PAPP103 - S07: Demographic models: stable population models

Lotka's stable population equations (cont.)

Lotka’s third equation: the characteristic equation of a stable population

The number of births in each age group is the product of the age-specific fertility rate and the size of the age group, so that the total number of births is the sum of births in each age group:

$B_{0} = P_{x, 0} \cdot f_{x} = \sum_{15}^{50} B_{0} \cdot L_{x} \cdot e x p {- r (x + 0.5)} \cdot f_{x}$

Dividing both sides of this equation by the number of births, gives

$1 = \sum_{15}^{50} e x p {- r (x + 0.5)} \cdot L_{x} \cdot f_{x}$

Lotka’s third equation

Lotka’s third equation expressed in terms of five-year age groups

$1 = \sum_{15}^{50} e x p {- r (x + 2.5)} \cdot {}_{5}L_{x} \cdot {}_{5}f_{x}$

Lotka’s third equation is the only one which shows the relationship between the growth rate r, the age-specific fertility patterns f_x, and the mortality pattern as expressed in the L_x function of the life table. However it does not give us a straightforward way of finding an expression for r. Even though r is the only unknown in this equation (as we "know" the values of the life table function and the fertility rates) we cannot easily "solve" the equation as it is not possible to re-write this equation as a simple expression of the form "r = . . . ."