learning, monetary policy and asset prices · learning, monetary policy and asset prices* marco...

Luiss Lab of European Economics LLEE Working Document no. 48

Learning, Monetary Policy and Asset Prices

Marco Airaudo, Salvatore Nisticò, Luis-Felipe Zanna

April 2007

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© Copyright 2007, Marco Airaudo, Salvatore Nisticò, Luis-Felipe Zanna Freely available for downloading at the LLEE website (http://www.luiss.it/ricerca/centri/llee)

Email: [email protected]

Learning, Monetary Policy and Asset Prices*

Marco Airaudo°, Salvatore Nisticò§, Luis-Felipe Zanna#

LLEE Working Document No.48 April 2007

Abstract

We explore whether monetary policy should respond to stock-price fluctuations in a New-Keynesian economy with Non-Ricardian agents. In this framework, risky equities are net wealth and affect the dynamics of both aggregate consumption and inflation. We show that in this framework the optimal discretionary monetary policy under Rational Expectations is a Taylor-type interest rate rule responding to the expectations of both aggregate inflation and a stock-price index. We then study the determinacy and the stability under adaptive learning of the Minimum State Variable (MSV) representation of the fundamental Rational Expectation Equilibrium (REE) for both the optimal monetary policy rule and for generic forward-looking Taylor-type interest rate rules that systematically respond to stock price expectations. Our results show that a Central Bank trying to achieve macroeconomic stability by responding to the expected stock price can induce multiple sunspot-driven equilibria and undermine the asymptotic convergence of the economy to the fundamental REE. We provide analytical conditions for the MSV-REE to be unique and stable under adaptive learning. Furthermore, we show that the problem of Indeterminacy/E-instability is more likely to occur when product markets are more competitive (and therefore stock dividends lower on average) and the Non-Ricardian agents have a longer planning horizon. We conclude by showing that explicitly targeting stock price stability makes Optimal Monetary Policy more likely to be E-unstable. JEL classification: E4, E5 Key words: Learning; Expectational Stability; Interest Rate Rules; Multiple Equilibria; Determinacy, Stock Prices.

(*) The authors have bene.ted from comments and conversations with James Bullard, Efrem Castelnuovo, Urban Jermann, Tommaso Monacelli, Francesco Sangiorgi, Eric Schaling, Harald Uhlig, and seminar participants at the Society for Computational Economics 2006, the Learning Week 2006 at the St. Louis Fed, the University of Padova and Collegio Carlo Alberto. All errors remain ours. The views expressed in this paper are solely the responsability of the authors and should not be interpreted as re.ecting the view of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System.

Marco Airaudo acknowledges the .nancial support from the Modigliani Fellowship of the Associazione Borsisti "Marco Fanno" (Medio Credito Centrale). (°) Department of Economics and Finance "G. Prato", University of Turin, Corso Unione Sovietica 218 bis, Torino, Italy. Collegio Carlo Alberto, Via Real Collegio 30, Moncalieri (TO), 10024, Italy. Phone: +39 0116705271. Email: [email protected] (§) Università “G. D’Annunzio” in Pescara and Libera Universita.Internazionale degli Studi Sociali (LUISS) "Guido Carli", Via Tommasini 1, Rome, Italy. Email: [email protected] (#) Board of Governors of the Federal Reserve System, 20th Street and Constitution Avenue, NW, Washington, D.C., 20551. Phone: (202)452-2337. Fax: (202)736-5638. E-mail: [email protected]

1 Introduction

The strong shift of private savings towards the stock market suggested that the 1990�s boom in consumption

was �nanced by heavily relying on stock market performances. The Federal Reserve appeared seriously

concerned about the links between �nancial and real stability, as well as the perils that irrational exuberance

might have exerted over consumers�and investors�con�dence and thereby on real activity. In this context,

a lively debate started in the economic literature aimed at de�ning the appropriate response of monetary

policy to large swings in the stock prices. Although economists have recognized the need to include some

form of market incompleteness/�nancial friction into standard New-Keynesian models in order to have a

signi�cant stock market channel, so far there has not been any agreement on whether an explicit response

to stock prices by monetary policy could be bene�cial.1

On the one hand, the "�nancial accellerator" model of Bernanke and Gertler (1999, 2001) claims that

since the macroeconomic relevance of stock price dynamics relies on its links with in�ation, a �exible in�a-

tion targeting approach is su¢ cient to achieve both price and �nancial stability. They even suggest that a

response to stock prices might induce perverse outcomes in the dynamics of real output. On the other hand,

Cecchetti et al. (2000, 2002, 2003) recommend that a central bank recognizing a stock price bubble in the

market should de�nitively respond to it. Non-fundamentals-driven booms/busts in the stock market do in

fact create ine¢ cient cycles in investments and real activity (and therefore welfare) via their impact on the

costs of external �nancing. Although their argument is theoretically sound, the design and implementation

of a monetary policy that reacts to � bubbles" is a di¢ cult task. As pointed out by Cogley (1999), given

the underlying uncertainty about the true model for the economy, speculative bubbles can in fact be ob-

servationally equivalent to movements in unobserved fundamentals. In the real world, central bankers face

a harder dilemma: responding or not responding to stock prices with very limited information on whether

they are driven by fundamentals or by bubbles. In this respect, some empirical works have provided evidence

suggesting that, no matter if optimal or not, some major Central Banks seem to have responded to asset

price �uctuations over the last few years.2

The lack of consensus in the literature on whether a central bank should or should not move the short

term nominal interest rate in response to asset prices �uctuations is what motivates our paper. We challenge

this issue from a broad perspective. We present a New-Keynesian DSGE model with an explicit stock market

channel and solve the Optimal Monetary Policy problem of a central bank under discretion. We show that

the optimal interest rate rule under Rational Expectations should respond to stock price expectations. Next,

1Bernanke and Gertler (1999,2001) and Cecchetti et al. (2000,2002,2003) are the most cited works on this issue. Other

interesting contributions Chadha, Sarno and Valente (2004), Di Giorgio and Nisticò (2006), Dupor (2005), Faia and Mona-

celli (2005), Filardo (2000), Goodhart and Hofmann (2002), Gruen, Plumb and Stone (2005), Ludvigson and Steindel (1999),

Miller, Weller and Zhang (2001), Mishkin (2001), and Schwartz (2002).2See, for instance, Rigobon and Sacks (2003) and Chadha et al. (2003).

2

we extensively study the equilibrium determinacy and the stability under adaptive learning (E-stability)

properties of the optimal rule and a general class of expectational Taylor rules.

Our analysis departs the Bernanke-Gertler�s approach in two respects. First, rather than focusing on

the supply-side e¤ects of asset price volatility given by their �nancial accelerator, we present a model where

stock price �uctuations have direct wealth e¤ects on the demand for consumption. We consider an economy

populated by Non-Ricardian agents, adapting the stochastic �nite-lifetime framework of Blanchard (1985)

and Yaari (1965) to a New-Keynesian monetary economy with risky equities. In this set-up, agents face a

constant probability of dying in every-period and therefore do not spread the e¤ect of stock price swings over

the entire in�nite future. Stock market booms (or bursts) induce then a larger increase (decrease) in current

consumption with respect to the standard in�nitively-lived case. By making the dynamics of aggregate

�nancial wealth relevant for current aggregate consumption as well as for current in�ation, this demand-side

channel introduces some real e¤ects of stock prices volatility. The appealing feature of our model is its

tractability. As the death probability gets closer to zero, our model approaches the case of in�nitively lived

Ricardian agents that is typically studied in the literature. We are then able to provide analytical results

for the in�nitively-lived case as well.3

Second, in the analysis of the log-linear approximation of this DSGE model we slightly depart from

Rational Expectations, following the line of Evans and Honkapohja (2001). Namely we assume that agents are

not endowed with Rational Expectations at the outset but form forecasts using recursive learning algorithms

on observed data. Boundedly-rational agents can over/underestimate the impact of fundamental shocks �

such as productivity or demand shocks � on in�ation, output and more notably the stock price because

of imperfect knowledge on how the economy works and/or what monetary policy rules the central bank is

adopting.4 A deviation of agents�expectations from the fundamental REE might then induce expectations-

driven endogenous cycles and therefore introduce non-fundamental aggregate instability.

In this Non-Ricardian New-Keynesian economy, we study whether the Central Bank should include

the stock price index in its interest rate rule. We study the desirability of such a rule from both the

"optimal monetary policy" and the "determinacy/expectational stability" point of views. Relating to the

second criterion, we consider a rule to be desirable if : 1) it delivers a unique equilibrium under Rational

Expectations; 2) the Minimal State Variable (MSV) representation of the fundamental Rational Expectations

Equilibrium (MSV-REE) is learnable in the E-stability sense de�ned by Evans and Honkapohja (2001). In

3Altissimo et al. (2005) documet strong evidence of wealth e¤ects on consumption from �nancial assets holdings for some

major industrialized economies. Their computation of the marginal propensity to consume out of �nancial wealth relies on a

calibrated version of the Blanchard-Yaari model.4We do not actually distinguish between these two possible sources of "non-rationality", namely, on the one side, the limited

knowledge of the aggregate economy decision making process and of the market clearing mechanism, and, on the other side,

the lack of transparency of monetary policy conduct by the central bank. See Eusepi (2005) on this issue.

3

particular we try to assess whether a systematic response to stock prices by the central bank induces the

economy to converge to the MSV-REE, when agents make persistent forecast errors due to non-rational

expectations.5

Our key results are the followings. First of all, we show that with Non-Ricardian agents the optimal

discretionary monetary policy under Rational Expectations is a Taylor-type interest rate rule whereby the

short term nominal interest rate responds to both expected in�ation the expected stock-price. In this sense,

our model identi�es a demand-side channel that rationalizes why Central Banks should pay attention to

�uctuations in asset prices. However, if the economy was populated by in�nitively lived Ricardian agents it

would not be optimal for the central bank to respond to stock prices.

Second, we show that not all forward-looking interest rate rules responding to stock price expectations

are able to deliver a unique and learnable REE. On the contrary, by responding positively to stock price

�uctuations the central bank might actually induce multiple sunspot-driven REE and, even worse, impede

convergence to the fundamental REE under persistent expectational errors by agents. We show that the

MSV-REE is more likely to be E-unstable the lower is the market power of the price-setting �rms, i.e. the

lower the average dividends, and the smaller are the wealth e¤ects from stock price �uctuations, i.e. the

longer the planning horizon of the Non-Ricardian agents.

