A Perturbational Approach for Approximating Heterogeneous Agent Models
📄 Summarized from the full manuscript · Human-reviewed for faithfulness before publication
In brief
Perturbation methods let economists quickly approximate models around their steady state, but doing this for households that differ from each other usually hits a wall: the population's distribution is infinite-dimensional, and standard techniques blow up in size as that distribution grows richer. This paper shows equilibrium responses of any order can instead be pinned down by simple, single-number derivatives solving small, easily-computed equations -- however large or kinked the population is. That matters: applied to a classic savings-and-borrowing economy, the method beats the leading rival at a basic approximation, and extends to welfare from stabilization policy, uncertainty spikes, and portfolio choice, none of which basic approximation alone can address.
What this paper finds — and why it matters
This paper develops a perturbational technique for approximating equilibria of a wide class of discrete-time heterogeneous-agent (HA) models with complex, possibly infinite-dimensional state spaces, including occasionally binding borrowing constraints that create kinks in policy functions. Its central insight is that traditional perturbation – which requires computing derivatives of policy functions with respect to every dimension of the state, an object whose size explodes with the dimension of that state – can instead be reformulated as finding a small number of scalar “directional derivatives”: values of policy-function derivatives evaluated along economically meaningful directions, such as the deterministic path traced out by a one-time (“MIT”) shock, or a direction capturing precautionary motives from anticipated risk. Because directional derivatives are always one-dimensional regardless of the underlying state’s dimensionality, and because the paper analytically derives the linear systems and recursive coefficient formulas that characterize them (using, among other inputs, generalized functions to handle kinks from borrowing constraints), the resulting method preserves the speed, simplicity and flexibility of classical representative-agent perturbation techniques (as in DYNARE) while remaining tractable at any approximation order in HA settings. Compared with the sequence-space-Jacobian method of Auclert, Bardóczy, Rognlie and Straub (2021), the paper’s approach is modestly faster at first order and, unlike that method, extends cleanly to second order and beyond, to models with stochastic volatility, to transition dynamics away from steady state, and to portfolio-choice problems in which risk premia and second-moment properties matter for even first-order dynamics. Illustrating the method on a Krusell-Smith-style economy with capital adjustment costs, the paper shows first-order approximation takes well under a second and second-order approximation only a few seconds more; that second-order terms materially change simulated aggregate paths relative to both the first-order and representative-agent counterparts, driven mostly by a precautionary-savings channel; that welfare from countercyclical labor-tax stabilization is exactly zero at first order but has an interior optimum at second order, with the popular “histogram method” for discretizing the distribution generating a quantitatively different (and, the paper shows analytically, generically incorrect) optimum at second order; that a five-times spike in aggregate uncertainty (calibrated to the VIX) generates aggregate and cross-sectional welfare losses substantially amplified relative to a representative-agent benchmark, concentrated among asset-poor households; and that letting households optimally choose between risky capital and safe debt materially changes the first-order response of aggregate capital relative to forcing households to hold a fixed portfolio.
Summary of a classic paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.
Questions & answers
Q1. What is the paper’s core reformulation of perturbation theory?
The paper shows that “aggregate equilibrium responses can be characterized using directional derivatives – values of policy function derivatives evaluated in suitably chosen directions,” so that “the directional derivative of any policy function remains scalar, ensuring its dimension is one regardless of the state variable’s dimensionality” (Introduction, p. 4). The first-order approximation corresponds to a directional derivative along the deterministic path following a one-time, unanticipated aggregate shock (“MIT shock”); second-order approximations use directional derivatives reflecting compounded MIT shocks at different dates plus a direction capturing precautionary motives from risk. All of these directional derivatives solve linear equations whose coefficients the paper derives analytically.
Q2. Why do occasionally binding borrowing constraints pose a special problem for perturbation, and how does the paper handle it?
