Research Discussion Paper – RDP 2025-02 Boundedly Rational Expectations and the Optimality of Flexible Average Inflation Targeting

There is agreement among economists that expectations are central to monetary policy design. However, there is disagreement over how to model expectations, and prescriptions about the optimal conduct of monetary policy can be sensitive to assumptions about expectation formation. We formulate a general framework for expectation formation that nests or approximates alternative expectation theories. We use this general environment to derive optimal target criteria under commitment for a welfare-maximising central bank given different assumptions about the constraints that it faces, such as an inability to observe contemporaneous economic conditions or the zero lower bound (ZLB) on nominal interest rates. We characterise the dependence of optimal policy prescriptions on features of the expectation formation process and provide recommendations that are independent of how expectations are formed.

We find that if a central bank can credibly commit to a path for the policy rate, then the optimal target criterion for a wide range of expectation formation theories is always a form of flexible average inflation targeting (FAIT). We show that the average in FAIT should be a weighted average of current and past inflation with declining weights over time. Monetary policymakers should respond most forcefully to current inflation, while also committing to partially ‘make up’ for past inflation misses. But the weighted average inflation target is also flexible because when a policymaker's ability to achieve this target is in question, it should seek to influence expectations more aggressively than usual. A policymaker should commit to stronger-than-usual make-up policy when they are unable to control the output gap to implement the current target, such as when they have imperfect information about the state of the economy or the ZLB binds. In addition, they should pre-emptively deviate from the target to steer expectations favourably when they anticipate future constraints, such as drifting inflation expectations or the ZLB. Especially with respect to the ZLB constraint, policy is a ‘use it or lose it’ proposition for all the expectations theories that we capture.

Central to understanding the advantage of a FAIT policy is recognising a symmetry between (i) the history dependence that a central bank should optimally commit to when agents' expectations are near the full information rational expectations (FIRE) benchmark, and (ii) the history dependence in agents' own expectations that emerges under adaptive learning. In the former case, it is optimal to manage agents' expectations by committing to make up for past misses to targets by setting current policy based on a weighted average of past outcomes. In the latter case, it is optimal to respond preemptively to expected future outcomes by influencing agents' expectations, which are themselves a weighted average of past outcomes.^[1] In both cases, optimal policy requires flexible deviations from the weighted average target to more aggressively influence expectations when policy is constrained. Therefore, despite the different rationales for policy in each case, the optimal policy shares several key features.

Our agnostic approach to expectations allows us to derive a more broadly applicable optimal policy framework, encompassing much of the existing literature's results, while also characterising novel environments not yet explored. Specifically, we model expectations as a weighted average of adaptive learning and FIRE, where the rational expectations take into account the learning expectations, and where the desired path of the policy rate is credible and known to all. We show that by varying the weight placed on rational expectations relative to learning, and by varying the speed of the learning, that we capture relevant dimensions of several types of expectation formation models within a standard New Keynesian economy. The expectation models include adaptive learning as in Evans and Honkapohja (2001) and Preston (2005), behavioral expectations as in Gabaix (2020), and key elements of level-k reasoning as in Angeletos and Lian (2018), Farhi and Werning (2019), Angeletos and Huo (2021) and Evans, Gibbs and McGough (forthcoming). Each of these different expectations models lies as a special case that we can recover or closely approximate within our more general setting, and allows for characterisation of optimal policy in novel settings.

The general target criteria that we derive constitute a formal theory for a FAIT policy framework. We start from the primitive that monetary policy has a statutory mandate to stabilise inflation and real activity. Policy, therefore, seeks to minimise a loss function over squared deviations in inflation from a target and squared deviations in the output gap, which captures the welfare theoretic loss function for the representative household. The general target criterion we derive for this loss function consists of two distinct pieces: 1) a weighted-average inflation target (WAIT), and 2) prescriptions for flexible deviations from WAIT that change based on the speed of learning or the constraints faced by the central bank. In the weighted average part of FAIT, the weights that a policymaker places on current versus past observations are determined by how rational private sector expectations happen to be and decline geometrically over time. When there is learning, optimal policy requires flexible deviations from what is implied by WAIT alone. The effect of shocks and policy today may be extrapolated far into the future beyond their natural or intended duration. This creates the rationale for deviating from the policy implied by past data alone to pre-empt expected deleterious drifts in agents' expectations caused by shocks, or to pre-emptively generate favourable drifts in expectations through policy.

Flexible deviations from WAIT are also required when information constraints or the ZLB prevent the central bank from implementing its desired level of demand in the economy. In these situations, again, the central bank should be flexible by more aggressively steering expectations. It should commit to stronger make-up policy than usual, and it should act pre-emptively to influence expectations if it expects the ZLB might bind in the future. This is the rationale behind forward guidance policy in Eggertsson and Woodford (2003), where under FIRE the promise of future policy alleviates the constraints on policy today. We show that this type of policy is desirable regardless of how inflation and output expectations are formed.

We demonstrate the robustness of the shared features of optimal policy by running two horse races comparing a range of different inflation and output gap policy targets. We first show that we can approximate the optimal target criteria using simple weighted-average criteria over past inflation and output gap outcomes, which are superior to price-level targeting, inflation targeting, or arithmetic average inflation targeting using a fixed window when agents are boundedly rational. We then compare how these competing criteria perform when policymakers do not know how expectations are formed. Specifically, we choose a single calibration for each different target criteria to maximise average welfare across different economies with expectations ranging from FIRE to full adaptive learning. We find that simple implementations of FAIT that target geometrically declining averages of inflation (WAIT) and the output gap (which captures some aspects of flexibility required in optimal policy) are the most robust, generating similar losses to the fully optimal policy regardless of how expectations are formed.

1.1 Literature review

We take a novel approach to the study of FAIT by using target criteria. Much of the policy work on this topic has focused on the study of modified interest rate rules that include averages of past inflation as arguments. For example, papers cited in strategy reviews by the Federal Reserve (e.g. Arias et al 2020), European Central Bank (e.g. Work Stream on the Price Stability Objective 2021) and Bank of Canada (e.g. Dorich, Mendes and Zhang 2021) have taken this approach.

There are several drawbacks to interest rate rule-based analysis. Svensson and Woodford (2005) argue that interest rate rules are a fragile and non-transparent way of specifying a policy framework.^[2] Interest rate rules are model specific and the mapping between policy framework and an interest rate rule is not clear cut. In some cases, different rules can generate the same equilibrium outcomes. For example, Eskelinen, Gibbs and McClung (2024) show that unconditional optimal policy in the standard New Keynesian model can be implemented with an interest rate rule that responds to a lagged interest rate term, similar to interest rate smoothing, or to a weighted average of past inflation, similar to some average inflation targeting specifications in the literature. In other cases, slightly different rules generate drastically different equilibria. For example, Honkapohja and McClung (2024a) point out several examples of the fragility of interest rate rule specification that try to capture average inflation targeting (AIT) under rational expectations and adaptive learning. They show that small changes to the number of periods included in the inflation average can lead to drastically different policy outcomes. It is therefore difficult to draw robust conclusions about policy frameworks from a comparison of interest rate rules.