Third, we show how a speci�c taste for stock-price stability in the Central Bank preferences makes the E-

stability of the optimal monetary policy REE more di¢ cult to obtain. In this respect we obtain a result that

quite di¤ers from the well-known work of Evans and Honkapoja (2003). They claim that a rule responding

to the subjective (non-Rational) expectations of economic agents is always able to stabilize the economy

around the optimal REE. We show that in a regime of stock price targeting this is not true.

Our work is related to the previous contributions of Bullard and Mitra (2002, 2005), Bullard and Schaling

(2002) and the more recent work by Carlstrom and Fuerst (2006). Bullard and Mitra (2002,2005) show

that even if it is possible to design interest rate rules that lead to a (locally) unique Rational Expectation

Equilibrium, that equilibrium is not necessarily learnable in the Evans and Honkapoja (2001) sense. In

some circumstances, small expectational errors by economic agents can make that equilibrium unachievable.

Bullard and Schaling (2002) are the �rst to highlight the possibly adverse consequence of responding to the

stock-price index in a standard monetary model, of the type of Bullard and Mitra (2002). Carlstrom and

Fuerst (2006) fully develop the idea of Bullard and Schaling (2002) in a more complete model. Their key

result is that by responding systematically to the stock-price index the central bank can make the targeted

REE expectationally unstable under a rule that would otherwise be stabilizing, absent any response to the

stock price itself. Although insightful, their model is an in�nitively lived representative agent model where

5We do not address the issue learnable sunspot equilibria. In this sense, our work is similar in spirit to Bullard and Mitra

(2001,2003) and Airaudo and Zanna (2005).

4

stock price �uctuations have no real e¤ects whatsoever. Responding to stock prices in that economy is never

optimal. In contrast, we analyze the importance of reacting to stock prices �uctuations in a model that has

a demand-side channel through which stock market �uctuations can have real e¤ects.

The paper is organized as follows. In Section 2 we present the theoretical model. Section 3 presents the

log-linear reduced form equilibrium on which the analysis is based. Section 4 derives the optimal interest rate

rule under discretion, assuming that agents are endowed with Rational Expectations. Section 5 presents the

analysis of equilibrium determinacy and E-stability for both general forward-looking interest rate rules and

for the optimal rule derived in Section 4. We provide analytical results for both the case of Non-Ricardian

and of Ricardian agents. Section 6 concludes.

2 A Structural Model with Wealth E¤ects.

This section outlines the key features of the theoretical model. We adapt the stochastic �nite-lifetime model

of Blanchard (1985) and Yaari (1965) to a New-Keynesian production economy with risky equities, following

Nisticò (2005). Two key features characterize our economy. First of all, because agents are Non-Ricardian,

stock-price �uctuations have real e¤ects on individual demand as the dynamics of aggregate �nancial wealth

a¤ect consumption and therefore in�ation. Second, microfounded cost-push shocks à-la Steinsson (2003)

induce a non-trivial trade-o¤ between in�ation and output stabilization.

2.1 The Demand-Side

The demand-side of the economy is a discrete-time stochastic version of the perpetual youth model introduced

by Blanchard (1985) and Yaari (1965). The economy is populated by an inde�nite number of cohorts of

Non-Ricardian agents who survive between any two subsequent periods with constant probability 1 � .

Assuming that birth and death rates are the same, and that total population has size 1, in each period

exactly a fraction of the alive population dies and a cohort of size of newborns enters the economy.6

Lifetime utility of the representative agent of age j at time 0 is given by

E0

1Xt=0

�t(1� )thlnCj;t + ln(1�Nj;t)

i(1)

where �; 2 [0; 1]; and the instantaneous utility is assumed to be log-separable between consumption (Cj;t)6We interpret the concepts of � living" and � dying" in the economic sense of being or not being operative in the markets,

a¤ecting economic activity through the individual decision-making process. In this perspective, the expected life-time 1=

is interpreted as the e¤ective decision horizon. For other recent discrete-time versions of the perpetual youth model see,

among others, Annicchiarico etl. al. (2004), Cardia (1991), Chadha and Nolan (2001,2003), Cushing (1999), Leith and Wren-

Lewis (2000), Leith and vonThadden (2004), Piergallini (2004) and Smets and Wouters (2002).

5

and leisure time (1�Nj;t).7 Future utility is discounted because of impatience (the intertemporal discount

factor �) and uncertain lifetime (the probability of survival between any two subsequent periods, (1 � )).

The possibly non-rational expectation operator E0 accounts for the uncertainty characterizing our economy:

the stochastic lifetime and the various fundamental shocks.

We consider a cashless economy - i.e. money is only a unit of account - whereby consumers have access to

two types of �nancial assets: state-contingent bonds issued by the government and equity shares issued by

monopolistically competitive �rms, to which they also supply labor.8 At the end of period t the representative

agent of age j holds a portfolio of contingent claims with one-period ahead stochastic nominal payo¤Bj;t+1

in period t + 1 - which he discounts according to the stochastic discount factor Fjt;t+1 - as well as a set

of equity shares issued by each intermediate good-producing �rm, Zj;t+1(i), whose real price at period t is

Qt(i). Therefore, his resources are given by current nominal net labor income (WtNj;t � PtTj;t), and the

nominal �nancial wealth j;t carried over from the previous period, including the nominal pay-o¤s on the

contingent claims, Bj;t; and the "price plus dividend" on each share of the equity portfolio, Qt(i) +Dt(i):

j;t �1

1�

"Bj;t + Pt

Z 1

0

�Qt(i) +Dt(i)

�Zj;t(i) di

#: (2)

As in Blanchard (1985), �nancial wealth j;t also pays o¤ the gross return on the insurance contract that

redistributes among survived consumers the �nancial wealth of the ones who have left the market. Total

personal �nancial wealth is therefore accrued by a factor of 11� .

9

The optimization problem faced at time 0 by the representative agent of age j is therefore to maximize

(1) subject to a sequence of budget constraints of the following form:

PtCj;t + EtfFjt;t+1Bj;t+1g+ PtZ 1

0

Qt(i)Zj;t+1(i) di �WtNj;t � PtTj;t +j;t; (3)

and to a No-Ponzi game condition

limk!1

Et

nFjt;t+k(1� )

kj;t+k

o= 0: (4)

The �rst-order conditions imply the following relationships:

Cj;t =Wt

Pt(1�Nj;t) (5)

PtQt(i) = Et

nFjt;t+1Pt+1

hQt+1(i) +Dt+1(i)

io; (6)

7The assumption that the utility function is logarithmic is necessary in order to retrieve time-invariant parameters charac-

terizing the equilibrium conditions. See Smets and Wouters (2002) for a non-stochastic framework with CRRA utility.8The analysis would not be qualitatively a¤ected by including real money balances into the utility function, as long as

preferences are log-separable. See Piergallini (2004).9Perfect competition and free entry into the insurance market imply that for each unit of wealth left at the insurance �rm

each agent will receive 11� units conditional on his survival.

6

(1 + rt)Et�Fjt;t+1

= 1 (7)

PtCj;t =1

�(j;t + hj;t); (8)

Equation (5) is the standard consumption-leisure trade-o¤. Equation (6) is the pricing equation for an equity

share of �rm i; while equation (7) is the related arbitrage condition determining the risk-free nominal interest

rate rt: Finally, equation (8) - which comes from forward-iteration of (3) together with the limiting condition

(4) - describes nominal individual consumption as a linear function of total nominal �nancial and human

wealth. In this last expression, hj;t � Et�P1

k=0 Fjt;t+k(1 � )k(Wt+kNj;t+k � Pt+kTj;t+k)

de�nes human

wealth as the expected stream of future disposable labor income, discounted by the individual stochastic

discount factor and conditional upon survival, while � � [1� �(1� )]�1 is the reciprocal of the marginal

propensity to consume out of �nancial and human wealth, common across cohorts.10

As shown in Nisticò (2005), the linearity of the �rst order conditions allows us to easily aggregate across

cohorts by averaging the corresponding generation-speci�c counterparts.11 The demand-side of our economy

can then be reduced to the following set of equations: a pricing equation for each wholesale stock i,

PtQt(i) = Et

nFt;t+1Pt+1

hQt+1(i) +Dt+1(i)

io; (9)

an aggregate labor supply equation

Ct =Wt

Pt(1�Nt); (10)

and the aggregate Euler Equation for consumption

(�� 1)PtCt = Et�Ft;t+1t+1

+ (1� )�Et

�Ft;t+1Pt+1Ct+1

; (11)

where aggregate �nancial wealth is

t �"Bt + Pt

Z 1

0

�Qt(i) +Dt(i)

�Zt(i) di

#(12)

Notice that the �rst term in (11) represents the �nancial wealth e¤ect, which fades out as the probability

of exiting the market goes to zero.

2.2 The Supply-Side and In�ation Dynamics.

The supply-side of the economy consists of two in�nitively-lived sectors: a retail sector operating in perfect

competition to produce the �nal consumption good, and a wholesale sector hiring labor from the households

to produce a continuum of di¤erentiated intermediate goods.10A more detailed derivation of these conditions can be found in Nisticò (2005) or obtained from the authors upon request.11Since the probability of surviving each period is (1� ) and the size of each newborn cohort was set to , the aggregated level

across cohorts for each generation-speci�c variable Xj;t is computed as the weighted average Xt �Ptj=�1 (1 � )t�jXj;t,

for all X = C;N;B; T; h; Z(i);F .

7

In the retail sector the �nal consumption good Yt is produced out of the intermediate goods through the

following CRS technology:

Yt =

"Z 1

0

Yt(i)(�t�1)=�tdi

#�t=(�t�1);

where �t > 1 is the inverse of the intratemporal elasticity of substitution between intermediate goods and

re�ects the degree of competition in the market for inputs. Following Steinsson (2003) and Ireland (2004)

we assume that �t follows a log-stationary stochastic process:

ln �t = (1� ��) ln �+ �� ln �t�1 + �t

�� 2 (0; 1) ; �t �WN�0; �2�

�In equilibrium, this translates into a cost-push shock entering the New-Keynesian Phillips Curve (NKPC)

which induces a trade-o¤ between in�ation and output stabilization in optimal monetary policy design.

