From Introduction to Social Macrodynamics. Secular Cycles and Millennial Trends by Andrey Korotayev, Artemy Malkov, and Daria Khaltourina. Moscow: Editorial URSS, 2006. Pp. 5–36.
Introduction: Millennial Trends^{1}
In the first part of our Introduction to Social Macrodynamics (Korotayev, Malkov, and Khaltourina 2006a) we have shown that more than 99% of all the variation in demographic, economic and cultural macrodynamics of the World System over the last two millennia can be accounted for by very simple general mathematical models. Let us start this part with a summary of these findings, along with some relevant new findings that we obtained after the first part of this Introduction had been published. This summary is intended for all those interested in patterns of social evolution and development, including those who are not familiar with higher mathematics. Accordingly, we have included some basic material that mathematically sophisticated readers may wish to skip over lightly or entirely ignore.
In 1960 von Foerster, Mora, and Amiot published, in the journal Science, a striking discovery. They showed that between 1 and 1958 CE the world's population (N) dynamics can be described in an extremely accurate way with an astonishingly simple equation:^{2}
,

(0.1)

where N_{t} is the world population at time t, and C and t_{0} are constants, with t_{0} corresponding to an absolute limit ("singularity" point) at which N would become infinite.
Parameter t_{0} was estimated by von Foerster and his colleagues as 2026.87, which corresponds to November 13, 2006; this made it possible for them to supply their article with a publicrelations masterpiece title – "Doomsday: Friday, 13 November, A.D. 2026".^{3}
Note that the graphic representation of this equation is nothing but a hyperbola; thus, the growth pattern described is denoted as "hyperbolic".
Let us recollect that the basic hyperbolic equation is:
.

(0.2)

A graphic representation of this equation looks as follows (if k equals, e.g., 5) (see Diagram 0.1):
Diagram 0.1. Hyperbolic Curve Produced by Equation
The hyperbolic equation can also be written in the following way:
.

(0.3)

With x_{0} = 2 (and k still equal to 5) this equation will produce the following curve (see Diagram 0.2):
Diagram 0.2. Hyperbolic Curve Produced by Equation
As can be seen, the curve produced by equation (0.3) at Diagram 0.2 is precisely a mirror image of the hyperbolic curve produced by equation (0.2) at Diagram 0.1. Now let us interpret the Xaxis as the axis of time (taxis), the Yaxis as the axis of the world's population (counted in millions), replace x_{0} with 2027 (that is the result of just rounding of von Foester’s number, 2026.87), and replace k with 215000.^{4} This gives us a version of von Foerster's equation with certain parameters:
.

(0.4)

In fact, von Foerster's equation suggests a rather unlikely thing. It "says" that if you would like to know the world population (in millions) for a certain year, then you should just subtract this year from 2027 and then divide 215000 by the difference. At first glance, such an algorithm seems most unlikely to work; however, let us check if it does. Let us start with 1970. To estimate the world population in 1970 using von Foerster's equation we first subtract 1970 from 2027, to get 57. Now the only remaining thing is to divide 215000 by the figure just obtained (that is, 57), and we should arrive at the figure for the world population in 1970 (in millions): 215000 ÷ 57 = 3771.9. According to the U.S. Bureau of the Census database (2006), the world population in 1970 was 3708.1 million. Of course, none of the U.S. Bureau of the Census experts would insist that the world population in 1970 was precisely 3708.1 million. After all, the census data is absent or unreliable for this year for many countries; in fact, the result produced by von Foerster's equation falls well within the error margins for empirical estimates.
Now let us calculate the world population in 1900. It is clear that in order to do this we should simply divide 215000 million by 127; this gives 1693 million, which turns out to be precisely within the range of the extant empirical estimates (1600–1710 million).^{5}
Let us do the same operation for the year 1800: 2027 – 1800 = 227; 215000 ÷ 227 = 947.1 (million). According to empirical estimates, the world population for 1800 indeed was between 900 and 980 million.^{6} Let us repeat the operation for 1700: 2027 – 1700 = 337; 215000 ÷ 337 = 640 (million). Once again, we find ourselves within the margins of available empirical estimates (600–679 million).^{7} Let us repeat the algorithm once more, for the year 1400: 2027 – 1400 = 627; 215000 ÷ 627 = 343 (million). Yet again, we see that the result falls within the error margins of available world population estimates for this date.^{8} The overall correlation between the curve generated by von Foerster's equation and the most detailed series of empirical estimates looks as follows (see Diagram 0.3):
Diagram 0.3. Correlation between Empirical Estimates of World
Population (in millions, 1000 – 1970) and the Curve
Generated by von Foerster's Equation

