Transparency, Expectations Anchoring and the. Inflation Target

Transcription

1 Transparency, Expectations Anchoring and the Inflation Target Guido Ascari Anna Florio Alessandro Gobbi University of Oxford Politecnico di Milano June 204 Università Cattolica del Sacro Cuore Abstract In a standard new Keynesian framework, a higher inflation target unanchors expectations, as feared by former Fed Chairman Bernanke. It does so both asymptotically, because it shrinks the E-stability region when a central bank follows a Taylor rule, and in the transition phase, because it slows down the speed of convergence of expectations. Moreover, the higher the inflation target, the more policy should respond to inflation and the less to output to guarantee E-stability. Hence, a policy that increases both the inflation target and the monetary policy response to output would be reckless. Moreover, we show that transparency is an essential component of the inflation targeting framework and it helps anchoring expectations. However, the importance of being transparent diminishes with the level of the inflation target. Keywords: Trend Inflation, Learning, Monetary Policy, Transparency. JEL classification: E5. The authors thank Klaus Adam, Efrem Castelnuovo, Martin Ellison, Takushi Kurozumi, and Seppo Honkapohja for helpful comments. Ascari thanks the MIUR for financial support through the PRIN 09 programme. The usual disclaimer applies. Address: Department of Economics, University of Oxford, Manor Road, Oxford OX 3UQ, United Kingdom. guido.ascari@economics.ox.ac.uk Address: Department of Management, Economics and Industrial Engineering, Politecnico di Milano, via Lambruschini 4/B, 2056, Milan, Italy. Fax: anna.florio@polimi.it Department of Economics and Finance, Università Cattolica del Sacro Cuore, via Lodovico Necchi 5, 2023 Milan, Italy. alessandro.gobbi@mail.polimi.it

2 Introduction Following the Great Recession Blanchard et al. (200) proposed to increase the central bank s inflation target in order to deal with the problem of the zero lower bound on interest rates. More recently, Ball (203) and Krugman (204) have forcefully supported this view. In various speeches, former Fed Chairman Ben Bernanke contrasted this argument by claiming that a higher inflation target could unanchor inflation expectations. The New Keynesian literature has convincingly shown that price stability should be the goal of monetary policy even taking into account the perils of hitting the zero lower bound (e.g. Coibion et al., 200; Schmitt-Grohé and Uribe, 200). However, these papers cannot address Bernanke s concern about the possibility that a higher inflation target could unanchor inflation expectations. A natural environment to study such an issue would entail agents who use adaptive learning to form expectations, as suggested by Bernanke himself. 2 In this paper we therefore consider a New Keynesian macromodel with trend inflation and learning to answer the following research question: Would it be more difficult for the central bank to stabilize inflation expectations at higher values of the inflation target? We thus investigate the link between inflation expectations under adaptive learning and the level of the inflation target. We characterize how the set of policy rules that guarantees stability of the rational expectation equilibria (REE) under learning dynamics (E-stability) changes with the inflation target. This would allow us to address questions such as: If the central bank targets a higher inflation level, does it need to respond more aggressively to inflation to stabilize expectations? Moreover, a higher inflation target does not only affect the possibility that expectations asymptotically converge to the REE (that is, the E-stability region of the parameter In this context, raising the inflation objective would likely entail much greater costs than benefits. Inflation would be higher and probably more volatile under such a policy, undermining confidence [...]. Inflation expectations would also likely become significantly less stable. Bernanke s remarks at the 200 Jackson Hole Symposium. 2 What is the right conceptual framework for thinking about inflation expectations in the current context? [...] Although variations in the extent to which inflation expectations are anchored are not easily handled in a traditional rational expectations framework, they seem to fit quite naturally into the burgeoning literature on learning in macroeconomics. [...] In a learning context, the concept of anchored expectations is easily formalized. Bernanke s speech at the NBER Monetary Economics Workshop, July 2007.

3 space), but it also affects the transition phase because it impacts on the speed at which convergence occurs. Following Ferrero (2007), we will then analyse how a higher inflation target influences the speed of convergence of expectations under E-stability. From a policy perspective, this is an important point, often neglected to focus mainly on asymptotic results (i.e., E-stability). A fast convergence means that the economy, and thus its dynamics, would always be very close to the REE, while a slow convergence implies that economic dynamics would be dominated by the transitional dynamics under learning. As such, the speed of convergence could well be another important criteria to judge a given policy. Finally, another main component of an inflation targeting framework is the communication strategy. We aim to capture this element by distinguishing between transparency and opacity, based on Preston (2006): a central bank is said to be transparent if agents know the policy rule and use this information in their learning process. 3 If not, it is said to be opaque. Thus, for each level of the inflation target, we study whether the ability to anchor expectations differs under transparency and under opacity. That is, does a central bank that fixes a higher level of inflation target need to be more transparent? The novelty of the paper is therefore to put all the above ingredients together, to study how the inflation target influences the learning process both asymptotically (i.e., E-stability) and in the transition phase (i.e., the speed of convergence) and how the transparency of monetary policy affects the relationship between trend inflation and learning. We first analyse the standard specification of the New Keynesian model, that is, assuming a zero inflation steady state. This analysis allows analytical results that provide revealing insights for the case of a positive inflation target, where only numerical results are possible. Economic agents do not have rational expectations but rather form their forecasts by using recursive learning algorithms. This part of the paper is similar to 3 The second major element of best-practice inflation targeting (in my view) is the communications strategy, the central bank s regular procedures for communicating with the political authorities, the financial markets, and the general public. [...] Most inflation-targeting central banks have found that effective communication policies are a useful way, in effect, to make the private sector a partner in the policymaking process. To the extent that it can explain its general approach, clarify its plans and objectives, and provide its assessment of the likely evolution of the economy, the central bank should be able to reduce uncertainty, focus and stabilize private-sector expectations. Bernanke s speech at the Annual Washington Policy Conference of the National Association of Business Economists, Washington, D.C., March

