optimal monetary policy and financial stability in a non-ricardian

OPTIMAL MONETARY POLICY ANDFINANCIAL STABILITY IN A NON-RICARDIANECONOMY

Salvatore NisticòSapienza Università di Roma

AbstractI present a model with Discontinuous Asset Market Participation (DAMP), where all agents arenon-Ricardian, and where heterogeneity among market participants implies �nancial-wealth effectson aggregate consumption. The implied welfare criterion shows that �nancial stability arises as anadditional and independent target, besides in�ation and output stability. Evaluation of optimal policyunder discretion and commitment reveals that price stability may no longer be optimal, even absentinef�cient supply shocks: some �uctuations in output and in�ation may be optimal as long as theyreduce �nancial instability. Ignoring the heterogeneity among market participants may lead monetarypolicy to induce substantially higher welfare losses. (JEL: E12, E44, E52)

Keywords: DAMP, Optimal Monetary Policy, Perpetual Youth, Financial Stability, DSGE Model,Asset Prices, LAMP.

1. Introduction

The burst of the “dotcom” bubble in 2001 and the recent global �nancial meltdownhighlighted that developments in asset markets can be of great relevance for thebusiness cycle, and revived a theoretical debate about the desirability that central banksbe directly concerned with �nancial stability. To evaluate this concern from a welfareperspective, I develop a tractable dynamic stochastic general equilibrium model of anon-Ricardian economy populated by heterogeneous agents that stochastically cyclebetween zones of activity and inactivity in asset markets.

The editor in charge of this paper was Fabio Canova.

Acknowledgments: I thank the editor and four anonymous referees for their comments on previous versionsof the paper. I also wish to thank Pierpaolo Benigno, Nicola Borri, Efrem Castelnuovo, Andrea Colciago,Anastasios Karantounias, Federico Nucera, Giorgio Primiceri, Gianluca Violante, Joseph Zeira, seminarparticipants at LUISS Guido Carli, Università di Milano “Bicocca”, Università “Cattolica” di Milano, theECB and the CEF2011, San Francisco, for helpful comments and discussions. I also wish to thank theEconomics Department at NYU for kind hospitality in the spring semester 2009, during which this projectwas initiated. Any remaining errors are, of course, my sole responsibility. PRIN2008 �nancial support isgratefully acknowledged. An earlier version of this paper was circulated under the title “Optimal monetarypolicy and stock-price dynamics in a non-ricardian DSGE model”, CASMEF Working Paper n.1107,November 2011. Address: Sapienza Università di Roma, Dipartimento di Scienze Sociali ed Economiche,piazzale Aldo Moro 5, 00185 Rome, Italy.

E-mail: [email protected]

Journal of the European Economic AssociationPreprint prepared on 9 March 2016 using jeea.cls v1.0.

Nisticò Optimal monetary policy and �nancial stability in a non-ricardian economy 2

In particular, the economy is populated by ex-ante identical in�nitely-lived agentsthat face discontinuous asset market participation (DAMP): at the beginning of eachperiod, each agent �nds out her current status in asset markets. If inactive, she behavesas a “rule-of-thumber”, consuming only her disposable labor income. If active, shetrades in asset markets to smooth consumption, but subject to a �nite planning horizon,as in each period she faces a constant probability of switching to the inactive type.This economy is therefore non-Ricardian for two reasons: “rule-of-thumbers” arenon-Ricardian because they do not smooth consumption in asset markets, “marketparticipants” because they face a �nite planning horizon, as in a “perpetual-youth”model.

The model therefore generalizes the limited asset market participation (LAMP)framework by introducing an additional layer of agent heterogeneity in the economy– besides “active” versus “inactive” – related to the cross-sectional distribution of�nancial wealth among market participants.1 This speci�c type of heterogeneity givesrise to �nancial-wealth effects on aggregate consumption of the same kind as thosearising in perpetual-youth models, and makes asset-price dynamics non-neutral forthe equilibrium allocation.2 The resulting DAMP framework therefore nests as polarcases two other popular models extensively used to study monetary business cycles:i) the standard “representative-agent” model, where infinitely-lived identical agentsare active in asset markets at all times, and ii) the discrete-time “perpetual-youth”(PY) model, where all agents are active at all times, but they are finitely lived andthus heterogeneous. The latter model, in particular, provides a natural theoreticalcharacterization of the behavior of active agents in a DAMP economy: they face apositive probability of being replaced by previously inactive agents, they discountfuture utility to account for both impatience and transition probabilities, they havean incentive to enter an insurance contract à la Blanchard (1985) in order to smoothconsumption while market participants.

Analyzing welfare-maximizing policy and the implied trade-offs in the DAMPmodel, thus, requires to deal with a social welfare function that aggregates the utility�ows of an in�nite set of heterogenous agents, as in PY models.3 One contributionof this paper is to show how to treat the heterogeneity among market participants tocharacterize a second-order approximation of social welfare in terms of aggregatevariables only: the cross-sectional consumption dispersion among active agents isshown to be proportional to squared aggregate �nancial wealth. The resulting criterionrequires �nancial stability to be an additional and independent target of a welfare-maximizing policy maker, besides in�ation and output stability. The relative weights onthe three targets re�ect the two layers of heterogeneity of this economy: between agent

1. For monetary policy analyses within the LAMP framework see, among others, Galí et al (2007) andBilbiie (2008).

2. For an analysis of the interplay between monetary policy and asset prices in a perpetual-youthframework see, among others, Nisticò (2012), Airaudo et al (2007), Castelnuovo and Nisticò (2010).

3. This complication is one reason why, to my knowledge, no normative analysis in a linear-quadraticframework is available within perpetual-youth models.



types and within the set of market participants. As a consequence, an endogenous trade-off arises between output and price stability on the one hand, and �nancial stability onthe other: price stability may no longer be an optimal monetary policy regime. Evenconditionally only on productivity shocks, some �uctuations in in�ation and the outputgap may be optimal as long as they increase �nancial stability.

From a methodological perspective, the model developed in this paper builds ontwo strands of the literature. The �rst one was initiated by the seminal papers ofYaari (1965) and Blanchard (1985), which introduced and characterized the PY modelin a continuous-time framework. Since then, discrete-time baseline versions have beenwidely used in the literature to study a variety of topics, both in closed and open-economy models,4 or as building blocks of more elaborate frameworks, to account fordemographic factors.5 In contrast to these studies, all of which use the perpetual-youthstructure to characterize the implications of �nite lifetimes for economic policy, thispaper shows that it captures quite naturally the transient nature of the “active” agenttype – while the agent’s lifetime itself is in�nite – and describe its optimal behavior.

The second strand, initiated by the empirical observation of Campbell andMankiw (1989) and the theoretical suggestions of Mankiw (2000), features anincreasing number of New-Keynesian models formally characterizing limited assetmarket participation as the interplay between two different types of agents in theeconomy: optimizing (Ricardian) agents smoothing consumption over an in�nitehorizon through asset-market trading, and hand-to-mouth (non-Ricardian) agentsconsuming instead their labor income period by period. As also discussed byBilbiie (2008), these LAMP models essentially build into the dynamic New-Keynesianframework the exogenous segmentation in asset markets typical of the literatureanalyzing the “liquidity effects” of monetary policy, such as Alvarez, Lucas andWeber (2001). Accordingly, in these models such partition of the economy is “static”:an agent of a given type stays of that type forever. This paper, in contrast, allowsfor a stochastic transition between the two agent types analogous to that impliedby the endogenously-segmented market structure discussed in Alvarez, Atkeson andKehoe (2002).6

Indeed, I interpret the transition between agent types – that in the DAMP modelis taken as exogenous – as a reduced-form representation of the interaction among (i)a costly decision to participate in asset markets, (ii) some liquidity constraint and (iii)idiosyncratic uncertainty about individual income. This interaction – microfoundedby Alvarez, Atkeson and Kehoe (2002) – implies that some agents optimally decide topay the participation cost and engage in asset-market trading to smooth consumptionintertemporally, while the remaining ones decide to be inactive and simply consume

4. See, among others, Ganelli (2005), Di Giorgio and Nisticò (2013), and Leith et al. (2011).

5. See, among others, Ferrero (2010).

6. A similar Markov-switching mechanism between agent types is used by Curdia and Woodford (2010)and Woodford (2010) to introduce heterogeneity in spending opportunities among ex-ante identicalhouseholds, and provide a related case for a monetary-policy concern for �nancial stability, mainly relatedto �uctuations in credit spreads.



their own endowment (previously sold for cash). In every period, idiosyncratic shocksto individual endowments make it optimal, for some agents, to revise their decisionand switch between the zones of activity and inactivity, thereby implying stochastictransition between states.

