ' (1� ��) = (1 + �)By comparing (3.10) with (3.2) one immediately notices that under the more gen-eral preferences (3.9) the threshold value of � is lower (higher) than under logutility if � > 0 (< 0). The intuition is that while making aggregate labor sup-ply more elastic, a positive � also makes equilibrium hours more elastic to wagechanges since it makes consumption of non-asset holders more responsive to thewage. In general however, the di¤erence induced by having 6= 1 on the thresholdvalue of � is quantitatively negligible19. In view of the relative innocuousness ofthese assumptions, we shall continue using the log-CRRA utility function in the re-mainder, since it preserves constant steady-state hours and hence allows analyzingpermanent technology shocks.

4. The Inverted Taylor Principle: Determinacy properties of interestrate rules

In this Section we study determinacy properties of simple interest rate rules20.Not surprisingly, the implications of LAMP and the insights outlined above for thisissue are dramatic. Formally, assume �rst that nominal interest rates are set as afunction of expected in�ation as e.g. in CGG (2000). This speci�cation providessimpler (sharper) determinacy conditions, and captures the idea that the centralbank responds to a larger set of information than merely the current in�ation rate:

(4.1) rt = ��Et�t+1 + "t

where "t is the non-systematic part of policy-induced variations in the nominal rate.The dynamic system for the zt � (xt; �t)0 vector of endogenous variables and the�t � ("t � r�t ; ut)

0 vector of disturbances is obtained by replacing (4.1) into (3:7)and (3.6) as:

Etzt+1 = �zt +�t;

where coe¢ cient matrices are given by:

� =

�1� ��1��1� (�� 1) ��1��1 (�� 1)

��1� ��1

�;(4.2)

=

��1 ��1��1 (�� 1)0 ��1

�:

Since both in�ation and output gap are forward-looking variables, determinacyrequires that both eigenvalues of � be outside the unit circle. The determinacy

19Even evaluating the di¤erence between the threshold values corresponding to = 0 and100 respectively, one obtains under the baseline parameterization (for values of ' = 0:5; 1; 5; 10respectively): 0.13;0.09; 0.03; 0.01.

20For analytical simplicity we abstract from inertia (interest rate smoothing) but this exten-sion should be straightforward.


properties of rule (4.1) are emphasized in Proposition 2 (the proof is in AppendixB).

Proposition 1. The Inverted Taylor Principle: Under policy rule (4.1)there exists a locally unique rational expectations equilibrium (i.e. the equilibriumis determinate) if and only if:

Case I: When � > 0 : �� 2�1; 1 + � 2(1+�)�

�;

Case II: When � < 0 : �� 2�1 + � 2(1+�)� ; 1

�\ [0;1) :

Case I corresponds to the standard �Keynesian� case and is a generalizationof the Taylor principle (Woodford (2001)): as in the full-participation case, thecentral bank should respond more than one-to-one to increases in in�ation21. CaseII is the �non-Keynesian economy�. In this case, the Central Bank should follow anInverted Taylor Principle: only passive policy is consistent with a unique rationalexpectations equilibrium. Obviously, the condition for the Inverted Taylor Principleto hold is the same as the one causing a change in the sign of �; as in (3.2).

4.1. Intuition: sunspot equilibria with LAMP. An intuitive explanationof Proposition 1 is in order. We will discuss why in cases covered by the Propositionsunspot shocks have no real e¤ect, whereas in the opposite cases they lead toself-ful�lling expectations. In Section 7 below we will compute sunspot equilibriaformally. Note that by substituting the rule (4:1) into (3.7) we obtain aggregatedemand as a function of expected in�ation:

(4.3) xt � Etxt+1 = ��1 (�� 1)Et�t+1:Suppose for simplicity and without losing generality that a sunspot shock hitsin�ationary expectations. In a Non-Keynesian economy (� < 0) a non-fundamentalincrease in expected in�ation generates an increase in the output gap today if thepolicy rule is active (�� > 1) as can be seen from (4.3). By the Phillips curvein�ation today increases, validating the initial non-fundamental expectation. Thisis not the case in the Keynesian economy (� > 0), since an active rule generates afall in output gap and (by Phillips curve logic) actual in�ation, contradicting initialexpectations.

How does a passive policy rule ensure equilibrium determinacy when � < 0?A non-fundamental increase in expected in�ation causes a fall in the real interestrate, a fall in the output gap today by (4.3) and de�ation, contradicting the initialexpectation that are hence not self-ful�lling. At a more micro level, the transmissionis as follows. The fall in the real rate leads to an increase in consumption of assetholders, and an increase in the demand for goods; but note that these are nowpartial e¤ects. To work out the overall e¤ects one needs to look at the componentof aggregate demand coming from non-asset holders and hence at the labor market.The partial e¤ects identi�ed above would cause an increase in the real wage (anda further boost to consumption of non-asset holders) and a fall in hours. Increaseddemand, however, means that (i) some �rms adjust prices upwards, bringing abouta further fall in the real rate (as policy is passive); (ii) the rest of �rms increaselabor demand, due to sticky prices. Note that the real rate will be falling along the

21It should also not respond �too much�, which is a well-established result for froward-lookingrules �rst noted by Bernake and Woodford (1997). Note that this upper bound is decreasing inthe share of non-asset holders.


entire adjustment path, amplifying these e¤ects. But since this would translate intoa high increase in the real wage (and marginal cost) and a low increase in hours, itwould lead to a fall in pro�ts, and hence a negative income e¤ect on labor supply.The latter will then not move, and no in�ation will result, ruling out the e¤ects ofsunspots. This happens when asset markets participation is limited �enough�in away made explicit by (3.2).

If the policy rule is instead active (�� > 1) sunspot equilibria can be con-structed. The shock to in�ationary expectations leads to an increase in the realrate and in aggregate demand by (4.3). This generates in�ation and makes theinitial expectations self-ful�lling. At a micro level, transmission is as follows: con-sumption of asset holders increases due to the real rate increase, which impliesa rightward shift of labor supply, and hence a fall in wage and increase in hours.Consumption of non-asset holders also falls one-to-one with the wage, and hence ag-gregate demand falls by more than it would in a full-participation economy. Firmswho can adjust prices will adjust them downwards, causing de�ation, and a furtherfall in the real rate. Firms who cannot adjust prices will cut demand, causing a fur-ther fall in the real wage and a small fall in hours (since labor supply is inelastic).But this will mean higher pro�ts (since marginal cost is falling), and eventuallya positive income e¤ect on labor supply of asset holders. As labor supply startsmoving leftward, demand starts increasing, its increase being ampli�ed by the sen-sitivity of non-asset holders to wage increases. The economy will establish at apoint on the wage-hours locus consistent with the overall negative income e¤ect onlabor supply of asset holders, i.e. with higher in�ation and real activity. Hence,the initial in�ationary expectations become self-ful�lling.

