Consumption externalities and monetary policy with limited asset market participation.

Airaudo, Marco ; Bossi, Luca

I. INTRODUCTION

In the benchmark, New-Keynesian (NK) model used for monetary policy analysis, a necessary (and often sufficient) condition for the equilibrium to be locally determinate is that the short-term nominal interest rate (the policy instrument) responds more than proportionally to inflation. A monetary policy rule satisfying this criterion is said to be active in the terminology of Leeper (1991), or, equivalently, to satisfy the Taylor principle. (1)

Despite being one of the most celebrated results in the monetary policy literature, the Taylor principle has been proven not to be robust to departures from the benchmark setting. (2) In particular, Gali et al. (2004) and Bilbiie (2008) show that this principle might fail if the economy is characterized by limited asset market participation (LAMP), whereby a share of agents, also called rule-of-thumb consumers or non-Ricardian, is assumed to have no access to financial markets and to just entirely consume their labor income in every period. More specifically, if the fraction of non-Ricardian agents passes a certain threshold, equilibrium determinacy requires the interest rate rule's response to inflation to be either sufficiently above or sufficiently below one--that is, sufficiently active or sufficiently passive--under a contemporaneous timing of the Taylor's rule (Gali et al. 2004) and strictly below one--that is, passive--under a forward-looking timing (Bilbiie 2008). (3) Given the significant degree of LAMP documented in the data, these results cast some doubts on the stabilizing properties of monetary policy rules following the Taylor principle. (4)

Our article extends the NK-LAMP model of Bilbiie (2008) by introducing consumption externalities, whereby households preferences are assumed to depend on consumption by a reference group or "the Joneses." We study how those spillovers impact the relationship between the Taylor principle and equilibrium determinacy, and assess whether, for suitable calibrations, our model under indeterminacy can generate aggregate volatilities of sizes comparable to those observed in the pre-Volcker era in the United States without necessarily requiring a violation of the Taylor principle, as in Clarida et al. (2000) or Lubik and Schorfheide (2004). We also investigate how the interaction between limited participation and externalities affects the model's response to a structural total factor productivity (TFP) shock.

Several articles provide empirical support for the existence of various types of consumption externalities in consumer preferences. Fuhrer (2000), using time series data for the United States, "rejects the hypothesis of no habit formation with tremendous confidence" (p. 367) and his appropriately modified monetary policy model can then match the hump-shaped response of consumption to various aggregate shocks. Luttmer (2005) and Dynan and Ravina (2007) report evidence of a positive relationship between self-reported happiness and relative income--that is, the difference between individual income and income earned by neighboring households--suggesting that this could be caused by people having utility functions that depend also on relative (and not just absolute) consumption. Bowles and Park (2005) investigate the importance of Veblen effects--that is, the desire to emulate the consumption behavior of the rich--by looking at individuals' allocation of time between labor and leisure. They provide evidence of a positive relationship between income inequality and hours worked by low-income earners. Maurer and Meier (2008) find evidence of moderate consumption externalities by estimating peer-group effects in intertemporal consumption choices. Finally, using dynamic panel data models, Korniotis (2010) estimates Euler equations with data for the 48 continental U.S. states over 30 years and find supporting evidence for external habit formation of the nature we model here. (5)

We highlight an interesting interaction between the degree of LAMP in the economy and the extent of consumption externalities. We find that increasing the strength of externalities enlarges the parametric range of LAMP where the simple Taylor principle fails to deliver a locally unique equilibrium if household exhibit keeping-up-with-the-Joneses (KUJ) preferences. (6) More specifically, if the KUJ consumption externality is sufficiently strong, the simple Taylor principle fails to be sufficient for equilibrium determinacy even if the degree of LAMP is low, that is, even if a large share of households are Ricardian. One of the nice features of the model we present is that the main results about determinacy and multiplicity of equilibria can be obtained both analytically and numerically.

Similar to Gali et al. (2004) and Bilbiie (2008), the intuition behind the failure of the Taylor principle is related to the occurrence of an Inverted Aggregate Demand Logic (IADL) of monetary policy transmission. That is, the possibility that an increase in the short-term interest rate induces an expansion rather than a contraction in aggregate demand. (7) The introduction of consumption externalities lowers the threshold share of non-Ricardian households--that is, the minimum degree of LAMP--necessary for the IADL to appear.

Consumption externalities can in fact magnify the link between the labor and the financial markets, which is the source of a IADL of monetary policy transmission. The intuition is as follows: suppose that households believe that, for no fundamental reason, inflation will increase. In the presence of LAMP, the resulting increase in the real interest rate (generated by an active policy response to inflation) generates two contrasting effects on Ricardian consumption. On the one hand, by the inter-temporal substitution effect, it should decrease. On the other hand, by the downward pressure on wages and subsequent increase in corporate dividends (income effect), it should increase. In the benchmark LAMP model (without externalities), the income effect appears stronger when the share of non-Ricardian households is sufficiently large and/or the labor supply elasticity is sufficiently low, as both cases lead to a larger drop in real wages and a larger increase in dividends. If that is the case, with Ricardian consumption eventually increasing, so will the real wage, and consequently non-Ricardian consumption. Since inflation is demand-driven, the initial belief-driven increase in inflation becomes self-fulfilling. Consumption externalities reinforce the income effect. In particular, in their attempt to keep up with the Joneses, non-Ricardian household will initially work less (following the interest rate driven cut in demand), thus creating a larger drop in aggregate demand and real wages, and therefore a larger increase in dividends. For the same reason, once Ricardian consumption increases, non-Ricardians will supply more labor, which in turn will put further upward pressure on demand and eventually on inflation.

We find that the output and inflation volatilities predicted by our model under indeterminacy are of similar order of magnitude of those observed during the 1960-1979 period in the United States. This result is attained for a policy rule specification which is mildly active, a degree of LAMP which is consistent with what observed in the data before the 1980s and intermediate levels of consumption externalities. Moreover, it appears to be robust to alternative parameterizations of labor elasticity (spanning from the low micro-based to the higher macro-based estimates) and it is not significantly affected by the volatility assumed for the non-fundamental belief shock. In this sense, our NK-LAMP model with consumption externalities can explain the macroeconomic instability of the 1970s without requiring a violation of the Taylor principle on behalf of the monetary authority.

Another important and surprising consequence of introducing consumption externalities in the NK-LAMP framework is that our model does well in capturing the stylized joint response of aggregate hours worked and real wages to a positive technology shock. Both of those variables in the presence of a mild consumption externality rise when we introduce a 1% increase in TFP. This is in line with the evidence put forth in Christiano et al. (2003) and Chari et al. (2008) and, to the best of our knowledge, is a completely novel result for NK models without capital. A similar behavior of aggregate hours was put forth by Altig et al. (2011) in a NK model with firm-specific predetermined capital. In that context, capital cannot be adjusted instantly and costlessly across firms. Those authors find that in that framework aggregate hours worked, investment, and consumption rise in response to a technology shock, but these increases are only marginally statistically significant. Our result is driven by a demand channel: we do not have capital and the economic mechanism in this model depends on how consumption externalities affect the labor and financial market equilibrium.

The intuition for this result is simple: a positive innovation shifts labor demand up and real wages increase as well. The Ricardian households benefit from this occurrence not only because their labor income increases, but also because, ceteris paribus, their profits will increase. Hence, they may or may not increase their hours worked (depending on whether the income effect is stronger than the substitution effect for leisure), but they certainly end up consuming more. The only choice that non-Ricardian agents have to expand their consumption due to the externality effect is, instead, to work more. This increase is so substantial to drive up total hours worked in the economy and even offset the potential decrease in the hours worked by Ricardian agents.

An article that is complementary to ours is Motta and Tirelli (2012). They are interested in studying fiscal policy in a LAMP NK model with consumption externalities. Their key finding is that automatic fiscal stabilizers can smooth the undesirable effects of LAMP (i.e., the presence of indeterminacy). They provide a numerical exercise showing that the threshold value for the share of non-Ricardians needed to obtain indeterminacy is negatively related to the strength of habits. However, their model is different from ours along several dimensions: first, as said, their main focus is on fiscal policy whereas ours is monetary policy. Second, they consider internal habits rather than external habits as we do: this affects the degree of volatility that their model is capable of generating and introduces an economic transmission mechanism that is very different from what we have. (8) Third, the presence of habits in their model make the inflation-output trade-off (i.e., the slope of the Phillips curve) steeper. In our model, aggregate consumption externality (ACE) prompts a flatter curve worsening the central banker stabilization task. This theoretical implication of our model is more in line with very recent empirical evidence that seems to indicate that the PC has indeed flattened in the data in recent years (Coibion and Gorodnichenko 2011). Lastly, we prove analytically and show numerically that the indeterminacy region boundaries are affected by the interaction of three parameters: strength of habits, share of non Ricardian agents, and the Frisch elasticity. (9)

The rest of the article is organized as follows. Section II presents the model. Section III defines the aggregate equilibrium. Section IV derives analytical conditions for equilibrium determinacy. Section V provides a quantitative evaluation of the analytical results on indeterminacy for a calibrated version of our economy, and assesses the role of LAMP and consumption externalities for the impulse responses to a fundamental shock. In this section, we also investigate whether, for suitable calibrations, our model under indeterminacy can generate output and inflation volatilities of sizes comparable to those observed in the pre-Volcker era in the United States without requiring a violation of the Taylor principle. Section VI presents some extensions to the benchmark setting by embedding an alternative specification of the Taylor rule and real wage rigidities. Section VII concludes.