An alternative interpretation might focus on the feature that in�nitely-livedagents take intertemporal consumption decisions over a �nite planning horizon. Amicrofounded rationale for such implication is provided by McKay et al. (2015),within a heterogenous-agent model addressing the forward-guidance puzzle. In onespeci�cation of their model economy, uninsurable idiosyncratic income risk makesagents randomly switch between employment and unemployment, with the latterfurther implying a binding liquidity constraint. In this environment, employed agentsdiscount the future more than implied by their time-discount rate, as they also take intoaccount the probability of suddenly becoming unemployed, whereby they would beunable to borrow and forced to consume only their home production. This effectivelyshortens the agents’ planning horizon, despite their in�nite lifetime, thus implyinga “discounted euler equation” analogous to that arising in PY models. Under thisinterpretation, the transition probabilities in the DAMP model would capture thetransition in and out of unemployment, where the latter also implies inactivity in assetmarkets.

A third possible interpretation might regard the DAMP model as an approximatedescription of consumers’ behaviour over the life cycle, in an economy wherefinitely-lived agents face (i) borrowing constraints, (ii) hump-shaped expected incomepro�les and (iii) uninsurable idiosyncratic labor-income risk, as in Gourinchas andParker (2002).7 Under this interpretation, the inactive agents are those in the earlystage of their life cycle – not smoothing consumption over time because of self-imposed borrowing constraints and only barely saving for precautionary motives –while the active agents are those in the late stage of their life cycle – whose behaviouris instead consistent with the permanent-income hypothesis. Accordingly, under thisinterpretation, the transition probabilities in the DAMP model would capture theexpected duration of the two stages of the life cycle.8

As the three interpretations above suggest, therefore, the DAMP model is inprinciple suitable to address different kinds of issues (with respect to the one analyzedin this paper), in economies where idiosyncratic uncertainty implies occasionallybinding liquidity constraints for some agents.9

More broadly, this paper is also related to a large literature spurred by the stock-market burst of 2001, evaluating to what extent a central bank is (and should be)

7. I thank an anonymous referee for pointing out this interpretation of the DAMP model.

8. Involving low-frequency cycles, however, this interpretation makes the model naturally better suitedto address other issues than a monetary policy concern for �nancial stability, which instead focuses on abusiness-cycle frequency.

9. Clearly, different interpretations might imply substantially different calibrations of the relevanttransition probabilities. In particular, please refer the Online Appendix for a discussion of how the mainresults would change by adopting calibrations more in line with the other two interpretations.



concerned with stock-price dynamics. This literature analyzes the implications formonetary policy of �uctuations in asset markets that affect real activity through botha supply-side channel – induced by the �nancial accelerator, as in Bernanke andGertler (1999) and Cecchetti et Al. (2000)10 – as well as a demand-side one – through�nancial-wealth effects induced by heterogeneity among finitely-lived agents, as in DiGiorgio and Nisticò (2007), Nisticò (2012), Airaudo et al (2015). The discontinuousparticipation to �nancial markets of this paper introduces an analogous demand-sidechannel in a more general class of models, where agents are heterogenous both betweenactive and inactive types and within the active type. The resulting framework is used tocomplement the positive analysis in Nisticò (2012) and Airaudo et al (2015), and theempirical analysis in Castelnuovo and Nisticò (2010), with a complete welfare analysisof the links between monetary policy and �nancial stability.

The paper is structured as follows. Section 2 characterizes the DAMP economy,and derives the welfare-based loss function as a second-order approximation of socialwelfare. Section 3 analyzes the stabilization trade-offs and the optimal monetary policydesign. Section 4 concludes.

2. The DAMP Economy

Consider an economy populated by a continuum of in�nitely-lived agents in theinterval [0, 1]. A mass .1 � #/ of these agents is �nancially inactive, consuming herlabor income period by period; I also refer to these agents as “rule-of-thumbers”.The remaining mass # of agents trades in asset markets to save/borrow and smoothconsumption; I refer to these agents as “market participants”, or “active”. The LAMPliterature (e.g. Galí et al., 2007 and Bilbiie, 2008) typically assumes that a given agentcannot change type: once inactive, always inactive. I generalize this framework byallowing for stochastic transition between these two types of agent. Each agent’s typeevolves over time as an independent two-state Markov chain: each period, each agentlearns whether or not she is going to participate in asset markets, where the relevantprobability is only dependent on the agent’s current state. In particular, with probability� 2 Œ0; 1/, a market participant becomes inactive, while with probability 1 � � sheremains active. Analogously, with probability % 2 Œ0; 1�, a “rule-of-thumber” becomesactive, while with probability 1� % she remains inactive.

De�ne a “cohort” as the set of agents experiencing a transition between types in thesame period: given the continuum of agents populating the economy, the size of eachindividual agent is negligible with respect to her cohort. By the law of large numbers, ineach period a mass #� of “market participants” becomes inactive, and a mass .1� #/%of “rule-of-thumbers” becomes active. Accordingly, the size of each cohort declinesdeterministically: at time t , the mass of agents that became active at time j and havenot yet switched type ismp;t .j /� #�.1� �/t�j , while the mass of agents that became

10. See also Bernanke and Gertler (2001), Gilchrist and Leahy (2002), Carlstrom and Fuerst (2001).



inactive at time k and have not yet switched type is mr;t .j / � .1 � #/%.1 � %/t�k ,where subscript p denotes market participants and r rule-of-thumbers.

I focus on an economy where the population shares of the two agent types remainconstant over time, in which therefore %.1� #/D #�: at each time, the mass of activeagents leaving �nancial markets is equal to the mass of rule-of-thumbers joining them.It follows that, in general, # and � must satisfy the following restriction:

#� � 1� #; (1)

requiring the inactive set to be at least large enough to provide a replacement for allactive agents switching type, so that all agents are in�nitely lived and population isalways equal to 1.11

Since they face a positive probability of becoming inactive, each cohort of “marketparticipants” has a different stock of accumulated wealth, depending on its longevityin the type. The model therefore introduces an additional layer of agents heterogeneity:agents are not only heterogenous between the sets of active and inactive – as inthe LAMP framework – but they also are within the set of active – as in a PYmodel. This additional layer of heterogeneity implies, in equilibrium, a �nancial-wealth effect on aggregate consumption of the same kind as that arising in a PYmodel. Characterizing the set of active agents, therefore, requires to keep track of thecross-sectional distribution of �nancial wealth. The rule-of-thumbers, instead, are allidentical, as they all consume their entire labor income, regardless of their longevityas inactive. Accordingly, the economy-wide aggregate level of generic variable X iscomputed as the mass-weighted average

Xt � #Xp;t C .1� #/Xr;t D #

tXjD�1

�.1� �/t�jXp;t .j /C .1� #/Xr;t ; (2)

where Xp;t �PtjD�1.mp;t .j /=#/Xp;t .j / D

PtjD�1 �.1 � �/

t�jXp;t .j / is theaverage value of X across all “market participants”, j 2 .�1; t � indexes the discreteset of cohorts of the latter at time t , and Xr;t is the average level across all “rule-of-thumbers” (which is the same for each of them).

Notice that, although lifetime is infinite for each agent, each “market participant”takes intertemporal consumption decisions subject to a planning horizon which isinstead finite, thereby considering that she will still be active after n periods withprobability .1� �/n.12 Thus, the problem of a market participant is isomorphic to that

11. Violating restriction (1) implies that either i) population size is constant but agents are not in�nitelylived, as a mass #.1C �/� 1 of agents actually dies each period, or ii) all agents are in�nitely lived butpopulation is not constant, as a mass #.1C �/� 1 of new agents is born, thereby increasing populationsize to #.1C �/ > 1 and reducing the relative size of the set of active agents to Q# D 1=.1C �/, whichsatis�es (1) with equality. In both cases, %D 1.

