Input-Output Matrix

From Open Risk Manual


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]


The transaction matrix is usually denoted as Z

Single Region Case

If there are n sectors in an economy the matrix reads:

    Z & = 
      Z_{11} & Z_{12} & \cdots & Z_{1n} \\
      Z_{21} & Z_{22} & \cdots & Z_{2n} \\
      \vdots  & \vdots  & \ddots & \vdots  \\
      Z_{n1} & Z_{n2} & \cdots & Z_{nn}
  • 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 Z^{pq}_{ij} 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.


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.

See Also

Further Resources



  1. R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009
  2. 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.