Standard perturbation assumes policy functions are sufficiently differentiable at the steady state, an assumption “violated in many staple HA models due to the presence of occasionally binding borrowing constraints, which induce kinks in policy functions,” and these kinks are themselves endogenous, depending on the state of the economy (Introduction, p. 4). The paper shows such kinks “can be explicitly incorporated into analysis using generalized functions” – functions with delta-function-like components – letting it characterize the effects of kinks in policy functions and of mass points in the invariant distribution on equilibrium responses to aggregate shocks.
Q3. What is the paper’s central critique of the “histogram method” commonly used to discretize the distribution’s law of motion?
The paper shows that discretizing the transition probability kernel first and then perturbing (the standard approach following Reiter 2009) “generically misses some of the second-order order terms and does not converge to the correct second-order expressions even as the grid size shrinks to zero,” because “the histogram method locally linearizes the [law of motion] for the aggregate distribution, which misses terms capturing second-order responses of the [law of motion] to the first-order changes in policy functions” (section 6, pp. 32-35). By contrast, the paper’s approach derives exact analytical expressions for the theoretical (continuous) distribution and its law of motion first, and only discretizes afterward, guaranteeing convergence to the true solution as the grid shrinks.
Q4. How does the method’s first-order speed compare with the sequence-space-Jacobian approach of Auclert et al. (2021)?
In a calibrated Krusell-Smith economy with capital adjustment costs, the paper’s method takes 0.44 seconds to compute all first-order terms needed to simulate aggregates and compute ergodic moments, versus 0.51 seconds for the authors’ own implementation of the sequence-space-Jacobian method of Auclert et al. (2021) under the same calibration (section 7.1, Table 1, pp. 35-37). The paper attributes the modest speed gain to using exact analytical derivative expressions rather than the numerical differentiation that the sequence-space-Jacobian approach relies on; computing the additional terms needed for a full second-order approximation (curvature and precautionary/risk-adjustment terms) adds about 3 further seconds.
Q5. What does the paper find is the main driver of the gap between first- and second-order simulated paths?
Decomposing the second-order correction into a “curvature” term (reflecting nonlinearities embedded in aggregate production and investment technologies) and a “precautionary” term (reflecting household savings behavior near the borrowing constraint), the paper finds the curvature term is small in its calibration – consistent with aggregate policy functions being close to linear in neoclassical-growth-style models – while “this precautionary motive accounts for virtually all of the difference between the [first-order] and [second-order] lines” in simulated capital paths (section 7.2.1, pp. 37-39). The strength of this precautionary channel is governed by household risk aversion, the volatility of aggregate shocks, and the mass of households near the borrowing constraint.
Q6. What does the welfare application (Section 7.2.2) show about stabilization policy?
Extending the framework to compute a consumption-equivalent welfare measure Δ(τ_Θ) under alternative countercyclical labor-tax rules, the paper finds “the welfare gain is zero in the representative agent economy (Ricardian equivalence) and also to the first-order of approximation in the HA economy (certainty equivalence),” but at second order there is “a meaningful welfare tradeoff” with an interior optimum at τ_Θ = -0.84* – i.e., raising taxes by 0.84 percentage points for every one-point fall in TFP (section 7.2.1-7.2.2, pp. 39-40, Figure 3). Using the histogram method instead to compute the same welfare object yields a substantially different optimal cyclicality parameter of -1.04, illustrating that the second-order inconsistency identified in Q3 is quantitatively material for a genuinely second-order economic question.
Q7. What does the paper find about the welfare effects of a spike in aggregate uncertainty?
Calibrating a stochastic-volatility extension to the CBOE Volatility Index (VIX) over 1990-2023 and simulating a one-time five-times increase in the standard deviation of TFP, the paper finds aggregate welfare falls by 0.53% on impact, an effect “substantially amplified relative to the representative agent counterpart,” with individual-level welfare losses ranging from about 0.94% to 0.20% of consumption across the asset distribution and concentrated among asset-poor households closest to the borrowing constraint (section 7.2.3, pp. 40-41, Figure 4).
Q8. How does the paper’s portfolio-choice extension matter for first-order dynamics, and why is that surprising?
