Power Series Approximation

From Open Risk Manual

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

Formula


\begin{align}
(I - A)^{-1} & =I + A + A^2 + \ldots  + A^{n} + \ldots = \lim_{n \rightarrow +\infty} \sum_{k=0}^{n} A^{k}
\end{align}

The power series converges only under certain conditions:

  • A ≥ 0 (coefficients matrix contains only non-negative terms)
  • The system produces more output than it requires inputs N(A) < 1
  • |I − A| > 0
  • More generally, the Hawkins-Simon conditions

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 computation power or specialized software required to evaluate the expansion

References