Learning About Inflation Measures for Interest Rate Rules
  • 1 https://isni.org/isni/0000000404811396, International Monetary Fund

Contributor Notes

Author’s E-Mail Address: ma639@drexel.edu, fzanna@imf.org

Empirical evidence suggests that goods are highly heterogeneous with respect to the degree of price rigidity. We develop a DSGE model featuring heterogeneous nominal rigidities across two sectors to study the equilibrium determinacy and stability under adaptive learning for interest rate rules that respond to inflation measures differing in their degree of price stickiness. We find that rules responding to headline inflation measures that assign a positive weight to the inflation of the sector with low price stickiness are more prone to generate macroeconomic instability than rules that respond exclusively to the inflation of the sector with high price stickiness. By this we mean that they are more prone to induce non-learnable fundamental-driven equilibria, learnable self-fulfilling expectations equilibria, and equilibria where fluctuations are unbounded. We discuss how our results depend on the elasticity of substitution across goods, the degree of heterogeneity in price rigidity, as well as on the timing of the rule.

Abstract

Empirical evidence suggests that goods are highly heterogeneous with respect to the degree of price rigidity. We develop a DSGE model featuring heterogeneous nominal rigidities across two sectors to study the equilibrium determinacy and stability under adaptive learning for interest rate rules that respond to inflation measures differing in their degree of price stickiness. We find that rules responding to headline inflation measures that assign a positive weight to the inflation of the sector with low price stickiness are more prone to generate macroeconomic instability than rules that respond exclusively to the inflation of the sector with high price stickiness. By this we mean that they are more prone to induce non-learnable fundamental-driven equilibria, learnable self-fulfilling expectations equilibria, and equilibria where fluctuations are unbounded. We discuss how our results depend on the elasticity of substitution across goods, the degree of heterogeneity in price rigidity, as well as on the timing of the rule.

I. Introduction

The review by Klenow and Malin (2010) of the microeconomic literature of price setting concludes that goods are highly heterogeneous with respect to their degree of price stickiness. At one extreme are goods that change prices every quarter or even once a month, including food, energy, and airfares; and at the other extreme are services that change prices much less often than once a year.1 This microeconomic evidence seems to be confirmed by macroeconomic estimates. Bouakez et al. (2009), for example, estimate a multi-sector New-Keynesian model for the U.S. and cannot reject the hypothesis that some sectors—typically non-services—display flexible prices. In contrast, they also find that services is by far the most price-rigid sector in the U.S. economy.

This evidence may raise questions regarding the inflation measure that should be the target for monetary policy, including interest rate rules. In particular, central bankers may wonder whether monetary policy should respond to an inflation measure that puts some weight on inflation of flexible-price goods or, instead, respond to a measure mainly based on the inflation of sticky-price goods. In fact, monetary policy discussions on the recent oil and food price shocks can be cast in these terms: should monetary policy respond to a headline inflation that includes the inflation of energy and food (flexible-price goods)? or should it respond to a core inflation measure that excludes the inflation of these goods?2

In this paper, we provide answers to these questions by selecting the inflation measure of an interest rate rule that is more prone to avert macroeconomic instability. By this we mean that the appropriate inflation measure of the rule should (i) shield the economy against fluctuations driven by self-fulfilling expectations3—Keynes’ animal spirits—and (ii) ensure that even if agents are boundedly rational—in the sense of adaptive learning as proposed by Evans and Honkapohja (2001)—they can still learn the monetary policy targeted equilibrium so the economy will converge to it.4

Specifically, we develop a two-sector New-Keynesian DSGE model to study the determinacy and learning stability (E-stability) properties of Rational Expectations Equilibria (REE) for interest rate rules responding to inflation measures that differ in their degree of price stickiness.5 In the model, both sectors feature nominal price rigidities but one of them—which we call the flexible-price sector—faces low rigidities in comparison with the other sector—which we refer to as the sticky-price sector. If the aggregate inflation of the rule is expressed as a weighted average of the inflations of these two sectors, then a myriad of inflation measures can be considered by varying, for instance, the weight on the flexible-price inflation. By allowing the central bank pick this weight, we see the inflation measure as a policy variable, while focusing on rules that satisfy the Taylor principle; that is, active rules whereby the response coefficient to inflation is larger than one.

We find that active rules responding to an inflation measure which puts some weight on the flexible-price inflation (a headline measure) are more prone to generate macroeconomic instability than rules that exclusively respond to sticky-price inflation (a core measure). As a result, policy makers should target the inflation measure of the truly sticky-price sectors.

Our conclusions follow from the analysis of several policy rule timings. We find that forward-looking rules that assign some positive weight to the expected flexible-price inflation are more prone to induce fundamental-driven equilibria that are not learnable and self-fulfilling expectations equilibria of the “sunspot” type that are learnable. Backward-looking rules that put some weight on the past flexible-price inflation, on the other hand, can be also destabilizing, as they are more susceptible to generate explosive equilibria for which economic fluctuations are potentially unbounded. Only contemporaneous rules appear to guarantee a unique and learnable equilibrium, for any inflation measure; that is, even if the rule puts some weight on the current flexible-price inflation. But across these different timings, the safe choice to rule out potential sources of instability is to respond exclusively to the sticky-price inflation.

The intratemporal elasticity of substitution of goods in agents’ preferences—in particular whether the goods are Edgeworth substitutes, neutral or complements—plays a key role in our results. Such elasticity determines the degree of comovement of the two sectoral inflation rates and, hence, affects the equilibrium response of the real interest rate to an adverse inflationary shock under different inflation measures. For instance, we show that equilibrium determinacy and learnability conditions are generally more stringent under the assumption of complementarity; and we identify a threshold for the weight on expected flexible-price inflation beyond which any active rule responding to the headline inflation always induces the non-learnability of the fundamental REE.

Our results hold under two different cases regarding the heterogeneity of price setting. First, we consider a strongly dichotomous economy with a fully flexible-price sector—which faces no price rigidities—and a sticky-price sector. This case allows us to derive analytical results and a straightforward intuition. In addition, this economy is analogous, on one hand, to the framework of Aoki (2001) and subsequent work in closed economies on optimal policy;6 while, on the other hand, it resembles the models in the extensive small open economy literature, such as Galí and Monacelli (2005), which typically feature a sharp distinction between a fully flexible-price traded good (or foreign good) and a sticky-price non-traded good (or domestic good).7 The second case is a less heterogeneous economy with two sticky-price sectors. We calibrate this economy to match roughly the micro evidence by Bils and Klenow (2004) and assess the quantitative relevance of our theoretical results.

Of particular interest is the result that by having some weight on the expected flexible-price inflation, a rule may lead to a non-learnable fundamental-driven equilibrium. The intuition is straightforward in the dichotomous economy, where the expected flexible-price inflation is negatively correlated to current sticky-price output. As a result, a greater weight on flexible-price inflation makes the rule respond more negatively to current sticky-price output. This response, however, goes against the policy prescriptions of “leaning against the wind” by Taylor (1999a)—nominal interest rates should be adjusted positively, and more than one-for-one, in response to inflation, and positively to current output. In the two-sector model, expected deviations of the sticky-price inflation from the REE may be reinforced by a rule that responds positively to expected flexible-price inflation, and therefore negatively to current sticky-price output. This is because this rule may bring about a decrease in the real interest rate which adds to sticky-price inflationary pressures, pushing the economy away from a targeted equilibrium. So the equilibrium may become non-learnable.

