🇬🇧 English · 🇪🇸 Español

Producer and purchaser accounting in the trade model

Producer/purchaser ("consistent") accounting is the only accounting the trade model offers; accounting="consistent" is the default, and any other value raises ValueError. This example runs as written on the hand-balanced two-country table used by the reference check below:

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

# Two countries (A, B), one good, final uses C and I, values in one currency unit.
Z = np.array([[10., 12.], [8., 14.]])                      # intermediate sales
F = np.array([[30., 8., 20., 20.], [15., 15., 50., 18.]])  # final sales to A-C, A-I, B-C, B-I
production_tax = np.array([4., 6.])
final_tax = np.array([2., 1., 3., 2.])
value_added = Z.sum(1) + F.sum(1) - Z.sum(0) - production_tax
data = np.vstack([np.hstack([Z, F]), np.r_[production_tax, final_tax],
                  np.r_[2*value_added/3, np.zeros(4)], np.r_[value_added/3, np.zeros(4)]])
calibration = calibrate_trade_model(data, ns=1, nc=2, nfd=2, country_codes=["A", "B"],
                                    sector_codes=["GOOD"])

# Tariff MULTIPLIERS (1 + rate), indexed (origin node, use, destination country):
# A charges 20% on imports from B, B charges 10% on imports from A.
intermediate_tariff_multipliers = np.ones((2, 1, 2))
intermediate_tariff_multipliers[1, 0, 0] = 1.2
intermediate_tariff_multipliers[0, 0, 1] = 1.1
final_tariff_multipliers = np.ones((2, 2, 2))
final_tariff_multipliers[1, :, 0] = 1.2
final_tariff_multipliers[0, :, 1] = 1.1

result = solve_trade_equilibrium(
    calibration,
    tau=intermediate_tariff_multipliers,
    tau_fd=final_tariff_multipliers,
    accounting="consistent",
    tol=1e-9,  # absolute, in the table's value units
)
assert result.converged
print(result.metadata["account_residuals"])

tol bounds absolute residuals in the calibration's value units. The attainable floor is roughly 1e-16 times the largest output value: tol=1e-9 suits the table above (values in tens), while the frozen OECD 3-region x 3-sector fixture in million USD (outputs up to 6.4e7) bottoms out near 5e-8; use 1e-5 to 1e-7 there, and rescale the tolerance when you change the currency unit. Consistent accounting needs a calibration whose final-use coefficients and expenditure shares are nonnegative: a table with negative final-use cells, or with a country whose investment share turns negative after the foreign-saving closure, is rejected with a ValueError. The bundled OECD table (load_icio_data()), native ingestion with package_mrio_to_calibration_result and the frozen OECD fixture all qualify.

The accounting uses the complete tariff schedule, homogeneous factor costs and the explicit closure below, and reports the effective conventions in metadata.

Prices, quantities and fiscal receipts

Let i index an originating country/sector, j a producing destination node, k a final-use category, and c a destination country. Prices are relative to baseline producer prices, which equal one. There are no iceberg or distribution margins in this model; these would require separate resource and valuation equations.

Item	Definition
Producer/border invoice price	p_i
Intermediate purchaser price	p_i τ^Z_ij
Final composite cost before local tax	Q_kc = Σ_i a^F_ikc p_i τ^F_ikc
Final composite purchaser price	P_kc = Q_kc / (1 − δ_kc)
Bilateral producer value	p_i × delivered quantity
Import duty	(τ − 1) × p_i × delivered quantity

Tariffs use the exporting good's price for both intermediate and final deliveries. They are paid once by the buyer and rebated once to the importing country. The local final tax is applied to tariff-inclusive expenditure. Domestic entries in the import-duty schedule must equal one. Subsidized imports are allowed through positive multipliers below one.

For a final-use expenditure E_kc, deliveries are

C_kc = E_kc / P_kc
F_ikc = a^F_ikc C_kc
local final tax = δ_kc E_kc

Thus producer value + import duties + local final tax equals purchaser expenditure in every category/country. The δ calibration is source final tax divided by actual source final purchases including that tax. Financial saving is excluded from this denominator. The calibration's stored tax_fd has a different denominator, so the share is recovered from the calibration table (or reconstructed from baseline budget coefficients when the table is absent).

Production combines calibrated input requirements (Leontief or CES sourcing) with Cobb–Douglas value added. The unit value-added cost is

v_j = (r_c / α_j)^α_j (w_c / (1 − α_j))^(1 − α_j) / β_j
(1 − t_j) p_j = v_j + intermediate purchaser cost per unit
production-tax receipt = t_j p_j Y_j
L_j = (1 − α_j) v_j Y_j / w_c
K_j = α_j v_j Y_j / r_c

These equations are homogeneous in nominal prices. A charge calibrated as a share of output revenue must collect t_j p_j Y_j, rather than t_j Y_j. The original raw OECD TLS row aggregates product taxes and subsidies; modeling its production-column amount as this output-revenue charge is an explicit calibration assumption. The source does not identify a full bilateral product or VAT tax schedule. This mode supplies a coherent reduced model, not a claim that those tax institutions have been independently identified.

These distinctions follow standard price and fiscal accounting principles: purchaser versus basic-price valuation in UN national accounts and production/duty receipts and a numeraire in the GAMS standard CGE example. Those sources motivate the conventions; the quantitative reference here is the separate two-country derivation below, not a replication of that GAMS model.