In contrast, target criteria are general. They do not depend on the statistical properties or the number of economy-wide shocks that the central bank faces. They depend only on how beliefs are formed and on the constraints the central bank encounters when implementing its desired policy. In addition, policy is specified as a target for endogenous variables, that is, a set objective for the evolution of economic outcomes policymakers hope to achieve. Target criterion are inherently less constraining than interest rate rules because they allow (subject to the ZLB) any choice for the path of policy rate necessary to achieve their goals. Communication of policy in such a framework is, therefore, forward looking and data dependent, which reflects the way policymakers actually communicate.

Optimal target criteria for both unconstrained and constrained policymakers have been widely studied under FIRE. The theoretical justification for inflation targeting (e.g. Giannoni and Woodford 2005), price level targeting (e.g. Giannoni 2014) and inflation forecast targeting (e.g. Svensson and Woodford 2005) policy frameworks rest on this work. In addition, Eggertsson and Woodford (2003) extend optimal target criteria to the case of optimal policy at the ZLB, and others have studied optimal policy under imperfect central bank information (e.g. Clarida, Galí and Gertler 1999; Woodford 2010). We extend these works by studying a more general model of expectations. We show that some conclusions from the FIRE analysis are knife edge. Small departures from the FIRE assumption imply a new optimal policy framework: FAIT.

Optimal target criteria for an unconstrained policymaker have been studied in the adaptive learning literature, such as in Molnár and Santoro (2014), Eusepi et al (2018) and Eusepi, Giannoni and Preston (2024). These papers establish that optimal policy shares some features with the optimal target criterion from the FIRE analysis, in which it is optimal to engineer periods of price level overshooting or make-up policy in certain circumstances. We extend these works by approximating them within our general model of expectations and by deriving optimal target criteria in settings with imperfect information and the ZLB. We show that the similarity in optimal policy under both adaptive learning and FIRE carries over to other general forms of bounded rationality, and can be captured by a FAIT policy framework.

A related set of studies derive optimal target criteria in the unconstrained case under other kinds of deviations from FIRE. Gabaix (2020) and Benchimol and Bounader (2023) derive optimal policy when agents have myopic expectations and over-discount the future. In Dupraz and Marx (2023), agents have finite planning horizons and are similarly myopic, but also learn about long-run outcomes, which adds a backward-looking component to expectations. Gasteiger (2014) and Gasteiger (2021) analyse optimal policy in a heterogeneous expectations setting, where some agents have rational expectations and others are adaptive.^[3] We can approximate the key elements of these models for expectations within our general framework, and extend the optimal policy analysis by deriving optimal target criteria in the constrained case, when the central bank has imperfect information or the ZLB can bind.

Outside of the unconstrained case, there are several papers that explore optimal policy at the ZLB in non-FIRE settings. Eusepi, Gibbs and Preston (2024) and Evans et al (forthcoming) both study optimal forward-guidance policy when agents must learn about the general equilibrium implications of policy. The former takes an adaptive learning approach, while the latter combines adaptive learning with level-k thinking. Dupraz, Le Bihan and Matheron (2024) analyse make-up policy when agents have finite planning horizons. Budianto, Nakata and Schmidt (2023) allow for both FIRE and a Gabaix (2020) style of myopic expectations, and compare delegated loss functions for a central bank optimising under discretion. In contrast to these papers, we derive optimal target criteria under commitment analytically, which shows that a particular kind of FAIT framework can capture the features of optimal policy at the ZLB that this prior work explores.

Our general framework also lets us approximate other models for expectations that feature some kind of myopia and a backward-looking learning process, for which optimal target criteria have not been derived. For example, Angeletos and Lian (2018), Farhi and Werning (2019) and Angeletos, Huo and Sastry (2021) all propose models with some combination of level-k reasoning, heterogeneous private information, and/or learning. Importantly, we characterise the unconstrained, imperfect central bank information, and ZLB cases under all of these different expectations assumptions. Because our framework nests a range of different specifications, we are also able to find the robustly optimal policy framework, in circumstances where policymakers do not know the true model for expectations.

Finally, there is a related adaptive learning literature that examines whether optimal target criteria derived under FIRE generate equilibria that are learnable. This literature finds that it depends on how the optimal target criteria is implemented. Optimal target criteria do not imply unique interest rate rules. Evans and Honkapohja (2003), Preston (2006), Orphanides and Williams (2007) and Evans and McGough (2010) show that different interest rate rules that react to different endogenous quantities, which implement the same equilibrium under FIRE, may have different stability properties under least squares learning. Honkapohja and Mitra (2020) study price level targeting under learning and varying credibility. The optimal targeting criteria we study here, however, are derived taking learning into account as a constraint, making expectational stability an endogenous concern of the policymaker. Our robust policy analysis finds versions of the FAIT framework that ensure expectational stability even when the policy rule is misspecified.

We study the standard New Keynesian environment of Woodford (2003) with sticky prices and monopolistically competitive firms. However, we depart from the standard environment by assuming that expectations need not be rational and may encompass a wide range of beliefs formation strategies. To accommodate this generality, we log-linearly approximate aggregate output and inflation dynamics as

where x_t is the output gap, ${\pi }_t$ is inflation, i_t is the nominal interest rate controlled by the central bank, $r_t^n$ is an exogenous real rate of interest (which could reflect movements in demand and productivity), u_t is a cost-push shock, and ${\widehat{𝔼}}_t$ represents potentially non-rational expectations. Equation (1) is the infinite-horizon IS curve, which captures household demand under arbitrary beliefs. Under FIRE, this equation collapses to the familiar two-period representation. However, the form here distinguishes the effect of longer-term income and real interest rate expectations on consumption decisions, which are critical when the central bank is constrained in the present.^[4]

For Equation (2), the Phillips curve, we use the common two-period approximation, which approximates the effect on pricing decisions of expected inflation and output gaps in all future periods using just the one-period-ahead inflation expectation. Appendix C.1 provides the full infinite-horizon form of the Phillips curve (also called the anticipated utility solution in the adaptive learning literature; under FIRE, it is equivalent to the two-period form), and Appendices C.2 to C.5 replicate all of our main results in that setting. We choose to use the two-period approximation for the Phillips curve in the main analysis because the key results are not sensitive to the choice (in contrast to the choice of IS curve representation) and are easier to interpret. It also allows us to more closely nest/approximate the reduced-form Phillips curve formulations of the myopia and anchoring model of Angeletos and Huo (2021), the hybrid New Keynesian Phillips curve, and older works in the adaptive learning literature such Evans and Honkapohja (2003) and Molnár and Santoro (2014).

Beliefs of the private sector. We put forward a general notion of belief formation. We do not intend for our formulation to be a theory of expectation formation on its own; however, many papers have done so using similar approaches, such as in Brock and Hommes (1997), Branch and McGough (2009) and Cole and Martínez-García (2023). Our goal is to write beliefs in such a way that for specific calibrations we recover distinct expectation theories put forward in the literature.

We assume that some fraction $\lambda \in$ [0,1] of agents possess FIRE, and a fraction 1 – $\lambda$ are adaptive learners, in that they use a statistical model to forecast future outcomes by extrapolating from past observations.^[5] Aggregate expectations of future inflation and output gaps are

where ${𝔼}_t$ is the FIRE forecast and ${𝔼}_t^l$ denotes the learners' forecast.

