# Input-Output Matrix

## 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 ${\displaystyle Z}$

### Single Region Case

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

{\displaystyle {\begin{aligned}Z&={\begin{bmatrix}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}\end{bmatrix}}\end{aligned}}}
• 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 ${\displaystyle Z_{ij}^{pq}}$ 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.