2  The projection model

2.1  An overview

The sequence of calculations is set out in Figure 1. The first stage in projecting social expenditures is the production of demographic projections for the size of the population, together with its distribution by age and gender. To achieve this requires forecasts of the underlying trends in fertility, mortality and net migration.[7] Estimates of the rates of labour force participation are then combined with age and gender specific unemployment rates to generate the size of the workforce. When this is multiplied by the average productivity per worker an estimate is obtained of GDP.

Combining the social expenditures per capita with the projected population by age and gender leads to the estimate of total social expenditure built up from the expenditure on each category for each of the age and gender groups. The resulting social expenditures are finally expressed as a share of projected GDP. In moving through the sequence of calculations, a random draw from the distribution of each variable is made, and the resulting set of input values is used to compute the ratio of social expenditure to GDP for a given forecast year. This process is repeated 5000 times for each year, to produce a distribution of the social expenditure ratio for each year from 2001 to 2051. The process therefore also generates distributions of the population by age and gender, as well as for each of 14 categories of social expenditure.

The uncertainty about each variable is reflected in the standard deviation of its distribution. The values used below are based on an analysis of past trends and the variability in fertility, mortality, migration, male/female birth ratios, labour force participation rates, unemployment rates and major categories of social expenditure. The resulting estimated standard deviations are used, and it is assumed that these remain the same in the future.[8]

The projections are entirely mechanical in that they do not rely on economic models of fertility, mortality, migration, labour force participation and so on. Instead, various exogenous age and gender specific rates are used and no allowance is made for possible feedback effects, which may for example be generated by general equilibrium changes in wage rates.[9]

2.2  Population projections

Population projections are obtained using the standard social accounting, or cohort component, framework.[10] There are
N = 100(single year) age groups, and no one is assumed to survive beyond the age of N. The square matrix of flows, f_ij , from columns to rows, has N - 1 non-zero elements which are placed on the diagonal immediately below the leading diagonal. The coefficients, a _ij , denote the proportion of people in the jth age who survive in the country to the age i, where p_j is the number of people aged j and

(1)    

where only the a _i+1,i for i= 1,..., ,N - 1 are non-zero. This framework applies to males and females separately, distinguished by subscripts m and f. Hence the matrices of coefficients for males and females are A_m and A_f respectively. Let p_m,t and p _f,t represent the vectors of male and female populations at time t, where the i-th element is the corresponding number of the aged i. The N-element vectors of births and immigrants are represented by b and m respectively, with appropriate subscripts; only the first element of b is non-zero, of course. The number of people existing in one year consists of those surviving from the previous year, plus births, plus immigrants. Hence the forward equations corresponding to this framework are:

(2)    P_m,t+1 = A_mP_m,t + b_m,t + m_m,t

(3)    P_f,t+1 = A_fP_f,t + b_f.t + m_f,t

Given population age distributions in a base year, and information about the relevant flows, equations (2) and (3) can be used to make projections. In general the matrices A_m and A_f, along with the births and inward migration flows, vary over time. Changes in the matrices can arise from changes in either mortality or outward migration.

Suppose that c_i represents the proportion of females of age i who give birth per year. Many elements, for young and old ages, of the vector, c, are of course zero, and in general the c_is vary over time. Suppose that a proportion, δ, of all births are male, and define the N-element vector τ as the column vector having unity as the first element and zeros elsewhere. Then births in any year can be represented by:

(4)    b_m = δτc'p_f

(5)    b_f = (1- δ)τc'p_f

where c′ is the transpose of the vector c, that is, the column vector written as a row. The b vectors contain only one non-zero element. Equations (2) to (5) can thus be used to make population projections, for assumed migration levels.

Figure 1 - An overview of the key relationships in the model

Note: The shadowed boxes represent input data.