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This article is about the economic model. For the computer interface, see Input/output.

In economics, an input-output model uses a matrix representation of a nation's (or a region's) economy to predict the effect of changes in one industry on others and by consumers, government, and foreign suppliers on the economy. Wassily Leontief (1905-1999) is credited with the development of this analysis. Francois Quesnay developed a cruder version of this technique called Tableau économique. Leontief won the Nobel Memorial Prize in Economic Sciences for his development of this model. And, in essence, Léon Walras's work Elements of Pure Economics on general equilibrium theory is both a forerunner and generalization of Leontief's seminal concept. Leontief's contribution was that he was able to simplify Walras's piece so that it could be implemented empirically. The International Input-Output Association[1] is dedicated to advance knowledge in the field input-output study, which includes "improvements in basic data, theoretical insights and modelling, and applications, both traditional and novel, of input-output techniques."

Input-output depicts inter-industry relations of an economy. It shows how the output of one industry is an input to each other industry. Leontief put forward the display of this information in the form of a matrix. A given input is typically enumerated in the column of an industry and its outputs are enumerated in its corresponding row. This format, therefore, shows how dependent each industry is on all others in the economy both as customer of their outputs and as supplier of their inputs. Each column of the input-output matrix reports the monetary value of an industry's inputs and each row represents the value of an industry's outputs. Suppose there are three industries. Column 1 reports the value of inputs to Industry 1 from Industries 1, 2, and 3. Columns 2 and 3 do the same for those industries. Row 1 reports the value of outputs from Industry 1 to Industries 1, 2, and 3. Rows 2 and 3 do the same for the other industries.

While most uses of the input-output analysis focuses on the matrix set of interindustry exchanges, the actual focus of the analysis from the perspective of most national statistical agencies, which produce the tables, 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.

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 has 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[2](SNA): the replacement for the current 1993 SNA standard is pending. 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 data was collected--typically as much as 5-7 years after. Moreover, the economic "snapshot" the benchmark version of the tables provide of the economy's cross-section are taken only once every few years, at best. Although many developed countries estimate input-output accounts annually and with much greater recency.

[edit] Usefulness

In addition to studying the structure of national economies, input-output economics has been used to study regional economies within a nation, and as a tool for national and regional economic planning. Indeed a main use of input-output analysis is for measuring the economic impacts of events as well as public investments or programs as shown by IMPLAN and RIMS-II. But 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.

[edit] Key Ideas

Leontief's text remains one of the best expositions of input-output analysis. Nonetheless, two books--a rather fundamental one by William Miernyk[3] and another by Ronald E. Miller and Peter D. Blair--probably have greater international currency. The latter is presently being rewritten and re-released, this time by Cambridge University Press.

Consider the production of the ith sector. We may isolate (1) the quantity of that production that goes to final demand, $c i$ , (2) to total output, $x i$ , and (3) flows $x i j$ from that industry to other industries. We may write a transactions tableau

**Table: Transactions in a Three Sector Economy**
Economic Activities	Inputs to Agriculture	Inputs to Manufacturing	Inputs to Transport	Final Demand	Total Output
Agriculture	5	15	2	68	90
Manufacturing	10	20	10	40	80
Transportation	10	15	5	0	30
Labor	25	30	5	0	60

or

$\begin{matrix} x_{11} + x_{12} + x_{13} + c_{1} & = & x_{1} \\ x_{21} + x_{22} + x_{23} + c_{2} & = & x_{2} \\ x_{31} + x_{32} + x_{33} + c_{3} & = & x_{3} \\ x_{41} + x_{42} + x_{43} + c_{4} & = & x_{4} \end{matrix}$

Note that in the example given we have no input flows from the industries to 'Labor'.

We know very little about production functions because all we have are numbers representing transactions in a particular instance (single points on the production functions):

$\begin{matrix} x_{1} & = & f(x_{11}, x_{12}, x_{13}, x_{14}) \\ x_{2} & = & g(x_{21}, x_{22}, x_{23}, x_{24}) \\ \vdots & = & \vdots \end{matrix}$

The neoclassical production function is an explicit function

Q = f (K, L),

where $Q =$ Quantity, $K =$ Capital, $L =$ Labor,

and the partial derivatives ( $\partial Q/ \partial K = f_K > 0 ; \partial Q/ \partial L = f_L > 0$ ) are the demand schedules for input factors.