4 Bullard and Mitra (2002a), but it adds the analysis of the speed of convergence (as in Ferrero, 2007) and the distinction between transparency and opacity (as in Preston, 2006). We then investigate the effects of the communication strategy on the learnability of the REE. More precisely, whether a transparent central bank is better able to anchor inflation expectations, and which features of the economy and of the monetary policy rule affect the different ability of anchoring expectations under transparency and under opacity. The main results of this section are as follows: (i) transparency helps anchoring expectations both asymptotically (that is, the E-stability region is wider under transparency than under opacity) and in the transition phase (the speed of convergence of expectations to REE is generally higher under transparency than under opacity); (ii) responding to the output gap is important to stabilize expectations: it is necessary to guarantee E-stability under opacity and to increase the speed of convergence both in the case of opacity and transparency; (iii) a pure inflation targeting central bank needs to be transparent to anchor inflation expectations, but it anyway exhibits a low speed of convergence; (iv) the more flexible are prices, the more transparency is valuable. Our results, thus, substantiate the claim that transparency is an essential component of the inflation targeting approach to monetary policy. We then turn to the analysis of a New Keynesian model with positive steady state inflation and adaptive learning, to tackle the main research question of the paper. We compute, for different values of the inflation target, both the E-stability and the determinacy regions in case of transparency and in case of no communication by the central bank. The main result of the paper is consistent with Bernanke s statement: a higher inflation target tends to destabilize expectations both asymptotically, because it shrinks the E-stability region for a given Taylor rule, and in the transition phase, because it slows down the speed of convergence of expectations to the REE. This paper deals with two points raised by Ball (203). Firstly, Ball claims: We have learned from recent experience that 4% inflation is better than 2%, because of the zero bound problem. Why can t policymakers explain this, raise inflation to 4%, and keep it there? However, we show that the higher the inflation target, the less effective 3

5 transparency is: so it would be more difficult to anchor expectations even exploiting transparency. Secondly, according to Ball, the transitional period of learning brought about by an increase in the central bank s target is definitely a period of higher uncertainty but nobody has demonstrated that this transition would harm the economy significantly. We show that a higher inflation target tends to lower the speed of convergence of expectations hence, in any case, this transitional period would last longer. What is more, the higher the inflation target, the more the policy should be hawkish with respect to inflation in order to stabilize expectations, while it should not respond too much to output. This result questions arguments often presented by some distinguished economists that urged the Fed to increase the inflation target and, contemporaneously, ease monetary policy to respond to the surge in unemployment. 4 Our findings suggest that this policy would indeed be reckless and unwise, as Bernanke recently put it. 5 The paper is structured as follows. After a brief account of the related literature (Section.), Section 2 presents the model and the methodology employed. Section 3 contains the results both for the zero inflation target case (Section 3.) and for positive trend inflation (Section 3.2). Some robustness checks are illustrated in Section 4, and Section 5 concludes.. Related literature Our paper is strictly linked to the seminal paper by Bullard and Mitra (2002a) and the more recent contributions by Ferrero (2007), Eusepi and Preston (200), and Kobayashi and Muto (20). Bullard and Mitra (2002a) analyse the determinacy and learnability of simple monetary policy rules in a standard New Keynesian model approximated around the zero inflation steady state. Bullard and Mitra (2007) enrich their previous results introduc- 4 E.g., see the recent article by Paul Krugman, 29 April 202 in the New York Times ( moc.semityn.www). 5 I guess the question is, does it make sense to actively seek a higher inflation rate in order to achieve a slightly increased pace of reduction in the unemployment rate? The view of the committee is that that would be very reckless. Bernanke, FOMC Press Conference transcript, 25th of April 202, 4

6 ing monetary policy inertia in the same model and showing how it helps to produce learnability of the rational expectation equilibrium. We are basically following their approach which builds on Evans and Honkapohja (200). We generalize their analysis by considering: i) transparency versus opacity (Eusepi and Preston, 200); ii) the speed of convergence (Ferrero, 2007); iii) positive average inflation, based on the model in Ascari and Ropele (2009) (Kobayashi and Muto, 20). However, we do also provide analytical results for the standard case of zero steady state inflation. Based on Marcet and Sargent (995), Ferrero (2007) studies the speed of convergence of expectations to the REE under E-stability and determinacy in a simple New Keynesian framework. We follow his approach and we add to his contribution the analysis of the impact of a positive inflation target and of transparency versus opacity. Following Preston (2006), we study two possible communication strategies by the central bank. Preston (2006) distinguishes two cases: one where agents know the monetary policy rule, call it the transparency case (TR), and the other where they are forced to infer the interest rate by learning it adaptively, call it the opacity case (OP). We analytically characterize the conditions for E-stability under TR and OP in the case of zero trend inflation. As far as we know, this is the first paper that analyses the difference between TR and OP in an Euler equation learning context. Note that, employing (as they do) a contemporaneous monetary rule, we need to assume agents base their expectation on period t information, because there would be no difference between TR and OP if agents decisions were based on current expectations. The intuition we gained through this analysis proved to be very useful for the case of positive inflation when numerical results are the only option. It follows that our analysis is linked to Preston (2006) and also to the more recent, and related, contribution by Eusepi and Preston (200). These papers do not consider neither the case of positive inflation neither the speed of convergence, as we do. Moreover, they employ what Honkapohja et al. (20) call the infinite horizon approach due to Preston (2005, 2006). Preston derives this model under arbitrary subjective expectations and finds that the model s equations depend on long-horizon expectations that is, on forecasts into 5