Our work distinguishes itself from the New-Keynesian literature on interest rate rules and macroeconomic instability by focusing on the inflation measure as the potential source of instability. Benhabib et al. (2001), Carlstrom and Fuerst (2001), Clarida et al. (2000), and Woodford (2003), among others, discuss extensively how the rule’s degree of aggressiveness towards inflation (i.e., active versus passive rules) and the timing of inflation (i.e., expected, contemporaneous or past inflation) may affect the extent to which rules induce endogenous fluctuations driven by self-fulfilling expectations. Bullard and Mitra (2002), Eusepi (2007), Evans and Honkapohja (2003), Evans and McGough (2005), Llosa and Tuesta (2008), and Preston (2006), among others, investigate the importance of similar features in the context of learning. But despite their seminal contributions, all these works do not consider the measure of inflation as a central bank policy choice that may affect both the learning and determinacy of equilibrium.

Our work also distinguishes from Carlstrom et al. (2006a), which is an important exception in the literature. By studying rules that respond to contemporaneous inflation, Carlstrom et al. (2006a) claim that the measure of inflation is irrelevant to ensure equilibrium determinacy, as long as the Taylor principle holds. Our work goes beyond theirs in two aspects. First, it tests the robustness of their claim by analyzing forward-looking and backward-looking rules. Second, it adds learnability as an extra criterion to select the inflation measure. Both aspects are crucial for revealing that, for a degree of heterogeneous price stickiness consistent with micro data, the inflation measure seems to be relevant for equilibrium determinacy and learnability. In this regard, our results suggest that the Taylor principle does not necessarily apply either at the sectorial level or at the aggregate level.

Finally by using different selection criteria, our work differs from the optimal monetary policy literature that selects the measure of inflation based on welfare analysis. Aoki (2001), Benigno (2004) and Mankiw and Reis (2003) recommend to target the inflation with the higher degree of price stickiness from an optimal targeting rule perspective. Our work supports a similar conclusion from an instrumental rule perspective (Taylor rules) and under the different criteria of equilibrium determinacy and learnability. The recent works by Anand and Prasad (2010), Bodenstein et al. (2008), and Catao and Chang (2010) study the implications of large swings in world food and oil price swings for the inflation measure of instrumental rules. Here again the selection criteria is welfare-based, abstracting from the fact that the inflation measure might exacerbate macroeconomic volatility by inducing non-fundamental-driven equilibria and unstable learning dynamics.

The remainder of this paper is organized as follows. Section II describes the general two-sector New-Keynesian economy. Section III presents the main results by pursuing the equilibrium determinacy and learnability analyses for forward-looking, contemporaneous, and backward-looking rules that respond to different measures of inflation. Section IV tests and discusses the robustness of our results under some extensions of our framework, including an explicit response to output in the rule and a different timing of real money balances in utility. Finally, Section V concludes.

II. The Model

In this section we develop a New-Keynesian dynamic stochastic general equilibrium model of a closed economy with two goods.8 The economy is populated by a continuum of infinitively-lived household-firm units, each of whom derives utility from consuming, not working, and the liquidity services of money.9 The unit consumes two types of composite goods—which we call F and S and are made of a continuum of intermediate differentiated products, indexed by j ∈ [0, 1]. The unit also produces the j-th variety of the composite good F (from now on, good j of type F)and the j-th variety of the composite good S (from now on, good j of type S). That is, each production unit is split in two sub-units, specializing in a type-F and type-S good, respectively. We will often refer to these two sub-units as sector F and sector S.

A. The Household-Firm Unit

Each household’s intertemporal preferences are described by:

𝔼0t=0βt {ct1σ1σA [(htsF)1+φF1+φF+(htsS)1+φS1+φS]+(mt)1+φm1+φm},(1)

where 𝔼0 denotes the (potentially) subjective expectations operator conditional on information available at time 0, ct is aggregate consumption, htsF (htsS) denotes the labor effort supplied to produce good F (S), and mt=MtPt are the real money balances in terms of the Consumer Price Index (CPI) Pt to be defined below. In addition, β ∈ (0, 1) is the subjective discount factor, σ > 0 is the constant relative risk aversion coefficient, and φi corresponds to the inverse of the Frisch elasticity of the labor supplied to sector i, for i = F, S.10

Aggregate consumption is defined by the CES aggregator of the two consumption goods ctF and ctS,

ct=[η1ξ(ctF)ξ1ξ+(1η)1ξ(ctS)ξ1ξ]ξξ1,(2)

where ξ is the intratemporal elasticity of substitution between goods, while η is a parameter associated with the share of good F in aggregate consumption.11

In this set-up, good F and good S can be Edgeworth complements, neutral, or substitutes, depending on the intertemporal and intratemporal elasticities of substitution 1σ and ξ. To see this, define the utility of aggregate consumption as U=ct1σ1σ. Then by (2) the sign of 2UcFcS is determined by the sign of 1σξ. Therefore, the goods are complements when ξ<1σ, neutral if ξ=1σ, and substitutes when ξ>1σ. Furthermore, under the specification in (2), the CPI index corresponds to

Pt=[η(PtF)1ξ+(1η) (PtS)1ξ]11ξ,(3)

and the relative demands for both goods are given by

ctF=η (PtFPt)ξctandctS=(1η) (PtSPt)ξct,(4)

satisfying Ptct=PtFctF+PtSctS.

The relative price qt is defined as PtF/PtS and satisfies the law of motion given by:

qt=qt1(πtFπtS) ,(5)

where πti=Pti/Pt1i denotes the “gross” inflation for good i = F, S.

The two goods are made of a continuum of differentiated intermediate products and defined as

ctF=[01(ctFj)μF1μFdj]μFμF1andctS=[01(ctSj)μS1μSdj]μSμS1,

with μF > 1 (respectively μS > 1) denoting the intratemporal elasticity of substitution between any pair of goods of ctF (respectively ctS). The relative demands of the good j of type F and the good j of type S are then given by

ctFj=(PtFjPtF)μF  ctFandctSj=(PtSjPtS)μSctS,

for any j ∈ [0, 1].

The representative unit produces the intermediate goods j of type F and type S using simple linear production technologies:

ytFj=zthtFjandytSj=zthtSj,

with htFj and htSj denoting the labor hired. Both productions occur in a regime of monopolistic competition. We assume labor to be perfectly mobile within each sector, implying that the nominal wages are equal within sectors, i.e., WtFj=WtF and WtSj=WtS for any j ∈ [0, 1]. However, labor is completely immobile across sectors and, therefore, WtF and WtS can differ in equilibrium.12 The economy-wide aggregate technology shock zt is the only source of fundamental intrinsic uncertainty and behaves as:13

z^t=ψz^t1+vt,(6)

where z^t=ln(zt), ψ ∈ (0, 1), and vt ~ Normal(0, ϱ2).

The monopolist unit sets the prices PtFj and PtSj of the variety j of both intermediate goods it supplies, taking as given their respective aggregate demands (PtFjPtF)μFCtF and (PtSjPtS)μSCtS, where CtF and CtS denote the economy-wide demand levels for the composite goods F and S. Hence, output is demand-determined in both sectors:

zthtFjCtF(PtFjPtF)μF and zthtSjCtS(PtSjPtS)μSfor any j[0,1].(7)

We introduce nominal price rigidities as in Rotemberg (1982). When setting the price of a good j of type i, the unit is subject to a quadratic resource cost given by γi2(PtijPt1ijπ¯i)2PtiCti for i = F, S, with the adjustment cost coefficient γi measuring the degree of nominal rigidity in sector i: the higher γi the larger the degree of price stickiness in that sector. The difference between γF and γS will capture the different degree of price stickiness across sectors. Without loss of generality, we assume that γF < γS, i.e., in sector F prices are always relatively more flexible, but not necessarily completely flexible, than in sector S But note that our set-up embeds the case of fully flexible prices and perfect competition in sector F, by setting γF = 0 and μF → +∞.

