Euler and Süßmilch’s Population Growth Model

Abstract

In 1761, the German demographer Johann Peter Süßmilch published a simple population growth model that starts with a couple, in the eighth chapter of his book "Die göttliche Ordnung". With the help of the Swiss mathematician Leonhard Euler, he projected the population for 300 years. He demonstrated that after that time the population will be growing approximately geometrically. In this paper, the population projection of Euler and Süßmilch is reanalyzed using matrix algebra. Graphs and tables show the time series of the population and its growth rates. Age structures of selected years are presented. The solution of the projection equation is derived. It is shown that the projection model can be described by a geometric trend model which is superimposed by six cyclical components. In the long run, the population time series can be explained quite well by the sum of only two components, the trend component and one component with explosive cycles of a period of about 24 years. In the very long run, the influence of the cyclical component diminishes, and the series can be solely explained by its geometric trend component, as has been also recognized by Euler and Süßmilch.