Input-Output Model

From Open Risk Manual


An Input-Output Model (sometimes conflated with the more narrow term Input-Output Table or Input-Output Matrix) is a mathematical model constructed from observed economic data which provides information about the activity of industries (economic sectors) and/or regions that produce and consume goods and services.

The core component of such models, the Input-Output Matrix (or Table) is a means of presenting a detailed analysis of the process of production and the use of goods and services (products) and the income generated in that production. Such tables can be either in the form of:

Traditional Input-Output (IO) tables summarise the exchanges between major sectors of an economy.[1]

An IO table can be seen as either:

  • a Stastical Summary (of some aspects) of an economy or (with additional assumptions) or
  • an Actual model of how supply and demand and production and consumption are related (subject to Model Risk).

In the first type of use the term table might be more appropriate, in the second usage the term model highlights this ore specific usage


The basic IO table, includes

This set is usually provided by national statistical offices for the national economy. Input-Output accounts are constructed using annual periods but this is conventional, in princinple any period over which data can be collected is allowed. Interindustry relationships are derived from interindustry transaction tables in which the rows describe the composition of inputs required by a particular industry to produce its output. The data often concern industries in a specific geographic region (nation, state, county, etc.) and consist in monetary flows (but see also Physical Input-Output Tables.

  • Additional matrix (table) columns represent final demand, i.e. the sales to final markets (consumers, government, exports).
  • Additional matrix (table) rows represent non-industrial production inputs (labor, capital, etc.)1.


An Input-Output Model is fundamentally a system of linear equations where the Total Output of an industry is distributed through sales to other sectors and to Final Demand:

x_{i} & = z_{i1} + \ldots  z_{in} + f_i \\
      & = \sum_{j=1}^{n} z_{ij} + f_i


  • n is the number of industries in the modeled economy
  • f_{i} is the final demand of sector's i production
  • z_{ij} is the flow of primary input i to sector j (value added bloc of the IO table), the demand of sector j for inputs from sector i
  • x_i is the monetary value of the output of sector j (production value) during a period (year)

This system of equations expresses a form of market balance.


  • Input-Output Analysis can serve various analytical purposes such as impact analysis, productivity analysis, employment effects, analysis of the interdependence of structures and analysis of price change.[2]
  • In a corporate context an IO Model can be used for analysing the effects of a company's activities on upstream production (supply) industries along the value chain.
  • Intermediate goods incorporated in the production process can be allocated to different sectors of the economy (the supply links among the individual sectors can be shown)
  • On the basis of the structure of the goods and services actually required or obtained, the intermediate goods and services (tier 2 – tier n) statistically required for the production of the company’s production input (tier 1) can be calculated using IO analysis. As a result, the (statistically) required total input of goods and services is determined along the upstream stages of the value chain.
  • IO tables and models are widely used in the context of classical economic impact analysis (EIA) and in environmentally extended IO analyses ( EEIO). Input-output analysis can be used to allocate upstream emissions or downstream emissions systematically while avoiding the double-counting problem. In doing so, ultimate upstream responsibilities are allocated to the consumers of the final products (such as households) by using the Leontief inverse and ultimate downstream responsibilities are allocated to the providers of primary factors (such as workers, investors, land owners, etc.) by using the Ghosh inverse

Issues and Challenges

  • The construction of such models may suffer Aggregation Bias
  • Single-Region models focus only the production and trade within the borders of the region in question, multi-regional models are required to take into account the imports of commodities from outside the national boundaries. This extension raises data requirements and introduces the potential of inconsistent accounting.
  • Monetary Focus: Leontief’s original input-output framework (Leontief, 1936) conceived of industry production functions, which he frequently referred to as production “recipes,” as mea- sured in physical units, such as specifying the technical coefficients in tons of coal or bushels of wheat, as inputs, required per dollar’s worth of an industry’s output or per ton of steel output. However, the data collection requirements and a number of other constraints rendered implementation of the framework in physical units too unwieldy, certainly at the time and even today to a lesser extent. Hence, the basic methodology for input–output analysis evolved, in both theory and application, through measuring all quantities in value terms with implicit fixed prices, as detailed in earlier chapters.
  • Simplifying assumptions of fixed input requirements and constant relative prices, both implicit in the classical input-output framework, render implementation easier, although satisfying the data requirements can still be quite difficult. On the other hand, these assumptions limit the ability of the framework to deal very effectively with some fundamental features of a modern economic system, such as prices or elasticities.

See Also

Further Resources


  1. R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009
  2. United Nations, Handbook on Supply and Use Tables and Input Output-Tables with Extensions and Applications, 2018