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

yn = Bn * yn + Cn * yn+ x = [I - (I - Bn)^-1 * Cn]^-1 * (I - Bn)^-1 * x . ⁸                 (14)

Therefore: Ma = [I - (I - Bn)^-1 * Cn]^-1 * (I - Bn)^-1 = M3*M₂*M1.                                (15)

The accounting multiplier matrix is thus decomposed into multiplicative components, each

of which relates to a particular kind of connection in the system as a whole (Stone, 1985)⁹.

- The intragroup or direct effects matrix, which represents the effects of the initial exogenous
injection within the groups of accounts into which it had originally entered, i.e.:

M1 = (I - Bn)^-1.                                                  (16)

- The intergroup or indirect effects matrix, which represents the effects of the exogenous
injection into the groups of accounts, after its repercussions have completed a tour through
all the groups and returned to the one which they had originally entered. In other words, if
we consider “t” to be the number of groups of accounts (five in the present study):

M2= {I - [(I - Bn)^-1 * Cn ]^t}^-1.                                     (17)

- The extragroup or cross effects matrix, which represents the effects of the exogenous
injection, when it has completed a tour outside its original group without returning to it, or,
in other words, when it has moved around the whole system and ended up in one of the
other groups. Thus, if we consider “t” to be the number of groups of accounts:

M3 = {I + [(I - Bn)^-1 * Cn ] + [(I - Bn)^-1 * Cn ]² + ... + [(I - Bn)^-1 * Cn ]^t-1}       (18)

The decomposition of the accounting multiplier matrix can also be undertaken in an
additive form, as follows:

Ma = I + (M1 - I) + (M2- I) * M1 + (M3- I) * M2 * M1.                (19)

Where I represents the initial injection and the remaining components the additional effects
associated, respectively, with the three components described above (M1, M2 and M3).

Defourny and Thorbecke (1984) introduced an alternative to the above decomposition,
namely structural path analysis, which makes it possible to identify and quantify the links
between the pole (account) of origin and the pole (account) of destination of the impulses
resulting from injections. According to this technique, the accounting multiplier is considered

⁸ yn = An*yn + x = Bn*yn + Cn*yn + x ⇔ yn - Bn*yn = Cn*yn + x ⇔ yn = (I-Bn)^-1* Cn * yn + (I-Bn)^-1 *x ⇔ yn - (I-Bn)^-1 *
Cn * yn = (I-Bn)^-1 *x ⇔ [I - (I-Bn)^-1 * Cn] * yn = (I-Bn)^-1 *x ⇔ yn = [I - (I-Bn)^-1 * Cn]^-1 * (I-Bn)^-1 *x.

⁹ For a detailed deduction and explanation of these components, see, for example, Stone (1985, pp. 156-162);
Pyatt and Round (1985, pp. 192-197); Santos (1999, pp. 67-69).

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