Epilogue

One of the most striking features of the topics analyzed in the previous chapters is the breadth and depth of the economics involved in the analysis of population dynamics. The conventional perception that “demographic movements were largely exogenous to the economic system, and were to be left to sociologists and other non-economists” (Samuelson, 1976, p. 243) may be based on a conventional understanding of demography itself. Once we realize that modern individual fertility decisions may be affected by many economic variables, we can understand why demographic movements may be correlated with various economic indexes of the society. Once we shift our focus from the size and growth rate of the population to its economic characteristics, we realize that there is an abundance of topics for research and analysis. Moreover, once we perceive that the characteristic composition of the population is usually an aggregate result of various decisions by individuals, we find that our analysis is not confined to fertility-related economic variables. Thus, we are able to use the general framework to study the income distribution (chapters 4, 5), the attitude composition (chapter 8), the occupation structure (chapter 9), and the aggregate savings and pensions (chapters 11,12) of the population. The methodology adopted in this book is quite consistent: I emphasize the impact of individual decisions on the aggregate dynamics of demographic characteristics. As far as the steady state or dynamic fluctuations are concerned, the theory of stochastic processes is the basic tool necessary for the analysis. Other than the possible technical difficulty, there is nothing conceptually difficult in the modeling. But very often, the aggregate variables in question may feed back and influence individual decisions. In chapters 8 and 9, we see how the aggregate custom or occupational composition in the previous period affects individual decisions in the current period. These are in fact special cases and are easily dealt with. For many other economic variables, the micro-macro interaction involved is rather complex. There are several variables that may affect and also be affected by individual decisions.

Easterlin Cycles: Fertility and the Labor Market

I mentioned in chapter 7 that the fluctuation of human population can be summarized into three broad categories: the pretransitional, transitional, and posttransitional cycles. Among these three categories, the last one has caught the attention of most demographic economists in the past thirty years. The main reason for this unbalanced research attention is that the posttransitional cycles appear only in developed countries, where high-quality data are available for empirical research. The recent development of advanced mathematical tools also facilitates the analysis of posttransitional density-dependent population dynamics. In this chapter we will provide a summary of the theoretical and empirical analyses of the most typical population fluctuations in the posttransitional period: the so-called Easterlin cycles. The well-known Easterlin cycles, named after the pioneer work by Richard Easterlin (1961, 1980), describe the observed two-generation-long birth cycles in the twentieth-century United States and in several other developed countries. Easterlin believed that there were two features associated with the observed cycles: they are related to the labor market, and they are more or less “self-generating” (Easterlin, 1961). The first feature implies that a complete theoretical framework should characterize how people’s fertility behavior is affected by the labor market and how the labor market is affected by the fertility pattern. The second feature addresses whether the theoretical framework can generate a persistent fertility fluctuation. An ideal theoretical framework should embody both of these features, and an ideal empirical analysis should also be compatible with these features. We start the background introduction by studying a Malthusian model presented by Lee (1974). Let us consider an overlapping-generation framework in which each individual lives one or two periods. The first period is childhood, the second period is adulthood, and all surviving adults will be in the labor force. Lee wrote down the following two equations: . . . W(t) = f(L(t)), (10.1). . . . . . b(t) = g(W(t)), (10.2). . . where W(t) is the wage rate (at time t), L is the size of the adult age group, b is the crude birth rate, and f(.) and g(.) are functions with f'(.) < 0 and g'(.) > 0.