7 the entire infinite future. Eusepi and Preston (200) further employ Preston s infinite horizon approach (the only change being assuming decisions are made based on period t information) to analyse what happens to E-stability when the Taylor principle holds and the central bank employs a variety of communication strategies. Our paper, beside sharing with Eusepi and Preston (200) the assumption of lagged expectations, is like theirs devoted to disentangle the effects of central bank communication on learnability (and, we add, determinacy) of rational expectations equilibria. However, while they employ the infinite horizon approach, we use the more standard Euler equation approach of Evans and Honkapohja (200). In the former approach agents are assumed to make forecasts over the infinite future, while in the latter agents forecast only one period ahead. So the two approaches represents the two extreme cases of farsightedness. Honkapohja et al. (20) show that, in the context of a New Keynesian model, the Euler equation approach in Bullard and Mitra (2002b) is anyway consistent with Preston (2005), so that both the [Euler equation] and [infinite horizon] approaches are valid ways to study stability under learning in the New Keynesian setting. (Honkapohja et al., 20, p. 3). 6 So our analysis in a zero inflation steady state could be seen as a robustness analysis of the results in Eusepi and Preston (200) in the Euler equation context. 7 As theirs, our analytical results, though different, have similar implications and intuition but, being analytically simpler, they allow us not to confine the analysis to the case where the Taylor principle holds (as Eusepi and Preston, 200), but, rather, to fully characterize the E-stability regions. Moreover, and most importantly, our study departs from theirs by analysing what happens as trend inflation changes. This, obviously, calls for a more general model specification that allows for positive trend inflation. In this respect, our paper is also close to a recent contribution by Kobayashi and Muto (20) that studies expectational stability under trend inflation and we get results consistent with their findings. The analysis in Kobayashi and Muto borrows a New 6 See Honkapohja et al. (20) for a thorough discussion of the two approaches. See also Evans and Honkapohja (203). 7 The [Euler equation] and [infinite horizon] approaches to modeling agent s behavioural rule are not identical and lead to different detailed learning dynamics. Thus there is in general no guarantee that the convergence conditions for the two dynamics are identical. (Honkapohja et al., 20, p. 8). 6

8 Keyneasian Phillips curve (NKPC) formulation under trend inflation (see Sbordone, 2007; Cogley and Sbordone, 2008) and plugs it into an otherwise standard New Keynesian model, that is adding an Euler equation and a Taylor rule. This formulation coincides with a simplified version of the model in Ascari and Ropele (2009). Their model, thus, differs from ours because of some simplifying assumptions. The same assumptions (mainly an infinitely elastic labour supply) are made in the analytical part of the paper by Ascari and Ropele (2009). However, both our and Kobayashi and Muto s (20) analyses of the positive inflation case are numerical, so those assumptions are not really needed. This may not be innocuous, because the simplifying assumptions make price dispersion irrelevant for the dynamics of the model. 8 As a consequence, our model has a higher-order system of difference equations and this may affect the results. 9 Furthermore, with respect to Kobayashi and Muto (20), we offer two contributions: i) we study the effects of central bank s transparency on the anchoring of expectations, by distinguishing between the cases of TR and OP; ii) we analyse the effects of the inflation target on the speed of convergence of learning. Moreover, as said above, thanks to our analytical investigation of the case of zero trend inflation, we are able to provide intuition about the effects that trend inflation has on the E-stability regions and on the difference between TR and OP. Further differences between us and Kobayashi and Muto (20) are the assumption of lagged expectations, the analysis of the case of inertia in the interest rate rule and of indexation. Finally, two other works employ Ascari and Ropele s (2009) model under learning. In a very insightful paper Branch and Evans (20) study the dynamics of the model when there is a change in the long-run inflation target, and agents have only imperfect information about the long-run inflation target. They show that imperfect knowledge of the inflation target could generate near-random walk beliefs and unstable dynamics due to self-fulfilling paths. Imperfect information of inflation targets can thus generate instability 8 However, the dynamics of price dispersion is one of the main features of a model with positive trend inflation (with respect to one linearized around zero inflation) and changes its behaviour by adding a backward-looking dynamic equation. 9 Kurozumi (203) makes a similar point and develops an analysis of the E-stability region that, as in Kobayashi and Muto (20), allows for positive trend inflation. His results are coherent with ours. 7