Besides money Mt, the household-firm unit has access to a government bond Bt, which pays a nominal interest rate Rt − 1. The unit also pays taxes Ttg and receives labor subsidies from the government at the rates υF and υS.14 The budget constraint can then be written as follows:

Mt+Bt+Ptct+PtTtgRt1Bt1+Mt1+(1+υF)WtFhtsF+(1+υS)WtShtsS(8)+[PtFjzthtFjWtFhtFjγF2(PtFjPt1Fjπ¯F)2PtFCtF]+[PtSjzthtSjWtShtSjγS2(PtSjPt1Sjπ¯S)2PtSCtS],

where the right-hand side represents the sources of the unit’s wealth, including interest payments on government bonds, money holdings, labor wage income, and net profits from the production of both goods. This wealth is used to finance new money holdings, purchase government bonds, and pay for total consumption and taxes, as described by the left-hand side of the constraint.

The representative household-firm unit chooses the set of stochastic processes {ct,ctF,ctS,htsF,htsS,htFj,htSj,PtFj,PtSj,Mt,Bt}t=0 in order to maximize (1) subject to (7), (8) and a standard Non-Ponzi game condition on total assets, given initial holdings (B−1, M−1), and the set of stochastic processes {Rt,Pt,PtF,PtS,WtF,WtS,Ttg,CtF,CtS,zt}t=0. The relevant first order conditions are:

ctσ=βRt𝔼t(ct+1σπt+1),(9)
A(htsi)φictσ=(1+υi)WtiPt,fori=F,S,(10)

and

ctσCti{(PtijPti)μi[μi(1MCtiPtij)1]+γi(PtijPt1ijπ¯i)PtiPt1ij}(11)=β𝔼t [ct+1σπt+1Pt+1iCt+1iγi(Pt+1ijPtijπ¯i)Pt+1ij(Ptij)2],fori=F,S,

where πt+1Pt+1/Pt is the gross CPI inflation and MCtiWtizt is the nominal marginal cost of producing good j of type i.

The interpretation of these equations is straightforward. Condition (9) is the standard Euler equation for consumption ct. Equation (10) makes the marginal rate of substitution between labor allocated to the production of the type i good and consumption equal to the respective real subsidized wage. Equation (11) is the optimal price setting condition for a unit producing good j of type i. It simply states that, in every period, the unit sets a new price Ptij taking into account its impact on current profits, as reflected by the terms on the left-hand side of (11), and the impact on future profits, as captured by the term on the right-hand side of the same equation.

While the first order conditions (9)-(11) together with the appropriate transversality condition are necessary and sufficient if 𝔼t is assumed to be the expectations operator conditional on all available information (hence, rational), the same might not apply for more general settings. In this paper, learning takes the form of the “Euler Equation” approach by Evans and Honkapohja (2001), which remains the most common in the adaptive learning literature related to our work. More specifically, it is assumed that economic agents are boundedly-rational and choose based on the first order conditions (9)-(11), which describe the relevant marginal benefits and marginal costs associated with each economic decision.

We focus on a symmetric equilibrium in which the monopolistic units belonging to the same sector pick the same price. That is, PtFj=PtF and PtSj=PtS for any j ∈ [0, 1]. Given that they face the same wage rates WtS and WtF, and the same technology, they also demand the same amounts of labor, htFj=htF and htSj=htS for any j ∈ [0, 1]. Combining the first order conditions of this maximization problem with the clearing conditions in the labor markets (htsF=htF and htsS=htS) and in the goods markets (ctS=CtS=zthtSγS2(πtSπ¯S)2CtS and ctF=CtF=zthtFγF2(πtFπ¯F)2CtF), we obtain the New-Keynesian Phillips curves (NKPC):

γi(πtiπ¯i)πti=γiβ𝔼t[Ht,t+1i(πt+1iπ¯i)πt+1i]+μi(mctiμi1μi),fori=F,S,(12)

where πtiPtiPt1i, Ht,t+1i(ct+1ct)σ(Ct+1iCti)(πt+1iπ) and mctiWtiztPti.

B. The Government

The government issues two nominal liabilities (money and a domestic bond), provides labor income subsidsies, pays interest on its debt and receives revenues from seigniorage and lump-sum taxes. It follows a generic Ricardian fiscal policy, by setting taxes in order to satisfy its intertemporal budget constraint and a Non-Ponzi game condition.

Following Taylor (1993), we define monetary policy as an interest rate feedback rule. The government sets the gross nominal interest rate Rt as an increasing and continuous function of the deviation of a chosen measure of inflation πto, with respect to a target π¯o. Formally,

Rt=R¯[𝔼t(πt+koπ¯o)]ϕπwherek=1,0,1,ϕπ>1,(13)

R¯ is the interest rate target and 𝔼 is the expectation operator. For simplicity, we also assume that the targets π¯o and R¯ correspond to the steady-state levels of πto and Rt.

Some remarks are in order regarding the specification in (13). First, we focus on rules that satisfy the Taylor principle, also known as active rules. These are rules that respond to increases in inflation with a more than one-for-one increase in the interest rate, i.e., ϕπ > 1. And second, we allow for different policy rule timings, including a forward-looking rule that responds to the deviation of the expected future inflation from the target, 𝔼t(πt+1oπ¯o); a contemporaneous rule that reacts to the current inflation deviation (πtoπ¯o); or a backward-looking rule that responds to the past inflation deviation (πt1oπ¯o).

To underscore the role of the inflation measure of the rule as a policy variable, we define the inflation measure in the rule (13) as:

πto=ωπtF+(1ω)πtS,(14)

where in principle the government can freely pick the weight ω. This set-up is general enough to nest the cases of a rule responding to the most flexible-price inflation only (ω = 1), the stickiest price (core) inflation (ω = 0), or any arbitrary measure of headline inflation (ω ∈ (0, 1)), including the CPI inflation case of ω = η.

III. Equilibrium Determinacy and Stability under Learning

Given ϕπ > 1, the objective of our analysis is to assess under what conditions on the policy parameter ω, the economy displays a determinate (unique) and expectationally stable (E-stable) Rational Expectations Equilibrium (REE). Both equilibrium determinacy and E-stability are particularly desirable from the point of view of the policymaker. Under determinacy, the equilibrium dynamics are solely driven by fundamentals, ruling out the effects of extrinsic uncertainty such as noise, market sentiment and all other factors often referred to as “sunspots.” Under E-stability, even if boundedly rational, economic agents will eventually learn the fundamental REE by recursive least squares regressions on past data, and the economy will converge to the targeted equilibrium.

To derive our main results we consider two different price-setting cases embedded in our previously discussed set-up. First, we study an economy with the strong dichotomy of a fully flexible-price sector and a sticky-price sector. This case allows us to derive analytical results and, as mentioned in the Introduction, reflects the popular two-sector price setting assumption in both the optimal monetary policy literature in closed economies and the small open economy literature. Furthermore, we believe this case captures the essential message of the micro and macro evidence, namely: goods differ greatly in how frequently their prices change with some of them changing prices every quarter or more often. From a modeling perspective, this implies a non-zero measure of fully flexible-price sectors, especially if the model is calibrated to match quarterly data. The second case attempts to take the micro evidence more seriously and therefore calibrates the main set-up, with price-stickiness in both sectors, to the evidence provided by Bils and Klenow (2004). This helps us show that our main analytical results hold (numerically) for a general two-sector economy with a lower, but still realistic, degree of heterogeneous price stickiness across sectors.

A. The Methodology

We now provide a brief description of our methodology for the equilibrium determinacy and learning analyses. We derive, in particular, the E-stability conditions that a representation of an equilibrium needs to satisfy in order to be learnable.

