Input–Output Analysis

An economy is not a collection of isolated industries. Farms buy machinery and fuel. Manufacturers buy raw materials, electricity, transport, and business services. Transport companies buy vehicles and energy. Each industry's output becomes another industry's input.

Input–output analysis turns this production network into a consistent accounting system. Its central question is:

How much production is required throughout the economy to satisfy a given amount of final demand?

The answer includes more than the product purchased by the final user. It also includes all intermediate inputs needed to produce that product, the inputs needed to produce those inputs, and every subsequent round of production requirements.

Starting with a small transaction table, the framework builds step by step to the Leontief inverse. One two-sector economy is used throughout, so every equation can be connected to an observable economic flow. The final part then extends the same logic to a multiregional input–output model, where every industry is identified by both sector and country.

The origins of input–output analysis

Input–output analysis emerged from the need to describe an economy as a connected production system rather than a set of separate industries. In the 1930s, Wassily Leontief made this idea operational by organising detailed US data into tables showing which industries supplied inputs to which others (Leontief, 1936).

Leontief presented a fuller empirical application in The Structure of American Economy, 1919–1929, published in 1941. The work demonstrated that a system of interindustry transactions could be studied with a compact set of linear equations (Leontief, 1941).

In 1973, Leontief received the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel “for the development of the input-output method and for its application to important economic problems.” The award highlighted the method's ability to connect a change in final demand with the direct and derived production required across the economy (Royal Swedish Academy of Sciences, 1973).

Meet the example economy

Consider a deliberately simple economy with two sectors:

• Agriculture, which produces food and other agricultural products.
• Manufacturing, which produces machinery and manufactured goods.

Both sectors sell products to each other. They also sell products to final demand: uses that do not return to the production system as intermediate inputs during the accounting period. Final demand commonly includes household and government consumption, investment, and exports.

Figure 1. Primary inputs enter production, industries supply intermediate inputs to one another, and finished output reaches final demand. Within-sector transactions are recorded in the table below but omitted here to keep the production logic clear.

The distinction between intermediate use and final demand is essential. Fuel used by a farm during production is intermediate use. A tractor purchased as long-lived equipment is investment and therefore part of final demand in the input–output framework. Food bought by a household is final consumption. Input–output analysis starts from this separation and traces the production consequences of final use through the supplier network.

Reading an input–output table

The same economy can be written as a balanced transaction table. All values below are in an arbitrary monetary unit.

The orientation is:

Rows supply; columns use.

The first two rows record sector output sold; the Value added row records primary inputs used by the sectors. Its total of 120 is the sum across both sectors, not the output of one sector.

Reading across the Agriculture row:

• Agriculture supplies 20 units to itself.
• Agriculture supplies 10 units to Manufacturing.
• Agriculture supplies 70 units to final demand.
• Its total output is therefore 20 + 10 + 70 = 100.

Reading down the Agriculture column:

• Agriculture uses 20 units of agricultural products.
• Agriculture uses 30 units of manufactured products.
• Agriculture generates 50 units of value added.
• Its total input is therefore 20 + 30 + 50 = 100.

The same balance holds for Manufacturing. Total output equals total input because every unit of output is produced using either intermediate inputs or primary inputs. In this simplified table, the primary-input row is represented by value added: mainly compensation of labour, operating surplus, and taxes less subsidies on production.

This balance is not an optional modelling assumption. It is the accounting foundation of an input–output table. Statistical agencies construct and balance supply, use, and input–output tables within the national-accounts framework (Eurostat, 2008; United Nations, 2018).

From transactions to technical coefficients

To move from the table to equations, we group transactions, output and final demand into matrices and vectors. The notation preserves the same accounting relationships while making them scalable to many industries and countries.

Let:

• $Z$ be the matrix of intermediate transactions;
• $x$ be the vector of total output by sector;
• $y$ be the vector of final demand;
• $\mathbf{1}$ be the summation vector: a column vector of ones.

For the example:

In this notation, $Z$ records sales between producing sectors, $x$ records everything each sector produces, and $y$ records the part purchased for final use. We use $\mathbf{1}$ as a summation vector—a column of ones that adds each row of $Z$ into total intermediate sales.

The row balance of the transaction table is:

That is, each sector's total output equals its intermediate sales plus its sales to final demand. For Agriculture, $20+10+70=100$; for Manufacturing, $30+20+50=100$.