II. THE MODEL

In this article, we modify Bilbiie (2008) who studies a version of Gali et al. (2004) New-Keynesian setting without physical capital.

This framework allows us to obtain analytical results.

A. Households

Consider an infinite horizon economy populated by a continuum of households. A fraction [gamma] [member of] [0, 1) of the population does not hold any type of assets and simply consumes its labor income every period. We refer to these households as rule-of-thumb or non-Ricardian consumers. The remaining share 1 - [gamma] behaves like standard forward-looking Ricardian households: they optimally choose their infinite horizon consumption plan taking advantages of all intertemporal trading opportunities available. Our economy is "cashless" a la Woodford (2003), in the sense that money does not provide any liquidity service to households (it is simply a numeraire in which all prices are denominated).

Ricardian Households. Ricardian households choose their optimal consumption-leisure plans to maximize their expected discounted lifetime utility. The latter is given by:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the superscript o stands for "optimizer," [beta] [member of](0,1) is the discount factor, [chi square] > 0 is the inverse of the Frisch elasticity of labor supply, and [sigma] > 0. The functional specification of the consumption part of utility follows Gali (1994) and Alonso-Carrera et al. (2008). The term [X.sup.o.sub.t] represents consumption by the household's reference group (to be defined below) and captures the existence of consumption externalities in our economy. (10) The parameter [[theta].sub.o] indexes the importance of such externalities in utility.

A Ricardian household has access to two financial assets: state-contingent bonds, and risky equity. His budget constraint reads as follows:

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

At the end of period t, the household holds a portfolio of contingent claims with one-period ahead stochastic nominal payoff [B.sup.o.sub.t+1] (where [F.sub.t,t+1] is the appropriate stochastic discount factor), and a continuum of risky equity shares issued by monopolistically competitive firms operating in the productive sector, that is, [A.sup.o.sub.j,t] for j [member of] [0,1]. The real price of a share issued by the jth firm is denoted by [Q.sub.j,t]. The first two terms on the right-hand side of Equation (2) correspond to the nominal financial wealth carried over from the previous period. This includes the nominal payoffs on the contingent claims, [B.sup.o.sub.t], and the "price plus dividend" on each share of the equity portfolio, [Q.sub.j,t] + [D.sub.j,t] for j [member of] [0,1]. The household also accrues labor income [W.sub.t] [h.sup.o.sub.t]. The wage is equal across all agents in the economy due to the assumption of perfect substitutability between Ricardian and non-Ricardian hours worked in production.

Taking first order conditions with respect to the household's choice variables, after simple manipulation, we obtain the following relationships:

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the gross inflation rate and [R.sub.t] is the risk-less rate, defined as [R.sub.t] = [([E.sub.t][F.sub.t,t+1]).sup.-l]. Equation (3) equates the real wage to the marginal rate of substitution between consumption and leisure, and corresponds to the labor supply schedule of Ricardian agents. Equation (4) is the Ricardian household's Euler equation which establishes a relationship between individual consumption and the real interest rate. Equation (5) is the pricing equation for the jth firm's equity share, for j [member of] [0,1],

Non-Ricardian Households. Non-Ricardian households behave like hand-to-mouth consumers: they do not save, and consume all their disposable income in every period. In line with the existing literature, we assume that the functional form of their instantaneous utility is identical to that of Ricardian households. Non-Ricardian households define their consumption-leisure choice by solving a simple static optimization problem:

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to the budget constraint

(7) [P.sub.t][c.sup.rt.sub.t] = [W.sub.t][h.sup.rt.sub.t],

where the superscript rt stands for "rule-of-thumb consumer." (11) Manipulating the first order conditions with respect to consumption and labor yields the following labor supply schedule for the non-Ricardian agents:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Equations (3) and (8) imply the equalization of the marginal rate of substitution between consumption and leisure across all agents in the economy:

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Moreover, by combining (8) with the budget constraint (7), we obtain an expression for the optimal labor supply of non-Ricardian households:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In the model studied by Bilbiie (2008) where consumption externalities are absent ([[theta].sub.n] = 0), under a log-utility specification ([sigma] = 1). non-Ricardian households supply a constant amount of hours. This is not the case in our framework: even if [sigma] = 1, non-Ricardian hours may vary due to the externality term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This feature plays a key role in explaining how consumption externalities affect equilibrium determinacy under LAMP.

B. Consumption Externalities

We consider the following specification for consumption externalities.

DEFINITION 1. (Aggregate Consumption Externality [ACE]) Let [C.sup.o.sub.t] and [C.sup.rt.sub.t] be total consumption by, respectively, Ricardian and non-Ricardian households, and define aggregate consumption as [C.sub.t] = [gamma] [C.sup.rt.sub.t] + (1 - [gamma]) [C.sup.o.sub.t]. The consumption externality terms [X.sup.o.sub.t] and [X.sup.rt.sub.t], entering, respectively, (1) and (6) take the following functional form: [X.sup.o.sub.t] = [X.sup.rt.sub.t] = [C.sub.t].

Under the ACE set-up, the "Joneses" (i.e., the household's reference group for consumption purposes) correspond to the economy-wide average household in the economy. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] type- i household's instantaneous utility, for i = o, rt. Under the specification of preferences in (1) and (6), the marginal utility of private consumption is increasing with respect to consumption by the reference group ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). As we assume that [[theta].sub.i] > 0, we say that households exhibit KUJ preferences: as consumption by the Joneses increases, type i households wish to consume more as their marginal utility is higher.

C. Production

The supply-side of the economy is standard. It consists of two sub-sectors: a retail sector and a wholesale sector. The retail sector is perfectly competitive and produces the final consumption good [Y.sub.t] out of a continuum of intermediate goods through the following CRS technology:

(11) [Y.sub.t] = [[[[integral].sup.1.sub.0] [Y.sup.[epsilon]- 1/[epsilon].sub.j,t] dj].sup.[epsilon]/[epsilon]-1]

where [epsilon] > 1 is the intratemporal elasticity of substitution between any two varieties of intermediate goods. Prices in the retail sector are perfectly flexible. From Equation (11), the optimal demand for the intermediate good [Y.sub.j,t] is given by [Y.sub.j,t] =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

price of the final consumption good.

The wholesale sector is made of a continuum of firms indexed by j, for j [member of] [0,1] They act under monopolistic competition and are subject to nominal rigidities in price setting. The jth firm in the wholesale sector hires labor through a competitive labor market to produce the jth variety of a continuum of differentiated intermediate goods which are sold to retailers. Production follows a simple linear technology:

(12) [Y.sub.j,t] = [Z.sub.t][H.sub.j,t]

Aggregate total factor productivity [Z.sub.t] is stochastic. We assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] follows a standard AR(1) stationary process: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[rho].sub.z] [member of] (0,1) and [[??].sub.z,t] is a mean-zero iid disturbance.

Firms in the wholesale sector are subject to nominal rigidities: in every period r, the jth firm faces a Rotemberg-style quadratic resource cost to price changes given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [??] > 0 and n is steady state gross inflation. The firm's objective is given by:

03) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [F.sub.0,t], is the Ricardian agents' stochastic discount factor. Nominal marginal costs are equal across firms and defined as [MC.sub.t] [equivalent to] (1 - [tau]) [W.sub.t]/[Z.sub.t], The term [tau] [member of] (0,1) is a constant rate at which the government subsidizes labor costs. As discussed in Proposition 1 in Section III below, this subsidy will allow us to obtain an equitable steady state. Firms are assumed to pay lump sum taxes [P.sub.t][T.sub.t], which the government will use to finance the labor subsidy. (12) The maximization of Equation (13) subject to market demand [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], implies the following optimal price setting condition:

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Profits generated by each monopolistically competitive firm are distributed to Ricardian households as dividends, in proportion to their equity shares. For the jth firm, real dividends are equal to:

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

D. Monetary and Fiscal Policy

The fiscal authority levies lump-sum taxes on wholesale firms to finance their labor subsidy. Its budget constraint reads:

(16) [P.sub.t][T.sub.t] = [tau][W.sub.t][H.sub.t]

where [H.sub.t] = [[integral].sup.1.sub.0] [H.sub.j,t]dj is total labor employed in the wholesale sector. The subsidy rate [tau] is set to eliminate the real wage distortion coming from monopolistic competition, at the steady state. (13)

Monetary policy takes the form of a Taylortype interest rate rule whereby the short-term nominal interest rate is set in response to deviations of current (gross) inflation from its (target) steady state level [[PI].sup.*] :

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the policy coefficient [[phi].sub.[pi]] is assumed to be non-negative. (14) The term [R.sup.*] is the interest rate target, which is attained if [[PI].sub.t] = [[PI].sup.*].

Following the terminology of Leeper (1991), the interest rate rule (17) is said to be active--or equivalently to satisfy the Taylor Principle--if [[phi].sub.[pi]] > 1, that is, in response to a 1% increase in inflation the monetary authority raises the nominal interest rate by more that 1%. On the contrary, a rule featuring [[phi].sub.[pi]] < 1 is said to be passive.