The characterization of a Rational Expectation Equilibrium (REE) follows Blanchard and Kahn (1980). For each of the aforementioned rules we log-linearize and write the model as

Etx^t+1=Jx^t+Λz^t,(15)

where E is now the rational expectations operator, x^t is a s × 1 vector of endogenous variables, z^t is the technology shock, and J and Λ are conformable matrices. The characterization depends on the number of roots of J that lie outside the unit circle: if this number is equal to the number of non-predetermined variables, there is a unique equilibrium (determinacy); when this number is less than the number of non-predetermined variables, then there are multiple equilibria (indeterminacy); and if this number of roots is greater than the number of non-predetermined variables, then almost surely there exists no non-explosive solution to the model.

The learning analysis draws on Evans and Honkapohja (2001). Agents no longer are endowed with rational expectations but, instead, have adaptive rules to form expectations using recursive least squares updating. We focus, in particular, on E-stability as a learning criterion: a representation of an equilibrium is learnable if it is E-stable. The reason is that under E-stability, recursive least-squares learning is in general locally convergent to the REE. This is shown analytically by Marcet and Sargent (1989) and Evans and Honkapohja (2001) for models with a unique equilibrium and, through simulations, by Evans and McGough (2005) for a New-Keynesian model that displays “sunspot” equilibria.

We proceed to derive the conditions under which a REE is E-stable. To do so, we reduce the model associated with each of the interest rate rules to:

x^t=ϒ+Ω𝔼tx^t+1+Γx^t1+Πz^tandz^t=ψz^t1+vt,(16)

where x^t is a s × 1 vector of endogenous variables, z^t is the exogenous fundamental shock. ψ ∈ (0, 1) and vt ~ Normal (0, ϱ2). In addition, ϒ, Ω, Γ, and Π are conformable matrices of constants, while 𝔼t denotes in general (non-rational) expectations.

Consider first the E-stability conditions for a fundamental-driven REE. In this case we focus on the learnability of McCallum’s (1983) Minimal State Variable (MSV) solution. Therefore we assume agents follow a Perceived Law of Motion (PLM) x^t=𝒜+x^t1+𝒩z^t, where 𝒜, ℱ, and 𝒩 are conformable matrices. Iterating forward the PLM and using it to eliminate all the forecasts in the model, we obtain the implied Actual Law of Motion (ALM) x^t=𝕭[ϒ+Ω𝒜+Γx^t1+(Ω𝒩ψ+Π)z^t] with 𝕭 = (I−Ωℱ)−1. The PLM and ALM, in turn, define the T-mapping T (𝒜, ℱ, 𝒩) = (𝒜A, ℱA, 𝒩A), where 𝒜A = 𝕭 (ϒ+Ω𝒜), ℱA = 𝕭Γ, and 𝒩A = 𝕭(Ω𝒩ψ + Π).15 The fixed points of this mapping are the REE. Following Evans and Honkapohja (2001), we say that the MSV representation of a REE is E-stable if all the eigenvalues of the matrices of the derivatives

DT𝒜=(IΩ)1Ω,  DT=[(IΩ)1Γ][(1Ω)1Ω], and DT𝒩=ψ[(1Ω)1Ω],(17)

evaluated at the REE fixed point, have real parts less than one. On the contrary, the MSV solution is not E-stable if any of the eigenvalues has a real part larger than one.

To derive the E-stability conditions for stationary “sunspot” equilibria, we concentrate on Common Factor (CF) representations, as in Evans and McGough (2005). When x^t is a vector of non-predetermined variables, the PLM for the CF representation non-corresponds to x^t=𝒜+𝒩z^t+𝒢εt, where 𝒜, 𝒩, and 𝒢 are conformable vectors. εt, in turn, is a “sunspot” following εt = υεt–1 + ϖt, where |υ| < 1 and ϖt is an arbitrary martingale difference sequence. Then T-mapping associated with this PLM corresponds to T(𝒜,𝒩,𝒢) = (𝒜A,𝒩A,𝒢A), where 𝒜A = (ϒ + Ω𝒜), 𝒩A = (Ω𝒩ψ + Π), and 𝒢A = Ω𝒢υ. Using this mapping, the E-stability conditions can be stated as having all the eigenvalues of the matrices

DT𝒜=Ω,DT𝒩=ψΩ,andDT𝒢=υΩ,(18)

with real parts less than one.

B. The Strong Dichotomy: A Flexible-Price and Sticky-Price Economy

This section provides the results of the case in which sector F features perfect competition (μF → +∞) and fully flexible prices (γF = 0), while sector S keeps the standard New-Keynesian characteristics.16 For reasons of space, we present analytical results only for forward-looking rules. That is, for k =1 in equation (13). Our interest for rules responding to expectations is motivated by the empirical estimates in Clarida et al. (2000), and more recently in Cogley and Sargent (2005) and Boivin (2006).17

Since γF = 0 and μF → +∞, then equation (12) for i = F reduces to mctF=1. The rest of the equations of our set-up still hold. Without loss of generality, we focus on the case where the steady-state relative price q¯ is equal to one, and hence where π¯F=π¯S=π¯.18 We then proceed by log-linearizing the equations of the model around the steady state of the economy. Log-linearized variables will be denoted with a hat. For instance, log-linearized sticky-price consumption will be denoted as c^tS, where c^tS=log (ctSc¯S) and c¯S corresponds to its steady-state level.

Under a forward-looking rule, log-linearizing (13) and (14) yields:

r^t=ϕπ𝔼tπ^t+1o(19)

with

π^t+1o=ωπ^t+1F+(1ω)π^t+1S.(20)

On the demand side, by combining the (log-linearized) expressions for (5), the CES specification in (2), the CPI (3), equation (10), mctF=1, the definition of the relative price qt, and the market clearing condition ctF=zthtF, we obtain the following relationship between the flexible-price and sticky-price consumptions:

c^tF=ςc^tS+ϰ1z^t,(21)

where ς(1η)(1ξσ)(φF+ση)ξ+1η and ϰ1(1+φF)ξ(φF+ση)ξ+1η. Note that sign (ς) = sign (1 − ξσ). Hence, along any equilibrium path, the two consumptions are positively (negatively) related when the goods are Edgeworth complements (substitutes) i.e. ξ<1σ(ξ>1σ). When the goods are neutral (ξ=1σ, hence ς = 0), flexible-price consumption depends only on the aggregate technology shock.

Combining the log-linearized versions of the demand functions in (4) with (5) and (21), we find that:

c^tSc^t1S=1θ(π^tFπ^tS)+ϰ1θξ(z^tz^t1),(22)

where θσ+φF(φF+ση)ξ+1η>0. This equation captures, in part, the expenditure switching or substitution effect that will be crucial to understand our results: for a given sticky-price inflation, the growth rate of sticky-price consumption is positively related to flexible-price inflation.

The demand side characterization is completed by log-linearizing the Euler equation (9) to obtain:

c^t=𝔼tc^t+11σ(r^t𝔼tπ^t+1),(23)

with the (log) CPI inflation given by

π^t+1=ηπ^t+1F+(1η)π^t+1S.(24)

Using this and the log-linearized versions of (2) and (3), with equations (21) and (22) allows us to derive a log-linearized Euler equation for sticky-price consumption:

c^tS𝔼tc^t+1S=α (r^t𝔼tπ^t+1S)ϰ2z^t,(25)

with α ≡ [σ (ης + 1 – η) + ηθ]–1 > 0 and ϰ2(1ψ)ϰ1η(ασξ1ξ).

The supply side, on the other hand, is described by the log-linearized version of the NKPC (12):

π^tS=β𝔼tπ^t+1S+κmc^tS,(26)

with κ(μS1)γS(π¯S)2>0, which relates the current inflation of the sticky-price good π^tS to the inflation expectations 𝔼tπ^t+1S and the marginal cost of producing this good mc^tS. Furthermore, by combining the log-linearized versions of (2), (3), (10) for i = S, and ctS=zthtSγS2(πtSπ¯)2ctS with equations (5) and (21), we can express the real marginal costs in terms of log-linearized sticky-price consumption:

mc^tS=ϑc^tS+𝔥z^t,(27)

where ϑφS+1α>0 and (1+φS)ϰ1η (1σξξ). Replacing this into (26) yields:

π^tS=β𝔼tπ^t+1S+δc^tS+sz^t,(28)

where δκϑ > 0 and s ≡ κℌ < 0, which clearly shows that, in the model, an excess of demand for the sticky-price good is translated into sticky-price inflationary pressures.

