Solving heterogeneous-agent models by projection and perturbation

The paper targets heterogeneous-agent general-equilibrium models in which “the state vector includes the whole cross-sectional distribution of wealth,” an infinite-dimensional object, and argues that reducing this to a handful of summary statistics – the dominant strategy following Krusell and Smith (1998) – “can hardly serve as a general approach” (Introduction, pp. 1-2). It notes that in models “where firms follow (S,s)-policies (price setting, inventory holdings etc.), we expect that the cross-sectional distribution of the relevant variables enters the solution in an essential way, and the solution algorithm may have to keep track of a medium- or high-dimensional representation of this distribution” (p. 2), motivating a method that does not presuppose a low-dimensional summary is adequate.

The method “combines features of projection methods and of perturbation methods,” computing “a solution that is fully nonlinear in the idiosyncratic shocks, but only linear in the aggregate shocks” (Introduction, p. 2; Section 3, p. 10). Step one solves the stationary equilibrium of the economy with the aggregate shocks turned off (constant technology and tax rate) but the full idiosyncratic productivity-shock process retained, using a finite (“projection”) parameterization of the consumption function and the wealth distribution. Step two computes a first-order perturbation of that finite-dimensional system’s dynamics with respect to small aggregate shocks, exploiting the fact that “with our approach we can include a very detailed representation of the cross-sectional distribution in the state vector” – up to 1000 state variables in the paper’s examples (Introduction, p. 2).

The consumption/savings function is represented by a cubic spline through np+1 “knot points,” with extra points concentrated near the liquidity-constraint kink because “the consumption function has high curvature and is more difficult to approximate with splines there” (Section 3.1.1, pp. 7-9, esp. footnote 3). The cross-sectional wealth distribution is represented as a histogram: its support is truncated at a maximum level and split into nd small intervals (up to nd=1000 in the baseline, nd=5000 for an accuracy check), with the probability mass in each interval stacked into a vector p_t (Section 3.1.2, pp. 8-10). The steady state is found by (1) solving the household’s Euler equation by collocation at the spline knots for a guessed capital stock, (2) computing the stationary histogram as the fixed point p* = Π* p* of the resulting Markov transition matrix via inverse iteration, and (3) using one-dimensional root-finding (bisection or Brent’s method) to find the capital stock consistent with that histogram (Section 3.2, pp. 11-12).

Collecting the full finite-dimensional system – consumption-function spline parameters, distribution histogram, transfers, and the exogenous shocks – into a state vector Θ_t, the paper writes the model’s equilibrium conditions as H(Θ_{t-1}, Θ_t, η_t, ε_t)=0 and linearizes this system around the steady state Θ, computing the needed partial derivatives “by forward differencing” rather than analytic or automatic differentiation, which the paper calls “clearly not computationally efficient… but it turns out to be good enough”* (Section 3.3, pp. 12-13). The resulting linear system is solved using Christopher Sims’s (2001) method for linear rational-expectations models; with nd=1000 state variables “Sims’ package then needs about 30 minutes to solve the model” (p. 13), with most of the time spent reordering eigenvectors.

The test economy is a standard incomplete-markets saving model – a continuum of ex ante identical households facing uninsurable idiosyncratic labor-productivity shocks and a borrowing (liquidity) constraint, with Cobb-Douglas production and CRRA utility – augmented with a government that levies a tax on capital purely “to create some random redistribution of wealth” (Section 2, pp. 4-5; Section 2.2, p. 4). The tax rate follows an AR(1) process chosen i.i.d. in the exercises specifically “to create unpredictable short-run redistributions” (Section 4, p. 15), so that, unlike the technology shock, it shifts the shape of the wealth distribution itself rather than just its scale – deliberately constructing “a laboratory in which one can test a method that uses a high-dimensional representation of the distribution” (Section 5.3.1, p. 19).

Both, depending on the shock: with only the technology shock active, the paper confirms approximate aggregation almost exactly – “the one-step R2 is 0.9999998” for a forecast of future capital using only its first moment – but once the redistributive tax shock is introduced, “a forecast based on one moment explains very little of the variation in aggregate capital,” and “even if we include 4 moments into the VAR, the forecast error is not always small in relative terms” (Section 5.3.1, pp. 18-19). The paper is explicit that this is a property of the exercise, not a claim that the specific tax-shock design is empirically realistic: “I do not claim that the specification with the big tax shocks is realistic; the purpose of the exercise is only to create a laboratory” for testing a high-dimensional method (p. 19). It also cautions that its moment-forecast numbers are “an upper bound” on what a Krusell-Smith-style solution could achieve, not a direct measurement of that method’s own solution error (p. 19).

