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Joint sourcing, capital and labor transitions

puremacro.trade.reallocation solves an experimental two-country, two-sector competitive equilibrium with endogenous origin sourcing, capital accumulation and costly sector labor reallocation. Goods prices, consumption, factor prices and all three allocation margins are solved jointly. The defaults are synthetic assumptions, with pooled national household income.

This small model has its own explicit closure. The existing dynamic MRIO model has fixed sourcing and nationally mobile labor; conditional sourcing takes output and prices as given. Their returned paths cannot simply be combined into this equilibrium. There are no intermediate inputs, international migration, traded bonds, uncertainty or heterogeneous workers here.

Run the temporary tariff

AAA announces at date zero a 10% import tariff on sector S01 for dates 2–7. All other rates, including the terminal rates, remain zero.

import numpy as np
from puremacro.trade import (
    ReallocationEconomy, audit_reallocation, reallocation_stability,
    run_reallocation_horizon_ladder, solve_reallocation_steady_state,
)

economy = ReallocationEconomy()
initial = solve_reallocation_steady_state(economy)
assert reallocation_stability(initial)["stable_graph"]

def policy(horizon):
    tariffs = np.zeros((horizon, 2, 2))
    tariffs[2:8, 0, 0] = 0.10
    return tariffs

ladder = run_reallocation_horizon_ladder(economy, policy, initial=initial)
assert ladder.converged, ladder.to_frame()
run = ladder.accepted
assert run is not None and audit_reallocation(run)["accepted"]
print(ladder.to_frame())
print(run.welfare)
print(run.sector_frame().head())

The full example compares the same policy with each adjustment cost set to zero:

python -m puremacro.examples.trade_reallocation --output trade-example --lang en

It exports complete arrays (path-*.npz), national and sector accounts, period audit errors, welfare, horizon attempts, PNG/PDF figures, parameters, environment versions and source hashes. A failed horizon check remains in the record; an unresolved case makes the command exit unsuccessfully. The example tries 40, 80, 160 and, if needed, 320 dates. The API defaults stop at 160.

Economic quantities and timing

Capital \(K_{cs,t}\) is inherited. The household chooses current sector labor \(L_{cs,t}\), a current supplier recipe \(q_{cso,t}\), and next capital \(K_{cs,t+1}\). Initial labor and supplier ratios are inherited from an audited steady state. National labor supply \(\bar L_c\) is fixed. All dates are known initially.

Competitive firms produce \(Y_{cs}=Z_{cs}K_{cs}^aL_{cs}^{1-a}\), with \(w_{cs}=(1-a)p_{cs}Y_{cs}/L_{cs}\) and \(R_{cs}=ap_{cs}Y_{cs}/K_{cs}\). Each country's household owns its capital and pools both sectors' earnings.

The final composite is \(A_c=\prod_s(X_{cs}/b_{cs})^{b_{cs}}\), with positive weights summing to one. Sector goods combine origins using

\[ \left[\sum_o\alpha_{cso}(q_{cso}/\alpha_{cso})^\rho\right]^{1/\rho}=1, \qquad \rho=(\sigma-1)/\sigma. \]

Home and foreign weights are home_shares and 1-home_shares. These are preferences/technology weights, not fixed purchase shares. The code includes the Cobb–Douglas limit \(\sigma=1\). The choice coordinate is \(r_{cs}=\log[(q_{cs,foreign}/\alpha_{cs,foreign})/(q_{cs,home}/\alpha_{cs,home})]\). With importer tariffs applying only to foreign purchases,

\[ P_{cs}=\sum_o(1+\tau_{cso})p_{os}q_{cso},\quad P_c=\prod_sP_{cs}^{b_{cs}},\quad X_{cs}=b_{cs}P_cA_c/P_{cs},\quad Q_{cso}=X_{cs}q_{cso}. \]

Goods clear at every origin: \(Y_{os}=\sum_cQ_{cso}\). The producer price in BBB, S02 is the numeraire at every date.

Real adjustment costs and accounts

Let \(g_{cs}=K'_{cs}/K_{cs}-1\) and \(I_{cs}=K'_{cs}-(1-\delta)K_{cs}\). All adjustment costs consume national final goods:

\[ \Phi^K_{cs}=\frac{\kappa_K}{2}K_{cs}g_{cs}^2,\quad \Phi^L_c=\frac{\kappa_L}{2\bar L_c}\sum_s(L_{cs}-L_{cs,prev})^2,\quad \Phi^Q_{cs}=\frac{\kappa_Q}{2}\bar L_cb_{cs}(r_{cs}-r_{cs,prev})^2. \]

Absorption is \(A_c=C_c+\sum_s(I_{cs}+\Phi^K_{cs}+\Phi^Q_{cs})+\Phi^L_c\). Tariff revenue \(TR_c=\sum_{so}\tau_{cso}p_{os}Q_{cso}\) is rebated lump sum. The household budget is \(P_cA_c=\sum_s(w_{cs}L_{cs}+R_{cs}K_{cs})+TR_c\). Households take prices and rebates as given. Financial autarky implies balanced trade at producer prices each date. Investment is reversible and can be negative; capital, consumption, quantities, prices and marginal capital values must be positive.

