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The Trouble with Rational Expectations in Heterogeneous Agent Models: A Challenge for Macroeconomics

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Pith's one-line read This essay argues that rational expectations about equilibrium prices should be dropped from heterogeneous-agent macroeconomics because they force agents to forecast entire cross-sectional distributions.

desk verdict A clearly-written and honest research agenda that restates a standard point about the Master equation, but its central 'hard for us, therefore unrealistic for agents' inference remains undefended against the as-if reply. read the letter →

arxiv 2508.20571 v1 pith:BKWXADUQ submitted 2025-08-28 econ.GN q-fin.EC

Benjamin Moll This is my paper · ORCID
classification econ.GNq-fin.EC
keywords rationalexpectationsheterogeneousagentsMasterequationcurseofdimensionalitypricetemporaryequilibriumleast-squareslearningreinforcement
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This essay argues that rational expectations about equilibrium prices—interest rates, wages, and similar price-like variables—should be abandoned in heterogeneous-agent macroeconomics because the assumption is conceptually implausible and computationally prohibitive. Under rational expectations, a household forecasting tomorrow's interest rate must first forecast the entire cross-sectional distribution of income and wealth, since equilibrium prices depend on that distribution. The resulting Bellman equation, called the Master equation or 'Monster equation,' has an infinite-dimensional state and is extremely hard to solve even with frontier methods. The essay proposes replacing it with models in which agents forecast prices directly using subjective beliefs, and offers three criteria for disciplining those beliefs: computational tractability, consistency with empirical evidence, and endogeneity of beliefs to model reality. A sympathetic reader would care because this reorientation could let heterogeneous-agent models address aggregate non-linearities such as financial crises.

What carries the argument

The load-bearing object is the Master equation, a Bellman equation for the value function $V(x, G, z)$ whose state includes the infinite-dimensional cross-sectional distribution $G$ of idiosyncratic states; the paper nicknames it the 'Monster equation.' It is what shows that rational expectations about prices force agents to forecast distributions: because equilibrium prices are functions of $G$ and $G$ evolves via a Chapman-Kolmogorov equation, prices alone are not Markov, and the rational expectation in equation (10) must integrate over $G'$. The proposed alternative machinery is a subjective price-belief distribution $P(p'|\cdot)$, which lets agents solve lower-dimensional Bellman equations in prices and idiosyncratic states, sidestepping the curse of dimensionality.

What would settle it

Survey or laboratory evidence that ordinary households' forecasts of interest rates and wages are as accurate as the objective forecasts computed from the full cross-sectional distribution of wealth—rather than from a simple function of current prices—would undercut the paper's central implausibility claim. A more direct test: in an experimental economy with a known distribution process, if subjects' price forecasts match the rational-expectations benchmark better than a price-only autoregressive rule, the claim that agents do not forecast via distributions is falsified.

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Extended reading notes

Core claim

The paper's central claim is that in any generic heterogeneous-agent model, rational expectations about equilibrium prices imply that decision makers solve a dynamic program in which the cross-sectional distribution $G$ is a state variable, because prices satisfy $p_t = \mathcal{P}^*(G_t, z_t)$ and $G$ is the Markov state while prices are not. Equation (10) makes the mechanism explicit: the expectation in the household's Bellman equation is taken over future distributions $G'$, so forecasting prices requires forecasting the entire distribution. The author argues this is not just computationally hard for modelers but psychologically implausible for real households and firms, who do not directly care about the distribution at all. The constructive proposal is to replace the rational expectation with a subjective probability distribution over future prices, $P(p'|\cdot)$, and to discipline that distribution by three criteria. If the paper is right, the field should shift from solving Master equations to computing temporary equilibria with disciplined subjective price beliefs.

Load-bearing premise

The argument depends on treating the computational difficulty of the Master equation for professional economists as evidence that real households and firms cannot behave as if they solved it, which is exactly the inference the 'as if' defense of rational expectations denies.

