The Leslie model and population stability: an application

In many countries with high socio-economic development, demographic trends highlight meaningful changes in the population structure and size especially due to the joined effects of declining fertility rates and increasing life expectancies, particularly at older ages. This phenomenon leads mainly to a drastic increase in the ratio of the elderly to the young active population and undermines the welfare systems. Specifically, it puts hard pressure on pension systems where the current expenditure is paid by the current contributions in the light of a pay-as-you-go financing because it occurs that the former is increasing and the latter is decreasing due to the afore-said phenomenon. In this paper, first we analyse the population forecast of five high-income countries on the basis of the current values of fertility rates and survival probabilities. We apply the Leslie model in a form elaborated by one of the authors in a previous article written in collaboration. Our findings confirm that the considered populations numerically decrease according to the value of the Leslie dominant eigenvalue lower than one, while the percentage distribution by age stabilises itself proportionally to the Leslie dominant eigenvector. The structure of this distribution indicates the afore-said “reversal” of the ratio of the elderly to the young population. Therefore, also consistently with the criterion of Brauer-Solow, we estimate the adjustment rate of the fertility rates at all ages in the female Leslie matrix so that the dominant eigenvalue is equal to one and thus the population tends to stabilise in its size in addition to its percentage distribution by age. As it is clear, this adjustment also involves a change in the Leslie dominant eigenvector and a realignment of the ratio of the elderly to the young population on lower values.