Difference between revisions of "Input-Output Matrix"
Wiki admin (talk | contribs) |
Wiki admin (talk | contribs) |
||
| (10 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
== Definition == | == Definition == | ||
| − | The '''Industry Transaction Matrix''' ( | + | The '''Industry Transaction Matrix''' (or ''Transactions Table'') is the fundamental quantitative information used in [[Input-Output Analysis]]. It concerns the flow of products from each industrial sector (considered as a producer) to each of the sectors, itself and others (considered as consumers).<ref>R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009</ref> |
| − | == | + | == Formula == |
| − | + | The transaction matrix is usually denoted as <math>Z</math> | |
| − | |||
| − | The | ||
| − | == | + | === Single Region Case === |
| − | + | If there are n sectors in an economy the matrix reads: | |
:<math> | :<math> | ||
| − | \begin{ | + | \begin{align} |
| − | Z = | + | Z & = |
| − | \begin{ | + | \begin{bmatrix} |
Z_{11} & Z_{12} & \cdots & Z_{1n} \\ | Z_{11} & Z_{12} & \cdots & Z_{1n} \\ | ||
Z_{21} & Z_{22} & \cdots & Z_{2n} \\ | Z_{21} & Z_{22} & \cdots & Z_{2n} \\ | ||
\vdots & \vdots & \ddots & \vdots \\ | \vdots & \vdots & \ddots & \vdots \\ | ||
Z_{n1} & Z_{n2} & \cdots & Z_{nn} | Z_{n1} & Z_{n2} & \cdots & Z_{nn} | ||
| − | \end{ | + | \end{bmatrix} |
| − | \end{ | + | \end{align} |
</math> | </math> | ||
| + | |||
| + | * The entries of the matrix may denote either monetary values (in some defined currency) or physical (activity) values, e.g. volumes. | ||
| + | * The matrix is a flow matrix, hence values refer to a particular time period. | ||
| + | |||
| + | === Multi-Regional Case === | ||
| + | In the case of a [[Multiregional Input-Output Model]] where industrial sectors operate within distinct "regions" (e.g., countries) the transactions data are most conveniently expressed as a ''Tensor'' <math>Z^{pq}_{ij}</math> that captures exchanges between sector i and j located in regions p and q respectively. | ||
| + | |||
| + | In practice this tensor of rank four is represented as a "rolled-out" as a [[Partitioned Matrix]]<ref>Matricization (also known as unfolding or Mode-k flattening) is the process of reordering the elements of an N-th order array into a matrix.</ref> under the assumption that there is a global transactions matrix, in other words, regions and sectors are merely labels for production units that are qualitatively of similar nature. | ||
| + | |||
| + | == Usage == | ||
| + | This basic information from which an input-output model is developed is contained in an inter-industry transactions table. The rows of such a table describe the distribution of a producer’s output throughout the economy. The columns describe the composition of inputs required by a particular industry to produce its output. | ||
| + | |||
| + | The Transaction Matrix is of fundamental importance and may underpin alternative possible [[Input-Output Model | input-output models]]. | ||
| + | |||
| + | == See Also == | ||
| + | * [[Technical Coefficient Matrix]] | ||
| + | * [[Final Demand]] | ||
| + | * [[Total Output]] | ||
| + | |||
| + | == Further Resources == | ||
| + | * [https://www.openriskacademy.com/course/view.php?id=70 Crash Course on Input-Output Model Mathematics] | ||
| + | * [https://www.openriskacademy.com/course/view.php?id=64 Introduction to Input-Output Models using Python] | ||
| + | |||
| + | == Code == | ||
| + | * [https://github.com/konstantinstadler/pymrio pymrio] | ||
== References == | == References == | ||
| Line 28: | Line 51: | ||
{{#set:Has Formula = HAS_FORMULA}} | {{#set:Has Formula = HAS_FORMULA}} | ||
| + | {{#set:Has Code = HAS_CODE}} | ||
__SHOWFACTBOX__ | __SHOWFACTBOX__ | ||
Latest revision as of 15:39, 16 November 2023
Contents
Definition
The Industry Transaction Matrix (or Transactions Table) is the fundamental quantitative information used in Input-Output Analysis. It concerns the flow of products from each industrial sector (considered as a producer) to each of the sectors, itself and others (considered as consumers).[1]
Formula
The transaction matrix is usually denoted as 
Single Region Case
If there are n sectors in an economy the matrix reads:
- The entries of the matrix may denote either monetary values (in some defined currency) or physical (activity) values, e.g. volumes.
- The matrix is a flow matrix, hence values refer to a particular time period.
Multi-Regional Case
In the case of a Multiregional Input-Output Model where industrial sectors operate within distinct "regions" (e.g., countries) the transactions data are most conveniently expressed as a Tensor 
that captures exchanges between sector i and j located in regions p and q respectively.
In practice this tensor of rank four is represented as a "rolled-out" as a Partitioned Matrix[2] under the assumption that there is a global transactions matrix, in other words, regions and sectors are merely labels for production units that are qualitatively of similar nature.
Usage
This basic information from which an input-output model is developed is contained in an inter-industry transactions table. The rows of such a table describe the distribution of a producer’s output throughout the economy. The columns describe the composition of inputs required by a particular industry to produce its output.
The Transaction Matrix is of fundamental importance and may underpin alternative possible input-output models.