Based on this log-linearized set of equations, we can provide the following definition of a fundamental-driven equilibrium in this economy.

Given the stochastic process {z^t}t=0, an equilibrium is a set of stochastic processes {r^t,π^to,π^tF,π^tS,c^tS}t=0 satisfying a) the rule (19); b) the definition of full inflation (20); c) equation (22); d) the Euler equation for sticky-price consumption (25); and e) the New-Keynesian Phillips curve (28).19

Some Analytical Results. Proposition 1 provides the results for the equilibrium determinacy analysis. It states that conditions under which an active forward-looking rule delivers a locally unique REE crucially depend on the weight of the flexible-price inflation ω in the inflation measure π^to, and the types of goods considered (complements, neutral, or substitutes). It reveals that across these types, as the weight on the flexible-price inflation increases, the rule is more prone to induce multiple equilibria, even if the Taylor Principle holds (i.e., ϕπ > 1).

Let ϕ1(ω)δα+2(1+β)δα+2(1+β)τω>0 and ω*=1τ, where τ ≡ αθ. Consider the forward-looking interest rate in (19)-(20) with ϕπ > 1, and assume that agents have Rational Expectations, i.e., E = E.

a) For any ω ∈ (0, 1) when ξ1σ (substitutes or neutral), or any ω ∈ (0, ω*) when ξ<1σ (complements), the following hold: 1) there exists a unique local REE if 1 < ϕπ < ϕ1 (ω), and 2) there exist multiple local REE if ϕπ > ϕ1(ω).

b) For any ω ∈ (ω*, 1) when ξ<1σ (complements), there exist multiple local REE for any ϕπ > 1.

See the Appendix B. ■

The existence of multiple equilibria allows for the presence of stationary “sunspot” equilibria. This, in turn, raises the question of whether there are representations of these equilibria that are learnable. If this were the case, then economic agents would be more likely to coordinate their actions and push the economy towards these equilibria driven by “sunspots.” The following proposition shows that, in fact, there are learnable Common Factor (CF) representations, whose learnability clearly depends on the weight of the flexible-price inflation ω in the headline inflation π^to.

Let ϕ1(ω)δα+2(1+β)δα+2(1+β)τω>0, ϕ2(ω)1τω>0 and ω*=1τ. Consider the forward- looking rule in (19)-(20) with ϕπ> 1. For any ω ∈ (0, 1), when 1σξ (substitutes or neutral), or any ω ∈ (0, ω*), when 1σ>ξ (complements), the CF representation of stationary “sunspot” equilibria is learnable in the E-stability sense if ϕ1(ω) < ϕπ < ϕ2(ω).

See the Appendix B. ■

But even if one focuses on fundamental-driven equilibria, ignoring the multiple REE problem, the weight on the flexible-price inflation ω and the types of goods play key roles for the presence of unstable learning dynamics. The next proposition presents the results for the learnability of the MSV representation of fundamental-driven equilibria (from now on MSV-REE). It shows that, given the response coefficient ϕπ, putting higher weights ω makes a rule more prone to induce fundamental-driven equilibria that are not learnable in the E-stability sense.20

Let ϕ2(ω)1τω>0 and ω*=1τ. Consider the forward-looking interest rate rule in (19)-(20) with ϕπ > 1, and concentrate on fundamental equilibria.

a) For any ω ∈ (0, 1) when ξ1σ (substitutes or neutral), or any ω ∈ (0, ω*) when ξ<1σ (complements), the MSV-REE is 1) learnable in the E-stability sense if 1 <ϕπ < ϕ2 (ω), and 2) not learnable in the E-stability sense if ϕπ > ϕ2(ω).

b) For any ω ∈ (ω*, 1) when ξ<1σ (complements), the MSV-REE is not learnable in the E-stability sense for any ϕπ> 1,.

See the Appendix B. ■

Figure 1 puts together the results from Propositions 1, 2, and 3 It plots, for each type of goods, the combinations of policy parameters ω and ϕπ that deliver equilibrium determinacy or indeterminacy. It also shows the combinations for E-stable (learnable) or E-unstable (non-learnable) MSV-REE, as well as the combinations for E-stable CF representations of “sunspot” equilibria.

Figure 1:
Figure 1:

Regions of determinacy and expectational stability for forward-looking rules, when varying the response coefficient to inflation ϕπ and the weight on expected flexible-price inflation ω in the headline inflation.

Citation: IMF Working Papers 2010, 296; 10.5089/9781455211777.001.A001

In this figure, the one-sector New-Keynesian model analyzed in Bullard and Mitra (2002) corresponds to the vertical axis, where ω = 0, meaning that the rule responds solely to expected sticky-price inflation. In this case, and regardless of the goods type, equilibrium determinacy requires the active rule to satisfy an upper threshold ϕπ01+2(1+β)αδ, so that ϕπ<ϕπ0. According to Bullard and Mitra, this threshold takes very high values for realistic calibrations and, therefore, is non-binding for policy design. E-stability of the MSV solution, on the other hand, is ensured by any active rule (ϕπ > 1).

Some interesting results emerge for positive weights on the flexible-price inflation (ω > 0), since the determinacy and E-stability bounds, ϕ1(ω) and ϕ2(ω), are decreasing in this weight.

First, across different types of goods, putting more weight on the flexible-price inflation shrinks the range 1 < ϕπ < ϕ1(ω) under which a rule delivers a unique and learnable equilibrium. When ϕπ > ϕ1(ω), the rule induces multiple equilibria and, from the learning perspective, gives rise to the following instability problems. If ϕ1(ω) <ϕπ < ϕ2(ω), then the rule induces “sunspot” equilibria with learnable CF representations. While if ϕπ > ϕ2(ω) then the rule leads to fundamental equilibria with non-learnable MSV representations. This contrasts starkly with Bullard and Mitra’s results for active forward-looking rules, which suggest that the funtamental MSV solution is always learnable, even under multiple equilibria.

Second, the determinacy conditions are generally more stringent that those related to the E-stability of the MSV solution. That is, for any ω ∈ (0, 1) when ξ1σ, and for any ω ∈ (0, ω*) when ξ<1σ, we have that ϕ2(ω) > ϕ1(ω), and therefore determinacy implies E-stability, but not viceversa.

Third, the lower the elasticity of substitution across goods, the more stringent the determinacy and E-stability conditions. In fact, simple algebra shows that, for any ω ∈ (0, 1) when ξ1σ (substitute or neutral goods), and for any ω ∈ (0, ω*) when ξ<1σ (complement goods), both frontiers ϕ1(ω) and ϕ2(ω) are increasing in the elasticity ξ.

And fourth, for complement goods, increasing ω can be even more harmful. For these goods there is a threshold ω* beyond which any active rule leads to multiple equilibria and makes the MSV solution non-learnable.21

Some Intuition. We provide a simple intuition for our results in the context of complement goods, although a similar intuition can be derived for substitute or neutral goods. We analyze the adaptive learning process of economic agents following an initial deviation from the fundamental REE. Under adaptive learning, agents make forecasts using recursive least squares estimations on available data. Though we assume that they have the right functional form of the MSV solution, they might still overestimate or underestimate the impact of the technology shock on inflation and output. Then we explain how this initial deviation can be reinforced or offset by the policy rule.

