# 2.1 Stocks and Flows

The flows of individuals among defined benefit categories or 'states' from one period to the next can be recorded in a demographic or social accounting matrix. Suppose the accounting period is three months, so that information is available about the state occupied by each individual at the beginning of each quarter. There are m states. Let (for ) denote the number of people who move from state j to state i from period t to period t+1 (at the beginning of quarter t they are in state j and at the beginning of quarter t+1 they are in state i). These flows are placed in a matrix . Let denote the vector whose ith element is the number of people who enter state i from outside the labour market during period t (for example, those leaving full time education and inward migrants): these are referred to as 'inflows'. Similarly is the vector of exits from the system at the end of period t for various reasons, including migration and death: these are referred to collectively as 'outflows'. Finally, is the vector of stocks of individuals in each state, , at the start of period t.

The information about stocks and flows are placed in a social accounting table shown in Figure 1 (where for convenience, t subscripts have been omitted). A prime attached to a variable indicates transposition, so that, for example, is the column vector of exits from the system, rewritten as a row vector. In this table, the flows take place from columns to rows. These flows (number of individuals) can be converted into transition rates, , which denotes the proportion of individuals who started quarter t in state j and moved into state i by the beginning of quarter t+1. The precise timing of the movement is not recorded in this discrete time framework, so moves are effectively assumed to occur at the end of period t.[6] The transition proportions are therefore given by

In matrix terms, this can be written as , where the 'hat' indicates that the column vector forms the leading diagonal of a square matrix, with zeros in the off-diagonals. Hence .

Letting denote a vector of units, the sum of elements in the ith row of S is expressed as Si, and noting that , the 'closing stocks' are related to the 'opening stocks' and the flows according to:

Hence moving forward one period:

And:

The time profile of the stocks take a particularly simple form if the transition rates and inflows remain constant over time. Thus, becomes:

Here I denotes a unit matrix (a square matrix with a leading diagonal of units, and zeros elsewhere). The column sums of C are less than one and all the elements are non-negative. Hence, if the process continues long enough, and . The vector of equilibrium stocks is therefore:

And the time profile of stocks can easily be obtained from repeated use of the modified form of equation , whereby:

Furthermore, if again the transition rates remain constant, the average time spent in each state (along with its variance) is a simple function of the appropriate diagonal element of the C matrix. It can be shown that the average time in state j is while the variance of the time in the state is .

The outflows can be subdivided into a number of categories. For example, some individuals may leave the benefit system by, for example, migrating, dying, or moving into employment with sufficiently large earnings. Hence the (column) vector can be redefined as a K by m matrix where represents the number of individuals who move out of the benefit system from category j for reason k. The corresponding transition proportions are:

And if these are, as with the other transition proportions, assumed to remain constant at , the numbers in each category in each period are obtained simply from .

### Notes

• [6]In the discrete time framework, those who enter and leave a benefit category within the period are not recorded. The extent to which individuals may repeatedly enter and leave the benefit system is also not recorded here. It would be possible to redefine states to include some past transitions (allowing for dependence on the past), but this substantially increases the size of the coefficients matrix.
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