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

“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” (Abstract). The paper argues that rational expectations “imply that decision makers forecast equilibrium prices like interest rates by forecasting cross-sectional distributions,” which “leads to an extreme version of the curse of dimensionality,” and that the resulting computational difficulty is itself evidence against the assumption’s realism, not merely an implementation obstacle: “Macroeconomists are spending a lot of intellectual and computational horse power solving an unrealistically complex problem” (Introduction, pp. 1-2).

In a standard incomplete-markets heterogeneous-agent model (as in Aiyagari, 1994, and Krusell and Smith, 1998), equilibrium prices are functions of the cross-sectional distribution, p_t = P(G_t(x), z_t) (equation 9), so a household’s Bellman equation must include G_t as a state variable in order to forecast future prices – even though “households and firms do not ‘care about’ the cross-sectional distribution G_t, i.e. it does not enter their objective functions”* (Section 2.2, pp. 12-13). The paper writes out a two-period version explicitly (equation 10), noting the expectation over next period’s value function is taken “not only over future productivity realizations z’ but also over future distributions G’… because households understand that equilibrium prices at time t=1 depend on this distribution.” It stresses the curse of dimensionality “arises exclusively because G’ enters decision makers’ expectations and not because of the distribution evolving over time per se” – simulating the distribution forward given known policy functions is “computationally straightforward”; only forecasting it as part of an optimization is hard (Section 2.2, p. 14).

The paper argues the difficulty of solving the Master equation is itself evidence that real agents cannot be doing so: “It seems self-evident that households and firms do not forecast prices by forecasting cross-sectional distributions… If even our most advanced computational tools struggle with the ‘Monster equations,’ how can we justify the assumption that real-world households and firms solve the associated decision problems?” (Section 2.2, p. 14). It situates this in a long critical tradition, quoting Morgenstern (1935) on the “improbably high demands placed on the intellectual capacity of economic agents,” Manski (2004) on the near-impossibility of learning true objective probability distributions, and Adam and Marcet (2011) on the “enormous demands” rational expectations places on agents’ market knowledge (Section 2.2, pp. 15-16), while arguing the point “applies a fortiori” in the heterogeneous-agent case “because of the model economy’s high complexity.”

The paper distinguishes three approaches (Section 2.2, pp. 15-16): (1) “MIT shock” and linearization methods (e.g. Kaplan et al., 2018; Reiter, 2009; Ahn et al., 2018; Auclert et al., 2021), which impose certainty (or certainty equivalence) about prices and thereby “completely sidestep the key difficulty of price expectations,” at the cost of being unsuitable for questions “in which aggregate risk and non-linearities play a key role,” such as financial crises; (2) methods that solve the full Master equation globally (e.g. Schaab, 2020; Bilal, 2023; Gu et al., 2024), which have made “impressive computational advances” but remain, in the paper’s view, aimed at “an unrealistically complex problem” – the direct target of the paper’s critique; and (3) Krusell and Smith (1998)/Den Haan (1996)-style methods, which have decision makers forecast a small set of distributional moments rather than the whole distribution. The paper judges the moment-forecasting approach “lacks realism for similar reasons as Master equation methods,” conjecturing it “would fail on Criterion 2, consistency with empirical evidence” – except for the specific variant in which “the ‘Krusell-Smith moments’ are the prices themselves,” which the paper regards as “relatively more promising” and closest to its own proposal.

“The most natural alternative is to instead assume that [decision makers] forecast prices directly in some way” – replacing the rational expectation over future distributions with a subjective conditional probability distribution over future prices, P(p’|.), which may condition on current prices or other low-dimensional variables but never requires forecasting the full distribution (Section 3.1, pp. 17-18). Given such beliefs, “solving the model is not hard: given beliefs solve for actual prices and quantities such that households and firms maximize and markets clear” – what the paper calls a temporary equilibrium (Section 3.1, p. 18). The paper notes this differs from the Krusell-Smith approach (forecasting moments) as well as from rational expectations (forecasting the whole distribution), with the exception of the “prices-as-moments” Krusell-Smith variant, which shares its logic.