With \(m_t=\beta(C_{t+1}/C_t)^{-\gamma}\), \(q^K=1+\kappa_Kg\) and \(l_t=L_{1,t}/\bar L\), the interior conditions are

\[ q^K_t=m_t[R_{t+1}/P_{t+1}+1-\delta+\kappa_K(g_{t+1}+g_{t+1}^2/2)], \]

\[ (w_{1,t}-w_{2,t})/P_t-2\kappa_L(l_t-l_{t-1}) +2m_t\kappa_L(l_{t+1}-l_t)=0, \]

\[ (A_t/\bar L)\partial_{r_{s,t}}\log P_{s,t} +\kappa_Q(r_{s,t}-r_{s,t-1})-m_t\kappa_Q(r_{s,t+1}-r_{s,t})=0. \]

The sourcing derivative multiplies gross absorption, including investment and all costs. Setting each cost to zero gives, respectively, the frictionless capital Euler equation, equal sector wages, or static CES sourcing. General equilibrium welfare need not improve monotonically when a cost is removed.

Terminal approximation and acceptance

A separate audited steady state uses the final tariff. The continuation value is linear in terminal capital, with derivative \(u'(C^*)/\beta\); derivatives with respect to inherited labor and sourcing are zero at stationarity. Thus the last capital condition is \(q^K_{T-1}=(C^*/C_{T-1})^{-\gamma}\) and the last forward labor/sourcing cost terms vanish. This is a first-order tail approximation.

run.converged means the finite equations and independent audit passed. It does not establish tail accuracy. The default horizon ladder requires:

Quantity	Limit
Solver and independent accounting error	\(10^{-8}\)
Maximum change in the first 20 log/logit/sourcing coordinates	\(10^{-6}\)
Change in consumption equivalents	\(10^{-4}\) percentage points
Maximum terminal coordinate gap from the steady state	\(10^{-5}\)

The last three compare successive accepted finite solves; all policy prefixes must match and extensions must keep the terminal policy. Failed solves break the comparison chain. These are observed refinement checks, not global uniqueness or rigorous infinite-tail bounds.

The solver uses 16 unknowns per date and a sparse three-diagonal block Jacobian with complex-step derivatives. One redundant budget is replaced by the numeraire. audit_reallocation rebuilds both budgets, bilateral goods, production, factor payments, fiscal and external balances, adjustment resources and every optimality condition from the economic arrays. It does not use the retained residual. Numerical log/logit bounds are ±30; solutions near them are rejected. The evaluation budget counts SciPy nfev, excluding Jacobian work. See SciPy least squares.

reallocation_stability checks the local stationary companion pencil using ordered generalized Schur decomposition. A stable graph requires 16 stable roots, including zero modes for variables without lags, a well-conditioned projection and a small matrix-equation error. The default calibration has 10 active inherited directions. A failed terminal graph rejects a transition by default. require_stable_terminal=False permits an explicitly finite boundary experiment; the horizon ladder always keeps the guard. This diagnostic is local and does not prove household global optimality.

Welfare and the illustrative calibration

run.welfare reports national consumption equivalents relative to initial steady consumption. It includes the recorded discounted consumption stream, constant terminal consumption, and the linear terminal-capital correction. In normalized utility the correction is \(\beta^{T-1}(C^*/C_0)^{-\gamma}\sum_s(K_{s,T}-K_s^*)/C_0\). The log-utility and near-log limits are supported. The result marks tail_is_approximation=True, even after refinement passes.

The initially considered sourcing cost 1.0 failed the local terminal graph (15 stable roots instead of 16) and the horizon pilot. It was retained as rejected, and the illustrative default was explicitly changed to 0.1. No tolerance was relaxed, and no universal upper bound on sourcing costs is inferred. Other defaults are \(\beta=.96\), \(\delta=.08\), \(a=.3\), \(\gamma=2\), \(\sigma=4\), \(\kappa_K=4\), \(\kappa_L=5\), home weight .75, equal sector weights, and unit productivity and labor supply.

For the example, all four cases pass at 160 dates. The full-model 80-to-160 comparison has coordinate gap \(3.78\times10^{-8}\), welfare gap \(6.23\times10^{-7}\) percentage points and terminal gap \(1.45\times10^{-7}\). The illustrative national consumption equivalents are about +0.1233% for AAA and −0.1469% for BBB. These synthetic values are not empirical tariff recommendations. Sector frames report wages and earnings, but pooled national income does not identify worker-specific welfare.

Labor-adjustment research such as Artuc, Chaudhuri and McLaren motivates studying adjustment explicitly. This continuous pooled-household model is not a replication of their worker-choice model. Larger dimensions, MRIO intermediates, borrowing, worker heterogeneity, empirical estimation and global optimality remain outside this first implementation.