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Reduced local stability of a trade equilibrium (experimental)

reduced_stability asks one narrow question about a converged trade equilibrium: if relative factor prices moved in response to their own excess demand while producer prices, gross outputs and national fiscal budgets cleared instantly, would a small displacement die out or grow? The answer is a classification (stable, unstable, nonhyperbolic or degenerate) of the eigenvalues of a reduced Jacobian, together with the checks that justify it. It is evidence about that one adjustment process. It is never a uniqueness result, and every StabilityResult carries that qualification in qualification.

import numpy as np
from puremacro.trade import calibrate_trade_model, solve_trade_equilibrium
from puremacro.trade.stability import reduced_stability

# Two countries, one sector, no intermediates, no taxes, balanced trade.
# Rows: goods A, goods B, taxes, labour, capital; columns: 2 intermediate, 2 final uses per country.
y0, alpha = np.array([100., 150.]), np.array([.3, .4])
F = np.array([[60., 40.], [40., 110.]])          # origin x destination final deliveries
data = np.zeros((5, 6))
data[3, :2], data[4, :2] = (1-alpha)*y0, alpha*y0
for c in range(2):
    data[:2, 2+2*c:4+2*c] = F[:, c][:, None]*np.array([.7, .3])[None, :]
calib = calibrate_trade_model(data, ns=1, nc=2, nfd=2, country_codes=["A", "B"])
base = solve_trade_equilibrium(calib, accounting="consistent", tol=1e-12)

report = reduced_stability(calib, base)
print(report.summary())
print(report.to_dataframe())
print(report.maximum_real_eigenvalue)         # 0.2222... = 2/9, the closed-form unstable root
print(report.qualification)

The reduced Jacobian of this toy has a closed form (the test file derives it), and its unstable root is exactly 2/9. The instability is economic, not numerical: with fixed physical baskets, demand for a country's good is price inelastic, so raising its factor prices raises its revenue and its excess factor demand.

What is computed

Write the physical residuals of the model as r(x) in the puremacro unknown layout x = [log p; log y; log r; log w; T; XN] (n = ns*nc cells, nc countries). The unknowns are split into a slow block z = (log r, log w) and a fast block u = (log p, log y, T); the foreign balances XN are held fixed. The rows are split into the factor rows f (labour and capital gaps) and the fast rows g (goods clearing, zero profit, fiscal budgets). The current-account rows are not imposed on the fast block: they certify the converged state and nothing else.

At every factor-price vector the fast block is cleared, g(u(z), z) = 0, and the excess factor-demand ratios are e(z) = -f(u(z), z) / endowment (positive means demand above endowment). Their Jacobian is the Schur complement

J_F = f_z - f_u g_u^{-1} g_z,        A = -J_F / endowment

with rows ordered (capital, labour) to match the columns (log r, log w); factor_labels gives the order r[c0], ..., w[c0], .... A is built from a central finite-difference Jacobian of the consistent-accounting residuals (puremacro.trade._accounting.evaluate). Complex step is unavailable because the state unpacker casts to float. The steps on the level unknowns (T, and XN in the naive partition below) are scaled by national income, because those unknowns enter the model linearly in calibration units and an unscaled step would drown their effect in roundoff.

Foreign balances are re-denominated so that the map is homogeneous of degree zero in all factor prices, A 1 = 0. The default balance_units="world_factor_income" fixes them in units of sum_k (w_k L_k + r_k K_k). Under the consistent accounting the identity

p.goods + (1 - tax) y.(pp - p) + w.labour_gap + r.capital_gap + 1.budget = 0

holds at every x, so on the fast-cleared manifold the value-weighted excess factor demand W.e vanishes (W = (r K, w L)), and differentiating it gives W A + W*e = 0, which is (r K, w L) A = 0 at an equilibrium. Every budget row is evaluated directly and the identity is reported as a check (metadata['world_identity_error']).

The relative law d log(f_i / f_ref) / dt = e_i - e_ref for every factor price other than the reference (default: the wage of country 0) gives the (2 nc - 1)-dimensional matrix R = A[keep, keep] - A[ref, keep]. Because A 1 = 0, every reference yields a similar R, so the spectrum does not depend on the reference. Optional country-factor speeds replace A by diag(speeds) A in the law, so the spectrum of R is then that of diag(speeds) A without its zero.

The optional dense check (dense=True, the default) recomputes A independently: it re-solves the fast block at perturbed factor prices by chord iteration (at most 30 iterations, stopping when goods rows are below 1e-12 max(1, max y), zero-profit rows below 1e-12 max(1, max p) and budget rows below 1e-12 max(1, max income)) and differences the resulting excess-demand ratios. The re-solve checks the nonlinear fast-cleared map itself rather than repeating the linear elimination of the same Jacobian.

Classification, with sign band tol and rank threshold rank_tol (both 1e-7 by default):

outcome condition
degenerate R is numerically rank deficient: singular values at or below rank_tol * max(1, largest) (a structural null space, or a fold)
unstable max Re eig(R) > tol
stable max Re eig(R) < -tol
nonhyperbolic otherwise

The naive partition that keeps the current-account rows in the fast block together with XN (current_account="fast") is structurally degenerate: its reduced matrix has rank nc, whatever the units of the table. The diagnostic reports this instead of a sign.

