Master equation in discrete time — Huggett example

Lecture note link

The code solves the steady state of a discrete-time Huggett economy (idiosyncratic income, a single implicitly priced bond with 1/q = 1 + r) and computes the interest-rate impulse response to a wealth-redistribution shock three independent ways:

• FAME (discrete-time) — First-order Approximation to the Master Equation: solves for the Impulse Value v(x,ξ) = ∂V(x)/∂g(ξ) and the linearized law of motion T' + G. This is Discrete-time companion to the continuous-time master-equation framework of Bilal, Solving Heterogeneous Agent Models with the Master Equation.
• SSJ — Sequence-Space Jacobian / "fake-news" algorithm of Auclert, Bardóczy, Rognlie & Straub (2021).
• MIT — fully nonlinear perfect-foresight transition (ground-truth validation).

The three agree up to truncation and numerical-differentiation error, which cross-validates the master-equation solution.

Repository structure

path	description
huggett_fame.jl	Model library: parameters, Rouwenhorst income (via QuantEcon), EGM household solver, steady state, the FAME solver solve_fame, the redistribution_shock, and the fame_irf helper. Functions only.
huggett_ssj.jl	Sequence-space Jacobian: fake-news primitives, expectation vectors, Jacobian assembly, and ssj_irf. Functions only.
huggett_mit.jl	Nonlinear perfect-foresight (MIT-shock) transition mit_transition. Functions only.
toplevel_huggett.jl	Toplevel driver. Loads the three libraries, solves the model, computes the FAME/SSJ/MIT interest-rate IRFs for one shock, prints a comparison table, and saves the combined figure.
bench_ssj_fame.jl	Performance benchmark producing the summary table below.
figure/	Generated figures (huggett_r_response_kappa*.pdf, and the spatial outputs spatial_irf_compare.pdf, spatial_timing.pdf + spatial_timing.csv).
Spatial/	Multi-region extension (trade + migration); see the section below.

The document/ folder (LaTeX derivation KrusellSmith_DiscreteTime.tex + compiled PDF) and literature/ (reference papers) are kept locally but are git-ignored.

Running

julia toplevel_huggett.jl     # solve + compare + save figure/huggett_r_response_kappa*.pdf
julia bench_ssj_fame.jl       # performance table

Dependencies: LinearAlgebra, Printf, Random, Plots, QuantEcon (and BenchmarkTools for the benchmark). Tested on Julia 1.12.

Performance

Time to compute one interest-rate impulse response on the same steady state (n = 300 states; SSJ horizon T = 300; MIT horizon T = 120):

method	median time	memory	speedup	output
SSJ (sequence-space Jacobian)	15.9 ms	20 MiB	1750×	price-path IRF
FAME (master equation)	~0.4 s	1.4 GiB	70×	Impulse Value v + law of motion T'+G
MIT (nonlinear transition)	27.8 s	57 GiB	1×	full nonlinear path

Speedup is relative to the slowest method (MIT). SSJ is fastest because it needs only one backward iteration plus a single T×T solve; FAME returns the richer Impulse Value object (useful for welfare and stability analysis) at the cost of a matrix fixed-point iteration; MIT re-solves household policies and re-clears the market every period and is used only as a ground-truth check.

The shock

A genuine wealth redistribution: a smooth compression of the asset distribution toward its cross-sectional mean, a ↦ a + κ(ā − a) (savers → borrowers), spread back onto the grid with the model's lottery. It conserves both population and aggregate assets exactly (Σ h₀ = 0, Σ aᵢ h₀ᵢ = 0), so the bond market only re-prices a reshuffling of the fixed asset stock.

Spatial extension (multi-region Huggett with trade & migration)

The Spatial/ folder scales the same FAME-vs-SSJ comparison up to a richer model: a multi-region Huggett economy with savings, CES goods trade across regions, and logit (type-I EV) migration — the status-quo economy of Donald, Fukui & Miyauchi (2025, Optimal Dynamic Spatial Policy), specialized to infinitely-lived households. The aggregate state is the distribution over (assets a, region j); within each period J prices clear (the bond rate r and wages w_2,…,w_J, with w_1 the numeraire), while populations L_j and CES price indices P_j follow from the distribution.

path	description
Spatial/SpatialHuggett.jl	Shared model module: parameters/grids, CES trade block, EGM household problem with logit migration, the (a,j) transition operator, and the nested steady-state fixed point.
Spatial/spatial_fame.jl	FAME for the spatial model: static/household/continuation Jacobians, the price-impact operator Q = dp/dg, the GE kernel G and law of motion T'+G, and the coupled fixed point fame_full.
Spatial/spatial_ssj.jl	Sequence-space Jacobian (fake-news) for the spatial model, the GE system over (r_t, w_jt, L_jt), and a nonlinear perfect-foresight transition.
Spatial/spatial_irf_compare.jl	Overlays the SSJ and FAME IRFs of every region to a productivity shock (colour = region, solid = SSJ, dashed+markers = FAME). → figure/spatial_irf_compare.pdf.
Spatial/spatial_timing.jl	Times FAME vs SSJ linearization as the number of regions J grows, appending results to figure/spatial_timing.csv as they are computed.
Spatial/spatial_timing_plot.jl	Reads figure/spatial_timing.csv and draws the timing comparison (linear + log panels) → figure/spatial_timing.pdf, so the figure can be restyled without re-running the benchmark.

julia Spatial/spatial_irf_compare.jl   # per-region IRFs, FAME vs SSJ
julia Spatial/spatial_timing.jl        # benchmark -> figure/spatial_timing.{csv,pdf}
julia Spatial/spatial_timing_plot.jl   # redraw figure from the stored CSV

Scaling. Both methods linearize the same model around the same steady state and agree to truncation/numerical error. FAME's cost grows much faster in J: it forms J dense n×n migration Jacobians and does J n×n matmuls per fixed-point iteration (n = n_a·J), whereas SSJ solves a (1+2J)·T sequence-space system. On the benchmark grid (n_a = 80, T = 150) the FAME/SSJ linearization times were:

J	n	SSJ	FAME
2	160	0.07 s	0.36 s
5	400	0.74 s	3.3 s
10	800	5.2 s	20.5 s
20	1600	38.5 s	141.6 s

(spatial_timing.jl has a wall-clock budget that stops attempting FAME at larger J once a run exceeds it, recording NaN; SSJ is always timed.)

Notes

Claude Opus 4.8 was used in designing and implementing the code.