Self-Fulfilling Fluctuations in HANK Economies

Layer 1: Overview

Research question and motivation: A central tenet of monetary policy is that aggressively raising nominal rates more than one-for-one with inflation (the Taylor principle) nips self-fulfilling inflationary beliefs in the bud. That logic is built on Representative-Agent New Keynesian (RANK) models that abstract from inequality and incomplete markets. Acharya and Benhabib ask whether this central tenet survives in Heterogeneous-Agent New Keynesian (HANK) economies where idiosyncratic income risk is countercyclical, and they answer in the negative: no matter how aggressively monetary policy responds to inflation, such economies remain susceptible to self-fulfilling fluctuations (“endogenous demand shocks”).

Model setup: The paper builds an analytically tractable continuous-time HANK model. Tractability comes from quasi-linear preferences (linear in labor), which makes the economy block-recursive — aggregate output and inflation dynamics can be characterized independently of the wealth distribution. Households face a 2-state Poisson idiosyncratic productivity process (high ξh / low ξl, treating ξl loosely as “unemployment”), with the transition rate into the low state given by λl,t = λl·y^(−Θ); Θ > 0 makes risk countercyclical (Θ = 0 is acyclical). Firms are monopolistically competitive with a forward-looking (Rotemberg-type) Phillips curve. The baseline monetary rule is a simple inflation-targeting Taylor rule it = r + φπ·πt with φπ > 1, and crucially the model imposes NO effective lower bound, to distinguish the mechanism from liquidity-trap multiplicity (Benhabib-Schmitt-Grohé-Uribe 2001).

Key mechanism: With countercyclical risk, the “natural rate” r*(y) = ρ − σ·y^(−Θ) (defined Keynes-style as the real rate consistent with constant output, not the flexible-price rate) is endogenous and co-moves with output: dr*/dy = σΘy^(−(1+Θ)) > 0. A belief that output will fall raises perceived future risk, raises desired precautionary saving, and lowers the natural rate; if policy does not cut rates enough, real rate exceeds natural rate, spending falls, and the pessimistic belief is self-fulfilling.

Main results (with magnitudes/scope): (1) Local determinacy requires a cyclical-risk-augmented Taylor principle φπ > φ(Θ) = 1 + ρσγΘ/κ, valid only if risk is not too countercyclical, Θ < Θ* ≡ ρ/(σγ); if Θ > Θ* the targeted equilibrium is locally indeterminate for any finite φπ. (2) GLOBAL indeterminacy holds for ANY Θ > 0 and any finite φπ (Proposition 3): an untargeted steady state always coexists with the target, and depending on cyclicality, fluctuations take the form of a saddle connection (mildly countercyclical, Θ < Θ⋄), a stable limit cycle around the target (moderately countercyclical, Θ⋄ < Θ < Θ*), or local indeterminacy (highly countercyclical, Θ > Θ*). (3) Calibration (real rate 4%, γ⁻¹ = 2, λl = 0.013, ch/cl = 1.1 implying ξh/ξl = 1.23, φπ = 1.5) yields Θ⋄ ≈ 15.8 and Θ* = 31.08; empirical estimates from Bilbiie-Primiceri-Tambalotti (2023) put Θ in [21.98, 29.9] with mode 28.1 — comfortably in the moderately countercyclical region. At Θ = 28.1 the untargeted steady state has output about 6.5% below target, and the stable cycle has output-gap amplitude of roughly ±2.5% — magnitudes comparable to U.S./Euro-area post-Great-Recession gaps and U.S. business cycle fluctuations. (4) Policy fixes: a monetary rule that responds to the endogenous natural rate, it = r + φπ·πt + φr·(r*(xt) − r) with φπ > 1 and φr ≥ 1 (a “Taylor principle for natural rates”), delivers global determinacy (Proposition 4). Alternatively, a passive-monetary/active-fiscal regime (φπ < 1, φb ∈ [0,1)) eliminates all manifestations of indeterminacy via the Fiscal Theory of the Price Level (Proposition 5). Rules responding only to output, inertial rules, or escape clauses that merely remove the untargeted steady state (e.g., switching to strict inflation targeting if output falls below x̃ = −0.1) fail because the stable cycle survives.

Layer 2: Deep Dive

What is the central claim and how does it overturn the RANK benchmark?

In RANK (or HANK with acyclical risk), the Taylor principle φπ > 1 delivers both local AND global determinacy because the IS curve has no higher-order terms. In HANK with countercyclical risk, the natural rate r*(y) = ρ − σy^(−Θ) co-moves with output. This adds a stabilizing first-order term (−σγΘx) to the IS curve requiring a stronger response for local determinacy (φπ > φ(Θ)), and adds stabilizing higher-order terms that no finite φπ can overwhelm — producing global indeterminacy for any Θ > 0. So aggressive inflation-fighting alone cannot anchor the economy.

How is the ’natural rate’ defined here, and how does it differ from standard usage?