Key to develop our intuition is the following expression for the real interest rate:

r^t𝔼tπ^t+1S=(1ϕπ1τωϕπ)𝔼tπ^t+1S, with ταθ,(29)

which is derived from the rule and equations (22) and (25). Under complement goods τ > 1, as shown by Lemma 1 in the Appendix B.

Assume that boundely-rational agents expect sticky-price inflation to be higher than in the REE. If the central bank sets ω(1τ,1) then by (29) the real interest rate declines (with respect to the REE) for any ϕπ > 1. This stimulates sticky-price consumption, by (25), which in turn induces higher sticky-price inflation, by (28). Over time, this mechanism leads to upward revisions of both the expected sticky-price inflation and consumption. As a result, the rule does not correct the initial deviation from the REE and, instead, moves the economy further away from it. This unstable dynamics is not ruled out if the central bank sets ω[0,1τ) and adopts an overly aggressive response to inflation,ϕπ>1τω.

On the other hand, for ω[0,1τ) but 1<ϕπ<1τω, the initial deviation from the REE induces an increase in the real interest rate, thus slowing down consumption and putting downward pressure on sticky-price inflation. Agents would then correct downward their initial higher inflation forecast, which would move sticky-price consumption and the related inflation closer to the REE. By this self-correcting mechanism, agents will (eventually) make the economy converge to the fundamental REE.

Similarly, we could also construct a sunspot-driven equilibrium. However, in this case, the initial deviation from the fundamental REE would not be due to the agents’ bounded rationality, but to the presence of non-fundamental belief shocks. Only under the conditions spelled in part a1) of Proposition 1, such initial deviation would not be self-fulfilled and agents would gradually revert to the fundamental REE.

To complement this intuition, we link our results to those policy recommendations in Bullard and Mitra (2002) and Taylor (1999a) They advocate for “leaning against the wind” policies: nominal interest rates should be adjusted positively, and more than one-for-one, in response to inflation, and positively to current output. In our model, a rule that puts some positive weight on expected flexible-price inflation (ω > 0) is actually equivalent to a rule that reacts negatively to current sticky-price output y^tS. Too see this, note that from equation (22) and y^tS=c^tS the rule (19) can be written as: r^t=ϕπ𝔼tπ^t+1S+ϕy(𝔼ty^t+1Sy^tS) with ϕyϕπωθ. Then, for given ϕπ and θ, increasing the weight ω is equivalent to responding more aggressively to current sticky-price output y^tS, but in a negative way. Clearly this negative response goes against the “leaning against the wind” policies. As a result, this response may bring about a decrease in the real interest rate, reinforcing potential b sticky-price inflationary pressures and pushing the economy away from a targeted equilibrium.

C. The General Set-up: The Bils-Klenow Calibration

We focus now in the two-sector model with price rigidities in both sectors. From the log-linearization of the CPI index (3), the relative demands in (4), the relative price equation (5), the Euler Equation (9), the intratemporal conditions in (10) for i = F, S, the New-Keynesian Phillips curves in (12) for i = F, S, the rule specified in (13) and (14), as well as the definition mcti=WtiztPti for i = F, S, and the market clearing conditions, we obtain the following system:

c^t=𝔼tc^t+11σ[r^tη𝔼tπ^t+1F(1η)𝔼tπ^t+1S] ,(30)
π^t=ηπ^tF+(1η)π^tS(31)
π^tF=β𝔼tπ^t+1F+κF(φF+σ)c^tκF(1+ξφF) (1η)q^tκF(1+φF)z^t,(32)
π^tS=β𝔼tπ^t+1S+κS(φS+σ)c^t+κS(1+ξφS)ηq^tκS(1+φS)z^t,(33)
q^t=q^t1+π^tFπ^tS,(34)

and

r^t=ϕπω𝔼tπ^t+kF+ϕπ(1ω)𝔼tπ^t+kS,

where z^t follows (6) and κi=μi1γi(π¯)2 for i = F, S. The system consists, respectively, of an aggregate Euler equation, the definition of CPI inflation, two New-Keynesian Phillips curves, the law of motion of the relative price, and the interest rate rule.

As analytical results are no longer attainable, we resort to the numerical analysis of a calibrated version of our economy. Most of our parameters are set in accordance to the existing literature. The parameter A in (1) is picked such that in steady-state the household-firm unit allocates one third of its time to work. We assume equal steady-state gross inflation across sectors (π¯F=π¯S=π¯) and set the annual inflation equal to 3 percent. We pick μF= μS = 8, implying a steady state gross mark-up of 15% in both sectors, while σ = 1, i.e., a log-utility. Given σ, the cases of complement, susbtitute and neutral goods are captured by setting ξ equal to 0.5, 1.5, and 1, respectively. The remaining structural parameters are given the following values: β = 0.99, ψ = 0.78, and φF = φS = 2 (a Frisch elasticity of labor supplied in both sectors equal to 0.5).22

The key aspect of our calibration are the price adjustment cost coefficients γF and γS We calibrate them using three key facts from the micro-evidence on price setting by Bils and Klenow (2004): the share of firms with an expected price duration of at most one quarter, i.e., the more flexible-price side of the economy; the average quarterly frequency of price change; and the average expected quarterly duration of price rigidity.23 As extensively described in the Appendix C, we obtain η =0.4, γF =0.67 and γS =74. That is, our calibrated economy features a significant degree of sticky-price heterogeneity across the two sectors, with sector F featuring very low adjustment costs to price changes. Since prices in sector F are almost flexible, although not fully flexible, in the analysis to follow in this section we refer to the inflation of this sector as the flexible-price inflation. Table 1 summarizes our benchmark calibration.

Table 1:

Benchmark Calibration

article image

C.1. Forward-Looking Rules

Figure 2 shows that, under a forward-looking rule, the determinacy and E-stability results for a realistic calibration of the two-sticky-price-sector economy are qualitatively similar to the analytical results of the “strong dichotomy” case in Subsection B (see Figure 1). Across different types of goods, putting a higher weight on the the flexible-price inflation restricts the set of forward-looking rules that induce a unique and learnable fundamental REE. Moreover, there exist combinations of ϕπ and ω for which the economy displays learnable “sunspot” equilibria. In addition, when goods are complements, any active rule induces non-learnable fundamental-driven equilibria, if the weight ω is above a threshold ω*. With the strong dichotomy being a limiting case of the more general set-up and given the significant degree of sticky-price heterogeneity resulting from our calibration, the intuition provided above for the results should be unaltered.

Figure 2:
Figure 2:

Regions of determinacy and expectational stability for forward-looking rules. Parameters other than ϕπ and ω are set at baseline values shown in Table 1.

Citation: IMF Working Papers 2010, 296; 10.5089/9781455211777.001.A001

Note: ME = Multiple Equilibria, ES-CF = E-Stable Common Factor representation of a “sunspot” equilibrium, and MSV = Minimum State Variable representation of a fundamental equilibrium.

To give a sense of the quantitative importance of our results, Table 2 quantifies the determinacy and E-stability thresholds for the case of Edgeworth complementarity.

Table 2:

Forward-Looking Rules

Determinacy and E-stability Bounds for Complement Goods

article image
Note: ω* = 0.92. UE = Unique equilibrim. ME = Multiple Equilibria.ES-MSV = E-stable (learnable) MSV representation of a fundamental equilibrium.ES-CF Sunspot = E-stable (learnable) CF representation of a “sunspot” equilibrium.EU-MSV = E-unstable (non-learnable) MSV representation of a fundamental equilibrium.

In row a) we consider an active rule with a response to inflation ϕπ = 2, which is the lower bound to the estimate of the “degree of activism” in the post-Volcker period by Cogley and Sargent (2005). We find that multiple equilibria arise for ω > 0.44, while the fundamental MSV-REE ceases to be learnable for ω > 0.48. This says that, even if the Taylor principle of ϕπ > 1 holds, endogenous aggregate instability occurs for a weight on sector F inflation just slightly above the CPI weight of η = 0.4.

