introduction to social macrodynamics: secular cycles and millennial trends in africa

Andrey Korotayev

Daria Khaltourina

INTRODUCTION

TO SOCIAL MACRODYNAMICS:

Secular Cycles

and Millennial Trends

in Africa

Moscow: URSS, 2006

RUSSIAN STATE UNIVERSITY FOR THE HUMANITIES

Faculty of History, Political Science and Law

RUSSIAN ACADEMY OF SCIENCES

Institute for African Studies

Institute of Oriental Studies

Andrey Korotayev

Daria Khaltourina

INTRODUCTION

TO SOCIAL MACRODYNAMICS:

Secular Cycles

and Millennial Trends

in Africa

Moscow: URSS, 2006

ББК 22.318 22.18 60.5 66.0 This study has been supported by the Russian Foundation for Basic Research (projects ## 06–06–80503 and 04–06–80225), the Presidium of the Russian Academy of Sciences ("The Power and Society in History" Program), and the Russian Science Support Foundation

Korotayev Andrey, Khaltourina Daria Introduction to Social Macrodynamics: Secular Cycles and Millennial Trends in Africa. – Moscow: KomKniga. – 160 p. ISBN 5–484–00560–4 Human society is a complex nonequilibrium system that changes and develops con-

stantly. Complexity, multivariability, and contradictions of social evolution lead researchers

to a logical conclusion that any simplification, reduction, or neglect of the multiplicity of fac-

tors leads inevitably to the multiplication of error and to significant misunderstanding of the

processes under study. The view that any simple general laws are not observed at all with re-

spect to social evolution has become totally dominant within the academic community, espe-

cially among those who specialize in the Humanities and who confront directly in their re-

search the manifold unpredictability of social processes. A way to approach human society as

an extremely complex system is to recognize differences of abstraction and time scale be-

tween different levels. If the main task of scientific analysis is to detect the main forces acting

on systems so as to discover fundamental laws at a sufficiently coarse scale, then abstracting

from details and deviations from general rules may help to identify measurable deviations

from these laws in finer detail and shorter time scales. Modern achievements in the field of

mathematical modeling suggest that social evolution can be described with rigorous and suf-

ficiently simple macrolaws.

The first book of the Introduction (Compact Macromodels of the World System

Growth. Moscow: Editorial URSS, 2006) discusses general regularities of the World System

long-term development. It is shown that they can be described mathematically in a rather ac-

curate way with rather simple models. In the second book (Secular Cycles and Millennial

Trends. Moscow: Editorial URSS, 2006) the authors analyze more complex regularities of its

dynamics on shorter scales, as well as dynamics of its constituent parts paying special atten-

tion to "secular" cyclical dynamics. It is shown that the structure of millennial trends cannot

be adequately understood without secular cycles being taken into consideration. In turn, for

an adequate understanding of cyclical dynamics the millennial trend background should be

taken into account.

In this book the authors analyze the interplay of trend and cyclical dynamics in Egypt

and Subsaharan Africa.

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . 4

Introduction: Secular Cycles and Millennial Trends . . . . . . . . 5

Part I: Secular Cycles and Millennial Trends in Egypt . . . . . . . 37

Chapter 1 General Trends of Egyptian Demographic Dynamics in the 1

st – 18

th Centuries CE . . . . . . . . .

37

Chapter 2 Some Features of Medieval Egyptian Political-Demographic Cycles . . . . . . . . .

45

Chapter 3 ‛Abd al-Rah man ibn Khaldun's Sociological Theory as a Methodological Basis for Mathematical Modeling of Medieval Egyptian Political-Demographic Dynamics . . .

57

Chapter 4 Basic Model of Medieval Egyptian Political-Demographic Dynamics . . . . . . .

65

Chapter 5 Extended Model of Medieval Egyptian Political-Demographic Dynamics . . . . . . .

76

Chapter 6 Secular Cycles and Millennial Trends in Egypt: Preliminary Conclusions . . . . . . . . . . . .

89

Part II: Cyclical and Trend Dynamics in Postcolonial Tropical Africa . . 92

Chapter 7 Postcolonial Tropical Africa: Trends and Cycles . . . . . . . . . . . . . .

92

Chapter 8 Postcolonial Tropical Africa: Demographic Growth and Internal Warfare (with Natalia Komarova) . . . . . . . .