The authors follow Keynes (1936): r*(y) is the real interest rate consistent with output remaining constant at level y. This differs from the standard New Keynesian definition (the flexible-price real rate r = ρ − σ). The two coincide in RANK, in HANK with acyclical risk, and at the steady state y = 1 (r = r*(1)), but DIVERGE when risk is countercyclical: there are many natural rates r*(y) — one per output level — while there is a single flexible-price rate r = ρ − σ. The flexible-price rate never depends on endogenous output; r*(y) does.

What distinguishes this source of multiplicity from prior determinacy literature?

Three distinctions. (1) Versus Benhabib-Schmitt-Grohé-Uribe (2001b) liquidity-trap multiplicity: the paper purposely imposes NO effective lower bound, so the ELB is not the driver — countercyclical risk is. (2) Versus the local-determinacy HANK literature (Acharya-Dogra 2020, Bilbiie 2024, Auclert et al. 2023, Ravn-Sterk 2021): those papers show a stronger ‘cyclical-risk-augmented Taylor principle’ restores LOCAL determinacy; this paper shows that same condition cannot rule out GLOBAL indeterminacy. (3) Versus Benhabib-Eusepi (2005) / older RANK global-indeterminacy work that relied on money-in-utility, money-in-production, or capital: this model is cashless and capital is not a factor of production, so the mechanism is genuinely the countercyclical risk.

How does the paper relate to Ravn and Sterk (2021), the only other HANK global-indeterminacy paper?

Ravn-Sterk (2021) study a HANK economy with search frictions and find an additional ‘unemployment trap’ steady state (100% unemployment) alongside the target. This paper’s characterization (two steady states) is complementary, but goes further by providing a COMPLETE analytical characterization of the dynamics through which countercyclical risk generates indeterminacy, and by analyzing which policy designs eliminate it. A key novel point: indeterminacy manifests not only as a second steady state but also as a stable cycle around the target, so policies that only kill the untargeted steady state can fail.

Why isn’t eliminating the untargeted steady state sufficient for global determinacy?

Because under moderately countercyclical risk a stable limit cycle surrounds the targeted steady state independently of the untargeted steady state. The paper shows an escape-clause rule that switches to strict inflation targeting (π = 0) when output falls below x̃ = −0.1 (i.e., more than 5% below target) does eliminate the untargeted steady state, yet trajectories near the target still diverge locally and then converge to the surviving stable cycle, remaining bounded. Hence only policies that neutralize ALL non-fundamental equilibria — not just the untargeted steady state — guarantee global determinacy.

What is the proposed monetary-policy fix and its scope conditions?

A rule it = r + φπ·πt + φr·(r*(xt) − r) with φπ > 1 and φr ≥ 1 (Proposition 4) delivers global determinacy for any Θ > 0. The intuition is a ‘Taylor principle for natural rates’: by committing off-equilibrium to move the nominal rate at least one-for-one with endogenous natural-rate fluctuations, policy undoes the precautionary-saving impulse so pessimistic/optimistic beliefs cannot be confirmed. Setting φr = 1 makes the nominal rate perfectly track r*(xt), analogous to the optimal RANK response to exogenous demand shocks. It is also related to Holden’s (2024) robust real-interest-rate rule.

What is the fiscal-policy alternative and the mechanism?

A passive-monetary/active-fiscal regime (φπ < 1, φb ∈ [0,1), Proposition 5) eliminates the untargeted steady state and the stable cycle for any Θ > 0, yielding a unique globally determinate equilibrium converging to x = π = 0, b = b*. Mechanism is the Fiscal Theory of the Price Level: with active fiscal policy, taxes do not rise enough to stabilize debt, so the price level must adjust to keep the real value of debt equal to the present value of future primary surpluses. A permanent-recession (deflationary) belief would raise real debt and eventually violate the government budget constraint, so such beliefs cannot be self-fulfilling. Importantly, the paper assumes b* > 0 (positive steady-state primary surplus), distinguishing it from Kaplan et al. (2023), where multiplicity arises under persistent deficits.

Do other standard monetary rules rescue determinacy?

No. Appendices E.1 and E.2 show that adding an output-gap response (it = φπ·πt + φx·xt) or making the rule inertial/backward-looking can make LOCAL determinacy easier but cannot eliminate global indeterminacy: for any finite (φπ, φx) however large, or any degree of backward-lookingness (any α), the equilibrium remains globally indeterminate as long as risk is countercyclical. The reason is that none of these rules respond to the endogenous natural-rate fluctuations directly.

How robust are the results to the functional form of countercyclical risk?

Robust. Appendix E.4 generalizes λl,t = λl·Λ(γxt) for any non-negative, weakly decreasing analytic Λ. The untargeted steady state exists whenever risk is countercyclical locally (−Λ’(0) = Θ > 0), even if Λ is linear. The stable cycle exists if Λ is sufficiently convex locally (Λ’’(0) sufficiently positive). Crucially the conditions depend only on local behavior at x = 0, which is reassuring given the thin empirical evidence on how risk varies far from steady state. The authors argue convexity is plausible: the inflow rate into unemployment rises sharply in recessions but does not fall as sharply in expansions (Crump et al. 2019), and labor-flow asymmetries exceed GDP asymmetries (McKay-Reis 2008).