III. EQUILIBRIUM AND AGGREGATION

We focus on a symmetric equilibrium, whereby: (a) all price setting firms set the same price and produce the same quantities: [P.sub.j,t] = [P.sub.t] and [Y.sub.j,t] = [Y.sub.t] for j [member of][0, 1]; (b) for both Ricardian and non-Ricardian households, individual and own-group consumption coincide: [c.sup.o.sub.t] = [C.sup.o.sub.t], and [c.sup.rt.sub.t] = [C.sup.rt.sub.t]. Two consequences of this symmetry are that all price setting firms make the same profits, and distribute the same amount of real dividends: [D.sub.j,t] = [D.sub.t] for j [member of] [0, 1].

Price and output equalization together with (14), imply a non-linear Phillips curve:

(18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Letting [C.sup.i.sub.t] and [H.sup.i.sub.t] be, respectively, average consumption and average hours worked by type i households, for i = o, rt, the economy-wide aggregate consumption and hours worked are defined, respectively, as:

(19) [c.sub.t] = [gamma][c.sup.rt.sub.t] + (1 - [gamma])[C.sup.o.sub.t]

(20) [H.sub.t] = [gamma][H.sup.rt.sub.t] + (1 - [gamma])[H.sup.o.sub.t].

Market clearing in the goods market accounts for the real resource costs of price changes:

(21) [Y.sub.t] = [C.sub.t] + [??]/2[([[PI].sub.t] - [PI]).sup.2] [P.sub.t][Y.sub.t]

where [Y.sub.t] = [Z.sub.t][H.sub.t]. The fact that only a fraction 1 - [gamma] of economic agents holds financial assets and owns firms implies that

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where, without loss of generality, we have assumed that each firm's supply of equity shares is constant and equal to one.

Proposition 1 characterizes the steady state equilibrium of our economy.

PROPOSITION 1. (Steady State) Let [mu] = [epsilon]/[epsilon] - 1 be the gross mark-up in the wholesale sector. If the government subsidizes firms ' labor costs at rate [tau] = [mu] - 1/[mu] via lump-sum taxes on monopolistically competitive firms, there always exists an equitable steady state equilibrium whereby consumption and hours are equal across all agents in the economy: [C.sup.o] = [C.sup.rt] = C and [H.sup.o] - [H.sup.rt] = H Furthermore, this is the unique steady state for [[theta].sub.rt] < 1 and 0 [less than or equal to] [[theta].sub.o] < 1 if [[theta].sub.o] [not equal to] [[theta].sub.rt], and for any [theta] if [[theta].sub.o] = [[theta].sub.rt] = [theta].

Proof. See Appendix A1.

IV. EQUILIBRIUM DETERMINACY ANALYSIS

In this section, we provide analytical conditions for the (local) determinacy of a Rational Expectations Equilibrium (REE) in the presence of ACE. For this purpose, we linearize the economy around a steady state equilibrium. Given the results of Proposition 1. we guarantee that the steady state is unique and equitable by imposing the following assumption.

ASSUMPTION 1.

0 < [[theta].sub.o] = [[theta].sub.rt] = [theta] < 1.

As proved in Proposition 1, this assumption on the lack of heterogeneity in consumption externality across households is innocuous in terms of existence and uniqueness of an equitable steady state as long as [[theta].sub.o] > 0 (positive consumption externality). (15) Although [[theta].sub.o] = [[theta].sub.rt] = [theta] is enough for steady state equitability under the ACE set-up, we also impose the restriction [theta] < 1. This is sufficient to guarantee that, under all circumstances, an IADL of monetary policy transmission necessarily requires a certain degree of LAMP, and that it cannot arise exclusively because of the presence of consumption externalities. (16)

From the steady state consumption and labor equalization across all agents in the economy, the log-linearization of Equations (19) and (20) gives, respectively (17):

(23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From the non-Ricardian household's optimization problem, simple algebra provides expressions for labor supply and consumption by non-Ricardian agents:

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the real wage, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is derived from Equation (3):

(27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The description of the demand-side of the economy is completed by the log-linear version of the Ricardian household's Euler equation (4):

(28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

One key asymmetry in the transmission channel of monetary policy in this framework can be observed by looking at Equations (26) and (28): a change in the interest rate set by the monetary authority directly impacts consumption of the Ricardians through the term ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]): however, the same policy will affect the rule of thumb consumers indirectly via its impact on wages (with or without the ACE) and hours worked (with ACE).

The supply side consists of a linearized New-Keynesian Phillips curve coming from Equation (18) together with the definition of real marginal costs, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]:

(29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where k = [epsilon]-1/[??] The model is closed by assuming the ACE specification, and by letting the interest rate [[??].sub.t], be determined according to a linearized version of policy rule (30):

(30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We introduce the following assumption just for analytical tractability.

ASSUMPTION 2.

[sigma] = 1.

This, of course, implies log-preferences with respect to individual consumption, and, more importantly, makes hours worked by non-Ricardian households not a function of the real wage as in Bilbiie (2008). (18)

As shown in Appendix A2, under the ACE specification, the linearized equilibrium conditions can be reduced to the following linear system:

(31) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(32) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where

(33) [delta] [equivalent to] 1 - [gamma]/1 - [theta] - [gamma] [1 + [chi] + [theta]/1 + [chi] - 2[theta]], [eta] [equivalent to] [1 + [chi] - [theta]]

while [[psi].sub.y,z] and [[psi].sub.[pi],z] are defined in Appendix A2.

Equation (31) is a dynamic IS curve that describes the demand-side block of our economy. In particular, the composite parameter o reflects the responsiveness of current aggregate consumption (output) to changes in the ex ante real interest rate, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which characterizes the aggregate demand channel of monetary policy transmission. The following lemma states some key properties of [delta].

LEMMA 1. (Monetary Policy Transmission) Let [??] [equivalent to] 1 - [theta]/1 + [chi] + [theta]/1 + [chi] - 2[theta] and [[gamma].sup.*] [equivalent to] 1/1+[chi] where the latter corresponds to the value of [??] for [theta] = 0 (no-externality case) The composite parameter [delta] defined in (33) satisfies the following properties: a) [delta] [??] 0 for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. where i. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ii. [partial derivative][??]/[partial derivative][theta], < 0 with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] b) [partial derivative][delta]/[partial derivative][gamma] > 0 with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The proof of this lemma is omitted as it is a simple calculus exercise on (33). According to point (a) in Lemma 1, if the share of non-Ricardian households is sufficiently large (y > y), then [delta]<0. This, by Equation (31), implies that aggregate output (and consumption) responds positively (rather than negatively) to a monetary-policy-driven increase in the real interest rate. There is a IADL of monetary policy transmission in place, as in Gali et al. (2004) and Bilbiie (2008). However, in our framework, the threshold value [??] above which the IADL occurs is also a function of the consumption externality parameter [theta]. In particular, under KUJ preferences ([theta] > 0), increasing the importance of positive externalities lowers y, inducing the IADL to occur for milder degrees of LAMR In the limiting case of [theta] [right arrow] 1, [??] approaches zero, inducing the IADL for virtually any degree of LAMR Since the occurrence of an IADL is the key threat to the Taylor principle in a NK-LAMP economy, the results stated in Lemma 1 show that the importance of LAMP for monetary policy design is strengthened if households feature KUJ preferences.

Proposition 2 defines analytical conditions for local equilibrium determinacy under the monetary policy rule (30).

PROPOSITION 2. Assume that household utility is affected by aggregate consumption (ACE), that is, [X.sup.0.sub.t] = [X.sup.rt.sub.t] = [C.sub.t] in (1) and (6). Recall the definition of [??] from Lemma 1, and define the following thresholds:

(34) [bar.[phi]] [equivalent to] 1 - [beta]/[delta]k[eta] and [??] [equivalent to] - 2(1 + [beta]) + [delta]k[eta]/[delta]k[eta]

Suppose that the central bank adopts a contemporaneous interest rate rule like 30. Then, the equilibrium is locally determinate:

1) for [gamma] < [??] (SADL), if [[phi].sub.[pi]] > 1;

2) for [gamma] > [??] (IADL) if [[phi].sub.[pi]] < min {1, [bar.[phi]], [??]} or if [[phi].sub.[pi]] max 1. {1, [??]}.

Proof. See Appendix A3.

This proposition is isomorphic to Proposition 7 in Bilbiie (2008), and in line with the numerical results of both Gali et al. (2004) and Colciago (2011). What is novel here is how LAMP and consumption externalities interact to affect the equilibrium determinacy properties of the Taylor principle. Following the results stated in Lemma 1, the consumption externality changes the LAMP threshold [??]. In particular, by lowering it, a stronger KUJ externality makes case 2) in the proposition (i.e., determinacy requires either a reinforced Taylor principle or a passive response to inflation) more likely to occur.

The simple economic intuition as to why increasing the importance of positive consumption externalities in utility makes it harder for the Taylor principle to guarantee equilibrium determinacy in a LAMP economy is as follows. Suppose that households feature KUJ preferences with [theta] [member of] (0, 1). Assume that, due to a sunspot shock, current inflation [[??].sub.t], and next period expected inflation [E.sub.t][[??].sub.t+1], both increase. An active response to inflation by the central bank induces a real interest rate increase, which lowers consumption by the Ricardian agents (intertemporal substitution effect, SE). By its impact on the marginal rate of substitution between consumption and hours worked, the latter leads to a rightward shift of labor supply by the Ricardians, and, as a result, to a lower real wage. Non-Ricardian consumption starts dropping for two reasons: first, because of the lower real wage (as in a standard LAMP model without externalities); second, because, with aggregate consumption falling, they will also reduce hours worked in their attempt to keep up with the Joneses (see Equation [25] with [sigma] = 1 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). Facing a larger decrease in labor income, non-Ricardian consumption decreases by more with respect to a model where the consumption externality is absent. This negative aggregate demand effect implies a further negative impact on Ricardian consumption too. As we can see from the Euler Equation (28), with [[??].sub.t] going down, for a given increase in real interest rate increase, the cut in [[??].sup.o.sub.t] will be larger.

