2.3  Social expenditure and GDP

The per capita social expenditures are placed in a matrix with N rows and k columns, where there are k items of social expenditure. If this expenditure matrix is denoted S , then the i, jth element S_ij measures the per capita cost of the jth type of social expenditure in the ith age group. Suppose that the jth type of social expenditure per capita is expected to grow in real terms at the annual rate ψ_j in each age group. Then define g_t as the k-element column vector whose j th element is equal to
(1+ψ_j)^t-1. Aggregate social expenditure at time t,c_t ,is thus equal to:

(6)    

where, as above, the prime indicates transposition. In the projections reported below, expenditure per person in each category and age differs for males and females, so that (6) is suitably expanded.[11]

Projections of Gross Domestic Product depend on assumptions about five factors, including: initial productivity (defined as GDP per employed person); productivity growth; employment rates; participation rates; and the population of working age.

Total employment is the product of the population, participation rates and the employment rate. Employment is calculated by multiplying the labour utilisation rate by the labour force. If u_t is the total unemployment rate in period t, the utilisation rate is 1-U_t. The aggregate unemployment rate is calculated by dividing the total number of unemployed persons in period t, V_t, by the total labour force in that period, L_t. The value of V_t is in turn calculated by multiplying the age distribution of unemployment rates by the age distribution of the labour force, where these differ according to both age and sex.

Let the vectors U_m and U_f be the N-element age distributions of male and female unemployment rates. If the symbol ^ represents diagonalisation, whereby the vector is written as the leading diagonal of a square matrix with other elements equal to zero, the total number of people unemployed in period t is:

(7)    

The labour force in period t, L_t , is given by:

(8)    

Suppose productivity grows at the constant rate, θ. Then GDP in period t is calculated as the product of the utilisation rate,
1 - U_t = 1-Vt / L_t, the labour force, L_t, and productivity, so that:

(10)    

If the population age distribution, along with the gender and age specific participation and unemployment rates, are constant, the social expenditure to GDP ratio remains constant if all items of expenditure grow at the same rate as productivity; that is if θ = ψ_j for j = 1,...,k. From (10), the rate of growth of real GDP (g_y) is given by:

(11)     (1 + g_y) = (1 + θ)(1 + g_w)

where g_w is the rate of growth in the number of workers.

Many assumptions are required to make projections, and many potential interdependencies may exist, though they are not easy to model. For example, productivity may itself depend on social expenditures and the age distribution of workers. Furthermore, participation rates and population growth are likely to be interdependent. The changing age distribution is only one component of the ratio of aggregate social expenditure to GDP, and its influence may, for example, be substantially affected by other components. However, as mentioned above, such interdependencies are not modelled here.

2.4  A Stochastic specification

The above description of the projection model is deterministic in the sense that each component is assumed to be known with certainty. The future mortality, fertility and migration rates, along with the various unemployment and participation rates and growth rates of productivity and social expenditure costs, cannot be known with certainty. One way to allow for this uncertainty is to specify, for each appropriate variable, a distribution. Each observation is regarded as being drawn from the corresponding distribution. A large number of projections can be made, where each projection uses random drawings from each of the distributions. This exercise produces a ‘sampling distribution’ of the ratio of social expenditure to GDP.[12]

It is therefore necessary to specify the form of the various distributions. Consider a relevant variable, denoted by x. This could be, for example, an unemployment rate, a fertility rate for women of a given age, or an item of social expenditure. In some cases, indicated below, it is assumed to be normally distributed with mean and variance μ and σ respectively; that is, x ~ N(μ,σ). If r represents a random drawing from the standard normal distribution N(0,1), a simulated value of x, x_y, can be obtained using:

(12)   x_r = µ+rσ 

since (x = µ)/σ is N (0,1). In cases where the variable is assumed to be lognormally distributed as Λ(µ,σ²), where now µ and σ² refer to means and variances of logarithms, a corresponding draw can be obtained using:

(13)      x_r = exp(µ+rσ)

The use of lognormal distributions ensures that the random draws are always positive.[13]