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Population dynamics are the patterns of change over time in populations, ranging from fluctuations to long-term trends, and the underlying principles that govern these changes.

Population Fluctuations

All human populations exhibit fluctuations in their vital rates and consequent irregularities in their age distributions to a greater or lesser degree. Analyses of such fluctuations are of interest for many reasons – for historical understanding, as a basis for forecasting, for a deeper understanding of underlying social processes – but perhaps most intriguing is the possibility that they may afford some insight into more fundamental aspects of population dynamics and may illuminate the very process of demographic renewal. More specifically, we may be able to learn from the occurrence or absence of longer cycles whether a population is subject to negative feedback of a Malthusian sort and perhaps to place bounds on its sensitivity if it occurs. To Malthus (1798) it seemed obvious that populations would perpetually oscillate about equilibrium. This notion is taken seriously as an interpretation of the long swings in the fertility of many contemporary developed countries, as we discuss in more detail below.

Fluctuations may come about in three ways (or through combinations of these ways). First, they may simply be imposed on a series of births or deaths by fluctuations in some driving force such as prices or the weather. In this case, both the amplitude and the period of the fluctuation depend entirely on the driving series. Second, damped fluctuations may be created by the internal structure of a demographic process, as it responds to random and non-cyclic external shocks; in this case the cycles will die out if the external disturbance stops. The period of such cycles depends entirely on the nature of the renewal process, not on the driving force; however, the amplitude of the cycles depends on the amplitude (variance) of the disturbing force.

The third possibility is that limit cycles occur. Like the aforementioned cycles, these are generated by the internal structure of the reproductive process, but unlike them they are self-sustaining or ‘self-exciting’ and would continue indefinitely even in the absence of outside shocks. In this case, both the amplitude and the period depend only on the reproductive process. When a dynamic equilibrium is unstable, such that trajectories tend to explode away from the equilibrium path, then one of three things may happen: explosive fluctuations may lead to extinction; the non-repeating fluctuations of chaos cycles may occur; or the system may settle down to a limiting pattern of cycles, called limit cycles. There are many examples of animal populations exhibiting such behaviour. In human demography, it is a matter of controversy whether such cycles have ever actually occurred, but if they have it is presumably through the kind of mechanism proposed by Easterlin (1968), a sort of Malthusian cycle about equilibrium.

Imposed Cycles

There are well known non-seasonal cycles in fertility and mortality at or below the annual frequency (obstetricians avoid deliveries on Sundays; people have lower mortality just before elections, compensated for by increased mortality thereafter). Seasonality is strong in fertility, mortality, nuptiality and migration, particularly in traditional agricultural societies and in those less insulated by their dwellings from the variations of climate (in the extreme case of Bangladesh in the 1970s, for example, the seasonal peak in fertility was two to three times the seasonal trough). In the case of mortality, nuptiality and migration the causes of seasonal variation are fairly well understood to be rooted in identifiable biological, institutional and economic influences. In the case of fertility, the causes of seasonality are much less well understood.

There are also somewhat longer fluctuations in vital rates, in the range of 2–15 years. These have been quite thoroughly studied and found to be associated with business cycle indicators in the developed world and with the harvest cycle in preindustrial conditions. Lower agricultural prices and less unemployment are associated with higher fertility and nuptiality and with lower mortality, with lag patterns of response indicating that much of the variation is confined to changes in the timing of events. Fluctuations in temperature also are important, with colder winters and hotter summers raising mortality and reducing fertility, with an appropriate lag. In the case of mortality, exogenous epidemiological variation historically played a larger role (Wrigley and Schofield 1981). These relationships have continued to hold at least until a few decades ago in the developed countries and are still evident in the Third World countries where they have been investigated.

Much longer fluctuations in population variables are also visible in the historical record. Kuznets cycles, of 15–25 years, include a pro-cyclical response of migration, both internal and international. Some of the birth-rate series of 19th century Europe show signs of the Kondratieff cycle. But most striking are the waves lasting two or three centuries in the demography of Europe and of China, from at least the 12th century up to the 18th (Wrigley and Schofield 1981). These are evident in population growth rates and in mortality; their existence in fertility is problematic. The cause of these very long waves is not clear, although a case can be made for the influence of climatic variation and for the effects of intercontinental exchange of diseases through conquest or trade. Whatever their cause, such demographic fluctuations played a critical role in economic history, driving rents, wages and other relative prices, and possibly inflation. It is possible that such fluctuations were generated internally by the economic demographic system as Malthusian fluctuations about equilibrium; in the present state of knowledge, however, it appears more likely that the cycles were imposed.