9 in inflation rates. A related and very interesting work by Cogley et al. (200) studies optimal disinflation under learning. When agents have to learn about the new policy rule, then, the optimal disinflation policy is more gradual, and the sacrifice ratio much bigger, than under the case of TR. The optimal disinflation is gradual under OP because the equilibrium law of motion under learning is potentially explosive. However, they find that imperfect information about the policy feedback parameters, rather than about the long-run inflation target, is the crucial source of the explosiveness of the dynamics. 2 Model and Methodology 2. The Model The model we use is based on Ascari and Ropele (2009), that extends the basic New Keynesian (NK) model (e.g., Galì, 2008; Woodford, 2003) to allow for positive trend inflation. The details are presented in the Appendix. Log-linearizing the model around a generic positive inflation steady state yields the following equations: ŷ t = E t ŷ t+ E t (î t ˆπ t+ ) + r n t, () ˆπ t = β πe t π t+ + λ π Et [( + σ n ) ŷ t + σ n ŝ t ] + η π Et [(θ ) ˆπ t+ + ˆφ ] t+ + λ π ( + σ n ) u t, (2) ˆφ t = αβ π (θ ) Et [(θ ) ˆπ t+ + ˆφ ] t+, (3) ŝ t = ξ πˆπ t + α π θ ŝ t, (4) î t = φ π E t ˆπ t + φ y E t ŷ t, (5) where hatted variables denote percentage deviations from steady state, apart from ŷ t, which is the usual output gap term defined as deviation from the flexible price output level. The structural parameters and their convolutions (λ π, η π and ξ π ) are described in Table. r n t and u t are exogenous disturbance terms that follow the processes: r n t = ρ r r n t + ε r t and u t = ρ u u t + ε u t, where ε r t and ε u t are i.i.d. noises and 0 < ρ r, ρ u <. 8

10 The first equation is the standard Euler equation in consumption, and r n t is the stochastic natural rate of interest. The second and the third equations describe the evolution of inflation in presence of trend inflation, so they are the counterpart of the standard NKPC for the standard zero inflation steady state case, where u t is a mark-up shock. ˆφt is just an auxiliary variable (equal to the present discounted value of future expected marginal revenue) that allows the model to be written in a recursive way. The fourth equation describes the evolution of price dispersion, ŝ t. In contrast to the zero inflation steady state case, in presence of positive average inflation price dispersion affects inflation dynamics at first-order approximation and thus has to be taken into account. 0 The fifth equation is the simplest standard contemporaneous Taylor rule. We deviate from Ascari and Ropele (2009), following Evans and Honkapohja (200) and much of the related literature on learning, by assuming that agents have non-rational expectations, that we denote with E. Furthermore, we assume that expectations are formed on the basis of period t information set (see also Bullard and Mitra, 2002b). According to Evans and Honkapohja (200), this assumption is more natural in a learning context, since it avoids simultaneity between expectations and current values of endogenous variables. Of course, Ascari and Ropele s (2009) generalized model of inflation dynamics, described by equations (2), (3) and (4), encompasses the standard NKPC. Assuming zero trend inflation π =, then η π = ξ π = 0, thus both the auxiliary variable and the measure of relative price dispersion become irrelevant for inflation dynamics. Thus, the above equations turn just into the standard specification of the NK model (where 0 Kobayashi and Muto (20) do not take into account price dispersion, because they assume a simple proportional relationship between the marginal cost and the output gap. However, as shown in Ascari and Ropele (2009), this is not general and it requires the additional assumption of indivisible labour (i.e., σ n = 0). We have chosen to assume that expectations are conditional on information at time t. This avoids a simultaneity between expectations and current values of the endogenous variables which may seem more natural in the context of the analysis of learning. (Evans and Honkapohja, 200, p. 229). 9

11 Table : Parameters and basic symbols. Parameters β σ n θ α π φ π φ y Description Subjective discount factor Intertemporal elasticity of labour supply Dixit-Stiglitz elasticity of substitution Calvo probability not to optimize prices Central Bank s inflation target (or trend inflation) Inflation coefficient in the Taylor rule Ouput coefficient in the Taylor rule NKPC Coefficients [ α π (θ ) ][ αβ π θ ] λ π απ (θ ) η π θα π (θ ) ( π ) ξ π α π (θ ) β ( π ) [ α π (θ )] κ = λ ( + σ n )): ŷ t = E t ŷ t+ E t (î t ˆπ t+ ) + r n t, (6) ˆπ t = βe t ˆπ t+ + κe t ŷ t + κu t, (7) completed with the monetary policy rule (5). 2.2 Methodology We are interested in analysing determinacy, learnability, and speed of convergence under both zero and positive trend inflation. The determinacy results are obviously the same as in Ascari and Ropele (2009), so we will not comment on those and refer the reader to their article Learnability When agents do not possess rational expectations, the existence of a determinate equilibrium does not ensure that agents coordinate upon it. As from the seminal contribution of Evans and Honkapohja (200), we assume agents do not know the value of the structural parameters and, as such, they cannot form expectations using the true law of motion of the economy. Rather, they behave as econometricians and compute expectations ac- 0