NOTE: black markers correspond to empirical estimates of the world population by McEvedy and Jones (1978) for 1000–1950 and the U.S. Bureau of the Census (2006) for 1950–1970. The grey curve has been generated by von Foerster's equation (0.4).

The formal characteristics are as follows: R = 0.998; R^{2} = 0.996;
p = 9.4 × 10^{17} ≈ 1 × 10^{16}. For readers unfamiliar with mathematical statistics: R^{2} can be regarded as a measure of the fit between the dynamics generated by a mathematical model and the empirically observed situation, and can be interpreted as the proportion of the variation accounted for by the respective equation. Note that 0.996 also can be expressed as 99.6%.^{9} Thus, von Foerster's equation accounts for an astonishing 99.6% of all the macrovariation in world population, from 1000 CE through 1970, as estimated by McEvedy and Jones (1978) and the U.S. Bureau of the Census (2006).^{10}
Note also that the empirical estimates of world population find themselves aligned in an extremely neat way along the hyperbolic curve, which convincingly justifies the designation of the pre1970s world population growth pattern as "hyperbolic".
Von Foerster and his colleagues detected the hyperbolic pattern of world population growth for 1 CE –1958 CE; later it was shown that this pattern continued for a few years after 1958,^{11} and also that it can be traced for many millennia BCE (Kapitza 1992, 1999; Kremer 1993).^{12} Indeed, the McEvedy and Jones (1978) estimates for world population for the period 5000–500 BCE are described rather accurately by a hyperbolic equation (R^{2} = 0.996); and this fit remains rather high for 40000 – 200 BCE (R^{2} = 0.990) (see below Appendix 2). The overall shape of the world’s population dynamics in 40000 BCE – 1970 CE also follows the hyperbolic pattern quite well (see Diagram 0.4):
Diagram 0.4. World Population Dynamics, 40000 BCE – 1970 CE
(in millions): the fit between predictions of a hyperbolic
model and the observed data
NOTE: R = 0.998, R^{2} = 0.996, p << 0.0001. Black markers correspond to empirical estimates of the world population by McEvedy and Jones (1978) and Kremer (1993) for 1000–1950, as well as the U.S. Bureau of the Census (2006) data for 1950–1970. The solid line has been generated by the following version of von Foerster's equation:
.
A usual objection (e.g., Shishkov 2005) against the statement that the overall pattern of world population growth until the 1970s was hyperbolic is as follows. Since we simply do not know the exact population of the world for most of human history (and especially, before CE), we do not have enough information to detect the general shape of the world population dynamics through most of human history. Thus, there are insufficient grounds to accept the statement that the overall shape of the world population dynamics in 40000 BCE – 1970 CE was hyperbolic.
At first glance this objection looks very convincing. For example, for 1 BCE the world population estimates range from 170 million (McEvedy and Jones 1978) to 330 million (Durand 1977), whereas for 10000 BCE the estimate range becomes even more dramatic: 1–10 million (Thomlinson 1975). Indeed, it seems evident that with such uncertain empirical data, we are simply unable to identify the longterm trend of world population macrodynamics.
However, notwithstanding the apparent persuasiveness of this objection, we cannot accept it. Let us demonstrate why.
Let us start with 10000 CE. As was mentioned above, we have only a rather vague idea about how many people lived on the Earth that time. However, we can be reasonably confident that it was more than 1 million, and less than 10 million. Note that this is not even a guesstimate. Indeed, we know which parts of the world were populated by that time (most of it, in fact), what kind of subsistence economies were practiced^{13} (see, e.g., Peregrine and Ember 2001), and what the maximum number of people 100 square kilometers could support with any of these subsistence economies (see, e.g., Korotayev 1991). Thus, we know that with foraging technologies practiced by human populations in 10000 BCE, the Earth could not have supported more than 10 million people (and the actual world population is very likely to have been substantially smaller). Regarding world population in 40000 BCE, we can be sure only that it was somewhat smaller than in 10000 BCE. We do not know what exactly the difference was, but as we shall see below, this is not important for us in the context of this discussion.
