# Input-output model

In economics, an input-output model is a quantitative economic technique that represents the interdependencies between different branches of a national economy or different regional economies.[1]Wassily Leontief (1905-1999) developed this type of analysis and took the Nobel Prize in Economics for his development of this model.[1] Earlier 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 was both a forerunner and generalization of Leontief's seminal concept. Leontief's major contribution was to simplify Walras's formulation to make the computation feasible and empirically useful. Input-output analysis assumes that input proportions between different economic sectors are fixed and do not change significantly in the short-term. As a result, results calculated by this method should be regarded as approximations and not accurate predictions.

The International Input-Output Association is dedicated to advancing knowledge in the field of input-output study, which includes "improvements in basic data, theoretical insights and modelling, and applications, both traditional and novel, of input-output techniques."

## Understanding the input-output model

An understanding of the economy as consisting of linked sectors goes back to the French economist François Quesnay, and was fully developed by Léon Walras in 1874.[2] However, Wassily Leontief was the first to use a matrix representation of a national (or regional) economy. His 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. 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.

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## Usefulness

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 environmental input-output analysis (EIOA). 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 RIMS-II. 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.

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## Basic derivation

Say that we have an economy with $n$ sectors. Each sector produces a single homogeneous good, $x_i$. Assume that the $i$th sector, in order to produce 1 unit, must use $a_{ij}$ units from sector $j$. 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 $i$th sector $d_i$. Then we might write

$x_i = a_{1i}x_1 + a_{2i}x_2 + \ldots + a_{ni}x_n + d_i,$

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

$\vec{x} = A\vec{x} + \vec{d}$

which after re-writing becomes $(I - A)\vec{x} = \vec{d}$. If the matrix $I - A$ 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 $I - A$ are all positive (known as the Hawkins-Simon Condition), the required output vector $\vec{x}$ is non-negative.

### Example

Consider an economy with two goods, A and B. The matrix of coefficients and the final demand is given by

$A = \left(\begin{array}{cc} 0.5 & 0.2 \\ 0.4 & 0.1 \end{array}\right) \text{ and } d = \left(\begin{array}{c} 7 \\ 4 \end{array}\right).$

Intuitively, this corresponds to finding the amount of output each sector should produce given that we want 7 units of good A and 4 units of good B. Then solving the system of linear equations derived above gives us

$x = (I - A)^{-1} d = \left(\begin{array}{c} 19.19 \\ 12.97 \end{array}\right).$

For practical purposes it might be challenging to actually compute the inverse matrix, given that some input-output tables are in excess of hundreds of sectors.

### Further research

There are many interesting aspects of the Leontief system, and there is an extensive literature. There is the Hawkins-Simon Condition on producibility. There has been interest in disaggregation to clustered inter-industry flows, and the study of constellations of industries. A great deal of empirical work has been done to identify coefficients, and data have been published for the national economy as well as for regions. This has been a healthy, exciting area for work by economists because the Leontief system can be extended to a model of general equilibrium; it offers a method of decomposing work done at a macro level.

### Introducing transportation

Transportation is implicit in the notion of inter-industry flows. It is explicitly recognized when transportation is identified as an industry – how much is purchased from transportation in order to produce. But this is not very satisfactory because transportation requirements differ, depending on industry locations and capacity constraints on regional production. Also, the receiver of goods generally pays freight cost, and often transportation data are lost because transportation costs are treated as part of the cost of the goods.

Walter Isard and his student, Leon Moses, were quick to see the spatial economy and transportation implications of input-output, and began work in this area in the 1950s developing a concept of interregional input-output. Take a one region versus the world case. We wish to know something about interregional commodity flows, so introduce a column into the table headed “exports” and we introduce an “import” row.

 Economic Activities 1 2 … … Z Exports Final Demand Total Outputs 1 2 … … Z Imports

A more satisfactory way to proceed would be to tie regions together at the industry level. That is, we could identify both intra-region inter-industry transactions and inter-region inter-industry transactions. The problem here is that the table grows quickly.

Input-output is conceptually simple. Its extension to a model of equilibrium in the national economy is also relatively simple and attractive but requires great skill and high-quality data. One who wishes to do work with input-output systems must deal skillfully with industry classification, data estimation, and inverting very large, ill-conditioned matrices. Moreover, changes in relative prices are not readily handled by this modeling approach alone. Of course, input-output accounts are part and parcel to a more flexible form of modeling, Computable general equilibrium models.

Two additional difficulties are of interest in transportation work. There is the question of substituting one input for another, and there is the question about the stability of coefficients as production increases or decreases. These are intertwined questions. They have to do with the nature of regional production functions.

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## Measuring input-output tables

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):[3] 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 interindustry 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 below 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 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.

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## Input-output Analysis Versus Consistency Analysis

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 [1], a pupil of Leontief, introduced Consistency Analysis in his book 'Foundations of Economic Science' (ISBN 984320655X), 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 in one model and deals with money in an ideology-free manner. It deals with the flow of funds via the movement of goods.

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## References

1. ^ a b Thijs Ten Raa, Input-output economics: theory and applications: featuring Asian economies, World Scientific, 2009
2. ^ Elements of pure economics, L. Walras
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## Bibliography

• Dietzenbacher, Erik and Michael L. Lahr, eds. Wassilly 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.[2].
• 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.
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