12 cording to a reduced-form model whose parameters are estimated through recursive least squares using the data produced by the economy itself. If the REE is learnable, the recursive estimates will converge to the actual parameter values and the economy will display a dynamic behaviour that resembles, and eventually coincides with, the REE. In light of these considerations, learnability is an obviously desired feature of monetary policy. We apply multivariate E-stability results outlined in Evans and Honkapohja (200, Section 0.2.). Agents are assumed to share identical beliefs and to form forecast using a perceived law of motion (PLM) which has the same structure of the minimal state variable (MSV) solution of the system obtainable under rational expectations. 2 Each period, as additional data become available, they estimate the coefficients of their PLM and compute expectations that, once inserted into the equations of the model, give rise to the actual law of motion (ALM). We then ask whether the PLM converges to the MSV. In other terms, we want to know under which conditions agents are able to learn the REE of the system (see Appendix B for details) Transparency versus Opacity In defining OP and TR of monetary policy, we follow closely the work of Preston (2006) and Eusepi and Preston (200). We assume that the central bank is perfectly credible: the public believes and fully incorporates the central bank s announcements. Agents are uncertain about the economy (ˆπ and ŷ) and about the path of nominal interest rates (î). Communication by the central bank simplifies the agents problem in that it gives them information on how the monetary authority sets interest rates, that is, on the monetary policy strategy. Therefore: (i) under OP, the private sector has to make learning about the economy (ˆπ and ŷ) and about monetary policy (î); under TR, it needs to forecast just the economy but not the path of nominal interest rates, since the central bank announces 2 Given our model, the PLM will not contain any constant term. Adding a constant to the PLM leaves our main conclusions unaffected (results are available from the authors upon request). Moreover, with zero trend inflation the PLM does not contain lagged endogenous variables, which are present in both the PLM and MSV in the case of positive trend inflation.

13 its reaction function. 3 In case of TR, we incorporate the reaction function directly in the aggregate demand equation and the agents problem boils down to forecast inflation and output. This, as we will show, should help anchoring expectations by aligning agents beliefs with the central bank s monetary policy strategy Speed of Convergence As discussed by Ferrero (2007), the speed at which the expectations based on adaptive learning converge to their rational counterparts is important to characterize the dynamics of models with learning. Rapid convergence implies that agents will readily learn back the REE after the economy is hit by an exogenous shock. It follows that variables can be safely studied by focusing on their asymptotic behaviour, given by the MSV law of motion. Conversely, persistent departures from the REE may be quite common if the rate of convergence is slow, and the model will be dominated by its transitional dynamics. In this paper we analyse the speed of convergence under the assumption that a single REE exists and is E-stable, that is, we concentrate on the parametric region where determinacy and E-stability hold. For expectations to converge, E-stability requires that the Jacobian of the T -map evaluated at the REE has all eigenvalues smaller than in real part. In a similar fashion, the eigenvalues of the Jacobian directly determine the speed of convergence. Applying results from Benveniste et al. (990), Marcet and Sargent (995) provide a sufficient condition to ensure that parameters estimated with recursive least squares converge at root-t rate to a normal distribution centred at the REE. 4 More specifically, all the eigenvalues of the Jacobian are required to be less than /2 in real part. Hence, root-t convergence is granted only in a subset of the parametrizations that ensure E-stability. 3 Alternatively, TR can be defined as in Berardi and Duffy (2007). Under their specification, in the presence of TR the private sector adopts the correct forecast model: the structure of the PLM that coincides with the MSV solution, hence without the constant. Under OP, instead, they use an overspecified (with a constant) PLM. Incorporating even this specification in our model does not change significantly the results (available from the authors upon request). 4 In classical econometrics, root-t is the speed at which the mean of the distribution of the least square estimates converges to the true value of the parameters estimated. 2

14 When the largest real part is between /2 and, the estimates converge to the REE but no analytic results about the speed of convergence are available. According to Marcet and Sargent (995), as the effect of initial conditions fails to die out at an exponential rate, the estimates are intuitively expected to converge at a rate slower than root-t. Marcet and Sargent (995) suggest a numerical procedure based on Monte Carlo simulations for investigating the speed of convergence when the Benveniste et al. s (990) results do not apply (see also Ferrero, 2007; Ferrero and Secchi, 200). We leave a more detailed description of the procedure in Appendix B. To study the effects of policy and trend inflation on the speed of convergence, we will consider both the parametric conditions for root-t convergence and the results from the Monte Carlo procedure. 3 Results This section presents the main results of the paper. We first consider the standard case of a zero inflation target (i.e., zero inflation steady state), for which some analytical results are presented. We then move to the more general and realistic case of a positive inflation target. 3. Zero inflation target 3.. E-stability The economy is described by a standard NK model, characterised by the following equations: (5), (6) and (7). This model has been extensively studied in the literature, and Bullard and Mitra (2002b) provides us with the seminal contribution regarding learning in this setup. Here, we extend their analysis to the case of TR and OP, as defined above and in Preston (2006) and, in what follows, we mainly concentrate on the difference between these two cases. 3

15 The determinacy conditions are as in Ascari and Ropele (2009): φ π + φ y β κ >, (D) φ π + κ φ y > β κ. (D2) As known (see Woodford, 2003, p. 256), (D) is the long-run Taylor principle since ( β)/κ is the long-run multiplier of inflation on output in (7). So (D) can be interpreted as requiring that the long-run reaction of the nominal interest rate to a permanent change in inflation should be bigger than. By the same token, the second condition can be interpreted as requiring that the short-run reaction of the nominal interest rate should be bigger than the opposite of the long-run multiplier of inflation on output. The first natural question to ask is: Does TR allow a central bank to better anchoring inflation expectations with respect to OP? The answer is yes, as explained by the following proposition (proof in Appendix C.). 5 Proposition. The MSV solution is E-stable: (i) Under TR iff φ π + φ y βρ κ ( ) βρ > ρ ( ρ), (TR) κ φ y > ρ + βρ 2. (TR2) (ii) Under OP iff (TR) holds and φ y > (2 ρ) (Ψ(β, ρ) + κφ π), (OP) where Ψ(β, ρ) = (ρ + βρ 2) [(2 βρ) (2 ρ) κρ]. 5 Our conditions are slightly different from the ones in Bullard and Mitra (2002a) because our PLM does not have a constant term and we assume lagged expectations. This is also why our conditions involve the persistence parameter of the shock processes. In what follows, we will have conditions that should hold for both processes, so we should write ρ i for i = r, u. For the sake of brevity we just write ρ without subindex. 4