12. Indeed, a market participant anticipates that, upon becoming inactive – and regardless of theconsumption decisions taken while still active – she will behave as a rule-of-thumber and consume herentire labor income in each period unless she switches back to the active-type. Therefore, the relevant time



of an agent in a PY model à la Blanchard (1985). In particular, market participantsalso have an incentive to enter an insurance contract which grants them an extra-return on �nancial assets in exchange for the accumulated wealth at the time in whichthey are drawn to be replaced. Without such insurance contract active agents wouldneed to consume (pay back) the accumulated wealth (debt) entirely in the period inwhich they switch type; entering the contract, instead, allows them to insure againstthe idiosyncratic risk of replacement, and consume (pay back) their wealth (debt)smoothly over time while still active.13 Therefore, all agents in this economy arenon-Ricardian: rule-of-thumbers because they do not trade in asset markets, marketparticipants because they face a �nite planning horizon.

The resulting DAMP framework nests as polar cases most models extensively usedin the monetary policy New-Keynesian literature. If # 2 .0; 1/ and � D 0, the modelbecomes the standard LAMP speci�cation of Galí et al. (2007) and Bilbiie (2008),among others: all agents are in�nitely lived and a share of them are inactive in assetmarkets, but those agents will never become active. If # D 1 and � D 0 the modelinstead nests the standard “representative-agent” framework, where all agents are bothin�nitely lived and continuously active in asset markets. Finally, if # D 1 and � 2 .0; 1/– thus violating (1) – the model collapses to a standard “perpetual-youth” framework,where the transition probability � re�ects expected lifetime: all agents are active, butthey still face a positive probability of exiting the market (dying) and being replacedby newcomers (newborn).

2.1. The Decentralized Allocation

All agents in the economy have Cobb-Douglas preferences over consumption C andleisure 1�N :14

Us;t .h/ � logCs;t .h/C ı log.1�Ns;t .h//; (3)

horizon when taking intertemporal consumption decisions as a market participant is the expected durationof the “active regime”: 1=� . Analogously, the expected duration of the “inactive regime” is 1=%, but theeffective planning horizon for consumption decisions is 1 period, since a rule-of-thumber cannot transferresources over time through �nancial markets.

13. An alternative assumption would be to allow inactive agents to passively roll over their assets untilthey switch back into the active type. This would complicate the analysis considerably (as it would requireto keep track of the distribution of wealth within both sets of agents) without affecting the qualitativeimplication that the transition between agent types induces wealth-effects of �nancial instability onaggregate spending. See also Section 2.1.

14. Ascari and Rankin (2007) argue that PY models with endogenous labor supply should feature GHHpreferences, to avoid issues related to the negative labor supply of oldest generations. I acknowledgethis result – which might affect active agents in the DAMP model as well – but choose to use standardpreferences, for the sake of comparability.



for s D p; r and hD j; k.15 Regardless of their speci�c longevity as rule-of-thumbers,all inactive agents face the same static problem: they seek to maximize their periodutility Ur;t subject to the following budget constraint: Cr;t D WtNr;t ; where Wtdenotes the real wage. Solution to this problem yields a labor supply schedule

Nr;t D 1� ıCr;t

Wt; (4)

implying that, at equilibrium, inactive agents work a constant amount of hours:Nr;t D.1C ı/�1.

In addition to supplying labor services and demanding consumption goods,“market participants” also demand two types of �nancial assets: nominal state-contingent bondsB�tC1 � PtC1BtC1 and equity sharesZtC1.i/ issued by a continuumof monopolistic �rms, indexed by i 2 Œ0; 1� and selling at real price Qt .i/.

At time t0, agents that have been active since period j � t0 seek to maximizeEt0

®P1tDt0

ˇt�t0.1� �/tUp;t .j /¯; where future utility �ows are discounted to

account for both impatience (ˇ) and the probability of becoming inactive in the nextperiod (�). The maximization problem is subject to a sequence of budget constraints(expressed in real terms) of the form:

Cp;t .j /CEt

²Ft;tC1

PtC1

PtBtC1.j /

³C

Z 1

0

Qt .i/ZtC1.j; i/ d i

�1

1� ��t .j /CWtNp;t .j /� Tt .j /; (5)

where WtNp;t .j / � Tt .j / is real disposable labor income, Ft;tC1 is the stochasticdiscount factor for state-contingent assets, and the real �nancial wealth carried overfrom the previous period is

�t .j / �

Z 1

0

�Qt .i/CDt .i/

�Zt .j; i/ d i CBt .j /; (6)

including the pay-off on the equity portfolio and on the contingent claims. As discussedearlier, in order to smooth consumption over the duration of the active regime, “marketparticipants” enter an insurance contract à la Blanchard (1985), which grants them anextra-return �=.1 � �/ on �nancial assets in exchange for the accumulated wealth atthe time in which they are drawn to be replaced.

It is useful, at this point, to further characterize the partition of agents interactingin asset markets in each period t : call “old traders” those that are active since atleast one period (i.e. j � t � 1), and “newcomers” those that became active inthe current period (i.e. j D t ). The latter were previously inactive, and I thereforeassume – in analogy with Blanchard (1985) – that they enter with zero holdings

15. This Section describes only the main features of the model. For details, please refer to the OnlineAppendix. To save on notation, subscripts p and r are used only for consumption and hours worked, thatenter the problems of both types.



of �nancial assets.16 The different longevity among “market participants” implies anon-degenerate distribution of �nancial wealth across cohorts, and a related cross-sectional distribution of individual consumption. I assume that the public authorityis able to directly affect such distribution only in the steady state, by means of anappropriate redistribution scheme: Tt .j / D Tt C .j /Rt : The �rst component iscommon across cohorts, while the second one is a generation-speci�c transfer, whereRt is an appropriate scaling factor,17 and .j / de�nes whether the transfer is a fee – .j / > 0 – or a subsidy – .j / < 0.18

The nominal gross return .1C rt / on a safe one-period bond paying off one unitof currency in period t C 1 with probability 1 is de�ned by:

.1C rt /Et ¹Ft;tC1º D 1: (7)

Using the individual equilibrium conditions to solve equation (5) forward, andaggregating across all market participants, one can derive the equilibrium dynamicsof their average consumption:

Cp;t D��

ˇ.1� �/Et ¹Ft;tC1.1C �tC1/ .�tC1 � nc/º

C1

ˇEt®Ft;tC1.1C �tC1/Cp;tC1

¯; (8)

where 1 C �t � Pt=Pt�1, nc � � .t/.1 � �/�1 > 0 is the transfer received bynewcomers upon becoming active, and � � Œ1 � ˇ.1 � �/�=.1C ı/. The �rst termin the equation above captures the �nancial-wealth effect, and can be traced back tothe cross-sectional consumption dispersion, as measured by the wedge between theaverage consumption of old traders (C otp ) and that of newcomers (C ncp ):19

C otp;tC1 � Cncp;tC1 D

�

1� �.�tC1 �

nc/ ; (9)

where C otp;tC1 �PtjD�1 �.1 � �/

t�jCp;tC1.j /; and C ncp;tC1 � Cp;tC1.t C 1/: Thismechanism, therefore, implies a direct channel through which �nancial instabilitycan affect the real economy. An upward revision in �nancial-wealth expectations attime t implies that all active agents in t increase their current spending, so as tooptimally smooth their intertemporal consumption pro�le; at t C 1, however, a fractionof these individuals will be replaced by newcomers that were previously inactive,and whose consumption at t C 1 will therefore not be affected by the current value

16. I later discuss why this assumption, which considerably simpli�es the analysis, is qualitativelyinessential.

17. In particular, Rt � .1� �/�1 �Et®Ft;tC1.1C �tC1/

¯.

18. Such “participation fee” serves the only purpose of implementing a steady-state cross-sectionaldistribution of �nancial wealth that is consistent with the social planner allocation. Please refer to the OnlineAppendix for details. This redistribution scheme, clearly, satis�es

PtjD�1 �.1� �/

t�j .j /D 0.

