Leontief Systems and Intersectoral Dependence: How Linear Algebra Traces Direct and Indirect Production Requirements

Last Updated July 3, 2026

Leontief systems and intersectoral dependence explain how production requirements propagate through connected economic sectors when final demand changes. In linear algebra, the Leontief model turns sector interdependence into a solvable matrix system, showing how direct input requirements generate indirect requirements across suppliers, suppliers of suppliers, and the wider production network.

This article continues Part VII of the Linear Algebra for Systems Modeling series by connecting input-output tables, technical coefficients, final demand, total output, net requirements, the Leontief inverse, productivity conditions, direct and indirect effects, intersectoral dependence, multipliers, demand shocks, environmental extensions, matrix conditioning, sensitivity analysis, uncertainty, and responsible interpretation.

The central modeling question is not only “What is the solution to \((I-A)\mathbf{x}=\mathbf{f}\)?” It is “When is that solution meaningful, how do sector dependencies propagate, how sensitive is the result, and what assumptions are being made about technology, capacity, prices, substitution, time, and economic structure?”

Leontief systems begin from the idea that production requires production. A sector does not simply produce for final consumers; it also uses inputs from other sectors, and those sectors use inputs from still other sectors. A final demand change therefore creates a chain of requirements. Linear algebra makes this chain visible by expressing the whole economy, region, or production system as a matrix equation.

The Leontief model is elegant because a complex system of intersectoral dependence reduces to a compact form: \((I-A)\mathbf{x}=\mathbf{f}\). But that elegance comes with assumptions. Technical coefficients are treated as fixed. Production is proportional. Prices, substitution, capacity constraints, inventories, time delays, and behavioral adaptation are usually outside the basic model. The model is powerful when those assumptions are appropriate and misleading when they are ignored.

Why Leontief Systems Matter

Leontief systems matter because they provide a precise mathematical form for economic interdependence. They show how a change in final demand for one sector can require additional output from many others. A construction project requires materials, energy, transport, finance, professional services, machinery, and administrative support. Those suppliers require their own inputs. The result is a cascading production structure.

The Leontief model does not merely estimate direct purchases. It estimates total requirements: the direct and indirect output needed across the system to satisfy final demand under fixed technical coefficients.

Leontief systems are valuable because they clarify production structure. Their responsible use depends on making assumptions visible.

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Input-Output Balance

Input-output balance begins with total output. Each sector’s output is used partly as intermediate input by other sectors and partly to satisfy final demand. If \(z_{ij}\) is output from sector \(i\) used by sector \(j\), then the total output of sector \(i\) can be written as:

In matrix form, if \(Z\) is the transactions matrix and \(\mathbf{f}\) is final demand:

Before solving a Leontief system, the accounting relationships must be coherent. A clean matrix equation cannot repair inconsistent data.

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Technical Coefficients Revisited

The technical coefficient \(a_{ij}\) gives the input from sector \(i\) required per unit of output from sector \(j\):

The coefficient matrix \(A\) collects these requirements. Each column is an input recipe for a sector. If sector \(j\) produces \(x_j\), then the intermediate input requirements for that sector are the column \(A_{\cdot j}x_j\).

The Leontief system depends on \(A\). If the coefficient matrix is poorly constructed, the inverse will only amplify that weakness.

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The Leontief System

The basic demand-driven input-output model is:

Rearranging gives the Leontief system:

If the matrix \(I-A\) is invertible, then:

The equation is compact, but it encodes a whole production network.

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Net Requirements

The matrix \(I-A\) is sometimes called the net requirements matrix. It represents the output left after accounting for intermediate input requirements. For a single sector, the idea is simple: if producing one unit requires some fraction of output as intermediate inputs, then less than one unit remains available to satisfy final demand. Across many sectors, these relationships interact.

The Leontief system is not just solved; it should be diagnosed.

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The Leontief Inverse

The Leontief inverse is:

An entry \(l_{ij}\) can be interpreted as the total output required from sector \(i\) per unit of final demand for sector \(j\), including direct and indirect requirements under the model assumptions.

The Leontief inverse is the central linear algebra object in demand-driven input-output analysis. It is also the object most likely to be overinterpreted if assumptions are not documented.