4.2. Output stabilization may restore the Taylor Principle. We nowstudy whether a policy rule incorporating an output stabilization motive can makethe Taylor principle a good policy prescription even in a �non-Keynesian economy�where � < 0. Consider a rule of the form:

(4.4) rt = ��Et�t+1 + �xxt;

where�x is the response to output gap. Intuitively, we may expect this to restoreKeynesian logic since it e¤ectively reduces the slope of the planned expenditure line,by introducing a negative dependence of asset holders�consumption on current in-come (through the Euler equation). However, there are non-trivial interactionsamong parameters due to the nature of the price-adjustment equation as empha-sized by the following Proposition. Replacing (4:4) into (3:7) and (3.6), the �matrix becomes:

� =

�1� ��1

��1� (�� 1)� �x

��1��1 (�� 1)

��1� ��1

�Applying exactly the same method as in the proof of Proposition 1 it can be

shown that the determinacy conditions are as follows.

Proposition 2. (a) Under (4.4), there exists a locally unique rational expec-tations equilibrium if and only if:

Case I: When � > 0 : ��+1�� x > 1 and �� < 1+

1+�� (�x + 2�) (the �Taylor

Principle�)


Case II: When � < 0 : EITHER II.A: �x < �� (1� �) and �� + 1�� x < 1

and �� > 1 + 1+�� (�x + 2�) OR II.B: �x > �� (1 + �) and �� + 1��

� �x > 1 and�� < 1 +

1+�� (�x + 2�)

(b) The equilibrium is indeterminate regardless of �� if � < 0 and �x 2(�� (1� �) ;�� (1 + �)) :

Part (a) studies equilibrium uniqueness. Case I is the standard Taylor principlefor an economy where � > 0. In contrast to Proposition 1, in Case II the inversion ofthe Taylor Principle is now not granted. If either � is very large in absolute value (ahigh degree of limited participation �) or the response to output is low, we end upin case II.A and an instance of the Inverted Taylor Principle is observed. However,for moderate values of � and/or a high enough response to the output gap, theTaylor Principle is restored. Another way to put this is that for a given share ofnon-asset holders, the Taylor Principle is a good guide for policy only insofar as theresponse to output is high enough. The response to output, however, can generateperverse e¤ects if it is not high enough and participation in asset markets is verylimited. As part (b) of the Proposition shows, the equilibrium is indeterminate if�x is in a certain range, regardless of the magnitude of the in�ation response. Thisregion is increasing with the share of non-asset holders.

To assess the magnitude of the policy coe¢ cients needed for restoring the Tay-lor principle, consider a standard quarterly parameterization with r = 0:01 andan average price duration of one year, implying � = 0:75, for a �non-Keynesianeconomy�with � = 0:4 and ' = 2 (giving � = �0:11 and � = 0:228 ):The conditionsfor Case II.B are �x > 0:21 ; �� > 1�0:043�x; �� < 8: 728�x�0:920 16:The �gurebelow shows that as soon as the Central Bank responds to output, the Taylor prin-ciple is restored under the baseline parameterization for a large parameter region.However, this result should be taken with care, for the very dangers associated withresponding to output might outweigh potential bene�ts. As soon as the share ofnon-asset holders increases or labor supply becomes more inelastic, equilibrium ismore likely to become indeterminate for any in�ation response.

Fig. 4 : Policy parameter region whereby Taylor Principle is restored in a �LAMPeconomy�by responding to output under baseline parameterization.

Finally, in appendix B we show that a version of the Inverted Taylor Principleholds for a contemporary rule also. This is done to further illustrate the di¤erencesof our determinacy results from GLV (2003b), where there is a dramatic distinctionbetween forward-looking and contemporaneous rules. GLV do note (relying uponnumerical simulations and not as a general result) a result similar to our Proposition


1: namely, the Taylor principle may need to be violated for a forward-lookingrule if the share of �rule-of-thumb�consumers is high. But for a contemporaneousrule to be compatible with a unique equilibrium, they argue that the central bankshould respond to in�ation more strongly than in the full-participation economy(and indeed very strongly under some parameter constellations). The message ofour paper in what regards determinacy properties of policy rules is di¤erent: weprovide analytical conditions for an inverted Taylor principle to hold generically,independently on the policy rule followed, and as part of a general theme having todo with LAMP�s in�uence on the aggregate demand elasticity to interest rates. Ourresults for a simple Taylor rule have the same �avor as for a forward-looking rule:in the �non-Keynesian economy� the inverted Taylor principle holds �generically�(i.e. if we exclude some extreme values for some of the parameters) for a somewhatlarger share of non-asset holders than was the case under a forward-looking rule. Itis also the case, as in GLV, that a policy rule responding to current in�ation verystrongly would insure equilibrium uniqueness22. But the implied response (�� = 35under the baseline parameterization): (i) is much larger than any plausible empiricalestimate; (ii) would imply that zero bound on nominal interest rates be violated foreven small de�ations; (iii) would have little credibility. This is in contrast with GLV,who do not consider a possible inversion of the Taylor principle in their numericalanalysis of such rules, but instead argue that for a large share of non-asset holdersmaking the required policy response too strong under a Taylor rule, the centralbank should switch to a passive forward-looking rule.

5. Optimal monetary policy.

The above analysis suggests that, in an economy with limited participationin asset markets, the central bank following an active rule would leave room forsunspot-driven real �uctuations. The size of these �uctuations would depend uponthe size of the sunspot shocks (something impossible to quantify in practice), butthis would unambiguously increase the variances of real variables such as output andin�ation. If such variance is welfare-damaging, it is clear that such policies wouldbe suboptimal since sunspot �uctuations themselves would be welfare-reducing. Incontrast, in the same �non-Keynesian economy�, a passive rule would rule out such�uctuations and would be closer to optimal policy. But well beyond ruling outsunspot �uctuations, the presence of limited asset markets participation is likelyto modify the optimal response to fundamental shocks too. Our next task is tocharacterize optimal policy rules in the presence of non-asset holders.

The objective function is calculated as follows. Following Woodford (2003) weuse a second-order approximation to a convex combination of households�utilities,described in detail in appendix C, where we use the more general CRRA functionalform introduced in Section 3.3, without restricting the coe¢ cient of risk aversion to equal unity. We make a series of assumptions that allow us to use this second-order approximation techniques. Firstly, we assume that e¢ ciency of the steadystate is obtained by appropriate �scal instruments inducing marginal cost pricingin steady state (subsidies for sales at a rate equal to the stead-state net mark-up �nanced by lump-sum taxes on �rms). Since this policy makes steady-state

22This is not the case under a forward looking rule, since there, even in a standard full-participation economy too strong a response leads to indeterminacy - see Bernake and Wodford1997.


pro�t income zero, the steady-state is also equitable: steady-state consumptionshares of the two agents are equal, making aggregation much simpler. This ensuresconsistency with the model outlined above23.

Secondly, we assume that the social planner maximizes the (present discountedvalue of a) convex combination of the utilities of the two types, weighted by themass of agents of each type: Ut (:) � �UH (CH;t; NH;t)+[1� �]US (CS;t; NS;t). Thisis consistent with our view that limited participation in asset markets comes fromconstraints and not preferences, since in the latter case maximizing intertemporallythe utility of non-asset holders would be hard to justify on welfare grounds. How-ever, note that for the discretionary Markov equilibrium studied here, this choicemakes no di¤erence since terms from time t+1 onwards are treated parametricallyin the maximization and the time-t objective function is identical. The followingProposition shows how the objective function can be represented (up to secondorder) as a discounted sum of squared output gap and in�ation (the proof is inappendix C).