Input required per unit of output

Absolute transaction values depend on the size of each sector. To describe production technology, we convert them into technical coefficients. Define:

where $\hat{x}$ is a diagonal matrix with sector outputs on its diagonal. The coefficient $a_{ij}$ measures how many units of input from supplying sector $i$ are directly required to produce one unit of output in using sector $j$.

Because both sectors produce 100 units, dividing each column of $Z$ by the corresponding sector's output gives:

Each coefficient has a direct economic interpretation:

• $a_{11}=0.2$: producing one unit of agricultural output directly requires 0.2 units of agricultural input.
• $a_{12}=0.1$: producing one unit of manufacturing output directly requires 0.1 units of agricultural input.
• $a_{21}=0.3$: producing one unit of agricultural output directly requires 0.3 units of manufactured input.
• $a_{22}=0.2$: producing one unit of manufacturing output directly requires 0.2 units of manufactured input.

The columns describe production recipes. The first column is Agriculture's recipe; the second is Manufacturing's recipe.

The Leontief model

If $x$ units of output are produced, the industries directly require $Ax$ units of intermediate inputs. Total output must cover both those intermediate requirements and final demand:

In words: total production equals production used by industries plus production used by final consumers.

Move intermediate requirements to the left-hand side:

where $I$ is the identity matrix. The matrix $I-A$ separates the part of gross output that remains available to meet final demand after direct intermediate requirements have been covered.

Solving for output gives:

This equation answers the central input–output question: given final demand $y$, how much gross output $x$ must every sector produce?

The matrix

is called the Leontief inverse. Each element $l_{ij}$ measures the total output from sector $i$—direct and indirect—required per unit of final demand for sector $j$.

For the example:

Inverting this matrix gives the Leontief inverse:

For example, $l_{12}\approx0.1639$ means that one unit of final demand for Manufacturing requires about 0.164 units of agricultural output once all direct and indirect supplier rounds are included.

Applying the observed final demand reproduces the original output vector:

The model therefore returns exactly the gross outputs with which the balanced table began.

Direct and indirect requirements

The Leontief inverse can also be written as an infinite series:

This expression reveals the production chain hidden inside the inverse:

• $I$ is the output delivered directly to final demand.
• $A$ is the first round of inputs required by the producing sectors.
• $A^2$ is the input required to produce those first-round inputs.
• $A^3$ is the next supplier round.
• The remaining powers continue the chain, with each round becoming smaller in a productive system.

The stages can be read as follows:

The series is more than a mathematical identity. It explains why an apparently local change in demand propagates through the economy. A manufacturer raises output, purchases more agricultural and manufactured inputs, and the suppliers of those inputs increase their own purchases in turn. The next example applies these stages to a specific change in demand.

A demand-shock example

Suppose final demand for manufactured products increases by 10 units while final demand for agricultural products remains unchanged:

The initial purchase is 10 units of Manufacturing output. It also triggers additional production throughout the supplier network.

The total output response is:

Meeting the additional final demand therefore requires approximately:

• 1.64 units of additional Agriculture output;
• 13.11 units of additional Manufacturing output;
• 14.75 units of gross output in total.

The direct effect is the 10 units of Manufacturing output delivered to final demand. The remaining 4.75 units are indirect effects: 1.64 units of Agriculture and 3.11 additional units of Manufacturing used within the production chain.

Gross output rises by more than the initial 10-unit purchase because intermediate products are counted whenever they are produced and used. This is not 14.75 units of new income. Gross output records production transactions; value added measures the new income created during production.

Value added and employment multipliers

The same production response can be translated into other economic measures. Output multipliers show the production required across the supply chain; value-added and employment multipliers show the income and jobs associated with that production.

The value-added coefficients in the table are value added per unit of gross output:

For the Manufacturing demand shock, value added generated in each sector is:

The total is 10 units of value added: 0.82 in Agriculture and 9.18 in Manufacturing. By contrast, the output multiplier is $14.75 / 10 = 1.475$: each additional unit of final demand for Manufacturing requires 1.475 units of gross output across both sectors. In this closed, balanced example, the value-added multiplier is one because all final expenditure ultimately becomes value added.

Employment uses the same logic. If $q_i$ denotes employment per unit of output in sector $i$, then the employment associated with a demand change is $\Delta \mathrm{EMP}=\hat{q}L\Delta y$. Sectors with different labour intensity will therefore generate different employment multipliers even when their output multipliers are similar.