Leontief, the innovator of input-output analysis, uses a special production function which depends linearly on the total output variables $x i$ . Using Leontief coefficients $a i j$ , we may manipulate our transactions information into what is known as an input-output table:

$\begin{matrix} x_{11} & = & a_{11}x_{1} \\ x_{12} & = & a_{12}x_{2} \\ x_{13} & = & a_{13}x_{3} \\ x_{14} & = & a_{14}x_{4} \\ \vdots & = & \vdots \end{matrix}$

or

$\begin{matrix} x_{ij} & = & a_{ij}x_{j} \end{matrix}$

Now

$\begin{matrix} a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3} + a_{14}x_{4} + c_{1} & = & x_{1} \\ \vdots & = & \vdots \\ a_{41}x_{1} + a_{42}x_{2} + a_{43}x_{3} + a_{44}x_{4} + c_{4} & = & x_{4} \end{matrix}$

gives

$\begin{matrix} x_{1} - a_{11}x_{1} - a_{12}x_{2} - a_{13}x_{3} - a_{14}x_{4} & = & c_{1} \\ \vdots & = & \vdots \\ x_{4} - a_{41}x_{1} - a_{42}x_{2} - a_{43}x_{3} - a_{44}x_{4} & = & c_{4} \end{matrix}$

Rewriting finally yields

$\begin{matrix} (1-a_{11})x_{1} - a_{12}x_{2} - a_{13}x_{3} - a_{14}x_{4} & = & c_{1} \\ \vdots & = & \vdots \\ - a_{41}x_{1} - a_{42}x_{2} - a_{43}x_{3} + (1-a_{44})x_{4} & = & c_{4} \end{matrix}$

Introducing matrix notation, we can see how a solution may be obtained. Let

$x = \begin{pmatrix} x_{1}\\ \vdots \\ x_{4}\end{pmatrix};\qquad c = \begin{pmatrix} c_{1}\\ \vdots \\ c_{4}\end{pmatrix};$

$I = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix};\qquad A = \begin{pmatrix} a_{11} & \cdots & a_{14} \\ \vdots & \ddots & \vdots \\ a_{41} & \cdots & a_{44} \end{pmatrix};$

denote the total output vector, the final demand vector, the unit matrix and the input-output matrix, respectively. Then:

$\begin{matrix} Ax + c & = & x \\ (I-A)x & = & c \\ x & = & (I-A)^{-1}c \end{matrix}$

provided $(I - A)$ is invertible.

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.

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.

**Table: Adding Export And Import Transactions**
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.

[edit] Forecasting and/or Analysis Using Input-Output

**Table: Interregional Transactions**
Economic Activities	Ag	North Mfg	...	...	Ag	East Mfg	...	...	Ag	West Mfg	...	...	Exports	Total Outputs
North Mfg
...
...
Ag
East Mfg
...
...
Ag
West Mfg
...
...

**Table: Input-Output Model for Hypothetical Economy Total requirements from regional industries per dollar of output delivered to final demand**
Purchasing Industry	Agriculture	Transport	Manufacturer	Services
Selling Industry
Agriculture	1.14	0.22	0.13	0.12
Transportation	0.19	1.10	0.16	0.07
Manufacturing	0.16	0.16	1.16	0.06
Services	0.08	0.05	0.08	1.09
Total	1.57	1.53	1.53	1.34

[edit] 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[4], 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.

In a technical sense, input-output analysis can be seen as a special case of consistency analysis without money and without entrepreneurship and transaction cost.

[edit] 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.
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., 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.[5].
Polenske, Karen. Advances in Input-Output Analysis. 1976.
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.

[edit] See also

[edit] External links

Input-Output Analysis and Related Methods, San José State University
Doing Business project input/output tables for reforms

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