The contraction in aggregate demand will force the sticky-price firms to cut on labor demand, thus putting further downward pressure on the real wage (and, as a result, even lower consumption by the non-Ricardian). If the share of non-Ricardian consumers is high enough, the labor supply curve is steep enough (low elasticity), or the ACE is strong enough, the real wage contraction will be sufficiently sizable to determine a large increase in corporate profits, and hence on dividends distributed to the Ricardian agents. The resulting income effect (IE) will then offset the SE described above, and increase Ricardian consumption. As demand starts picking up, firms will hire more workers, which will lead to a progressive increase in the real wage. Non-Ricardian agents' consumption will increase as well, both because of the positive income effect (higher wage) and their desire to keep up with aggregate consumption. This further boost in demand will make firms hire even more, until, because of the substantial increase in the marginal costs of production, they will raise prices: the initial (sunspot-driven) increase in inflation is self-fulfilling. (19)

The ACE channel amplifies both the initial drop and the subsequent rebound in aggregate demand. On the one hand, during the initial contraction, for given degree of LAMP and given labor supply elasticity, the KUJ externality implies a larger increase in corporate dividends, and therefore, a larger IE on Ricardian consumption. On the other hand, during the expansion phase, it boosts non-Ricardian consumption, thus increasing the upward pressure on marginal cost (hence, on price setting) coming from demand. It therefore follows that, with a positive ACE channel at work, the economy might experience an IADL for lower degrees of LAMP.

Similarly to Bilbiie (2008), Proposition 2 argues that, when the IADL holds, equilibrium determinacy requires either a sufficiently active or a sufficiently passive response to inflation. The intuition for why a passive rule is desirable with an IADL follows the same logic (just a reversed argument), we used to construct a sunspot equilibrium under an active rule: namely, a passive rule generates a positive SE, but a potentially strong negative IE. A very active response to inflation counter-acts instead the initial sunspot-driven inflation by inducing a very strong negative SE (hence, a large drop in Ricardian consumption), which then makes it harder for the IE coming from corporate dividends to generate an overall increase in aggregate consumption.

V. QUANTITATIVE ANALYSIS

In this section, we evaluate quantitatively the performance of our model along three different dimensions. First, to better gauge the importance of the analytical results in Proposition 2, we provide a numerical appraisal of the extent of lack of equilibrium determinacy for a benchmark parameterization of the economy. Second, we check whether, for suitable calibrations, our model under indeterminacy can generate aggregate volatilities of sizes comparable to those observed in the pre-Volcker era in the United States without necessarily requiring a violation of the Taylor principle. Third, since the IADL channel stems from the interaction between labor and financial markets, we quantitatively assess how the degree of LAMP and the strength of consumption externalities affect the dynamic responses of endogenous variables in those markets to a structural shock on TFP.

A. (In)Deteminacy

Steady state gross inflation is fixed equal to unity. The discount factor [beta] is set equal to 0.99, so that our economy features an annual real interest rate of about 4%. We set the intratemporal elasticity of substitution across intermediate goods, e, equal to 6, as in the benchmark New-Keynesian model presented in Gali (2008). (20) This value implies a 20% steady state net mark-up in the wholesale sector. The labor disutility parameter [chi] is set equal to 1, which corresponds to a unit Frisch elasticity of labor supply, [[chi].sup.-1], at the steady state. (21) This value implies that, absent the externality, the threshold value across which the economy goes from a SADL to a IADL--that is, [[gamma].sup.*] as defined in Lemma 1--is equal to one-half. The price adjustment cost parameter A is chosen by taking advantage of the reduced-form equivalence between a Rotemberg-style and a Calvo-style approach to price stickiness. More specifically, we consider an economy with an average duration of price contracts equal to four quarters, which, under Calvo pricing, corresponds to a probability of no price change equal to 0.75. This is the most common calibration of price stickiness in the New-Keynesian literature, and the same used in Bilbiie (2008). We then compute the temporary elasticity of inflation to real marginal costs in the (linearized) Phillips curve: that is, [k.sub.Calvo] [equivalent to] (1 - [[beta].sub.p])(1 - p)/p = 0.085 for p = 0.75. For given p and [epsilon], we then set the adjustment cost parameter [??] such that [epsilon]-1/[??] = 0.085 This procedure gives [??] = 59. Table 1 summarizes the parameterization. To show how equilibrium determinacy is affected by both the degree of LAMP and the importance of consumption externalities, we do not assign [gamma] and [theta] to any specific value, but rather consider a wide range of parameterizations for both of them.

Figure 1 displays the determinacy and indeterminacy areas with respect to the rule's response to inflation, [[phi].sub.[pi]], and the share of non-Ricardian consumers, [gamma], for different degrees of positive consumption externalities (KUJ preferences). Absent the externality, the equilibrium is locally determinate (respectively, indeterminate) for combinations of [[phi].sub.[pi]] and [gamma] that fall within the white (respectively, grey) areas. In particular, for [gamma] smaller than 0.52, the Taylor principle guarantees local determinacy. As the share of non-Ricardian consumers passes such threshold, the central bank can rule out sunspot-driven equilibria by granting either a sufficiently active or a sufficiently passive response to inflation. (22) For instance, under an active rule, the response coefficient to inflation needs to be extremely aggressive: it has to be at above 4.5 if [gamma] = 0.55, and above 9 if [gamma] = 0.6! The introduction and strengthening of a positive aggregate consumption externality (larger 0) restricts the value range of y for which the simple Taylor principle applies: the figure shows that as [theta] monotonically increases from 0 to 0.95--with intermediate steps given by [theta] = 0.5 and [theta] = 0.75--the grey indeterminacy area increases as well adding portions of the area labeled as [D.sub.1] through [D.sub.3]. The quantitative impact is substantial: raising [theta] from 0 to 0.95, the threshold share [??] drops from 0.52 to 0.1. As a consequence, a larger [theta] also increases the required degree of aggressiveness toward inflation necessary to guarantee determinacy. Consider the case of [gamma] equal to 0.5. The minimum response to inflation goes from unity (for [theta] = 0) to about 6 (for [theta] = 0.5). (23)

[FIGURE 1 OMITTED]

Figure 2 displays the indeterminacy implications of consumption externalities from a different angle. Holding constant the policy response to inflation [[phi].sub.[pi]] = 2--which is consistent with the moderate degree of activism toward inflation observed in the United States in the post-Volcker period (see Cogley and Sargent 2005 and Boivin and Giannoni 2006, for instance)--this figure identifies the combinations of [theta] and [gamma] for which the equilibrium is locally determinate. These combinations also in turn depend on the labor disutility parameter, [chi]. The figure should be interpreted as follows: regardless of the three alternative parameterizations of the labor disutility parameter, [chi], considered, the equilibrium is always locally determinate in the region lying below the downward-sloping determinacy frontiers. Hence, if [chi] = 1, the equilibrium is determinate within the region [D.sub.1] + [D.sub.2] (white area, our benchmark parameterization). If [chi] = 2, equilibrium determinacy only arises within the region [D.sub.2]; and if [chi] = 1/3. the determinacy region expands to [D.sub.1] + [D.sub.2] + [I.sub.2]. The figure also clearly shows that the degree of LAMP, indexed by [gamma], below which the policy rule guarantees determinacy is strictly decreasing in 0. For this it suffices to consider the downward-sloping determinacy frontiers: let us examine the no-externality case of [theta] = 0. The equilibrium is determinate for [gamma] smaller than 0.53. Following the first downward-sloping determinacy frontier from the origin, the reader can see that this value of [gamma] drops to just above 0.1 when [theta] approaches the limiting value of 1. As the figure shows with the various downward-sloping determinacy frontiers, the negative impact of the externality parameter [theta] on the size of the determinacy region is common across alternative parameterizations of the (inverse) Frisch elasticity of labor supply [chi]

B. Macroeconomic Instability in the Pre-Volcker Era

A well-known fact, first documented by Clarida et al. (2000), is that during the 1970s, both output and inflation were remarkably volatile in the United States. In Table 2, we report their findings: between 1960 and 1979, the standard deviations of inflation and output were, respectively, 1.48% and 1.83% (the volatility of inflation was then about 80% of what observed for output).

[FIGURE 2 OMITTED]

Among the plausible explanations for such abnormal volatilities, the most prominent one in the literature underlines the possibility that, during those years (the so-called pre-Volcker era), the FED adopted monetary policies that were too accommodating toward inflation. (24) Clarida et al. (2000) and Lubik and Schorfheide (2004), for instance, suggest that the FED systematically violated the Taylor principle by adopting a passive monetary policy before 1979 (but then shifted to an active policy once Paul Volcker's term began) and show how this might have led to macroeconomic instability from self-fulfilling changes in expectations. Orphanides (2004) and Coibion and Gorodnichenko (2011) suggest instead that policy-makers misperception of key structural parameters in the 1970 could have been the source of the excessive volatility. This is in spite of the fact that, as they document, the FED appeared to have followed the Taylor principle also in the pre-Volcker's years. (25)

Motivated by the theoretical and numerical results presented in previous sections, we assess whether our model--in which active monetary policies can still lead to equilibrium indeterminacy--can generate both output and inflation volatility which are comparable to those observed in the United States in the pre-Volcker era.

