Evaluating macroeconomic outcomes under asymmetries: Expectations matter

Layer 1 — Overview

Research Question

This paper investigates whether and how assumptions about household and firm expectations alter the macroeconomic implications of asymmetries commonly embedded in macroeconomic models. Specifically, it asks: when a model features a nonlinearity — such as an asymmetric monetary policy rule or a nonlinear Phillips curve — do the longer-run average outcomes and the distributional properties of inflation and unemployment depend on whether agents have rational expectations (RE, accounting for the possibility of future shocks) versus perfect foresight (PF, not anticipating future shocks)?

Methodology

The paper works within a standard three-equation New Keynesian model comprising an IS curve (linking the unemployment gap to the policy rate and the natural rate of interest via Okun’s law with coefficient c ≈ 2), a forward-looking Phillips curve, and a monetary policy rule. The model is parameterized at a quarterly frequency with β = 0.99, κ = 0.01, φπ = 1.5, φu = −0.25, shock persistence ρ_rn = 0.9, and shock standard deviation σ_rn = 0.0025 (calibrated to match a 1-percentage-point standard deviation of the unemployment gap under the symmetric baseline rule).

The key methodological distinction is the specification of the expectations operator. Under RE, agents use the true stochastic transition matrix for the natural rate (approximated via the Rouwenhorst method with 105 grid points). Under PF, agents instead use a transition matrix that always places probability one on the steady-state value of the natural rate next period — i.e., they do not anticipate future shocks. The model is solved globally with a discrete state space projection (parameterized expectations) method, applied identically to RE and PF cases. The authors first derive analytical results in a simplified three-state environment and then present numerical results from 3,000 simulations of 1,000 periods each.

Two types of asymmetry serve as case studies: (i) an asymmetric monetary policy rule — the “Shortfalls rule” — under which the central bank does not tighten in response to a tight labor market (negative unemployment gap), in the spirit of the FOMC’s 2020 framework update; and (ii) a nonlinear (kinked) Phillips curve that steepens by a factor of three when the labor market is tight (unemployment gap < 0), consistent with empirical evidence in Smith, Timmermann, and Wright (2025).

Main Findings

The core finding is that the sign and magnitude of longer-run average outcomes under asymmetric macroeconomic environments can differ substantially — and can even reverse — depending on whether agents have rational expectations or perfect foresight.

For the Shortfalls rule, under PF the model implies a longer-run tradeoff: average unemployment gap is −0.32 percentage points and average inflation gap is +0.25 annualized percentage points relative to the symmetric Deviations rule. PF thus suggests policymakers can lower average unemployment at modest inflationary cost. Under RE, however, this apparent tradeoff disappears entirely: the average unemployment gap is essentially zero (−0.05 percentage points) while average inflation is elevated by approximately 1.02 annualized percentage points. The gap in average inflation outcomes between RE and PF thus exceeds one percentage point, and the labor market benefit implied by PF is absent under RE.

For the nonlinear Phillips curve (under a symmetric deviations rule with φu = 0), the results again diverge across expectations assumptions, and the direction of the effects reverses. Under PF, the kinked Phillips curve implies average inflation of +0.41 annualized percentage points and a near-zero unemployment gap (+0.30 percentage points). Under RE, the average inflation gap is essentially zero while the average unemployment gap rises to +0.63 percentage points — the opposite directional pattern from PF.

The mechanism driving the RE–PF divergence is the interaction between forward-looking price-setters and an inflation-stabilizing central bank. Under RE, anticipated future episodes in which the asymmetry may bind (e.g., the Shortfalls rule providing accommodation, or the Phillips curve steepening) cause firms to set higher prices today. The central bank responds to the resulting pickup in inflation expectations with tighter policy, generating a persistent contractionary offset. This channel is absent under PF because agents expect no future shocks.

Robustness

The main conclusions are robust across three extensions: (i) Bounded rationality (following Gabaix 2020, with m_br = 0.97): outcomes move toward the PF case, confirming that what matters is the degree to which agents internalize the probability of future shocks; (ii) Cost-push shocks instead of natural rate shocks: the RE–PF divergence under a Shortfalls rule is broadly similar in direction and magnitude to the baseline; (iii) Alternative shock specifications: the qualitative conclusions are maintained.

Crucially, under the symmetric Deviations rule the RE and PF solutions are identical in all cases, confirming that the divergence is specific to models with macroeconomic asymmetries, not an artifact of the solution method.

Layer 2 — Q&A

Q1: What is the central methodological claim about perfect foresight solutions in asymmetric models?

The paper argues that in macroeconomic models with asymmetries or nonlinearities, perfect foresight solutions — in which agents do not account for the possibility that future shocks may occur — can yield longer-run average outcomes and distributions that differ from their rational expectations counterparts in magnitude and potentially in sign. The paper is explicit that this is not a critique of PF methods per se, as PF is often necessary for estimating larger models; rather, the point is that researchers should check the robustness of conclusions about longer-run averages using simplified models solvable under both approaches.

Q2: How is the difference between RE and PF operationalized in the model?