From national IO to a multiregional model

The model developed so far describes industries inside one economy. That is sufficient when all intermediate production is treated as domestic or when imports are handled outside the endogenous production system. It is not sufficient for tracing modern supply chains.

A manufacturer in Slovakia may use components made in Germany, metals processed in Poland, energy produced elsewhere, and business services supplied from several countries. Each foreign supplier uses further domestic and imported inputs. To trace this network, the model must identify both which industry produces an input and where that industry is located.

A multiregional input–output model, or MRIO model, joins national input–output systems through bilateral trade flows. The OECD Inter-Country Input-Output tables are one such system: they provide a globally balanced view of interindustry flows within and between economies (Yamano et al., 2023).

The country–industry index

Suppose the model contains $R$ regions and $K$ industries in each region. The global system contains:

country–industry nodes. Instead of an index identifying only a sector, each row and column now has a compound index:

where $r$ identifies the producing region and $k$ identifies the industry. “Slovak motor vehicles” and “German motor vehicles” are therefore different nodes even though they belong to the same industry classification.

The global transaction matrix

Let $Z^{rs}$ be a $K\times K$ block of intermediate transactions from industries in supplying region $r$ to industries in using region $s$. The complete MRIO transaction matrix is:

The orientation remains exactly the same as in the small example: rows supply; columns use.

• A diagonal block such as $Z^{11}$ records domestic intermediate transactions within region 1.
• An off-diagonal block such as $Z^{12}$ records inputs produced in region 1 and used by industries in region 2.
• The opposite block $Z^{21}$ records inputs flowing in the other direction.

With two regions, Home $H$ and Foreign $F$, the structure reduces to:

Here, the first superscript identifies the origin of the input and the second identifies the country where it is used. The off-diagonal blocks are therefore the bridge between national production systems.

Final demand by destination

An MRIO model also preserves the destination of final demand. Let $y^{(s)}$ be the $N\times1$ vector of products purchased for final use in destination region $s$. Its rows still identify the country and industry where each product was made.

Collecting all destination vectors gives the final-demand matrix:

In a detailed database, each destination can contain separate columns for household consumption, government consumption, investment, and other final-demand categories. These categories can be retained or summed depending on the analytical question. Total global final demand is:

This distinction matters. The row of $Y$ tells us where a product was made; the column tells us where final demand occurs.

The same Leontief model at global scale

Once countries and industries are stacked into one system, the algebra does not change:

The difference is dimensional and economic, not conceptual. The matrix $A$ now contains domestic and cross-border input coefficients. The Leontief inverse $L$ traces every direct and indirect requirement across both industries and countries.

For final demand located in region $s$, the associated global output is:

Each element of $x^{(s)}$ answers: how much output from this particular country–industry is required, directly and indirectly, by final demand in region $s$?

What an MRIO table adds—and what it estimates

MRIO construction requires more than placing published national tables next to each other. National classifications, valuation conventions, time series, and bilateral trade statistics must be harmonised. Global exports and imports must balance, and bilateral flows must be allocated across using industries and final-demand categories. Some cells are therefore estimated even when the national totals are official statistics.

The OECD ICIO system combines national accounts, supply and use tables, input–output tables, bilateral trade data, and balancing procedures to produce one globally consistent system. This makes cross-border tracing possible, but results still inherit uncertainty from source data, sector aggregation, trade allocation, and balancing choices (Yamano et al., 2023).

Value Atlas indicators

The Country Profile uses the same multiregional production network to show how value, employment and emissions connect domestic production with final demand.

The common engine is the Leontief inverse $L$. What changes is the coefficient attached to the required output and the direction from which the production network is read.

Origin of Value Added

What it shows. For the country in the profile, the chart asks which domestic industries generate value and where the final demand supporting that value is located. A domestic industry can therefore be linked to demand at home or abroad, even when its output passes through several downstream producers before reaching the final user.

For final demand in country $p$, the calculation is:

The vector $y^{(p)}$ contains final demand located in country $p$. Multiplication by $L$ traces all output required along the supply chain, while $\widehat{v}$ converts that output into value generated by each country–industry node. Value Atlas then retains the nodes belonging to the country in the profile.

This is OECD's FDVA_BSCI indicator: the value of a final good or service is an accumulation of value generated by many industries in many countries (OECD, 2025). Value Atlas presents the remaining source-industry–destination-country relationship in two linked treemaps:

• Left — industry: where the selected country's value is generated.
• Right — market: where the final demand supporting it is located.
• Selection: the intersection between one producing industry and one final-demand market.