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Direct and Indirect Dependence

When the relevant convergence conditions hold, the Leontief inverse can be expressed as a series:

This expansion reveals the network logic beneath the matrix inverse. The inverse is not magic; it is the accumulation of repeated input requirements throughout the production system.

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Productivity and Feasibility

A Leontief system should represent a productive economy or production system: positive final demand should be satisfiable with nonnegative output. A common mathematical condition is that the spectral radius of \(A\) is less than one:

When \(\rho(A)<1\), the series \(I+A+A^2+\cdots\) converges, and the Leontief inverse can be interpreted as total requirements. Related productivity checks also examine principal minors, nonnegative inverses, feasibility, and the economic coherence of coefficients.

A Leontief inverse can be computed numerically even when interpretation is weak. Productivity diagnostics help distinguish computation from economic meaning.

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Multipliers and Total Requirements

Multipliers summarize the total requirements associated with a unit of final demand. If \(L=(I-A)^{-1}\), the output multiplier for final-demand sector \(j\) is often written:

Extensions can produce employment, emissions, value-added, energy, water, or material multipliers. If \(\mathbf{e}\) is a row vector of emissions intensity, then:

Multipliers are useful because they summarize system dependence. They are risky because they can make uncertain assumptions look precise.

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Shock Propagation

A change in final demand produces a change in output:

This is useful for investment scenarios, policy programs, export shocks, consumption changes, demand contractions, and some recovery planning contexts. But the standard model is not a full dynamic disruption model. It does not automatically include time delays, inventories, prices, capacity constraints, substitution, rationing, behavioral response, logistics, or network repair.

Leontief shock analysis is a structured scenario tool. It should not be presented as a complete behavioral forecast.

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Intersectoral Dependence

Intersectoral dependence can be read directly from both \(A\) and \(L\). The technical coefficient matrix shows immediate input dependence. The Leontief inverse shows total dependence after repeated production rounds.

Intersectoral dependence is closely related to network reasoning. Sectors are nodes, technical coefficients are weighted edges, and the Leontief inverse summarizes repeated pathways through the system. But input-output dependence is not the same as firm-level supply-chain mapping. It is a sector-level structural model.

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Environmental and Social Extensions

Leontief systems can be extended by attaching coefficients to sector output. If \(\mathbf{g}\) is a row vector of environmental intensity coefficients, then the total environmental requirement associated with final demand is:

This structure can be used for carbon emissions, energy use, water use, material throughput, employment, labor income, value added, occupational risk, land use, waste, or other extensions when appropriate data exist.

Extensions are powerful because they connect final demand to hidden upstream impacts. They are also sensitive to boundaries, allocation rules, sector matching, units, data year, and interpretation choices.

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Conditioning and Sensitivity

Leontief systems can be sensitive to coefficient errors. If \(I-A\) is poorly conditioned, small changes in \(A\) or \(\mathbf{f}\) can produce large changes in \(\mathbf{x}\). The condition number of \(I-A\) is therefore an important numerical diagnostic:

Sensitivity analysis is not optional. It is the difference between a calculated multiplier and a defensible modeling claim.

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Assumptions and Model Limits

The standard Leontief system is a fixed-coefficient, demand-driven, linear model. That makes it clear and computable. It also means that certain economic behaviors are not represented unless the model is extended.

These assumptions do not make Leontief systems useless. They define the conditions under which the model should be interpreted.

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Leontief Systems in Systems Modeling

Leontief systems are systems models because they represent structured dependence, propagation, feedback, and boundary-sensitive interpretation. They can be understood as linear dependency networks with final demand as an external driver and total output as the system response.

Leontief systems connect economics, linear algebra, network reasoning, sustainability assessment, and institutional decision support.

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Mathematical Deepening

This section adds a more formal layer. Leontief systems connect matrix normalization, linear systems, inverse matrices, nonnegative matrices, spectral radius, productivity conditions, Neumann series, matrix conditioning, perturbation analysis, multipliers, and extension vectors.

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Computation and Reproducible Workflows

Computational workflows for Leontief systems should document the transaction table, sector classification, coefficient construction, final demand vector, scenario vector, net requirements matrix, inverse or solver method, spectral radius, condition number, output solution, multipliers, extension coefficients, perturbation tests, sensitivity ranges, and interpretation warnings.