Budgets, foreign saving and the numeraire

All domestic taxes and import duties enter government receipts G_c. In the supported lump-sum closure, the solved transfer T_c equals G_c. Household income is I_c = w_c L_c + r_c K_c + T_c. Calibrated expenditure shares θ allocate it, with foreign saving B_c deducted from investment:

E_kc = θ_kc I_c − 1[k = investment] B_c
Σ_k E_kc + B_c = I_c
B_c = baseline foreign saving, fixed in numeraire units
Σ_c B_c = 0

Investment is category index 1, or index 0 for a single-category model. Native OECD ingestion first maps the six final uses to C/I/Cx. Final expenditure must be nonnegative in an accepted solution. The consistent mode rejects any negative final-use cell. On the native OECD 2019 table, I = GFCF + INVNT has 90 negative cells although every country's I total is positive; the count per category is recorded in calib.metadata["negative_final_demand_cells"]. Such a table needs aggregation or an explicit inventory model first. The FIGARO, EXIOBASE, WIOD and Eora loaders condense their final uses to C (final consumption), I (gross fixed capital formation) and Cx (changes in inventories and valuables, and EXIOBASE exports); unlike the OECD mapping, their Cx is not residents' purchases abroad. The mapping is in metadata["final_use_mapping"].

Fixing foreign saving is a substantive closure choice. Allowing all foreign balances to be endogenous while imposing national budgets leaves independent cross-country transfer choices unspecified. The coherent mode replaces that redundancy with baseline foreign balances. The first country/sector's producer price is fixed at one. Its redundant goods-clearing equation is omitted from the numerical square system, then checked independently before convergence can be reported. Realized exports minus imports must also recover B_c in every country, including the last one.

Because B_c is fixed in numeraire units by default (foreign_saving_units="numeraire"), the equilibrium is not homogeneous of degree zero in nominal prices when foreign saving is nonzero: entering the same table with a different country listed first changes the real allocation and Hicksian EV/CV (on the two-country welfare reference table, country A's EV is -27.55% of baseline consumption with A first and -45.99% with B first). Pass foreign_saving_units="world_income" to hold each country's baseline saving fixed as a share of world factor income, B_c = (B_c^0 / Σ_d (L_d + K_d)) Σ_d (w_d L_d + r_d K_d); the system is then homogeneous of degree zero and the results do not depend on the country order (the same example gives -36.95% for either order). The default is unchanged.

For full-size tables use solve_trade_equilibrium(method="equilibrated_newton"), which rescales the equations and unknowns before each Newton step. On failure its metadata["near_singular_country"] names the country whose direction of the Jacobian has become singular.

GDP at factor cost equals factor income. GDP at market prices adds domestic taxes and duties; it must equal final purchaser expenditure plus net exports valued at producer/border prices. Household and government budgets and all physical equilibrium equations are exposed in metadata["account_residuals"]. Solver success alone cannot certify convergence when these physical equations fail or demand is infeasible.

Reporting and solver scope

p_sol is the actual solved producer price. p_fd is Q; Pfd_final is P. c_fd is the actual composite quantity after foreign-saving allocation. Bilateral trade, imports and exports use producer values. Metadata exposes separate producer and purchaser value matrices, domestic-tax receipts and duties.

data_model_vf contains producer transaction values, local tax receipts and nominal factor payments. data_tariff_vf additionally includes import duties: production columns sum to producer output revenue, and final-use columns sum to purchaser expenditure. These tables contain nominal values throughout.

The reported cpi is a fixed-basket purchaser-price Laspeyres index divided by the wage relative to the baseline. metadata["nominal_cpi"] preserves the undeflated index. A supplied baseline must also use consistent accounting. This is an index, not Hicksian welfare or an expenditure-function derivation.

The mode supports NumPy, Leontief/CES intermediate sourcing and lump-sum rebates. Newton, sparse LU, Krylov, Broyden, SciPy hybrid/LM and full Keller continuation all solve the same equations. The Newton, sparse-LU, Krylov, Broyden and Keller solvers stop only when the solved residuals and the independently checked physical equations (the omitted goods equation and realized foreign balances) both meet tol, so a state the solver accepts is not rejected afterwards at the same tolerance; their finite-difference Jacobians scale the step of the value-unit transfer and foreign-balance unknowns with their magnitude, so the solve does not depend on the currency unit. SciPy hybrid/LM use SciPy's own termination test and are audited afterwards. A request for method="condensed" uses full Newton and records effective_method="newton". Quasi-condensed, other fiscal recycling schemes, accelerator backends and capacity penalties explicitly raise NotImplementedError, pending their own economic derivations.

Independent two-country check

The reference script uses a hand-balanced transaction table with two countries, one producing sector, two final uses, nonzero foreign saving, local taxes and heterogeneous tariffs. It derives the reference directly from the table, without calling puremacro's calibration, residual or GE routines inside the oracle.

With one sector per country, factor clearing fixes outputs. For each candidate relative producer price, intermediary quantities are fixed by Leontief inputs. Let h_kc be the share of final purchaser expenditure returned as local tax plus import duty. The oracle solves income algebraically:

I_c = [p_c Y_c − Σ_i p_i Z_ic − h_investment,c B_c]
      / [1 − Σ_k h_kc θ_kc]

It then clears one goods market with a scalar Brent root. The other goods market is checked separately. This provides independent prices, factor payments, deliveries and fiscal receipts at baseline and under a nonuniform tariff shock. The tests also check nominal homogeneity, the Jacobian's full rank after closure, all budget ledgers, currency scaling and the conserving OECD aggregation.

The reference code reproduces the check. The Hicksian consumption-welfare interface provides derived EV/CV and an endpoint attribution.