The available estimates of world population between 10000 BCE and 1 CE can, of course, be regarded as educated guesstimates. However, in 2 CE the situation changes substantially, because this is the year of the "earliest preserved census in the world" (Bielenstein 1987: 14). Note also that this census was performed in China, one of the countries that is most important for us in this context. This census recorded 59 million taxable inhabitants of China (e.g., Bielenstein 1947: 126, 1986: 240; Durand 1960: 216; Loewe 1986: 206), or 57.671 million according to a later reevaluation by Bielenstein (1987: 14).^{14} Up to the 18^{th} century the Chinese counts tended to underestimate the population, since before this they were not real census, but rather registrations for taxation purposes; in any country a large number of people would do their best to escape such a registration in order to avoid paying taxes, and it is quite clear that some part of the Chinese population normally succeeded in this (see, e.g., Durand 1960). Hence, at least we can be confident that in 2 CE the world population was no less than 57.671 million. It is also quite clear that the world population was substantially more than that. For this time we also have data from a census of the Roman citizenry (for 14 CE), which, together with information on Roman social structure and data from narrative and archaeological sources, makes it possible to identify with a rather high degree of confidence the order of magnitude of the population of the Roman Empire (with available estimates in the range of 45–80 million [Durand 1977: 274]). Textual sources and archaeological data also make it possible to identify the order of magnitude of the population of the Parthian Empire (10–20 million), and of India (50–100 million) (Durand 1977). Data on the population for other regions warrant less confidence, but it is still quite clear that their total population was much smaller than that of the four abovementioned regions (which in 2 CE comprised most of the world population). Archaeological evidence suggests that population density for the rest of the world would have been considerably lower than in the "Four Regions" themselves. In general, then, we can be quite sure that the world population in 2 CE could scarcely have been less than 150 million; it is very unlikely that it was more than 350 million.
Let us move now to 1800 CE. For this time we have much better population data than ever before for most of Europe, the United States, China^{15}, Egypt^{16}, India, Japan, and so on (Durand 1977). Hence, for this year we can be quite confident that world population could scarcely have been less than 850 million and more than 1 billion. The situation with population statistics further improves by 1900^{17} for which time there is not much doubt that world population this year was within the range of 1600–1750 million. Finally, by 1960 population statistics had improved dramatically, and we can be quite confident that world population then was within the range of 2900–3100 million.
Now let us plot the mid points of the above mentioned estimate ranges and connect the respective points. We will get the following picture (see Diagram 0.5):
Diagram 0.5
As we see the resulting pattern of world population dynamics has an unmistakably hyperbolic shape. Now you can experiment and move any points within the estimate ranges as much as you like. You will see that the overall hyperbolic shape of the longterm world population dynamics will remain intact. What is more, you can fill the space between the points with any estimates you find. You will see that the overall shape of the world population dynamics will always remain distinctly hyperbolic. Replace, for example, the estimates of McEvedy and Jones (1978) used by us earlier for Diagram 0.4 in the range between 10000 BCE and 1900 CE with the ones of Biraben (1980) (note that generally Biraben's estimates are situated in the opposite side of the estimate range in relation to the ones of McEvedy and Jones). You will get the following picture (see Diagram 0.6):
Diagram 0.6
As we see, the overall shape of the world population dynamics remains unmistakably hyperbolic.
So what is the explanation for this apparent paradox? Why, though world population estimates are evidently infirm for most of human history, can we be sure that longterm world population dynamics pattern was hyperbolic?
The answer is simple, for in the period in question the world population grew by orders of magnitude. It is true that for most part of human history we cannot be at all confident of the exact value within a given order of magnitude. But with respect to any timepoint within any period in question, we can be already perfectly confident about the order of magnitude of the world population. Hence, it is clear that whatever discoveries are made in the future, whatever reevaluations are performed, the probability that they will show that the overall world population growth pattern in 40000 BCE – 1970 CE was not hyperbolic (but, say, exponential or lineal) is very close to zero indeed.
Note that if von Foerster, Mora, and Amiot also had at their disposal, in addition to world population data, data on the world GDP dynamics for 1–1973 (published, however, only in 2001 by Maddison [Maddison 2001]), they could have made another striking "prediction" – that on Saturday, 23 July, A.D. 2005 an "economic doomsday" would take place; that is, on that day the world GDP would become infinite if the economic growth trend observed in 1–1973 CE continued. They also would have found that in 1–1973 CE the world GDP growth followed a quadratichyperbolic rather than simple hyperbolic pattern.
Indeed, Maddison's estimates of the world GDP dynamics for 1–1973 CE are almost perfectly approximated by the following equation:
,