The three criteria (Section 3.3, pp. 19 onward) are: (1) Computational tractability – the alternative “should actually simplify the computational solution of heterogeneous agent models with aggregate risk (relative to the Master equation approach),” which explicitly rules out “any approach that begins with rational expectations as a special case and then derives the subjective probability distribution… by merely ’twisting’ this objective distribution,” since that still requires solving the rational-expectations problem as an intermediate step; (2) Consistency with empirical evidence – beliefs should be disciplined by the empirical literature on expectations formation, including survey data; (3) Endogeneity of beliefs to model reality (the Lucas critique), split into Criterion 3a (beliefs should be approximately consistent with actual outcomes in familiar, stationary settings) and Criterion 3b (beliefs should still respond sensibly in unfamiliar settings and under policy counterfactuals). The paper frames these criteria as tools for “navigat[ing] the ‘wilderness of non-rational expectations’” (a phrase from Sargent, 2008), intended to narrow, not eliminate, the space of viable alternatives.

A temporary equilibrium at time t is defined as allocations and prices such that “(i) households and firms optimize given expectations of future variables (including future prices) that are specified in the model but that are not necessarily rational, (ii) markets clear at time t” (Section 4.1, p. 26), a concept traced to Hicks (1939) and Lindahl (1939), of which rational-expectations equilibrium is the special case with model-consistent beliefs. The paper emphasizes that computing a temporary equilibrium is computationally cheap because “there is no fixed point between price beliefs and actual equilibrium prices” – only “slightly more difficult than the computation of a stationary equilibrium” in a standard Aiyagari-style model. It links this to Adam and Marcet’s (2011) concept of internal rationality, in which decision makers derive policy functions from fully specified intertemporal optimization given (possibly non-rational) subjective beliefs about variables outside their control, which the paper argues avoids the logical inconsistencies that can arise from simply substituting a different expectations operator into rational-expectations first-order conditions (Section 4.1, p. 27, citing Preston, 2005).

To satisfy Criterion 2, the paper argues alternative approaches “should incorporate the findings from the large empirical literature on expectations formation,” citing documented departures from rational expectations such as pervasive “belief disagreement” across households, overreaction to idiosyncratic news paired with underreaction to aggregate news, and “cross-domain extrapolation” (Section 4.2, pp. 27-29). It highlights “temporary equilibrium with measured expectations” (citing Piazzesi and Schneider, 2016) as a concrete implementation that disciplines subjective price beliefs directly with survey data, while flagging a practical concern that survey responses can contain substantial “cognitive noise” – compression of stated probabilities toward 50:50, especially among less certain respondents (citing Enke and Graeber, 2023) – that calls for care in interpreting such data rather than abandoning it.

Least-squares learning (Section 4.3, tracing to Bray, 1982, Marcet and Sargent, 1989, and others) has decision makers update the parameters of a perceived, typically simpler, forecasting model using past data, rather than knowing the true rational-expectations law of motion; the paper notes this is “a special case of a more general set of stochastic approximation methods,” a family that also includes reinforcement learning, which it defines as “learning value functions of incompletely-known Markov decision processes” and credits with “impressive advances in artificial intelligence” (Introduction, p. 3; Section 4.3-4.4). The paper explicitly connects least-squares learning to the Krusell-Smith approach – both involve positing a law of motion for forecasting variables and estimating its coefficients from data – while noting learning approaches differ by updating beliefs in real time rather than assuming they have already converged to their fixed point.

The paper is explicit that it offers a diagnosis, not a worked-out solution: “I only know the problem, not the solution!” and states that “implementing any of these approaches in practice will entail its own computational, empirical, and conceptual challenges, so my case for their promise remains necessarily speculative” (Introduction, p. 4; Section 3.2, footnote area). In its conclusion, the paper warns that heterogeneous-agent macroeconomics “risks repeating the mistakes of the pre-financial-crisis representative-agent literature” if it continues to rely mainly on linear or locally linearized solution methods, quoting Blanchard’s (2014) retrospective criticism of pre-crisis linearity, and closes by urging that work on alternatives to rational expectations be paired with the development of efficient global (not just local) solution methods for heterogeneous-agent models (Conclusion, pp. 38-39).

The paper notes an explicit link between its proposed price-forecasting approach and the “sequence space” methods of Auclert et al. (2021), since both focus on a low-dimensional vector of equilibrium prices/variables rather than the full distribution, but distinguishes its own aim as handling “stochastic price sequences outside steady-state neighborhoods, i.e. a global rather than local solution” (Section 3.1, p. 18). It also notes that writing a Bellman equation with prices as state variables under full rational expectations (rather than subjective beliefs) does not work as a shortcut, “because prices p do not follow a Markov process” – a point the paper says it develops further in its discussion of why local, linearization-based methods cannot simply be patched into a recursive global solution (Section 3.1, p. 18, referencing Section 3.5).