Proposition 3. If the steady state of the model in Section 3 is e¢ cient theaggregate welfare function can be approximated by (ignoring terms independent ofpolicy and terms of order higher than 2):

Ut = �UCC2

"

Et

1Xi=t

��x2t+i + �

2t+i

;(5.1)

� � '+

1� � [1� � (1� ) (1 + ')]

":

Note that when � = 0 we are back to the standard case � = ('+ ) " : In thecase studied extensively in the rest of this paper ( = 1); the relative weight onoutput gap is � = 1+'

1�� " and is increasing in the share of non-asset holders. In

general, an increase in the share of non-asset holders leads to an increase in therelative weight on output (if � '

1+' ; which is empirically plausible)24: When �

tends to one, the implicit relative weight on output stabilization tends to in�nity(for ' > 0). Hence, the presence of non-asset holders modi�es the trade-o¤ facedby the monetary authority. The intuition for this result is simple. Since aggregatereal pro�ts can be written as Dt = [1� (MCt=Pt)�t]Yt; relative price dispersion�t (related here linearly to squared in�ation) erodes aggregate pro�t income forgiven levels of output and marginal cost. Given that only a fraction of (1� �)receives pro�t income, when this fraction falls the welfare-based relative weight onin�ation (price dispersion) also falls. In�ation becomes completely irrelevant forwelfare purposes when �! 1: since nobody holds assets, asset income need not bestabilized.

23Note, however, that since steady-state consumption shares are equal we do not need toassume increasing returns. Under these assumptions, the reduced-form coe¢ cients simply modi�yas follows: �o = 1 + ' and �o = 1� '�= (1� �) :

24When this condition is not ful�lled, so < '= (1 + ') ; the relative weight on output gapis decreasing in � and can even become negative when � > [(1� ) (1 + ')]�1 : We exclude thisparameter region on grounds of its being empirically irrelevant.


The optimal discretionary rule frot g10 is found by minimizing �Ut taking as

a constraint the system given by (3:6) and (3:7) and re-optimizing every period25.Note that by usual arguments this equilibrium will be time-consistent. This is,up to interpretation of the solution, isomorphic to the standard problem in CGG(1999). Hence, for brevity, we skip solution details readily available elsewhere andgo to the result:

(5.2) xt = ��

��t:

When in�ation increases (decreases) the central bank has to act in order to contract(expand) demand. Assuming an AR(1) process for the cost-push shock Etut+1 =�uut for simplicity, we obtain the following reduced forms for in�ation and outputfrom the aggregate supply curve:

(5.3) �t = ��ut; xt = ��ut;

where� ��2 + � (1� ��u)

��1: Since � is generally increasing in �, in an economy

with limited asset market participation optimal policy results in greater in�ationvolatility and lower output gap volatility than in a full participation economy (� >0). Optimal policy in this case requires more output stabilization at the cost ofaccommodating in�ationary pressures.

Substituting the expressions given by (5:3) into the IS curve, we obtain theimplicit instrument rule consistent with optimality26:

rot = r�t + �o�Et�t+1;(5.4)

�o� =

�1 +

��

�

1� �u�u

�:

Some of the results obtained in a full-participation economy carry over: from theexistence of a trade-o¤ between in�ation and output stabilization, to convergenceof in�ation to its target under the optimal policy (e.g. CGG (1999) ). Also, realdisturbances a¤ect nominal rates only insofar as they a¤ect the Wicksellian interestrate, as discussed for example by Woodford (2003 p:250). There is one importantexception however, emphasized in the following Proposition.

Proposition 4. In a non-Keynesian economy (� < 0) the implied instrumentrule for optimal policy is passive �o� < 1. The optimal response to in�ation isdecreasing in the share of non-asset holders @�o�=@� < 0 and changes from passiveto active as � changes sign.

This Proposition shows the precise way in which the central bank has to changeits instrument in order to meet the targeting rule (5:2): contract demand whenin�ation increases, but move nominal rates such that the real rate decreases when� is negative. This happens because, as explained previously, real interest rate cutsare associated with a fall in current aggregate demand when the slope of the IScurve is positive (� < 0).

Finally, when cost-push shocks are absent (and there is no in�ation-outputstabilization trade-o¤), the �exible-price allocation can be achieved by having the

25To keep things simple, we focus on the discretionary, and not fully optimal (commitment)solution to the central banker�s problem. This case can be argued to be more realistic in practice,as do CGG (1999).

26A positive policy response to in�ation requires � � �� 1��u�u

:


nominal rate equal the Wicksellian rate at all times rot = r�t ; as in the standardmodel (e.g. Woodford, Ch. 4). However, note an important di¤erence with respectto the baseline model: when � < 0 this policy can also be consistent with a uniquerational expectations equilibrium27. To see this note that such a policy is equivalentfrom an equilibrium-determinacy standpoint to an interest rate peg, i.e. an interestrate rule with �� = 0. From Proposition 1, Case II, one can easily see that �� = 0leads to equilibrium determinacy if and only if 1 + � 2(1+�)� � 028:However, theability of the central bank to achieve full price stability as the unique equilibriumapplies to the simple model assumed here and relies upon the ability/willingnessof the bank to monitor the natural rate of interest and match its movement one-to-one by movements in the nominal rate. Moreover, the natural interest rate cansometimes be negative. All these caveats suggest that this result is unlikely to havemuch practical relevance.

6. The e¤ects of shocks and cyclical implications

In this section we go back to the simple instrument rule and compute an-alytically the e¤ects of fundamental and sunspot shocks under determinacy andindeterminacy, allowing for the change in sign of � due to LAMP. Our interest inthis exercise is twofold. First, it might be of interest in itself to understand thee¤ects of shocks in a determinate non-Keynesian economy. One obvious historicalcandidate for such a case is the pre-Volcker period in the U.S.; it is fairly well es-tablished (see e.g. CGG (2000), Lubik and Schorfheide (2004)) that the responseof monetary policy in that period implied a (long-run) response to in�ation of lessthan one. But if participation in asset markets was so limited that � < 0, thiswould imply that policy was consistent with a unique equilibrium and close to opti-mal. Hence, we are able to assess the e¤ects of fundamental shocks, an impossibletask under indeterminacy29. Secondly, there is the mirror image of the above ar-gument. Estimates of policy rule coe¢ cients in the post-1980 era for the U.S. andother industrialized countries indicate a response of nominal rates to in�ation largerthan one (see e.g. CGG (2000) ). Coupled with the possibility of a non-Keynesianeconomy (� < 0), this would instead imply indeterminacy. Hence, it may be ofinterest to assess the e¤ects of various (fundamental and sunspot) shocks in an in-determinate equilibrium. This may be relevant for countries with underdeveloped�nancial markets that nevertheless pursue an active policy.

6.1. Determinacy. The equilibrium under determinacy (i.e. when Proposi-tion 1 holds) is particularly easy to calculate when all shocks �t have zero persistence(see appendix for derivations). �Expectation errors��t � (xt � Et�1xt; �t � Et�1�t)

0

27In the baseline model, the bank needs to commit to respond to in�ation by ful�lling theTaylor principle rot = r

�t + ��t; �� > 1 in order to pin down a unique equilibrium.