(0.5)

where G_{t} is the world GDP (in billions of 1990 international dollars, in purchasing power parity [PPP]) in year t, С = 17355487.3 and t_{0 }= 2005.56 (see Diagram 0.7):
Diagram 0.7. World GDP Dynamics, 1–1973 CE (in billions of 1990 international dollars, PPP): the fit between predictions of a quadratichyperbolic model and the observed data
NOTE: R = .9993, R^{2} = .9986, p << .0001. The black markers correspond to Maddison's (2001) estimates (Maddison's estimates of the world per capita GDP for 1000 CE has been corrected on the basis of Meliantsev [1996, 2003, 2004a, 2004b]). The grey solid line has been generated by the following equation:
.
Actually, as was mentioned above, the best fit is achieved with С = 17355487.3 and t_{0 }= 2005.56 (which gives just the "doomsday Saturday, 23 July, 2005"), but we have decided to keep hereafter to integer numbered years.
The only difference between the simple and quadratic hyperbolas is that the simple hyperbola is described mathematically with equation (0.2):
,

(0.2)

whereas the quadratic hyperbolic equation has x^{2} instead of just x:
.

(0.6)

Of course, this equation can also be written as follows:
.

(0.7)

It is this equation that was used above to describe the world economic dynamics between 1 and 1973 CE. The algorithm for calculating the world GDP still remains very simple. E.g., to calculate the world GDP in 1905 (in billions of 1990 international dollars, PPP), one should first subtract 1905 from 2005, but than to divide С (17355487.3) not by the resultant difference (100), but by its square (100^{2} = 10000).
Those readers who are not familiar with mathematical models of population hyperbolic growth should have a lot of questions at this point.^{18} How could the longterm macrodynamics of the most complex social system be described so accurately with such simple equations? Why do these equations look so strange? Why, indeed, can we estimate the world population in year x so accurately just by subtracting x from the "Doomsday" year and dividing some constant with the resultant difference? And why, if we want to know the world GDP in this year, should we square the difference prior to dividing? Why was the hyperbolic growth of the world population accompanied by the quadratic hyperbolic growth of the world GDP? Is this a coincidence? Or are the hyperbolic growth of the world population and the quadratic hyperbolic growth of the world GDP just two sides of one coin, two logically connected aspects of the same process?
In the first part of our Introduction to Social Macrodynamics we have tried to provide answers to this question and these answers are summarized below.
However, before starting this we would like to state that our experience shows that most readers who are not familiar with mathematics stop reading books (at least our books) as soon as they come across the words – "differential equation". Thus, we have to ask such readers not to get scared with the presence of these words in the next passage and to move further. You will see that it is not as difficult to understand differential equations (or, at least, some of those equations), as one might think.
To start with, the von Foerster equation, , is just the solution for the following differential equation (see, e.g., Korotayev, Malkov, and Khaltourina 2006a: 119–20):
.

