RDP 2025-02: Boundedly Rational Expectations and the Optimality of Flexible Average Inflation Targeting 1. Introduction

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.

Footnotes

Eusepi and Preston (2018) and Eusepi, Giannoni and Preston (2018) also point to this feature of adaptive learning models relative to rational expectations models when analysing unconstrained optimal policy. We show how this insight generalises to several novel environments. [1]

There has been a gradual evolution of the language describing different characterisations of monetary policy. Svensson and Woodford define a monetary policy rule broadly as a prescribed guide for monetary policy. Target criterion and interest rate rules are two different ways of specifying a policy rule in their framework. In more recent treatments in the literature, there is no distinction made between interest rate rules and policy rules. The two are synonymous. We adopt the term ‘policy framework’ to capture the broader notion of a general policy that may be implemented with a target criterion or an interest rate rule. [2]

Di Bartolomeo, Di Pietro and Giannini (2016) and Hagenhoff (2021) also consider a heterogeneous expectations framework, but their focus is on how heterogeneity in expectations across agents affects price and consumption dispersion, altering the micro-founded loss function. We are instead interested in capturing how a broad range of different expectations models affect optimal aggregate outcomes, so we assume the standard loss function. [3]