The sole technical distinction lies in the specification of the conditional expectations operator Et. Under RE, this operator uses the true stochastic Markov transition matrix for the natural rate (P^RE), which assigns positive probability to all feasible future states. Under PF, agents use a degenerate transition matrix (P^PF) that assigns probability one to the mean value of the natural rate next period regardless of the current state — effectively, agents expect no future innovations. The same global solution method (discrete state space projection with 105 Rouwenhorst grid points) is applied to both, so differences in equilibrium outcomes are entirely attributable to the expectation specification.

Q3: What are the analytical results for the Shortfalls rule in the simplified three-state model?

In the simplified environment with the natural rate taking three equiprobable values (low, steady-state, high) and no persistence, the analytical solution shows that under PF the average unemployment gap is −Δ/(1 + φπκ) < 0 and the average inflation gap is Δκ/(1 + φπκ) > 0, where Δ parameterizes the degree of additional accommodation in the high-demand state. Under RE, the average unemployment gap is exactly zero and the average inflation gap is Δ/(φπ − 1) > 0. The inflation gap under RE exceeds that under PF by Δ(1 + κ)/[(φπ − 1)(1 + φπκ)] > 0, and the unemployment gap under RE exceeds that under PF by Δ/(1 + φπκ) > 0. Thus, PF spuriously implies an exploitable long-run tradeoff that vanishes under RE.

Q4: What are the analytical results for the nonlinear Phillips curve in the simplified model, and how do the directions of the effects compare to the Shortfalls rule case?

Under PF with a nonlinear (kinked) Phillips curve, the average inflation gap is positive (= Δpc > 0) while the average unemployment gap is zero. Under RE, the signs reverse: the average unemployment gap is positive (= Δpc/κ > 0) and the average inflation gap is zero. The difference is ūRE − ūPF = Δpc/κ > 0 and π̄RE − π̄PF = −Δpc < 0. This sign reversal relative to the Shortfalls rule case illustrates that the directional error introduced by PF is not uniform but depends on the specific asymmetry — the key feature is always the absence, under PF, of the forward-looking price-setting channel interacting with monetary policy.

Q5: What is the quantitative magnitude of the RE–PF divergence in the numerical model for the Shortfalls rule?

In the fully parameterized numerical model (Table 2), under a Shortfalls rule the average inflation gap is 1.02 annualized percentage points under RE versus 0.25 annualized percentage points under PF — a difference of roughly 0.77 percentage points. The average unemployment gap is −0.05 percentage points under RE versus −0.32 percentage points under PF — a difference of 0.27 percentage points. The paper also notes that model-implied averages for inflation and nominal interest rates “under perfect foresight can easily differ by at least one percentage point from their rational expectations counterparts.”

Q6: How do the simulated distributions differ between RE and PF under a Shortfalls rule?

Under PF, the simulated distributions of unemployment and inflation gaps exhibit a pronounced kink near the steady-state value (zero gap), reflecting the asymmetric treatment of expansions and contractions. Under RE, the distributions are substantially more symmetric, shifted to the right for inflation (mean of 1.0 versus 0.25 under PF). Standard deviations of the unemployment and inflation gaps are somewhat larger under PF (1.42 and 1.10, respectively) than under RE (1.33 and 1.03), because under RE the contractionary force from inflation expectations moderates the amplitude of fluctuations. These distributional differences have direct implications for how policymakers interpret the risks associated with state-contingent policies.

Q7: What is the role of the forward-looking pricing–central bank interaction in generating RE–PF differences?

The key mechanism is as follows: under RE, the possibility that the asymmetry may bind in the future (e.g., a positive demand shock triggering more accommodation under the Shortfalls rule, or a tight labor market steepening the Phillips curve) causes forward-looking firms to raise prices today in anticipation of future inflation. This increase in current inflation leads the central bank — whose mandate includes inflation stabilization — to raise policy rates, generating a contractionary offset even when the economy is not currently in the high-demand state. Under PF, agents do not form these anticipatory expectations, so this channel is entirely absent, and the asymmetry affects outcomes only when it directly binds.

Q8: Does the RE–PF divergence arise under a symmetric Deviations rule?

No. The paper shows analytically and numerically that when the monetary policy rule is symmetric (the Deviations rule, responding equally to deviations above and below target), the RE and PF solutions are identical. Unemployment and inflation gaps are both zero on average under either expectations assumption, and the policy rate gap is essentially zero (0.01 annualized percentage points) in both cases. This equivalence result confirms that the RE–PF divergence is not an artifact of the solution method or parameterization but is specifically generated by the interaction between an asymmetry and agents’ forward-looking behavior.

Q9: What do the bounded rationality results imply about the mechanism?

The extension following Gabaix (2020), with a myopia parameter m_br = 0.97, produces results that lie between the full-RE and PF cases: the adoption of the Shortfalls rule yields average unemployment of −0.26 percentage points (intermediate between RE’s −0.05 and PF’s −0.32) and average inflation of 0.62 annualized percentage points (between RE’s 1.02 and PF’s 0.25). This gradient confirms that the key driver is the extent to which agents internalize the probability of future shocks: the more forward-looking agents are, the more strongly the anticipatory pricing channel operates and the less favorable (and more inflationary) the apparent policy tradeoff becomes.