We assume that adaptive learners forecast inflation and the output gap using an unobserved components state-space model, which they estimate using a steady-state Kalman filter. Inflation and the output gap are assumed to be driven by an unobserved persistent component, with an autoregressive coefficient of $\rho$, and an iid component. Learners use observations up to period t – 1 to estimate the persistent component, where ${\omega }_{t-1}^z,z\in \left\{\pi ,x\right\}$, denotes period t's state estimate. Learners use this estimate to forecast future variables $z_T,T\ge t+1$. Their expectations for ${\pi }_T$ and $x_T,T\ge t+1$, are therefore given by

where the steady-state gain coefficient g reflects their beliefs of the relative variance of the perceived unobserved persistent and transitory shocks in the state-space model. This procedure implies the updating rules

We assume for simplicity that the parameters relevant for beliefs – $\lambda ,\rho$ and g – are the same for both households and firms, and for both inflation and the output gap.

For expectations of the nominal interest rate, we assume that everyone has rational expectations over its path.^[6] This is equivalent to assuming that the policymaker communicates its expected policy rate path each period, and that path is believed by all agents. The distinction between the formation of nominal interest rate expectations and the formation of output gap and inflation expectations is common in the existing literature on make-up policy.^[7] If the policymaker communicates its expected path for the nominal interest rate, then this path is immediately available to households and firms, whereas determining the future output gap and inflation implications of that path involves much more sophisticated general equilibrium reasoning. Even if the central bank's communication of its interest rate expectations is not perfect, the effect of interest rate expectations on household decisions will often flow through fixed rates and asset prices, which are determined by sophisticated financial market participants who are highly forward looking and likely to form similar expectations to the central bank.^[8]

Beliefs of the policymaker. We assume throughout that the central bank possesses FIRE and seeks to maximise the standard welfare function

which is a quadratic approximation to household welfare when $\alpha =\kappa /\theta .$ When choosing policy, the policymaker takes household decisions and beliefs as constraints:

The policymaker's problem. The optimal policy problem is to choose the sequence ${\left\{{\pi }_t,x_t,{\omega }_t^{\pi },{\omega }_t^x,i_t\right\}}_{t=0}^{\infty }$ that maximises the welfare function (9) subject to the constraints (10) to (13) imposed by private sector equilibrium behaviour. We assume that the policymaker is credible and pursues optimal policy from a timeless perspective, in that they do not seek to exploit the exogeneity of the initial conditions. Instead, the policymaker in the initial period acts according to the rule that would have been optimal had they committed to it in advance.^[9]

Nesting/approximating other models. The pay-off for the particular way we model beliefs is in nesting the structural equations implied by most of the prominent expectations theories in the literature. For example, when

• FIRE $\left(\lambda =1\right)$ : the system reduces to structural equations implied by full information rational expectations:

• myopia and level-k (0 < $\lambda$ < 1 and g = 0): the system reduces to a model that is ‘over-discounted’ such as in Angeletos and Lian (2018) or Gabaix (2020):

• hybrid New Keynesian Phillips curve (0 < $\lambda$ < 1 and g = 1): the system reduces to a model with a hybrid New Keynesian Phillips curve, or one that features elements of myopia and anchoring as in Angeletos and Huo (2021):

• adaptive learning ($\lambda$ = 0 and 0 < g < 1): the system reduces to a model of adaptive learning:

By nesting these competing assumptions, we can compare optimal policy within a shared framework.

We start with the unconstrained optimal policy problem given by Equations (9) to (13). The policymaker is free to choose any policy today and into the future that they would like. Appendix A.1 contains the full derivations for all results; here we summarise the key equations to build intuition.

Monetary policy in this case is a powerful tool even with bounded rationality because the path of interest rates is common knowledge. If the central bank only faces the demand shock, then it is able to perfectly stabilise the economy as in the standard FIRE case.

Result 1 (IS curve irrelevance). The IS curve is not a constraint when the path of interest rates is known and credible. Demand shocks (and other shocks that do not affect the gap between the efficient and flexible-price levels of output, such as productivity shocks) may be perfectly offset with current and promised changes in the interest rate, generating no welfare loss.

The trade-off for the central bank, therefore, lies with ‘cost-push’ shocks (i.e. shocks that do affect the gap between the efficient and flexible-price levels of output).

To characterise optimal policy for cost-push shocks, we take the first-order conditions with respect to ${\pi }_t$ and x_t for the central bank's policy problem and eliminate the Lagrange multiplier on the Phillips curve:

where L is the lag operator and ${\mu }_t{}^{\omega \pi }$ is the Lagrange multiplier on the updating rule for learners' inflation expectations. The term ${\mu }_t{}^{\omega \pi }$ represents the benefit of increasing learners' inflation expectations and is determined by the first-order condition with respect to ${\omega }_t^{\pi }$:

Solving Equation (15) for ${\mu }_t^{\omega \pi }$ gives

where $\text{Ω}\equiv \frac{{\beta }^2{\zeta }_2^{-1}}{\rho \beta \left(1-g\right)+\left(1-\lambda \right)g{\rho }^2{\beta }^2}$ and ${\zeta }_1$ and ${\zeta }_2$ are the roots of the characteristic polynomial associated with Equation (15).^[10]

Result 2 (Unconstrained optimal target criterion). The optimal target criterion is

which is composed of two distinct parts:

1. a weighted-average inflation target (WAIT)
2. pre-emptive policy actions based on expected future trade-offs between inflation and output gaps due to drifts in learning beliefs.

The WAIT term represents a commitment to the forward-looking rational agents. The policymaker promises to adjust the output gap in response to a weighted average of past inflation. Shocks today that generate misses in the inflation target imply future policy adjustments, lessening the equilibrium impact of any shock.

The pre-emption term represent the management of backward-looking adaptive learning expectations and is present if g > 0. The policymaker forms a belief about how inflation expectations are likely to evolve in the future given current and past misses in inflation, which are propagated through the adaptive learning component of beliefs. Drifts in inflation change the trade-off between future inflation and output gaps. Additional policy today can pre-empt future undesirable movements in this trade-off, or engineer more favourable ones.

A key feature of the optimal target criterion (17) is that it encapsulates the target criteria from a range of different primitive assumptions about how agents form beliefs as special cases. Table 1 summarises the target criterion for the special cases discussed in the previous section. Optimal policy in this case is WAIT + pre-emption. The table, however, includes a fixed belief case of $\lambda$ = 0 and g = 0, which is of interest because it recovers the same target criterion that is implied under FIRE for discretionary policy. This case reveals that so long as expectations can be influenced, that is, g > 0 and/or $\lambda$ > 0, it makes sense to deviate from the standard FIRE optimal discretion policy, that is, from inflation targeting.

3.1 Mechanics of optimal policy

To see how policy works in practice in response to a shock, let the cost-push shock follow a first-order autoregressive process such that E_tu_T depends only on u_t for $T\ge t$. Recall from Equation (16) that the Lagrange multiplier on learners' inflation expectations ${\mu }_t^{\omega \pi }$ is a function of past and current expectations of inflation at different horizons. In Appendix A.1, we show that, in the optimal equilibrium solution, this multiplier ${\mu }_t^{\omega \pi }$ can be written

for some positive constants a_x, $a_{\omega }$ and a_u. Sensibly, optimal policy aims to lower learners' inflation expectation when inflation itself is high, when their expectations ${\omega }_t^{\pi }$ are high, or when the cost-push shock is positive. This desire for lower inflation though is tempered by a desire to close the output gap, which gives rise to the positive relationship with x_t. Using Equation (18), we can write the optimal target criterion (17) (in equilibrium) as

where $\gamma \equiv \frac{\lambda }{1-\left(1-\lambda \right)\rho g{a}_x\frac{\kappa }{\alpha }},\varphi \equiv \frac{1+\left(1-\lambda \right){\rho }^2{g}^2{a}_{\omega }}{1-\left(1-\lambda \right)\rho g{a}_x\frac{\kappa }{\alpha }},v_{\omega }\equiv \frac{\left(1-\lambda \right){\rho }^{2}g\left(1-g\right)a_{\omega }}{1+\left(1-\lambda \right){\rho }^2{g}^2{a}_{\omega }}$ and $v_u\equiv \frac{\left(1-\lambda \right)\rho g{a}_u}{1+\left(1-\lambda \right){\rho }^2{g}^2{a}_{\omega }}$.