The chart does not follow the gross sales route of a particular exported product. It attributes value to the place where it is created and to the final market that ultimately supports it. In the Value Atlas pipeline, the coefficient is $v_i=(VA_i+TLS_i)/x_i$, where $TLS$ denotes taxes less subsidies. This closes the primary-input total to final demand; throughout the interface, the measure is labelled value added for readability.

Origin of Employment

What it shows. This view asks the same origin-and-destination question, but counts jobs instead of monetary value: in which domestic industries is employment recorded, and which countries' final demand supports it?

The production requirements are unchanged. Only the coefficient attached to output changes:

Here, $e_i=EMP_i/x_i$ is employment per unit of output in country–industry $i$. The result attributes jobs to the industries where people work and connects those jobs to the final demand requiring their output, directly or through downstream production.

• Left — industry: where employment is recorded within the selected country.
• Right — market: where the final demand supporting those jobs is located.
• Selection: the jobs in one domestic industry associated with one final-demand market.

Employment is a count, not a monetary measure such as GDP or value added. Before calculating the indicator, Value Atlas aggregates the ICIO table to the leaf-sector classification available in the employment source. The employment coefficients and Leontief requirements are therefore calculated on the same sector structure.

Environmental Footprint

What it shows. The environmental views distinguish two accounting perspectives. Production-based emissions are assigned to the resident industries and households that release them. Consumption-based emissions are reassigned through domestic and international production networks to the final demand that ultimately drives them.

The country-profile decomposition combines the OECD principal footprint total with the ICIO production network. It separates supply-chain emissions by final-demand sector and by the economy where emissions are physically released. Direct household emissions, such as fuel burned in private vehicles or homes, are shown separately and assigned to the consuming economy.

All greenhouse gases are expressed in CO₂ equivalent. Totals use million tonnes of CO₂ equivalent (Mt CO₂e); per-capita views use tonnes of CO₂ equivalent per person. These are model-based accounts under the residence principle, not territorial inventories prepared under the UN climate-reporting framework.

What it shows. Supply Chain reverses the perspective. Starting from a selected industry in the profile country, it asks where the primary inputs embodied in that industry's production originate. The result includes immediate suppliers, their suppliers and every subsequent upstream round represented by $L$.

For a selected country–industry $j$, a unit vector $e_j$ isolates that target. Value Atlas traces its upstream output requirements and converts them into primary inputs:

Each element of $d_j$ identifies how much primary input originates in one supplier country–industry for one unit of the selected target. The coefficient is $v=(VA+TLS)/x$. For the default view, Value Atlas rescales this pattern to the selected industry's own $VA+TLS$ total and normalises the result to shares. The chart therefore describes primary-input dependency, not gross output or the value of direct purchases.

The marimekko reads from top to bottom and then from left to right:

• Row height: the supplier industry's share of the total dependency.
• Block width within a row: the supplier country's share within that industry.
• Block area: the supplier country–industry's share of the full dependency.

Selecting a country highlights its contribution across all supplier industries. Because the Leontief inverse includes indirect requirements, a country can appear even when the selected industry does not purchase from it directly. The Whole economy view applies the same logic across all industries in the profile country, weighted by their primary inputs.

Data and interpretation. Origin of Value Added and Supply Chain use OECD TiVA/ICIO data for 2022; Origin of Employment uses OECD TiM/ICIO employment data through 2022. Its historical series uses the legacy release for 1995–2009 and the OECD TiM 2025 revision for 2010–2022. The OECD 2025 release covers 1995–2022, 80 economies, a Rest of the World aggregate and 50 industries. Recent ICIO years partly rely on extrapolated harmonised supply-use tables. These indicators are model-based accounting decompositions under observed input structures and sector aggregation. They are not observed firm-to-firm transactions or causal forecasts (OECD, 2025).

Data sources and coverage

Every interactive chart shows a permanently visible source line. It identifies the exact dataset edition used by that chart, its displayed time and economy coverage, and the unit or price basis. The counts differ because Value Atlas does not fill missing observations merely to force a common country set.

OECD Inter-Country Input-Output tables and TiVA

The production and value-added views use the OECD Inter-Country Input-Output tables, 2025 edition. The regular system covers 1995–2022, 80 economies, a Rest of the World aggregate and 50 economic activities. Monetary flows are reported at current prices in USD. The Global Production chart therefore contains 81 geographic entries: 80 named economies plus Rest of World.