19. Please refer to the Online Appendix for details.



of �nancial wealth. Thus, higher asset prices today, signaling an expected increasein future �nancial wealth, raise current consumption more than it does the expecteddiscounted future level.20

Equation (9) shows that the magnitude of the wealth effect depends upon twofactors. First, higher rates of replacement � , for given �uctuations in �nancial wealth,imply a larger heterogeneity among active agents. Second, stronger �uctuations inaggregate �nancial wealth translate a given heterogeneity among active agents into alarger wedge between the average consumption of old traders and that of newcomers.Moreover, it can be shown that this wedge distorts the decentralized equilibriumallocation with respect to the socially optimal one, in which the cross-sectionaldistribution of individual consumption is egalitarian and invariant to aggregatestates.21 In this respect, equation (9) shows that, even if the economy starts from suchsocial planner allocation, �uctuations in �nancial wealth induced by aggregate shocksmake the decentralized cross-sectional consumption distribution endogenously deviatefrom the socially optimal one. This is at the heart of the normative implications thatwill be derived in the next sections. In particular, if either (i) there is no heterogeneityin market participants (� D 0) or (ii) aggregate �nancial wealth is stable at nc , thewedge in average consumption between old traders and newcomers is closed, and thecross-sectional distribution of consumption is egalitarian.

The supply-side of the economy is standard New-Keynesian. A competitive retailsector packs a continuum of intermediate differentiated goods Yt .i/ into a �nal goodYt , using a CES technology. A monopolistic wholesale sector produces the formerout of labor services, using the linear technology Yt .i/ D AtNt .i/, where At is anaggregate productivity index, following At D A1��aA

�at�1 exp."at /, with �a 2 Œ0; 1�

and "at � N.0; �2a /. Aggregating across �rms and using the equilibrium input demand

coming from the retail sector yields exp.�pt /Yt D AtNt ; in which Nt �R 10 Nt .i/ d i

is the aggregate level of hours worked and

�pt � log

Z 1

0

�Pt .i/

Pt

��"di (10)

is a (second-order) index of price dispersion over the continuum of intermediategoods-producing �rms, where " denotes the price-elasticity of demand for input i and�� "=."� 1/ the corresponding degree of market power. As in most related literature,I assume that the government subsidizes employment at the constant rate � , to offset –in the steady state – the monopolistic distortion. Monopolistic �rms are subject to the

20. Equation (9) clari�es that, for a �nancial-wealth effect to arise, the assumption that “newcomers”enter with zero-holdings of �nancial assets is actually inessential. What is essential is that the average asset-holdings of “newcomers” differ from that of “old traders”. Since “newcomers” were previously inactive,this is generally the case, as they were not able to adjust it over time to smooth consumption in response toaggregate shocks, as done instead by “old traders”. Accordingly, a level of �nancial wealth that is passivelyrolled over while inactive – as for example assumed in Alvarez et al. (2002) – would also imply a �nancial-wealth effect of the same kind, while making analytical derivations considerably more cumbersome (sinceone would have to keep track of the distribution of wealth across “rule-of-thumbers” as well).

21. Please refer to the Online Appendix for details.



“Calvo pricing”, with 1� � denoting the probability for a �rm of having the chance tore-optimize in a given period. When able to set its price optimally, each �rm seeks tomaximize the expected discounted stream of future dividends, otherwise they keep theprice unchanged. Equilibrium in this side of the economy implies the familiar New-Keynesian Phillips Curve, relating current in�ation to future expected in�ation andcurrent marginal costs.

Averaging across all agents in the economy yields the aggregate per-capita levelof hours worked Nt � #Np;t C .1 � #/Nr;t D #Np;t C .1 � #/=.1 C ı/ andconsumption Ct � #Cp;t C .1� #/Cr;t : The �scal authority runs a balanced budget,�nancing the employment subsidies through lump-sum taxes to active agents: #Tt D�WtNt : In equilibrium, all output is absorbed by private consumption (Yt D Ct ), thenet supply of state-contingent assets is nil, and the stock of outstanding equity sharesis 1, for each monopolistic �rm. As a consequence, aggregate pro�t income (enjoyedby active agents only) is: #Dt D Yt � .1� �/NtWt :

2.2. The Optimal Steady State and the Linear Model

In order to use a linear-quadratic approach to optimal policy analysis, here I de�ne the“optimal steady state” as one that is consistent with the social planner allocation. Inparticular, the Online Appendix shows that the latter implies (in addition to aggregateef�ciency) an egalitarian cross-sectional distribution of individual consumption thatis invariant to aggregate states.

The decentralized allocation characterized in the previous section, thus, deviatesfrom the socially optimal one because of three distortions. First, a static distortiondue to monopolistic competition implies a lower level of supplied output, both in theshort and in the long run. As usual, this distortion can be corrected by an appropriateemployment subsidy � such that .1C �/.1 � �/ D 1. Second, a dynamic distortiondue to nominal rigidities implies a positively-sloped supply schedule, price dispersionacross different �rms and ultimately a lower level of short-run equilibrium output (thisdistortion fades away in a zero-in�ation steady state). Third, an analogous dynamicdistortion is due to the interaction in �nancial markets of heterogeneous agents, whichimplies that the cross-sectional consumption distribution across households changesendogenously in response to any aggregate shock affecting aggregate �nancial wealth,unlike in the social planner allocation. This third distortion is speci�c to the frameworkwith �nancial-wealth effects and can be corrected in the steady state by an appropriateredistribution scheme .j /, for all j . To understand the analogy between the twodynamic distortions above, consider an economy starting from a steady state consistentwith the social planner allocation, and it is hit by an aggregate shock. In the sameway in which the aggregate shock triggers an inef�cient response of the relative-pricedistribution across heterogeneous (in terms of price duration) �rms, it also triggers ananalogously inef�cient response of the cross-sectional consumption distribution acrossheterogeneous (in terms of asset holdings) consumers: social welfare is reduced by



endogenous consumption dispersion in an analogous way in which it is by endogenousprice dispersion.22

A socially optimal steady state can be implemented by a redistribution scheme .j /, such that, for all j , .j / D �.j / � .1 � �/�: As a consequence, the steady-state interest rate satis�es the familiar ˇ.1C r/ D 1, and all agents supply the sameamount of hours worked Np.j / D Nr D N D .1C ı/�1 and consume the same levelof consumption Cp.j / D Cr D C D Y D A.1C ı/�1; for all j .

First-order approximation around such optimal steady state yields:23

yt D EtytC1 C

‚Et!tC1 �

1

‚.rt �Et�tC1 � �/C

1�‚

‚Et�atC1 (11)

!t D ˇEt!tC1 � .1� ˇ/1C ' � �

�.yt � at /� ˇ.rt �Et�tC1 � �/C .1� ˇ/at

(12)

�t D ˇEt�tC1 C �.yt � at /; (13)

where ' � N=.1 � N/ D 1=ı denotes the steady-state labor-leisure ratio, alsocapturing the elasticity of the real wage with respect to aggregate hours worked,� � ��1.1 � �/.1 � �ˇ/.1 C '/ is the slope of the Phillips Curve, � the steady-state real interest rate, and =‚ the elasticity of output with respect to expected future�nancial wealth, where

� �1� ˇ.1� �/

.1� �/.1C ı/

1

1� ˇ

�

1C �� 0

and ‚ � 1� '.1� #/=# .24

The partition of consumption between active and inactive agents is �nallydetermined by

cp;t D ‚yt C .1�‚/at (14)

cr;t D .1C '/yt � 'at : (15)

2.2.1. The Flexible-Price and Socially Optimal Allocations. The �exible-priceequilibrium (FPE) arises in this model when the probability of having to charge lastperiod’s price goes to zero (� D 0). In this case, denoting variables in the FPEwith an upper bar, real marginal costs are constant and the optimal employmentsubsidy implements the ef�ciency condition NWt D At . Accordingly, labor-market

22. One notable difference is that the former dispersion only affects the distributional implications of theequilibrium allocation, while the latter affects also aggregate ef�ciency.