Equation (19) reveals that optimal policy adjusts the output gap in response to an exponentially weighted average of current and past inflation, ${\omega }_t^{\pi }$, and cost-push shocks, where the weight on lag j is ${\gamma }^j$. In addition, relative to FIRE, the central bank increases its response to current inflation while also responding to learners' inflation expectations ${\omega }_{t-1}^{\pi }$ – pre-emption – because both affect future inflation due to drifting inflation expectations. The central bank also must respond separately to the cost-push shock u_t (beyond its effect on current inflation). How aggressive or accommodative policy is with respect to past inflation, expectations, or the shock depends on $\lambda$ and g, that is, how quickly these interventions show up in expectations.

Figure 1 shows the optimal response to a cost-push shock as $\lambda$ is varied compared to policy under an inflation-targeting (IT) regime ($\gamma =v_{\omega }=v_u=0$ and $\varphi =1$). IT here is equivalent to the optimal targeting criteria implied under rational expectations discretion.

Optimal policy for a wide range of $\lambda$ s implies initially a muted response followed by a more aggressive policy in the medium run compared to IT. In the high $\lambda$ cases, policy reflects the make-up quality of WAIT. Higher inflation now means a higher interest rate in the future even as inflation falls. The expectation of this policy is capitalised into current expectations blunting the impact of the shock and lessening the need to initially raise interest rates aggressively.

In the low $\lambda$ cases, policy reflects the different way a shock propagates when there is learning. Backward-looking agents' expectations are less sensitive to the shock on impact because their expectations are anchored to the past. This means the shock has less of an impact initially, which allows for a muted interest rate response by policymakers. However, the learning process endogenously propagates the shock as expectations are revised. This leads to a growing inflation impulse over time because of rising ${\omega }_t^{\pi }$. The optimal policy response to this growing impulse is a higher interest rate in the medium run to constrain inflation and output. The optimal policy, therefore, for a wide range in $\lambda$ follows the same prescription: muted on impact and more aggressive over the medium run. This is true even though the rationale for why the central bank should enact such a policy changes as $\lambda$ is varied.

Optimal policy only departs from this pattern when $\lambda$ is close to zero. Here, the WAIT term is nearly absent and pre-emption is the dominant feature of policy. Stronger policy is required in these cases initially to blunt the propagation of the shock in beliefs. This represents the flexibility required by optimal policy in FAIT. If a policymaker expects expectations to drift, that is, become unanchored, in response to a shock, then more aggressive policy is required now.

For moderate values of $\lambda$, however, the pre-emption motive is secondary to WAIT. Figure 2 illustrates how pre-emption affects policy when $\lambda$ = 0.5 as we vary g making expectations more or less responsive to the shock. The figure shows WAIT ($\gamma =\lambda ,\varphi =1$ and $v_{\omega }=v_u=0$) compared to optimal policy. The difference between WAIT and optimal policy is pre-emption. The WAIT policy delivers lower interest rates and real rates in each case. Pre-emption on the other hand increases the current response as g increases to counter the greater influence of the shock in causing drifts in learned expectations.

There are of course other scenarios where even with moderate values of $\lambda$ pre-emption becomes more central. For example, when shocks are very persistent or the central bank expects to be constrained in the future such as when the ZLB binds, which we explore in Section 4.

3.2 The optimal average inflation target

The continued importance of WAIT relative to pre-emption shown in Figure 2 when $\lambda$ is small is surprising. The reason it occurs is that the expectations of adaptive learners in our set-up evolve as a weighted average of current and past inflation,

Pre-emption, therefore, is an optimal response to a weighted average of past inflation. It turns out that that weighted average implied by the pre-emption component of optimal policy in equilibrium shares many features with the original WAIT component of optimal policy. In fact, it is possible to rewrite optimal policy as a simple WAIT policy with different weights that share a similar geometric decay over time.

To illustrate, substitute the weighted-average expression above for adaptive learning beliefs into the rewritten optimal target criterion (19) and rearrange:

where $\Psi \equiv 1-\frac{\rho g}{\rho \left(1-g\right)-\gamma }{v}_{\omega }$. The average inflation measure that a policymaker should target is a weighted average of two exponentially weighted averages of current and past inflation.

Figure 3 illustrates the similarities in the weights by plotting the term in front of $\pi$ from Equation (20) for different values of $\lambda$ and g. On the left side, we plot the optimal weights for different values of $\lambda$ when g = 0 and ${\omega }_t^{\pi }$ = 0. This calibration reflects models with myopia like Gabaix (2020) and it has no pre-emption component to policy. The high weight placed on current inflation reflects $\alpha$, the welfare weight on the output gap, which here is set to the welfare theoretic value of $\kappa /\theta$. Higher values of $\alpha$ scale these weights profiles down while lower values raise them. When $\lambda$ = 1, we have the optimal policy under FIRE: price level targeting. Equal weight is given to all past inflation outcomes. When $\lambda$ = 0, there is no value to commitment and we get the rational expectations discretion outcome.

On the right side of Figure 3, we add a positive gain (g > 0) activating the pre-emption term. The optimal target criterion again responds to a weighted average of inflation. Pre-emption causes the weights to be higher depending on the specific level of g and $\lambda$. However, the general profile of the weights remains the same with a geometric decline over time. Optimal policy tracks a qualitatively similar weighted average of inflation when most agents have FIRE expectations and when most do not. In Section 5, we consider whether there is a single calibration for this weighted average that performs well regardless of how expectations are formed.

In addition, it is important to note that the profiles of optimal weights are very different to the arithmetic averages that many researchers have studied when modeling average inflation targeting such as in Eo and Lie (2020) or Honkapohja and McClung (2024b). Arithmetic averages imply uniform weights over a fixed number of lags and zero thereafter. We show in Section 5 that arithmetic averages do a poor job approximating the optimal weights, while simple geometric averages approximate optimal weights well, even when the geometric weights are significantly misspecified relative to the optimal weights.

We now turn to the case where the policymaker faces a constraint that prevents it from implementing a desired level of the output gap period by period. We consider two ways in which this might occur: 1) imperfect information, and 2) the zero lower bound on nominal interest rates. Again, Appendix A.2 provides detailed derivations, while we focus here on the key equations and intuition.

4.1 Imperfect central bank information

Let $I_t^{CB}$ denote the central bank's information set in period t and let I_t denote the full information set. Let z_T|t denote the best estimate of any variable z_T conditional on $I_t^{CB}$, while ${𝔼}_t{z}_T$ continues to denote the expectation conditional on the full information set I_t. We now assume

In other words, the central bank does not possess all information in the current period necessary to implement optimal policy. The nominal interest rate i_t must be set each period based on incomplete information $I_t^{CB}$, that is,

For our purposes, it does not matter what information is missing. The key is that any missing information relevant to policy represents a constraint on current policy choices.

