RDP 2025-02: Boundedly Rational Expectations and the Optimality of Flexible Average Inflation Targeting 2. The Model

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.

Footnotes

This is especially important because we are going to make different assumptions about the formation of output gap and interest rate expectations. In the two-period ‘Euler equation’ representation, nominal interest rate expectations do not even appear. See Preston (2005) for a detailed discussion of the two representations. [4]

We focus on how forward- or backward-looking monetary policy should be to achieve optimal aggregate outcomes and do not consider distributional consequences that may occur due to heterogeneity in expectations. We maintain a representative agent assumption where the representative decision-maker takes a weighted average of forecasts from two different models: the correct structural model, and a reduced-form model which is re-estimated each period. Therefore, the representative agent assumption is maintained. This is the approach taken in Gibbs (2017) and Gibbs and Kulish (2017). [5]

Note that we still allow for drift in longer-term real interest rates due to drifting beliefs about future inflation. [6]

Farhi and Werning (2019) assume that households perfectly observe future policy rates, but deduce the consequences for inflation and output gap expectations using level-k reasoning. Similarly, Dupraz et al (2024) model households with finite-planning horizons, but who save and borrow based on financial prices determined by intermediaries with fully rational expectations over nominal interest rates. Quantitative evaluations of make-up policy by Federal Reserve Board staff have have used versions of their semi-structural model (FRB-US) in which financial market participants have model-consistent expectations, even while households and firms may not be rational (Bernanke, Kiley and Roberts 2019; Hebden et al 2020). [7]

There are clearly interesting questions about what occurs when the central bank lacks credibility over the path of interest rates that it plans to implement. See, for example, Eusepi et al (2018) and Eusepi, Giannoni and Preston (2024). Our results represent a best case scenario with respect to central bank credibility regarding the policy rate path. In Appendix B.1, we discuss how our results might be affected if policy rate expectations are not entirely rational. [8]

In Appendix B.2, we use a simple two-period model to explore what optimal policy under discretion would look like with our hybrid specification for expectation formation. [9]