28In terms of deep parameters, this condition translates into � �h1 + 1

1+FY'(1��)(1��)(1+�)(1+��)

i=h1 + 1

1+�'i:

29This avenue is explored in a companion paper (Bilbiie and Straub (2004)), discussed inmore detail in the Introduction, where we estimate the model in this paper for the pre-Volckerand post-Volcker samples. We �nd that asset market participation changed generating a switch inthe sign of the slope of the IS curve, and equilibrium was determinate in the pre-Volcker samplecharacterized by a passive policy rule. We show that fundamental shocks can explain stylizedfacts pertaining to the Great In�ation.


are determined exclusively by fundamental shocks (and sunspot shocks would haveno e¤ect on dynamics) by �t = ��1�t, namely:

(6.1) �t = ��1�1�

�("t � r�t ) +

�01

�ut

The initial impact on output and in�ation is also given by the same expression.Since both roots are eliminated under determinacy, there is no endogenous persis-tence. There are sharp di¤erences for the two sub-cases identi�ed above, showingasymmetric e¤ects of some shocks depending on the sign of �. In the Keynesiancase identi�ed in Proposition 1, the e¤ects are of usual sign, but of di¤erent mag-nitude than in the full-participation case. A policy-induced interest rate cut oran increase in the natural rate of interest (coming e.g. from shocks to technologygrowth) increase both the output gap and in�ation. These e¤ects are stronger, thehigher the share of non-asset holders � (and hence the higher ��1). This can beeasily understood using Figure 1 or the WN-LS diagram in Figure 3. One-timecost-push shocks have no e¤ect on the output gap, and increase in�ation one-to-one; this is only because the interest rate rule responds to expected future in�ation,whereas a one-time shock increases only in�ation today. In the non-Keynesian caseof Proposition 1, and in contrast to the standard case, a monetary contraction(positive "t) has expansionary e¤ects, and causes in�ation. This follows directlyby the mechanism discussed in detail above. An increase in the natural rate ofinterest driven by technology results in a recession and de�ation. It is clear thenthat a policy response increasing the nominal rate by more than the natural rate"t > r�t increases both output and in�ation, whereas when it falls short of doingso, it has de�ationary e¤ects, and causes a fall in output. These e¤ects diminishas � tends to 1, since ��1 tends to zero. Cost-push shocks have the same e¤ectsregardless of the sign of � due to the zero-persistence assumption. However, in thepresence of persistence the magnitude of the responses to these shocks will dependon �, since in that case the roots of the system matter for dynamics; but the sign ofthe response is unaltered: cost-push shocks always generate an increase in in�ationand a fall in the output gap, since marginal cost increases (see Bilbiie and Straub(2004) for some simulations).

Cyclical implications.While evidence overwhelmingly suggests that pro�ts are procyclical (see Rotem-

berg and Woodford (1999)), the mechanism underlying our results might seem torely on countercyclical pro�ts. In this subsection we brie�y show that this is notnecessarily the case. First, note that the condition for pro�ts to be procyclicaldd=dy > 0 is (see Appendix): dx < ��(1+�)du

�(1+�)+(1+�)du��dy�; where we dropped time

subscripts, and the right-hand side term is exogenous and depends on technology.When shocks have zero persistence, we know the solution for output gap from (6:1)as: dx = ��1d ("� r�) < ��(1+�)du

�(1+�)+(1+�)du��dy�: Without a cost-push shock, this

condition becomes ��1d ("� r�) < [�= (� (1 + �)� �)] dy�: We can hence see anexample whereby ��1 < 0 satis�es this condition and leads to procyclical pro�ts.Namely, if the shock to technology is such that dy� > 0; but the policy responseis such that d" < dr� pro�ts are always procyclical in the non-Keynesian economy,and countercyclical otherwise. Alternatively, the same is true if there is no shockto technology (dy� = 0) and d" < 0:


Moreover, we can assess how relative pro�t cyclicality depends on the degreeof asset market participation30. We take two economies with di¤erent participationrates and ask under what conditions does one have more procyclical pro�ts thanthe other (denoted by superscript 0):

dddy >

dd0

dy0 ,dmcdy < dmc0

dy0 , dy�

dy > dy�

dy0 , dx < dx0: For the zero-persistence

case again this becomes: ��1d ("� r�) > ��10 d ("� r�) : Hence, if d ("� r�) > 0,the condition for dd=dy > dd0=dy0 is ��1 > ��10 . This means that in case the shockscon�guration is such that the policy response exceeds the natural rate, a Keynesianeconomy has more procyclical pro�ts the higher its � (��1 is increasing in �). A non-Keynesian economy ��1 < 0 has always less procyclical pro�ts than a Keynesianone (but getting more and more procyclical as � increases). However, when thepolicy response falls short of the natural rate, the opposite holds: ��1 < ��10 .A non-Keynesian economy will hence always have more procyclical pro�ts thana Keynesian one. Finally, note that if the source of �uctuations is only a cost-push shock the condition is du

dy < dudy0 so dy > dy0: This can easily be the case

when the shock persistence is di¤erent than zero (note that with zero persistencethe response of output is zero). For instance, under the optimal policy solutioncalculated in (5.3), the response of output to a cost shock is always larger (lessnegative) when � is larger; this implies that pro�ts are more procyclical the largeris �:Similar reasoning applies to the cyclicality of real wage, noting that moreprocyclical pro�ts imply less procyclical real wage.

6.2. Indeterminacy. In this case one of the roots q� will be inside the unitcircle. Sunspot shocks have real e¤ects, and the responses to fundamental shockschange too in a way made explicit below. We con�ne ourselves to the case wherebythe smaller root is inside the unit circle and the larger one is greater than one, i.e.q� 2 (�1; 1) and q+ > 1: This can be shown to be the case if either (i) � > 0; �� < 1or (ii) � < 0; �� > 131. Since in this case there is one-dimensional indeterminacy,the stability condition for (D:1) modi�es: expectation errors are not spanned byfundamental shocks, but by both fundamental and sunspot shocks.

We can apply the results in Proposition 1 in Lubik and Schorfheide (2003) tosolve for the full solution set for the expectation errors. This is described in somedetail in the Appendix, and the solution is:

�t = ��1

d2

��q+1� q+

�("t � r�t ) +

1

d2

��1 (1� q+)

(1� q+)�q� � ��1

� �ut ++1

d

�q+ � 1�q+

�(M1�t + &

�t ) ;(6.2)

where M1 is an arbitrary 2�2 matrix, d > 0 is de�ned in the Appendix and &�t is areduced-form sunspot shock, which can be interpreted as a belief-induced increase

30This is especially important, since an economy with limited participation can imply moreprocyclical pro�ts than an economy with full participation without necessarily implying procyclicalpro�ts. That is because in a standard sticky-price model pro�ts can be strongly counter-cyclical,unless one introduces labor hoarding, variable utilization, or other features meant to break thelink between markup and pro�ts (see Rotemberg and Woodford 1999).