We compute predicted volatilities of output and inflation for which the model economy displays equilibrium indeterminacy despite an active response of monetary policy to inflation. For this purpose, we consider two alternative degrees of LAMP: [gamma] = 2/3 and [gamma] = 3/4. Those figures are within the range of available estimates for the share of hand-to-mouth consumers in the United States before the mid-1980s. (26) As far as the strength of consumption externalities, we let [theta] [member of] [0,0.5] which is consistent with the empirical findings by Maurer and Meier (2008). The remaining structural parameters are as in the benchmark calibration of Table 1. Our conjecture is that monetary policy did not necessarily experience any regime switch (from passive to active) in 1979, but rather that the FED responded actively to inflation also in the pre-Volcker era. To this end, we consider three alternative values for the FED stance on inflation, [[phi].sub.[pi]]: 1.05, 1.25, and 1.5, all of which lead to equilibrium indeterminacy under the benchmark calibration. (27)

We let the economy be hit by both fundamental shocks (e.g., shocks to TFP and to the policy rate) and non-fundamental shocks (sunspot shock). Our parameterization of the TFP process defined in Section II.C follows Arias et al. (2007). Their estimates for the auto-regressive coefficient, [[rho].sub.z], and the standard deviation of the TFP innovation before 1983 are, respectively, 0.95% and 0.75%. (28) The interest rate shock is assumed to be i.i.d. with standard deviation equal to 0.2, which is consistent with empirical estimates. We model the sunspot shock as a belief shock to inflation, as in Lubik and Schorfheide (2003). The belief shock is assumed to be a i. i. d. sunspot shock to the forecast error on inflation, with standard deviation equal to 0.0075%, that is, one-tenth of the standard deviation assumed for the TFP disturbance. We impose a zero correlation between the belief and the TFP shocks, that is, the belief shock to inflation is a pure sunspot. We solve the model under indeterminacy following the methodology introduced by Lubik and Schorfheide (2003).

The panels in Figure 3 display absolute and relative predicted volatilities. Few interesting results emerge. First, for all parameterizations considered, inflation appears to be less volatile than output, with a relative volatility around 80% (as in the data) for the case of a mildly active response to inflation ([[phi].sub.[pi]] = 1.05) and a sizable externality ([theta] = 0.5). Second, absolute volatilities are mildly increasing with respect to the consumption externality. Third, a more active response to inflation by the policy-marker has a significant impact on the volatility of inflation, but almost no effect on output. Table 3 reports some numerical values for the graphical results displayed in Figure 3. Despite the assumed low volatility for the belief shock, overall, our model under indeterminacy performs rather well when compared to the data.

For each pair of values ([gamma], [theta]), the table displays the predicted standard deviation of inflation ([[sigma].sub.[pi]]), of output ([[sigma].sub.y],) and the relative standard deviation ([[sigma].sub.[pi]]) of output ([[sigma].sub.y]) obtained from the model under indeterminacy. Predicted standard deviations are Monte-Carlo averages (for N = 10. 000) of standard deviations computed for simulated series of 80 observations, which corresponds to the length of the period 1960-1979 at quarterly frequency.

C. Response to a TFP Shock

How does the model respond to a TFP shock? We find that both aggregate hours worked and real wage increase in our model. The opposite behavior of hours with respect to a technology shock is put forth in Francis and Ramey (2005) in a model with adjustment cost in investment and internal habits. Also in NK models with LAMP and without firm specific capital, a positive technology innovation generates a declines in total labor input as in Furlanetto and Seneca (2012). Hence, having ACE and LAMP is clearly crucial to generate a positive response of hours worked to a TFP shock. Contrary to Furlanetto and Seneca (2012), in our setting the expansion of aggregate consumption as a byproduct of the technology shock is not curbed by rule of thumb consumers, but rather enhanced. To this end, nesting the externality in the preferences is crucial because, without this assumption, our numerical exercise shows that the response of total hours worked to a technological innovation would be negative as in the standard NK model.

To pursue this analysis, we set the share of non-Ricardian consumers [gamma] equal to 0.3 and the response coefficient to inflation in the policy rule [[phi].sub.[pi]] equal to 2. The remaining parameters are as in Table 1, while the persistence of the TFP shock [[rho].sub.z] is set to 0.95. We compute impulse responses under two alternative parameterizations: a NK-LAMP model without externalities ([gamma] = 0.3, [theta] = 0); and a NK-LAMP with KUJ-ACE ([gamma] = 0.3, [theta] = 0.25). Under both parameterizations considered, the model economy features a locally determinate equilibrium, such that the impulse responses to the TFP shock are uniquely defined. Figure 4 displays the results of the analysis. (29)

[FIGURE 3 OMITTED]

We consider a 1% positive shock to TFP. In the no externality benchmark set-up, a positive shock to technology raises the marginal productivity of labor, which, through its negative impact on marginal costs, lowers inflation. Given the active monetary policy rule, both the nominal and real interest rate decrease, creating an incentive for the Ricardian households to consume more. With non-Ricardian households' labor supply schedule shifting leftward and the labor demand schedule by firms shifting rightward (to satisfy the additional demand), their equilibrium real wage increases, and so does consumption. Similar to a benchmark full-participation New-Keynesian model, (Ricardian) hours are lower in equilibrium. (30)

[FIGURE 4 OMITTED]

One very compelling consequence of introducing positive consumption externalities in this setting is a sign switch in the response of aggregate hours to the technology shock, from negative to positive (lower-left panel in Figure 4). As aggregate consumption increases, both Ricardian and non-Ricardian households work extra hours to keep up with the Joneses, leading to an overall increase in hours worked. This is in line with the evidence put forth in Chari et al. (2008) and, to the best of our knowledge, is a completely novel result for NK models without capital. This result, which is driven by how consumption externalities affect the labor and financial market equilibrium, is important because typically in NK models a positive technology innovation generates a declines in total labor input. Since the way hours worked and real wages react to technological innovations are at the core of the Real Business Cycle (RBC) literature as the main driving force of aggregate fluctuations, we interpret this finding as a nice opportunity to reconcile the NK and RBC traditions on this particular issue. Along the expansion, aggregate output as well as the consumption levels by both household types increase (with consumption by non-Ricardians rising by more than consumption by Ricardians). Through the Phillips curve, the large jump in output leads to a lower decrease in inflation and, as a consequence, in the nominal and real interest rates with respect to the no-externality case. Interestingly, consumption externalities also weaken the positive stock price response to the TFP shock. The externality-driven boost in consumption attenuates the demand increase for risky stocks occurring in a boom. Since, in equilibrium, the supply of stocks is constant (and normalized to one) and Ricardian household are consuming more and do not have other ways of saving, all share prices increase by less and so does the stock price index when compared to the no externality case.

VI. EXTENSIONS AND ROBUSTNESS

In this section, we present results for some natural extensions of our benchmark model: in particular, we focus on forward-looking Taylor rules, and real wage rigidities and how those added features affect the indeterminacy region.

A. Forward-Looking Taylor Rule

We now consider a forward-looking policy rule, whereby the interest rate is a set as a function of expected future inflation. In log-linearized form, we have that:

(35) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proposition 3 provides analytical results.

PROPOSITION 3. (ACE and Forward-Looking Rule) Assume that household utility is affected by aggregate consumption (ACE), that is, [X.sup.o.sub.t] = [X.sup.rt.sub.t] = [C.sub.t] in (1) and (6). Recall the definitions of [delta] and [eta] from (33), off from Lemma 1, and let [[phi].sup.f] [equivalent to] 1 + 2(1+[beta])/[delta]k[eta]. If the central bank follows a simple forward-looking Taylor rule like (35), then the equilibrium is locally determinate:

a) for [gamma] [member] [0, [??]) if 1 [[phi].sub.[pi]] < [[phi].sup.f];

b) for [gamma] [member of] ([??]. 1) if max {[[phi].sup.f], 0} < [[phi].sub.[pi]] < 1.

The timing of the policy rule matters for equilibrium determinacy. This is clear by comparing the results of Propositions 2 and 3. For instance, determinacy is more likely to emerge under a contemporaneous than a forward-looking rule since active forward-looking rules leading to determinacy need also to satisfy an upper bound [[phi].sup.f], which becomes more binding as the degree of LAMP increases (i.e. [partial derivative][[phi].sup.f]/[partial derivative][gamma] < 0 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). (31)

The introduction of consumption externalities affects equilibrium determinacy in two ways. First, as seen in Lemma 1, the threshold value [??] is strictly decreasing in [theta] and smaller than the no-externality threshold, [[gamma].sup.*], if the externality is positive. Second, for given [gamma] < [??], the upper bound [[phi].sup.f] is strictly decreasing in 0: increasing the importance of positive consumption externalities in preferences restricts the range of active (forward-looking) interest rate rules for which the equilibrium is locally unique. Furthermore, in the limiting case of 0 [right arrow] 1, we have that [??] [right arrow] 0 (see Lemma 1) and also [[phi].sup.f] [right arrow] 1 : a rule satisfying the Taylor principle always leads to sunspot-driven equilibria for any degree of LAMP. We summarize these results in the following Corollary.

COROLLARY 1. A larger [theta] restricts the value range of [gamma] for which equilibrium determinacy is attained under an active forward-looking Taylor rule.

