Global Nonlinear Solutions in Sequence Space and the Generalized Transition Function

The RTM “utilizes the equilibrium’s ergodicity: if a simulated path of the aggregate shock is long enough, all possible equilibrium outcomes are realized somewhere on the path. Then, the conditional expectation at each period can be completely characterized by identifying the periods of each possible future state realization and combining the corresponding time-specific value functions” (as described in the paper’s framing of its own method; Introduction and Related literature, pp. 4-5). Unlike Marcet (1988), Den Haan and Marcet (1990), and Krusell and Smith (1997, 1998), which “use parametric expectations or laws of motion” and “face challenges with nonlinear dynamics due to difficulties in correctly specifying the law of motion,” the RTM “translates this expectation-formation problem into the sequence space… without requiring a moment-based summary or embedding distributions into the value function state space” (Related literature, p. 4).

“Reiter (2010) is the closest to the RTM in its treatment of the endogenous state distributions in heterogeneous-agent models,” introducing “a reference distribution, characterized by a few moments, which is updated across iterations of the simulation” to compute conditional expectations of future value functions without a full law of motion; “However, Reiter’s approach remains anchored in the state space, as it categorizes aggregate distributions via their moments,” and so “is still susceptible to the curse of dimensionality,” while the RTM “enables sharp identification of distributional states by matching across simulation periods” instead (Related literature, p. 4).

“While also operating in sequence space, the RTM differs fundamentally from Auclert et al. (2021), which achieve remarkable computational efficiency through sequences of Jacobians. Their approach enables rapid likelihood-based estimation but requires perfect foresight. The RTM, in contrast, handles aggregate uncertainty while maintaining computational efficiency” (Related literature, p. 5), and further “computes aggregate allocations and market-clearing prices directly on the simulated path without requiring law of motion specifications,” in contrast to perturbation and linearization methods generally (Reiter 2009; Boppart et al. 2018; Ahn et al. 2018; Bhandari et al. 2023).

“The generalized transition function (GTF)… captures short-run stochastic equilibrium dynamics over any possible exogenous state paths. Each GTF is a sub-path of the recursive competitive equilibrium (RCE), which nests generalized impulse response functions (GIRF)… and stochastic growth path[s]” (Introduction, pp. 2-3). Because the RTM’s global solution already computes the RCE, the GTF “immediately computes” as part of that solution rather than requiring separate high-order approximation machinery, opening up analysis of “endogenous interaction between the growth and the business cycle components” and “state-dependent shock responsiveness… obviating the need for analyses based on different steady states” (p. 3).

“The economy displays endogenous fragility: when a larger share of households are hand-to-mouth, moderate shocks generate disproportionately severe downturns. This helps explain why similar exogenous shocks can produce dramatically different outcomes across time” (Introduction, p. 3). Because the RTM solves the model globally and without assuming a fixed law of motion, it can capture how the state-dependent severity of a downturn depends on the prevailing share of constrained households – a form of nonlinearity that would be missed by any method linearized around a single steady state.

“Uncertainty dampens short-run growth when capital adjustment is subject to irreversibility. This highlights a novel interaction between uncertainty and growth that is typically overlooked in macro models” (Introduction, p. 3), which the paper connects to the “wait-and-see” mechanism of Bloom et al. (2018) but extends “into the economic growth context in the stochastic environment” using the GTF’s short-run stochastic growth path (Related literature, p. 6) – i.e., the paper is not merely showing uncertainty delays investment timing, but that it measurably slows the pace of short-run growth itself when adjustment is irreversible.

“Fiscal multipliers vary strongly with the distribution of hand-to-mouth households, confirming the mechanism in Kaplan and Violante (2014) within the fully nonlinear RCE framework,” with the mechanism “operat[ing] through labor supply elasticities that vary endogenously with the share of hand-to-mouth households” (Introduction, pp. 3, 6). On portfolios: “portfolio adjustment differs starkly across the wealth distribution: wealth-poor households maintain highly leveraged risky positions and rebalance aggressively over the business cycle, frequently hitting borrowing constraints – unlike wealth-rich households” (p. 3), connecting to the heterogeneous-portfolio-choice literature (Fagereng et al. 2017; Bayer et al. 2019; Auclert et al. 2024, 2025) but distinguished by embedding the mechanism in a fully nonlinear, globally-solved recursive competitive equilibrium.

“The response of the risk premium to a TFP shock is highly state dependent and can even change sign depending on the distribution of wealth portfolios” (Introduction, p. 3). Because sign-switching, state-dependent responses are inherently nonlinear phenomena invisible to a solution linearized around a single steady state, capturing this result specifically requires the RTM’s global, non-perturbative treatment of the model’s equilibrium dynamics across the full range of possible wealth distributions.

The paper is explicitly cross-cited by, and complementary to, these other methods: Auclert, Rigato, Rognlie and Straub (2026) group “alternative simulation-based methods (Lee 2025)” alongside neural-network-based approaches (Kase, Melosi and Rottner 2022; Han, Yang et al. 2021, i.e. DeepHAM) and structural reinforcement learning (Yang, Wang, Schaab and Moll 2025) as a genuinely “global” literature they view as “highly complementary” to their own perturbative certainty-correspondence approach. Distinctively, this paper’s RTM neither trains neural networks (unlike DeepHAM and Kase-Melosi-Rottner) nor learns policies via simulated reward-maximization (unlike structural reinforcement learning); instead it works directly with the sequence-space history of a single long simulation and matches realized states across time, making it, in the paper’s own framing, sharply distinguished from “moment-based state approaches” and “functional approximations” alike (Related literature, pp. 4-5).