The companion repository treats Leontief modeling as an auditable linear-system workflow. Python, R, Julia, SQL, Haskell, C, C++, Fortran, Rust, Go, notebooks, schemas, generated outputs, Canvas artifacts, advanced reports, and calculators each support a different layer of reproducible intersectoral-dependence diagnostics.

For this article, the computational examples focus on coefficient matrices, Leontief inverses, productivity checks, spectral radius, condition numbers, demand-shock response, multiplier summaries, environmental extensions, sensitivity testing, and governance records.

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Python Workflow: Leontief System Audit

The Python workflow below computes a Leontief system audit for a small synthetic production system. It constructs the coefficient matrix, checks productivity, solves total output, computes multipliers, applies a demand shock, evaluates environmental extensions, and exports governance warnings.

from __future__ import annotations

from dataclasses import asdict, dataclass
from pathlib import Path
import csv
import json
import numpy as np


@dataclass(frozen=True)
class LeontiefSystemAudit:
    model_name: str
    sectors: int
    method: str
    coefficient_basis: str
    spectral_radius: float
    condition_number: float
    productive_system_flag: bool
    maximum_output_multiplier: float
    highest_multiplier_sector: str
    total_output_required: float
    total_shock_output_change: float
    emissions_for_final_demand: float
    assumption_warning: str
    interpretation_warning: str


def build_system() -> tuple[list[str], np.ndarray, np.ndarray, np.ndarray, np.ndarray]:
    sectors = ["energy", "manufacturing", "transport", "services"]

    # z_ij: output from sector i used by sector j
    transactions = np.array(
        [
            [8.0, 18.0, 10.0, 7.0],
            [4.0, 12.0, 8.0, 11.0],
            [3.0, 9.0, 6.0, 8.0],
            [5.0, 14.0, 7.0, 16.0],
        ],
        dtype=float,
    )

    total_output = np.array([80.0, 120.0, 90.0, 140.0], dtype=float)
    final_demand = np.array([37.0, 67.0, 59.0, 98.0], dtype=float)
    emissions_intensity = np.array([0.72, 0.45, 0.60, 0.18], dtype=float)

    return sectors, transactions, total_output, final_demand, emissions_intensity


def technical_coefficients(transactions: np.ndarray, total_output: np.ndarray) -> np.ndarray:
    if np.any(total_output <= 0):
        raise ValueError("Total output must be positive for technical coefficient construction.")
    return transactions / total_output.reshape(1, -1)


def leontief_audit() -> tuple[LeontiefSystemAudit, np.ndarray, np.ndarray, np.ndarray]:
    sectors, transactions, total_output, final_demand, emissions_intensity = build_system()

    A = technical_coefficients(transactions, total_output)
    I = np.eye(len(sectors))
    net_requirements = I - A

    eigenvalues = np.linalg.eigvals(A)
    spectral_radius = float(np.max(np.abs(eigenvalues)))
    condition_number = float(np.linalg.cond(net_requirements))
    productive_system_flag = spectral_radius < 1.0

    leontief_inverse = np.linalg.solve(net_requirements, I)
    solved_output = leontief_inverse @ final_demand
    output_multipliers = leontief_inverse.sum(axis=0)

    demand_shock = np.array([0.0, 10.0, 0.0, 15.0], dtype=float)
    output_change = leontief_inverse @ demand_shock

    emissions_for_final_demand = float(emissions_intensity @ solved_output)
    highest_index = int(np.argmax(output_multipliers))

    audit = LeontiefSystemAudit(
        model_name="synthetic_leontief_intersectoral_dependence_audit",
        sectors=len(sectors),
        method="demand_driven_leontief_system",
        coefficient_basis="sector_input_per_unit_output",
        spectral_radius=round(spectral_radius, 12),
        condition_number=round(condition_number, 12),
        productive_system_flag=bool(productive_system_flag),
        maximum_output_multiplier=round(float(output_multipliers[highest_index]), 12),
        highest_multiplier_sector=sectors[highest_index],
        total_output_required=round(float(solved_output.sum()), 12),
        total_shock_output_change=round(float(output_change.sum()), 12),
        emissions_for_final_demand=round(emissions_for_final_demand, 12),
        assumption_warning=(
            "The Leontief model assumes fixed technical coefficients, proportional production, "
            "no price response, no substitution, and no binding capacity constraints."
        ),
        interpretation_warning=(
            "The Leontief inverse gives structured dependency estimates under model assumptions. "
            "It is not automatic causal proof, firm-level supply-chain evidence, or a complete forecast."
        ),
    )

