# Symmetric Input-Output Tables

In the ESA 2010, the product-by-product input-output table is the most important symmetric input-output table and this table is described here. However, it should be noted that a few countries in the EU prefer to compile industry-by-industry tables.

The product-by-product input-output table is compiled by converting the supply and use tables, both at basic prices. This involves a change in format, i.e. from two asymmetric tables to one symmetric table. The conversion can be divided into three steps:

1. allocation of secondary products in the supply table to the industries of which they are the principal products;
2. rearrangement of the columns of the use table from inputs into industries to inputs into homogeneous branches (without aggregation of the rows);
3. aggregation of the detailed products (rows) of the new use table to the homogeneous branches shown in the columns, if appropriate.

Step 1 involves transfers of outputs in the form of secondary products in the supply table as additions to the industries for which they are principal, and removals from the industries in which they were produced.

Step 2 is more complicated, as the basic data on inputs relate to industries and not to each individual product produced by each industry. The kind of conversion to be made here entails the transfer of inputs associated with secondary outputs from the industry in which that secondary output has been produced, to the industry to which they principally belong.

In case the conversion was started from rectangular tables step 3 involves the aggregation of the products in the new use table to the industries that generate them according to step 1.

These amendments start from data based on LKAUs (Local Kind of Activity Units).

The symmetric input-output table should be accompanied by at least two tables:

• a symmetric matrix showing the use of imports; and
• a symmetric input-output table for domestic output.

The latter table is used in calculating the cumulated coefficients, by the Leontief-inverse. This matrix is the inverse of the difference between the identity matrix I and the matrix of technical coefficients. By multiplication of the Leontief-inverse with the vector of final demand the vector of total output by product can be compiled.