Editorial extensions

If this is right

  • If the argument is correct, the heterogeneous-agent research program should redirect effort from solving Master equations toward models with disciplined subjective price beliefs.
  • Such models can be computed as temporary equilibria, which are only modestly harder than stationary equilibria, because there is no fixed point between beliefs and actual prices at the level of the individual decision problem.
  • Candidate belief models must satisfy three criteria: tractability, consistency with survey and experimental evidence on expectations, and approximate consistency with model-generated price dynamics to retain some immunity to the Lucas critique.
  • Departing from rational expectations may allow heterogeneous-agent models to study aggregate non-linearities—financial crises, boom-bust cycles—that are currently out of reach.
  • Least-squares learning and reinforcement learning are both stochastic approximation algorithms, so convergence results from that theory can be used to analyze learning about equilibrium prices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • My inference: the argument cuts against Krusell-Smith style moment forecasting as well, since forecasting moments of the distribution to forecast prices is conceptually similar to forecasting the distribution itself; the paper hints at this but stops short of a full rejection.
  • My inference: if tractability is taken seriously as a criterion, it rules out any behavioral model that is defined as a 'twist' on rational expectations, such as diagnostic expectations or cognitive discounting, because those still require agents to compute the rational-expectations benchmark.
  • My inference: a testable extension would be to elicit household interest-rate and wage forecasts under different hypothetical aggregate states and compare them with the objective forecasts implied by a solved Master equation; the paper's claim predicts large, systematic gaps.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This essay argues that rational expectations about equilibrium prices in heterogeneous-agent macro models are unrealistic and should be replaced. The argument is that, under rational expectations, households and firms must forecast equilibrium prices by forecasting the entire cross-sectional distribution of idiosyncratic states, which makes the distribution a state variable in individual Bellman equations. The paper formalizes this in the two-period example of Eq. (10), shows in Section 3.5 that equilibrium prices are generically non-Markov even in a finite-state setting, and reviews why this leads to an extreme curse of dimensionality in the resulting 'Master equation.' The constructive part proposes that agents instead forecast prices directly using subjective beliefs P(p'|·), subject to three criteria: computational tractability, consistency with empirical evidence, and endogeneity of beliefs to model reality (Lucas-critique immunity). Sections 4.1-4.5 discuss temporary equilibrium and internal rationality, survey-expectation approaches, least-squares learning, reinforcement learning, and heuristics as candidate replacements. The paper explicitly states in footnote 7 that it proposes criteria rather than a concrete alternative, and describes several of its own candidates as speculative.

Significance. If the central thesis were established, the paper would justify a reorientation of the heterogeneous-agent research program away from solving Master equations and toward models with disciplined subjective price beliefs. The paper has real strengths: Eq. (10) correctly places the future distribution G' inside the expectation, Section 3.5 gives a clean finite-state proof that prices are not Markov, and the taxonomy of equilibrium concepts in Appendix B (REE, SCE, RPE, CEE, IREE, TE) is useful and well-referenced. The paper is also commendably honest about the limits of its constructive proposal. However, the significance is conditional: the paper does not establish that any alternative belief process satisfies the three criteria simultaneously, and it does not engage the standard 'as if' defense of rational expectations. As a programmatic challenge the essay is valuable; as a demonstration that rational expectations 'should be replaced' it remains incomplete.

major comments (3)
  1. [Section 2.2, 'The unrealism of rational expectations'] The paper's central inference from computational intractability for modelers to behavioral implausibility for agents is not defended against the standard Lucasian 'as if' reply. The text asks: 'If even our most advanced computational tools struggle with the Monster equation, how can we justify the assumption that real-world households and firms solve the associated decision problems?' But rational expectations is an equilibrium consistency condition, not a claim about the algorithms agents use internally. The manuscript does not discuss whether decentralized mechanisms—selection, evolutionary arguments, or simple learning heuristics—could produce approximately RE-consistent behavior without any agent literally solving a Master equation. Indeed, Section 4 later appeals to such mechanisms, but the connection to this objection is never made. As written, the computational difficulty supports a pragmatic research-priority claim, not the unconditional thesis of the abstract that rational expectations 'is unrealistic and should be replaced.' The authors should either weaken the thesis to a conditional challenge or directly address the as-if defense in the context of the full-distribution state.
  2. [Footnote 7 and Sections 3.3-4.5] The constructive half of the thesis requires the existence of a subjective belief process P(p'|·) satisfying Criteria 1, 2, and 3 simultaneously. Footnote 7 concedes 'I only know the problem, not the solution!', and Section 4 explicitly acknowledges that temporary equilibrium with measured expectations may fail Criterion 3b for policy counterfactuals (Section 4.2), that least-squares learning can be slow or non-convergent (Section 4.3), and that RL beliefs are conjectured to be 'close to' rational, which would make Criterion 2 hard to satisfy (Section 4.4). These admissions are honest, but they mean the paper's central claim rests on an unproved existence conjecture. The paper should be reframed as an explicitly conditional challenge, or it should provide at least one worked example of a belief process satisfying all three criteria in a nontrivial heterogeneous-agent model.
  3. [Section 3.5, last paragraph] The claim that tracking prices rather than the distribution converts the agents' problem into a POMDP does not by itself deliver Criterion 1. In a POMDP, the optimal policy is a function of the posterior belief about the hidden Markov state—which is again an N-dimensional or infinite-dimensional object. Moreover, the price observation is generated by the same high-dimensional latent state. The paper does not explain how the POMDP formulation avoids the curse of dimensionality unless agents are assumed to use drastically simplified, low-dimensional belief-updating rules. This gap is load-bearing because 'computational tractability' is the first criterion and the main promised payoff of Section 3.1. The paper should either specify the class of approximate belief-updating rules that make the POMDP tractable or state explicitly that tractability is only conjectural under the POMDP interpretation.
minor comments (4)
  1. [Section 3.3, Criterion 2] Typo: 'evidendence' should be 'evidence' in the paragraph beginning 'The second criterion is...'.
  2. [Section 4.4] Typo: 'macroeonomies' should be 'economies' in the sentence 'Heterogeneous-agent economies (and indeed real-world macroeonomies)...'.
  3. [Section 4.4, Figure 1] If the manuscript is published in its current form, permission to reproduce Figure 1 from Sutton and Barto (2018) must be secured; the figure appears to be a direct reproduction from a copyrighted textbook.
  4. [Section 5] The conclusion states that developing alternatives 'holds two main promises' and that realism and simplicity 'could be a rare case in which the two go hand in hand.' This is fine as a conjecture, but the conditional phrasing could be carried into the abstract, which currently states the thesis unconditionally.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the essay's claims are argumentative and self-contained; its formal implications are derived from the model, and the author's self-citations are not load-bearing.