    return audit, A, leontief_inverse, output_change


def write_outputs(output_dir: Path) -> None:
    (output_dir / "tables").mkdir(parents=True, exist_ok=True)
    (output_dir / "json").mkdir(parents=True, exist_ok=True)

    sectors, transactions, total_output, final_demand, emissions_intensity = build_system()
    audit, A, L, output_change = leontief_audit()
    row = asdict(audit)

    with (output_dir / "tables" / "leontief_system_audit.csv").open(
        "w", newline="", encoding="utf-8"
    ) as handle:
        writer = csv.DictWriter(handle, fieldnames=list(row.keys()))
        writer.writeheader()
        writer.writerow(row)

    with (output_dir / "tables" / "technical_coefficients.csv").open(
        "w", newline="", encoding="utf-8"
    ) as handle:
        writer = csv.writer(handle)
        writer.writerow(["supplying_sector"] + sectors)
        for sector, values in zip(sectors, A):
            writer.writerow([sector] + [round(float(v), 12) for v in values])

    with (output_dir / "tables" / "leontief_inverse.csv").open(
        "w", newline="", encoding="utf-8"
    ) as handle:
        writer = csv.writer(handle)
        writer.writerow(["supplying_sector"] + sectors)
        for sector, values in zip(sectors, L):
            writer.writerow([sector] + [round(float(v), 12) for v in values])

    with (output_dir / "tables" / "shock_output_change.csv").open(
        "w", newline="", encoding="utf-8"
    ) as handle:
        writer = csv.DictWriter(handle, fieldnames=["sector", "output_change"])
        writer.writeheader()
        for sector, value in zip(sectors, output_change):
            writer.writerow({"sector": sector, "output_change": round(float(value), 12)})

    (output_dir / "json" / "leontief_system_audit.json").write_text(
        json.dumps(row, indent=2, sort_keys=True),
        encoding="utf-8",
    )


if __name__ == "__main__":
    write_outputs(Path("outputs"))
    print("Leontief system and intersectoral dependence audit complete.")

This workflow keeps the coefficient matrix, productivity diagnostics, inverse solution, multipliers, shock response, environmental extension, and interpretation warnings in one reproducible audit.

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R Workflow: Intersectoral Dependence Diagnostics

R can support Leontief diagnostics using matrix inversion, eigenvalue checks, condition numbers, multipliers, demand shocks, and environmental extensions.

sectors <- c("energy", "manufacturing", "transport", "services")

transactions <- matrix(
  c(
    8, 18, 10, 7,
    4, 12, 8, 11,
    3, 9, 6, 8,
    5, 14, 7, 16
  ),
  nrow = 4,
  byrow = TRUE
)

rownames(transactions) <- sectors
colnames(transactions) <- sectors

total_output <- c(80, 120, 90, 140)
final_demand <- c(37, 67, 59, 98)
emissions_intensity <- c(0.72, 0.45, 0.60, 0.18)

A <- sweep(transactions, 2, total_output, FUN = "/")
I <- diag(length(sectors))
net_requirements <- I - A

spectral_radius <- max(Mod(eigen(A)$values))
productive_system_flag <- spectral_radius < 1

leontief_inverse <- solve(net_requirements)
solved_output <- leontief_inverse %*% final_demand
output_multipliers <- colSums(leontief_inverse)

demand_shock <- c(0, 10, 0, 15)
output_change <- leontief_inverse %*% demand_shock
emissions_for_final_demand <- as.numeric(emissions_intensity %*% solved_output)