Under perfect competition and �exible prices, the optimal demand for the intermediate good Yt(i) and the

�nal good price Pt are, respectively:

Yt(i) =

"Pt(i)

Pt

#��tYt; (13)

Pt =

"Z 1

0

Pt(i)1��t di

#1=(1��t): (14)

The �rms in the wholesale sector operate in monopolistic competition to produce a continuum of dif-

ferentiated perishable intermediate goods out of hours worked, according to the following linear production

function

Yt(i) = AtNt(i); (15)

at = (1� �a)a+ �aat�1 + �a;t

at � lnAt; �a 2 (0; 1) ; �a;t �WN�0; �2a

�In choosing the optimal level of labor demand, each �rm i enters a competitive labor market and seeks to

minimize total real costs subject to the technological constraint (15). Real marginal costs are MCt =Wt

AtPt

and therefore equal across �rms. We introduce nominal rigidities following Calvo�s (1983) staggered price

setting: each �rm in the wholesale sector optimally revises its price with probability 1�� in any given period

t: The optimal price is set to maximize the expected stream of future dividends taking into account that it

will be charged till period t + k with probability �k. Given the common technology, perfect labor mobility

and the fact that both shocks at and �t are aggregate, each revising �rm will set the same optimal price.

Therefore, the solution to the optimal price setting problem implies the familiar log-linear version of the

8

New-Keynesian Phillips Curve (NKPC) for aggregate in�ation12 :

�t = e�Et�t+1 + (1� �)(1� �e�)�

mct + ut; (16)

where e� � �1+ and � 1��(1� )(1� )

PC is a factor proportional to the steady state real �nancial wealth

to consumption ratio, PC :

13 The marginal costs mct take the familiar form mct = wt � pt � at while the

cost-push shock ut is de�ned as

ut � (1� �) (1� �)(1� �e�)

�ln(�t=�) (17)

ut = ��ut�1 + "u;t; "u;t � iid�0; �2u

�where � � �

��1 is the steady state gross mark-up (or the reciprocal of the steady state real marginal costs).

Notice that a negative shock to the degree of competition - i.e. a higher �t and therefore a higher ut �raises

the market power and the desired markup of wholesalers; it then has a direct and positive e¤ect on in�ation:

all price setting �rms raise prices more than they would have done if the shock had not occurred.

2.3 The Equilibrium

Aggregating across �rms and using equation (13) we obtain aggregate output:

AtNt = Yt

Z 1

0

�Pt(i)

Pt

��tdi = Yt�t;

where Nt �R 10Nt(i) di is the aggregate level of hours worked and �t �

R 10

�Pt(i)Pt

��tdi is an index of price

dispersion over the continuum of intermediate goods-producing �rms.14 . Clearly, in equilibrium, the aggre-

gate stock of outstanding equity for each intermediate good-producing �rm must equal the corresponding

total amount of issued shares, which, without loss of generality, we normalize to 1, i.e.RjZt(i) = 1 for

all i 2 [0; 1]. De�ning total real dividend payments and the aggregate real stock-price index as the simple

integration over the continuum of �rms - i.e. Dt �R 10Dt(i) di and Qt �

R 10Qt(i) di - the demand-side of the

economy in equilibrium is given by the following aggregate resource constraints

Yt = Ct +Gt = Ct +$tYt; (18)

PtYt = NtWt + PtDt; (19)

the labor supply

Ct =Wt

Pt(1�Nt); (20)

12Lower case letters denote log-deviations from the respective steady state. Since at the steady state prices are constant, the

steady state in�ation rate is zero. See Appendix A.2.13See Appendix A.1.14The price dispersion index �t will not appear in the log-linearized equilibrium since its log is of second-order.

9

the aggregate Euler equation

(�� 1)Ct = Qt + (1� )�Et�Ft;t+1�t+1Ct+1

; (21)

and two pricing equations

Qt = Et

nFt;t+1�t+1

hQt+1 +Dt+1

io(22)

1 = (1 + rt)Et�Ft;t+1

(23)

Equation (21) de�nes the dynamic path of aggregate consumption, which is clearly a¤ected by the

dynamics of the stock-price Qt de�ned in equation (22).15 Equations (22) and (23) then determine the

1-period risk-less nominal interest rate rt:

Et

nFt;t+1�t+1

hQt+1+Dt+1

Qt

ioEt�Ft;t+1

= (1 + rt) (24)

Notice that a model with identical in�nitely-lived Ricardian consumers is a special case of the one dis-

cussed here, and corresponds to a zero-probability of death, = 0. In this case, the stock price Qt disappears

from equation (21), and the latter collapses to the standard Euler Equation for aggregate consumption.

We consider a balanced-budget �scal policy equilibrium such that the aggregate supply of state-contingent

bonds is always zero, i.e. Bt = 0 for any t: We assume that government expenditure is a stochastic fraction

$t of total output, Gt = $tYt; and that it is entirely �nanced entirely by lump-sum taxation, i.e. Gt = Tt.16

3 The Log-Linearized Equilibrium

We are interested in the equilibrium dynamics around the unique non-stochastic zero-in�ation steady state

of the economy. In an arbitrarily small neighborhood of such steady state, the demand-side is described (to

15Equation (22) derives from the aggregation across �rms of the individual asset price condition (6).16 It might well be that other forms of �scal policy rules could have some impact on the dynamics of stock prices. We leave

this issue for future research.

10

�rst-order) by the following log-linear relationships:17

yt = ct + gt (25)

yt = at + nt (26)

wt � pt = ct + 'nt (27)

ct =1

1 + Etct+1 +

1 + qt �

1

1 + (rt � Et�t+1 � ~�) (28)

qt = ~�Etqt+1 +�1� ~�

�Etdt+1 � (rt � Et�t+1 � ~�) (29)

dt =Y

Dyt �

WN

PD(nt + wt � pt); (30)

where (1 + r)�1 = ~� � �1+ , ' �

N1�N is the inverse of the (steady-state) Frisch elasticity of labor supply,

and we have de�ned gt � � log�1�$t

1�$�.

The log-linear supply-side is given by the NKPC (16) where, by (25)-(27), real marginal costs mct =

wt � pt � at are equal to:

mct = (1 + ')(yt � at)� gt (31)

In the limiting case of full price �exibility � for � ! 0 � and in the absence of ine¢ cient shocks to

the markup, all �rms would set their price as a constant markup over nominal marginal costs: Pnt (i) =

�PtMCnt = Pt.18 Along this equilibrium, real marginal costs would match their long-run level at each point

in time, such that mcnt = 0. Imposing this condition on equation (31) allows us to retrieve the equation for

the natural level of output, de�ned as the (log) level of output prevailing under no nominal rigidities and no

ine¢ cient shocks:

ynt = at +1

1 + 'gt: (32)

De�ning the output gap by xt � yt � ynt , real marginal costs are:

mct = (1 + ')xt: (33)

By combining (25)-(33), the equilibrium economic dynamics are described by the following linear system

17 In what follows lower-case letters denote log-deviations from the steady state: xt � log(Xt=X). Note that, (1 + rt) being

the gross interest rate, rt is (to �rst order) the actual net interest rate. The log-deviation of the gross interest rate from its

steady state is therefore rt � ~�, where we set ~� � log(1 + r) = � log ~�. The unique steady state is computed in the Appendix.18Henceforth we label the value that each variable takes in this benchmark equilibrium as natural (or potential), and denote

it by a superscript n.

11

in deviations from the �exible price equilibrium:

xt =1

1 + Etxt+1 +

1 + st �

1

1 + (rt � Et�t+1 � rrnt ) (34)

st = ~�Etst+1 � �Etxt+1 � (rt � Et�t+1 � rrnt ) (35)

�t = ~�Et�t+1 + �xt + ut; (36)

where � ��1� ~�

��1+'��1 � 1

�and � � (1��)(1��~�)

� (1 + '), rrnt is the real interest rate that would prevail

in the "zero in�ation - zero output gap" benchmark equilibrium, and st � qt � qnt is the log-deviation of the

stock-price from the �exible price equilibrium level.19

The local equilibrium dynamics are therefore entirely described by a forward-looking IS equation (34),

the pricing equation for equities (35) and the Neo-Keynesian Phillips Curve (NKPC) (36). The key element

of the model is the term 1+ st entering (34). With Non-Ricardian agents, > 0 and therefore > 0;

the stock price dynamics given by (35) a¤ect the output gap via the real wealth e¤ects of equity holdings.

This does not occur in an in�nitively-lived representative agent economy since ; = 0: Furthermore, by

the properties of spelled in Appendix A.1, the larger the real wealth e¤ects (i.e. higher and therefore

a shorter planning horizon of the agents) the stronger the impact of the current stock price, and the lower

the impact of the real interest rate on the output gap. As the probability of exiting the market next period

goes up, the current consumption of Non-Ricardian agents gets more sensitive to the current value of their

�nancial wealth while their incentive for intertemporal consumption smoothing diminishes.

4 Optimal Monetary Policy

In this section, we lay out the solution to the Optimal Monetary Policy problem under discretion for a

Central Bank subject to the reduced form equilibrium system (34)-(36). We show that, as long as there are

wealth e¤ects from holding equities, the optimal Taylor-type interest rate rule should respond not only to

expected in�ation but also to the expectations of the stock price index. The problem nests the standard

optimal monetary policy problem under discretion of Clarida et al. (1999): if the economy was populated by

in�nitively lived Ricardian agents, the optimal interest rate rule should respond only to expected in�ation.

19 rrnt is what Woodford (2003) calls the Wicksellian Natural Rate of Interest. To �nd the expressions for rrnt and q

nt we

need to solve (28) and (29) with respect to these two variables, along an equilibrium path with zero in�ation and yt = ynt ,

using conditions (25), (27) and (30), together with a no-bubble condition for the stock price. After simple algebra:

rrnt = ~�+ Et�at+1 �'

1 + 'Et�gt+1 +

�g(1� ~�)1 + � ~��g

gt (37)

qnt = ynt �1 + � �g

1 + � ~��ggt (38)

12

Suppose the central bank�s objective is to minimize the following:

1

2Et

1Xi=0

�i��2t+i + �

xx2t+i�

(39)

where �x is the relative weight for the volatility of the output gap and � is the policymaker�s discount rate.20

In order to characterize the optimal policy solution, we rewrite (34)-(36) as:

xt =1

1 + Etxt+1 +

1 + st �

1

1 + (rt � Et�t+1 � rrnt ) (40)

st =~�

1 + Etst+1 + xt �

1 + �

1 + Etxt+1 (41)

�t = ~�Et�t+1 + �xt + ut; (42)

The problem of optimal discretionary policy corresponds to a sequence of static problems in which the policy

maker chooses the nominal interest rate rt; in order to obtain the values of xt; �t, and st that minimize12

��2t+i + �

xx2t�+Gt, subject to st = xt+Rt and �t = �xt+ut+Nt: Here Gt; Rt; and Nt denote remainder

terms that are treated as given.21 The �rst order condition of this problem implies the typical short-run

in�ation-output trade-o¤:

�xxt + ��t = 0: (43)

The Central Bank then sets the optimal nominal interest rate rt in order to satisfy (43) together with (40)-

(42). Similarly to Clarida et. al (1999) and Evans and Honkapoja (2003), under Rational Expectations, the

solution to the optimal policy problem has the following MSV representation:

�t = a1 + d1ut; xt = a2 + d2ut; st = a3 + d3ut: (44)

where aj ; dj j = 1; 2; 3 depend on the structural parameters.22

The following Proposition shows that the presence of Non-Ricardian agents sensibly alters the optimal

monetary policy prescription we would get from a standard New-Keynesian model.