OECD Trade in Employment

Employment measures combine the ICIO production system with OECD employment-by-industry data. The OECD Trade in Employment 2025 edition is used for the recent series; legacy observations are retained where required for historical coverage. Country-profile availability can begin later for economies without a complete historical employment series.

OECD Greenhouse Gas Footprints

The emissions views use the principal indicators from the OECD Greenhouse Gas Footprints 2025 edition. The local series covers 1995–2022 and 80 economies. The Production vs Consumption chart uses all 80. The GDP scatter displays 78: Taiwan lacks the required observations in the WDI extract and São Tomé and Príncipe is excluded from the displayed scatter. Production and consumption accounts follow the residence principle and include direct household emissions.

World Bank World Development Indicators

The GDP scatter uses World Bank World Development Indicators GDP in current US dollars (NY.GDP.MKTP.CD) and total population (SP.POP.TOTL). GDP per person is GDP divided by population; population also determines bubble area. Taiwan is absent from this matched scatter because the WDI extract used by Value Atlas does not contain the required GDP and population observations.

Value Atlas transformations

Values shown in the charts are derived local snapshots rather than live API calls. Value Atlas joins country codes, aggregates published industries into the interface's sector groups, calculates shares and per-capita measures where stated, and rounds values only for display. The chart-specific source line is authoritative for the observations currently visible; this page explains the shared accounting transformations and limitations.

Assumptions and limitations

Input–output analysis is powerful because it makes production interdependence explicit. Its results must nevertheless be interpreted within the model's assumptions.

Fixed input structures

The coefficients in $A$ are treated as fixed. If Manufacturing requires 0.1 units of agricultural input per unit of output in the observed table, the model applies that ratio to the scenario being analysed. Firms are not assumed to substitute automatically toward other inputs when relative prices or availability change.

Linear production and constant returns

Doubling output doubles all intermediate requirements. The model does not represent economies of scale, threshold effects, or nonlinear production relationships.

No capacity constraints

The quantity model calculates the output that would be required to satisfy final demand. It does not test whether industries have enough workers, machinery, finance, energy, or time to produce it.

No behavioural price response

The standard quantity model does not model inflation, changes in wages, consumer substitution, or firms' pricing decisions. Separate price input–output models exist, but they answer a different question.

Average sector technology

Each sector is represented by an average production recipe. Differences between firms, products, production sites, and technologies within the same sector are hidden by aggregation.

Accounting dependency is not automatic causality

The Leontief inverse traces dependencies implied by an observed accounting structure. A scenario result is conditional on that structure remaining applicable. It is not, by itself, a causal forecast of what firms and consumers will do after a real-world shock.

Notation

References

Eurostat. (2008). Eurostat Manual of Supply, Use and Input-Output Tables. Office for Official Publications of the European Communities. ISBN 978-92-79-04735-0. Publication page.

OECD. (2025). Guide to OECD Trade in Value Added (TiVA) Indicators, 2025 edition. Directorate for Science, Technology and Innovation.

Leontief, W. W. (1936). Quantitative input and output relations in the economic system of the United States. The Review of Economics and Statistics, 18(3), 105–125. https://doi.org/10.2307/1927837.

Leontief, W. W. (1941). The Structure of American Economy, 1919–1929: An Empirical Application of Equilibrium Analysis. Harvard University Press. Bibliographic record.

Miller, R. E., & Blair, P. D. (2009). Input–Output Analysis: Foundations and Extensions (2nd ed.). Cambridge University Press. https://doi.org/10.1017/CBO9780511626982.

Royal Swedish Academy of Sciences. (1973, October 18). The Prize in Economic Sciences 1973: Press release. NobelPrize.org. https://www.nobelprize.org/prizes/economic-sciences/1973/press-release/.

United Nations Statistics Division. (2018). Handbook on Supply, Use and Input-Output Tables with Extensions and Applications. United Nations. Publication record.

Yamano, N., Alsamawi, A., Webb, C., Cimper, A., Zürcher, C., & Chiapin Pechansky, R. (2023). Development of the OECD Inter-Country Input-Output Database 2023. OECD Science, Technology and Industry Working Papers, No. 2023/08. OECD Publishing. https://doi.org/10.1787/5a5d0665-en.