The optimal policy problem is to choose the sequence ${\left\{{\pi }_t,x_t,{\omega }_t^{\pi },{\omega }_t^x,i_t\right\}}_{t=0}^{\infty }$ that maximises the welfare function (9) subject to the constraints (10) to (13) imposed by private sector equilibrium behaviour and the informational constraint (21).^[11]

The first-order condition of the policymaker's problem with imperfect information with respect to i_t is

where ${\mu }_t^i$ is the Lagrange multiplier on the information constraint (21). Condition (22) shows that the benefit of lowering the nominal interest rate is given by the benefit of raising the output gap in the current period, and in previous periods through the fact that agents anticipate future policy responses. Condition (23) shows that the central bank should try to set the nominal interest rate so that there is no benefit to increasing or decreasing it, given the information on hand. The main difference from the unconstrained case is that demand shocks are no longer irrelevant.

Result 3 (IS curve relevance). The central bank cannot perfectly offset shocks to the IS curve when $I_{t-1}\subset I_t^{CB}\subset I_t$.

In the unconstrained case, when $I_t^{CB}=I_t$, then $u_t^i={\mu }_{t|t}^i=0$ and so $u_t^x=0$. A shock to the IS curve is irrelevant. When $I_t^{CB}\subset I_t$, the best the central bank can do is ${\mu }_{t|t}^i=0$, and so ${\mu }_t^i\ne 0$ and $u_t^x\ne 0$ in general, which makes the IS curve a constraint on policy.

To see how this affects optimal policy, we take the expectation of (22) conditional on $I_t^{CB}$ and using ${\mu }_{t-j|t}^x={\mu }_{t-j}^x$ for all $j\ge 1$ gives

which implies that the central bank commits to an i_t that balances the estimated costs of an off-target output gap in period t, ${\mu }_{t|t}^x$, against the reduced losses in the periods before t due to expectations of $i_t,-{\displaystyle {\sum }_{j=1}^t{\mu }_{t-j}^x}$. Equation (24), therefore, describes a new intertemporal trade-off, where the central bank cannot set the desired interest rate it would like, ex post, while considering ex ante how its corrections for any errors it makes interact with private sector expectations.

Taking the first-order conditions with respect to ${\pi }_t$ and x_t, eliminating the Lagrange multiplier on the Phillips curve, and substituting in ${\mu }_t^x=\sigma \left({\mu }_t^i-{\mu }_{t-1}^i\right)$ from (22) gives

which generalises Equation (14).

The information constraint implies three additional terms. The new term in ${\mu }_t^i$ on the left-hand side is zero in expectation and represents the extent to which a shock causes the central bank to ‘miss’ its intended target for the output gap. The first new term on the right-hand side – in ${\mu }_{t-1}^i$ – is an additional commitment requirement of optimal policy. It indicates how the central bank should respond to past misses caused by unexpected shocks, that is, additional make-up policy. The commitment to respond to misses in the future shifts current expectations, thereby dampening the costs of the shock. In this way, commitment under imperfect information acts as a kind of ‘automatic stabiliser’. Finally, the Lagrange multipliers on learners' inflation and output gap expectations are now

The new components in these expressions, that is, the terms in ${\mu }_t^i$, dampen the extent of optimal make-up policy, by accounting for its effect on learners' future expectations.

Substituting the Lagrange multipliers (26) and (27) into (25), we can derive the optimal target criterion under information constraints.

Result 4 (Information constrained optimal target criterion). Recall from Equation (17) that the optimal target criterion under full information is ${\tilde{\pi }}_t=0$, where ${\tilde{\pi }}_t$ is defined as

Then the optimal target criterion under imperfect central bank information is

with $\Theta \equiv \frac{\left(1-\beta \right)\beta }{1-\beta \rho }+\text{Ω}\frac{1}{\beta \left(1-\beta \rho \right)}\frac{\kappa }{\sigma }-\text{Ω}\frac{\lambda }{\beta }\frac{1}{1-{\zeta }_2^{-1}\lambda }\frac{\kappa }{\sigma }$ and

The optimal target criterion under imperfect information (28) generalises the unconstrained target criterion (17). It differs from (17) in two ways: (i) the central bank does its best to implement it given its information set $I_t^{CB}$ (i.e. it can only control ${\tilde{\pi }}_{t|t}$, not ${\tilde{\pi }}_t$), and (ii) its target for ${\tilde{\pi }}_{t|t}$ is not always zero, but adjusts based on past errors in implementing optimal policy – ${\text{Δ}}_t$ – due to the information constraint.

The optimal target criterion though remains a flexible average inflation target. The difference here is in the definition of flexibility. The central bank still retains the same WAIT and pre-emption incentives but now must commit to make up for any errors in implementing policy that arise from the information constraint.

Make-up policy works through three channels represented by the three terms in the first line of Equation (29). The first term reflects the effect on inflation expectations of a commitment to make up for an error. Rational forward-looking agents respond today to the expected make-up policy they anticipate will occur, which dampens the impact of the shock. The second term represents the effect of this commitment on output gap expectations, which affect the current output gap through the permanent income hypothesis with (1 – $\beta$) representing the marginal propensity to consume from a change in current period income. The final term represents the effect of commitment on nominal interest rate expectations. Committing to offset misses shifts the expected nominal yield curve and further ameliorates the effect of any unexpected shock. The first two terms are both multiplied by $\lambda$, which signifies their dependence on forward-looking expectations. By contrast, the third term is multiplied by one since all agents incorporate the optimal path of interest rates into their beliefs.

The second line of Equation (29) reflects the change in the cost and benefit to make-up policy when there are learning expectations. Policy actions, like shocks, are instantiated into learning agents' beliefs when output and inflation respond and must be accounted for when implementing policy. To aid intuition, consider the case when prices are fixed, $\kappa$ = 0. The optimal target criterion reduces to

where $x_t^{*}$ is

with ${\text{Δ}}_t\equiv x_t^{*}-x_t$. The central bank should set its best estimate of the output gap $x_{t|t}$ equal to the moving target $x_t^{*}$, which is determined by the previous period's miss, ${\text{Δ}}_{t-1}$. The extent to which the central bank should commit to offset past misses depends on the strength of: (i) the forward-looking income expectations channel, given by $\frac{\lambda }{\beta }\left(1-\beta \right)$; (ii) the nominal yield curve channel, given by the 1 in the parentheses; and (iii) the effect of learners' drifting income expectations, given by $\left(1-\lambda \right)g{\rho }^2\frac{\left(1-\beta \right)\beta }{1-\beta \rho }$. This last term lowers the response to a given miss as g rises or $\lambda$ falls. Make-up policy itself is a source of variation in this economy when agents are learning, which generates further drifts in beliefs that must be pre-empted.^[12]

4.1.1 Mechanics of optimal policy: imperfect information

Figure 4 shows how optimal policy compares with IT in response to an unexpected cost-push shock under the same calibration explored in Figure 1. In contrast to the unconstrained case, the optimal policy is now more aggressive than IT policy. IT policy ‘lets bygones be bygones’ and does not respond to the failure to recognise the shock on impact, whereas optimal policy seeks to ‘make up’ for this miss. The anticipation of the aggressive policy acts as an automatic stabiliser when the shock occurs. Forward-looking agents understand the aggressive response is coming and that inflation will be lower in the future, which lowers inflation today. When $\lambda$ is high, optimal policy is only a little different from the unconstrained case. This is because even under perfect information, optimal policy required only a small contemporaneous response, with a larger and persistent subsequent response. The information friction is therefore not too costly and only a small increase in make-up policy is required. When $\lambda$ is low, the information friction makes more of a difference. With perfect information, the required contemporaneous response was large and the role of make-up policy was smaller. Imperfect information prevents the contemporaneous response, but introduces a substantial role for make-up policy because policymakers now want to influence interest rate expectations, which are forward looking even when $\lambda$ = 0. The optimal policy prescription therefore ends up being similar across $\lambda$ in the imperfect information case.