Row b) looks at the consequences of a positive ω from another angle. We fix its value below ω* = 0.92. For instance, we set ω = η and identify the determinacy/E-stability bounds with respect to ϕπ. We obtain multiple equilibria for ϕπ > 2.18 and a non-learnable MSV-REE for ϕπ > 2.3. But for ω > 0.92 no active rule is stabilizing.

C.2. Other Timings for the Policy Rule

Figure 3 displays the results for the case of the contemporaneous rule r^t=ϕππ^to. In this case, and regardless of the types of goods (complements, substitutes, or neutral), the measure of inflation does not affect both equilibrium determinacy and E-stability. As long as the rule satisfies the Taylor principle of ϕπ > 1, there is a unique and learnable equilibrium. This result is reminiscent of that by Carlstrom et al. (2006a), although they exclusively focus on determinacy, while here we complement their analysis by showing the learnability of the MSV representation.24

Figure 3:
Figure 3:

Regions of determinacy and expectational stability for contemporaneous rules. Parameters other than ϕπ and ω are set at baseline values shown in Table 1.

Citation: IMF Working Papers 2010, 296; 10.5089/9781455211777.001.A001

Note: MSV = Minimum State Variable representation of a fundamental equilibrium.

A different type of instability occurs under the backward-looking rule r^t=ϕππ^t1o. Responding to past flexible-price inflation can induce explosive equilibria, that is, the economy diverges from the steady state equilibrium and give rise to unbounded fluctuations. Figure 4 shows that raising ω makes the backward-looking rule more prone to induce these equilibria, in particular for the case of Edgeworth complementarity. However, when the equilibrium is determinate, it is also learnable.25

Figure 4:
Figure 4:

Regions of determinacy and expectational stability for backward-looking rules. Parameters other than ϕπ and ω are set at baseline values shown in Table 1.

Citation: IMF Working Papers 2010, 296; 10.5089/9781455211777.001.A001

Note: MSV = Minimum State Variable representation of a fundamental equilibrium.

These results for other timings for the policy rule together with the results for forward-lookig rules suggest an important policy lesson to rule out potential sources of instability: to be on the safe side, policy makers should consider targeting the inflation of the sectors with a higher degree of price stickiness.

C.3. Quantitative Sensitivity

We conclude this Section by investigating the quantitative effect of varying the price stickiness heterogeneity across sectors as well as the share η of flexible-price goods. Varying η allows us to study the effects of responding to CPI inflation.

Aggregate Price Stickiness and Its Heterogeneity The variety of empirical estimates from the micro-based studies motivates us to assess how equilibrium determinacy and E-stability depend on different degrees of aggregate price rigidity and its heterogeneity across sectors. We focus on the average frequency of price change f—a measure of aggregate price rigidity—and the relative standard deviation (also known as, the coefficient of variation) of the sectoral frequencies fF and fS, which we take as a measure of the degree of heterogeneity and denote by Δf ∈ (0, 1). More formally, we define

f=ηfF+(1η)fS and Δf=η(fFf)2+(1η) (fSf)2f.(35)

Concentrating on the frequencies of price change (fi for i = F, S) rather than an average adjustment costs (γi for i = F, S), is motivated by the fact that the available empirical evidence is about the frequency and not the cost of price changes. Clearly, this choice does not affect the interpretation of the results. There is a one to one mapping between these measures, as explained in the Appendix C. A higher fi implies a lower γi.

In this assessment, we consider f ∈ [0.2, 0.8], which includes the value f = 0.52 used in our benchmark “Bils-Klenow” calibration. Then, for each possible pair (f, Δf), we set fF and fS such that they satisfy the equations in (35). Once fF and fS are identified, we compute the corresponding adjustment cost parameters γF and γS following the same calibration procedure explained in the Appendix C. We consider a forward-looking rule and an economy of equally-sized sectors, i.e., η = 0.5, and set ω = η.26 All remaining parameters are as in Table 1. For reasons of space, we only present the results for the case of Edgeworth complementarity, i.e., ξ = 0.5.

Several interesting results emerge (see Figure 5). First of all, for a given average frequency of price change just above 0.45, endogenous aggregate instability arises when sticky-price heterogeneity is sufficiently high. Second, this upper threshold on heterogeneity is strictly decreasing in the aggregate level of rigidity: the more flexible prices are on average (i.e., the higher is f) the lower is the minimum degree of heterogeneity (i.e., the lower is Δf) above which aggregate instability arises. Third, the equilibrium is always unique and E-stable, independently from the degree of heterogeneity, if the economy is sufficiently sticky in aggregate terms (low f).27

Figure 5:
Figure 5:

Regions of determinacy and expectational stability when varying f and Δ f. Goods are complements ξ = 0.5 and the rule is forward-looking with ϕπ = 2 and ω = η = 0.5. All remaining parameters, except for γF and γS, are as in Table 1.

Citation: IMF Working Papers 2010, 296; 10.5089/9781455211777.001.A001

Note: CF = Common Factor representation of a “sunspot” equilibrium and MSV = Minimum State Variable representation of a fundamental equilibrium.

The Share of Flexible-Price Goods and the Response to CPI Inflation Next, we assess how the results depend on the share of flexible price goods in aggregate consumption η. Note that, in contrast to ω, the share η affects the steady state, including the relative price q¯. Therefore, it is not longer possible to focus on the steady state where q¯=1. As shown in the Appendix A the (log) CPI inflation, which affects the Euler equation (30), is redefined as π^t=ηFπ^tF+ηSπ^tS where ηF=η[η+(1η)(1q¯)1ξ]1 and ηS=(1η) [ηq¯1ξ+(1η)]1.

Figure 6 presents the results of varying the share η, assuming the goods are complements and the rule is forward-looking. As η decreases the threshold value for ω, beyond which aggregate instability arises, decreases as well. As a result, a rule responding to the expected headline inflation π^t+1o=ωπ^t+1F+(1ω)π^t+1S is more prone to induce stability problems, the smaller is the share of flexible price goods. This figure also shows that in our model responding to the CPI inflation π^t, i.e. ω = η (which corresponds to the 45 degree line) does not guarantee determinacy and/or E-stability.

Figure 6:
Figure 6:

Regions of determinacy and expectational stability when varying ω and η. Goods are complements ξ = 0.5 and the rule is forward-looking with ϕπ = 2. All remaining parameters are as in Table 1.

Citation: IMF Working Papers 2010, 296; 10.5089/9781455211777.001.A001

Note: CF = Common Factor representation of a “sunspot” equilibrium and MSV = Minimum State Variable representation of a fundamental equilibrium.

But what are the effects of responding to CPI inflation while varying the share η? Figure 7 shows the determinacy and E-stability areas with respect to ϕπ and η for complement goods and the policy rule r^t=ϕπ𝔼tπ^t+1. The Figure shows that the results are to some extent qualitatively similar to those obtained for a rule responding to expected headline inflation (see Figure 2). In particular, when responding to CPI inflation, forward-looking rules are more suceptible to non-learnable fundamental-drive equilibria, the higher the share η. However, note that there is no threshold for this share beyond which any active rule induces these instability problems, as in the case of responding to the headline inflation.

Figure 7:
Figure 7:

Regions of determinacy and expectational stability for a forward-looking rule that responds to the CPI inflation. Goods are complements ξ = 0.5. All remaining parameters are as in Table 1.

Citation: IMF Working Papers 2010, 296; 10.5089/9781455211777.001.A001

Note: CF = Common Factor representation of a “sunspot” equilibrium and MSV = Minimum State Variable representation of a fundamental equilibrium.