(0.8)

This equation can be also written as:
,

(0.9)

where .
What is the meaning of this mathematical expression, ? In our context dN/dt denotes the absolute population growth rate at some moment of time. Hence, this equation states that the absolute population growth rate at any moment of time should be proportional to the square of population at this moment.
Note that by dividing both parts of equation (0.9) with N we will get the following:
,

(0.10)

Further, note that is just a designation of the relative population growth rate. Indeed, as we remember, dN/dt is the absolute population growth rate at a certain moment of time. Imagine that at this moment the population (N) is 100 million and the absolute population growth rate (dN/dt) is 1 million a year. If we divide now (dN/dt = 1 million) by (N = 100 million) we will get 0.01, or 1%; which would mean that the relative population growth rate at this moment is 1% a year.
If we denote relative population growth rate as r_{N}, we will get a particularly simple version of the hyperbolic equation:
.

(0.10')

Thus, with hyperbolic growth the relative population growth rate (r_{N}) is linearly proportional to the population size (N). Note that this significantly demystifies the problem of the world population hyperbolic growth. Now to explain this hyperbolic growth we should just explain why for many millennia the world population's absolute growth rate tended to be proportional to the square of the population.
We believe that the most significant progress towards the development of a compact mathematical model providing a convincing answer to this question has been achieved by Michael Kremer (1993), whose model will be summarized next.
Kremer's model is based on the following assumptions:
1) First of all he makes "the Malthusian (1978) assumption that population is limited by the available technology, so that the growth rate of population is proportional to the growth rate of technology" (Kremer 1993: 681–2).^{19} This statement looks quite convincing. Indeed, throughout most of human history the world population was limited by the technologically determined ceiling of the carrying capacity of land. As was mentioned above, with foraging subsistence technologies the Earth could not support more than 10 million people, because the amount of naturally available useful biomass on this planet is limited, and the world population could only grow over this limit when the people started to apply various means to artificially increase the amount of available biomass, that is with the transition from foraging to food production. However, the extensive agriculture also can only support a limited number of people, and further growth of the world population only became possible with the intensification of agriculture and other technological improvements.
This assumption is modeled by Kremer in the following way. Kremer assumes that overall output produced by the world economy equals
,

(0.11)

