Appendix A: The Projection Model

Population projections are obtained using a social accounting, or cohort component, framework.[16] There are N = 100 (single year) age groups. 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 j th age who survive in the country to the age i, where p_j is the number of people aged j and

            (A1)

where only the a_i+1,i, for i = 1,..., N - 1 are non-zero. Males and females are distinguished by subscripts m and f, so that matrices of coefficients are A_m and A_f. Let p_m,t and p_f,t denote vectors of male and female populations at t, where the i-th element is the corresponding number of age i. The vectors of births and immigrants are b and m respectively. The forward equations:

            (A2)

           (A3)

In general the matrices A_m and A_f, along with the births and inward migration flows, vary over time.

Suppose that c_i represents the proportion of females of age i who give birth per year. In general the c_i values vary over time. Suppose that a proportion, δ, of 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 are:

           (A4)

           (A5)

where c′ is the transpose of the vector c. Equations (A2) to (A5) can be used to make population projections, for assumed migration levels.

The per capita social expenditures are placed in a matrix, S, with N rows and k columns, where there are k items of social expenditure and the i, j th element s_ij is the per capita cost of the j th type of social expenditure in the i th age group. Suppose the j th social expenditure 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 t,C_t, is thus:

           (A6)

Expenditure per person in each category and age differs for males and females, so that (A6) is suitably expanded.[17]

Projections of Gross Domestic Product depend on: initial productivity (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, V_t, by the total labour force, L_t. The value of V_t is 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 vectors U_m and U_f be the age distributions of male and female unemployment rates. If ^∧ represents diagonalisation (a vector is written as the leading diagonal of a square matrix with other elements zero) unemployment in period t is:

      (A7)

The labour force, L_t, is:

      (A8)

If productivity grows at the rate, θ, GDP t is the product of the utilisation rate, 1 - U_t = 1 - V_t/L_t, the labour force, Lt, and productivity, so that:

            (A9)