23. Let ´t � log.Zt=Z/ for ´ D y;!;a; rt is the net nominal interest rate and �t the net in�ationrate.

24. As shown by Bilbiie (2008),‚ relates the average participation rate to �nancial markets# to the slopeof the forward-looking IS curve, with the threshold #� D '=.1C '/ determining whether the economybehaves according to the standard (# > #�) or the inverted (# < #�) aggregate demand logic. In the restof the analysis I will restrict attention to the case (# > #�).



clearing implies the equilibrium level of potential output: Nyt D at ; and the equilibriumallocation can be derived as the solution to the system:

Nyt D at (16)

N�t D 0 (17)

Nrr t D �CEt�atC1 C Et N!tC1 (18)

N!t D ˇEt N!tC1 � ˇ. Nrr t � �/C .1� ˇ/at : (19)

Thus, the FPE level of �nancial wealth is proportional to aggregate productivity

N!t D1� ˇ�a

1� ˇ�a.1� /at ; (20)

and the real interest rate supporting such allocation is

Nrr t D �CEt�atC1 C1� ˇ�a

1� ˇ�a.1� / �aat : (21)

This allocation features aggregate ef�ciency because nominal rigidities are shutdown and the static distortion related to monopolistic competition is corrected by theemployment subsidy. It is not, however, consistent with a cross-sectional consumptiondistribution that is invariant across aggregate states, as instead implied by the sociallyoptimal allocation. Indeed, while the heterogeneity between the sets of active andinactive agents fades away, as implied by (14)–(16), the heterogeneity within theset of market participants does not, and in fact responds to �uctuations in �nancialwealth, which still affect the wedge in average consumption between newcomers andold traders. As a result, the �exible-price allocation is suboptimal with respect to thesocially optimal one. With the benchmark speci�cation of the model (in which � D 0)the DAMP economy shares the same natural level of output,25 but the natural levels ofreal interest rate and �nancial wealth are generally different. In the benchmark setups,the natural rate of interest fully accommodates the pressures on output coming fromproductivity shocks ( Nrr t D �CEt�atC1), and �nancial wealth mirrors the evolutionof productivity ( N!t D at ). Equation (21), instead, shows that in the DAMP economythe natural interest rate leans partially against productivity shocks, to absorb the�uctuations in �nancial wealth that would otherwise distort the dynamics of aggregateconsumption.

The socially optimal allocation – characterized by both aggregate ef�ciency andoptimal cross-sectional consumption distribution – can be de�ned as an equilibrium inwhich not only the output gap and in�ation rate are zero, but also �nancial wealth is

25. Since in the �exible-price equilibrium the Phillips Curve is vertical, the equilibrium level of realactivity is determined by the supply side of the economy only, for which �uctuations in �nancial wealthare neutral.



stable at the steady-state level (!et D 0):

yet D Nyt D at (22)

�et D 0 (23)

!et D 0: (24)

As implied by equation (9), when �nancial wealth is stable at nc , the cross-sectionalconsumption dispersion is zero, and the distributional ef�ciency of the equilibriumallocation is restored. However, unlike in the FPE, there is no real interest-rate paththat supports this equilibrium allocation.

Proposition 1. In a decentralized DAMP economy with an active stock market, thereis no real interest rate dynamics supporting the socially optimal allocation.

Proof. Impose yt D at and �t D !t D 0 on system (11)–(13). Then equation (11)implies rret D �CEt�atC1, while equation (12) implies rret D �C Œ.1� ˇ/=ˇ�at ,which are the same only as long as at D 0 for all t . �

2.3. The Welfare-Based Loss Function.

Given Proposition 1, it becomes useful to evaluate the welfare implications of targetingthe FPE allocation instead of the socially optimal one. To this aim, I now derivethe monetary-policy loss function as a second-order approximation of social welfarearound the optimal steady state.26 De�ne social welfare at t0 as

Wt0 � Et0

´1XtDt0

ˇt�t0�#Up;t C .1� #/Ur;t

�µ; (25)

where Up;t �PtjD�1 �.1 � �/

t�jUp;t .j / is the average period utility of activeagents, Ur;t �

PtkD�1 %.1 � %/

t�jUr;t .k/ D Ur;t that of “rule-of-thumbers”, andUs;t .h/ is de�ned in (3), for s D p; r and h D j; k.27

Given the ef�ciency condition 'ı D 1 implied by the optimal employment subsidy,and ignoring terms independent of policy and of higher order, a second-order Taylor

26. This approach satis�es the conditions that deliver a valid second-order approximation of expectedwelfare when evaluated using private-sector equilibrium conditions that are approximated only to �rstorder. See Woodford (2003).

27. The de�nitions of Up;t and Ur;t are consistent with the assumption that different longevities aretreated equally by the social planner. Please refer to the Online Appendix for details. Analogous de�nitionsof social welfare are adopted by Curdia and Woodford (2010), in a model with Markov-Switching agent-types, and by Bilbiie (2008), among others, in the benchmark LAMP framework.



expansion of the aggregate period�t utility for active agents can be written as

Up;t D cp;t �1

2varj cp;t .j /� np;t C

1

2varjnp;t .j /�

1C '

2Ej�np;t .j /

2�

D cp;t � np;t �1C '

2n2p;t �

1C '

2'varj cp;t .j /; (26)

in which the last equality uses E.´2/ D E.´/2 C var.´/ and a �rst-orderapproximation of the individual labor supply, to relate the cross-sectional variance ofhours worked to that of consumption: '2varjnp;t .j / D varj cp;t .j /. Since inactiveagents are homogeneous and supply a constant amount of hours worked, a second-order approximation of their period�t utility is simply Ur;t D cr;t . Using the resourceconstraint and a second-order approximation of the aggregate production function, onethen obtains the period�t social utility as a function of output gap xt , price dispersionacross �rms�pt and consumption dispersion across active agents�cp;t � varj cp;t .j /:

#Up;t C .1� #/Ur;t D �.1C '/

2#

�x2t C

2#

1C '�pt C

#2

'�cp;t

�: (27)

Social welfare thus displays an additional second-order term, compared to theLAMP framework, which is related to the cross-sectional dispersion in consumption�cp;t : were all active agents homogenous, consuming in every period the same amountof goods, the cross-sectional variance would be zero, and the LAMP model wouldarise. To understand the implications of this additional term, it is useful to recall thepartition of the set of active agents in “newcomers” and “old traders”. Indeed, usingthe law of total variance on such partition, one can characterize the evolution over timeof the cross-sectional consumption dispersion across all active agents in the economyas:28

�cp;t D .1� �/�cp;t�1 C

�

1� ˇ

�

1C �!2t : (28)

Moving from any arbitrary initial level �cp;t0�1, which is independent of policiesimplemented from t D t0 onward, I can therefore write the discounted value over allperiods t > t0 as

1XtDt0

ˇt�t0�cp;t D �

.1C ı/.1� ˇ/.1C �/

1XtDt0

ˇt�t0!2t (29)

Finally, the time�t0 conditional expectation of the discounted stream of futureperiod social losses yields the welfare-based loss function, as a share of steady-stateaggregate consumption:29

Lt0 � �Wt0 D.1C '/

2#Et0

´1XtDt0

ˇt�t0�x2t C ˛��

2t C ˛!!

2t

�µ: (30)

28. For details on the derivation, please refer to the Online Appendix.

29. The second equality in (30) uses the familiar lemmata: �pt � "

2vaript .i/, andP1

tDt0ˇ t�t0vaript .i/D

�.1��/.1��ˇ/

P1tDt0

ˇ t�t0�2t .



In the loss function above, ˛� � #"=� denotes the relative weight on in�ationstability and ˛! � #2 �=Œ.1C '/.1� ˇ/.1C�/� � 0 the one on financial stability.As discussed in Bilbiie (2008), the relative weight on in�ation is a fraction # of thatarising in the standard Ricardian model – i.e. the ratio between the price-elasticity ofdemand " and the slope of the Phillips Curve � – since in�ation variability reducespro�t income, which is only enjoyed by market participants. The additional weighton �nancial stability is instead peculiar to the DAMP framework, re�ecting the factthat �uctuations in aggregate �nancial wealth affect the cross-sectional consumptiondistribution across active agents and make it deviate from the socially optimal one.30

3. Stabilization Trade-offs and the Optimal Policy

The loss function (30) shows that cross-sectional consumption dispersion is induced by�nancial instability and is a source of welfare loss, and implies that the �exible-priceallocation is therefore not a proper target for welfare-maximizing monetary policy.

