Distribution of aggregate income in Portugal from 1995 to 2000 within a SAM (Social Accounting Matrix) framework. Modeling the household sector

as a “global influence”, which is decomposed into a series of “total influences”. These, in
turn, are decomposed into “direct influences” multiplied by the “path multiplier”:

ma _i = I^G = ∑ⁿ I^T = ∑ⁿ I^D *Mp                      (20)

maji=I_(i→j)=_p∑₌₁I_(i→j)p =_p∑₌₁I_(i→j)p Mp                          (20)

Where:

maji is the (j,i)^th element of the Ma (accounting multipliers) matrix, which quantifies the
full effect of a unitary injection xj on the endogenous variable yj;

I₍^G_{i→ j)} is the Global Influence of pole i on pole j;

p is the n^th elementary path - the arc linking two different poles, oriented in the direction of
the expenditure, located between i and j, with i being the pole of origin of the
elementary path 1 (the first) and j the pole of destination of the elementary path n
(the last);

T

I_{(i→ j)} is the Total Influence transmitted from i to j along the elementary path p;

I^D is the Direct Influence of i on j transmitted along the elementary path p, which
(i→ j)_p

measures the magnitude of the influence transmitted between its two poles through
the average expenditure propensity;

Mp is the Multiplier of the path p, or the path Multiplier, which expresses the extent to
which the influence along elementary path p is amplified through the effects of
adjacent feedback circuits¹⁰:

Mp = ^δΔP                                   (21)

where: ∆ = the determinant of matrix ∣ I-A_n I of the structure represented by the SAM,
∆p = the determinant of the submatrix of ∣ I-A_n ∣ obtained by removing the row
and the column associated with the poles of the elementary path p.

¹⁰ A circuit is a path for which the first pole (pole of origin) coincides with the last pole (pole of destination)
(Defourny and Thorbecke, 1984, p. 119).

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