IV. Robustness and Discussion

In this Section, we discuss the implications of relaxing some of the assumptions of our set-up, including (i) rules that, besides inflation, also respond to output; (ii) different timings for money in the utility function; and (iii) the approach to learning.

Extended Interest Rate Rules The destabilizing consequences of considering a flexible-price inflation measure persist if we augment those rules with an explicit response to aggregate activity. In particular, for any given coefficient ϕπ, the Appendix D shows that a positive response to (expected) output lowers the upper bound on ω above which multiple equilibria arise and the fundamental MSVREE ceases to be stable under learning. So aggregate instability is more pervasive. This outcome is in line with the intuition we provided for the analytical results of Propositions 1-3, as well as the numerical results of Subsection C. As shown, a higher ω implies a more negative implicit response to current sticky-price output, which undermine determinacy and E-stability.

Cash-In-Advance Timing for Money Our model embeds a Cash-When-I’m-Done (CWID) timing where end-of-period money balances, net of current income and consumption, enter the utility function. This is the most frequent timing used in the New-Keynesian literature. But how would our results change if we adopted a Cash in Advance (CIA) timing? that is, if the money that is available to satisfy consumption is the cash left after visiting the bonds market, but before entering the goods market?

In the Appendix D, we show that the determinacy and learnability conditions for a forward-looking rule under the CWID timing are identical to those for a contemporaneous rule under the CIA timing. This is because the CWID Euler equation under a forward-looking rule is identical to the CIA Euler Equation with a contemporaneous rule. This isomorphism was first pointed out by Carlstrom and Fuerst (2001) in a one-sector New-Keynesian model and in the context of equilibrium determinacy. We show that it survives in our two-sector model and, more importantly, that it also applies to the learnability conditions.

Therefore, the graphical characterizations of our results in Figure 1 and 2 are also valid for a contemporaneous rule under the CIA timing: that is, active contemporaneous rules responding to inflation measures that include some flexible-price inflation are more prone to induce indeterminacy and non-learnable fundamental equilibria than rules that exclusively respond to sticky-price inflation.28

The Euler Equation Approach versus the Infinite Horizon Learning Approach We have adopted the “Euler Equation” learning approach of Evans and Honkapohja (2001), as it remains the most common in the literature on learning and policy rules, including the seminal work of Bullard and Mitra (2002). By selecting this approach, we have assumed that our “boundedly rational” agents make decisions based the a simple marginal costs - marginal benefits analysis captured by the first order conditions obtained under rational expectations. In other words, agents were only required to make one-period-ahead forecasts when making decisions about variables such as consumption, labor and price setting.

The infinite horizon (IH) approach proposed by Preston (2006) is an alternative approach to model learning. Preston suggests that when agents have subjective beliefs about the evolution of aggregate state variables, the decision rules for infinitely-lived optimizing agents should depend on infinite horizon expectations. Hence, in equilibrium, the Euler equation and the New-Keynesian Phillips curve should feature expectations over the infinite future sequences of inflation, real activity and the policy rate. As agents do not know others’ beliefs and preferences, they cannot compute the exact law of motion of aggregate variables and therefore long-horizon forecasts are “irreducible” to a one-step-ahead prediction.

We infer, however, that our results might still hold under the IH approach provided some plausible conditions. Our inference is based on Honkapohja et al. (2003). They argue that even under the IH approach, if each agent actually forms the same forecasts—given that everyone is assumed to rely on the same econometric model and to have access to the same historical data—then ex-post all agents behave identically, leading to a symmetric equilibrium. But in a symmetric equilibrium with all agents (correctly) inferring that the market would clear at any point in time, the IH approach collapses to the “Euler equation” approach to learning. For more general conditions, there are no equivalence convergence results between these two approaches. In this regard, it would be interesting to pursue an extensive IH learning analysis of our model, taking into account that in our multiple-sector economy, agents would need to form long-horizon expectations not only of future marginal costs and sectoral inflations, but also of future relative prices. This goes beyond the scope of this paper and we are currently pursuing it as a separate research project.

More generally, once the assumption of rational expectations is dropped, there are other alternative ways to model expectation formation. All that is required is that the latter satisfy standard probability laws at the individual level. For instance, Schorfheide (2005) and Svensson and Williams (2007) assume Bayesian learning to account for the possibility of, respectively, policy shifts and model uncertainty. More recently, Adam and Marcet (2010) provide a general discussion about the minimal set of requirements for subjective expectations to be consistent with individual optimization.

V. Conclusions

We develop a two-sector closed economy model with heterogeneous price stickiness across sectors, to show that the inflation measure included in an interest rate rule is relevant for ensuring equilibrium determinacy and learnability a la Evans and Honkapohja (2001). We find that active rules responding to inflation measures that put some weight on the inflation of the sector with low price stickiness are more suceptible to induce aggregate instability in the economy than rules that respond exclusively to the inflation of the sector with high price stickiness. The sources of instability are related to the presence of equilibria driven by fundamental shocks that are not learnable, equilibria driven by self-validating beliefs that are learnable, and equilibria with unbounded fluctuations.

Our results may provide some guidance for the discussion on the appropriate inflation measure for the monetary policy stance in the context of energy price shocks. Should central banks respond to a headline measure that includes the inflation of flexible energy prices? or, instead, should they react to a core measure that includes mainly the inflation of sticky prices? The results of Bodenstein et al. (2008) advocate for core inflation as they find that, for fundamental-driven equilibria, rules responding to a forecast of headline inflation induce greater volatility in core inflation and the output gap than rules reacting to a forecast of core. Our results for forward-looking rules support this policy recommendation. But our explanation is different: by responding to an inflation measure that includes flexible-price inflation, a rule may induce greater volatility associated with learnable self-fulfilling expectations equilibria of the “sunspot” type, and fundamental-driven equilibria that are not learnable by the agents.

While we have focused on price rigidity, there are additional sources of sectoral heterogeneity that one might want to embed into a multi-sector New-Keynesian model. In Airaudo and Zanna (2010), we allow sectors to differ both because of price stickiness and the impact of policy rates on marginal costs of production, i.e., the cost channel of Ravenna and Walsh (2006).29 We show that the inflation measure adopted by the central bank is relevant for equilibrium determinacy and E-stability both under a forward-looking and a contemporaneous interest rate rule. Another issue is the “durability” of goods. Barsky et al. (2007) show that the aggregate consumption share and the degree of price rigidity of durable goods have strong implications for the monetary policy transmission in New-Keynesian models. Whether durability affects our determinacy and E-stability results is an interesting question to explore.

Learning About Inflation Measures for Interest Rate Rules
Author: Luis-Felipe Zanna and Mr. Marco Airaudo
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    Regions of determinacy and expectational stability for forward-looking rules, when varying the response coefficient to inflation ϕπ and the weight on expected flexible-price inflation ω in the headline inflation.

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    Regions of determinacy and expectational stability for forward-looking rules. Parameters other than ϕπ and ω are set at baseline values shown in Table 1.

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    Regions of determinacy and expectational stability for contemporaneous rules. Parameters other than ϕπ and ω are set at baseline values shown in Table 1.

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    Regions of determinacy and expectational stability for backward-looking rules. Parameters other than ϕπ and ω are set at baseline values shown in Table 1.

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    Regions of determinacy and expectational stability when varying f and Δ f. Goods are complements ξ = 0.5 and the rule is forward-looking with ϕπ = 2 and ω = η = 0.5. All remaining parameters, except for γF and γS, are as in Table 1.

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    Regions of determinacy and expectational stability when varying ω and η. Goods are complements ξ = 0.5 and the rule is forward-looking with ϕπ = 2. All remaining parameters are as in Table 1.

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    Regions of determinacy and expectational stability for a forward-looking rule that responds to the CPI inflation. Goods are complements ξ = 0.5. All remaining parameters are as in Table 1.