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

- L = leakages from endogenous into exogenous accounts

- X = injections from exogenous into endogenous accounts

From Table 4, it can be written that

yn= n + x                                        (1)

yx = l + r                                              (2)

The amount that the endogenous accounts receive is equal to the amount that they spend.
In other words, in aggregate terms, total injections from the exogenous into the endogenous
accounts, i.e. the column sum of “x”, are equal to total leakages from the endogenous into the
exogenous accounts, i.e. considering i’ to be the unitary vector (row), the column sum of “1”
is: x * i^’ = l * i’.                                                                                                             (3)

a) Deduction of the accounting multipliers

In the structure of Table 4, if the entries in the N matrix are divided by the corresponding
total expenditures, a corresponding matrix (squared) can be defined of the average
expenditure propensities of the endogenous accounts within the endogenous accounts or of
the use of resources within those same accounts. Calling this matrix An, it can be written that

An = N*ÿn -1                                                  (4)

N = An* ÿn                                            (5)

Considering equation (1), yn= An*yn + x                                             (6)

Therefore, yn = (I-An)^-1* x = Ma * x.                                                        (7)

We thus have the equation that gives the total receipts of the endogenous accounts (yn), by
multiplying the injections “x” by the matrix of the accounting multipliers:

Ma = (I-An)^-1.                                                   (8)

On the other hand, if the entries in the L matrix are divided by the corresponding total
expenditures, a corresponding matrix (non squared) can be defined of the average expenditure
propensities of the endogenous accounts within the exogenous accounts or of the use of
resources from the endogenous accounts within the exogenous accounts. Calling this matrix
Al, it can be written that

Al= L*ÿn^-1                                                       (9)

L = Al* ÿn                                                 (10)

Considering equation (2), yx = Al*yn + r                                             (11)

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