In the model assumptions we
2.1. Forward and backward linkages
As originally stated by Leontief in his groundbreaking papers (1936, 1941), the total output of a country can be expressed as the sum of intermediate input and final demand:equation(1)x=Ax+yx=Ax+ywhere x is the n × 1 vector of total output, y is the n × 1 vector with the each BIBR-1048 representing final demand—households, governments, capital and exports—and AA is the n × n matrix of input coefficients as follows:equation(2)A= Aij = XijXj
When solved for total output, this equation yields:equation(3)x=(I−A)−1y=Lyx=(I−A)−1y=Lywhere I is the identity matrix of the same order as A, L = (I − A)−1 is the well-known Leontief inverse, and its element lij represents the increase of the output in industry i due to a unit increase of final demand in industry j . Then, the sum of the elements in thejthjth column of the Leontief inverse matrix,equation(4)l.j=∑i=1nlijmeasures the total output from all sectors generated from one unit of final demand of sector j's output. Thus, l.j reflects the backward linkage of sector j (see Rasmussen (1956)).