Proposition 2. In a decentralized DAMP economy with an active stock market,a welfare-maximizing policy maker should target the socially optimal allocation.Financial stability (!t D 0) becomes an explicit target of welfare-maximizing monetarypolicy, in addition to output and inflation stabilization (xt D �t D 0). The welfare-relevance of financial stability (˛!) is higher when financial-wealth effects ( ) arestronger, the market power of monopolistic firms (�) is higher and/or the averageasset-market participation rate (#) is higher.

An interesting implication of Propositions 1 and 2 is that a welfare-relevantstabilization tradeoff endogenously arises, even absent inef�cient supply shocks:since there is no interest-rate dynamics supporting the socially optimal allocation,minimization of the loss function (30) implies that some �uctuations in in�ation andthe output gap will be optimal in order to reduce �nancial instability.

We can see this formally, by evaluating optimal policy in this framework alongtwo dimensions: the optimal response of the DAMP economy to a productivity shock,and the expected welfare loss implied by a speci�c policy regime, captured by theunconditional mean of the period loss in (30):

E¹�Utº D.1C '/

2#

�var.xt /C ˛�var.�t /C ˛!var.!t /

�: (31)

The complete system of equilibrium conditions for the private sector, constrainingthe optimal policy, can be expressed in terms of the relevant gaps from the socially

30. Notice that an analogous (and even stronger) role for �nancial stability arises in the welfare criterionof a perpetual-youth economy, which is nested in the DAMP framework under the parameterization# D 1.



optimal allocation:

xt D EtxtC1 C

‚Et!tC1 �

1

‚.rt �Et�tC1 � Qrr t / (32)

!t D ˇEt!tC1 � .1� ˇ/1C ' � �

�xt � ˇ.rt �Et�tC1 � Qrr t /C .1� ˇ�a/at

(33)

�t D ˇEt�tC1 C �xt ; (34)

in which Qrr t � � C Et�atC1. As usual in this class of models, I can determinethe optimal path of the nominal interest rate residually, and derive �rst the optimalallocation in terms of the other endogenous variables. In this case, the constraints thatare relevant for optimal policy can be given the following reduced form:

�t D ˇEt�tC1 C �xt ; (35)

!t D ˇ.1� /Et!tC1 C �xt � ˇ‚EtxtC1 C .1� ˇ�a/at (36)

in which � � Œ� � .1 � ˇ/.1 C '/��1 � ˇ.1 � ‚/. The IS schedule (32) – orequation (33) – can then be used to back out the equilibrium path of the nominal interestrate consistent with the optimal policy.

To understand the tradeoffs arising in this economy, consider a transitoryproductivity shock (�a D 0) and solve equations (35) and (36) forward:

�t D �

1XkD0

ˇkEtxtCk; (37)

�xt D !t C

�1�

�.1� /

‚

� 1XkD1

�ˇ‚

�

�kEt!tCk � at : (38)

Equation (37) shows the familiar result that in�ation depends on the current aswell as expected future output gaps. Absent inef�cient supply shocks, this equationclari�es that targeting the �exible-price allocation at all times is consistent with pricestability, regardless of productivity shocks: no tradeoff exists between in�ation andoutput-gap. Equation (38), in contrast, implies that a stabilization tradeoff does existbetween real and �nancial stability: conditional on a positive productivity shock, theeconomy will necessarily experience either a fall in the output gap (and thereby alsoin�ation), or an increase in �nancial wealth. In the standard Ricardian economy (or thebenchmark LAMP model), these tradeoffs are meaningless from a welfare perspectivebecause !t does not enter the loss function, and are trivially resolved because �nancialstability is not a concern for the policy maker. In a DAMP economy, instead, !t is anexplicit argument of the welfare criterion, and these tradeoffs therefore require explicitconsideration from welfare-maximizing policy makers. The forward-looking nature ofin�ation and asset prices, moreover, suggests that such tradeoffs can be improved bycredibly committing to a speci�c state-contingent path for �nancial wealth and/or theoutput gap, so that the central bank can reduce the welfare losses associated to anyshock.



3.1. Optimal Policy under Discretion and Full Commitment

One way of dealing with the above tradeoffs is for the central bank to chose in everyperiod the optimal level of all target variables, so as to minimize the period welfareloss. The optimal policy under discretion therefore minimizes the period-loss function

.1C '/

2#

�x2t C ˛��

2t C ˛!!

2t

�such that !t D �xt CK!;t and �t D �xt CK�;t , where K�;t and K!;t collect theterms related to expectations about the future and exogenous state variables, that thecentral bank cannot affect in this policy regime. The solution to this problem requiresthe following optimal targeting rule:

xt C ˛��t C ˛!�!t D 0: (39)

Using the above relation in the reduced system (35)–(36), yields the optimal state-contingent path for the in�ation rate, output gap and �nancial wealth:

�dt D �ˆ�at (40)

xdt D �ˆxat (41)

!dt D ˆ!at ; (42)

where

ˆ� �˛!��.1� ˇ�a/

AC ˛!BC C; ˆx �

˛!�.1� ˇ�a/2

AC ˛!BC C; ˆ! �

AAC ˛!BC C

;

with ˆ! 2 Œ0; 1�, A � .1� ˇ�a/.1C ˛��2 � ˇ�a/, B � .1� ˇ�a/.��‚ˇ�a/� andC � ˇ�a.1C ˛��2 � ˇ�a/.

In the case with no real effects of �nancial instability (� D D ˛! D 0) –regardless of the share of active agents # – optimal monetary policy under discretionrequires price stability (ˆ� D 0), output-gap stability (ˆx D 0) and that all volatilitybe borne by �nancial wealth (ˆ! D 1). This is the familiar case of the optimality ofprice stability in an economy with no inef�cient supply shocks, in which the �exible-price allocation is ef�cient. The picture changes in a DAMP economy:

Proposition 3. In a decentralized DAMP economy with an active stock market, pricestability is not an optimal policy regime from a welfare perspective.

Indeed, given convexity of the welfare-based loss function, the optimaldiscretionary monetary policy requires the volatility implied by productivity shocksto be shared among the three targets. Accordingly, following a positive productivityshock, �nancial wealth will rise less than under price stability (ˆ! < 1), and bothin�ation and the output gap will decline (ˆ� ; ˆx < 0).

An alternative way of dealing with the stabilization tradeoff is to commit to a state-contingent path for the target variables. If the commitment is credible, there is room



for improving the tradeoff by anchoring expectations, as in the standard framework. Inthis case, the monetary policy problem at time t0 is to minimize the loss function (30)such that the constraints (35)–(36) are satis�ed for all t . The optimality conditions forthis policy problem under full commitment are:

xt D ��2;t �‚�2;t�1 C ��1;t (43)

˛��t D �1;t�1 � �1;t (44)

˛!!t D .1� /�2;t�1 � �2;t ; (45)

in which �1;t and �2;t are the Lagrange multipliers associated to constraints (35)–(36), respectively, and �1;t0�1 D �2;t0�1 D 0. In the benchmark LAMP case with� D ˛! D 0, equations (43)–(45) imply �2;t D 0 for all t , and therefore price stabilityis still the optimal policy. In a DAMP economy, instead, �2;t will generally differ fromzero, which makes the trade-off discussed earlier relevant for output and in�ation, andimplies that Proposition 3 applies also under full commitment.

Figure 1 provides a graphical illustration of such proposition, and shows animpulse-response simulation exercise to display the policy implications for the DAMPeconomy. The �gure shows the dynamic response of selected variables – underalternative policy regimes – to a transitory, one-standard-deviation productivity shock,with standard deviation �a D 0:5%. Table 1 then compares the cyclical and welfareimplications of the alternative policy regimes, reporting the equilibrium standarddeviations (in percentage points) of selected variables and the implied welfare losses(in basis points of steady-state aggregate consumption).