20Variances are with respect to some targets that without loss of generality we assume equal to zero. If �x = 0 the policy

maker becomes a pure in�ation targeter. We consider an in�nitively lived policy maker whose discount rate is simply �. In a

later section we consider the case of a Central Bank which explicitly cares about stock price stability by adding the stock price

volatility in the loss function. We show that stock price targeting can have some major implications for the E-Stability of the

optimal monetary policy REE.21Loosely speaking Expectations of future variables are taken as given since they can not be a¤ected by any current policy

announcement due to lack of credibility.22We provide analytical expression for these coe¢ cients in a later Section, while studying the determinacy and E-stability of

the Optimal Monetary Policy.

13

Proposition 1 Given the objective (39) and the stationary cost-push shock ut (17), in an economy populated

by Non-Ricardian agents, the optimal discretionary monetary policy under Rational Expectations is a Taylor-

type interest rate rule rt = rrnt + �RE� Et�t+1 + �

REs Etst+1 with the following features:

1. the rule entails always a positive response to the expected stock-price, i.e. �REs > 0; furthermore, when

� 2 (1; 2 + ') ; it is also active with respect to expected in�ation, i.e. �RE� > 1;

2. the higher the real wealth e¤ects from stock-price �uctuations, i.e. the higher , the higher should be

the responses to both expected in�ation and the expected stock-price.

Proof. See Appendix.

Should the Central Bank respond to stock prices if agents were in�nitively lived? The answer is: No.

When agents are Ricardian, �uctuations in current or expected stock-prices do not have any impact on

aggregate demand via the intertemporal Euler Equation. We are therefore back to the standard optimal

monetary policy case studied in Clarida et al. (1999): the Central Bank can contract demand by simply

raising the real interest rate, i.e. by responding actively to expected in�ation.

Corollary 2 In a standard New-Keynesian economy with in�nitively lived Ricardian agents the optimal

discretionary monetary policy under Rational Expectations is a Taylor-like interest rate rule responding

solely (and actively) to expected in�ation.


5 Determinacy and E-stability Analysis

This sections examines conditions for the determinacy of Rational Expectation Equilibrium (REE) and the

stability under adaptive learning (E-stability) of the fundamental REE when monetary policy takes the form

of a Taylor-type interest rate rule. Proposition 1 motivates us to pursue the analysis for a rule responding to

the expectations of both in�ation and the stock price. However, in order to get a broader prespective on the

implications of Taylor rules within our framework, we consider both the optimal interest rate rule de�ned in

Proposition 1, as well as more general forward-looking Taylor rules.

We start with the latter and characterize analytically the Determinacy/E-stability (D-ES) frontier in the

space of response coe¢ cients to expected in�ation, ��, and the expected stock price, �s: We show that,

other things being equal, the lowest value of �� guaranteeing D-ES is strictly increasing in �s : the higher

the response to the expected stock price, the more aggresive the rule should be towards expected in�ation.

Furthermore, we show that for any given (��; �s) D-ES is more likely to occur the shorter the planning

horizon of the Non-Ricardian agents and the bigger the average market power in the wholesale sector.

14

Next, we prove that the optimal interest rate rule of Proposition 1 always induces indeterminacy of REE

and E-instability of the fundamental REE. We conclude by showing that the Expectations-based optimal

interest rate rule proposed by Evans and Honkapoja (2003) can induce D-ES depending on whether the

Central Bank explicitly targets stock price stability or not.

5.1 Forward-looking Interest Rate Rules

Our methodology closely follows Bullard and Mitra (2002). Suppose that the Central Bank adopts a Taylor-

type interest rate rule whereby the short term nominal interest rate responds to the private sector�s expec-

tations of the output gap, in�ation and the stock price. Consider the following log-linear speci�cation:

rt = e�+ �xEtxt+1 + ��Et�t+1 + �sEtst+1 (45)

where rt � e� is the log-deviation of the nominal interest rate from its steady state, with e� = log (1 + r) ;

and the expectation operator Et is not necessarily rational.23 Let z0t � [xt; �t; st] and #0t � [at; gt; ut] be two

vectors containing, respectively, all the endogenous variables and all the fundamental shocks of our economy.

Substituting (45) into the equilibrium system (34)-(36) we obtain:

zt = K +Etzt+1 + �#t (46)

where K = 0, � is a conformable matrix including the coe¢ cients on the structural shocks and

�

266641� �1+ � �x 1� ��

1+ e� � �s

��1� �1+ � �x

� e� + � (1� ��) ��

1+

e� � �s�� x 1� �� e� � �s

37775 (47)

The determinacy of equilibrium analysis employs the standard procedure of Blanchard and Khan (1980).

Since none of the three endogenous variables is predetermined, under Rational Expectations the equilibrium

is (locally) determinate if and only if all eigenvalues of have real parts within the unit circle.24

When studying the stability under adaptive learning of the Minimum State Variable representation of

the fundamental Rational Expectation Equilibrium (MSV-REE), we follow Evans and Honkapohja (1999,

2001) and assume that agents are no longer endowed with rational expectations at the outset.25 Rather,

they form expectations adaptively by recursive least squares updating on the data generated by the system.

In particular, we focus on the concept of E-stability developed by Evans and Honkapoja (2001) as a criterion

23As we consider a zero-in�ation steady state, r stands both for the nominal and real steady state interest rate.24See also Farmer (1999).25The Minimum State Variable representation of the fundamental REE is the one developed in McCallum (1983) and Uhlig

(1999).

15

of learnability of an equilibrium: under some general conditions, the notional time concept of expectational

stability governs the local convergence of real time adaptive learning algorithms. In other words, if a REE

is E-stable, boundedly-rational agents will be able to converge towards it (i.e. to learn it) via their recurvie

least-squares learning algorithm.

The MSV representation of the fundamental REE of system (46) takes the following form:

zt = h+ P#t (48)

Under Rational Expectations, agents are assumed to know the true probability distribution of the endogenous

variables of the model, i.e. given the stochastic processes for the shocks in #t, they know the correct

representation of the fundametal REE both qualitatively (the linear form in (48)) and quantitatively (the

values of h and P ). We introduce bounded-rationality by assuming that agents form expectations based on

the correct linear form but need to estimate the values of h and P from past data.26 Agents have an initial

Perceived Law of Motion (PLM) zt = bh+ bP#t; which they use to compute the 1-period ahead expectationsentering system (46). This generates an Actual Law of Motion (ALM):

zt = K +hbh+ bPEt#t+1i+ �#t

= K +hbh+ bP�#ti+ �#t

= K +bh+ h bP�+ �i#twhere � � diag

��a; �g; ��

�: We can then de�ne a mapping T from the PLM to the ALM:

T�bh; bP� = �K +bh; bP�+ ��

The �xed points of this mapping correspond to the MSV-REE of our model, i.e. the MSV-REE is the (h; P )

solution to:

h (I � ) = K

P � P� = �

We say that an equilibrium described by the MSV representation is E-stable if the T -mapping is stable at

the equilibrium in question, i.e. the matrix di¤erential equation d(h;P )d�n

= T (h; P ) � (h; P ) in notional time26This is a very mild form of deviation from Rational Expectations. Agents know what are the relevant fundamentals driving

the economy, but do not know (or are unable to compute) their quantitative impact. We could think of various reasons for such

imperfect knowledge. For instance, an agent might not observe directly other agents�preferences, or might not understand how

markets clear, etc..A plausible explanation could also be the lack of transparency in monetary policy. If the monetary policy

rule followed by the Central Bank is not publicly observable, agents can not possibly know both h and P .

16

�n is locally asymptotically stable at the REE solution�h; P

�. As shown in Evans and Honkapoja (2001),

this requires the following matrices to have all eigenvalues with real parts less than 1:

DTh (h; P ) =

DTP (h; P ) = �0

Since all fundamental shocks are stationary, � has all roots with real parts less than one. A necessary and

su¢ cient condition for ES is therefore that all eivenvalues of have real parts less than 1, or, equivalently,

that all eigenvalues of � I have roots with negative real part. It is enough that one eigenvalue of � I

has a positive real part for the MSV-REE to be E-unstable.27

5.1.1 Non-Ricardian Agents

Due to the complexity of system (46)-(47), it is not possible to derive clear-cut analytical conditions for

equilibrium determinacy when 2 (0; 1). Figure 1 below provides a numerical charactetization of the D-ES

frontier for a calibrated version of our economy. On the other hand, under some mild assumptions, we can

identify necessary and su¢ cient conditions for the E-stability of the MSV-REE.

Proposition 3 Suppose that the central bank follows a forward-looking interest rate rule like (45) and that

�� > 1, i.e. the rule is active with respect to expected in�ation. Furthermore assume that �x � 0, � 2

(1; 2 + ') and that the steady state interest rate satis�es r >q

1+ .

1. When � 2 (0; ] ; the MSV solution zt = h+ P#t is E-stable for any �� > 1 and any �s � 0.

2. When � > ; the MSV solution zt = h+P#t is E-stable for any �� > 1 when �s 2�0;(1�e�+ )�x+(1�e�+�)

��

�;

and for any

�� > 1 +e� � 1�

1�e� + �1� e� + + 1� e��

1 + � e��s + e� � 1� �x (49)

when �s >(1�e�+ )�x+(1�e�+�)

�� :


From Proposition 3 we should take out two key points. First, a positive response to stock price expecta-

tions can undermine the stability under learning of the MSV-REE. The lower bound on �� de�ned in (49)

is clearly stricly increasing in the response coe¢ cient �s: This somehow contrasts with the optimal policy

27A fundamental part in the learnability analysis consists of making explicit what agents know when they form their forecasts.

In the E-stability analysis literature it is common to assume that when agents form their expectations Etzt; they do not know

zt: In this paper this assumption may be inconsistent with the assumptions that we use to derive the equations of the model.

Henceforth for the learnability analysis we will assume that when forming expectations agents know yt:

17

result of Proposition 1 which states that with Non-Ricardian agents it is optimal to respond to stock prices.