Figure 5 displays the equilibrium determinacy and indeterminacy regions with respect to the rule's responsiveness to inflation, [[phi].sub.[pi]], and the share of non-Ricardian consumers, [gamma], for progressively higher values of [theta], under the ACE specification. Increasing the strength of positive consumption externalities moves to the left the threshold [??] identified in Proposition 3, thus restricting the range of y for which active forward-looking rules can induce equilibrium determinacy. For instance, [gamma] is equal to, respectively, 0.5, 0.4, 0.28, and 0.08 for 0 equal to, respectively, 0, 0.5, 0.75, and 0.95.32 Moreover, a larger [theta] also makes the upper-bound [[phi].sup.f] more binding.

B. Real Wage Rigidity

Colciago (2011) shows that the introduction of nominal wage rigidity can restore the Taylor principle as a necessary and sufficient condition for equilibrium determinacy in the benchmark LAMP of Gali et al. (2004). (33) By dampening fluctuations in real wages, the nominal rigidity reduces the upward shift in aggregate demand due to non-Ricardian consumption, making a sunspot-driven surge in real activity more unlikely to be self-fulfilling. Here, we analyze the strength of this channel under consumption externalities.

To capture the essence of the mechanism highlighted by Colciago (2011) without losing analytical tractability, we introduce wage stickiness along the lines of Blanchard and Gali (2007, 2010); that is, we assume that real wages respond sluggishly to labor market conditions, as a result of un-modeled labor market imperfections. More specifically, we assume that the real wage paid to workers is a weighted average of a notional wage--which we pose equal to the real wage occurring in a perfectly flexible labor market--and a wage norm (using the terminology of Hall, 2005)--for which we take the fully efficient real wage occurring in steady state. (34) Letting [w.sup.f.sub.t] be the real wage in a friction-less environment and [w.sup.e] its efficient steady state level, the new real wage wt is given by the following expression:

(36) [w.sub.t] = [([w.sup.f.sub.t]).sup.1-[xi]] [([w.sup.e]).sup.[xi]]

where, because of the labor subsidy to firms, we have that [w.sup.e] = 1 (see the proof of Proposition 1). The parameter [xi] [member of] [0, 1] is an index of real wage rigidity in the economy: for 4 = 0, the real wage is fully flexible and [w.sub.t]/[p.sub.t] = MRS" = MRS" as in (9); for [xi] = 1, the real wage is constant as in the canonical model of Hall (2005) and equal to its steady state efficient level. (35)

[FIGURE 5 OMITTED]

With this specification, we obtain an aggregate IS curve and a New-Keynesian Phillips curve which are isomorphic to equations (31) and (32), but where the composite parameters [eta] and [delta] are replaced, respectively, by [[eta].sup.w] = (1 - [xi])/[gamma](1-[xi]) + (1- [gamma]) and

(37) [[delta].sup.w] [equivalent to] 1 - [gamma]/(1 - [gamma] [theta]/1 + [chi])[1 - [gamma](1 + [chi](1 - [xi]))]. - [theta][(1 - [gamma]).sup.2]

As previously discussed, the key source of indeterminacy in a LAMP model is the occurrence of a IADL of monetary policy transmission, which is implied by a sign switch (from positive to negative) in the responsiveness of aggregate activity to the ex ante real interest rate. In a no-externality flexible-real-wage economy ([theta] = 0 and 4 = 0), the switch occurs at [gamma] = 1/1+[chi]. If externalities are absent but the real wage is instead determined by (36), the occurrence of a IADL is less likely since this requires [gamma] > [[gamma].sub.w] [equivalent to] 1/1 + [chi](1 - [xi]) where the threshold [[gamma].sub.w], is strictly increasing in [xi] and tends to unity as [xi] [right arrow] 1. The economic intuition is along the lines of Colciago (2011): as sticky real wages are less responsive to real activity, so is non-Ricardian consumption.

Although it remains valid, this mechanism seems to be less effective once consumption externalities are introduced. For [theta] > 0, the threshold share above which a IADL appears--that is, where [[delta].sup.w] goes from positive to negative--lies somewhere between [[gamma].sup.*] (the no-externality flexible-real-wage case) and [[gamma].sub.w] (the no-externality sticky-real-wage case): that is, a positive consumption externality counteracts the beneficial consequences of a wage rigidity for equilibrium determinacy. The intuition is simple: while the hourly wage becomes less responsive to a boom in real activity, non-Ricardian households supply more hours to production to keep up with the Joneses, so that their overall labor income can still display a significant increase.

[FIGURE 6 OMITTED]

Figure 6 displays the determinacy/indeterminacy regions with respect to the degree of LAMP and the degree of real wage rigidity, for different values of [theta], keeping the policy coefficient [[phi].sub.[pi]] equal to 2. The figure shows that positive consumption externalities weaken the (determinacy) benefits of having wage rigidities for almost all parameterizations of For instance, suppose that [xi] = 0.5. Without externalities, the equilibrium is indeterminate for [gamma] > 0.65. This value drops to just above 0.4 if 0 = 0.95. Only in the limit (and rather implausible) case of full real wage rigidity, equilibrium determinacy is not affected by consumption externalities.

VII. CONCLUSIONS

The introduction of limited asset market participation into an otherwise standard New-Keynesian economy makes active interest rate rules--that is, monetary policy rules whereby the short-term interest rate responds more than proportionally to inflation (Taylor principle)--prone to induce aggregate instability in the form of self-fulfilling sunspot equilibria due to equilibrium indeterminacy.

In this article, we study the impact of consumption externalities on the indeterminacy problem generated by LAMP. We find that increasing the strength of externalities enlarges the parametric range of LAMP where the Taylor principle fails to deliver a locally unique equilibrium if household exhibit KUJ preferences. For instance, under KUJ preferences, if the parameter measuring the importance of consumption externalities in preferences is sufficiently large, the simple Taylor principle fails to be sufficient for equilibrium determinacy, for (almost) any degree of LAMP in the economy. Under these circumstances, restoring equilibrium determinacy requires the central bank to grant an extremely and unrealistically large response to inflation.

Our quantitative analysis shows that, for given level of LAMP, there exists a wide range of parametrizations for the Frisch elasticity of labor supply and the degree of consumption externality for which even a mildly active policy rule leads to an indeterminate equilibrium. This result seems to suggest that the misperception of key structural parameters on behalf of policy-makers, rather than policy mistakes (i.e., a passive response to inflationary pressure), could have been the source of excessive volatility in the 1970s. We find that our model-predicted volatilities under indeterminacy mimic very well those observed in the preVolcker period. For instance, for a Frisch elasticity equal to unity (a calibration often used in the macro literature) and an intermediate degree of externality, the predicted volatilities of inflation and output are about 80% of that observed in the data, while the relative volatility exactly matches the empirical evidence.

The introduction of consumption externalities also helps to generate a positive response of aggregate hours worked to TFP, a result which is in line with the empirical findings by Christiano et al. (2003) and Chari et al. (2008). This is typically not the case in a benchmark New-Keynesian model (where externalities are absent) unless one introduces additional real rigidities such as firm-specific capital or capital adjustment costs. Since the way hours worked and real wages react to technological innovations are at the core of the Real Business Cycle (RBC) literature as the main driving force of aggregate fluctuations, we interpret this finding as a nice opportunity to reconcile the NK and RBC traditions on this particular issue.

APPENDIX

A1. PROOF OF PROPOSITION 1: STEADY STATE EQUILIBRIUM

Proof. We consider a zero inflation (i.e., [[PI].sup.*] = [PI] = 1) nonstochastic steady state equilibrium. From the Euler equation of Ricardian agents (4)--and for any specification of the externality--we obtain the steady state (nominal and real) interest rate: R = [[beta].sup.-1].

As standard, the Phillips curve (18) implies that real marginal costs are equal to the inverse of the gross-markup in the retail sector: MC/p = [epsilon]-1/e[epsilon] [equivalent to] 1/[mu]. This implies that at the steady state the real wage is equal to W/P = 1/(1-[tau])[mu]. We assume that the subsidy rate x is set by the fiscal authority to offset the distortion coming from monopolistic competition, that is, X is chosen such that the steady state real wage is equal to one (the marginal productivity of labor), as it would occur in a friction-less economy. This gives [tau] = [mu]-1/[mu].

With W/P = 1. the steady state Ricardian and the non-Ricardian households budget constraints become, respectively:

(Al) [C.sup.o] = [H.sup.o] + [D.sup.o] and [C.sup.rt] = [H.sup.rt]

where from (22) and the steady state version of (15) [D.sup.o] = D/(1-[gamma]) = (1 - MC/P)Y - T/(1-[gamma]) From the fact that M C/P = 1/[mu], the government balanced-budget (16) and the definition of the labor subsidy [tau] = [mu]-1/[mu], it easily follows that steady state aggregate dividends are zero: D = 0. From (A1), we then have that [C.sup.o] = [H.sup.o] and [C.sup.rt] = [H.sup.rt].

By these last two equalities, at the steady state, the labor supply conditions (3) and (8) are then equivalent to:

(A2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(A3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consider the ACE set-up, for which [X.sup.o] = [X.sup.rt] = C. Given the definition C [equivalent to] [gamma][C.sup.rt] + (1 - [gamma])[C.sup.o], equations (A2) and (A3) can be written as follows:

(A4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(A5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose that [[theta].sub.o] = [[theta].sub.rt] = 0. Dividing (A4) by (A5), it immediately follows that [C.sup.o] = [C.sup.rt], for any [theta].