Lineage Extinction and Inheritance Patterns

Perhaps the most basic biological instincts of all creatures are to survive and to produce offspring. In ancient times, poor hygienic environment and occasional widespread epidemics obviously gave people strong reasons to worry about the possible extinction of their own lineage. But to transform such a worry into a mathematical problem, it is helpful if the upper class of the society, which has the ability and the leisure to think about the problem on an abstract level, also feels the possibility of such an extinction. Indeed, this was the case in eighteenth-century western Europe. The development of the theory of branching processes in fact started with the calculation of the probability of family surname extinction. Mode (1971) argued that one of the reasons for the decay of family names was that “physical comfort and intellectual capacity were necessarily accompanied by a diminution in fertility.” This statement, that parents choose to have fewer children because they want increase their enjoyment of life, seems to be a more suitable characterization of the argument of Becker’s (1991) contemporary household economics. Others, such as Chu and Lee (1994), argued that it was the scourges of war and famine that were responsible for the major rises of mortality in ancient history. Whatever the cause of lineage extinction, as a large proportion of family surnames continued to die out, Francis Galton (1873), one of the founders of the theory of branching processes, presented his concern with lineage extinction on an abstract level: The problem of family surname extinction concerns part (i) of problem 4001; part (ii) is about the distribution of surnames (types), which was the focus of chapter 4 of this book. Problem 4001 did not attract much attention until Agner Krarup Erlang became interested in this problem because his mother’s surname, Krarup, was about to become extinct. Erlang arrived at the solution below. First, we remove the restriction in problem 4001 that a man can have at most five sons, and let pk be the probability of having k surviving male children.

Sex Preferences and Two-Sex Models

The demographic models I reviewed in previous chapters are all one-sex models, in which the sex referred to is usually the female. This setting can be justified if we assume either that the life-cycle vital rates (as functions of state variables) for both sexes are the same or that the population dynamics are determined by one sex alone, independent of the possibly relative abundance of the other sex. However, at least for human population, neither assumption is valid. The ratio of newborn girls and newborn boys is close to one, but is less than one for almost all countries in the world. The age-specific mortality rates of women are also lower than those of men worldwide. This is called sexual dimorphism in the demography literature. Such a dimorphism makes the study of two-sex models indispensable. If we look at the male and female vital rates, we find that the differences are small. Despite this small difference, population dynamics derived solely from male vital rates and those derived solely from female vital rates will show ever-increasing differences with the passage of time. Furthermore, because the intrinsic growth rates derived from male and female lines, respectively, are distinct, we cannot avoid the undesirable conclusion that, if we do not incorporate males and females in a unified model, eventually the sex ratio will become either zero or infinity, which is never the case in reality. This is the inconsistency we have to overcome while dealing with population models with two sexes. Another technical difficulty with two-sex modeling has to do with the irreducibility of the state-transition matrix. I mentioned in chapters 2 and 3 that in an age-specific one-sex model, because people older than a particular age, say β, are not fertile anymore, the age group older than β is an absorbing set; hence, our focus of population dynamics can be restricted to the age set [0, β]. This is why we can transform the n × n Leslie matrix to a Lolka renewal equation. In a two-sex model, however, there does not exist a common upper bound for the reproduction of both sexes, for a male older than β can marry a female younger than β and become fertile again.

Income-Specific Population Models: Steady States and Comparative Dynamics

I mentioned in chapter 1 that the standard new household economics model of fertility, derived and modified by Becker (1960), DeTray (1973), Willis (1973), and later followers, emphasized the parental choices and tradeoffs between the quantity and the quality of their children. As Becker (1960) pointed out, one motivation for the new household economics approach to fertility decisions is to construct a demand-side household-decision structure to replace Malthus’s out-of-date supply-side population theory. The fertility decision theory along these lines has been called by Schultz (1981, 1988) and Dasgupta (1995) the demand-side demography theory. One difference between the demand-side demography theory and the classical Malthusian theory is that the former approach emphasized the static decision of a micro agent, whereas the Malthusian theory described the macro dynamic pattern of the population. Thus, from a theoretical point of view, the development of the demand-side demography lacks a macro dynamic counterpart. In this chapter I shall establish a macro dynamic population theory based on a fairly general version of Becker’s and others’ static setup of fertility demand. Once we shift our focus to the household fertility decision, it is natural that the household economic variables that affect female fertility decisions, such as her wages, family income, or the opportunity cost of babysitting, will become important explanatory variables of aggregate demographic patterns. Given that the fluctuation of mortality is no longer significant in recent years and that human fertility decisions are largely affected by the above-mentioned household economic variables, then in order to explain the aggregate pattern of population movement, it is natural to classify people by these economic variables rather than by ages. This is another motivation for the derivation of a non-age-specific stable population theory. As we focus upon the macro dynamic implications of Becker’s micro static fertility decision model, it is convenient to ignore sex differences and suppress the age structure of a person by assuming that everyone lives two periods, young and old. This is very much the same as the one-sex Samuelsonian (1958) overlapping-generation model: individuals who remain in the parental household are called young; they become old when they form their own families.