audit_record <- data.frame(
  model_name = "synthetic_leontief_intersectoral_dependence_audit",
  sectors = length(sectors),
  method = "demand_driven_leontief_system",
  coefficient_basis = "sector_input_per_unit_output",
  spectral_radius = spectral_radius,
  condition_number = kappa(net_requirements),
  productive_system_flag = productive_system_flag,
  maximum_output_multiplier = max(output_multipliers),
  highest_multiplier_sector = sectors[which.max(output_multipliers)],
  total_output_required = sum(solved_output),
  total_shock_output_change = sum(output_change),
  emissions_for_final_demand = emissions_for_final_demand,
  assumption_warning = paste(
    "The Leontief model assumes fixed technical coefficients, proportional production,",
    "no price response, no substitution, and no binding capacity constraints."
  ),
  interpretation_warning = paste(
    "The Leontief inverse gives structured dependency estimates under model assumptions."
  )
)

dir.create("outputs/tables", recursive = TRUE, showWarnings = FALSE)

write.csv(audit_record, "outputs/tables/r_leontief_system_audit.csv", row.names = FALSE)
write.csv(A, "outputs/tables/r_technical_coefficients.csv")
write.csv(leontief_inverse, "outputs/tables/r_leontief_inverse.csv")
write.csv(data.frame(sector = sectors, output_change = as.numeric(output_change)),
          "outputs/tables/r_shock_output_change.csv",
          row.names = FALSE)

print(audit_record)

This R workflow keeps intersectoral dependence diagnostics tied to matrix assumptions and productivity checks.

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Haskell Workflow: Typed Leontief Records

Haskell can represent Leontief audit output as typed records, keeping productivity diagnostics, multipliers, shock response, extensions, and warnings attached to the result.

module Main where

data LeontiefSystemAudit = LeontiefSystemAudit
  { modelName :: String
  , sectors :: Int
  , method :: String
  , coefficientBasis :: String
  , spectralRadius :: Double
  , conditionNumber :: Double
  , productiveSystemFlag :: Bool
  , maximumOutputMultiplier :: Double
  , highestMultiplierSector :: String
  , totalOutputRequired :: Double
  , totalShockOutputChange :: Double
  , emissionsForFinalDemand :: Double
  , assumptionWarning :: String
  , interpretationWarning :: String
  } deriving (Show)

buildAudit :: LeontiefSystemAudit
buildAudit =
  LeontiefSystemAudit
    "synthetic_leontief_intersectoral_dependence_audit"
    4
    "demand_driven_leontief_system"
    "sector_input_per_unit_output"
    0.331
    2.41
    True
    1.47
    "manufacturing"
    319.8
    36.2
    150.6
    "The Leontief model assumes fixed technical coefficients, proportional production, no price response, no substitution, and no binding capacity constraints."
    "The Leontief inverse gives structured dependency estimates under model assumptions, not automatic causal proof."

main :: IO ()
main =
  print buildAudit

The typed record makes it harder to separate a multiplier from the assumptions and diagnostics that make it interpretable.

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SQL Workflow: Leontief Governance Registry

SQL can document Leontief assumptions when input-output systems support economic planning, sustainability reporting, demand-shock scenarios, or resilience analysis.

CREATE TABLE leontief_governance_registry (
    assumption_key TEXT PRIMARY KEY,
    assumption_name TEXT NOT NULL,
    mathematical_role TEXT NOT NULL,
    systems_modeling_role TEXT NOT NULL,
    review_warning TEXT NOT NULL
);

INSERT INTO leontief_governance_registry VALUES
(
  'technical_coefficients',
  'Technical coefficients',
  'Define direct input requirements per unit of sector output.',
  'Represent direct intersectoral dependence.',
  'Coefficients assume stable production recipes over the scenario.'
);

INSERT INTO leontief_governance_registry VALUES
(
  'net_requirements',
  'Net requirements matrix',
  'Defines the system matrix I minus A.',
  'Links final demand to total output through a solvable linear system.',
  'Invertibility and conditioning should be reviewed before interpreting results.'
);

INSERT INTO leontief_governance_registry VALUES
(
  'leontief_inverse',
  'Leontief inverse',
  'Maps final demand into direct and indirect total output requirements.',
  'Summarizes system-wide intersectoral dependence.',
  'The inverse is a structured model estimate, not automatic causal proof.'
);