Second, the destabilizing e¤ect of �s depends on two key structural parameters of the model: the steady

state mark-up � in the wholesale sector - driving steady state dividends - and the length of the expected

planning horizon of the Non-Ricardian agents, i.e. the inverse of the death probability : Via their impact

on the two composite parameters � and ; both � and determine whether we fall in either one of the two

cases considered in Point 1. and 2. of Proposition 3.28 The next Corollary provides some details on the

impact of � and on E-stability.

Corollary 4 A rule that that responds to the expected stock price is more prone to induce E-instability of the

MSV-REE the lower is the average mark-up in the wholesale sector � and the longer the planning horizon

of the Non-Ricardian agents (i.e. the lower the death probability ):


We assess the likelihood of Indeterminacy/E-instability (I-EI) by plotting the relevant Determinacy/E�

Stability (D-ES) frontier in the space of policy coe¢ cient (��; �s), for a calibrated version of our economy.

CALIBRATION

Figure 1 displays the response coe¢ cient �s on the horizontal axis and the respose coe¢ cient �� on the

vertical axis. The plot corresponds to the case in Point 2 of Proposition 3 where the E-stability frontier with

respect to �� is stricly increasing in �s: The white area denotes the case of D-ES, while the dotted/shaded

area the case of I-EI. We notice the following features. First, the D-ES frontier is strictly increasing with

respect to �s for any �s > 0: This is because for our calibration the threshold value (1�e�+ )�x+(1�e�+�)

��

of �s is basically zero. Second, once �s > 0; the Taylor-Principle with respect to expected in�ation does no

guarantee D-ES. Third, when the MSV-REE is E-stable it is also the unique REE.29

The reason why a positive response to stock prices could push the economy away from the MSV-REE

has to do with the fact that, in equilibrium, dividends and the output gap are negatively correlated, and

that the magnitude of this correlation depends - among others - on the steady state mark-up �: From the

log-linearized equilibrium one can easily obtain:

dt =�� (2 + ')(1 + ') (�� 1)mct + at +

gt1 + '

(50)

=�� (2 + ')(�� 1) xt + at +

gt1 + '

28Notice that, from (49), a more aggressive response to the output gap is able to push down the lower bound on the response

to in�ationary expectations, thus enlarging the E-stability region in the (�� ; �s) plane.29From a theoretical point of view, if the model is purely forward-looking (no lagged terms in the system) then we have that

1) if the REE is determinate then it is also E-stable; 2) if the MSV-REE is E-stable it is not necessarily the unique REE. We

conjecture that for any given �s a su¢ ciently large �� would induce equilibrium indeterminacy while leaving the MSV-REE

still E-stable.

18

For � 2 (1; 2 + ') ; dividends depend negatively on real marginal costs, and therefore on the output gap.

Given that ' = [� (1�$)]�1 ; it easily follows that the lower the mark-up � the bigger the drop in dividends

for a given increase in the output gap. From (50), we can see that a rule that responds positively to expected

stock prices is somehow isomorphic to a rule that responds negatively to expected future output gaps, which

is exactly the opposite of what one should do to achieve stabilization.

The way an initial deviation of agents�expectations from the MSV-REE (either due to a sunspot shock

or to an initial mis-perception of the equilibrium law of motion) propagates throughout the system is the

following. Suppose agents� in�ationary expectations positively deviate from the fundamental REE, and

consider an interest rate rule that always responds actively (i.e. more than proportionally) to expected

in�ation. When no reaction is granted to expected stock prices, the short run dynamics of the real interest

rate depend solely on in�ation expectations. The active policy rule pushes up the real interest rate and

therefore drags down both the output gap and in�ation, thus mitigating the initial upward deviation. Along

the adjustment path, the real interest rate gradually reverts to its long run level, bringing down in�ation

and the output gap even further. What happens to stock prices depends on the magnitude of the composite

parameter � in (35). On the one hand, stock prices jump downwards because of the initial hike in the real

interest rate. On the other hand, the decrease in the expectations of the output gap pushes up dividends

and therefore stock prices as well. However, as real interest rates, the output gap and in�aiton eventually

converge to their steady state, so do stock prices.

Now suppose the active Taylor rule also responds positively to expected stock prices and engeneer the

same upward revision in in�ationary expectations. Similarly to the previous case, the active rule will drag

down both in�ation and the output gap, via downward revision in their expectations. However, the stock

19

price dynamics now become relevant. If the steady state mark-up � (and therefore �) is su¢ ciently low,

stock prices move up, as the progressive fall in the output gap expectations more than o¤sets the negative

impact from the initial real interest rate hike. This triggers an even higher increase in the real interest rate,

thus further drops in in�ation and the output gap, and therefore an additional accelleration in stock prices.

As the mechanism continues, the economy moves farther away from the MSV-REE.

Three factors can contribute to mitigate the upward-spiralling stock price dynamics. The �rst one is a

shorter planning horizon. Along the transition, higher stock prices have a positive impact on the output

gap via the term 1+ st in (34). As the probability of dying increases (and so does ), this positive e¤ect

can dominate the negative impact from the real interest rate increase. The output gap would then not fall

that dramatically, and stock prices would not spiral upwards. The second possibility is a higher steady

state mark-up. A higher mark-up implies a lower elasticity of stock prices to output gap expectations, as

� is strictly decreasing in �: This would contribute to make both in�ation, the output gap and the stock

price go down after a real interest rate increase. Third, the central bank could simply be more aggressive

towards in�ation. As the latter (and its expectations) move down along the adjustment, this would reduce

the increase in the interest rates triggered by raising stock prices, thus reducing the e¤ect on the output gap

and stock prices themselves.

5.1.2 Ricardian Agents

Now suppose = 0: Our economy then collapses to a standard in�nitively lived representative agent economy

with a New-Keynesian supply side. As discussed in a previous section, when agents are Ricardian stock price

�uctuations have no wealth e¤ect whatsoever on consumption. In Proposition 1 and Corollary 2 we have

then shown that in this case the optimal rule should not respond to stock price expectations at all. But

what if the central bank does it anyway? This is the case studied in Carlstrom and Fuerst (2006), although

they focus on contemporaneous interest rate rules.

Proposition 5 Suppose that = 0 and � 2 (1; 2 + ') ; and the the central bank follows a forward-looking

interest rate rule like (45).

1. There exists a unique Rational Expectation Equilibrium if and only if

�x < 1 + ��1 (51)

� (�� 1) + (1 + �)�x + �s�(1 + �) (�� 1) + (1� �) (1 + ')

�� 1

�< 2 (1 + �) (52)

� (�� 1) + (1� �)�x > (1� �)�2 + '� �� 1

��s (53)

20

2. The MSV-REE zt = h+ P#t is E-Stable if and only if

� (�� 1) + (1� �)�x > (1� �)�2 + '� �� 1

��s (54)


The results of Proposition 5 and Proposition 3 are qualitatively similar: given �� 0 and �x � 0,

a forward-looking rule including responding to the expected stock price, �s > 0, is more prone to induce

equilibrium indeterminacy and E-instability of the MSV-REE solution with respect to a rule where �s = 0.

As a matter of fact, notice that for �s = 0 the conditions for local determinacy (51)-(53) and the condition

for E-stability (54) collapse to the local determinacy and E-Stability conditions spelled, respectively, in

Proposition 4 and 5 of Bullard and Mitra (2002). Similarly to their paper, a given pair (��; �s) is more likely

to induce E-stability the more responsive the rule is to the output gap:

Comparing the E-stability condition (54) with the set of determinacy conditions (51)-(53) of Proposition

5, we can see that equilibrium determinacy implies E-stability of the MSV representation of that equilib-

rium, as it is common in purely forward-looking RE linear systems. The opposite does not hold: for some

parametrizations we can have an E-stable MSV solution although there exists a continuum of sunspot-driven

REE.

Furthermore, by writing (54) as

�� > 1 +1� ��

2 + '� �� 1 �s �

1� ��

�x

we can see that E-Stability implies a lower bound on the response coe¢ cient to expected in�ation which

depends positively on �s as long as � 2 (1; 2 + ') ; and negatively on �x: Moreover, for given �s and �x, the

higher the degree of market power in the wholesale sector 1� - i.e. the higher the average mark-up � - the

lower the minimum response to in�ation required for E-stability.30

A closer inspection of the conditions in Point 1 of the Proposition shows that, given a response to the

output gap satisfying condition (51), for local determinacy we need the response to expected in�ation �� to

lie within a range whose lower and upper bounds are, respectively, linearly increasing and linearly decreasing

in �s for any �s � 0: The following Corollary shows that an excessive response to the stock price gap makes

the REE always indeterminate.

Corollary 6 Assume that �x < 1 + ��1: Then the REE is always indeterminate if �s �

1+�(1��x)1+� :

Proof. See Appendix.30The analysis also highlight an additional stabilization e¤ect played by the wealth e¤ects. We notice that if we set �s = �x = 0

in (54), then being active with respect to in�ation is necessary and su¢ cient for E-stability. However, the same restrictions in

(49) imply that �� 1 >e��1� 1�e�+�1�e�+ , which is to say that a suitable rule that is passive towards in�ation can still deliver

E-stability.

21

5.2 Optimal Interest Rate Rules under Discretion

Proposition 1 has shown that for the standard loss function used in the literature optimal monetary policy

under discretion is a Taylor-type interest rate rule responding to both expected in�ation and the expected

stock-price. As we have done for general forward-looking rules, the question is whether such optimal rule

can induce both equilibrium determinacy and ES of the fundamental REE. Here we generalize our analysis

and consider the case of a Central Bank that also targets stock price stability. We model that by considering

the following modi�ed objective:

1

2Et

1Xi=0

�i��2t+i + �

xx2t+i + �ss2t+i

�(55)

where �s is the relative weight given to stock price volatility. Our analysis will show that the value of the

preference parameter �s has some major implications for the determinacy and E-stability of the optimal

monetary policy under Rational Expectations.

The Central Bank chooses rt in order to minimize (55) subject to the reduced form system (40)-(42). With

stock price volatility in the objective, the �rst order condition of the problem becomes �xxt+��t+�sst = 0:

To obtain the optimal interest rate rule under Rational Expectations, assume that the solution takes the

following form: �t = a1 + d1ut, xt = a2 + d2ut and st = a3 + d3ut:31 Substituting this result into (40), the

RE-optimal fundamentals-based interest rate rule responds to all the shocks hitting the economy:

rt = [(d1 + d2)�� (1 + ) d2 + d3]ut + rrnt

By combining the latter with (40)-(42) we obtain the expectational system zt = Ko + AoEtzt+1 +

Uout;where zt = [xt; �t; st]0; Ko = 03�1;

Ao =

266641�

1+ (1 + �) 1 ~�1+

�h1�

1+ (1 + �)i e� + � � ~�

1+

�� 1 ~�

37775 ;and Uo is a 3�1 vector that is omitted since it is not required for the following analysis. In the next

Proposition we consider the representative agent case and show that if the central bank implements the

RE-optimal monetary policy then the MSV-REE is never E-stable and therefore not learnable: the economy

will never converge to the REE.

