Input–output model

Francois Quesnay had developed a cruder version of this technique called Tableau économique, and Léon Walras's work Elements of Pure Economics on general equilibrium theory also was a forerunner and made a generalization of Leontief's seminal concept.[2]

Alexander Bogdanov has been credited with originating the concept in a report delivered to the All Russia Conference on the Scientific Organisation of Labour and Production Processes, in January 1921.[3] This approach was also developed by L. N. Kritsman and T. F. Remington, who has argued that their work provided a link between Quesnay's tableau économique and the subsequent contributions by Vladimir Groman and Vladimir Bazarov to Gosplan's method of material balance planning.[3]

Wassily Leontief's work in the input–output model was influenced by the works of the classical economists Karl Marx and Jean Charles Léonard de Sismondi. Karl Marx's economics provided an early outline involving a set of tables where the economy consisted of two interlinked departments.[4]

Leontief was the first to use a matrix representation of a national (or regional) economy.

The model depicts inter-industry relationships within an economy, showing how output from one industrial sector may become an input to another industrial sector. In the inter-industry matrix, column entries typically represent inputs to an industrial sector, while row entries represent outputs from a given sector. This format, therefore, shows how dependent each sector is on every other sector, both as a customer of outputs from other sectors and as a supplier of inputs. Sectors may also depend internally on a portion of their own production as delineated by the entries of the matrix diagonal.[5] Each column of the input–output matrix shows the monetary value of inputs to each sector and each row represents the value of each sector's outputs.

Say that we have an economy with ${\displaystyle n}$ sectors. Each sector produces ${\displaystyle x_{i}}$ units of a single homogeneous good. Assume that the ${\displaystyle j}$th sector, in order to produce 1 unit, must use ${\displaystyle a_{ij}}$ units from sector ${\displaystyle i}$. Furthermore, assume that each sector sells some of its output to other sectors (intermediate output) and some of its output to consumers (final output, or final demand). Call final demand in the ${\displaystyle i}$th sector ${\displaystyle d_{i}}$. Then we might write

${\displaystyle x_{i}=a_{i1}x_{1}+a_{i2}x_{2}+\cdots +a_{in}x_{n}+d_{i},}$

or total output equals intermediate output plus final output. If we let ${\displaystyle A}$ be the matrix of coefficients ${\displaystyle a_{ij}}$, ${\displaystyle \mathbf {x} }$ be the vector of total output, and ${\displaystyle \mathbf {d} }$ be the vector of final demand, then our expression for the economy becomes

${\displaystyle \mathbf {x} =A\mathbf {x} +\mathbf {d} }$

which after re-writing becomes ${\displaystyle \left(I-A\right)\mathbf {x} =\mathbf {d} }$. If the matrix ${\displaystyle I-A}$ is invertible then this is a linear system of equations with a unique solution, and so given some final demand vector the required output can be found. Furthermore, if the principal minors of the matrix ${\displaystyle I-A}$ are all positive (known as the Hawkins–Simon condition),[6] the required output vector ${\displaystyle \mathbf {x} }$ is non-negative.

Because the input–output model is fundamentally linear in nature, it lends itself to rapid computation as well as flexibility in computing the effects of changes in demand. Input–output models for different regions can also be linked together to investigate the effects of inter-regional trade, and additional columns can be added to the table to perform environmentally extended input–output analysis (EEIOA). For example, information on fossil fuel inputs to each sector can be used to investigate flows of embodied carbon within and between different economies.

The structure of the input–output model has been incorporated into national accounting in many developed countries, and as such can be used to calculate important measures such as national GDP. Input–output economics has been used to study regional economies within a nation, and as a tool for national and regional economic planning. A main use of input–output analysis is to measure the economic impacts of events as well as public investments or programs as shown by IMPLAN and Regional Input–Output Modeling System. It is also used to identify economically related industry clusters and also so-called "key" or "target" industries (industries that are most likely to enhance the internal coherence of a specified economy). By linking industrial output to satellite accounts articulating energy use, effluent production, space needs, and so on, input–output analysts have extended the approaches application to a wide variety of uses.

The mathematics of input–output economics is straightforward, but the data requirements are enormous because the expenditures and revenues of each branch of economic activity have to be represented. As a result, not all countries collect the required data and data quality varies, even though a set of standards for the data's collection has been set out by the United Nations through its System of National Accounts (SNA):[15] the most recent standard is the 2008 SNA. Because the data collection and preparation process for the input–output accounts is necessarily labor and computer intensive, input–output tables are often published long after the year in which the data were collected—typically as much as 5–7 years after. Moreover, the economic "snapshot" that the benchmark version of the tables provides of the economy's cross-section is typically taken only once every few years, at best.