Introduction

The classical model of Malthus ([1783] 1970) can be described as a supply-side, dynamic, macro theory of population. It is a supply-side theory because Malthus did not emphasize the role of individual demand decisions on population-related variables. Individuals in ancient times certainly had their preferences for children and marriage; but Malthus assumed that such preferences would by and large be checked by natural constraints and that only when families had sufficient incomes would their preferences for children and marriage be effectively revealed (Schultz, 1981). In ancient times, when the hygienic environment and medical technology were primitive, and when production technology and administrative capacity changed relatively slowly, the natural checks on human growth and fertility almost always dominated the dynamics of population; the demand-side scenario, which originates from individual preferences, never played a significant role. The Malthusian theory is mainly a dynamic one because it describes why a population would have an equilibrium size that corresponds to the subsistence level of income and why a population would converge to such an equilibrium. Malthus argued that when a population size is larger than the equilibrium size, the per capita income will fall because of diminishing returns. This fall will be followed by an increase in mortality and a reduction in population growth rate, which in turn will drive the population size down to the equilibrium. When a population size is smaller than the equilibrium, the adjustment mechanism works in the opposite way, and the size increases toward the equilibrium. The conventional Malthusian model also largely ignores differences in decisions made by individual families; hence a set of macro variables becomes the only focus, thereby making a simple dynamic analysis possible. As Samuelson (1976) pointed out, although most classical economists, such as Adam Smith, David Ricardo, and John Stuart Mill, considered population analysis part of economics, by the early twentieth century most economists had decided that demographic movements were largely exogenous to the economic system and should be left to sociologists and other noncconomists for discussion. A more active alternative for economists is to modify the conventional Malthusian theory to allow it to be compatible with contemporary population practices and issues.

Demographic Transition and Economic Development

Demographic transition refers to a shift in reproductive behavior from a state of high birth and death rates to a state of low birth and death rates. This transition takes place because of advances in agricultural technology and medical science or improvement in hygiene environment, all of which result in corresponding declines in the mortality rate. In this first phase of the demographic transition, population growth rises because the decline in mortality rate has not been coupled with any significant change in parents’ fertility decisions. Then, in the second phase of the transition, parents begin to reduce their fertility as they realize that their ideal number of children can be more easily achieved with fewer births. The widespread use of contraceptive techniques facilitates parents’ attempts to reduce fertility, which in turn causes a decline in the population growth rate. Eventually, the population growth rate converges to a new level, which may be higher or lower than in the pretransitional stage. To facilitate comparison, we can use figure 11.1 to characterize the time and process of the transition. In figure 11.1, Tα marks the apparent starting point of a continuous decline in mortality. Tβ, which normally occurs later than Tα, refers to the time at which the fertility rate begins to decline. Tγ, is the point of lasting return, with an average rate of natural increase equal to or less than that of the period preceding the date of Tα. The convention is to define D = Tγ - Tα as the duration of the transition period. Chesnais (1992) separated the observations of world demographic transition into several types. The first type includes developed countries in Europe and Japan; the second type consists of countries with immigrant European populations, such as the United States, Australia, and Argentina; late-developing countries, such as India. South Korea, and Jamaica, belong to the third type. For countries of the first type, the mortality decline process is closely related to the development of medical technology, which was gradual and spread out over time; hence, the demographic transition is also long. Late-developing countries and those with large immigrant populations were able to adopt the already-developed medical technology from the advanced countries at one time.