Figure 5 shows the dynamics of optimal policy versus IT for an unexpected demand shock (i.e. an increase in the natural rate $r_t^n$). In the full information case, this shock is perfectly offset and has no effect on the output gap or inflation. Information frictions, however, change the prediction. Demand shocks generate inflation. As in the cost-push shock case, optimal policy is now more aggressive than required under IT, which causes inflation to undershoot following the shock. Once again, under imperfect information, optimal policy is similar for both high and low $\lambda$ cases.

4.1.2 Optimal average inflation target: imperfect information

Like the unconstrained case, a FAIT policy is robust because we can capture most of WAIT, preemption and the additional make-up policy using only a weighted average of past outcomes. In contrast to the unconstrained case, though, optimal policy requires tracking a weighted average over both inflation and the output gap.

To illustrate, we substitute ${\text{Δ}}_{t-1}\equiv {\pi }_{t-1}^{*}-{\tilde{\pi }}_{t-1}$ into Equation (29), rearrange, and then substitute the result into Equation (28) to arrive at

where ${𝔼}_{t|t}$ denotes the expectation conditional on $I_t^{CB}$. This formulation makes clear that the ‘moving target’ described in Equations (28) and (29) is equivalent to trying to set some weighted average of current and past ${\tilde{\pi }}_t$ equal to zero. Since the central bank cannot implement the full information optimal target criterion ${\tilde{\pi }}_t=0$ in every period, it instead tries to implement it on average over time.

As in the full information case, we can combine the optimal policy conditions with the equilibrium conditions to evaluate the expectation terms in Equation (30) (which are inside ${\tilde{\pi }}_t$). This gives us a description of optimal policy just in terms of past inflation and output gaps, akin to Equation (20). Assuming again that the cost-push is a first-order autoregressive process,^[13] Appendix A.2 shows that, following an analogous approach to the derivation of Equation (20), we can write optimal policy under imperfect information in equilibrium as

where $Q_x\left(L\right),Q_{\pi }\left(L\right)$ and $Q_u\left(L\right)$ are power series in L and are defined in Appendix A.2. The coefficients in the power series Q_x and $Q_{\pi }$ give the precise weight that the central bank places on current and past values of the output gap and inflation when implementing optimal policy.

Figure 6 plots these weights for different values of $\lambda$ and g. The top row mirrors the left panel of Figure 3. It shows the case where there is no updating of learning beliefs (g = 0), which reduces the model to one featuring myopia as in Gabaix (2020). Optimal policy requires increasing weights in this case. This reflects the requirement of additional make-up policy. The bottom row of the figure shows the g $\ge$ 0 case. Additional make-up policy is required but it is moderated by the need to adjust for drifts in expectations. Importantly, when g is high, the optimal weights eventually decline in a similar way to the unconstrained case. It is this declining feature of the weights that makes FAIT a robust framework across a wide range of scenarios. We show in Section 5 that tracking weighted averages of inflation and output with declining weights captures most of the benefits of optimal policy.

4.2 Optimal policy with the zero lower bound

Optimal policy with the zero lower bound is very similar to optimal policy under imperfect information. Both constraints mean that the central bank sometimes cannot set the output gap it wants, in which case it cannot implement the full information optimal target criterion. As with imperfect information, the central bank should respond to this ZLB constraint by using commitment to influence the forward-looking expectations that enter the IS equation – that is, households' expectations over future output, inflation, and nominal interest rates. The key difference from the imperfect information case is that losses caused by a binding ZLB are not zero in expectation, even in the presence of mean zero shocks drawn from a symmetric distribution. Therefore, the pre-emptive channel of policy has a more significant role. The central bank should seek to pre-emptively raise learners' output and inflation expectations if it expects the ZLB to bind in the future. This is the use it or lose it character of optimal policy when confronted with the ZLB. It is optimal to move expectations pre-emptively before the constraint binds.

To derive optimal policy with the ZLB, we replace the information constraint (21) with the lower bound constraint

The optimal policy problem is then to choose the sequence ${\left\{{\pi }_t,x_t,{\omega }_t^{\pi },{\omega }_t^x,i_t\right\}}_{t=0}^{\infty }$ that maximises welfare (9) subject to the constraints (10) to (13) imposed by private-sector equilibrium behaviour and the lower bound constraint (32).

The first-order conditions are the same as in the imperfect information case, except that Equation (23) is replaced with the complementary slackness condition

Just as ${\mu }_t^i$ previously represented the extent to which imperfect information constrained the central bank, it now represents the losses caused by the ZLB. It is positive when the ZLB is binding and zero otherwise.

Condition (25) is the same as before, but the expressions for ${\mu }_t^{\omega \pi }$ and ${\mu }_t^{\omega x}$ in conditions (26) and (27) take a more general form, because ${𝔼}_t{\mu }_T^i$ is no longer necessarily equal to zero for all $T\ge t+1$:

If ${𝔼}_t{\mu }_T^i=0$ for all $T\ge t+1$, then conditions (34) and (35) collapse to conditions (26) and (27). The second line of (34) indicates how the central bank can use learners' inflation expectations to lower the real interest rate when the ZLB binds. The first term in the parentheses represents the effect of raising learners' inflation expectations on the inflation expectations of the rational forward-looking agents. This term, therefore, has both pre-emptive and history-dependent aspects. The second term in the parentheses represents the effect of shifting learners' inflation expectations on future real interest rates. Similarly, (35) represents the effect of shifting learners' output expectations for future output gaps.

Conditions (25), (33), (34) and (35) define optimal policy with the ZLB constraint.

Result 5 (Optimal target criterion with ZLB). Recall that the optimal target criterion under full information and without the ZLB is ${\tilde{\pi }}_t=0$, where we defined

Then the optimal target criterion when the ZLB can bind is

where ${\pi }_t^{*}\equiv \frac{1}{1+\left(1-\lambda \right)g{\rho }^2\Theta }\left(P\left(L\right){\text{Δ}}_{t-1}+{𝔼}_t{P}_1\left(L^{-1}\right){\text{Δ}}_{t+1}+P_2\left(L\right){𝔼}_{t-1}{P}_3\left(L^{-1}\right){\text{Δ}}_t\right)$ and $\Delta \equiv {\pi }_t^{*}-{\tilde{\pi }}_t$

where

is the same as Equation (29) and

Optimal policy commits to responding to past misses (see P(L)), exactly as in the information constraint case, and it adjusts the target ${\pi }_t^{*}$ in response to expected future target misses (P₁(L^–1)) and past expectations of target misses (P₂(L) and P₃(L^–1)). The net effect is even more make-up policy is promised.