Proposition 7 Assume that = 0. Then for the RE-optimal interest rate rule under discretion rt =

[(d1 + d2)�� (1 + ) d2 + d3]ut + rrnt ; the representation �t = a1 + d1ut; xt = a2 + d2ut; st = a3 + d3ut

of a REE is not learnable in the E-stability sense.31The exact values of aj ; dj for j = 1; 2; 3 are obtained from the Method of Undetermined Coe¢ cients. Although their speci�c

value is not relevant for our analysis, we report them in Appendix ??.

22


For the more general case of 2 (0; 1), it is hard to derive clear analytical results. We conjecture that

the fundamentals-based optimal rule will still induce E-instability.32 It is worth pointing out that for any

value assigned to the Determinacy/E-stability properties of the fundamental optimal policy rule do not

depend on the relative weights assigned by the monetary authority to the output gap, �x, and the stock price

index, �s. In other words, a fundamentals-based optimal rule would induce E-instability independently from

the relative prefereces of the Central Bank for "in�ation targeting", "output stabilization" and "�nancial

stability". This contrasts with the results that we will derive for the Expectations-based optimal rule.

In general the E-instability problem of the fundamentals-based rule is due to the fact that the policy

maker implicitly assumes that the agents in the economy have perfectly rational expectations at every point

in time. But what if expectations are not rational? Then it seems natural to base the interest rate rule

on expectations of these private agents. This corresponds to the expectations-based optimal rule of Evans

and Honkapoja (2003). To derive this rule we assume that the policy maker can observe these private

expectations. Thus the optimal rule can be derived by combining (40)-(42) and the optimal condition (43).

After some tedious algebra we obtain:

rt = �0 + ��Et�t+1 + �sEtst+1 + �xEtxt+1 + �uut + rrnt (56)

where

�� = 1 +~��

�2 + �x + �s;

�s =~�

1 +

� +

�s

�2 + �x + �s

�;

�x = 1� 1 + �

1 +

� +

�s

�2 + �x + �s

�;

�0 = 0, and �u =�

�2 + �x + �s:

This rule, by construction, implements the optimal discretionary policy in every period and for all values

of private expectations. To pursue the Determinacy/E-stability analysis for this rule we combine it with

(40)-(42) to obtain the system zt = Ke +AeEtzt+1 + Ueut;where zt = [xt; st; �t]0; Ke = 03�1;

32A fundamental-based rule is somehow equivalent to interest rate pegging, since the central bank instrument does no respond

to any of the endogenous variables. It is well known that these types of rules are very much prone to induce indeterminacy,

and therefore E-instability.

23

Ae =

266641+�1+ �

r � ~��2+�x+�s � ~�

1+ �r

� 1+�1+ �r e� �1� �2

�2+�x+�s

�� ~�

1+ �r

� 1+�1+ (1� �

r) � ~��2+�x+�s

~�1+ (1� �

r)

37775 ; (57)

and �r = �s

�2+�x+�s , Uo is a 3�1 vector that is omitted since it is not required for the following analysis.33

In the following Proposition we show that if the central bank does not have any concern for stock price

stability - namely �s = 0 in its loss function - then under the optimal expectations-based interest rate rule

the MSV-REE is always E-stable: the economy converges almost surely to the REE corresponding to the

optimal policy under discretion.

Proposition 8 Consider the expectations-based optimal rule (56), such that the equilibrium dynamics are

given by zt = Ke+AeEtzt+1+Ueut with Ae de�ned in (57). If the central bank is not concerned about stock

price stability, i.e. �s = 0, then under rule (56) the REE of the optimal policy under discretion is always

E-stable.


Di¤erently from the fundamentals-based rule, the E-stability results for the expectations-based rule seem

to depend on the relative weight assigned to stock price �uctuations �s in the central banker�s loss function.

As the analysis becomes rather complicated, we stress our point by presenting the results from a numerical

simulation on a calibrated version of our model. The plot shows that as the relative weight �s passes a

certain threshold the expectations-based rule is not able to deliver the desired E-stability. Although the

results at this point are only numerical, we summarize our �ndings in the following proposition.

Proposition 9 Consider the optimal expectations-based interest policy rule rt = �0+��Et�t+1+�sEtst+1+

�xEtxt+1+�uut+rrnt : The higher �

s > 0; the more prone is the rule to induce indeterminacy and E-instability

of the REE of the economy.

This E-instability result for the expectations-based rule clearly contrasts with Evans and Honhapohja

(2003). They study the E-stability properties of fundamentals-based and expectations-based interest rate

rules for a typical in�nite horizon representative agent model in which there are no wealth e¤ects. In their

model an expectations-based rule responding to expected future in�ation and expected future output-gap

is able to guarantee determinacy and E-stability. This is not the case in our model given that, as we

showed before, forward-looking rules that respond to expected future stock prices are more prone to deliver

indeterminacy and E-instability.

The last Proposition still conveys this message in the context of the optimal expectations-based rule under

discretion. The intuition is straightforward. As �s increases then the response coe¢ cient to expected in�ation33As a matter of fact if we assume rational expectations in (56) matrix Ae becomes identical to Ao:

24

in (56), �� = 1 +~��

�2+�x+�s ; decreases converging to one, whereas the response coe¢ cient to stock prices,

�s =~�

(1+ )

� + �s

�2+�x+�s

�; increases. Both e¤ects make the rule more prone to induce indeterminacy

and E-instability, as we studied for forward-looking rules. This analysis also suggests that to guarantee

E-stability under the optimal expectations-based interest rule, it is su¢ cient to put zero weight on stock

price stability.

This result seems rather interesting from the point of view of the ongoing debate on the links between

"price stability" and "�nancial stability". Our results point out that although it might be optimal to include

stock price expectations in the policy rule (even if �s = 0 still �s > 0), there should not be any speci�c

concern for stabilizing stock price �uctuations on behalf of the central bank. The optimality of responding

to the stock market has only to do with delivering the best mix between in�ation and output stabilization.

An interesting and intuitive result is that when �s = 0, for a reasonable calibration, the optimal response

to the expected stock price gap should be higher the higher the real wealth e¤ects. This can be seen by

noticing that @�s@ > 0 if r < 2�� :

Combined with what obtained in Proposition 3, these results point out that, with positive real wealth

e¤ects, being active with respect to in�ation and not responding at all to stock prices is not an optimal

policy, although it surely delivers E-stability of the REE. A forward-looking rule that is consistent with the

optimal policy allocation might requires a positive response to stock prices as long as > 0:

25

6 Conclusions

In this paper we study whether the central bank should react to development in the stock market by moving

its instrumental interest rate in response to movements in a stock price index. We consider a tractable model

where stock price �uctuations have real (demand-side) e¤ects. Our economy is an extension of the stochastic

overlapping generation framework of Blanchard (1985) and Yaari (1965) to a New-Keynesian monetary

economy with risky equities. In our set-up, agents are Non-Ricardian and face a constant probability of dying

(or exiting the market) in every-period. They are therefore unable to spread the e¤ect of current (or expected)

stock prices swings over the entire in�nite future. Since their (expected) planning horizon is �nite, stock

market booms (or bursts) imply optimally a larger increase (decrease) in current consumption with �respect

to the standard in�nitively-lived case. By making the dynamics of aggregate �nancial wealth relevant for

current aggregate consumption as well as for current in�ation, this demand-side channel introduces some real

e¤ects of stock prices volatility. We believe this framework is well suited to capture the common wisdom that

the 1990�s boom in consumption was �nanced by heavily relying on stock market performances. An appealing

feature of our model is its tractability. It is in fact general enough to nest also the case of in�nitively lived

Ricardian agents that is typically studied in the literature. We therefore provide some results for that case

as well.

Our key results are the followings. First, we show that, under discretion, the optimal monetary policy

rule when agents have Rational Expectations is a Taylor-type interest rate rule responding to both expected

in�ation and the expected stock-price. In this sense, we provide some rationale for why Central Banks

should pay close attention to asset price �uctuations when setting their instrumental interest rate. When

the planning horizon of the Non-Ricardian agents shortens - implying stronger wealth e¤ects from equity

holdings - the optimal response to stock price �uctuations should increase.

Next we study the problem of equilibrium determinacy and stability under adaptive learning (E-stability)

for a general class of forward-looking interest rate rules, including the optimal rule. We show that in order

to induce a unique and E-stable REE the rule should satisfy a reinforced Taylor principle, whereby the lower

bound on the response coe¢ cient to in�ation is an increasing function of the response to stock prices. As it

turns out, these expectational rules are more prone to generate aggregate instability the lower the degree of

monopolistic competition in the wholesale sector and the shorter the planning horizon of the Non-Ricardian

agents. We conclude by highlighting the destabilizing e¤ects of explicit stock-price targeting.

26

A Appendix

A.1 The Steady State

We derive some key equilibrium conditions at the non-stochastic steady state. Some of them will be useful

to calibrate the model to the US economy.

First of all, at the zero in�ation steady state, the aggregate Euler Equation is:

� (1 + r) = 1 +

�1

1� � ��

PC

which di¤ers from the standard � (1 + r) = 1 since 2 (0; 1) : In order to get the steady state real interest

rate r we need an expression for the real �nancial wealth to consumption ratio, PC :

From the market clearing condition Y = C+G and the assumption on government expenditureGt = $tYt,

we get that C +$Y: The consumption to output ratio is simply CY = 1�$:

As in standard New-Keynesian models, the steady state real marginal costs are simply MC = ��1 where

� � ��1 : Combining this with the steady state conditions C =

WP (1�N), Y = AN and MC = W

AP ; we get

the steady state value of hours worked N = 11+�(1�!) . From steady state hours we easily get a value of the

inverse of the steady state Frisch elasticity of labour supply, ' � N1�N = 1

�(1�$) :Then, steady state output

and consumption are, respectively:

Y =A

1 + � (1� !)

C =A (1� !)

1 + � (1� !)

Given the steady state pricing equation (1 + r)F = 1; equations (9), (12) and (19) become:

Q+D = (1 + r)Q

P= Q+D

D = Y�� 1�

By combining these last three equations with the expression for steady state consumption C written above,

after some simple algebra, we obtain the real �nancial wealth to consumption ration PC =

��1�

1�!1+rr . We can

then substitute the latter into the steady state Euler Equation to get an implicit function for the the steady

state real interest rate r:34

34Notice that with Non-Ricardian agents, the real interest rate r is a function of some key structural parameters of the model,

such as the mark-up and the average non-fundamental shock.