Two error sources are checked separately: discretization error and state-space-reduction error. The average absolute Euler-equation residual (expressed as a relative consumption error) is on the order of 10^-7, though the maximum residual – concentrated near the liquidity-constraint kink where the consumption function is most curved – is about three orders of magnitude higher, around 10^-4 (Section 5.2, Table 3, pp. 17-18). Comparing the invariant distribution computed with nd=1000 versus nd=5000 intervals, the mean of the wealth distribution “is pinned down quite precisely” while higher moments (skewness, kurtosis) are harder to pin down exactly, though the paper judges accuracy “quite good” given that even minimal transition-matrix differences can compound over an infinite horizon (p. 17). For the reduced, spline-basis solution, the absolute 10-year-ahead forecast error of aggregate capital is “always around 10^-6 or lower” for spline bases of dimension 100, “better than the 4-moment forecast by at least one order of magnitude, often more” (Section 5.3.2, p. 20).

Rather than carrying the full nd-dimensional histogram as a state variable, the paper approximates deviations of the distribution from steady state by p_t ≈ p + B b_t for a chosen basis matrix B (e.g., splines) and a much smaller coefficient vector b_t, replacing the exact transition equation with a weaker projected version (Section 3.4.1, pp. 14-15).* For a further reduction, the paper simulates the model, applies principal-components analysis to the b_t coefficients that actually arise, and uses the leading principal components as a new, even lower-dimensional basis; it reports that “using 5 components gives somewhat better results than 10; using only 3, accuracy would drop by an order of magnitude,” while “using 10 or 30 principal components makes essentially no difference” (Section 5.3.2, p. 20), and cautions that this PCA-based shortcut “will not always work” and is offered mainly as a proof of concept that the linear solution can itself be used to search for good low-dimensional state aggregations (p. 20).

The method is explicitly a first-order (linear) approximation in the aggregate shocks only – “if we are satisfied with a first-order perturbation in the aggregate state variables, as we are in this paper, we can stop at this point” – and the paper flags this as best suited to settings “where individual shocks are much bigger than aggregate shocks, a typical situation in economic applications,” while noting that “even in cases where nonlinearity in aggregate shocks is essential, the linear solution is a useful first step” (Section 3.4.2, p. 15; Introduction, p. 3). It also notes unresolved theoretical ground: existence of the underlying continuum-of-agents stationary equilibrium is not proven for this exact model (“It has not yet been proven that a stationary equilibrium in Θ_t exists”), so the numerical procedure is offered as something that “can probably be seen as a ‘constructive proof’” only for the discretized, approximate economy (Section 2.5, p. 7). The paper’s own suggested next steps – testing the method on asset pricing, and computing genuinely nonlinear-in-aggregate-shocks solutions once a low-dimensional state representation is found adequate – are left to future work (Section 6, p. 20).

The paper positions itself between two existing strands: moment-reduction methods in the tradition of den Haan (1997) and Krusell and Smith (1998), which “try to represent the cross-sectional distribution of wealth by a very small number of statistics,” and full local-perturbation methods such as Kim, Kim, and Kollmann (2005) and Preston and Roca (2006), which “perturb the solution around the deterministic steady state, with no aggregate or idiosyncratic shocks and a degenerate cross-sectional distribution,” a purely local approach the paper says has trouble “handling of inequality constraints, for example liquidity constraints” (Introduction, pp. 1-2). It also notes a close contemporary, Algan, Allais, and Haan (2006), which “uses a flexible parameterization of the cross-sectional distribution and makes heavy use of projection methods,” and credits Bohacek and Kejak (2005) as similar in spirit to its own steady-state step, though “many details of the implementation differ” (Introduction, p. 2, footnote 1). The paper’s distinguishing feature relative to all of these is retaining full nonlinearity in the idiosyncratic dimension while accepting linearity only in the aggregate dimension, which is what permits the high-dimensional (nd up to 1000-5000) distributional state space.