The terms P₁(Z^–1) and P₃(L^–1) capture the insurance nature of optimal policy when agents are learning. Eusepi, Gibbs and Preston (2024) coin the term the insurance principle, which says that when the duration of the ZLB is uncertain, and agents are learning, that larger front-loaded forward guidance promises are optimal. These policies are too stimulatory when the shock is short-lived, but highly effective when the shock is long-lived because of the positive contribution they make to agents' drifting beliefs. The expectations operator present in these two terms captures the idea that policy must be set according to the expectation of how long the ZLB binds, which may not match the ex post realised duration of the shock. The more learning that is present (lower $\lambda$), and the faster learning happens (higher g), the more the central bank should respond now to these expectations of future binds of the ZLB.

To more clearly see how the terms in Result 5 relate to make-up and insurance policies, consider again the case where prices are fixed, $\kappa =0$. Optimal policy implies

where

with ${\text{Δ}}_t\equiv x_t^{*}-x_t$. The first line is identical to the target criterion in the imperfect information case with fixed prices. The second line represents the additional requirement of optimal policy to lift agents' expectations pre-emptively when the ZLB is expected to bind, or continue to bind. Policy should move in anticipation of a binding of the ZLB to raise expectations to pre-empt the losses that are expected to accrue while the ZLB binds.

Figure 7 compares optimal policy to IT in the presence of the ZLB.^[14] First, note that even before the shock hits, optimal policy maintains a small positive output gap and inflation rate (except when g is very small). This is the pre-emptive component of optimal policy: given that the ZLB could bind with some probability in any period, it is optimal to ensure that learners' inflation and output gap expectations are somewhat above target. For larger g, shocks have greater influence over learners' expectations, and so it is optimal to ‘run the economy hotter’. Optimal policy also holds the policy rate at the ZLB for longer than IT, generating a positive output gap and inflation overshoot. This is the make-up component. As a result of these two components, the downturn when the shock hits is much smaller, even though there is some welfare loss before and after the shock. Interestingly, even though optimal policy involves a more expansionary stochastic steady state, the policy rate is higher, because inflation expectations are higher.

Figure 8 shows how the pre-emptively higher stochastic steady-state inflation target varies with $\lambda$, g, and the mean neutral rate, rⁿ.^[15] As expected, the inflation target is higher when g is higher, which requires more pre-emptive policy, and when rⁿ is lower, which increases the likelihood of hitting the ZLB. When g is high, the inflation target is highest for intermediate values of $\lambda$, reflecting the amplification that can occur between forward-looking and backward-looking expectations through optimal pre-emptive policy.

The optimal target criteria we have derived require complex considerations on the part of the central bank where policy must respond to competing objectives in precise ways. The target criteria also requires a knowledge of the underlying expectation formation process in order to implement. In this section, we ask how robust are the policy lessons we draw from the optimal criteria. In other words, how flexible can you be as a flexible average inflation targeter?

To answer this questions, we first look at how closely simple weighted-average inflation targeting schemes can perform relative to the optimal target criteria, and how do these perform against known target criteria like inflation or price-level targeting. Second, we study how these simplified optimal target criteria perform when a central bank does not know how expectations are formed and must calibrate their target criterion to maximise the average welfare for a range of different belief formation processes.

We allow for cost-push shocks and imperfect central bank information in this analysis so the policy cannot perfectly offset all demand and productivity shocks. We consider this to be the most general and realistic case.^[16] Appendices D.1 and D.2 provide results for the unconstrained case, where cost-push shocks are the only source of losses.^[17]

5.1 Approximating optimal policy with simple target criteria

Table 2 presents a list of simple target criteria that capture key features of the optimal target criteria in the unconstrained and constrained cases, as well as other target criteria commonly studied in the literature. The list includes inflation targeting, price-level targeting, and simple average inflation targeting rules (‘k-yr AIT’). Informed by the analytical results in the previous sections, we also include rules that incorporate some combination of an IT or weighted AIT component (‘WAIT’); a weighted average output-gap targeting component (‘WAXT’); and a preemptive component that is a forward-looking weighted average of inflation forecasts (‘preemptive’).

For each of the target criteria listed in Table 2, we find the optimal parameters that minimise welfare loss and compare it to the loss achieved under the fully optimal target criteria. For the calibration, we set $\sigma$ = 1 and $\kappa$ = 0.1 (equivalent to a Calvo stickiness probability of around 0.7). For the expectation formation parameters, we use a baseline of $\lambda$ = 0.5, g = 0.08 and $\rho$ = 0.95. Finally, we set the persistence of the shocks to ${\rho }_u={\rho }_{rn}$ = 0.5 with the standard deviation of the neutral rate shock at ten times the standard deviation of the cost-push shock (similar to Giannoni (2014)). For simplicity, we let $\beta \to 1$, so that the loss function is $\mathcal{L}=\text{Var}\left({\pi }_t\right)+\alpha \text{Var}\left(x_t\right)$, with $\alpha =\kappa$/7.87.

Table 3 presents the results. The gains from optimal policy relative to IT are large, at 36.5 per cent of the loss under IT. These gains can be almost entirely achieved by following a rule that responds to a weighted average of both past inflation and past output gaps (the ‘WAIT + WAXT’ rule). Significant gains come from making up for past output gaps, because imperfect information prevents the central bank from perfectly controlling the output gap. Recall that lagged output gap terms enter the optimal target criterion under imperfect information, Equation (28) (as with the ZLB), unlike the unconstrained case, Equation (17). The decay parameters in the WAIT and WAXT are 0.94 and 0.89, so substantial weight is placed on inflation and output outcomes in the past. The PLT rule outperforms IT. Simple AIT rules with short windows perform worse than the weighted average rules.

The results support the idea that the average in FAIT should be a weighted average. Importantly, though, the simple target criteria all require a strong response to current inflation, which is seen by the optimised values of $\Psi$ and $\gamma$. Simple AIT rules require large $\Psi$ values as well to compensate for the 1/k weight applied to current inflation. However, it is not enough to offset the wrong type of average.

5.2 Robust optimal policy under parameter uncertainty

The previous results show that simpler targeting rules can approximate the optimal target criteria. However, the simple targeting rules require the policymaker to know $\lambda$, g and $\rho$. Here we study what happens when the policymaker does not know these values and instead must choose a target criterion that is optimised to maximise welfare on average when $\lambda$, g and $\rho$ can take on any value in economically meaningful ranges, which we approximate with a grid. In particular, we consider the values: $\lambda \in$ {0,0.25,0.5,0.75,1}, g $\in$ {0,0.01,0.08,0.25,1} and p $\rho \in$ {0.95,0.99}.^[18]

Table 4 shows the average welfare and the optimised parameter values for each target criterion. In addition, we show the average loss that would be achieved if the optimal criterion was implemented at each point of the grid, which represent the minimum loss achievable. WAIT rules perform best, with decay parameters of around 0.75 to 0.87. Making up for past output gaps as well via a WAXT component substantially lowers the loss. The PLT rule significantly outperforms IT. The 5-yr and 2-yr AIT rules always perform poorly.

To show what lies behind the average losses in the table, Figure 9 plots the loss under a selection of target criteria from Table 4 for different values of the expectation formation parameters $\lambda$ and g. The parameters in the policy rules are held fixed at the robustly optimal values set to minimise average loss and not optimised for each $\lambda$ and g shown in the figure. The dashed line shows the optimal target criteria optimised at each point to show how close the simple rules with the average coefficients can come to the fully optimal policy.