In the numerical illustration, the key parameters to calibrate are the share of savers# and the turnover rate �. As a baseline calibration, I use # D 0:8 and � D 0:17. Theformer is the estimated value reported by Bilbiie and Straub (2013) for the Great-Moderation sample (i.e. post 1984); the latter is chosen to imply an elasticity ofaggregate consumption to stock-market wealth ( =‚) of about 0.15, as documentedby Ludvigson and Steindel (1999) for the Great-Moderation sample, and is consistentwith the turnover rate in U.S. �nancial portfolios reported by Cella et al (2013), whichimplies an average investment horizon of about 6 quarters.31

The circled line in the Figure 1 displays the case of In�ation Targeting, underwhich both in�ation and the output gap can be fully stabilized – given the absenceof inef�cient supply shocks – and all the volatility induced by the productivity shock

31. Please refer to the Online Appendix for an extensive sensitivity analysis of the main results tothe calibration of these key parameters. The baseline calibration � D 0:17 is also consistent with thereplacement rate and the �nancial-wealth effects documented by Castelnuovo and Nisticò (2010) withinan empirical version of the perpetual-youth speci�cation of this model, estimated with post-WWII USdata. The rest of the model is calibrated using convention or previous studies: the time discount factor isˇ D 0:99, to imply a steady-state annualized real interest rate of 4%, the Calvo parameter � D 0:75,implying an average price duration of 4 quarters, the average price-markup is � D 20%, consistentlywith Bilbiie and Straub (2013), among others, and the elasticity of real wages to aggregate hours is' � N=.1�N/ D 0:3, consistently with Rotemberg and Woodford (1997), Galí et al (2007), and withthe average hours worked for the US in the Great-Moderation sample.



0 2 4 6−0.1

−0.075

−0.05

−0.025

0

0.025

0.05Inflation (%)

0 2 4 6−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05Output Gap

0 2 4 62

2.5

3

3.5

4

4.5Interest Rate (%)

0 2 4 6−0.1

0

0.1

0.2

0.3

0.4

0.5Financial Wealth

Inflation Targeting

0 2 4 60

0.005

0.01

0.015

0.02

0.025Cons. Dispersion

Discretion

0 2 4 6−0.1

0

0.1

0.2

0.3

0.4

0.5Productivity Shock

Commitment

Figure 1. Response of selected variables to a transitory, one-standard-deviation productivity shock.Circled line: In�ation Targeting. Dashed line: optimal policy (Discretion). Solid line: optimal policy(Full Commitment).

is borne by �nancial wealth and the interest rate. These dynamics are the same as thosethat would arise in the standard Ricardian economy – in which � D 0 and # D 1 – andin the benchmark LAMP framework – in which � D 0 and # 2 .0; 1/.32 However, inthose frameworks �uctuations in �nancial wealth do not have distributional effects andthe welfare loss is therefore zero (see �rst column of Table 1). In the DAMP economy,instead, �uctuations in �nancial wealth do affect the cross-sectional consumptiondispersion among savers �ct – evolving according to (28) – which in this case isdetrimental from a welfare perspective. The baseline calibration, indeed, implies aweight on �nancial stability of the same order of magnitude as that on the outputgap: ˛! D 1:18. As a result, under In�ation Targeting, the welfare loss implied bya productivity shock with standard deviation of 0.5% – evaluated using equation (31)– is about 0.239bp of steady-state aggregate consumption, as reported in the secondcolumn of Table 1.

The dashed line in Figure 1 shows instead the dynamic response of the DAMPeconomy under the optimal discretionary policy. In this case, equations (40)–(42)implies that monetary policy �nds it optimal to cut in�ation by about 10bp and outputgap by about 20bp, in order to reduce by about 35% the short-run response of �nancialwealth to the shock. This requires cutting the nominal interest rate about 75bp less

32. Indeed, equations (20) and (21) show that, when �a D 0, In�ation Targeting implies the same pathfor interest rates and �nancial wealth as in the RA setup.



Table 1. Implied volatilities and welfare losses conditional on a productivity shock.

Benchmark DAMP

IT IT Disc Comm

In�ation 0.000 0.000 0.024 0.019Output Gap 0.000 0.000 0.210 0.232

Financial Wealth 0.500 0.500 0.319 0.292Interest Rate 0.500 0.500 0.306 0.234

Cons. Dispersion 0.000 0.063 0.025 0.022

Welfare Loss 0.000 0.239 0.152 0.138

Note: Implied volatilities are standard deviations in percentage points; Welfare Losses are bases points of steady-state aggregate consumption. Standard deviation of productivity shock is 0.5%. IT: In�ation Targeting; Disc:Discretion; Comm: Full Commitment.

than under In�ation Targeting, and allows to reduce the response of cross-sectionalconsumption dispersion by over a half, throughout the transition. From the cyclicalperspective, as reported by the third column of Table 1, the effect is that the volatilityimplied by the productivity shock is now “shared” among the three welfare-relevantvariables: the standard deviation of in�ation and the output-gap raises from zero to,respectively, 0.024% and 0.210%, while that of �nancial wealth drops from 0.5% to0.32%. Given convexity of the welfare criterion, this reduces the welfare loss to about0.152bp of steady-state aggregate consumption: price stability in this environment issubstantially suboptimal, as it implies a welfare loss about 57% larger than under theoptimal discretionary policy.

The general implication is reinforced under full commitment, as implied by thesolid line in Figure 1 and the fourth column of Table 1. As to the dynamic responseof the economy, the commitment takes two forms. On the one hand monetary policycommits to keep �nancial wealth higher than steady state for some time even afterthe shock has reverted back to mean, through persistently lower nominal interestrates (which on impact fall more than 1 percentage point less than under In�ationTargeting); this achieves a smaller response of �nancial wealth on impact and a lowerconsumption dispersion throughout the transition. On the other hand, it also commitsto an output boom after the shock has returned to steady state, in order to dampen theshort-run effect of the shock on the in�ation rate, which now falls by about 6bp only.Interestingly, note that the short-run response of the output gap in this case is slightlystronger than under discretion: the tradeoff between �nancial and real variables is suchthat even full commitment to a speci�c state-contingent path for the target variablesis not enough to dampen the short-run effects of the shock on all of them, unlike inthe standard analysis of commitment with inef�cient supply shocks. From the cyclicalperspective, exploiting the effect of the credible commitment on expectations enablesthe central bank to further reduce the welfare loss to 0.138bp of steady-state aggregateconsumption.

Overall, therefore, the optimal response of the DAMP economy – whether underdiscretion or commitment – requires non-negligible deviations from the corresponding



response under the In�ation Targeting regime, that would instead be optimal in aneconomy with no �nancial-wealth effects (whether Ricardian or LAMP). In particular,In�ation Targeting implies welfare losses up to 73% higher than under optimal policy,and a variability of consumption dispersion up to three times higher (as implied by the�fth row of Table 1).

3.2. Optimal Policy with Inefficient Supply Shocks

Until now I have considered only productivity shocks to show that a meaningfultradeoff for monetary policy arises in a DAMP economy even when there are noinef�cient supply shocks that break the link between in�ation and the welfare-relevantoutput gap. Here I extend the analysis to the case in which such shocks exist, to show –perhaps not surprisingly, at this point – that also in the presence of the familiar in�ation-output tradeoff, ignoring the role of �nancial-wealth effects (whether for welfare or forthe propagation mechanism) may have non-trivial implications.

In particular, consider an environment in which the market power of monopolistic�rms (�) is subject to exogenous stochastic disturbances. In this case, the PhillipsCurve is subject to a cost-push shock ut , capturing the stochastic deviations of �exible-price output from potential one:

�t D ˇEt�tC1 C �xt C ut ; (46)

where cost-push shocks follow ut D �uut�1C "ut , with �u 2 Œ0; 1� and "ut �N.0; �2u/.

Also in this case, the targeting rules (39) and (43)–(45) characterize the optimalpolicy under discretion and commitment, respectively. The welfare losses implied byalternative policy regimes, however, now depends on the cyclical features of bothshocks.

Consider, as before, purely transitory shocks: a productivity shock at with standarddeviation �a D 0:5%, and a cost-push shock ut with standard deviation �u D 0:08%.33

I �rst evaluate the relevance of �nancial-wealth effects for the monetary-policy lossfunction, by comparing the dynamic and cyclical implications of four alternative policyregimes. The �rst two regimes are the optimal policy under discretion and commitment,respectively. The other two regimes consider instead the case in which the central bankdisregards the welfare implications of �nancial-wealth effects and therefore adopts theloss function that would arise in the standard LAMP setup, as in Bilbiie (2008):

Lt0 D.1C '/

2#Et0

´1XtDt0

ˇt�t0�x2t C ˛��

2t

�µ; (47)

33. The standard deviations of productivity and cost-push shocks are calibrated such that the cyclicalimplications of the model are consistent with empirical evidence along two dimensions: i) cost-push shocksare about 15% as important as productivity shocks in explaining the variance of real output, and ii) thestandard deviation of the cross-sectional variance of log-consumption is about 0.02 (consistently withKrueger and Perri, 2006). Consistency with the evidence above is evaluated with respect to a policy regimein which the central bank follows a standard Taylor Rule with response coef�cients to in�ation and outputgap equal to 1.5 and 0.5, respectively.



