Difference between revisions of "Technical Coefficient Matrix"
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== Definition == | == Definition == | ||
| − | A '''Technical Coefficient Matrix''' is the collection of | + | A '''Technical Coefficient Matrix''' (also known as the ''Technology Matrix'', ''Direct Coefficients'' or ''Direct Requirements Matrix'') is the collection of [[Input-Output Coefficient | input-output coefficients]] . |
| + | |||
| + | This matrix is a central element of an [[Input-Output Model]]. It is derived by dividing the [[Input-Output Matrix]] (Z) by the vector of [[Total Output]] (x), thereby generating a normalized representation of economic structure that is independent of the units used to quantify exchanges (whether those are monetary or physical units). | ||
| + | |||
| + | The input coefficients for all of the intermediate inputs and all value added components add up to one (100%). | ||
== Formula == | == Formula == | ||
| − | + | The technical coefficients matrix is often in IO literature denoted as the <math>A</math> matrix and it is given by the multiplication of Z with the diagonalised and inverted industry output x: | |
| + | |||
| + | :<math> | ||
| + | \begin{align} | ||
| + | A & = Z\hat{x}^{-1} \\ | ||
| + | A_{ij} & = \frac{Z_{ij}}{x_j} | ||
| + | \end{align} | ||
| + | </math> | ||
| + | |||
| + | In the case of a [[Multiregional Input-Output Model]] the generalized expression is<ref>R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009</ref>: | ||
| + | |||
| + | :<math> | ||
| + | \begin{align} | ||
| + | A & = Z\hat{x}^{-1} \\ | ||
| + | A^{pq}_{ij} & = \frac{Z^{pq}_{ij}}{x^{p}_j} | ||
| + | \end{align} | ||
| + | </math> | ||
== Issues and Challenges == | == Issues and Challenges == | ||
| − | “Accounting coefficient” would be a better term than “technical coefficient”, since what we have are monetary amounts and not quantities. However, the term “technical coefficient” is generally used.<ref>Understanding National Accounts F.Lequiller, D. Blades, OECD 2014</ref> | + | * “Accounting coefficient” would be a better term than “technical coefficient”, since what we have are monetary amounts and not quantities. However, the term “technical coefficient” is generally used.<ref>Understanding National Accounts F.Lequiller, D. Blades, OECD 2014</ref> |
| + | |||
| + | == 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] | ||
== See Also == | == See Also == | ||
Latest revision as of 15:34, 16 November 2023
Definition
A Technical Coefficient Matrix (also known as the Technology Matrix, Direct Coefficients or Direct Requirements Matrix) is the collection of input-output coefficients .
This matrix is a central element of an Input-Output Model. It is derived by dividing the Input-Output Matrix (Z) by the vector of Total Output (x), thereby generating a normalized representation of economic structure that is independent of the units used to quantify exchanges (whether those are monetary or physical units).
The input coefficients for all of the intermediate inputs and all value added components add up to one (100%).
Formula
The technical coefficients matrix is often in IO literature denoted as the 
matrix and it is given by the multiplication of Z with the diagonalised and inverted industry output x:
In the case of a Multiregional Input-Output Model the generalized expression is[1]:
Issues and Challenges
- “Accounting coefficient” would be a better term than “technical coefficient”, since what we have are monetary amounts and not quantities. However, the term “technical coefficient” is generally used.[2]