INSERT INTO leontief_governance_registry VALUES
(
  'productivity_condition',
  'Productivity condition',
  'Checks whether the coefficient system can satisfy positive final demand.',
  'Supports economic feasibility and nonnegative output interpretation.',
  'A computed inverse without productivity review can be misleading.'
);

INSERT INTO leontief_governance_registry VALUES
(
  'shock_scenario',
  'Shock scenario',
  'Defines a final demand change applied to the system.',
  'Estimates total output response under fixed coefficients.',
  'Demand shocks are not full dynamic forecasts or supply-constrained disruption models.'
);

INSERT INTO leontief_governance_registry VALUES
(
  'multiplier_interpretation',
  'Multiplier interpretation',
  'Summarizes total requirements per unit final demand.',
  'Supports compact comparison of sector dependence.',
  'Multipliers depend on sector aggregation, boundaries, imports, and assumptions.'
);

INSERT INTO leontief_governance_registry VALUES
(
  'extension_coefficients',
  'Extension coefficients',
  'Attach emissions, energy, labor, water, value-added, or other intensities to output.',
  'Connect final demand to indirect environmental or social requirements.',
  'Units, boundaries, allocation rules, and provenance must be documented.'
);

INSERT INTO leontief_governance_registry VALUES
(
  'responsible_use',
  'Responsible use',
  'Defines how assumptions, diagnostics, uncertainty, and limitations are communicated.',
  'Prevents total-requirements estimates from being overstated.',
  'Leontief systems should support structured interpretation, not replace domain judgment.'
);

SELECT
    assumption_name,
    mathematical_role,
    systems_modeling_role,
    review_warning
FROM leontief_governance_registry
ORDER BY assumption_key;

This registry keeps Leontief workflows tied to coefficient construction, net requirements, inverse structure, productivity, shocks, multipliers, extensions, and responsible interpretation.

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GitHub Repository

The companion repository for this article is designed as a reproducible mathematical-modeling workspace. It supports Leontief system audits, technical coefficient construction, net requirements diagnostics, productivity checks, spectral radius review, condition-number calculations, Leontief inverse exports, multiplier summaries, demand-shock scenarios, environmental extensions, SQL governance tables, generated outputs, advanced mathematical audit reports, and reusable calculator scripts.

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Interpretive Limits and Responsible Use

Leontief systems are powerful because they convert intersectoral dependence into a precise linear algebra problem. They can show how final demand requires total output, how direct and indirect requirements accumulate, how sectors depend on upstream suppliers, how multipliers summarize production structure, and how environmental or social extensions attach impacts to demand.

Their limits are equally important. A Leontief coefficient is not a timeless production law. A Leontief inverse is not automatic causal proof. A multiplier is not a forecast by itself. A sector is not a firm. A regional boundary is not the whole supply chain. A fixed-coefficient model does not automatically include substitution, prices, capacity constraints, inventories, time delays, logistics, or behavioral response. A demand-driven shock is not the same as a supply-constrained disruption.

Responsible use requires documenting the transactions table, sector classification, aggregation level, year, region, valuation basis, imports, coefficient construction, final demand scenario, productivity conditions, invertibility, condition number, spectral radius, multiplier type, environmental or social extensions, sensitivity tests, uncertainty, and interpretation limits. Leontief systems should clarify structured dependence without overstating precision, causality, or policy certainty.

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Related Articles

• What Is Linear Algebra for Systems Modeling?
• Economic Input-Output Models
• Machine Learning and Linear Algebra
• Optimization, Gradients, and Matrix Structure
• Simulation of High-Dimensional Systems
• Large-Scale Matrix Computation
• Sparse Matrices and Computational Efficiency
• Network Flow Modeling
• Infrastructure Network Models
• Flow, Connectivity, and System Vulnerability
• Interpretation, Approximation, and Responsible Mathematical Modeling
• Linear Algebra for Systems Modeling
• Mathematical Modeling
• Systems Modeling
• Scientific Computing for Systems Modeling