16 Figure visualizes how the five conditions above define the relevant regions for determinacy and E-stability in both cases of TR and OP in the space (φ π, φ y ) implied by Proposition. To grasp the main results, it is instructive to confine the analysis to the positive quadrant for (φ π, φ y ). To facilitate the reader, we highlight the main implications of this proposition, by using a numbered list of results. φ Y D Indeterminacy and E-stability under TR and OP κ φy > ( φπ) β Determinacy and E-stability under TR and OP φ OP > Ψ( β ρ) + κφ 2 ρ ( ) y, π TR -βρ -βρ φπ + φy > ρ ( ρ) κ κ β Determinacy and E-stability under TR E-unstable under OP TR2 D2 φy ρ ρβ φ > β κφ y φ Π > + 2 π Figure : Determinacy and E-stability regions under TR and OP from Proposition. Result. Transparency helps anchoring inflation expectations. Let s φ π, φ y 0. If the rational expectation equilibrium is determinate (i.e., (D) holds), then it is always learnable under TR, while this is not true under OP. This follows immediately from the fact that for φ y 0 the long-run Taylor principle (D) implies (TR) and, further, (TR2) is always satisfied. Note that, if the determinacy conditions hold, then the equilibrium is learnable under TR, but the contrary is not true, in contrast to Bullard and Mitra (2002b): E-stability does not imply determinacy. 6 Moreover, while determinacy implies E-stability under TR, this is not true under OP. The learnability region of the parameter space is thus smaller under OP than under TR. This 6 Again this is because we do not use the constant term in the PLM (see Section 4). 5

17 retraces a similar result in Preston (2006). Using his infinite horizon approach, Preston shows that under OP the Taylor principle is not sufficient for E-stability. However, even when agents form expectations just one-period ahead, requiring them to learn the policy rule is an important source of instability. Hence, the difference between Bullard and Mitra s (2002a) and Preston s (2006) results are not due to the different assumption regarding the forecast horizon of the expectation process under learning. We have shown that under OP the standard Euler equation approach delivers a similar result as in Preston (2006): the condition for E-stability are more stringent under OP. Besides, Preston (2006) already shows that TR yields Bullard and Mitra s (2002a) result that the Taylor principle is necessary and sufficient for E-stability. 7 So it seems that the two approaches deliver similar results. No matter the forecast horizon of the expectation process under learning, TR delivers E-stability if the REE is determinate, while OP generates more instability and requires more stringent conditions. Moreover, Figure clearly displays an important result: transparency is an essential part of the inflation targeting framework. Under pure inflation targeting, the interest rate rule responds only to inflation. In that case, OP would lead either to E-instability or to indeterminacy, depending if on whether the Taylor principle is satisfied or not. Thus, we can state: Result.2 A pure inflation targeting central bank needs to be transparent to anchor inflation expectations. As Eusepi and Preston (200) in the infinite horizon case, we were able to obtain an analytical expression for the E-stability condition under OP. This allows us to uncover the features of the monetary policy rule that make the equilibrium learnable under OP. For example, does responding more aggressively to inflation help to anchor inflation expectations under OP? Quite surprisingly, the answer is no. Result.3 Under OP, monetary policy should respond to the output gap. Let s φ π, φ y 0. If the rational expectation equilibrium is determinate (i.e., (D) 7 In our case, it is only sufficient, again because of different modelling assumptions about expectations formation. 6

18 and (D2) hold), it is learnable under OP iff φ y > (2 ρ) (Ψ(β, ρ) + κφ π). For the equilibrium to be learnable under OP, condition (OP) implies a lower bound on φ y that increases with φ π. In other words, an aggressive response to inflation can destabilize expectations under OP, unless it is counteracted by an increase in the response to output. Intuitively, with higher inflation expectations, agents fail to anticipate higher real rates under OP, even if the opaque central bank follows the Taylor Principle. As a result output rises leading to an increase in inflation that validates the initial surge in inflation expectations. The intuition for this result in well-explained by Eusepi and Preston (200, p ) in an infinite horizon framework. This is due to the fact that policy responds not to current, but to expected variables. 8 A strong response to expected inflation then tends to destabilize the economy, since monetary policy is responding too much and too late. The central bank can stabilize inflation expectations by responding relatively more to expected output, which is a more leading indicator of inflation. This result has two main implications. First, OP may be costly, since the optimal policy literature generally suggests that it is suboptimal to respond to the output gap. 9 Second, under OP it is not true that determinacy implies E-stability, as claimed by McCallum (2007). This finding echoes similar results in Preston (2006), Bullard and Mitra (2007), and Eusepi and Preston (200) in the infinite horizon framework. Finally, note there is a region of the policy parameter space (φ π, φ y ) where despite the rational expectation equilibrium being indeterminate, it is learnable under both TR and OP. And in this particular region, the Taylor principle is not satisfied. In general: Result.4 The Taylor principle is not a necessary condition for E-stability neither under TR nor under OP. Having analysed how the E-stability property depends on the policy parameters, the 8 Indeed, if we use contemporaneous expectations, there is no difference between the OP and TR case. See the robustness Section 4. 9 This is generally true for optimal policy in a simple NK model Woodford (see 2003). Moroever, Schmitt-Grohé and Uribe (2006) show it in the context of a medium-scale DSGE NK model à la Christiano et al. (2005) or Smets and Wouters (2003). 7