Q10: What are the results for the nonlinear Phillips curve in the numerical model?

Under the numerically calibrated nonlinear Phillips curve model (Panel B.3 of Table 3, with the slope increasing by a factor of three when the unemployment gap is negative), the average unemployment gap under RE is 0.63 percentage points versus 0.30 under PF, and the average inflation gap under RE is essentially zero (0.01 annualized percentage points) versus 0.41 under PF. The authors note that “the average outcomes for both unemployment and inflation can differ by roughly 0.3 to 0.4 percentage points between rational expectations and perfect foresight” in this case.

Q11: What is the paper’s advice for researchers who must use perfect foresight methods?

The paper explicitly states that PF methods remain valuable, especially for estimating or simulating larger models with heterogeneity at the micro level where RE solutions are computationally prohibitive. The authors recommend that researchers relying on PF to solve larger models “check the robustness of their conclusions on longer-run averages and the distribution of outcomes using simplified models which can be solved under both perfect foresight and rational expectations.” To support this, the authors provide multiple versions of code for solving simple macroeconomic models under various asymmetries and expectations assumptions.

Q12: How does the paper position its contribution relative to prior work on RE vs. PF in asymmetric models?

The paper acknowledges that Adam and Billi (2007) and Nakov (2008) previously documented that, at the zero lower bound, households’ anticipation of future ZLB episodes leads to lower average inflation — an RE–PF difference in the spirit of this paper’s findings. However, the paper’s contribution is to show that the sign and quantitative implications of a given asymmetry can change depending on the expectations assumption, and to systematically characterize this sensitivity across multiple types of asymmetry (asymmetric policy rules and nonlinear Phillips curves). The paper also categorizes the existing literature by expectations assumptions in Table A.1, showing that many papers examining macroeconomic asymmetries use only one approach.

Key Concepts

Shortfalls Rule: A monetary policy rule, motivated by the FOMC’s 2020 Statement on Longer-Run Goals and Monetary Policy Strategy, under which the central bank responds only to shortfalls of employment from its maximum level — i.e., it does not tighten policy in response to a tight labor market (negative unemployment gap) during an expansion. Formally, it = φπ πt + φu ut when ut ≥ 0 (labor market slack), and it = φπ πt only when ut < 0 (labor market tight). Contrasts with the symmetric Deviations rule that responds to deviations of employment in both directions.

Deviations Rule: A symmetric monetary policy rule in which the central bank responds to the unemployment gap regardless of its sign — tightening in expansions and easing in contractions. Serves as the baseline against which the Shortfalls rule is compared, and as the case in which RE and PF solutions are identical.

Perfect Foresight (PF) Equilibrium: An equilibrium in which agents solve their optimization problems assuming that no future shocks will occur — they expect all endogenous variables to converge to their longer-run (steady-state) values next period, regardless of the current state. In the paper’s notation, the PF transition matrix P^PF assigns probability one to the mean state next period. In linear models, PF and RE yield identical outcomes; in models with asymmetries, they diverge.

Rational Expectations (RE) Equilibrium: An equilibrium in which households and firms correctly account for the full stochastic distribution of future shocks in forming their expectations. Agents use the true Markov transition matrix P^RE for the natural rate process. This allows forward-looking pricing behavior to incorporate the possibility that the economy may enter states in which asymmetries bind in the future.

Nonlinear (Kinked) Phillips Curve: A Phillips curve in which the slope coefficient κ̃t is state-contingent, increasing when the unemployment gap is negative (labor market is tight). In the paper’s numerical implementation, the slope triples (κ̃ = 3κ) when ut < 0, consistent with empirical evidence in Smith, Timmermann, and Wright (2025) on structural breaks in the Phillips curve. The nonlinearity generates an asymmetric inflationary response: a given level of unemployment produces more inflation when the labor market is tight than when it is slack.

Stochastic Steady State: The equilibrium to which the economy converges in the absence of additional shocks, taking into account the stochastic nature of the environment (i.e., accounting for the possibility of future shocks). Used as the initial condition for computing impulse response functions under RE. Contrasts with the deterministic steady state (zero gaps), which serves as the initial condition under PF.

Parameterized Expectations (Global Solution) Method: The numerical solution algorithm used in the paper to solve for equilibrium policy functions for unemployment and inflation gaps over the state space. Implemented identically for RE and PF cases, differing only in the transition matrix used. Applied with 105 Rouwenhorst grid points for the natural rate. The paper shows this method is orders of magnitude faster than the more common shooting algorithm (0.04 seconds vs. 10.8 seconds) while yielding identical policy functions.

Bounded Rationality (Gabaix 2020): An extension of the baseline model in which agents discount the influence of future expectations by a myopia parameter m_br ∈ (0, 1), applied to both the IS curve and the Phillips curve. The parameter m_br = 0.97 (following McKay, Nakamura, and Steinsson 2017) limits the degree to which distant future states affect current decisions. Produces outcomes intermediate between full RE and PF, confirming that the key dimension of variation is the extent to which agents internalize the probability of future shocks.