5.4 Open and closed input-output models

Basic input-output model

Input-output analysis is a framework for studying how economic sectors depend on each other. Every industry both uses outputs from other industries and supplies its own output to them. By mapping these flows with linear equations, you can trace how a change in one sector affects the whole economy.

Wassily Leontief developed this framework and earned the Nobel Prize in Economics for it. The core idea: represent an economy as a system of linear equations describing the flows of goods and services between industries.

Structure of input-output tables

An input-output table is organized as a matrix where both rows and columns represent industries or sectors.

• Rows show how a given industry's output is distributed across all other sectors
• Columns show what inputs a given industry requires from every other sector
• The table also includes intermediate consumption (goods used up in production), final demand (household consumption, government spending, exports), and value added (wages, profits)

A key accounting rule holds everything together: for each sector, total output must equal total input. This balance condition is what makes the math work.

Leontief production function

The Leontief production function assumes each industry needs inputs in fixed proportions. You can't substitute more steel for less aluminum; the recipe is rigid. This gives you constant returns to scale and no input substitution.

Mathematically:

$x_i = \min\left(\frac{z_{i1}}{a_{i1}}, \frac{z_{i2}}{a_{i2}}, \ldots, \frac{z_{in}}{a_{in}}\right)$

where $x_i$ is the output of industry $i$, $z_{ij}$ is the amount of input from industry $j$, and $a_{ij}$ is the technical coefficient (the fixed amount of input $j$ needed per unit of output $i$).

This is a simplification. Real industries do substitute between inputs as prices change. But the fixed-proportions assumption makes the model linear and solvable, which is the whole point.

Technical coefficients matrix

Technical coefficients capture the direct input requirements per unit of output. You calculate each one by dividing the flow from industry $i$ to industry $j$ by the total output of industry $j$:

$a_{ij} = \frac{z_{ij}}{x_j}$

Collect all these coefficients into a matrix $A$, and you have the foundation for the entire input-output model. Each column of $A$ tells you the "recipe" for one unit of that industry's output.

One important assumption: these coefficients are treated as stable over time. In a rapidly changing economy (say, one undergoing a tech revolution), this assumption can break down.

Open input-output models

An open model treats final demand as exogenous, meaning it's determined outside the model. You plug in a final demand vector, and the model tells you how much each industry needs to produce. This is the most common setup for economic impact studies.

Exogenous final demand

Final demand includes everything that isn't consumed as an intermediate input by another industry:

• Household consumption
• Government spending
• Exports
• Investment

In the model, these are collected into a vector $f$. Because $f$ is set externally, you can change it to simulate different scenarios: What happens if government spending increases by 10%? What if exports drop?

Output determination

The core equation of the open model is:

$x = Ax + f$

This says: total output $x$ equals intermediate demand ($Ax$, what industries need from each other) plus final demand $f$. Rearranging:

$(I - A)x = f$

where $I$ is the identity matrix. To solve for $x$, you need to invert $(I - A)$.

Leontief inverse matrix

The solution to the open model is:

$x = (I - A)^{-1}f$

The matrix $(I - A)^{-1}$ is called the Leontief inverse (or total requirements matrix). This is where the real analytical power lives.

Each element of $(I - A)^{-1}$ tells you the total output required from industry $i$ to deliver one unit of final demand from industry $j$. "Total" here means it captures both the direct input needs and all the indirect rounds of production that ripple through the supply chain.

Closed input-output models

A closed model takes some components of final demand and makes them endogenous, meaning they're determined within the model itself. Most commonly, the household sector gets "closed in": household spending depends on household income, which depends on production, which depends on household spending. This feedback loop is what distinguishes closed from open models.

Endogenous final demand

Instead of treating household consumption as a fixed external number, the closed model makes it a function of income generated within the economy. This means the technical coefficients matrix expands to include an additional row and column representing the household sector.

The payoff: you can now capture induced effects (the extra economic activity generated when workers spend their wages) alongside direct and indirect effects.

Household sector inclusion

In the closed model, households play a dual role:

• As producers: they supply labor to industries (represented by a new row showing labor inputs)
• As consumers: they purchase goods and services (represented by a new column showing consumption patterns)

This creates an income-consumption feedback loop. Industries pay wages, households spend those wages, that spending becomes demand for industries, which pay more wages, and so on. The model is more complex but captures a more realistic picture of how economies actually work.

Multiplier effects

Multipliers measure the total impact on output from a one-unit change in exogenous final demand.

• Open model multipliers capture direct + indirect effects
• Closed model multipliers capture direct + indirect + induced effects

Closed model multipliers are calculated using the expanded Leontief inverse:

$(I - A^*)^{-1}$

where $A^*$ is the augmented technical coefficients matrix that includes the household sector. Because of the additional feedback loop, closed model multipliers are always larger than their open model counterparts.

Mathematical formulation

The math behind input-output models relies on linear algebra. Here's how the pieces fit together.

Matrix notation

The open model in compact form:

$x = Ax + f \quad \Longrightarrow \quad (I - A)x = f$

For the closed model, the matrices expand to include endogenous sectors, but the structure is identical. This compact notation is what makes it feasible to work with models containing dozens or hundreds of industries.

System of linear equations

Written out equation by equation, the model says that for each industry $i$:

$x_i = \sum_{j=1}^{n} a_{ij}x_j + f_i \quad \text{for } i = 1, 2, \ldots, n$

Each equation balances total output of industry $i$ against the sum of what all other industries demand from it (intermediate demand) plus what final consumers demand from it.