19 next obvious question is: Which structural parameters of the economy do affect the different ability of anchoring expectations under TR and under OP? That is, which features of the economy do make TR more necessary to anchor inflation expectations? First note that the slope of the NKPC (7), κ, is the key parameter to interpret determinacy. It is immediate to see that determinacy is more likely, the lower the slope of the NKPC (see condition (D)). Recall that κ = λ( + σ n ) and λ depends (inversely) on α. Hence, it follows that the higher the degree of price stickiness (higher α) or the lower the elasticity of the marginal disutility from working (σ n ), then the lower is κ, and the larger is the determinacy region. D and TR φ y φ> π Indeterminate and E-unstable under TR and OP Determinate and E-stable under TR and OP OP φ> κφ y π Determinate and E-stable under TR E-unstable under OP φ π Figure 2: The importance of the slope of the Phillips curve, κ. Similarly, κ is also the key parameter that affects the E-stability conditions. To clearly see this assume β = ρ =. 20 Then, the determinacy and E-stability conditions under TR collapse to the standard Taylor principle, because (D) and (TR) become φ π >, while 20 This is just for the sake of exposition without loss of generality. The same argument applies for values 0 < β, ρ <. 8

20 (D2) and (TR2) are always satisfied in the positive quadrant. The E-stability condition under OP (OP) simply reduces to φ y > κφ π. As evident from Figure 2, it follows that the higher the slope of the Phillips curve, the greater the difference between the E-stability regions under TR and OP. Hence: Result.5 The higher the slope of the Phillips curve, the smaller the E- stability region under OP. That is, the smaller the subset of feasible policies in the parameter space that yields E-stability for an opaque monetary policy. The degree of price rigidity is one of the key parameters in determining the slope of the Phillips curve. So one interesting implication is that TR is very important to stabilize expectations for central banks acting in countries with flexible prices (high κ). On the contrary, if the degree of price rigidity is high (a lower κ), then there is not much gain to be transparent, in terms of ability of anchoring expectations, so a central bank may choose to be opaque. Figure 3 shows the difference between the E-stability regions under OP calibrating the degree of price rigidity α respectively to 0.35, as estimated for the US in Christiano et al. (2005), and to 0.9, as estimated for the Euro Area in Smets and Wouters (2003). 2 The grey area represents the indeterminacy region, while the blue line and the red line delimit the E-stability regions for the cases of TR and OP, respectively. In the left panel, the OP line is steep, and the equilibrium is E-unstable for the values of (φ π, φ y ) usually considered in the literature. Only relatively low values of φ π and extreme values of φ y would guarantee an E-stable equilibrium. In the right panel, instead, the OP line appers very flat, almost horizontal, and lies below zero. 22 A determinate equilibrium would be E-stable under OP (as well as under TR). It is very important to get the intuition of why this is happening, since this will help also understanding the results in the next section. Recall that an opaque central bank needs to respond to output because responding to inflation can destabilize expectations 2 Remaining parameters are calibrated with the rather standard values used by Ascari and Ropele (2009): σ n = ; θ = ; β = 0.99; ρ r = ρ u = Note that κ affects both the slope and the intercept term Ψ(β,ρ), through Ψ(β, ρ). If α = 0.35, then Ψ(β,ρ) (2 ρ) Ψ(β,ρ) = 0.008, while if α = 0.9, then (2 ρ) = (2 ρ)

21 4 α = α = 0.9 E stable under TR and OP 3 3 φ y 2 φ y 2 E stable under TR, E unstable under OP E stable under TR and OP φ π φ π Figure 3: E-stability regions and price stickiness (indeterminacy region: grey area; transparency: blue line; opacity: red line). (see Result.3). However, this potentially destabilizing effect under OP depends on policy effectiveness. In a New Keynesian model, a change in the interest rate affects inflation, through output, via the Euler equation. It follows that the effectiveness of monetary policy rests on the link between output and inflation in the NKPC. If the latter is weak, then monetary policy is not very effective. Thus, the E-stability region under OP enlarges for higher values of price rigidities (low κ) and, for a given φ π, monetary policy could respond less to output to deliver E-stability. Note that in the simple case of β = ρ =, then κ determines the lower bound of the relative weight φ y /φ π necessary to ensure E- stability under OP. So, the less effective monetary policy is, the more the central bank can be opaque, simply because what it does matters less. For the same reasons, the difference between TR and OP decreases with the slope of the Phillips curve. The less monetary policy is effective, the less knowing or not knowing the path of the interest rate makes a difference. Intuitively, at the limit, if prices are completely rigid and κ = 0, monetary policy can not influence inflation, there is no advantage in knowing the policy and no difference between TR and OP. On the contrary, the interest rate path is a very valuable information when policy is very effective in affecting inflation. 20