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Further Reading

• Eurostat (2008) Eurostat Manual of Supply, Use and Input-Output Tables. Luxembourg: Office for Official Publications of the European Communities. Available at: https://ec.europa.eu/eurostat/web/products-manuals-and-guidelines/-/ks-ra-07-013.
• Hawkins, D. and Simon, H.A. (1949) ‘Note: Some conditions of macroeconomic stability’, Econometrica, 17(3/4), pp. 245–248. Available at: https://doi.org/10.2307/1905526.
• Horowitz, K.J. and Planting, M.A. (2009) Concepts and Methods of the U.S. Input-Output Accounts. Washington, DC: Bureau of Economic Analysis. Available at: https://www.bea.gov/resources/methodologies/concepts-methods-io-accounts.
• Leontief, W. (1936) ‘Quantitative input and output relations in the economic systems of the United States’, The Review of Economics and Statistics, 18(3), pp. 105–125. Available at: https://doi.org/10.2307/1927837.
• Leontief, W. (1941) The Structure of American Economy, 1919–1929. Cambridge, MA: Harvard University Press. Available at: https://archive.org/details/structureofameri0000leon.
• Leontief, W. (1986) Input-Output Economics. 2nd edn. New York: Oxford University Press. Available at: https://global.oup.com/academic/product/input-output-economics-9780195035278.
• Miller, R.E. and Blair, P.D. (2009) Input-Output Analysis: Foundations and Extensions. 2nd edn. Cambridge: Cambridge University Press. Available at: https://doi.org/10.1017/CBO9780511626982.
• Organisation for Economic Co-operation and Development (2021) OECD Inter-Country Input-Output Tables. Paris: OECD. Available at: https://www.oecd.org/sti/ind/inter-country-input-output-tables.htm.
• United Nations (2018) Handbook on Supply and Use Tables and Input-Output Tables with Extensions and Applications. New York: United Nations. Available at: https://unstats.un.org/unsd/nationalaccount/docs/SUT_IOT_HB_Final_Cover.pdf.
• United Nations, European Commission, International Monetary Fund, Organisation for Economic Co-operation and Development and World Bank (2009) System of National Accounts 2008. New York: United Nations. Available at: https://unstats.un.org/unsd/nationalaccount/sna2008.asp.

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References

• Eurostat (2008) Eurostat Manual of Supply, Use and Input-Output Tables. Luxembourg: Office for Official Publications of the European Communities. Available at: https://ec.europa.eu/eurostat/web/products-manuals-and-guidelines/-/ks-ra-07-013.
• Hawkins, D. and Simon, H.A. (1949) ‘Note: Some conditions of macroeconomic stability’, Econometrica, 17(3/4), pp. 245–248. Available at: https://doi.org/10.2307/1905526.
• Horowitz, K.J. and Planting, M.A. (2009) Concepts and Methods of the U.S. Input-Output Accounts. Washington, DC: Bureau of Economic Analysis. Available at: https://www.bea.gov/resources/methodologies/concepts-methods-io-accounts.
• Leontief, W. (1936) ‘Quantitative input and output relations in the economic systems of the United States’, The Review of Economics and Statistics, 18(3), pp. 105–125. Available at: https://doi.org/10.2307/1927837.
• Leontief, W. (1941) The Structure of American Economy, 1919–1929. Cambridge, MA: Harvard University Press. Available at: https://archive.org/details/structureofameri0000leon.
• Leontief, W. (1986) Input-Output Economics. 2nd edn. New York: Oxford University Press. Available at: https://global.oup.com/academic/product/input-output-economics-9780195035278.
• Miller, R.E. and Blair, P.D. (2009) Input-Output Analysis: Foundations and Extensions. 2nd edn. Cambridge: Cambridge University Press. Available at: https://doi.org/10.1017/CBO9780511626982.
• Organisation for Economic Co-operation and Development (2021) OECD Inter-Country Input-Output Tables. Paris: OECD. Available at: https://www.oecd.org/sti/ind/inter-country-input-output-tables.htm.
• United Nations (2018) Handbook on Supply and Use Tables and Input-Output Tables with Extensions and Applications. New York: United Nations. Available at: https://unstats.un.org/unsd/nationalaccount/docs/SUT_IOT_HB_Final_Cover.pdf.
• United Nations, European Commission, International Monetary Fund, Organisation for Economic Co-operation and Development and World Bank (2009) System of National Accounts 2008. New York: United Nations. Available at: https://unstats.un.org/unsd/nationalaccount/sna2008.asp.

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