Difference between revisions of "Power Series Approximation"
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== Definition == | == Definition == | ||
| − | '''Power Series Approximation''' in the context of IO analysis is the representation of a [[Leontief Inverse Matrix]] as an infinite sum of powers of the [[Technical Coefficient Matrix]] | + | '''Power Series Approximation''' in the context of IO analysis is the representation of a [[Leontief Inverse Matrix]] as an infinite sum of powers of the [[Technical Coefficient Matrix]]. <ref>R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009</ref> |
== Formula == | == Formula == | ||
| Line 10: | Line 10: | ||
The power series converges only under certain conditions: | The power series converges only under certain conditions: | ||
| − | * A ≥ 0 (coefficients matrix contains only non-negative terms) | + | * <math>A ≥ 0</math> (coefficients matrix contains only non-negative terms) |
| − | * The system produces more output than it requires inputs N(A) < 1 | + | * The system produces more output than it requires inputs <math>N(A) < 1</math> |
| − | * |I − A| > 0 | + | * The Leontief matrix is non-singular, <math>|I − A| > 0</math> |
| − | * More generally, the Hawkins-Simon conditions | + | * More generally, the Hawkins-Simon conditions apply |
== Usage == | == Usage == | ||
| Line 19: | Line 19: | ||
== Issues and Challenges == | == Issues and Challenges == | ||
| − | * For large matrices there may be significant | + | * For large matrices there may be significant computational power or specialized software required to evaluate the expansion |
| + | |||
| + | == Further Resources == | ||
| + | * [https://www.openriskacademy.com/mod/page/view.php?id=800 Crash Course on Input-Output Model Mathematics] | ||
== References == | == References == | ||
Revision as of 13:53, 18 September 2023
Definition
Power Series Approximation in the context of IO analysis is the representation of a Leontief Inverse Matrix as an infinite sum of powers of the Technical Coefficient Matrix. [1]
Formula
The power series converges only under certain conditions:
- Failed to parse (lexing error): A ≥ 0 (coefficients matrix contains only non-negative terms)
- The system produces more output than it requires inputs
- The Leontief matrix is non-singular, Failed to parse (lexing error): |I − A| > 0
- More generally, the Hawkins-Simon conditions apply
Usage
The power series approximation makes it more explicit that the Leontief inverse matrix represents value chain interrelations at an ever increasing expansion of the interacting entities required to produce an output.
Issues and Challenges
- For large matrices there may be significant computational power or specialized software required to evaluate the expansion
Further Resources
References
- ↑ R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009