Next, consider the more general case of [[theta].sub.o] [not equal to] [[theta].sub.rt], and assume that [[theta].sub.o], [[theta].sub.rt] < 1. Furthermore, let q [equivalent to] [C.sup.rt]/[C.sup.o]. By combining (A4) and (A5), after simple manipulation, we obtain the following condition:

(A6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[omega].sub.i] [equivalent to] [sigma][[theta].sub.i]/[chi]+[sigma](1 - [[theta].sub.i]) for i = o, rt. Let LHS(q) and RHS(q) denote, respectively, the left- and the right-hand sides of (A6). Simple calculus shows that LHS(q) is strictly decreasing in q with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any [[theta].sub.rt] < 1. Meanwhile, RHS(q) is strictly increasing in q with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and RHS(q) = 1 for every q if [[theta].sub.o] = 0. It follows that if 0 [less than or equal to] [[theta].sub.o] < 1 and [[theta].sub.rt] < 1. there exists a unique value q such that LHS([q.sup.*]) = RHS([q.sup.*]). Moreover, as LHS(1) = RHS(1) = 1, it must be that q = 1, that is, at the unique steady state, we must have that [C.sup.o] = [C.sup.rt]. On the other hand, if [[theta].sub.o] < 0 and for any [[theta].sub.rt], < 1, both LHS(q) and RHS(q) are strictly decreasing in q. independently from the sign of [[theta].sub.rt]. Although q = 1 remains a solution to (A6), it is not necessarily the unique one.

A2. REDUCED-FORM SYSTEM UNDER ACE

From Definition 1, we obtain

(A7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

such that the Ricardian households' Euler equation (28) takes the following form:

(A8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

To derive the aggregate counterpart of (A8), we need to express Ricardian consumption [[??].sup.o.sub.t] as a function of aggregate consumption [[??].sub.t]. By combining equations (26) and (A7), together with the assumption [sigma] = 1, equation (23) for aggregate consumption can be written as follows:

(A9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By the market clearing condition, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and equation (25), we can use (24) to find an expression for hours worked by Ricardian agents:

(A10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The latter, together with (27), and simple manipulation, gives us an expression for the the real wage, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]:

(A11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Substituting (All) into (A9), it is possible to write the term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] entering (A8) as a function of aggregate consumption:

(A12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

(A13) [delta] [equivalent to] 1 - [gamma]/1 - [theta] - [gamma][1 + [chi + [theta]/1 + [chi] - 2[theta]]

Plugging (A12) into (A8), we obtain the economy-wide Euler equation for aggregate consumption:

(A14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Letting [[psi].sub.y,z] [equivalent to] [delta] [gamma][chi]/1 - [gamma] ([[rho].sub.z] - 1) and imposing the market clearing condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we obtain equation (31) in the text.

Next, consider the New-Keynesian Phillips curve (29). From (All) and (A 12), the real wage [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be written as a function of aggregate consumption and TFP:

(A15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(A16) [eta] [equivalent to] [1 + [chi] - [theta]].

Assumption 1 guarantees that [eta] is always positive. Plugging (A 15) into (29), and letting [[psi].sub.[pi],z] [equivalent to] -k (1 + [chi]), we obtain equation (32).

A3. PROOF OF PROPOSITION 2: DETERMINACY

Under the policy rule [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the system (31) and (32) can be written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [gamma] is a conformable matrix whose specification is not needed for the determinacy analysis.

As both variables are non-predetermined, the equilibrium is locally determinate if and only if both eigenvalues of the Jacobian matrix J are outside the unit circle in the complex plane. Let Tr(J) and Det(J) denote, respectively, the trace and the determinant of matrix J. Necessary and sufficient conditions for local determinacy are either

CASE A. i. Det (J)> 1, ii. 1 - Tr(J) + Det(J)>0, and iii. 1 + Tr(J) + Det(T) > 0

CASE B. i. 1 - Tr(J) + Det(J) < 0 and ii. 1 + Tr(J) + Det (J) < 0.

Suppose that [gamma] < [??] = 1 - [theta]/1 + [chi] + [theta]/1 + [chi] - 2[theta]. From its definition in

(33) and the properties spelled out in Lemma 1, we have that [delta] > 0. On the one hand, it is straightforward to check that, in this case, the determinacy conditions of case B are never satisfied. On the other hand, simple algebra shows that conditions i, ii, and iii of case A are, respectively, equivalent to:

(A17) [[phi].sub.[pi]][delta]k[eta] > [beta] - 1

(A18) [delta]k[eta]/[beta] ([[phi].sub.[pi]] - 1) > 0

(A19) - [delta]k[eta][phi][pi] < 2 (1 + [beta]) + [delta]k[eta].

By Assumption 1, [eta] > 0 such that conditions (A17)-(A19) hold if and only if [[phi].sub.[pi]] > 1.

Next, suppose that [gamma] > [??]. By Lemma 1, we have now that [delta] < 0. Consider the determinacy conditions for case A and let

[bar.[phi]] [equivalent to] 1 - [beta]/[delta]k[eta] and [??] [equivalent to] - 2(1 + [beta]) + [delta]k[eta]/[delta]k[eta].

Conditions (A17) and (A19) reduce to the following inequality:

(A20) [[phi].sub.[pi]] < min {1, [bar.[phi]], [??]}.

Consider now the conditions for case B. Using the definition of [??]. conditions B.i and B.ii can be written as follows:

(A21) [delta]k[eta]/[beta] ([[phi].sub.[pi]] - 1) < 0 and [phi][pi] > [??].

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By the inequalities in (A21), equilibrium determinacy obtains if [[phi].sub.[pi]] < min {1, [bar.[phi]], [??]} or if [[phi].sub.[pi]] > max {1, [??]}.

A4. PROOF OF PROPOSITION 3: FORWARD-LOOKING TAYLOR RULE

Proof. The local equilibrium dynamics are described by a linear system made of equations (31) and (32). together with the interest rate rule (35). After simple algebra, the equilibrium system becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[gamma].sub.f] is a conformable matrix whose specification is not needed for the determinacy analysis.

As both variables are non-predetermined, the equilibrium is locally determinate if and only if both eigenvalues of the Jacobian matrix J are inside the unit circle in the complex plane. Let V (e) = [e.sup.2] - Tr(J)e + Det (J) e be the characteristic polynomial of J. where Tr(J) and Det(J) denote, respectively, its trace and its determinant. Necessary and sufficient conditions for both roots of P(e) = 0 (i.e., both eigenvalues of J) to be inside the unit circle are i) [absolute value of (Det(J))] < 1; ii) P(+l) > 0 and P(-1) < 0. Straightforward algebra shows that Det(J) = [beta] [member of] (0, 1) and

P (+1) = k[eta][delta] ([[phi].sub.[pi]] - 1) P(-1) = 2(1 + [beta]) + k[eta][delta] (1 - [[phi].sub.[pi]]).

Let [[phi].sup.f] = 1 + 2(1+[bet])/k[eta][delta]. Given the definition of 8 and q in (33), by the properties spelled in Lemma 1, it immediately follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can then conclude that both P (+1) > 0 and P (-1) < 0 hold (hence, the equilibrium is locally determinate) if and only if

1 < [[phi].sub.[pi]] < [[phi].sup.f] when [gamma] [member of] [0, [??]) max {[[phi].sup.f], 0} < [[phi].sub.[pi]] < when [gamma] [member of] ([??], 1) ABBREVIATIONS ACE: Aggregate Consumption Externality IADL: Inverted Aggregate Demand Logic KUJ: Keeping Up with the Joneses LAMP: Limited Asset Market Participation NK: New-Keynesian RBC: Real Business Cycle TFP: Total Factor Productivity

doi: 10.1111/ecin.12366

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(1.) Bullard and Mitra (2002) provide a detailed analysis of this issue under different timing for the policy rule.

(2.) See, for instance, Bullard and Singh (2008) and De Fiore and Liu (2006) for an open economy environment. Surico (2008) for a closed economy subject to a cost channel, and Airaudo and Zanna (2012) for a closed economy with heterogeneous price rigidities across sectors.

(3.) An interest rate rule is said to be contemporaneous (forward-looking) if it responds to current (expected future) endogenous variables.

(4.) Various empirical studies using both aggregate and micro data confirm that a large fraction of the U.S. population-ranging from 30% to 60%-does not behave according to the permanent income hypothesis. Bilbiie (2008) provides a concise review of the empirical literature.

(5.) Ghiglino and Goyal (2010) and Frank, Levine, and Dijk (2014) build up a model of expenditure cascades based on the theory of networks, showing how social interactions shape equilibrium prices, savings rates and income inequality.

(6.) The definition of KUJ preferences is as in Dupor and Liu (2003) and Chen and Hsu (2007). KUJ require the marginal utility of individual consumption (relative to leisure) to be increasing with respect to consumption by the reference group.

(7.) The benchmark New-Keynesian framework-where all agents are Ricardian-features instead a Standard Aggregate Demand Logic, increasing the interest rate always induces a contraction in aggregate demand.

(8.) In their model internal habits increase the sensitivity of wages to employment (output) and partially offset real wage rigidities thus giving rise to the indeterminacy region. In our model the sensitivity of wages to employment is decreased by the presence of ACE, but because of the KUJ effect, non-Ricardians respond to an increase (decrease) in aggregate consumption by working more (less) which translates into larger (smaller) non-Ricardian consumption. It is a self-fulfilling mechanism.