31For the rest of the parameter regions where there is indeterminacy we would have q+ 2(�1; 1) and q� < �1; but this can be shown to imply very restrictive conditions on the deepparameters and the policy rule coe¢ cient.


in output and/or in�ation of undetermined size. First thing to note is that a posi-tive realization of this shock will increase output and in�ation no matter whether� 7 0 since q+ > 1 as established above. This conforms our intuitive constructionof sunspot equilibria when discussing determinacy properties of interest-rate rules.On the other hand, the e¤ects of fundamental shocks become ambiguous, and de-pend crucially upon the choice of the M1 matrix:Unfortunately, there is nothing topin down a choice for this matrix, which captures a well-known problem of inde-terminate equilibria - the e¤ects of fundamental shocks cannot be studied withoutfurther restrictions. Two leading possibilities to restrictM1 are suggested by Lubikand Schorfheide: orthogonality and continuity.

Under orthogonality, the two sets of shocks are orthogonal in their contri-bution to the forecast error, and hence M1 = 0 in (6.2). The e¤ect of a cost-pushshock is of the same sign under either scenario, as it is independent of �: a positiverealization of this shock would increase in�ation (since (1� q+)

�q� � ��1

�> 0)

and decrease output (q+ > 1). The e¤ects of policy shocks, and of shocks to thenatural rate of interest, are again di¤erent depending on which case we consider. Ineither case, the overall e¤ect on in�ation and the output gap depends on whetherthe policy response "t is stronger or weaker than the variation in the natural rater�t . In the Keynesian case, an interest rate increase larger than the natural rate"t > r�t decreases output under its natural level but causes in�ation as 1� q+ < 0(this is also found by Lubik and Schorfheide for a contemporaneous rule). In the�Non-Keynesian economy�, the same shock boosts output and causes de�ation.

In order to preserve continuity of the impulse responses to the fundamentalshock when passing from determinacy to indeterminacy,M1 can be chosen such thatit makes the response to the fundamental shocks �t under indeterminacy identical tothose under determinacy (6.2), implying that the partial e¤ect of sunspot shocks &�ton expectation errors �t is d

�1 � q+ � 1; �q+�0. While continuity is an attractive

feature, there is nothing to insure that the M1 takes exactly the form necessary toget this result.

7. Redistribution restores Keynesian logic.

The mechanism of all the previous results relies on the interaction betweenlabor and asset markets, namely income e¤ects on labor supply of asset holdersfrom the return on shares. This hints to an obvious way to restore Keynesianlogic relying on a speci�c �scal policy rule that shuts o¤ this channel: tax divi-dend income and redistribute proceedings as transfers to non-asset holders. Wefocus on the non-Keynesian case whereby in the absence of �scal policy � < 0: Tomake this point, consider the following simpli�ed �scal rule: pro�ts are taxed atrate �Dt and the budget is balanced period-by-period, with total tax income �

Dt Dt

being distributed lump-sum to all non-asset holders. We focus on the case wherepro�ts are zero in steady-state. The balanced-budget rule then is �Dt Dt = �LH;twhich around the steady state (both pro�ts and transfers are shares of steady-state GDP, lH;t � LH;t=Y ) is approximately: �lH;t = �Ddt. Replacing thisinto the new budget constraint of non-asset holders we get cH;t = wt +

�D

� dt.Asset holders� consumption will then be given by (substituting the expressionfor pro�ts): cS;t = (1� �)�1

�1� �D�= (1 + �)

�yt +

��D � �

�= (1� �)

�wt: The

wage-hours locus is then obtained following the same method as before: wt =

28 FLORIN O. BILBIIE�1� �D

��1 �1 + '+ �

�1� �D

��nt: Finally, consumption of asset holders as a func-

tion of total output is:

cS;t = ��yt; where �� =1

1� �D

�1� �D �

1 + �+�D � �1� �

'

1 + �

�:

For a given � there exists a minimum threshold for the tax rate such that�� > 0 when in the absence of such �scal policy � < 0: This threshold is (note that'= (1� �)� � > 0 where � < 0):

1 > �D > 1� 1 + '

(1� �)�1 '� �.

The necessary tax rate is higher, the more inelastic is labor supply and thehigher the share of agents with no assets. The intuition for this result is straight-forward: a higher tax rate on asset income eliminates some of the income e¤ect ofdividend variation on asset holders�labor supply.

8. Conclusions

The above analysis has shown how limited asset markets participation (LAMP),by changing aggregate demand�s sensitivity to interest rates nonlinearly, changesmonetary policy prescriptions likewise. Despite their insensitivity to real interestrates, non-asset holders a¤ect the sensitivity of aggregate demand to interest ratessince these agents are oversensitive to real wage variations. Real wages are relatedto interest rates through the labor supply decision of asset holders and the waythis interacts with their asset holdings (through income e¤ects and intertemporalsubstitution). Their asset income, in turn, consists of dividend income and isalso related to real wages which are equal to marginal costs. Therefore, non-assetholders and asset holders interact through the interdependent functioning of laborand asset markets. These interactions can either strengthen (if participation is not�too�limited) or overturn the systematic link between interest rates on aggregatedemand. The latter case occurs if the share of non-asset holders is high enoughand/or and the elasticity of labor supply is low enough. This is the main mechanismidenti�ed by this paper to make monetary policy analysis dramatically di¤erentwhen compared to a standard full-participation case whereby aggregate demand iscompletely driven by asset holders. This paper develops an analytical frameworkincorporating the foregoing insight and uses it to study in detail the dynamics of asimple general equilibrium model, the determinacy properties of interest rate rulesand optimal, welfare-based, monetary policy. Its aim is to make a contribution tothe literature emphasizing the role of LAMP in shaping macroeconomic policy andhelping towards a better understanding of the economy. In that respect, we justseek to add to a new developing literature analyzing the role of non-asset holdersin macroeconomic dynamic general equilibrium models (see Mankiw (2000), Galí,Lopez-Salido and Valles (2002) or Alvarez, Lucas and Weber (2001)).

Our results have clear normative implications. In a nutshell, central bank pol-icy should be pursued with an eye to the aggregate demand side of the economy.Empirical results on consumption behavior and asset market participation, on theone hand, and labor supply elasticity and the degree of monopoly power in goodsmarkets, on the other, would become an important part of the policy input. Whilethe degree of development of �nancial markets may well make this not a concernin present times in the developed economies, central banks in developing countries


with low participation in �nancial markets might �nd this of practical interest. Thetheoretical results hinting to such policy prescriptions are related to limited partici-pation beyond a certain threshold making the economy behave in a �non-Keynesian�way. Namely, the IS curve changes sign, and an Inverted Taylor Principle appliesgenerally: the central bank needs to adopt a passive policy rule to ensure equilib-rium uniqueness and rule out self-ful�lling, sunspot-driven �uctuations. Moreover,optimal time-consistent monetary policy also requires that the central bank movenominal rates such that real rates decline (thereby containing aggregate demand).The e¤ects and transmission of shocks are also radically modi�ed.

In this paper we modelled limited asset market participation in a very simpleway, and were as a result able to isolate and study its implications on monetary pol-icy analytically in the same type of framework used in standard, full-participationanalyses. This simplicity (shared with the rest of the literature), while justi�ed ontractability grounds, also implies that many realistic features have been left out. Forinstance, one could try to break the link between asset markets participation andconsumption smoothing behavior, which this paper assumed. Another importantextension would endogenize the decision to participate in asset markets. Lastly, anempirical assessment of limited participation, its dynamics and implications at themacroeconomic aggregate level, is in our view a necessary step for understandingbusiness cycles, which we pursue in current work.