Extending the Krusell-Smith model to let households choose between risky capital and risk-free debt, the paper shows that portfolio choice matters “even for a first-order approximation of aggregates” – comparing the first-order response of aggregate capital to a TFP shock when households hold their optimal portfolios versus when they are forced to hold identical (non-optimized) portfolios, the response is “larger” under optimal portfolios (section 7.2.4, pp. 41-42, Figure 5), even though determining the optimal portfolio itself is inherently a second-order problem (since in a purely deterministic economy all assets are risk-free and portfolio shares are indeterminate).
Q9. How does this paper relate to other higher-order heterogeneous-agent solution methods covered in this reading list, such as Bilal (2023) and Auclert, Rigato, Rognlie and Straub (2026)?
The paper positions Bilal (2023) and related continuous-time “mean field game” approaches as complementary, sharing “the use of linear operators over infinite-dimensional spaces to analytically characterize the exact derivatives,” while noting those papers “do not consider economies in which policy functions have kinks that are functions of endogenous states, or settings with heteroskedastic shocks or portfolio problems” (section 6, p. 34) – precisely the features this paper’s discrete-time, directional-derivative approach is designed to handle. Auclert, Rigato, Rognlie and Straub (2026), written afterward, in turn describes this paper as providing “a hybrid of state- and sequence-space approaches” that “derives a second-order Volterra expansion for heterogeneous-agent models,” while extending the underlying logic to a third order “certainty correspondence” that (unlike this paper’s approach) requires no explicit manipulation of derivatives of the equilibrium function at all.
Key terms in this paper
Definitions below follow the paper's own usage.
- Directional derivatives
- The paper's key reformulation: instead of computing the full derivative of a policy function with respect to every dimension of a (possibly infinite-dimensional) state, compute only its derivative along a single, economically meaningful direction -- such as the direction traced out by a one-time, unanticipated ("MIT") shock, or a direction capturing precautionary motives from anticipated risk. Because a directional derivative is always scalar, "its dimension is one regardless of the state variable's dimensionality" (Introduction, p. 4), which is what allows the method to scale to rich heterogeneous-agent state spaces.
- Generalized functions (for kinked policy functions)
- Functions with delta-function-like components (as opposed to ordinary, everywhere-differentiable functions) that the paper uses to explicitly represent kinks in policy functions induced by occasionally binding borrowing constraints, and the associated mass points in the invariant distribution -- kinks that are themselves endogenous functions of the state and that violate the differentiability standard perturbation methods assume (Introduction, p. 4, fn. 2; the authors note this use of "distribution" is unrelated to probability distributions).
- Histogram-method inconsistency at second order
- The paper's finding that the standard "histogram method" used to discretize the transition probability kernel of the invariant distribution (following Young 2010) -- while adequate at first order -- "generically misses some of the second-order order terms and does not converge to the correct second-order expressions even as the grid size shrinks to zero," because it locally linearizes the distribution's law of motion and so misses second-order responses of that law of motion to first-order changes in policy functions (section 6, pp. 32-35, Appendix C).
- Curvature versus precautionary second-order terms
- The paper's demonstration (section 7.2.1) that, in a mapping from individual assets to a basis function of the generalized-moment-style approximation, aggregate policy responses to a one-time shock decompose into a curvature term (nonlinearity in aggregate technology/investment, found to be quantitatively small) and a precautionary term reflecting household saving behavior near the borrowing constraint (found to account for "virtually all" of the difference between first- and second-order simulated paths in the calibrated model).
- Welfare from countercyclical stabilization policy
- A worked application (section 7.2.2) in which the framework is extended to compute welfare, expressed as a consumption-equivalent change Δ(τ_Θ), under alternative countercyclical labor-tax rules τ_Θ. Welfare gains are exactly zero at first order (certainty equivalence / Ricardian equivalence) but strictly positive at second order in the heterogeneous-agent economy, with an interior welfare-maximizing tax-cyclicality parameter -- a result the paper shows is quantitatively distorted if the (first-order-consistent but second-order-inconsistent) histogram method is used instead.