Demographic Models and Branching Processes

Abstract

All models describing the dynamic pattern of human population have two common features. First, the human population is usually divided into several types, and second, each type has a type-specific stochastic reproduction rate. The traditional literature of demography has been dominated by the age-specific models of Lotka (1939) and Leslie (1945,1948), where the type refers to the age of an individual and the type-specific reproduction rates refer to the age-specific vital rates in a life table, It has been shown that, mathematically, these age-specific models can be analyzed in a more general framework, namely, the multitype branching process. Most demography researchers, however, do not bother to pursue properties of the general branching process. They prefer to follow Lotka’s (1939) age-specific renewal equation approach in proceeding with their analysis because that renewal equation is technically convenient, whereas the steady-state and dynamic properties of a general branching process are usually much more difficult to derive. Although the analytical convenience of the age-specific models has facilitated the research on age-related topics, it also tends to obscure the fact that the age-specific model is merely a special kind of branching process. When female fertility becomes a decision variable of the family and the fertility-related family decision problems expand, these age-specific models are often unworkable. Despite the difficulties inherent in applying the traditional age-specific models to these decision dimensions, researchers still hesitate to go back to the general, but more difficult, branching process for solutions. This is perhaps why, as we mentioned in chapter 1, the demand-side theory of demography has not made much progress in describing the macro aggregate pattern of the population. In this chapter, I separate the discussion into the age-specific branching process and general branching processes. I show that the steady states and ergodic properties of these models can both be established under some regularity conditions. Although the material in this chapter is mostly a reorganization of previously established mathematical results, I believe that my summary is systematic and will be helpful to most readers. All the results summarized will be used in later chapters, but aspects of branching processes that are irrelevant to our purposes will not be discussed.