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LIMITED ASSET MARKETS PARTICIPATION AND MONETARY POLICY. A-1

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A-2 FLORIN O. BILBIIE

Table 1. Model Summary

Euler equation, S EtcS;t+1 � cS;t = rt � Et�t+1Labor supply, S 'nS;t = wt � cS;tBudget constraint S cS;t = (wt + nS;t) +

11��dt:

Labor supply, H 'nH;t = wt � cH;tBudget constraint, H cH;t = wt + nH;tProduction function yt = (1 + �)nt + (1 + �) atReal marginal cost mct = wt � at;Real pro�ts dt = �mct + �

1+�yt;

Phillips curve �t = �Et�t+1 + mct; � (1� �) (1� ��) =�Labor mkt. clearing nt = �nH;t + (1� �)nS;tAggregate cons. ct = �cH;t + (1� �) cS;t:Monetary policy rt = ��Et�t+1 + �xxt + "t:Note: By Walras�law the resource constraint yt = ct also holds

Appendix A. Limited participation due to transaction costs: anexample.

The purpose of this Appendix is to show that our assumption on limited partic-ipation in asset markets can be supported by the presence of heterogenous transac-tion costs. Our model draws on a large literature on asset-pricing with transactioncosts, and its purpose is merely to show how for a given level of the non-participationrate �; there exists a structure of costs that supports it. Such costs have beenemphasized by many as one important explanation for the observed participationstructures (see, e.g., Vissing-Jorgensen (2003) for empirical estimates). This liter-ature originates with Cochrane [1989] ; who showed that the foregone utility gainsfrom consuming one�s income as opposed to following a permanent-income decisionrule are likely to be very small (10 cents to 1 dollar per quarter); consequently, thelower bound on transaction costs preventing an agent from following a permanent-income rule (e.g. participating in asset markets) is likely to be small. He andModest (1995) study the role of market frictions, including proportional transac-tion costs, in reconciling asset market equilibrium with data on consumption andasset returns. Using data from the Survey of Consumer Finances, Mulligan andSala-i-Martin (2002) estimate that the median per-period transaction cost for anyinterest-bearing asset is 111 dollars per year.

Suppose that each time the household goes to the asset market it has to paya proportional transaction cost which is household-speci�c kj : The optimality con-dition under proportional transaction costs is (where the second inequality holdsfor situation in which the household sells the asset short, i.e. brings consumptionforward - see also He and Modest):�

1 + kj�> Et

h�jt;t+1R

at+1

i> 1

(1 + kj)

where �jt;t+1 � �PtUC (Cj;t+1) = [UC (Cj;t)Pt+1] is the stochastic discount factor ofhousehold j and Rat+1 is the expected gross return of asset a: Suppose for simplicitythat there are two types of agents: one type indexed by S faces no transaction costskS = 0 while the other indexed by H has to pay a proportional cost kH = k for


each transaction. Such an extreme, bimodal distribution of costs is not necessary,but makes aggregation simpler and is enough for our point. Notably, in equilibriumno cost will be paid since agents who participate face a cost of zero, and agentswho would have to pay the cost choose not to pay it. As in the model of Section2, consumption of type-S agents obeys a standard Euler equation, which for theriskless bond reads:

(A.1)1

Rt= Et

��St;t+1

�Type-H agents face the following optimality condition:

(1 + k)1

Rt> Et

��Ht;t+1

�> 1

(1 + k)

1

Rt

Substituting forRt from (A:1) we have: 1+k > Et��Ht;t+1

�=Et

��St;t+1

�> (1 + k)�1 ;

which with log utility becomes:

1 + k >CH;tEt

hC�1H;t+1

iCS;tEt

hC�1S;t+1

i > 1

1 + k

We will look for a minimum level of the proportional transaction cost �k that makesthe measure of households at a corner solution holding no assets (for which the aboveis a strict inequality) be precisely �. Assuming log-normality and homoskedasticity,we can approximate to second order this lower bound on costs by

(A.2) �k ��Et�cS;t+1 � Et�cH;t+1 + 12 ��2H � �2S�

�� > 0;where �2j = var (cj;t+1 � Etcj;t+1) and lowercase letters denote logs. Note thatsince consumption growth is stationary, �k is bounded above (applying the triangleinequality to (A:2)).

Equation (A:2) can be compared to data as follows. Given an observed non-participation rate � and time series on consumption of asset-holders and non-assetholders cS;t; cH;t , one can compute the value of the cost that would explain thisparticipation structure. This value can then be compared to actual transactioncosts. However, measurement issues abound related both to classifying householdsrelative to their asset-holding status and to �nding an appropriate measure of thecost. Indeed, most costs that prevent an agent hold any asset at all are likely to benon-pecuniary and related to information imperfections, time spent understandingthe way asset markets work, etc. An alternative route is to solve for the momentsinvolved in (A:2) in a dynamic general equilibrium framework as functions of fun-damental shocks, for a given level of non-participation �: Then, for the assumed� there exists a minimum level of the cost �k rationalizing it that can be found bysolving (A:2) :

Finally, note that as regards shareholding, the cost that prevents householdsparticipate from the stock market needs to be larger than �k; due to the exis-tence of an equity premium. In the framework of Section 3, the Euler equa-tion for shares of agents who face a zero cost is 1 = Et

�RAt+1�

St;t+1

�; where

RAt+1 � (Vt+1 + Pt+1Dt+1) =Vt is the gross return on shares. Agents facing a cost

kA choose not to hold shares H;t = 0 i¤ 1 + kA > Et�RAt+1�

Ht;t+1

�>�1 + kA

��1:


Taking second-order approximations under assumptions of joint conditional lognor-mality and homoskedasticity as above yields a lower bound for the transaction costin the stock market:

kA ��Et�cS;t+1 � Et�cH;t+1 + 12 ��2H � �2S�+ �AS � �AH

��where �Aj = cov

�rAt+1 � EtrAt+1; cj;t+1 � Etcj;t+1

�can be computed from the gen-

eral equilibrium model, as before. The bottomline is that a certain level of non-participation � can be rationalized by proportional transaction costs �k; kA.

Appendix B. Proof of Proposition 1

Necessary and su¢ cient conditions for determinacy are as follows (given inWoodford (2003), Appendix to Chapter 4). EitherCase A: (Aa) det� > 1; (Ab) det��tr� > � 1 and (Ac) det� + tr� > � 1 or Case B: (Ba) det� � tr� < � 1 and(Bb) det� + tr� < � 1: For our forward -looking rule case, the determinant andtrace are:

det� = ��1 > 1(B.1)

tr� = 1 + ��1 � ��1��1� (�� 1)

Imposing the determinacy conditions in Case A above (where Case B can be ruledout due to sign restrictions), we obtain the requirement for equilibrium uniqueness:

��1 (�� 1) 2�0;2 (1 + �)

�

�This implies the two cases in Proposition 1: Case I: � > 0; �� 2

�1; 1 + � 2(1+�)�

�;

which is a non-empty interval; Case II: � < 0; �� 2�1 + � 2(1+�)� ; 1

�. Notice that (i)

1+ 2� (1 + �) =� < 1 so the interval is non-empty; (ii) 1+ 2� (1 + �) =� > 0 impliesinstead that we can rule out an interest rate peg, whereas a peg is consistent witha unique REE for 1 + � 2(1+�)� < 0: The last condition instead holds if and only if

� ��1 + 1

1+�'(1��)(1��)(1+�)(1+��)

�=�1 + 1

1+�'��1 + 1

1+�'��1

.