where G is output, T is the level of technology, N is population, 0 < α < 1 and r are parameters.^{20} With constant T (that is, without any technological growth) this equation generates Malthusian dynamics. For example, let us assume that α = 0.5, and that T is constant. Let us recollect that N^{0.5} is just √N. Thus, a four time expansion of the population will lead to a twofold increase in output (as √4 = 2). In fact, here Kremer models Ricardo's law of diminishing returns to labor (1817), which in the absence of technological growth produces just Malthusian dynamics. Indeed, if the population grows 4 times, and the output grows only twice, this will naturally lead to a twofold decrease of per capita output. How could this affect population dynamics?
Kremer assumes that "population increases above some steady state equilibrium level of per capita income, m, and decreases below it" (Kremer 1993: 685). Hence, with the decline of per capita income, the population growth will slow down and will become close to zero when the per capita income approaches m. Note that such a dynamics was actually rather typical for agrarian societies, and its mechanisms are known very well – indeed, if per capita incomes decline closely to m, it means the decline of nutrition and health status of most population, which will lead to an increase in mortality and a slow down of population growth (see, e.g., Malthus 1978 [1798]; Postan 1950, 1972; Abel 1974, 1980; Cameron 1989; Artzrouni and Komlos 1985; Komlos and Nefedov 2002; Turchin 2003; Nefedov 2004 and Chapters 1–3 below). Thus, with constant technology, population will not be able to exceed the level at which per capita income (g = G/N) becomes equal to m. This implies that for any given level of technological development (T) there is "a unique level of population, n," that cannot be exceeded with the given level of technology (Kremer 1993: 685). Note that n can be also interpreted as the Earth carrying capacity, that is, the maximum number of people that the Earth can support with the given level of technology.
However, as is well known, the technological level is not a constant, but a variable. And in order to describe its dynamics Kremer employs his second basic assumption:
2) "High population spurs technological change because it increases the number of potential inventors…^{21} In a larger population there will be proportionally more people lucky or smart enough to come up with new ideas" (Kremer 1993: 685), thus, "the growth rate of technology is proportional to total population".^{22} In fact, here Kremer uses the main assumption of the Endogenous Technological Growth theory (Kuznets 1960; Grossman and Helpman 1991; Aghion and Howitt 1992, 1998; Simon 1977, 1981, 2000; Komlos and Nefedov 2002; Jones 1995, 2003, 2005 etc.). As this supposition, to our knowledge, was first proposed by Simon Kuznets (1960), we shall denote the corresponding type of dynamics as "Kuznetsian",^{23} while the systems in which the "Kuznetsian" populationtechnological dynamics is combined with the "Malthusian" demographic one will be denoted as "MalthusianKuznetsian". In general, we find this assumption rather plausible – in fact, it is quite probable that, other things being equal, within a given period of time, one billion people will make approximately one thousand times more inventions than one million people.
This assumption is expressed by Kremer mathematically in the following way:
.

(0.12)

Actually, this equation says just that the absolute technological growth rate at a given moment of time is proportional to the technological level observed at this moment (the wider is the technological base, the more inventions could be made on its basis), and, on the other hand, it is proportional to the population (the larger the population, the higher the number of potential inventors).^{24}
In his basic model Kremer assumes "that population adjusts instantaneously to n" (1993: 685); he further combines technology and population determination equations and demonstrates that their interaction produces just the hyperbolic population growth (Kremer 1993: 685–6; see also Podlazov 2000, 2001, 2002, 2004; Tsirel 2004; Korotayev, Malkov, and Khaltourina 2006a: 21–36).
Kremer's model provides a rather convincing explanation of why throughout most of human history the world population followed the hyperbolic pattern with the absolute population growth rate tending to be proportional to N^{2}. For example, why will the growth of population from, say, 10 million to 100 million, result in the growth of dN/dt 100 times? Kremer's model explains this rather convincingly (though Kremer himself does not appear to have spelled this out in a sufficiently clear way). The point is that the growth of world population from 10 to 100 million implies that human technology also grew approximately 10 times (given that it will have proven, after all, to be able to support a ten times larger population). On the other hand, the growth of a population 10 times also implies a 10fold growth of the number of potential inventors, and, hence, a 10fold increase in the relative technological growth rate. Hence, the absolute technological growth rate will grow 10 × 10 = 100 times (as, in accordance with equation (0.12), an order of magnitude higher number of people having at their disposal an order of magnitude wider technological basis would tend to make two orders of magnitude more inventions). And as N tends to the technologically determined carrying capacity ceiling, we have good reason to expect that dN/dt will also grow just by about 100 times.
In fact, Kremer's model suggests that the hyperbolic pattern of the world's population growth could be accounted for by the nonlinear second order positive feedback mechanism that was shown long ago to generate just the hyperbolic growth, known also as the "blowup regime" (see, e.g., Kurdjumov 1999; Knjazeva and Kurdjumov 2005). In our case this nonlinear second order positive feedback looks as follows: the more people – the more potential inventors – the faster technological growth – the faster growth of the Earth's carrying capacity – the faster population growth – with more people you also have more potential inventors – hence, faster technological growth, and so on (see Diagram 0.8):
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