Closures and the result fields

argument meaning
balance_units "world_factor_income" (default, reference-invariant spectrum), "reference_factor_price" (homogeneous, but the spectrum moves with the reference whenever balances are nonzero, by 3.5e-2 on the synthetic fixtures), "numeraire" (the model's own closure, balances fixed in numeraire units; not homogeneous, reported with a warning). Not applicable with current_account="fast"
reference, reference_factor country and factor ("w" or "r") of the reference price
current_account "certificate" (default) or "fast" (naive, degenerate)
tau, tau_fd, tauf, tauf_fd tariff schedules; results carry theirs in metadata and explicit values must agree
speeds optional positive adjustment speeds, length 2 nc
step, fd_order finite-difference step for log unknowns, at most 1e-2 (levels T and XN are scaled by income; they enter linearly), and stencil order 2 or 4
dense, dense_step, dense_tol independent check by re-solving the fast block at perturbed factor prices (dense_step at most 1e-2); raises StabilityError on disagreement
tol, rank_tol sign band for the classification and relative singular-value threshold for the rank, set separately so that a wide sign band does not turn ordinary equilibria into degenerate ones
homogeneity_tol, equilibrium_tol threshold for the homogeneity and Walras identities, and for accepting the state as an equilibrium of its own equations

Where an identity is exact for the evaluated map, its failure is an error, not a caveat. Under the consistent accounting homogeneity is exact (with balance_units other than "numeraire", or in the naive partition) and so are the Walras and world identities; a violation beyond homogeneity_tol (world identity: 1e-10) can only come from finite-difference error, for example a step that is too small or too large, and raises StabilityError.

StabilityResult exposes classification, maximum_real_eigenvalue, eigenvalues (sorted by real part), reduced_jacobian (A), relative_matrix (R), rank, homogeneity_error (max|A 1| / max|A|), walras_error (max|W A + W*e| / max|W_i A_ij|, relative), fast_block_condition (after row and column equilibration, because the raw block mixes log unknowns with transfer levels in calibration units), equilibrium_residual and dense_check_error (both absolute), the adjustment, closure and qualification texts, reference, factor_labels, and metadata (eigenvectors, singular values, absolute spectrum of A, the world-identity check, the partition and all tolerances; its arrays are read-only). to_dataframe() lists the eigenvalues; mode_loadings(k) gives the right eigenvector of mode k on the factor-price labels (zero at the reference); to_markdown(), to_latex() and to_typst() render the table.

What the tests establish and what they do not

Verified in tests/test_trade_stability.py (tolerances stated there; observed values in parentheses):

  • closed-form reduced Jacobian of the two-country toy for all three balance denominations and both references, atol 1e-8 (2.3e-10 with the default second-order step 1e-6; 1.8e-13 with fd_order=4, step=1e-3);
  • Schur complement versus an independent fsolve re-solve of the fast block on the two-country fixture, 1e-8;
  • the built-in chord re-solve check on every fixture, 1e-8 (1.1e-9 to 2.8e-9);
  • homogeneity and the value-weighted Walras identity, 1e-8 (below 2e-9), also at an off-equilibrium point of the fast-cleared manifold where W A alone is not zero;
  • reference invariance of the spectrum under world factor income units (1.8e-10) and its documented dependence under reference factor price units;
  • adjustment speeds: 8 random draws in [1/4, 4] on the three-country fixture, where the relative spectrum equals the spectrum of diag(speeds) A without its zero (1e-8; observed at most 2.2e-10) and the classification stays unstable;
  • mode loadings are right eigenvectors of the law modulo the constant vector (1e-8; observed 5e-16);
  • degenerate naive partition (rank nc) on both synthetic fixtures, on the three-country table with flows multiplied by 1e6 and on the OECD 3x3 fixture in million USD; A itself is unchanged by that rescaling (1e-8);
  • the sign band and the rank threshold act separately;
  • eigenvalue continuity across 1e-3 and 1e-2 tariff changes;
  • Euler tatonnement on the nonlinear fast-cleared map: the excess-demand norm falls along the stable eigenvector and grows along the unstable one;
  • the OECD 3x3 fixture and the Hicksian two-country CES model.

Observations recorded by those tests, all closure dependent: the two synthetic puremacro fixtures classify unstable under fixed-coefficient sourcing (max Re +0.245 and +0.239); the maximum real eigenvalue falls monotonically with the intermediate substitution elasticity and turns negative at sigma = 1 (2c2s: +0.245, +0.114, -0.016, -0.278 at sigma 0, 0.5, 1, 2); the OECD 3x3 fixture does the same (+0.145, +0.050, -0.022 at sigma 0, 0.5, 1).

Not established: a local stability result is conditional on the specified relative-wage adjustment process; it is not a uniqueness result, and numerical verification does not establish empirical identification, global uniqueness, or stability under every possible adjustment process. The classification is never a uniqueness certificate, says nothing about basins of attraction, and does not estimate how factor prices actually adjust. Only the lump-sum fiscal closure without capacity margins or revenue recycling is supported. The finite-difference Jacobian costs 2 (2 n + 3 nc) residual evaluations, which is fine at a few hundred country-sectors and impractical at 45-sector native sizes without an analytic Jacobian.