The standard New Keynesian model with an occasionally binding effective lower bound (ELB) on the nominal interest rate is incoherent — it admits no rational-expectations solution — when demand shocks are large or persistent, and incomplete — it admits multiple solutions — in adjacent regions. Ascari, Mavroeidis and McClung (2023) showed that two deviations from full-information rational expectations restore coherence where the rational model fails: restricted-perceptions equilibria, in which agents forecast with misspecified but self-confirming rules, and bounded-rationality equilibria with fixed cognitive discounting; under incompleteness, the E-stability criterion selects one equilibrium. The same paper explicitly left open the completion of this program: whether other self-confirming or boundedly rational concepts (consistent-expectations, level-k, social-memory) restore coherence and what their general existence and learnability conditions are; whether the bounded-rationality fix survives when agents choose attention optimally (the paper's Appendix B.2 finds numerically that it fails for large shocks); analytical existence and uniqueness conditions for rational-expectations equilibria under continuously distributed persistent shocks (untreated beyond the i.i.d. case);

Background

The New Keynesian (NK) model is the workhorse framework of monetary policy analysis. It consists of three expectational equations: an aggregate-demand (Euler) equation relating the output gap — the deviation of output from its flexible-price level — to expected future output and the real interest rate; an expectational Phillips curve relating inflation to expected future inflation and the output gap; and a Taylor rule that sets the nominal interest rate as an increasing function of inflation. Under full-information rational expectations (RE), agents know the model and forecast with its true probability law, and solutions are sought in the minimum-state-variable (MSV) class, i.e. decision rules expressible as functions of the model's basic state variables.

Since the financial crisis, policy rates in several economies have spent long spells at the zero lower bound (ZLB, in practice an effective lower bound, ELB): nominal interest rates cannot fall materially below zero, so whenever the Taylor-rule-implied rate is negative the policy rate is stuck at the bound and the model becomes piecewise linear, switching between a normal regime and a bound regime. In that environment the rational model suffers two pathologies, in terminology borrowed from the literature on switching simultaneous-equations systems: incoherence, meaning that no MSV solution exists at all, and incompleteness, meaning that multiple MSV solutions exist (up to 2k2^k solutions when the exogenous shock follows a kk-state Markov chain). Multiplicity includes expectations-driven liquidity traps, in which self-fulfilling pessimism pushes the economy to the bound even without a large shock, as in Mertens and Ravn (2014). Ascari and Mavroeidis (2022) established that incoherence arises exactly when negative demand shocks are too large in magnitude or too persistent, which implies that a fundamentals-driven RE liquidity trap must be short-lived under standard calibrations — at odds with the decade-long ZLB episodes observed in Japan, the Euro Area and the US. Relatedly, Mavroeidis (2021) shows that the occasionally binding bound creates serious identification problems for estimating such models at the ZLB.

Ascari, Mavroeidis and McClung (2023) asked whether deviations from full-information RE mitigate these pathologies, in a version of the model that nests several assumptions about expectation formation. They analyzed two. First, restricted-perceptions equilibria (RPE): agents forecast with deliberately misspecified, simple subjective models (for example, constant forecasts of future inflation), and the equilibrium requires these beliefs to be self-confirming — the forecasting model's implied moments match what agents actually observe along the equilibrium path, so agents never see evidence of their misspecification; this is the self-confirming equilibrium idea applied to misspecified beliefs. Second, bounded-rationality equilibria (BRE) with cognitive discounting in the sense of Gabaix (2020): attention parameters below one discount future output and inflation in the Euler equation and the Phillips curve, so agents are less forward-looking; here these parameters are held fixed. The findings: with persistent demand shocks, RPE exist precisely in a region where the RE equilibrium does not, so non-rationality restores coherence, and under fixed strong myopia the BRE exists for any shock; combining both relaxes the existence condition further. Under incompleteness, the E-stability criterion — the standard device of the adaptive-learning literature for asking whether an equilibrium could be learned, defined as local stability of the dynamics in which agents recursively re-estimate their subjective forecasting models by least squares on past data — selects a unique equilibrium. The authors also show the same weakening of forward-lookingness that resolves incoherence is the mechanism behind proposed resolutions of the forward-guidance puzzle, the finding that promised future policy has implausibly large current effects in RE models.