However, many developed countries estimate input–output accounts annually and with much greater recency. This is because while most uses of the input–output analysis focus on the matrix set of inter-industry exchanges, the actual focus of the analysis from the perspective of most national statistical agencies is the benchmarking of gross domestic product. Input–output tables therefore are an instrumental part of national accounts. As suggested above, the core input–output table reports only intermediate goods and services that are exchanged among industries. But an array of row vectors, typically aligned at the bottom of this matrix, record non-industrial inputs by industry like payments for labor; indirect business taxes; dividends, interest, and rents; capital consumption allowances (depreciation); other property-type income (like profits); and purchases from foreign suppliers (imports). At a national level, although excluding the imports, when summed this is called "gross product originating" or "gross domestic product by industry." Another array of column vectors is called "final demand" or "gross product consumed." This displays columns of spending by households, governments, changes in industry stocks, and industries on investment, as well as net exports. (See also Gross domestic product.) In any case, by employing the results of an economic census which asks for the sales, payrolls, and material/equipment/service input of each establishment, statistical agencies back into estimates of industry-level profits and investments using the input–output matrix as a sort of double-accounting framework.

Despite the clear ability of the input–output model to depict and analyze the dependence of one industry or sector on another, Leontief and others never managed to introduce the full spectrum of dependency relations in a market economy. In 2003, Mohammad Gani, a pupil of Leontief, introduced consistency analysis in his book Foundations of Economic Science, which formally looks exactly like the input–output table but explores the dependency relations in terms of payments and intermediation relations. Consistency analysis explores the consistency of plans of buyers and sellers by decomposing the input–output table into four matrices, each for a different kind of means of payment. It integrates micro and macroeconomics into one model and deals with money in a value-free manner. It deals with the flow of funds via the movement of goods.

ابونوری, اسمعیل, فرهادی, & عزیزاله. (2017). آزمون فروض تکنولوژی در محاسبه جدول داده ستانده متقارن ایران: یک رهیافت اقتصاد سنجی. پژوهشهای اقتصادی ایران, 21(69), 117–145.

• Dietzenbacher, Erik and Michael L. Lahr, eds. Wassily Leontief and Input–Output Economics. Cambridge University Press, 2004.
• Isard, Walter et al. Methods of Regional Analysis: An Introduction to Regional Science. MIT Press 1960.
• Isard, Walter and Thomas W. Langford. Regional Input–Output Study: Recollections, Reflections, and Diverse Notes on the Philadelphia Experience. The MIT Press. 1971.
• Lahr, Michael L. and Erik Dietzenbacher, eds. Input–Output Analysis: Frontiers and Extensions. Palgrave, 2001.
• Leontief, Wassily W. Input–Output Economics. 2nd ed., New York: Oxford University Press, 1986.
• Miller, Ronald E. and Peter D. Blair. Input–Output Analysis: Foundations and Extensions. Prentice Hall, 1985.
• Miller, Ronald E. and Peter D. Blair. Input–Output Analysis: Foundations and Extensions, 2nd edition. Cambridge University Press, 2009.
• Miller, Ronald E., Karen R. Polenske, and Adam Z. Rose, eds. Frontiers of Input–Output Analysis. N.Y.: Oxford UP, 1989.[HB142 F76 1989/ Suzz]
• Miernyk, William H. The Elements of Input–Output Anaysis, 1965.Web Book-William H. Miernyk Archived 26 November 2019 at the Wayback Machine.
• Polenske, Karen. Advances in Input–Output Analysis. 1976.
• Pokrovskii, Vladimir N. Econodynamics. The Theory of Social Production, Springer, Dordrecht, Heidelberg et cetera, 2011.
• ten Raa, Thijs. The Economics of Input–Output Analysis. Cambridge University Press, 2005.
• US Department of Commerce, Bureau of Economic Analysis . Regional multipliers: A user handbook for regional input–output modeling system (RIMS II). Third edition. Washington, D.C.: U.S. Government Printing Office. 1997.
• Eurostat Eurostat manual of supply, use and input-output tables. Office for Official Publications of the European Communities, 2008.

• International Input–Output Association
• Input–Output Accounts Data, Bureau of Economic Analysis
• Input–Output Analysis and Related Methods Archived 5 May 2021 at the Wayback Machine, San José State University
• Doing Business project input/output tables for reforms
• Energy Economics. Input–Output Analysis: Lecture – 6 and Lecture 7 – two introductory videos on Input–Output methodology with a focus on energy economics from IIT Kharagpur.