Solving for equilibrium output

For the open model, the solution is:

$x = (I - A)^{-1}f$

Here's the step-by-step process:

1. Construct the technical coefficients matrix $A$ from the input-output table

2. Form the matrix $(I - A)$

3. Verify that $(I - A)$ is invertible (it will be as long as the economy is productive)

4. Compute $(I - A)^{-1}$

5. Multiply by the final demand vector $f$ to get equilibrium output $x$

For closed models, the same steps apply but with the augmented matrix $A^*$. For very large systems, iterative methods may be used instead of direct inversion.

Economic interpretation

Interdependence of industries

The Leontief inverse makes interdependence visible and quantifiable. A change in demand for automobiles doesn't just affect auto plants; it ripples through steel, glass, rubber, electronics, and dozens of other sectors. The model quantifies the strength of these linkages, revealing which sectors are most tightly connected to the rest of the economy.

This helps identify potential bottlenecks (sectors whose disruption would cascade widely) and strategic industries worth targeting for development.

Direct vs. indirect effects

These three types of effects are central to interpreting input-output results:

• Direct effects: the immediate output change in the industry where demand shifts (e.g., more cars demanded → more car production)
• Indirect effects: the subsequent output changes in supplying industries (more steel, more rubber, etc.)
• Induced effects (closed models only): the additional activity from workers spending their increased income

The total effect is the sum of all three. Open models capture only direct and indirect effects; closed models add induced effects on top.

Backward and forward linkages

These measure how connected an industry is to the rest of the economy, but in different directions:

• Backward linkages measure how much an industry stimulates its suppliers. Calculated from the column sums of the Leontief inverse $(I - A)^{-1}$.
• Forward linkages measure how important an industry is as a supplier to others. Calculated from the row sums of the Ghosh inverse matrix.

Industries with strong backward and forward linkages are considered "key sectors." Policymakers often target these for investment because stimulating them has outsized effects on the broader economy.

Applications and limitations

Economic planning

Governments use input-output models to support planning and policy decisions:

• Identifying key sectors for development or investment
• Simulating the economy-wide effects of different policy scenarios
• Coordinating infrastructure projects with industrial policy
• Allocating resources across sectors

The models provide a systematic framework for thinking about how different parts of the economy connect, which is valuable even when the precise numbers carry some uncertainty.

Impact analysis

Impact analysis is one of the most common applications. Typical use cases include:

• Evaluating the economic effects of a major investment or factory closure
• Estimating job losses from a natural disaster
• Quantifying the ripple effects of a new government spending program
• Supporting cost-benefit analysis of public projects

The model quantifies direct, indirect, and (in closed models) induced effects on output, employment, and income.

Static vs. dynamic models

Traditional input-output models are static: they represent the economy at a single point in time with fixed technical coefficients.

Dynamic models extend the framework by incorporating:

• Changes in technology over time
• Investment functions and capital accumulation
• Evolving production techniques

Static models are simpler and require less data, making them practical for many applications. But they can't capture long-term structural change. Dynamic models address this at the cost of significantly greater complexity and data requirements.

Extensions and variations

Regional input-output models

These adapt the national framework to a specific geographic area. Regional models must account for the fact that a region's production technology and trade patterns differ from the national average. Techniques like location quotients help estimate regional technical coefficients from national data.

Regional models also incorporate inter-regional trade flows, making them useful for assessing the local impact of events like a plant closure or a new infrastructure project.

Environmental input-output analysis

This extension adds environmental data (resource use, emissions, waste) to the standard economic model. By linking pollution and resource consumption to industry output, you can trace the environmental consequences of changes in final demand.

For example, you could estimate how much additional carbon is emitted economy-wide when demand for a particular product increases. This supports sustainable production planning and can be combined with life cycle assessment for comprehensive environmental impact studies.

Social accounting matrices

A social accounting matrix (SAM) expands the input-output framework to include more detailed accounts for:

• Factors of production (labor, capital)
• Institutions (households, firms, government)
• Capital transactions and transfers

SAMs provide a richer picture of income distribution and economic structure. They're particularly useful in developing countries for analyzing poverty, inequality, and the social impacts of economic policies.

Empirical considerations

Data collection and compilation

Building an input-output table requires extensive data on inter-industry transactions, typically drawn from national accounts, economic censuses, and industry surveys. Practical challenges include:

• Reconciling data from multiple sources
• Capturing informal sector activity
• Handling confidential business information

Increasingly, administrative records and large-scale datasets supplement traditional survey data.

Updating input-output tables

Input-output tables take years to compile, so they're often outdated by the time they're published. Economists use updating techniques like the RAS method (a biproportional balancing technique) to project older tables forward using more recent aggregate data.

There's always a trade-off: more frequent updates mean less detailed tables, while highly detailed tables take longer to produce.

Accuracy and reliability issues

Several sources of error affect input-output results:

• Aggregation bias: grouping diverse industries into a single sector hides important differences
• Fixed coefficients assumption: technical coefficients change as technology and prices evolve
• Data errors: missing values, estimation procedures, and balancing adjustments all introduce uncertainty

Sensitivity analysis helps assess how robust your results are to these issues. When presenting input-output findings, it's important to communicate the uncertainties involved rather than treating the numbers as precise predictions.