Difference between revisions of "Input-Output Matrix"
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== Formula == | == Formula == | ||
+ | The transaction matrix is usually denoted as <math>Z</math> | ||
=== Single Region Case === | === Single Region Case === | ||
− | + | If there are n sectors in an economy the matrix reads: | |
:<math> | :<math> | ||
\begin{align} | \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{align} | \end{align} | ||
</math> | </math> | ||
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=== Multi-Regional Case === | === Multi-Regional Case === | ||
− | In the case of a [[Multiregional Input-Output Model]] where industrial sectors operate within distinct "regions" (e.g., countries) the transactions | + | 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 == | == Usage == | ||
− | This basic information from which an input-output model is developed is contained in an | + | 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]]. | The Transaction Matrix is of fundamental importance and may underpin alternative possible [[Input-Output Model | input-output models]]. | ||
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* [[Final Demand]] | * [[Final Demand]] | ||
* [[Total Output]] | * [[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 == | == Code == |
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.