Properties of accounting matrices

8.126 Each entry in an aggregate matrix such as Table 8.19 can be considered as the grand total of a submatrix in which categories of transactors involved at either end of the set of transactions under consideration are presented. A very useful option in a matrix presentation of accounts is that different types of transactors and groupings thereof can be selected in each account, without giving up the coherence and the integration of the complete accounting system. This means that one may apply 'multiple actoring and multiple sectoring', by choosing in each account a unit and a classification of units which are most relevant to the set of economic flows under consideration.

8.127 In principle, each account can be broken down in two rather different ways: by subdividing the total economy into groups of units, or by assigning the categories of transactions shown in an account to various sub-accounts. For instance, a subdivision of the total economy in the first five accounts could run as follows:

  1. distinguish products in the goods and services account and classify these by product groups;
  2. distinguish local kind-of-activity units in the production account and classify these by industries;
  3. distinguish institutional units in the accounts for the primary and secondary distribution of income and for the use of income and classify these by institutional (sub)sectors.
8.128 These subdivisions have two major consequences. First, for all categories of transactions distinguished in a single cell of these accounts it becomes clear which group of paying units has exchanged what with which group of receiving units. Secondly, the interrelations among various economic flows are revealed through detailed cross- classifications. For instance, in the example given in the previous paragraph, a simple circular flow of income is presented, at a meso-level, through the following mappings:
  1. submatrix (3,2) shows which institutional sub-sector receives net value added from which industries;
  2. submatrices (4,3) and (5,4) show which institutional sub-sector receives primary income and disposable income from which institutional sub-sector (naturally, in the distribution of income accounts and in the use of income account different classifications can be applied, and then these submatrices are no longer diagonal);
  3. submatrix (1,5) shows which product group is consumed by which institutional sub-sectors; and
  4. submatrix (2,1) shows which industry makes which product groups.
8.129 When compiling such a matrix, it is convenient to start by designing an accounting structure which is relevant to the applications envisaged. Subsequently, in each account the most appropriate units and classifications of units are selected. However, in practice it will be an interactive process. Suppose, for instance, that there is a transaction category for which only total receipts and payments of transactors (the row and column totals of a submatrix) are known, and not who paid whom (the interior structure of the submatrix). This problem can be solved by the insertion of an undivided, dummy account.

8.130 Among the general properties of a matrix presentation of accounts are the following:

  1. a detailed matrix presentation is suitable for mathematical treatment using matrix algebra; this can also be of help when balancing the accounts;
  2. a detailed matrix presents a simultaneous breakdown of interrelated transactions by paying and receiving units; as a consequence, it is an appropriate format to reveal, at a meso-level, interrelations among economic flows; this includes those flows which involve two different types of units (e.g. final consumption expenditure on various categories of goods and services by a number of household sub-sectors);
  3. for a set of accounts giving a breakdown of transactions by paying and receiving units, a matrix presentation is more concise than other methods of presentation; the payment by one unit and the receipt by another unit involved in each transaction are represented by a single entry.
8.131 An aggregate matrix for the total economy can serve as a reference table for subsequent, more detailed tables. As soon as the reader is then introduced to a detailed presentation of parts of the system (supply and use tables, sector accounts, etc.), the relation of the detailed submatrices to the aggregate matrix should be clear through a system of codes. The matrix format is particularly advantageous if it is not possible or desirable to show an equally detailed classification in all accounts of the system.

8.132 The matrix presentation is a suitable tool for exploring the flexibility of the system. For instance, one may further elaborate on the interrelations between the social and economic aspects of the system to arrive at a Social Accounting Matrix. The SAM approach is set out and illustrated in the next subsection of this chapter.