22 3..2 Speed of convergence We now study the speed of convergence in the zero inflation target case. If the Jacobian of the T -mapping (i.e., DT c ( c), see Appendix B.2) has all the eigenvalues with real part lower than /2 then there is convergence at root-t speed, which is the fastest possible given least square learning. Otherwise, if the largest eigenvalue of DT c ( c) has real part above /2, the speed of convergence is expected to be lower, and one needs to resort to numeric simulations in order to explore it, as suggested by Marcet and Sargent (995). In any case, the slope of the T -map represented by the largest real part of the eigenvalues of the Jacobian DT c ( c) is intuitively the main determinant of the speed of convergence. So we will first provide some analytical results about the slope of the T -map, and then confront them with the outcome of simulations. Proposition 2 establishes the conditions for root-t convergence in the case of TR (proof in Appendix C.2). Proposition 2 (Transparency and Root-t Convergence). In the case of TR and zero trend inflation, the real part of the largest eigenvalue of the derivative of the T -map is lower than z iff (i.a) φ y > ρ( + β) 2z (8) when [ρ(β ) + φ y ] 2 < 4κ(φ π ρ) holds; (i.b) (ρ(β ) + φ y ) 2 + 4κ(ρ φ π ) < 2z ρ( + β) + φ y (9) when [ρ(β ) + φ y ] 2 4κ(φ π ρ) holds. Expectations converges to the REE at the rate root-t when (8) and (9) are evaluated at z = 0.5. Corollary. Assume φ y, φ π 0. If ρ = 0, the real part of the largest eigenvalue is always negative, so there is root-t convergence. More generally, assuming β, there exists a value ρ such that root-t convergence obtains for ρ < ρ. 2

23 The corollary follows from the two conditions above and it highlights the role of ρ. The persistence of the shock process affects the speed of convergence of expectations to REE: the higher is the persistence, the slower is generally the convergence, and in particular it always exists a sufficient low value of ρ that guarantees root-t convergence for any possible parameter combination in the E-stability and determinate region of the parameter space. More interesting, however, are the effects of the policy parameters and of price stickiness on the real part of the largest eigenvalue given by the following proposition. Proposition 3 (Transparency, Convergence and Policy). In the case of TR and zero trend inflation, the largest eigenvalue, i.e. λ, of the Jacobian of the T -map from PLM to ALM (i.e., DT c ( c) in Appendix C.) is real iff = [ρ(β ) + φ y ] 2 + 4κ(ρ φ π ) = (φ y, φ π ) 0 and complex otherwise. Moreover: (i) Effect of φ π : if λ is real, it is strictly decreasing with φ π ; if λ is complex, its real part does not depend on φ π ; (ii) Effect of φ y : if λ is real, λ φ y 0 φ π ρ; if λ is complex, its real part is strictly decreasing with φ y ; (iii) Effect of κ: if λ is real, λ κ 0 φ π ρ; if λ is complex, its real part does not depend on κ. The implications of the last two propositions are visualized in the left panel of Figure 22

24 4, 23 that displays the combinations of the policy parameters that deliver the same largest eigenvalue in the case of TR. The dark blue region features root-t convergence. It is enclosed by a horizontal line and a positively-sloped line, which respectively correspond to conditions (8) and (9) evaluated for z = The two conditions cross at a point that implies that DT c ( c) has four coincident real eigenvalues. Hence, the locus of point where (8) and (9) cross is the set of points such that the discriminant of the characteristic equation associated with DT c ( c) is equal to zero, that is (φ π, φ y ) = This defines the couples (φ π, φ y ) where the largest eigenvalue is at a minimum, that is the set of values of φ y that minimizes the real part of the largest eigenvalue for any given value of φ π. Before hitting the locus (φ π, φ y ) = 0, a higher response of monetary policy to inflation, i.e., higher φ π, decreases the largest eigenvalue, and hence tends to increase the speed of convergence. When it hits the locus (φ π, φ y ) = 0, further increases in φ π have no effects on the real part of the eigenvalue (point (i) in Proposition 3). The effect of a higher response to output, i.e., higher φ y, is instead ambiguous: a higher φ y decreases the real part of the eigenvalue below the (φ π, φ y ) = 0 locus, because in our calibration φ π > ρ, while it increases the real part of the eigenvalue above the (φ π, φ y ) = 0 locus (point (ii) in Proposition 3). The right panel of Figure 4 displays the case of OP. Since the real part of the eigenvalue is calculated just in the E-stability region and this region is smaller under OP, this is reflected into a more acute angle of the iso-eigenvalue curves. The two panels convey the same message. Recall from Figure the shape of the region that is both E-stable and determinate in the two cases of OP and TR. When the policy is such that the economy is close either to the E-stability frontier (lower frontier on the right) or the determinacy frontier (upper frontier on the left), then the real part of the T -map eigenvalue is approaching 23 Countour plots are based on the usual parameter calibration (see footnotes 2 and 27). In particular, in view of the corollary, it is worth stressing that we assume quite persistent shock processes with ρ = Appendix C.2 shows that if condition (8) holds then (9) can be written as: ( 2ρβ) [ + 2 (φ y ρ)] + 4κ(φ π ρ) > 0, which implies a linear relationship between φ π and φ y. 25 (A) and (A2) intersect at (φ π = ρ+(ρβ z) 2 /κ, φ y = ρ(+β) 2z) which is on the (φ y, φ π ) = 0 locus. 23