Robust Real Rate Rules

Q1. What is the real rate rule, and why does it achieve determinacy without requiring the aggregate Euler equation?

The real rate rule i_t = r_t + φπ_t (φ > 1) combined with the Fisher equation i_t = r_t + E_t[π_{t+1}] immediately gives E_t[π_{t+1}] = φπ_t, whose unique non-explosive solution is π_t = 0 for all t; the proof is complete at this step, requiring no information about how households form expectations or optimize intertemporally. Standard Taylor-rule determinacy proofs rely on the aggregate Euler equation to close the system — the IS curve determines aggregate demand as a function of the real interest rate; deviation from determinacy arises when the Euler equation-Phillips curve system allows self-fulfilling expectation spirals. The real rate rule bypasses this entirely: the Fisher equation alone pins down the inflation path. The Fisher equation is a no-arbitrage condition between nominal and real bonds; it holds as long as two “deep-pocketed, fully-informed, rational agents” can trade both types of bonds — a condition that does not require aggregate household rationality, representative agent assumptions, or any specific consumption theory. Hand-to-mouth households, heterogeneous expectations, learning dynamics, and non-Ricardian fiscal regimes all leave the Fisher equation intact as long as some agents are pricing both asset classes. The consequence is that the Euler equation in the three-equation NK model becomes residual under the real rate rule: it determines the path of real interest rates given already-determined inflation and output gap, but plays no role in choosing among inflation equilibria.

Q2. What does the real rate rule imply about causation between inflation and the output gap?

Under the real rate rule, the Phillips curve operates in reverse relative to standard models: inflation is determined first (by the Fisher equation and the monetary rule), and the Phillips curve then determines the output gap as a residual; cost-push and demand shocks cannot amplify or dampen inflation variance under the rule. In the standard three-equation NK model with a mark-up shock ζ_t (law of motion ζ_t = ρ_ζ ζ_{t-1} + ε_{ζ,t}) and cost-push shock ω_t, the output gap under the real rate rule is x_t = −ζ_t/(κ(φ − ρ_ζ)) − ω_t/κ — a closed-form solution determined entirely by shocks, where the Euler equation does not appear. Inflation is π_t = 0 at all t (zero target): shocks affect the output gap but not inflation. Under an augmented rule that also responds to the output gap (i_t = r_t + φ_π π_t + φ_x x_t), determinacy still holds as long as a Phillips curve linking inflation and the output gap exists and the Taylor principle φ_π > 1 holds — providing additional policy degrees of freedom without sacrificing robustness. The decoupling of inflation from the Phillips curve is consistent with the empirical finding of Dotsey, Fujita, and Stark (2018) that the Phillips curve ceased to forecast inflation after 1984 — compatible with the hypothesis that the Fed’s post-Volcker behavior moved toward more real-rate-rule-like rules, giving the Fisher equation stronger anchor over inflation.

Q3. How does global stability under learning extend the determinacy result beyond local uniqueness?

Equilibrium determinacy is a local result (unique bounded solution near the target); the real rate rule additionally provides global stability under learning — even if financial market participants start with prior beliefs far from zero, the learning process converges to π_t = 0, preventing self-fulfilling sunspot equilibria from taking hold in the first place. The proof (Appendix D, using Gautschi’s inequality) establishes that the mapping from current beliefs to future beliefs is a contraction in the appropriate norm: since E_t[π_{t+1}] = φπ_t with φ > 1 drives realized inflation to zero, agents who update beliefs based on observed prices will progressively correct any initial error. This contrasts with Taylor rules, which are only locally determinate — an economy that starts at a non-zero sunspot inflation level may remain there if the sunspot is self-fulfilling. The global stability result also provides a response to the Cochrane (2022) critique that indeterminate equilibria under standard Taylor rules are “everywhere”: under the real rate rule, the only globally stable equilibrium is the target. The interest rate smoothing variant (Section 1.5) — fully smoothed real rate rule, θ > 0 — provides additional robustness: it requires agents to believe only that the central bank responds positively to inflation (not that φ > 1 specifically), and still generates identical inflation dynamics; this is more credible as a commitment device because the specific magnitude of φ cannot be directly observed.

Q4. How can the real rate rule implement arbitrary inflation dynamics, including optimal policy?

With a time-varying inflation target π_t, the modified rule i_t = r_t + φ(π_t − πt) implements any target inflation path determinately: the Fisher equation gives E_t[π{t+1} − π*_{t+1}] = φ(π_t − π*_t), whose unique solution is π_t = π*_t for all t, so realized inflation tracks the announced target exactly.** The CB must announce π*_t each period; this announcement may respond to the output gap, cost-push shocks, or any other variable. For example, to stabilize inflation while accommodating a cost-push shock, the CB sets π*t as a function of ω_t; realized inflation then follows this target, and the Phillips curve determines the output gap residually. There are two constraints: (1) the CB must be able to compute a reasonable approximation to E_t[π*{t+1}] — achievable via inflation futures, inflation swap markets, or an internal forecasting model; (2) the target path itself must not be explosive (a target that amplifies its own past realizations would generate explosive equilibria). Under these constraints, the paper formally proves (Appendix E.5) that real rate rules with time-varying targets can replicate the outcomes of any other monetary regime. This implies: (a) real rate rules can implement Ramsey-optimal policy, attaining the highest possible welfare; (b) it is empirically impossible to test whether a central bank is following a general real rate rule — any observed inflation and interest rate dynamics are consistent with some choice of π*_t. The Smets-Wouters (2007) estimated rule for the US illustrates: at the posterior mode, the correlation between the rule component z_t and the real interest rate r_t is 0.63, with both variables having standard deviation 0.46%, suggesting the Fed is already approximately two-thirds of the way toward a simple robust real rate rule.