Cycles Arising from the Internal Age and Temporal Structure of Reproduction

A characteristic pattern of delay between an event and its recurrence can act as a filter which creates quasi-cyclic behaviour in the series of events when the timing is subject to continual random perturbation. In this way, the typical spacing of a mother’s births two to three years apart tends to generate cycles of this length, as was first pointed out by Yule (1906). Such cycles are visually discernible in many birth and fertility series and show up in the empirical power spectra.

More importantly, the typical delay between a woman’s own birth and the time she herself gives birth to female children leads to cycles of 25–35 years, or the approximate length of a generation, when fertility is randomly perturbed (see Coale 1972; Lee 1974). This may be shown as follows. Let B(t) be the number of births in year t, and let φ(a) be the expected number of births to each of these births at age a, net of mortality (φ(a) is known as the ‘net maternity function’). φ(a) typically rises from zero at an age around 15 years to a peak in the twenties and declines again to zero at around age 45; its mean, μ, is the mean age at child-bearing and falls between 25 and 35 years depending on the population. The renewal process is written:

$$ B(t)=\sum\limits \varphi (a)\;B\left(t-a\right) $$
(1)

where the sum is taken over the reproductive years. Such a process will settle down to a stable exponential growth path if the characteristic roots of φ lie within the unit circle. But as the B series converges to this growth path from an irregular past, it will fluctuate, and the fluctuations can be characterized by further examination of the characteristic roots of φ. There will generally be one real root, describing the steady state growth rate, and the others will come in pairs of complex conjugates. The pair with the largest modulus is the only one of substantive interest; it will describe a damped oscillation with length roughly equal to the mean age of child-bearing, μ. Any initially distorted age distribution, if subsequently subjected to fixed vital rates described by ϕ, will generate a birth sequence which moves in waves one generation long as it converges towards exponential growth.

The argument can easily be generalized to cover the case of a population whose net maternity function is subject to constant stochastic disturbance of any autocovariance structure; the age structure of reproduction, described by the mean values of φ, will amplify variation in the neighbourhood of frequencies corresponding to cycle length μ, leave them unchanged at higher frequencies, and attenuate them in the neighbourhood of cycle length 2μ. Thus, a population in a random environment will tend to exhibit cycles one generation long or to superimpose these on whatever pattern of variation is forced on it by the environment. The birth series of many pre-industrial populations, particularly at the parish level, indeed do reveal such waves; whether the mechanism described above suffices to account for them has not yet been established empirically.

Some scholars have seen a major economic influence in such population waves, but this view now appears exaggerated; waves generated in this way are generally quite mild; they have low amplitude, and they damp fairly rapidly following an identifiable disturbance.

Cycles Arising from Economic-Demographic Interaction

Interest in dynamic economic–demographic models of population renewal, stressing fluctuations arising from age distributions, was prompted by the long ‘cycle’ in US fertility, with a trough in the 1930s, a peak in the late 1950s, and a trough in the 1970s. A number of scholars, most notably Easterlin, suggested around 1960 that the fertility fluctuations might reflect the economic conditions faced by young labour market entrants, conditions which in turn were worse for large cohorts and better for smaller ones. This insight led them to forecast correctly the sharp decline in fertility occurring in the 1960s, as larger cohorts aged into the labour market. Easterlin (1968) developed a detailed theory, buttressed by extensive empirical investigation, leading to a tentative prediction of self-generating demographic cycles two generations long, as small birth cohorts had high fertility and gave birth to large cohorts, who in turn reared small cohorts, and so on. Such cycles are known as ‘Easterlin cycles’. A considerable empirical literature has since appeared on the subject, lending considerable support at the aggregate time series level in the United States and some other countries, but very little at the micro level.

I now briefly review the theoretical literature on economic–demographic cycles. The account of the renewal process given above implicitly assumed that net maternity at time t, φ(a, t), was independent of the population age distribution at time t, or equivalently of the preceding series of births. But it is entirely possible that this is not so. Suppose, for example, that a Malthusian model is appropriate, such that fluctuations in labour supply lead to inverse fluctuations in wages, and that fertility depends positively on the wage level. This leads to different dynamic possibilities and a modified renewal equation.

Suppose that the net maternity function, φ(a), depends on some set of economic variables, let us say wages for concreteness. Suppose that these in turn depend on some set of economic variables, Z, which are independent of age distribution, as well as on the current population age distribution, which thus in conjunction with Z determines wages. If mortality is constant and the population closed to migration, as we here assume, then the current age distribution is completely determined by past births. We can then write:

$$ B(t)=\sum\limits {\varphi}^{\ast}\left[B(t),Z(t)\right]\;B\left(t-a\right), $$
(2)

where B(t) denotes the vector of past births; this replaces the purely demographic renewal eq. (1) introduced above (Lee 1974).