(9.) The latter parameter is another important difference with our setting because in their model both type of agents work the same amount of time. Hence, the IRF of their model to a TFP shock would not be similar to what we find.

(10.) Note that, as our analysis is conducted by log-linearizing around the steady state, our results are qualitatively robust to alternative functional forms for the externality term.

(11.) Similar to (1), [X.sup.rt.sub.j] is the consumption externality entering non-Ricardian utility and [[theta].sub.rt] its index.

(12.) Our analysis would be effectively unchanged if we instead assumed that that lump-sum taxes were levied on Ricardian households only. Although they would equally hold numerically, some of results would not be attainable in analytical form if we assumed that taxes were also levied on non-Ricardian consumers.

(13.) See Appendix A1.

(14.) To better highlight the impact of consumption externalities on the sufficiency of the Taylor principle for determinacy, we do not consider policy rules that also respond to output. Section 6.1. studies the case of a forward-looking rule.

(15.) Relaxing Assumption 1 may affect the parameter ranges for which the dynamic equilibrium is locally (in)determinate because the output elasticity to the real interest rate in the IS curve would depend on the difference between [[theta].sub.o] and [[theta].sub.rt]. Assumption 1 is also motivated by the fact that, to the best of our knowledge, there is no empirical evidence on cross-household heterogeneity in consumption externality.

(16.) A similar assumption is present both in Gali (1994) and Alonso-Carrera et al. (2008). In Gali (1994), this is sufficient to guarantee the existence of a symmetric equilibrium in a static representative agent framework. Alonso-Carrera et al. (2008) require this condition to make utility concave at the social level.

(17.) From now on, a "hat" on top of a variable will denote its log-deviation from the respective steady state value.

(18.) If [sigma] = 1, utility takes the following form: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] As the last term depends only on [X.sup.i.sub.t] (which is taken as given by the individual household), the first order and equilibrium conditions can be obtained by simply setting [sigma] = 1 in all conditions derived under the more general setup with [sigma] > 0. This assumption is made for analytical tractability, but our results are robust to any realistic calibration of [sigma].

(19.) Alternatively, one could construct a sunspot equilibrium building on an initial non-fundamental increase in aggregate activity, similar to that done in Gali et al. (2004) and Colciago (2011). Consider a sunspot shock leading to an increase in aggregate activity [[??].sub.t]. This generates two effects. First, from equation (26) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], non-Ricardian consumption?" [[??].sup.rt.sub.t] increases. This is made possible through a higher real wage per hour worked (the nominal wage increases as firms hire more workers to face a larger demand, while, due to the assumed nominal rigidities, prices do not fully adjust), and an increase in labor supply. The latter is driven by the term [theta]/1 + [chi] [[??].sub.t]: non-Ricardian households work harder to keep up with the Joneses. Second, by the Phillips curve (32), the aggregate boom raises current inflation, which, under an active policy rule, induces a higher real interest rate [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The latter lowers the marginal utility of current consumption for the Ricardian agents, or, equivalently, through the Ricardian Euler equation (28) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it implies a drop in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Absent the externality, this would imply an equivalent drop in Ricardian consumption [[??].sup.o.sub.t]. Here instead, because of a positive [theta] and the initial sunspot-driven increase in c., Ricardian consumption [[??].sup.o.sub.t] may drop less (or even increase). As under KUJ preferences non-Ricardian consumption increases by more while Ricardian consumption decreases by less, the minimum share of non-Ricardian households for which aggregate consumption [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] increases is lower with respect to the no-extemality case--making the initial sunspot-driven boom self-fulfilling.

(20.) Available estimates for e vary from slightly above 2 (from the micro-estimates of the industrial organization and international trade literature) to about 10 (from the macroeconomic literature).

(21.) Empirical estimates of the Frisch elasticity of labor supply vary considerably in the literature, depending on 1) whether they are based on micro or macro data, and 2) whether they refer to the intensive or the extensive margin. Chetty et al. (2011) and Keane and Rogerson (2011) try to reconcile the existing estimates from the micro and the macro literature.

(22.) These results are in line with that found by Gali et al. (2004) and Bilbiie (2008). The LAMP studied by Gali et al. (2004) includes physical capital. Our framework instead assumes that labor is the only input to production, similar to Bilbiie (2008). However, the latter features a fixed cost to production, whose ratio to total output is assumed to be equal to the steady state net mark-up in order to equalize consumption across all agents in the economy at the steady state. This makes the threshold for the share of non-Ricardian households marking the switch from a SADL to a IADL in Bilbiie's externality-free paper equal to 1/1 + [chi]/(1 + [mu]). compared to 1/1 + [mu] in our framework.

(23.) The introduction of an ACE appears to have a minimal or no impact at all on the determinacy area corresponding to a passive interest rate rule (lower-right corner of Figure 1).

(24.) See Primiceri (2006) for a summary of existing views on this matter, and a novel explanation.

(25.) Orphanides (2004) argues that the FED's estimates of potential output (and therefore the output gap) were seriously biased during the 1970s, leading to excessively activist interest rate policies. Coibion and Gorodnichenko (2011) evaluate the likelihood of indeterminacy due to misperception of the true level of trend inflation.

(26.) Campbell and Mankiw (1989) estimate that, during the period 1954-1986, for about 50% of the U.S. households consumption perfectly tracked current (not permanent) disposable income. Poterba and Samwick (1995) report that, before 1983, only 20% of the U.S. households had some form of direct participation in the stock market.

(27.) As Coibion and Gorodnichenko (2011) point out. regardless of whether they are larger or smaller than one, empirical estimates of the FED's response coefficient to inflation often come with large standard errors. This makes the issues of whether the Taylor principle was violated or not still unsettled. Sims and Zha (2006) together with several other papers cited therein do not find strong support against stability of policy coefficients across 1979.

(28.) These estimates are similar to those used by Collard and Dellas (2007) to explain the great inflation in the 1970s.

(29.) For any realistic parametrization of [[rho].sub.z]. we have not found any significant role played by the degree of LAMP.

(30.) Non-Ricardian hours are constant.

(31.) This is a direct consequence of the properties of 5 stated in Lemma 1.

(32.) For [theta] [right arrow] 1. the determinacy area A4 is empty.

(33.) The same result would hold in the LAMP model without physical capital studied by Bilbiie (2008).

(34.) Colciago (2011) models the nominal wage rigidity through a Calvo price setting mechanism, which gives rise to a wage Phillips curve. Although the same approach could be adopted in our framework, this would make the attainment of analytical results rather cumbersome without adding any additional insight.

(35.) Other formulations of real wage rigidities assume the wage norm to be equal to the past wage [w.sup.rt.sub.t-1], such that [w.sup.rt.sub.t] corresponds to an exponentially-decaying weighted average of the infinite stream of past flexible real wages: that is, [w.sup.rt.sub.t] = [([w.sup.f.sub.t]).sup.1-[xi]] [([w.sup.rt.sub.t- 1]).sup.[xi]] (*). Uhlig (2007), Krause and Lubik (2007) and Christoffel and Linzert (2010), among others, evaluate the quantitative performance of such specification along different dimensions. Our results do not hinge on the assumption of a constant wage norm, but would also obtain under the specification (*). TABLE 1 Benchmark Parameterization [beta] [sigma] [epsilon] [chi] [??] 0.99 1 6 1 59 TABLE 2 Observed Volatilities: U.S. Data 1960-1979 (Quarterly) [[sigma].sub.[pi]] = [[sigma].sub.y] = [[sigma].sub.[pi]]/ 1.48 1.83 [[sigma].sub.v] = 0.81 Source: Clarida et al. (2000). TABLE 3 Model-Predicted Volatilities under Indeterminacy [[phi].sub.[pi]] [gamma] [theta] = 0 [[sigma].sub.[pi]] [[sigma].sub.y] 1.05 0.75 1.19 1.49 0.66 1.25 1.28 1.25 0.75 0.77 1.35 0.66 0.62 1.36 1.5 0.75 0.6 1.38 0.66 0.44 1.46 [[phi].sub.[pi]] [gamma] [theta] = 0 [theta] = 0.25 [[sigma].sub.[pi]]/ [[sigma].sub.[pi]] [[sigma].sub.y] 1.05 0.75 0.79 1.24 0.66 0.99 1.24 1.25 0.75 0.61 0.83 0.66 0.48 0.65 1.5 0.75 0.46 0.65 0.66 0.32 0.47 [[phi].sub.[pi]] [gamma] [theta] = 0.25 [[sigma].sub.y] [[sigma].sub.[pi]]/ [[sigma].sub.y] 1.05 0.75 1.59 0.78 0.66 1.4 0.9 1.25 0.75 1.47 0.59 0.66 1.5 0.45 1.5 0.75 1.54 0.45 0.66 1.62 0.31 [[phi].sub.[pi]] [gamma] [theta] = 0.5 [[sigma].sub.[pi]] [[sigma].sub.y] 1.05 0.75 1.32 1.71 0.66 1.24 1.56 1.25 0.75 0.9 1.64 0.66 0.7 1.7 1.5 0.75 0.71 1.74 0.66 0.51 1.85 [[phi].sub.[pi]] [gamma] [theta] = 0.5 [[sigma].sub.[pi]]/ [[sigma].sub.y] 1.05 0.75 0.78 0.66 0.82 1.25 0.75 0.59 0.66 0.43 1.5 0.75 0.43 0.66 0.29