27

� (1 + r) = 1 +

�1

1� � �� 1

�

1� !1 + r

r(58)

One can easily show that a solution r > 1�� always exists, and that - by the Implicit Function Theorem

- r is strictly increasing in the death probability and the steady state mark-up � : @r@ > 0 and @r@� > 0:

De�ning � 1��(1� )(1� )PC ; from equation (58) we have that at the steady state = � (1 + r) � 1; or, as

oftern written in the main text e� (1 + r) = 1: Hence also is strictly increasing in both and � : @ @ > 0

and @ @� > 0:

Another key composite parameter of our analysis is �: From the equation of the stock price Q we have

that YQ =r��1 : Given the latter and

e� (1 + r) = 1, we easily obtain that � � e� 1+'� YQ �

�1�e�� is equivalent

to:

� =

�1 + '

�� 1 � 1�

r

1 + r

Using (58) to substitute r1+r out of the latter equation, we obtain that � =

1�$

�1

1� � ��2+'�� : Given

that ' and are, respectively, strictly decreasing and strictly increasing in �; it easily follows that � is

strictly decreasing in � : @�@� < 0: Since � is a¤ected by the probability only via the interest rate r; we can

also see that � is strictly increasing in : @�@ =�1+'��1 � 1

�1

(1+r)2@r@ > 0:

A.2 Calvo�s (1983) Staggered Price Setting

Given real marginal costsMCt; the wholesale sector �rm i faces the following dynamic price setting problem

at time t:

maxPt(i)

Et

( 1Xk=0

�kFt;t+kYt+k(i)�Pt(i)� Pt+kMCt+k

�);

subject to the constraint coming from the demand for intermediate goods of the retail sector (13). The

�rst-order condition for the solution of the above problem implies that all �rms revising their price at time

t will choose a common optimal price level, P �t , set according to the following log-linear rule:

p�t � ln(P �t =P ) = (1� �~�)Et

( 1Xk=0

(�~�)k(mct+k + pt+k � (�� 1) ln(�t+k=�)));

whose recursive form reads:

p�t = �~�Etp�t+1 + (1� �~�)(mct + pt)� (�� 1)(1� �~�) ln(�t=�):

Finally, from the de�nition of the general price level (14) and considering that all �rms revising their

price at t (a fraction (1 � �) of all �rms) choose the same price P �t and that all �rms keeping the price

constant (a fraction �) charge last period�s general price level, we obtain the New Keynesian Phillips Curve

(NKPC) (16).

28

A.3 Fundamentals form of the RE-optimal policy rule

With stock price volatility in the objective, the �rst order condition of the problem becomes �xxt + ��t +

�sst = 0: Assume that the solution takes the following form: �t = a1+d1ut, xt = a2+d2ut and st = a3+d3ut:

Then the Rational Expectations Et�t+1 = a1 + d1��ut; Etxt = a2 + d2��ut; and Etst+1 = a3 + d3��ut can

be combined with (40)-(42) and the �rst order condition to obtain a1 = a2 = a3 = 0 together with

d3 = ��h1� ��

1+ (1 + �)i

�1� ~��

� h�x�1� ~��

1+

�+ �s

�1� �� 1+�1+

�i+ �2

h1� ~��

(1+E)(1+ )

id1 =

�x � ��sd3�x�1� ~��

�+ �2

d2 = � 1

�x(�d1 + �

sd3) ,

A.4 Lemmata and Proofs

Lemma 1 Recall the de�nition of in (47) and let J � � I. Furthermore assume that �� > 1 and

r >q

1+ : If S2 (J) > 0 holds for some parametrization, then S2 (J)Tr (J)�Det (J) < 0 holds as well.

Proof. To simplify the notation let � � [1�e�+�]��s[�� ]1+ ; such that we can write:

S2 (J) =�e� � 1� hTr (J)� �e� � 1�i+�� (1� ��)�1 + e�

1 +

�

Det (J) =�e� � 1��+ � (1� ��)

"1�

e�1 +

#Substituting � out of the �rst equation using the second, after some manipulation, we get::�e� � 1�S2 (J) = �e� � 1�2 hTr (J)� �e� � 1�i+Det (J)� � (1� ��) e�2

1 +

From this we obtain:

S2 (J)Tr (J)�Det (J) = S2 (J)hTr (J)�

�e� � 1�i (59)

+

��e� � 1�2 hTr (J)� �e� � 1�i� � (1� ��) e�2

1 +

�Given the de�nition of Tr (J) in (60) and the assumption of �� > 1; we have that

hTr (J)�

�e� � 1�i < 0;such that S2 (J)

hTr (J)�

�e� � 1�i < 0 as long as S2 (J) > 0. Now consider the second term within bracketsin (59). By simple algebra one can show that this term is negative if

��e� � 1�2 � e�2 1+

�� 0: Using the

steady state condition � (1 + r) = 1 + ; this last inequality is equivalent to assuming a lower bound on the

steady state interest rate r; i.e. r >q

1+ :

29

A.5 Proofs of the Propositions

A.5.1 Proof of Proposition 1

Proof. It is straightfoward to show that the MSV representation of the optimal fundamental REE takes

the form of simple linear rules �t = a1 + d1ut, xt = a2 + d2ut and st = a3 + d3ut; whereby the coe¢ cients

(aj ; dj)3j=1 depend on the structural parameters.

35 We can then combine the optimality condition (43) with

the fact that under Rational Expectation Et�t+1 = ��t to substitute (41) into (40), and then solve for the

optimal nominal interest rate rt as a function of rrnt , expected in�ation and the expected stock-price:

rt = rrnt + �RE� Et�t+1 + �

REs Etst+1

where

�RE� � 1 +�

(1 + )�x��[1� �� + (1 + ��)]

�REs =

(1 + )2 �

First of all, recall from the steady state analysis in Appendix A.1 that � 2 (1; 2 + ') implies that � > 0:

Given the stationarity of ut; i.e. �� 2 (0; 1), it then easily follows that �RE� > 1 always: the optimal interest

rate rule satis�es the Taylor-principle with respect to expected in�ation. From the steady state we also have

that � 1��(1� )(1� )PC =

�1

(1� ) � �� 1

�

1�!1+rr � 0 if and only if � 0: Then, in an economy populated

by Non-Ricardian agents, i.e. 2 (0; 1), the optimal rule responds positively to the expected stock-price,

�REs > 0:

From simple di¤erentiation of both �RE� and �REs with respect to we obtain:

@�RE�@

=�

�x��

�� (1 + �)

(1 + )2 > 0

@�REs@

= �1� (1 + )

3 > 0

where the second inequality always holds when the steady state interest rate is below 1, given that =

� (1 + r) � 1 for � 2 (0; 1) : Combining these results with the fact that is strictly increasing in (see

Appendix A.1 again) we obtain Point 2 in the Proposition.


Proof. From our discussion, for the MSV-REE to be E-stable all roots of matrix J � � I need to have

negative real parts. By the Routh Theorem, all roots of J have negative real parts if and only if the following

35Since the reduced form equilibrium system (34)-(36) is completely forward-looking and does not include any constants we

actually have aj = 0 for j = 1; 2; 3:

30

three conditions are all satis�ed: Det (J) < 0; T r (J) < 0 and S2 (J)Tr (J) �Det (J) < 0; where S2 (J) is

the sum of the 2x2 principal minors of J:36 After simple algebra we obtain that:

Tr (J) = �

1 + (1 + �) + 2

�e� � 1�� s � �x + � (1� ��) (60)

Det (J) =�e� � 1�

24 �1�e� + �� s (�� )+

1 +

35 (61)

+�e� � 1� 1�e� +

1 + �x + � (1� ��)

1�

e�1 +

!

S2 (J) = � (�� 1) 2�

e�1 +

!+ �1�e� + �� s (�� )

1 +

+

1� e� + 1�e� +

1 +

!�x

+�e� � 1��

1 + (1 + �) + e� � 1� �s� (62)

By Lemma 1, under the assumption of �� > 1 and r >q

1+ ; S2 (J) > 0 is a su¢ cient condition for

the third inequality of the Routh Theorem, S2 (J)Tr (J)�Det (J) < 0, to hold:

First, it is immediate that, as�e� � 1� < 0; then Tr (J) < 0 for any �s; �x � 0 when �� > 1 : the �rst

condition of the Routh Theorem then holds for any rule that is "active" with respect to expected in�ation

and that responds non-negatively to the expected stock-price and output gap. Next consider (61) and (62).

By simple algebra, both inequalities hold as long as the two following conditions are satis�ed:

�� 1 >e� � 1�

1�e� + �1� e� + + 1� e��

1 + � e��s + e� � 1� �x (63)

�� 1 > � �1�e� + ��+ �1� e�� (1 + ) h 1+�

1+ + 1� e�i�h2 (1 + )� e�i (64)

+

h�� S

i�h2 (1 + )� e�i�s �

�1� e�� (1 + ) + 1� e� �

�h2 (1 + )� e�i �x

where �S � +�1� e�� (1 + ) : In order to simplify the analysis we let �D and �S denote, respectively,

the right hand sides of (63) and (64), respectively. For the MSV-REE to be E-stable we therefore need to

de�ne appropriate conditions on �� and �s such that �� 1 > �D and �� 1 > �S :

36For a statement of the Routh Theorem see Gandolfo (1996).

31

As it turns out, E-stability depends on the the value of �; whereby � > 0 is always guaranteed by the

assumption � < 2 + '. Notice that, for any � > 0; both �D and �S take negative values if we set �s = 0:

Suppose that � 2 (0; ] : Given that �S > ; a close inspection of the right hand sides of (63) and (64)

reveals that in this case both �D and �S are strictly negative for any �s � 0: Therefore conditions Tr (J) < 0;

(63) and (64) simultaneously hold for any �� > 1 and any �s � 0: This proves Point 1 of the Proposition.

Now suppose that � 2� ; �S

i: In this case, �D strictly increases with respect to �s. It is then immediate

that condition (63) holds 1) for any �� > 1 if �s 2h0; �Ds

i; where �Ds �

(1�e�+ )�x+(1�e�+�) �� is the solution

to �D = 0 with respect to �s; 2) for �� > 1+ �D > 1 if �s > �Ds : Besides, since � � �S ; we clearly have that

�S < 0 for any �s � 0; then condition (64) always holds for any �� > 1 in this case.