0 2 4 6−0.1

0

0.1

0.2

0.3Inflation (%)

0 2 4 6−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05Output Gap

0 2 4 63.8

4

4.2

4.4

4.6

4.8

5Interest Rate (%)

0 2 4 6−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05Financial Wealth

CommitmentFlexible Inflation Targeting: Commitment

0 2 4 60

0.001

0.002

0.003

0.004

0.005Cons. Dispersion

DiscretionFlexible Inflation Targeting: Discretion

0 2 4 6−0.02

0

0.02

0.04

0.06

0.08Cost−push Shock

Figure 2. Response of selected variables to a transitory, one-standard-deviation cost-push shock.Dashed line: optimal policy (Discretion). Solid line: optimal policy (Full Commitment). Dash-dotted line: �exible in�ation targeting (Discretion). Dotted line: �exible in�ation targeting (FullCommitment).

with ˛� � #"=�, which is maximized under either discretion or commitment. Iwill refer to this case as “Flexible In�ation Targeting” regime, consistently withSvensson (1999). Figure 2 displays the optimal response of the economy to a onestandard-deviation cost-push shock under the four regimes, and the �rst four columnsof Table 2 report their cyclical and welfare implications.

Ignoring the welfare role of �nancial instability by minimizing loss (47) insteadof (30) induces non-negligible differences both in the short-run response of theeconomy to a cost-push shock and in the implied welfare losses. In particular, themonetary policy response is overly restrictive, implying higher nominal interest rates(in the order of 20 to 30bp) and lower output gap (in the order of 7 to 10bp) andin�ation (between 3 and 5bp), while consumption dispersion increases by about twiceas much, under both discretion and commitment. As a result, the welfare loss impliedby ignoring the welfare relevance of �nancial instability in the DAMP economy isabout 31% higher in the case of discretion, and about 41% higher in the case ofcommitment, as implied by Table 2.

A second dimension along which the relevance of �nancial-wealth effects can beevaluated is their role in the propagation of structural shocks with respect to the LAMPbenchmark. Indeed, comparing the loss under optimal policy in the DAMP model (�rsttwo columns of Table 2) with the corresponding loss in the LAMP framework (i.e.� D 0, last two columns) shows that the welfare costs associated to the additional



Table 2. Implied volatilities and welfare losses conditional on productivity and cost-push shocks.

DAMP Benchmark

Flexible ITDisc Comm Disc Comm Disc Comm

In�ation 0.067 0.055 0.052 0.045 0.052 0.045Output Gap 0.264 0.284 0.250 0.216 0.250 0.216

Financial Wealth 0.347 0.310 0.544 0.523 0.544 0.529Interest Rate 0.339 0.235 0.551 0.505 0.551 0.506

Cons. Dispersion 0.030 0.025 0.073 0.068 0.000 0.000

Welfare Loss 0.326 0.263 0.429 0.371 0.146 0.110

Note: implied volatilities are standard deviations in percentage points; Welfare Losses are basis points of steady-state aggregate consumption. Standard deviation of productivity shocks is 0.5%; standard deviations of cost-pushshocks is 0.08%. Flexible IT: Flexible In�ation Targeting; Disc: Discretion; Comm: Full Commitment.

distortion characterizing the DAMP economy are substantial, as they imply welfarelosses more than twice as high, under both discretion and commitment.

Therefore, the total welfare costs of ignoring the �nancial-wealth effects re�ectboth dimensions analyzed above – their role in the propagation of structural shocksand in the relevant welfare criterion – and can be evaluated by comparing the welfareloss implied by the Flexible In�ation Targeting regime in the DAMP economy (thirdand fourth columns in Table 2) with the loss under optimal policy in the benchmarkLAMP framework (last two columns): a policy maker that believes to be operating inan economy where �nancial developments are neutral for equilibrium allocation when,in fact, they are not, induces, on average, welfare losses that are about 3 times higherthan expected, under both discretion and commitment.

Overall, the quantitative relevance of these welfare implications – though theymay seem small in absolute terms – is rather sizeable in relative terms, and suggestspotentially important implications of ignoring the �nancial-wealth channel in thedesign of monetary policy.34

4. Conclusion

This paper presents a dynamic stochastic general equilibrium model of an economywhere agents participate to asset markets only discontinuously. The framework impliestwo layers of agents heterogeneity: between types (market participants versus rule-of-thumbers) and within the type of market participants (because of different longevities inthe type). This latter kind of heterogeneity implies �nancial-wealth effects on aggregateconsumption and makes all agents non-Ricardian.

34. Also in absolute terms, the reported welfare implications are in the same order of magnitude thatcharacterizes welfare exercises of this kind that have been used in the literature to rank alternative policyregimes. See, among others, Galí and Monacelli (2005). Please refer to the Online Appendix for a sensitivityanalysis of the implications above.



The discontinuous asset market participation (DAMP) model nests as specialcases most frameworks currently used to study monetary policy: (i) the standardrepresentative-agent model, where all agents are in�nitely-lived, homogeneous, andwith continuous access to asset markets (hence Ricardian); (ii) the LAMP model,where all agents are in�nitely-lived and heterogeneous only between groups, as somehave continuous access to asset markets (Ricardian) and some have no access at all(non-Ricardian); (iii) the perpetual-youth model, where all agents have continuousaccess to asset markets but they are �nitely-lived, and therefore heterogeneous(and non-Ricardian). With respect to the latter framework, one appealing feature ofthe DAMP model is that it implies the same kind of wealth-effects on aggregateconsumption, but within an in�nitely-lived-agent economy: the replacement rate inasset markets is fundamentally disconnected from expected lifetimes. The exogenoustransition between agent types captures, in reduced form, the effects of occasionally-binding borrowing (or liquidity) constraints induced by different kinds of idiosyncraticuncertainty: the DAMP model is thus �exible enough to address different issues relatedto agents’ heterogeneity, where the speci�c structural interpretation is re�ected in thecalibration of the relevant transition probabilities.

Within a DAMP economy with an active stock market – where the transitionbetween agent types is interpreted as a reduced-form representation of the interactionbetween a costly decision to participate in asset markets and idiosyncratic uncertaintyabout individual income – the paper then provides a formal normative analysisof monetary policy, deriving a welfare-based monetary-policy loss function andcharacterizing the interplay between optimal policy and �nancial stability.

A second contribution is the derivation of the welfare criterion per se: since thedemand side of the economy features an in�nite number of heterogenous agents, itimplies aggregation issues, when deriving a second-order approximation of socialwelfare. These issues arise also in standard perpetual-youth models, and are behind thelack of optimal policy analyses in a linear-quadratic framework within such models. Inthis respect, the contribution of the paper is to show how to deal with such heterogeneityand derive a quadratic welfare criterion as a function of aggregate variables only. Thederivation applies to both perpetual-youth and DAMP models.

The third contribution is heuristic. The paper shows that the discontinuity ofasset market participation implies an additional dynamic distortion in the model,which results in an additional term in the welfare criterion: while the social plannerallocation implies that the cross-sectional consumption distribution is invariant toaggregate shocks, in the decentralized allocation it endogenously responds to anyshock inducing �uctuations in aggregate �nancial wealth. Hence, �nancial stabilityappears in the welfare-based loss function and becomes an explicit additional targetfor welfare-maximizing policy makers, besides in�ation and output stability. Finally,in this economy an endogenous trade-off arises between real and �nancial stabilization,even in the absence of cost-push or �nancial shocks. The ultimate implication is thatprice stability is no longer necessarily optimal: given the quadratic form of the welfarecriterion, some �uctuations in output and in�ation may be optimal as long as they



reduce �nancial instability.35 Ignoring the heterogeneity of asset-market participantspotentially leads monetary policy to incur substantially higher welfare losses, underboth discretion and full commitment.

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