(1) Y=Z—AZ In equation (1), the term Y denotes a final demand vector, Z denotes a gross output vector, and A denotes a matrix of intermediate use coefficients, which can be calculated using data from IO tables. Equation (1) therefore defines final output with respect to intermediate input requirements.

The first matrix that is calculated is the widely used Leontief inverse matrix, also known as the multiplier matrix M. With some manipulation we arrive at the Leontief inverse matrix:

# �������<sup>��</sup> ����

The multiplier matrix M measures the inputs contained in a unit of final output. This M matrix contains both direct and indirect inputs. In particular, if we assign the sector indexes i,j to the M matrix, then a representative element of the M matrix, ���, gives the direct and indirect inputs (and thus receipts) of sector i linked to each unit (for example each dollar) of sector j receipts in the data. This implies real production activities measured by value of output. For our purposes, it provides a means to trace through these income flows, the flow of gross activity (and eventually labor value added) from intermediate to final goods and services, ostensibly across borders as well as sectors. Because linkages will vary by industry, each industry will be characterized by different multipliers.

Next, we need to calculate a diagonal matrix that measures the labor value added (compensation of employees) share of total gross output, which can be calculated using data from SAMs. To focus on labor value added, rather than gross activity, we note first that in terms of gross output values Z, some share of this involves labor value added within each sector. We define �<sup>�</sup> as the diagonal matrix indexed over i,j with diagonal elements equal to the compensation of employees’ shares of output Z. These shares can be split between types of workers (e.g. skilled/unskilled).

We then use M and �<sup>�</sup> to provide a breakdown of the flow of labor value added across activities in the form of the matrix V: Similar to the Leontief inverse matrix itself, the V matrix identifies the inputs of labor value added in each sector related to a unit of final demand.

Using the V matrix as a multiplier for exports and final outputs, one can obtain the total labor value added of exports and final outputs, respectively. If we multiply V by the diagonal matrix �<sup>�</sup> whose non-zero elements are the vector of final demand, the matrix yields a breakdown of economy-wide labor value added (both direct and indirect) (the primary component of Gross National Product on a source basis). Similarly, if we multiply V by the diagonal matrix �<sup>�</sup> whose non-zero elements are the national export vector, we can recover the labor value added content of exports X (both direct and indirect).

The G matrix and the H matrix thus give us the set of linkages, both direct and indirect, between value added across sectors for total domestic demand and export demand, respectively.

# _<u>Computing the jobs content of exports</u>_

We are also interested in the jobs content of exports. In order to compute that one would ideally need to split the compensation of employees between the number of jobs and the average wage. However this split 7