102

Appendix: Cyclical Dynamics and Mechanisms of Hyperbolic Growth

. 116

Bibliography . . . . . . . . . . . . . . . . . . . . . . 144

Acknowledgements

First and foremost, our thanks go to the Institute for Advanced Study, Princeton. Without the first author's one-year membership in this Institute this book could hardly have been written. We are especially grateful to the following professors and members of this institute for valuable comments on the first sketches of this monograph: Patricia Crone, Nicola Di Cosmo, John Shepherd, Ki Che Angela Leung, and Michael Nylan. We would also like to express our deepest gratitude to Elizabeth Sartain, Eleonora Fernandez and Nelly Hanna of the Arabic Studies Department, American University in Cairo, for their invaluable help and advice. We are also grateful to the Russian Science Support Foundation and the Russian Foundation for Basic Research for financial support of this work (projects ## 06–06–80503 and 04–06–80225). We would like to express our special gratitude to Robert Graber (Truman State University), Victor de Munck (State University of New York), Gregory Malinetsky and Sergey Podlazov (Institute for Applied Mathematics, Russian Academy of Sciences), Diana Pickworth (Aden University, Yemen), Antony J. Harper (New Trier College), Ahren La Londa (Beloit College), Duran Bell, Donald Saari, and Douglas R. White (University of California, Irvine) for their invaluable help and advice. We would also like to thank our colleagues who offered us useful comments and insights on the subject of this book: Ferida Atsamba and Vitalij Meliantsev (Institute of Asia and Africa, Moscow State University), Leonid Borodkin (Historical Informatics Laboratory, Moscow State University), Igor Sledzevski and Dmitri Bondarenko (Institute for African Studies, Russian Academy of Sciences), Robert L. Carneiro (American Museum of Natural History, New York), Henry J. M. Claessen (Leiden University), Dmitrij Chernavskij (Institute of Physics, Russian Academy of Sciences), Georgi and Lubov Derlouguian (Northwestern University, Evanston), Leonid Grinin (Center for Social Research, Volgograd), Alexander Kadyrbaev (Oriental Institute, Russian Academy of Sciences), Natalia Komarova (University of California, Irvine), Sergey Nefedov (Russian Academy of Sciences, Ural Branch, Ekaterinburg), Nikolay Kradin (Russian Academy of Sciences, Far East Branch, Vladivostok), Eginald Mihanjo and Abillah H. Omari (Centre for Foreign Relations, Dar es Salaam), Rifat Pateev (Russian Cultural Centre in Tanzania), and Peter Turchin (University of Connecticut, Storrs). We would also like to thank Tatiana Shifrina, the Director of "Khalturka-Design" Company, for the design of the cover of this monograph. Needless to say, faults, mistakes, infelicities, etc., remain our own responsibility.

Introduction:

Secular Cycles and Millennial Trends1

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 in-terested 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 popu-lation (N) dynamics can be described in an extremely accurate way with an astonishingly simple equation:

2

tt

CNt

0

, (0.1)

where Nt is the world population at time t, and C and t0 are constants, with t0 corresponding to an absolute limit ("singularity" point) at which N would be-come infinite. Parameter t0 was estimated by von Foerster and his colleagues as 2026.87, which corresponds to November 13, 2006; this made it possible for them to

1 This book is a translation of an amended and enlarged version of the third part of the following

monograph originally published in Russian: Коротаев, А. В., А. С. Малков и Д. А. Хал-турина. Законы истории: Математическое моделирование исторических макропроцессов (Демография. Экономика. Войны). М.: УРСС, 2005.

2 To be exact, the equation proposed by von Foerster and his colleagues looked as follows:

99.00 )( tt

CN t

. However, as has been shown by von Hoerner (1975) and Kapitza (1992, 1999),

it can be written more succinctly as tt

CNt

0

.

Introduction

6

supply their article with a public-relations masterpiece title – "Doomsday: Fri-day, 13 November, A.D. 2026".

3

Note that the graphic representation of this equation is nothing but a hyper-bola; thus, the growth pattern described is denoted as "hyperbolic". Let us recollect that the basic hyperbolic equation is:

x

ky . (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 x

y5

0

50

100

150

200

250

0 0.5 1 1.5 2

x

y

3 Of course, von Foerster and his colleagues did not imply that the world population on that day

could actually become infinite. The real implication was that the world population growth pattern that was followed for many centuries prior to 1960 was about to come to an end and be trans-formed into a radically different pattern. Note that this prediction began to be fulfilled only in a few years after the "Doomsday" paper was published (see, e.g., Korotayev, Malkov, and Khaltou-rina 2006a: Chapter 1).

