2 The projection model

This section provides a brief description of the projection model: further details are given in Appendix A.[5] Exogenous age-specific and gender-specific rates are used and, as mentioned above, no allowance is made for possible feedback effects, which may for example be generated by general equilibrium changes in price and wage rates, or endogenous policy responses.

The sequence of calculations is set out in Figure 1, where the grey boxes represent input data. The first stage is the production of projections for the size of the population, together with its distribution by age and gender. This requires projections of trends in fertility, mortality and net migration. Projected labour force participation rates are then combined with age and gender specific unemployment rates to generate the size of the workforce. This is multiplied by average productivity per worker to obtain GDP.

Figure 1: The Structure of the Model

Social expenditures per capita are combined with population (by age and gender) to obtain total social expenditure on each of 13 categories for each age and gender group. These categories are listed in Appendix Tables 4 and 5. The resulting total social expenditure is 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, as explained below. This process is repeated 5000 times, to produce a distribution of the social expenditure ratio for each projection year. The process therefore also generates distributions of the population by age and gender, as well as for each category of social expenditure.

Consider a relevant variable, Χ, which could be, for example, an unemployment rate, a fertility rate for women of a given age, or an item of social expenditure. In cases where the variable may take positive or negative values, it is assumed to be normally distributed. Where a variable is necessarily positive, and the distribution is positively skewed, the distribution is assumed to be lognormal.[6]

Where a variable is normally distributed with mean and variance μ and σ² respectively, Χ is distributed as N(μ,σ²) If r represents a random drawing from N(0,1), a simulated value, χ, can be obtained using χ = μ + r σ. In the lognormal case, μ and σ² refer to the mean and variance of logarithms. A random draw from a lognormal distribution is given by χ = exp (μ + r σ), where r is again a random N(0,1) variable.[7]

Each social expenditure projection is associated with its own demographic structure. The populations are necessarily derived using single-year age groups, but when calculating social expenditures and employment, some age grouping is necessary in view of the more limited data available for these variables.

The growth rates and standard deviations used in Creedy and Scobie (2005) were based on considerable information about past trends and the variability in several hundred fertility rates, mortality rates, migration rates, male/female birth ratios, labour force participation rates, unemployment rates and major categories of social expenditure. Their growth rates and standard deviations were estimated from the following regression:[8]

     log γ_t = α + β + u_t       (1)

Where γ_τ is per capita expenditure in the relevant category at time τ and u_t is an error term. This specification implies an estimated constant growth rate of . Furthermore, Var(log γ_t) = , so the standard deviations are derived from the estimated standard error of the regression, . The log-linear specification was found to provide a good fit to the historical data.[9]

In producing the projections reported below, data were obtained for 2010 relating to male and female populations, inward and outward migrants by single year of age, along with male and female unemployment rates by five-year age groups. The details along with data sources are given in Tables 1-3 in Appendix B. The 2010 social expenditure costs per capita, again within 5-year age groups, were also obtained for 13 categories and are reproduced in Tables 4 and 5 of Appendix B. The standard deviations for demographic components of the model were adapted from Creedy and Scobie (2005), in view of the lack of more recent data.