Population Size and Early Development

The Malthusian theory hypothesizes that the natural environment imposes various capacity constraints on human population growth and that population size has been and will be checked by these constraints. In such a classical theory, which was presumably motivated by observations of the ancient world, population might be the most important dynamic variable, although its role is rather passive: population is a variable that would be affected by, but would not affect, the environment. Boserup (1981), however, sees the role of population in the development of human economy as more consequential. She gave many persuasive examples that showed that, at least for the period up to the mid-twentieth century, population size might be a variable which actively spurred technological progress. This is also the viewpoint held by Lee (1986) and Pryor and Maurer (1982). After the Industrial Revolution, the role of population in economic dynamics, along with the reduction of mortality fluctuations and the increasing control of female fertility, evidently became secondary. The key variable that dominates the analysis of economic dynamics in the neoclassical growth theory along the lines of Solow (1956) is capital (or per capita capital). In Solow’s growth model, the role of population is minimal in the steady state: neither the level nor the growth rate of the steady-state per capita consumption has anything to do with the size of a population; only the steady-state per capita income level will be affected by the population growth rate. The growth pattern in the latter half of the twentieth century is markedly different. A key feature of our recent growth experience is the rapid innovation of new technologies. Modern growth theory has embraced the concept of increasing returns to explain such a unique growth pattern. However, various versions of the theory of increasing returns turn out to be necessarily linked to population. The hypothesis of learning by doing implies that growth in productivity is an increasing function of aggregate production, which is itself positively related to the size of population.

Occupation-Specific Population Models: Population and Dynastic Cycles

As Lee (1987) pointed out, vital rates of the human population are often determined by forces such as culture, institutions, technology, and individual rationality, forces that have little to do with density pressure or prior growth. Perhaps most people also expect “rational” human practice to weaken the biological responses of both fertility and mortality to density pressure, while strengthening the nonbiological response through institutional regulations. But can human institutional designs and rational responses really reduce the impact of natural checks? As we study the pattern of population dynamics in ancient China, we can provide some viewpoints different from the general opinion. The long-term relationship between human institutional designs and natural checks is discussed in chapter 14. The books by Ho (1959) and Chao and Hsieh (1988; hereinafter C&H) contain the most thorough research on the history of Chinese population. The data summarized in C&H have presented us with a time-population diagram, shown in figure 9.1. From this figure, as well as other related literature, the following “stylized facts” of population dynamics in Chinese history can be summarized: 1. Population declines often coincided with dynasty changes (C&H; Ho, 1959). 2. Population declines were often drastic in a rather short period of time. 3. Natural checks such as famine and epidemics did not independently reduce the population surplus (Ho, 1959); rather, population declines were often the direct and indirect results of internecine wars. 4. There are obvious peaks and troughs in the population data, but no regular cyclical patterns (C&H). The fact that no serious population decline appears to have been independently due to famines and epidemics seems to suggest a weak pattern of density-dependency for ancient Chinese populations, a pattern consistent with the observation of Lee mentioned in the beginning of this chapter. However, as noted by many historians (see, e.g., Ho, 1959, and C&H 1988), the frequent clashes between soldiers and rebellious peasants in Chinese history were often initiated by famine or density pressure. As such, the originally weak natural checks on population were often magnified by war, and such magnified “institutional checks” caused very drastic population changes.

Age-Specific Population Models: Steady States and Comparative Statics

Mainstream demographers studying the pattern of human population are used to classifying people by their ages. In the terminology of branching processes, the type space of the stochastic process is a subset of positive real numbers that characterize human ages. This chapter deals with this case and studies the corresponding steady states and comparative statics. I showed in chapter 2 that the dynamics of any type-specific population structure can be described by the equation Nt = QNt·l and that Q is block-decomposable in the age-specific case. The fact that the northeast block of Q being a zero matrix not only helps us derive the eigen-values and eigenvectors of Q but also helps us characterize the dynamic evolution of the birth size. Let Bt be the size of birth at period t, la = p1 × • • • × pa be the probability that a person can survive to age a, and ma be the average number of births per surviving member aged a. We see that the following accounting identity must hold: which is Lotka’s (1939) well-known renewal equation. is useful for deriving the steady-state age distribution. Given the assumption of a time-invariant fertility function mu, the total size of birth Bt, which is a linear combination of birth sizes of all fertile age groups, naturally grows at a constant rate in the steady state.