Q5. Why does the Taylor principle fail in richer models, and how does the real rate rule avoid those failures?

The Taylor principle (φ_π > 1) is sufficient for determinacy in the benchmark three-equation NK model with a representative rational agent, but it is neither necessary nor sufficient in richer environments: Bilbiie (2008) shows that with enough hand-to-mouth consumers, higher φ_π can destabilize the economy; Leeper-Leith (2016) shows that following the Taylor principle can generate explosive inflation under the fiscal theory when nominal debt is present. Bilbiie (2008, 2019) inverts the Euler equation for the representative rational household when hand-to-mouth agents dominate: the aggregate consumption Euler equation has a negative intertemporal substitution sign, making the system’s eigenvalues switch. With enough hand-to-mouth agents, φ_π > 1 actually generates explosive equilibria (indeterminacy flips). Under the real rate rule, the Euler equation is disconnected from inflation determination entirely — Bilbiie’s mechanism cannot operate because the inflation equation relies only on the Fisher equation, not on whether the Euler equation has positive or negative sign. Similarly, the paper’s Section 2 result on fiscal robustness: with long-maturity government debt (Appendix B), a stable inflation equilibrium always exists under the real rate rule regardless of whether fiscal policy is active or passive. This implies the fiscal theory of the price level (FTPL) cannot uniquely determine inflation under the real rate rule — there is always a stable solution — so FTPL determinations are not unique, which may be of independent theoretical interest. The proof uses the contracting property of the non-linear real rate rule in the fully non-linear model, showing the target gross inflation Π* is always a solution of the bond-pricing fixed-point equation and that it is approached from all starting points via iteration.

Q6. How is the real rate rule implemented in practice, and what are the policy implications for central bank design?

Implementation uses TIPS yields (Treasury Inflation-Protected Securities) or inflation swap markets as real-time signals for r_t; the central bank sets i_t = TIPS_yield_t + φπ_t without estimating the natural rate (r) or output gap, eliminating the key measurement error in standard rules.* The key operational advantage over standard Taylor-type rules: standard rules require estimating the natural rate r* (now known to be mismeasured; Holston-Laubach-Williams 2017 revisions) and the output gap (subject to large real-time revisions); the real rate rule bypasses both because r_t is directly observable from financial markets (it equals the TIPS yield to a risk premium). The CB must also compute E_t[π*_{t+1}] to set the time-varying target; inflation futures or swap markets provide a forward-looking market price for this purpose. The paper discusses Hall and Reis (2016) “indexed payment on reserve” rules, which use a different mechanism (central bank liability indexation) to achieve similar robustness goals but do not rely on the Fisher equation as directly. Adão, Correia, and Teles (2011) achieve related results via complete nominal bond indexation. The real rate rule is more transparent and simpler to communicate: the CB says “we will raise the policy rate one-for-one with the real rate plus respond to inflation with coefficient φ.” For a smoothed version, communicating “we respond positively to inflation” — without specifying exactly how much — is sufficient for determinacy, and arguably more credible as a commitment. Section 4 (not covered here) develops a ZLB-adapted version of the rule for zero lower bound episodes that rules out explosive inflation equilibria at the bound.

Key concepts

real rate rule : the monetary policy rule i_t = r_t + φπ_t (φ > 1), where r_t is the current real interest rate observed from TIPS or inflation swap markets; achieves equilibrium determinacy via the Fisher equation alone, without invoking the aggregate Euler equation, making it robust to heterogeneous agents, hand-to-mouth consumers, non-rational expectations, and active fiscal policy.

Fisher equation : the no-arbitrage condition i_t = r_t + E_t[π_{t+1}] linking the nominal policy rate, real rate, and expected inflation; in the context of the real rate rule, it is the only structural equation needed for determinacy; requires only two deep-pocketed rational agents to arbitrage between nominal and real bonds — not aggregate household rationality.

inflation decoupling : the property under the real rate rule that the Phillips curve determines the output gap residually given already-determined inflation, rather than operating as a transmission mechanism for cost-push or demand shocks into inflation; implies that only monetary policy shocks and Fisher equation shocks can move inflation — cost-push and demand shocks affect the output gap but not the price level.

Taylor principle failure : the result (Bilbiie 2008) that standard Taylor rules can fail to deliver determinacy in models with hand-to-mouth consumers or heterogeneous agents — because the inverted aggregate Euler equation can flip eigenvalue signs — and (Leeper-Leith 2016) that following the Taylor principle can generate explosive inflation under the fiscal theory of the price level with nominal debt; the real rate rule avoids both failures by relying on the Fisher equation rather than the Euler equation for inflation determination.

global stability under learning : the property that even if financial market participants start with beliefs far from the inflation target, the learning process converges to the target under the real rate rule, proven via a contraction argument using Gautschi’s inequality; stronger than local determinacy (which only guarantees uniqueness near the target), ruling out self-fulfilling sunspot equilibria from any starting point.

fiscal theory robustness : the paper’s finding that with long-maturity government debt, the real rate rule always implies a stable inflation equilibrium regardless of whether fiscal policy is active (non-Ricardian) or passive (Ricardian); equivalently, the fiscal theory of the price level cannot uniquely determine inflation under the real rate rule because a stable solution always coexists with any fiscal regime.