full rationale

The paper contains no fitted parameter masquerading as a prediction and no derivation that presupposes its conclusion. The formal step (Eq. 10) is built from the equilibrium price dependence P*(G,z), market clearing, and the definition of rational expectations: if prices depend on the distribution, RE agents must form expectations over G'. Section 3.5's non-Markovian-price argument is likewise derived from the Chapman-Kolmogorov equation and the price function. The author's self-citations (Ahn et al. 2018; Achdou et al. 2021; Kaplan et al. 2018) are used only to locate solution methods (MIT shocks, linearization, KF equations) and standard technique, not to establish the thesis. Two passages weaken the argument but are not circular: Section 2.2's inference from computational difficulty to behavioral implausibility ('If even our most advanced computational tools struggle with the Monster equation, how can we justify...') leaves the Lucasian 'as if' defense unanswered, and footnote 7 concedes that no concrete replacement satisfying Criteria 1-3 is provided ('I only know the problem, not the solution!'). These are gaps in support and validity, not reductions of the claim to its own inputs, so the appropriate circularity finding is zero.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper performs no estimation and introduces no free parameters; the numeric objects in the illustrative models (discount factor, depreciation rate, double-well potential in Appendix A) are standard calibration-free examples. The central argument rests on three substantive assumptions: the inference that algorithmic intractability for modelers implies psychological implausibility for agents; the standard result that equilibrium prices generically depend on the full distribution and are not Markov; and the admittedly unproven existence of a subjective belief process satisfying all three proposed criteria. No new entities are postulated.

assumptions (4)
  • domain assumption A decision problem whose solution is computationally intractable for a modeler is implausible as a description of real agents' behavior.
    Load-bearing link in the central argument, stated in Section 2.2: 'If even our most advanced computational tools struggle with the Monster equation, how can we justify the assumption that real-world households and firms solve the associated decision problems?' This equates algorithmic hardness with psychological implausibility, which the Lucasian 'as if' defense contests.
  • domain assumption Equilibrium prices generically depend on the entire cross-sectional distribution (Eq. 9) and are therefore not Markov.
    Stated as generic in Section 2.2 and proved for the finite-state case in Section 3.5. It is standard in the cited literature (Krusell and Smith 1998; Rios-Rull 1997), so it is a safe, well-supported premise.
  • ad hoc to paper There exists a subjective price-belief process P(p'|.) that satisfies Criteria 1, 2, and 3 simultaneously.
    Required for the constructive proposal in Section 3.1 and Eq. (11). The paper explicitly disclaims it: footnote 7 'I only know the problem, not the solution!' and Section 4 'my case for their promise remains necessarily speculative.' If false, the claim 'should be replaced' loses its constructive support.
  • standard math Standard mathematical tools: Chapman-Kolmogorov equation (14), stochastic approximation convergence (Ljung 1977; condition (18)), TD-learning convergence (Jaakkola et al. 1993; Tsitsiklis 1994).
    Invoked in Sections 3.5, 4.3, and 4.4 as background mathematical tools; none are derived in the paper and none are central to the thesis itself.

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Pith. "Pith review of The Trouble with Rational Expectations in Heterogeneous Agent Models: A Challenge for Macroeconomics." pith.science (2026). https://pith.science/paper/BKWXADUQ

@misc{pith2026250820571,
  author       = {Pith},
  title        = {Pith review of: The Trouble with Rational Expectations in Heterogeneous Agent Models: A Challenge for Macroeconomics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BKWXADUQ}},
  note         = {Machine review of arXiv:2508.20571}
}
read the original abstract

The thesis of this essay is that, in heterogeneous agent macroeconomics, the assumption of rational expectations about equilibrium prices is unrealistic and should be replaced. Rational expectations imply that decision makers forecast equilibrium prices like interest rates by forecasting cross-sectional distributions. This leads to an extreme version of the curse of dimensionality: dynamic programming problems in which the entire distribution is a state variable ("Master equation" a.k.a. "Monster equation"). Frontier computational methods struggle with these infinite-dimensional Bellman equations, making it implausible that real-world agents solve the associated decision problems. These difficulties also limit the applicability of the heterogeneous-agent approach to central questions in macroeconomics -- those involving aggregate risk and non-linearities such as financial crises. This troublesome feature of the rational expectations assumption poses a challenge: what should replace it? I outline three criteria for alternative approaches: (1) computational tractability, (2) consistency with empirical evidence, and (3) (some) immunity to the Lucas critique. I then discuss several promising directions, including temporary equilibrium approaches, incorporating survey expectations, least-squares learning, and reinforcement learning.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.