Social Accounting Matrix

Definition

The Social Accounting Matrix is a type of Input-Output Model that aims to represent in more detail monetary flows in an economy, including in particular households and labour income.

Usage

The Social Accounting Matrix (SAM) has become used increasingly in the last years as a general equilibrium data system linking, among other accounts, production activities, factors of production and institutions (companies and households).

As such, it captures the circular interdependence characteristic of any economic system among

• production,
• the factorial income distribution (i.e. the distributionof value added generated by each production activity to the various factors), and
• the income distribution among institutions and, particularly, among different socio-economic household groups.

SUTs and IOTs provide a detailed picture of the structure of the economy but they do not show the interrelationship between GVA, final uses and incomes. By extending SUTs with the institutional sectors accounts, the entire Circular Flow Of Income can be shown in a SAM.

Formula

Given a SAM matrix T, the element $t_{ij}$ represents the amount of money from state j to state i.

The vector of totals, y, is such that $y_j = \sum_i t_{ij} = \sum_i t_{ji}$ , this is, rows and columns sum equal.

This is because the total amount of money received by sector i has to be equal to the total amount spent.

The SAM coefficient matrix A is defined as $a_{ij} = t_{ij} / y_{j}$ .

By construction, the SAM coefficient matrix and vector of totals y hold y = Ay, and considering normalized y vector, we have too $yi = yi / \sum_j y j, y = Ay$.

Observe that A cannot be considered a stochastic, or conditional probability, matrix, this is, in general <math>\sum_j a_{ij} = \sum_j t_{ij} /y j = 1 <math>.

However, the transposed matrix A^T is a stochastic matrix.

The variables used in defining the basic national economic accounts using the SNA conventions.

The transactions reflected in the table are the following:

• C = total consumption of goods and services in the economy
• I = total investment in capital goods
• X = total exports of goods and services
• G = government expenditures
• Q = total income generated in the economy
• D = depreciation or consumption of capital goods
• H = income generated overseas
• S = total private savings
• M = total imports of goods and services
• O = transfers of money overseas
• L = net lending of resources from overseas
• T = total direct taxation of consumers
• B = total government deficit spending

The row and column sums of this matrix constitute a set of macroeconomic accounting balance equations, which also corresponded to a set of accounting T accounts corresponding to each major set of economic activity:

• Production Account: Q + M = C + I + X + G
• Consumption Account: C + S + O + T = Q + D + H
• Capital Accumulation Account: I + D + L + B = S
• Balance of Payments Account: X + H = M + O + L
• Government Account: G = T + B