The results in Figure 9 reflect the nesting of different expectations assumptions within our approach. The IT rule and PLT rules perform similarly when $\lambda$ is low (when make-up commitments have little effect on inflation or output gap expectations) or g is high (when the sensitivity of learners' beliefs increases the cost of make-up commitments relative to their benefit).^[19] When $\lambda$ is high, make-up commitments are more powerful and the IT rule performs poorly.

In contrast, the WAIT rules generally perform well for all $\lambda$ and g, and including a WAXT component consistently lowers the loss for all parameters (except for g = 1, when it has little effect). It is important to reiterate that this is the case even though we have fixed the coefficients of the simple WAIT and WAXT rules to their robustly optimal values in Table 4 so it is a fair comparison with the IT and PLT rules. Only in the optimal policy benchmark do we allow the policy rule parameters to change as we vary $\lambda$ and g. The performance of the fixed-coefficient simple WAIT and WAXT rules is still remarkably close to the optimal benchmark, which illustrates the robustness of this form of FAIT.

5.3 Discussion

The reason that simple FAIT target criteria are robust is that once $\lambda$ and g are in the interior of their theoretical ranges, and expectations are both backward looking and forward looking, the features of optimal policy are the same. Optimal policy requires a weighted-average inflation target and some flexible component that captures pre-emption and make up. Pre-emption appears to be satisfied in most cases simply by placing high weight on current inflation and output realisations as found in the optimised $\Psi$ and $\gamma$ coefficients in the WAIT + WAXT rules, while additional make-up policy is captured by including the WAXT term.^[20]

It is widely recognised that rational expectations is a strong assumption and many researchers and policymakers have worried that policy recommendations that rely on it may not be robust. A large literature has shown that this concern is warranted in a variety of settings. We show that, in a framework encompassing a broad set of theories of expectation formation, flexible average inflation targeting is a robust policy strategy. Whether the central bank is unconstrained, faces information constraints or the ZLB, it is optimal to target a weighted average of inflation.

In addition, the optimal target criteria we derive give clear guidance on when policy should be flexible and deviate from the policy implied by the weighted average of inflation alone. If expectations drift in ways that are inconsistent with policymakers' objectives, then pre-emptive actions should be taken, with more agressive policy now to re-anchor beliefs. If targets are missed today, then policy should do more to make up for the misses in the future. If targets are expected to be missed in the future because of the constraint of the ZLB, then policy should do more today to insure itself against the possibility of a prolonged period of constrained actions. This guidance summarises what the flexible in FAIT should mean.

A.1 Unconstrained optimal policy

The unconstrained optimal policy problem is to maximise (9) subject to (10) to (13). The first-order conditions for this problem are

From (A5), we have ${\mu }_t^x$ = 0. This is Result 1. When policy is unconstrained, the IS curve is not a binding constraint. The central bank can generate whatever output gap it wants.

Substituting ${\mu }_t^x$ = 0 into (A4) gives ${\mu }_t^{\omega x}$ = 0. Learners' output gap expectations enter only the IS curve. Therefore, when policy is unconstrained, there is no reason for the central bank to try to influence learners' output gap expectations.^[21]

Combining (A1) and (A2) to eliminate ${\mu }_t^{\pi }$ gives Equation (14) in the main text. Similarly, using (A1) to eliminate ${\mu }_t^{\pi }$ in (A3) gives Equation (15). The remainder of the derivation for unconstrained optimal target criterion (17) is in the main text.

A.1.1 Derivation of Equation (19)

First, we rewrite ${\mu }_t^{\omega \pi }$ in terms of expected output gaps by substituting (14) into the second right-hand side term in (15) and iterating forward to get

Then, inverting the optimal target criterion (17) gives

The equilibrium system of (10), (12) and (A6) has the following minimum state variable solution

for some constants $a_x,a_{\omega }$ and a_u, where the second equality uses the law of motion of ${\omega }_t^{\pi }$.

A.2 Constrained optimal policy

Suppose we add to (10) to (13) some constraint on the policy rate. This could be an information constraint (21), or a ZLB constraint (32). For this optimal policy problem, the first four first-order conditions, (A1) to (A4), remain the same. But instead of (A5), we have (22), reproduced here:

where ${\mu }_t^i$ is the Lagrange multiplier on the policy rate constraint.^[22]

Using (A1) to eliminate ${\mu }_t^{\pi }$ from (A2) to (A4) and then substituting in $\sigma \left(1-L\right){\mu }_t^i={\mu }_t^x$ from (22) gives (25), (34) and (35), which we reproduce here:

Now substitute (34) and (35) into (25) and collect all the non-$u_t^i$ terms in ${\tilde{\pi }}_t$, which is defined in Result 4 in the main text. This gives

where $\text{Θ}$ and P(L) are defined in Result 4 and P₁(L^–1), P₂(L) and P₃(L^–1) are defined in Result 5. Finally, define

So we have

To get the optimal target criterion under imperfect information, Result 4, take the expectation of (A7) with respect to the central bank's information set. Since ${\text{Δ}}_t$ is just a multiple of ${\mu }_t^i$, (23) implies ${\text{Δ}}_{t|t}$ = 0 and ${𝔼}_t{\text{Δ}}_{t+k}$ = 0 for any $k\ge 1$.

To get the optimal target criterion with the ZLB, Result 5, just recognise that ${\text{Δ}}_t$ is non-zero if and only if the ZLB is binding.

A.2.1 Derivation of Equation (31)

Start the optimal policy conditions (25), (26) and (27). Together with the Phillips curve (10) and the updating rule for learners' inflation expectations (12), these five equations determine the optimal equilibrium paths of $\left\{{\pi }_t,x_t,{\omega }_t^{\pi },{\mu }_t^{\omega \pi },{\mu }_t^{\omega x}\right\}$ in terms of the paths of ${\mu }_t^i$ and the cost-push shock u_t.

As in the derivation of Equation (19), we can rewrite ${\mu }_t^{\omega \pi }$ in (26) in terms of expected output gaps instead of expected inflation:

Then combining (27) and (A8) and using the minimum state variable solution to evaluate expectations terms, we can write

where a_x, $a_{\omega }$ and a_u are the same constants as in (18), and $a_{\mu ,1}$ and $a_{\mu ,2}$ are new constants. Combining with the updating rule for learners' inflation expectations and using (21) gives the following description of optimal policy

Or, simply ${𝔼}_{t|t}\left(Q_x\left(L\right)x_t+Q_{\pi }\left(L\right){\pi }_t+Q_u\left(L\right)u_t\right)=0$, where $Q_x\left(L\right),Q_{\pi }\left(L\right)$ and Q_u(L) are power series in L. The new parameters are

The first and third power series, Q_x(L) and Q_u(L), are two different weighted averages of two exponentially weighted averages of lags. The second power series, $Q_{\pi }\left(L\right)$, is a weighted average of these same two exponentially weighted averages of lags, plus a third exponentially weighted average of lags, that is,

where ${\xi }_1$ and ${\xi }_2$ are the roots of ${\xi }^2-\left(\gamma +{\rho }_0\right)\xi +{\rho }_1$ and ${\varphi }_{\pi 1},{\varphi }_{\pi 2},{\varphi }_x$ and $$${\varphi }_u$ are constants.