Does the multiplicity survive introducing predetermined variables?

Yes, with a caveat about jumps. The baseline has no predetermined variables, so the economy can instantaneously jump between steady states/onto the cycle. Appendix E.5 lets the fraction of ξl households vary (a predetermined state), Appendix E.2 uses a backward-looking rule (lagged inflation predetermined), and Section 4.2/Appendix D.1 add government debt. In all cases instantaneous jumps are ruled out, but global indeterminacy persists: transitions to the untargeted steady state or the stable cycle become GRADUAL (e.g., a slow rise in the ξl fraction alongside falling output and inflation) rather than instantaneous.

What are the headline calibrated magnitudes and how credible are they?

Calibration: real rate 4%, relative risk aversion γ⁻¹ = 2, transition rate λl = 0.013 (from Bilbiie-Primiceri-Tambalotti 2023), consumption drop at job loss ch/cl = 1.1 implying ξh/ξl = 1.23, and φπ = 1.5. This gives regime boundaries Θ⋄ ≈ 15.8 and Θ* = 31.08. The empirically estimated Θ lies in [21.98, 29.9] (mode 28.1), squarely in the moderately countercyclical region. At Θ = 28.1, the untargeted steady state has output ~6.5% below target (comparable to post-Great-Recession U.S./Euro-area gaps) and the stable cycle has output-gap amplitude ~±2.5% (comparable to U.S. business cycle fluctuations). The 10% consumption drop is within empirical estimates (Cochrane 1991: 24–27% lower growth; Ganong-Noel 2019: ~11%; Gruber 1997: 6.8% for food).

What are the policy implications and their caveats?

Central banks should monitor and react to private-sector beliefs about REAL activity (consumer confidence, perceived job-loss probability) as vigilantly as they monitor inflation expectations — ignoring real-activity beliefs can leave even inflation expectations unanchored. Because multiplicity does not stem from the ELB, it can afflict the economy even during a tightening cycle, and large rate hikes against inflation do NOT by themselves guarantee anchored expectations. Caveat/scope: the prescriptions hold in this stylized cashless, quasi-linear, no-aggregate-risk model; the precise cycle magnitude/periodicity and depth of the untargeted steady state depend on the full shape of Λ away from steady state, even though their existence depends only on local behavior.

What is the broader methodological lesson?

Local stability/determinacy analysis can be misleading: even when the targeted equilibrium is locally determinate, multiple bounded global equilibria can exist. Researchers using HANK models should check global, not just local, determinacy. Because linear models have no higher-order terms, local determinacy implies global determinacy there; but HANK with countercyclical risk is genuinely nonlinear, so the implication breaks.

Key Concepts

Natural rate of interest r(y)*: Defined Keynes-style (1936) as the real interest rate consistent with output remaining constant at level y; given by r*(y) = ρ − σy^(−Θ). Distinct from the flexible-price real rate. With countercyclical risk it is endogenous and rises with output (dr*/dy > 0), and there is one natural rate per output level.

Neutral rate of interest: The single flexible-price real interest rate r = ρ − σ in the model — the natural rate consistent with full-employment output y = 1, i.e., r = r*(1). It depends only on exogenous parameters, never on endogenous output.

Countercyclical risk (parameter Θ): Idiosyncratic income risk that rises when output falls, modeled via transition rate λl,t = λl·y^(−Θ). Θ > 0 means a ξh household is more likely to fall to the low-productivity (loosely ‘unemployment’) state when output is low; Θ = 0 is acyclical. Θ governs the strength of this cyclicality.

Endogenous demand shock: A self-fulfilling, non-fundamental fluctuation arising because a belief about future activity shifts desired precautionary saving, moves the endogenous natural rate, and — if policy does not offset it — confirms the original belief. Functions like an exogenous demand shock but is generated internally by countercyclical risk.

Global vs local determinacy: Local determinacy: the targeted steady state is the only bounded equilibrium in a small neighborhood (governed by first-order/eigenvalue terms). Global determinacy: it is the only bounded equilibrium starting from ANY point (governed also by higher-order terms). In this nonlinear HANK model local determinacy does NOT imply global determinacy.

Taylor principle for natural rates: The proposed fix: monetary policy must move the nominal rate at least one-for-one (φr ≥ 1) with endogenous fluctuations in the natural rate r*(x), in addition to responding to inflation (φπ > 1). This off-equilibrium commitment prevents beliefs about real activity from becoming self-fulfilling.

Risk-cyclicality regimes (mild / moderate / high): Mildly countercyclical (Θ ∈ (0, Θ⋄)): indeterminacy via a saddle connection to the untargeted steady state. Moderately countercyclical (Θ⋄ < Θ < Θ*): a stable limit cycle surrounds the target. Highly countercyclical (Θ > Θ* = ρ/(σγ)): the target is locally indeterminate for any finite φπ. Calibrated thresholds Θ⋄ ≈ 15.8, Θ* = 31.08.