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. Equation (1) therefore defines final output with respect to intermediate input requirements. With some manipulation we arrive the Leontief inverse matrix, also known as the multiplier matrix M. Z=(I—A)−1 Y=MY (2) The multiplier matrix M measures the inputs contained in a unit of final output. In particular, if we assign the sector indexes i, j to the A and M matrices, then a representative element of the M matrix gives the direct and indirect inputs (and thus the sector i receipts) 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 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. To focus on value added, we note first that in terms of gross output values Z, some share of this involves value added within each sector. We define as the diagonal matrix indexed over i,j with diagonal elements equal to the value added shares of output Z. We then use M to provide a breakdown of the flow of value added across activities in the form of the matrix V. V= M (3) Similar to the Leontief inverse matrix itself, the V matrix identifies the inputs of value added in each sector related to a unit of final demand. If we multiply V by the diagonal matrix whose non-zero elements are the vector of final outputs, the matrix yields a breakdown of economy-wide value added (the primary component of Gross National Product on a source basis). Similarly, if we multiply V by the diagonal matrix whose non-zero elements are the national export vector, we can recover the value added content of exports X (both direct and indirect). G=V (4) H=V (5) The G matrix and the H matrix give us the set of linkages, both direct and indirect, between value added across sectors.
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. Equation (1) therefore defines final output with respect to intermediate input requirements. With some manipulation we arrive the Leontief inverse matrix, also known as the multiplier matrix M. Z=(I—A)−1 Y=MY (2) The multiplier matrix M measures the inputs contained in a unit of final output. In particular, if we assign the sector indexes i, j to the A and M matrices, then a representative element of the M matrix gives the direct and indirect inputs (and thus the sector i receipts) 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 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. To focus on value added, we note first that in terms of gross output values Z, some share of this involves value added within each sector. We define as the diagonal matrix indexed over i,j with diagonal elements equal to the value added shares of output Z. We then use M to provide a breakdown of the flow of value added across activities in the form of the matrix V. V= M (3) Similar to the Leontief inverse matrix itself, the V matrix identifies the inputs of value added in each sector related to a unit of final demand. If we multiply V by the diagonal matrix whose non-zero elements are the vector of final outputs, the matrix yields a breakdown of economy-wide value added (the primary component of Gross National Product on a source basis). Similarly, if we multiply V by the diagonal matrix whose non-zero elements are the national export vector, we can recover the value added content of exports X (both direct and indirect). G=V (4) H=V (5) The G matrix and the H matrix give us the set of linkages, both direct and indirect, between value added across sectors.
EVAD_Documentation_FrancoisManchinTomberger2014.docx