The paper's analysis, however, is deliberately partial, and its own conclusion flags what remains. The existence results are closed-form because the demand shock is a two-state Markov chain; the only analytical treatment of non-discrete shocks in the literature, Mendes (2011), assumes i.i.d. shocks, and the paper states (Appendix B.7, footnote 35) that no paper provides existence and uniqueness conditions for the RE equilibrium under shocks that are both persistent and continuously distributed, while its own analytical existence criterion covers only restricted-perceptions equilibria under continuous shocks. The BRE restoration of coherence depends on the cognitive discount factors being fixed exogenously: Appendix B.2, building on an unpublished working paper, shows numerically that when households and firms choose their attention optimally subject to attention costs, they pay full attention in deep recessions and a solution ceases to exist for large negative shocks, so whether bounded rationality resolves incoherence hinges on whether the discount factors are predetermined. And other popular non-rational expectation concepts — consistent-expectations equilibria (CEE), in which agents' perceived law of motion matches the sample moments of the actual data; stochastic consistent-expectations equilibria (SCEE), the extension of this moment-matching fixed point to a stochastic actual law of motion, in which the perceived linear forecasting rule must match the unconditional moments of the noisy data; level-k reasoning, in which agents reason a finite number of iterated steps about others' decisions; and social-memory frictions, in which agents lose information about distant aggregate history — were excluded from the analysis, with their general existence and learnability properties in the lower-bound setting left open. No resolution of these open items is known in the literature to date; the closest related work proposes alternative fixes for the same pathologies — truncated Markov chains, simple fiscal policy, and algorithms for simulating multiple equilibria — rather than completing this coherence program.

Problem Statement

The problem is to complete the existence-and-uniqueness theory for the three-equation New Keynesian model with an occasionally binding effective lower bound on the nominal interest rate, in the formulation of Ascari, Mavroeidis and McClung (2023), by establishing provable conditions on the model's primitives (discount factor, degree of price rigidity, shock persistence, magnitude and support, Taylor-rule coefficients, attention-cost parameters) for components (1)–(3) below, together with the empirical direction proposed by the source paper's conclusion as component (4).

  1. The expectation-formation concepts excluded from the source analysis — consistent-expectations equilibria, stochastic consistent-expectations equilibria, level-k reasoning, and social-memory frictions: do equilibria under these concepts exist, and are they learnable, in the parameter regions where the rational-expectations model is incoherent? That is, do they, like restricted-perceptions and fixed-attention bounded-rationality equilibria, restore coherence, and under what general conditions? The four named concepts are treated jointly as the class of non-rational expectation concepts excluded from the source analysis: the question is the general existence-and-learnability characterization for this class — which of these concepts restore coherence and under what conditions.

  2. The robustness of the bounded-rationality resolution to endogenous attention: when agents choose their cognitive discount factors optimally subject to attention costs rather than having them fixed exogenously, under what conditions do equilibria exist and when are they unique — completing the partial, purely numerical analysis of Appendix B.2 of the source paper, which finds non-existence for large negative shocks in one calibration?

  3. The rational benchmark under realistic shocks: necessary and sufficient (or sharp sufficient) conditions for existence and uniqueness of rational-expectations equilibria in this model when the demand shock is continuously distributed and persistent (for example an AR(1) process) — the case excluded by the source paper's discrete Markov-chain analysis and treated analytically in the literature only for i.i.d. shocks.

  4. The empirical counterpart, proposed by the source paper's conclusion: calibrate or estimate the non-rational equilibria in medium-scale, empirically relevant DSGE models with the lower bound actually binding, and assess their fit to historical zero-lower-bound episodes relative to the rational-expectations benchmark.

Components (1)–(3) jointly serve one scientific objective — a complete characterization of when the lower-bound New Keynesian model is coherent (equilibria exist) and complete (equilibria are unique) across expectation-formation assumptions and shock processes — with component (4) assessing the quantitative relevance of the resulting fixes. The model equations, equilibrium definitions, and the E-stability learnability criterion are those fixed by the source paper.

Components (1)–(3) each ask for provable conditions on the model's primitives that establish existence, and where applicable uniqueness, of the corresponding equilibria; component (2) is an analytical characterization generalizing the numerical findings of Appendix B.2 rather than further numerical analysis. The three components are independent: establishing conditions for one does not by itself resolve the others. Component (4) is the empirical extension proposed by the source paper's conclusion.

References

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