The renewal process will have an exponential equilibrium growth path, B*(t) = B exp(nt), which satisfies (2) for all t. For simplicity, suppose that Z is such that n = 0, so that the equilibrium path is stationary. It is helpful to consider the process of proportional deviations about this equilibrium path, denoted b(t). Let φ(a) be the value of φ*[ ] evaluated at equilibrium. In this case, the sum of φ(a) over all a, known as the net reproduction rate or NRR, is unity when evaluated at the equilibrium age distribution. Let Γ(a) be the elasticity of the NRR with respect to the size of age group a, or equivalently with respect to births a years previously, B (t – a); these elasticities are readily derived from the original function φ. Then the renewal process for fluctuations about the equilibrium growth path of births is simply:

$$ b(t)=\sum\limits \left[\varphi (a)+\Gamma (a)\right]b\left(t-a\right). $$
(3)

The smaller the effect of the current age distribution on fertility (Γ), the more the population renewal process resembles the purely demographic version of (1). In any event, exactly the same procedures can be used to study the dynamic behaviour of birth fluctuations in this model as were used previously.

The first step is to check the characteristic roots to assess stability. If the oscillations of the process tend to explode away from the equilibrium growth path, then a different kind of analysis, discussed below, is called for. If the roots indicate that oscillations are damped, then the analysis of dynamic behaviour in the neighbourhood of equilibrium will be informative.

We can now consider specifications of the model which have been proposed in the literature. The first is the simplest Malthusian model, in which all age groups in the labour force are assumed to be perfect substitutes in production, and fertility at each age is assumed to be negatively related to the size of the potential labour force, through an hypothesized effect on wages. In this case, Γ(α) = β k(a), where β is independent of age, and expresses the sensitivity of response (elasticity of the net reproduction rate with respect to labour force size at equilibrium), while the k(a) depend only on mortality conditions and equilibrium age specific labour supply and are therefore easily calculated from data at hand. Depending on values of β, this model will generate cycles ranging from one generation (as in the purely demographic model) to a century and a half or more. For β = 7.5, which is the empirical estimate from US data, 1917–1973, a cycle corresponding to the observed time path of births may be produced (Lee 1974; Wachter 1991).

Another model which is often used makes the fertility of a birth cohort depend only on the size of the cohort and makes it independent of all other age group sizes. The simplest form of this specification leads to:

$$ b(t)=\left(1{-} \alpha \right)\sum\limits \varphi (a)\;b\left(t{-} a\right), $$
(4)

where α is the elasticity of each age’s fertility with respect to cohort size. For α less than 1, there is a generation-long cycle; for α greater than 1 but less than 2, there is a damped two-generation cycle, and for α greater than 2 an explosive two-generation cycle occurs.

Specifications reflecting other degrees of substitutability of age groups of labour could of course be tried. Easterlin typically has used a ratio of younger to older workers to drive fertility (this could be derived from a CES model with two age groups of labour as separate factors, for example). The general expression can be used to explore dynamics under a wider variety of specifications. For example, the burden of supporting the elderly retired population might lead to a reduction in fertility; this would be expressed as a suitable negative Γ(α) for a = 65 and over. If couples were led to desire larger families when they observed other couples’ children, then Γ(α) would be positive for ages zero to ten.

When the cyclic behaviour near equilibrium is found to be explosive, then we need to consider behaviour further from equilibrium, at which point nonlinearities become important (unless, of course, the behaviour is truly linear, in which case population extinction results). Dynamic behaviour can be ‘chaotic’, an endless series of non-repeating fluctuations; for many models, however, limit cycles will occur, with amplitude and period determined not by the pattern of disturbances but rather by the functional relations themselves. Such cycles are observed in animal populations in laboratories and occasionally in the wild; in human populations their occurrence is conjectural: Samuelson (1976) considered a particular three-age group model leading to limit cycles.

Long-Term Population Trends and Economic Growth

Longer-term trends in population have also been viewed in the context of processes related to economic growth. Solow (1956) studied the behaviour of a population whose growth varied first positively and then negatively with respect to per capita income. Combining this study with his neoclassical growth model, he showed there was a stable low-level equilibrium at which per capita income was constant and population grew at the rate of technological progress, but also a second equilibrium at a high per capita income which was unstable. If the capital–labour ratio could be raised slightly above this equilibrium level, then per capita income would rise without limit while the population growth rate fell lower and lower.

In Solow’s approach, as in Malthus’s, technological progress was taken as exogenous. Boserup (1981) and others have suggested that larger denser populations would be more likely to experience technological progress in the long run, for reasons related to both the supply of innovations and the demand for them. She suggested that, combined with a Malthusian endogenous response of population growth to economic progress, an upward spiral of population growth and technological progress might occur, with positive feedback. A number of scholars have developed formal models of this process (Lee 1986; Kremer 1993), in a literature that overlaps slightly with the endogenous growth literature (Jones 2003).

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