When this condition is not ful�lled, we have 0 < 1 + � 2(1+�)k < 1; so there

still exist policy rules �� 2�1 + � 2(1+�)k ; 1

�bringing about a unique rational ex-

pectations equilibrium. But in this case an interest rate peg, and any policy rule

with too weak a response �� 2h0; 1 + � 2(1+�)k

�is not compatible with a unique

equilibrium.

B.1. Determinacy properties of a simple Taylor rule. We consider rulesof the form:

(B.2) rt = ��t + "t

Replacing this in the IS equation and using the same method as previously weobtain the following Proposition.

Proposition 5. An interest rate rule such as (B:2) delivers a unique rationalexpectations equilibrium if and only if:


Case I: If � > 0; �� > 1 (the �Taylor Principle�)Case II: If � < 0;

�� 2�0;min

�1; �

� � 1�

; ��2 (1 + �)

�� 1��

[�max

�1; �

�2 (1 + �)�

� 1�;1�

It turns out that the �inverted Taylor principle�holds in Case II for a somewhatlarger share of non-asset holders than was the case under a forward-looking rule.

Proof. Substituting the Taylor rule in the IS equation and writing the dy-namic system in the usual way for the zt � (yt; �t)0 vector of endogenous variablesand the �t � ("t � r�t ; ut)

0 vector of disturbances :

Etzt+1 = �zt +�t

The coe¢ cient matrices are given by:

� =

�1 + ��1��1� ��1

�� 1

��1� ��1

�and =

��1 0

0 ��1�

Determinacy requires that both eigenvalues of � be outside the unit circle. Notethat:

det� = ��1�1 + ��1��

�and tr�=1 + ��1

�1 + ��1�

�For Case A we have: (Aa) implies:

��1�� >� � 1�

(Ab) implies��1 (�� 1) > 0

(Ac) implies

��1 (1 + ��) >�2 (1 + �)

�The determinacy requirements are as follows. First, note that (Ab) merely requiresthat ��1 and (�� 1) have the same sign. Hence, we can distinguish two cases:

Case I: ��1 > 0; �� > 1. The standard case is encompassed here and the Taylorprinciple is at work as one would expect. The other conditions are automaticallysatis�ed, since both ��1�� and �

�1 (1 + ��) are positive, and��1� ; �2(1+�)� < 0:

Case II: ��1 < 0; �� < 1: Condition (Aa) implies (note that since � < 0

the right-hand quantity will be positive): �� < � ��1� : The third requirement for

uniqueness (Ac) implies: �� < ��2(1+�)� � 1: Since �� 0; this last requirement

implies a further condition on the parameter space, namely ��2(1+�)� � 1 � 0:

Overall, the requirement for determinacy when ��1 < 0 is hence:

(B.3) 0 � �� < min

�1; �

� � 1�

; ��2 (1 + �)

�� 1�

Case B, instead, involves ful�lment of the following conditions: (Ba) implies ��1 (�� 1) <0 and (Bb) implies ��1 (1 + ��) <

�2(1+�)� : In Case I, whereby ��1 > 0; these con-

ditions cannot be ful�lled due to sign restrictions (this is the case in a standardeconomy as in Woodford (2003), e.g.). In Case II however, the two conditionsimply:

(B.4) �� > max

�1; �

�2 (1 + �)�

� 1�


(B:3) and (B:4) together imply the following overall determinacy condition for thepolicy parameter:

�� 2�0;min

�1; �

� � 1�

; ��2 (1 + �)

�� 1��

[�max

�1; �

�2 (1 + �)�

� 1�;1��

To assess the magnitude of policy responses needed for determinacy as a func-tion of deep parameters, we can distinguish a few cases for di¤erent parameterregions (note that we are always looking at the subspace whereby ��1 < 0):

Share of non-asset holders Determinacy condition� < ��1 �� > 1

� 2��1; ��2

�� 2

h0; ��2(1+�)� � 1

�[ (1;1)

� 2��2; ��3

�� 2

h0; � ��1�

�[ (1;1)

� 2��3; ��4

�� 2

h0; � ��1�

�[��2(1+�)� � 1;1

�� 2

��4; 1

�� 2 [0; 1) [

��2(1+�)� � 1;1

�where

��i =

�1 +

1

1 + �'(1� �) (1� ��)

hi (�)

�=

�1 +

1

1 + �'

�h1 (�) = (1 + �) (1 + ��) ;h2 (�) = 1 + ��

2 + 2��;h3 (�) = 1 + ��2;h4 (�) = 1� ��2

Fig.4: Threshold value for share of non-asset holders making determinacyconditions closest to Inverted Taylor Principle.

We plot the last case � 2��4; 1

�;�� 2 [0; 1) [

��2(1+�)� � 1;1

�in Figure

4 above, where the region above the curve and below the horizontal line givesparameter combinations compatible with the above condition. The di¤erent curvescorrespond to di¤erent labor supply elasticities (' = 1 dotted line and ' = 10thick line). In view of usual estimates of � in the literature (e.g. 0.4-0.5 Campbelland Mankiw (1989)) we shall consider this case as the most plausible. Wheneverthese parameter restrictions are met, determinacy is insured by either a violationof the Taylor principle, or for a strong response to in�ation. However, note thatthe lower bound on the in�ation coe¢ cient then becomes very large (35: 433 underthe baseline), which is far from any empirical estimates. Indeed, the thresholdin�ation coe¢ cient is sharply increasing in the share of non-asset holders and inverse


elasticity of labor supply, as can be seen by merely di¤erentiating ��2(1+�)� �1 withrespect to these parameters.

Appendix C. Proof of Proposition 3: derivation of aggregate welfarefunction

Assume the steady-state is e¢ cient and equitable, in the sense that consump-tion shares across agents are equalized. This is ensured by having a �scal authoritytax/subsidize sales at the constant rate � and redistribute the proceedings in alump-sum fashion T such that in steady-state there is marginal cost pricing, andpro�ts are zero. The pro�t function becomes Dt (i) = (1� �) [Pt(i)=Pt]Yt(i) �(MCt=Pt)Nt(i) + Tt; where by balanced budget Tt = �Pt(i)Yt(i). E¢ ciency re-quires � = ��; such that under �exible prices P �t (i) = MC�t and hence pro�ts areD�t (i) = 0: Note that under sticky prices pro�ts will not be zero since the mark-up

is not constant. Under this assumption we have that in steady-state:

V 0 (NH)

U 0 (CH)=V 0 (NS)

U 0 (CS)=W

P= 1 =

Y

N;

where NH = NS = N = Y and CH = CS = C = Y:Suppose further that the social planner maximizes a convex combination of the

utilities of the two types, weighted by the mass of agents of each type: Ut (:) ��UH (CH;t; NH;t) + [1� �]US (CS;t; NS;t). A second-order approximation to typej0s utility around the e¢ cient �ex-price equilibrium delivers:

Uj;t � Uj (Cj;t; Nj;t)� Uj�C�j;t; N

�j;t

�=

= UCCj

�Cj;t +

1� 2

C2j;t + (1� ) c�j;tCj;t��

�VNNj�Nj;t +

1 + '

2N2j;t + (1 + ')n

�j;tNj;t

�+ t:i:p+O

�k � k3

�;

where variables with a hat denote log-deviations from the �ex-price level (or �gaps�)Ct � log Ct

C�t= ct � c�t , and variables with a star �ex-price values as above c

�t �

logC�t

C . Note that since UCCH = UCCS and VNNH = VNNS and using c�H;t =c�S;t = c�t (which holds since asset income in the �ex-price equilibrium is zero, aspro�ts are zero) we can aggregate the above into:

Ut = UCC

�Ct + (1� ) c�t Ct +

1� 2

��C2H;t + (1� �) C2S;t

��

�VNN�Nt + (1 + ')n

�t Nt +

1 + '

2

��N2

H;t + (1� �) N2S;t

��+ t:i:p+O

�k � k3

�Note that Ct = Yt and Nt = Yt +�t; where �t is price dispersion as in Woodford(2003) ; �t = log

R 10(Pt (i) =Pt)

�"di: Since UCC = VNN we can show that the

linear term boils down to:

UCChCt + (1� ) c�t Ct

i� VNN

hNt + (1 + ')n

�t Nt

i= �UCC

h�Yt + ( � 1) c�t Yt + Yt +�t + (1 + ')n�t Yt

i+O

�k � k3

�= �UCC [�t] +O

�k � k3

�


Quadratic terms can be expressed as a function of aggregate output. For that pur-pose, note that in evaluating quadratic terms we can use �rst-order approximationsof the optimality conditions (higher order terms would imply terms of order higherthan 2, irrelevant for a second-order approximation). Up to �rst order, we havethat CH;t = (1 + �) Wt; NH;t = �Wt and Wt = 'Nt+(1� ) Ct = ('+ ) Yt+'�tso:

C2H;t = (1 + ')2Y 2t +O

�k � k3

�N2H;t = (1� )2 Y 2t +O

�k � k3

�C2S;t =

1

(1� �)2[1� � (1 + ')]2 Y 2t +O

�k � k3

�N2S;t =

1

(1� �)2[1� � (1� )]2 Y 2t +O

�k � k3

�The aggregate per-period welfare function is thence, up to second order (ignoringterms independent of policy and of order larger than 2):

Ut = �UCC� � 12

��C2H;t + (1� �) C2S;t

�+1 + '

2

��N2

H;t + (1� �) N2S;t

�+�t

�= �UCC

�1

2

'+

1� � [1� � (1� ) (1 + ')] Y2t +�t

�The intertemporal objective function of the planner will hence be Ut=

P1i=t Ut+i:

This is consistent with our view that limited participation in asset markets comesfrom constraints and not preferences. In the latter case, maximizing intertemporallythe utility of non-asset holders would be hard to justify on welfare grounds. How-ever, note that for the discretionary (Markov) equilibrium studied here, this choicemakes no di¤erence since terms from time t+1 onwards are treated parametricallyin the maximization and the time-t objective function is identical and equal to Ut.By usual arguments readily available elsewhere (see Woodford (2003), Galí andMonacelli (2004)) we can express price dispersion as a function of the cross-sectionvariance of relative prices: �t = ("=2)V ari (Pt (i) =Pt) and the (present discountedvalue of the) cross-variance of relative prices as a function of the in�ation rateP1t=0 �

tV ari (Pt (i) =Pt) = �1P1t=0 �

t�2t . Hence, the present discounted values ofprice dispersion and the in�ation rate are related by

P1t=0 �

t�t = ("=2 )P1t=0 �

t�2tand the intertemporal objective function becomes (5:1) in Proposition 3 (reintro-ducing the notation Yt = xt).

Appendix D. Cyclical implications and sunspot equilibria

We follow the new method proposed by Lubik and Schorfheide (2003) to com-pute sunspot equilibria by decomposing expectation errors. The (3:7)-(3.6) systemcan be written, in terms of newly de�ned variables (for k = x; �) �kt � Etkt+1 andexpectation errors �kt � kt � Et�1kt :

�t = ��t�1 +�t + ��t

where �t � (�xt ; ��t )0 and �t � (�xt ; ��t )

0: The coe¢ cient matrices �; are still the

same given in (4.2). We replace � by its Jordan decomposition � = JQJ�1, de�ne


the auxiliary variables zt = J�1�t and rewrite the above model as

(D.1) zt = Qzt�1 + J�1"t + J

�1��t

The eigenvalues of � are:

q� =1

2

�tr��

q(tr�)

2 � 4 det ��;

(where the determinant and trace are det� = ��1 > 1; tr�=1+��1��1��1� (�� 1)).The corresponding eigenvectors are stacked in the J matrix:

J =

�1� (1� �q�)

1� (1� �q+)

1 1

�The equilibrium under determinacy is particularly easy to calculate since whenshocks have zero persistence the only stable solution is �t = 0; obtained for: �t+��t = 0: Hence, the expectation errors are determined exclusively by fundamentalshocks (and sunspot shocks would have no e¤ect on dynamics) by �t = ��1�t,namely:

(D.2) �t = ��1�1�

�("t � r�t ) +

�01

�ut

The condition for pro�ts cyclicality used in text is derived as follows, using theexpression for pro�ts and the relationship between marginal cost and output gap:dddy > 0,

�1+� �

dmcdy > 0, �dxdy +

dudy <

�1+� , �

�1� dy�

dy

�+ dudy <

�1+� , �dy

�

dy >

� � �1+� +

dudy

dy>0, dy < �(1+�)�(1+�)+(1+�)du��dy

�: In terms of output gap: dx <��(1+�)du

�(1+�)+(1+�)du��dy� where the right-hand side term is exogenous and depends on

technology.The stability condition in the case of indeterminacy is - see Lubik and Schorfheide

(2003), p. 278 (where [A]2: denotes the second row of the A matrix, attached tothe explosive component):�

J�1�2:�t +

�J�1�

�2:�t = 0

Straightforward algebra to calculate�J�1

�2:

=1

q+ � q��1 ��1 � q�

��J�1�

�2:

=1

q+ � q��q+ q+ � 1

�delivers the stability condition as:

��1 ("t � r�t ) +��1 � q�

�ut � �q+�yt + (q+ � 1) ��t = 0

Since only one root is suppressed, there is endogenous persistency of the e¤ects ofshock (which was not the case under determinacy).

Following Lubik and Schorfheide (2003) we compute a singular value decom-position of (q+ � q�)

�J�1�

�2::�

J�1��2:

= 1 ��d 0

�� q+

dq+�1d

q+�1d

�q+d

�d =

q(�q+)

2+ (q+ � 1)2


Using also � (q+ � q)�J�1

�2:=��1 ��1 � q�

�we get the full set of stable

solutions as described in text.

Limited Asset Markets Participation, Monetary Policy and ...

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