Now suppose that � > �S : In this case both �D and �S increase with respect to �s: First of all, notice

that, as for the previous case, (63) holds for any �� > 1 if �s 2h0; �Ds

iand only for �� > 1 + �D > 1 if

�s > �Ds : Second, by simple algebra one can show that condition (64) holds 1) for any �� > 1 if �s 2h0; �Ss

i;

where �Ss � �Ds�� S+ CONSTANT is the solution to �S = 0; 2) for �� > 1+�

S > 1 if �s > �Ss : Third, if we

consider both �D and �S as functions of �s; after some tedious algebra it is possible to show if r >q

1+ then

�D is always steeper than �S ; i.e. @�D

@�s> @�S

@�s> 0; for any �s � 0: By a simple graphical argument, one can

see that this result together with the fact that �Ss > �Ds (since � > �S > ); implies that �D > �S for any

�s � �Ds : It easily follows that the conditions for both (63) and (64) to hold are the same as those spelled

for the case of � 2� ; �S

i: This proves Point 2 of the Proposition.


Proof. Recall the de�nition of in (47). If = 0 - such that = 0 - then takes the following form

=

266641� �x 1� �� s

� (1� �x) � + � (1� ��) ��s��ra � �x 1� �� s

37775where �ra � (1� �)

�2+'��1

�is � for = 0: Clearly � 2 (1; 2 + ') implies �ra > 0: Since z0t � [xt; �t; st]

includes only non-predetermined variable, there is a unique REE if and only if all eigenvalues of have

real parts inside the unit circle. After simple algebra, the characteristic polynomial of can be written as

(� � e)�e2 + a1e+ a2

�= 0 where a1 = [�s � � � (1� �x)� � (1� ��)] and a2 = [� (1� �x)� �s (1 + �ra)] :

One eigenvalue is e1 = � 2 (0; 1) :We need to verify that both solutions to the quadratic equation f (e) = e2+

a1e+a2 = 0 belong to the unit circle as well. Necessary and su¢ cient conditions for e2; e3 2 (0; 1) are 1�a1+

a2 > 0; 1+ a1+ a2 > 0 and a2 < 1: (see Gandolfo, 1996). The third inequality holds for any parametrization

of the policy rule given that 1) a2 = e2e3 always; 2) Det () = e1e2e3 = � [� (1� �x)� �s (1 + �ra)] for

e1 = �; and that 3) � (1� �x) � �s (1 + �ra) < 1 for any �s; �x � 0 as �ra > 0: By simple algebra the

32

remaining two inequalities are equivalent, respectively, to conditions (52) and (53). These conditions can

also be written as

�� < 1 +(1 + �) (2� �x)

�� 2 + �

ra

��s

�� > 1�1� ��

+�ra

��s

Therefore, for any given �s � 0, there exist some �� satisfying both last inequalities only if 1+(1+�)(2��x)

� >

1� 1�� ; i.e. only if �x < 1 + �

�1: This conludes the proof of Point 1.

For the E-stability analysis, it is convenient to de�ne J � � I: When = 0; under the forward-looking

interest rate rule (45), J takes the following form:

J =

26664��x 1� �� s��x � � 1 + � (1� ��) ��s

��ra � �x 1� �� s � 1

37775From previous analysis we know that for the MSV-REE to be E-stable all roots of J need to have negative

real parts. From the Routh Theorem all roots of J have negative real parts if and only if the following three

conditions are all satis�ed: Det (J) < 0; T r (J) < 0 and S2 (J)Tr (J)�Det (J) < 0; where S2 (J) is the sum

of the 2x2 principal minors of J: First of all, by simple algebra, we have that:

Tr (J) < 0 if and only if � (�� 1) + �x > 2 (� � 1)� �s (65)

Det(J) < 0 if and only if � (�� 1) + (1� �)�x > �ra�s (66)

Now let � [�s�ra + � (1� ��) + (� � 1)�x] : Since S2 (J) = (� � 1) [Tr (J)� (� � 1)]�; it follows that:

S2 (J)Tr (J)�Det (J) = f(� � 1) [Tr (J)� (� � 1)]�gTr (J)� (1� �)

= [Tr (J)� (� � 1)] [(� � 1)Tr (J)�] (67)

If condition (66) holds then so does condition (65). As such under condition (66) we have both Det (J) < 0;

T r (J) < 0 and < 0: Moreover, (65) also implies that Tr (J) � (� � 1) < 0: If we put these results

together we have that (67) must hold too: the third condition of the Routh Theorem is satis�ed. Hence, we

can conclude that � (�� 1) + (1� �)�x > �ra�s is necessary and su¢ cient for all the roots of J to have

negative real parts and the MSV is E-Stable in the Evans and Honkapoja (2001) sense.


Proof. To prove this proposition it is su¢ cient to characterize the eigenvalues of the matrix Ao � I; where

Ao is de�ned in (??). Since = 0; then = 0: Using this and (??), we can derive the following expression

for the determinant:

33

Det(Ao � I) = � (1� �) :

Since � > 0 and � 2 (0; 1) then Det (Ao � I) > 0:But this means that there is at least one eigenvalue with

a positive real part. Then there is E-instability.


Proof. From the previous analysis, we know that the REE of the system zt = Ke + AeEtzt+1 + Ueut is

E-stable if and only if all eigenvalues of J � Ae � I have negative real parts. Once more we appeal to the

Routh Theorem to determine under what conditions this occurs. As �s = 0 implies that �r = 0 as well, J

takes the following form:

J =

26664�1 � ~��

�2+�x 0

0 e� �1� �2

�2+�x

�� 1 0

� 1+�1+ � ~��

�2+�x~�

1+ � 1

37775Recall that ~� 2 (0; 1) and that �x; ; � > 0: Since the matrix is block-triangular, it is immediate that one of

the eigenvalues is~�

1+ �1 < 0: To study the sign of the remaining two, we can restrict to the 2�2 submatrix

J22 =

24 �1 � ~��2+�x

0 e� �1� �2

�2+�x

�� 1

35Matrix J22 has both roots with negative real parts if and only if Det (J22) > 0 and Tr (J22) < 0: The second

inequality clearly always holds. Then, by simple algebra we have that Det (J22) = 1 � e� �1� �2

�2+�x

�> 0:

All three roots of J have therefore negative real parts. The expectations-based optimal interest rate rule

always induces E-stability of the optimal REE under discretion when �s = 0:

A.6 Proofs of the Corollaries

A.6.1 Proof of Corollary 2

Proof. The proof is trivial. Simply set = 0 such that = 0 as well. It follows from the Proof of

Proposition 1 that �REs = 0 and that �RE� � 1 + �(1��)�x��

> 1:


Proof. From Proposition 3 we know that whether a rule responding to stock price expectations can induce

ES or not depends on the sign of the inequality � T : We are going to show how the latter depends on the

average mark-up �:

34

First of all, recall the de�nition � � ~� 1+'�YQ � (1 � ~�): From the steady state equations Y

Q = r��1and

~� (1 + r) = 1; after some simple algebra, it easily follows that the inequality � T is equivalent to�1+'��1 � 1

�T 1+r

r [� (1 + r)� 1] (�): Now consider equation (58) which implicitly de�nes the steady state

interest rate r; for given structural parameters �; ; � and $: Other things being equal, by the Implicit

Function Theorem (IFT) on (58), we can consider r as a function of the average mark-up �; i.e. r (�) ; such

that r : (1; 2 + ') ! R++: From Appendix A.1, we have that ' � N1�N = 1

�(1�$) such that ' can also

be considered as a strictly decreasing function of �; i.e. ' (�) : Hence, inequality (�) can be written as an

implicit inequality with respect to � :�1 + ' (�)

�� 1 � 1�T 1 + r (�)

r (�)f� [1 + r (�)]� 1g (68)

Let and � denote, respectively the left and the right hand sides of (68); and consider both as functions

of the parameter �; for � 2 (1; 2 + ') : Notice that both are continuous in � for any � 2 (1; 2 + ') :37By

simple di¤erentiation, @@� < 0; with lim�!1+

= +1 and lim�!+1

= �1: Now consider �: First of all, notice

that by the IFT on equation (58) it follows that lim�!1+

� (1 + r) = 1; and that @r@� > 0: Hence, by continuity

of r with respect to � and by continuity of � with respect to r; we can infer that lim�!1+

�(r (�)) = 0 and, by

Chain Rule, that @�@� =@�@r

@r@� > 0: Given the properties of and � with respect to �; it follows that = �

necessarily for some �D 2 (1) 2 + '); such that T � if and only if � S �D: Hence, we can conclude that

� T if and only if � S �D:

Recalling the results in Proposition 3 we can infer that if the average mark-up in the wholesale sector,

� , is su¢ ciently low, then � > occurs and we are in a case where a positive response to the stock price

gap might generate E-instability - unless the rule is su¢ ciently active towards in�ation.

Next, we are going to show that the � < �D - and therefore � > - is more likely to hold the lower is

the wealth e¤ect of stock price �uctuations, i.e. the lower : To do that, consider the equation = � (68)

de�ning the threshold �D: By a simple application of the Implicit Function Theorem and by the Chain Rule

we have that:@�D

@ = �

@�@r

@r@

@�@r

@r@�D

� @@�D

< 0

where the sign is the consequence of the results stated at the beginning of the proof of this Corollary and of@r@ > 0 (by the IFT on (58)). Therefore we have that the lower is - i.e. the longer the planning horizon of

Non-Ricardian agents and the closer we get to the in�nitively lived agent case - the higher is the threshold

value �D; which make it more likely for � < �D and therefore for � > �D to hold: In other words, the

lower is the more likely Point 2 of Proposition 3, for which a "reinforced Taylor Principle" is required for

E-Stability.

37Actually � is continuos in r for any r > 0; while r is continuos in � for any � 2 (1; 2 + ') : Therefore � is continuos in �

for any � 2 (1; 2 + ') :

35


Proof. First of all let�s write the two determinacy conditions (52) and (53) as follows:

�� < 1 +(1 + �) (2� �x)

�� 2 + �

ra

��s

�� > 1�1� ��

+�ra

��s

Let h (�s) � 1+(1+�)(2��x)

� � 2+�ra

� �s and g (�s) � 1� 1�� + �ra

� �s: Notice that they are, respectively, linearly

decreasing and linearly increasing with respect to �s for any �s � 0: If �x < 1+ ��1 then 1+(1+�)(2��x)

� >

1� 1�� : This implies that h (�s) T g (�s) if and only if �s S

1+�(1��x)1+� ; where 1+�(1��x)1+� > 0 as �x < 1+�

�1:

Therefore, if �s �1+�(1��x)

1+� there is no �� that can simultaneously satisfy the two inequalities above. We

conclude that if the response to the stock price gap is excessively high the REE is always indeterminate for

any response to expected in�ation.

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