Editing Input–output model (section)

== Basic derivation ==
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.<ref>{{cite web |title=How to understand and solve Leontief input-output model (technology matrix) problems |url=https://bloomingtontutors.com/blog/how-to-understand-and-solve-leontief-input-output-model-technology-matrix-problems |publisher=Bloomington Tutors}}</ref> Each column of the input–output [[Matrix (mathematics)|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 <math> n </math> sectors. Each sector produces <math> x_i </math> units of a single homogeneous good. Assume that the <math> j </math>th sector, in order to produce 1 unit, must use <math> a_{ij} </math> units from sector <math> i </math>. 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 <math> i </math>th sector <math> y_i </math>. Then we might write

:<math>
x_i = a_{i1}x_1 + a_{i2}x_2 + \cdots + a_{in}x_n + y_i,
</math>

or total output equals intermediate output plus final output. If we let <math> A </math> be the matrix of coefficients <math> a_{ij} </math>, <math>\mathbf x </math> be the vector of total output, and <math>\mathbf y </math> be the vector of final demand, then our expression for the economy becomes
{{NumBlk|::|
:<math>
\mathbf{x} = A\mathbf{x} + \mathbf{y}
</math>
|{{EquationRef|1}}}}

which after re-writing becomes <math> \left(I - A\right)\mathbf{x} = \mathbf{y} </math>. If the matrix <math> I - A </math> 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 [[Minor (linear algebra)|minors]] of the matrix <math> I - A </math> are all positive (known as the [[Hawkins–Simon condition]]),<ref>{{cite book |last=Nikaido |first=H. |title=Introduction to Sets and Mappings in Modern Economics |location=New York |publisher=Elsevier |year=1970 |isbn=0-444-10038-5 |pages=13–19 }}</ref> the required output vector <math>\mathbf x </math> is non-negative.

=== Example ===

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

:<math>
A = \begin{bmatrix} 0.5 & 0.2 \\ 0.4 & 0.1 \end{bmatrix} \text{ and } \mathbf{y} = \begin{bmatrix} 7 \\ 4 \end{bmatrix}.
</math>

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

:<math>
\mathbf{x} = \left(I - A\right)^{-1} \mathbf{y} = \begin{bmatrix} 19.19 \\ 12.97 \end{bmatrix}.
</math>

=== Further research ===

There is extensive literature on these models. The model has been extended to work with non-linear relationships between sectors.<ref>{{Cite journal |last=Sandberg |first=I. W. |date=1973 |title=A Nonlinear Input-Output Model of a Multisectored Economy |url=https://www.jstor.org/stable/1914043 |journal=Econometrica |volume=41 |issue=6 |pages=1167–1182 |doi=10.2307/1914043 |jstor=1914043 |issn=0012-9682|url-access=subscription }}</ref> There is the Hawkins–Simon condition on producibility. There has been research on disaggregation to clustered inter-industry flows, and on the study of constellations of industries. A great deal of empirical work has been done to identify coefficients, and data has been published for the national economy as well as for regions. The Leontief system can be extended to a model of general equilibrium; it offers a method of decomposing work done at a macro level.

=== Regional multipliers ===
While national input–output tables are commonly created by countries' statistics agencies, officially published regional input–output tables are rare. Therefore, economists often use [[Economic base analysis|location quotients]] to create regional multipliers starting from national data.<ref>{{cite journal | url=https://www.tandfonline.com/doi/abs/10.1080/00343409512331349173?journalCode=cres20 | doi=10.1080/00343409512331349173 | title=On the Appropriate Use of Location Quotients in Generating Regional Input–Output Tables | date=1995 | last1=Flegg | first1=A. T. | last2=Webber | first2=C. D. | last3=Elliott | first3=M. V. | journal=Regional Studies | volume=29 | issue=6 | pages=547–561 | bibcode=1995RegSt..29..547F }}</ref> This technique has been criticized because there are several location quotient regionalization techniques, and none are universally superior across all use-cases.<ref>
Lehtonen, Olli & Tykkyläinen, Markku. [https://www.researchgate.net/publication/254336057_Estimating_Regional_Input_Coefficients_and_Multipliers_Is_the_Choice_of_a_Non-Survey_Technique_a_Gamble "Estimating Regional Input Coefficients and Multipliers: Is the Choice of a Non-Survey Technique a Gamble?"], 16 July 2007. Retrieved 29 May 2019.</ref>

=== 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 inter-regional commodity flows, so introduce a column into the table headed "exports" and we introduce an "import" row.

{| border="1" cellpadding="5" cellspacing="0" align="center"
|+'''Table: Adding Export And Import Transactions'''
|Economic Activities
| 1
| 2
| ...
| ...	
| Z
| Exports
| Domestic Final Demand
| Total Outputs
|-
|1
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|							
|-
|2
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|
|
|
|								
|-
|...	
|
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|							
|-
|...
|
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|								
|-
|Z
|
|
|
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|-								
|Imports	
|
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|							
|}

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 has been done successfully using high-quality data. One who wishes to work with input–output systems must deal with [[industry classification]], data estimation, and inverting very large, often ill-conditioned matrices. The quality of the data and matrices of the input-output model can be improved by modelling activities with digital twins and solving the problem of optimizing management decisions.<ref>{{cite journal |title=Leontev Input-Output Balance Model as a Dynamic System Control Problem |first=S. N. |last=Masaev |year=2021 |journal= Herald of the Bauman Moscow State Technical University. Series Instrument Engineering |volume=2 |issue=135 |pages=66–82 |doi=10.18698/0236-3933-2021-2-66-82 |s2cid=237889078 }}</ref> Moreover, changes in relative prices are not readily handled by this modelling approach alone. Input–output accounts are part and parcel to a more flexible form of modelling, [[computable general equilibrium]] models{{Efn|However, CGE models rely on economic production functions, such as CES functions, and are not suited to representing actual technologies in detail, whereas in IO there is no limit in resolution. Furthermore, the use of CES functions results in using a number of separability assumptions which can have strong implications on the assumed technology.<ref>Shinichiro Nakamura and Yasushi Kondo, Waste Input-Output Analysis: Concepts and Application to Industrial Ecology, Springer, 2009, Section 4.1</ref> }}.

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.

===Technology Assumptions===
To construct input-output tables from supply and use tables, four principal assumptions can be applied. The choice depends on whether product-by-product or industry-by-industry input-output tables are to be established.<ref>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.</ref><ref>''Eurostat manual of supply, use and input-output tables'', 2008, Eurostat. Office for Official Publications of the European Communities, (P.24)</ref>