The Theorem of Envelopment and Directives of Control in Resource Dependent Branching Processes

Abstract

The objective of this paper is to shed light on the implications of the Theorem of Envelopment of Societies in Resource Dependent Branching Processes (Bruss, Grenzen einer jeden Gesellschaft (in German; English summary). In: Jahresb. der Deut. Math.-Ver., vol. 116, pp. 137–152. Springer, Berlin, 2014; Bruss and Duerinckx, Ann. Appl. Probab. 25(1):324–372, 2015), with respect to real world conclusions. At the same, our interest is to attract interest to resource dependent branching processes as a way to model complicated growth processes. The theorem of envelopment displays a macro-economic phenomenon: Every human population modelled by a resource dependent branching process will, for whatever way it chooses policies to distribute resources in different generations, in the long run be bound to fluctuate between two extreme society forms. Understanding a phenomenon does not imply that one can turn this into a useful tool. Indeed, we only know some situations so far where this theorem is directly applicable to obtain specific advice how societies should control policies to reach certain objectives. One way to a better understanding to more general cases is to find parallels with related known results. This is what we try to do here, very much instigated by the suggestion of Prof. K.B. Athreya to indicate links which are visible in contributions presented by colleagues.