Secular Cycles and Millennial Trends

7

The hyperbolic equation can also be written in the following way:

xx

ky

0

. (0.3)

With x0 = 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 x

y

2

5

0

50

100

150

200

250

0 0.5 1 1.5 2

x

y

As can be seen, the curve produced by equation (0.3) at Diagram 0.2 is precise-ly a mirror image of the hyperbolic curve produced by equation (0.2) at Dia-gram 0.1. Now let us interpret the X-axis as the axis of time (t-axis), the Y-axis as the axis of the world's population (counted in millions), replace x0 with 2027 (that is the result of just rounding of von Foester’s number, 2026.87), and re-

Introduction

8

place k with 215000.4 This gives us a version of von Foerster's equation with

certain parameters:

tN t

2027

215000 . (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 popu-lation in 1970 (in millions): 215000 ÷ 57 = 3771.9. According to the U.S. Bu-reau 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 esti-mates (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

4 Note that the value of coefficient k (equivalent to parameter С in equation (1)) used by us is a bit

different from the one used by von Foerster. 5 Thomlinson 1975; Durand 1977; McEvedy and Jones 1978; Biraben 1980; Haub 1995; Modelski

2003; UN Population Division 2006; U.S. Bureau of the Census 2006. 6 Thomlinson 1975; McEvedy and Jones 1978; Biraben 1980; Modelski 2003; UN Population Di-

vision 2006; U.S. Bureau of the Census 2006. 7 Thomlinson 1975; McEvedy and Jones 1978; Biraben 1980; Maddison 2001; Modelski 2003;

U.S. Bureau of the Census 2006. 8 350 million (McEvedy and Jones 1978), 374 million (Biraben 1980).


9

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 corre-spond to empirical estimates of the world population by McEvedy and Jones (1978) for 1000–1950 and the U.S. Bu-reau of the Census (2006) for 1950–1970. The grey curve has been generated by von Foer-ster's equation (0.4).

The formal characteristics are as follows: R = 0.998; R2 = 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 inter-preted as the proportion of the variation accounted for by the respective equa-tion. 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

9 The second characteristic (p, standing for "probability") is a measure of the correlation's statisti-

cal significance. A bit counterintuitively, the lower the value of p, the higher the statistical signif-icance of the respective correlation. This is because p indicates the probability that the observed correlation could be accounted solely by chance. Thus, p = 0.99 indicates an extremely low sta-tistical significance, as it means that there are 99 chances out of 100 that the observed correlation is the result of a coincidence, and, thus, we can be quite confident that there is no systematic rela-tionship (at least, of the kind that we study) between the two respective variables. On the other hand, p = 1 × 10-16 indicates an extremely high statistical significance for the correlation, as it means that there is only one chance out of 10000000000000000 that the observed correlation is the result of pure coincidence (in fact, a correlation is usually considered as statistically signifi-cant with p < 0.05).

10 In fact, with slightly different parameters (С = 164890.45; t0 = 2014) the fit (R2) between the dy-namics generated by von Foerster's equation and the macrovariation of world population for CE 1000 – 1970 as estimated by McEvedy and Jones (1978) and the U.S. Bureau of the Census

Introduction

10

Note also that the empirical estimates of world population find themselves aligned in an extremely neat way along the hyperbolic curve, which convincing-ly justifies the designation of the pre-1970s 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 con-tinued for a few years after 1958,

11 and also that it can be traced for many mil-

lennia 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 (R2 = 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

5000

0

-5000

-10000

-15000

-20000

-25000

-30000

-35000

-40000

4000

3000

2000

1000

0

predicted

observed

NOTE: R = 0.998, R2 = 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

(2006) reaches 0.9992 (99.92%), whereas for 500 BCE – 1970 CE this fit increases to 0.9993 (99.93%) (with the following parameters: С = 171042.78; t0 = 2016).

11 Note that after the 1960s, world population deviated from the hyperbolic pattern more and more; at present it definitely is no longer hyperbolic (see, e.g., Korotayev, Malkov, and Khaltourina 2006a: Chapter 1).

12 In fact, Kremer asserts the presence of this pattern since 1 million BCE; Kapitza, since 4 million BCE! We, however, are not prepared to accept these claims, because it is far from clear even who constituted the "world population" in, say, 1 million BCE, let alone how their number could have been empirically estimated.


11

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:

tNt

2022

189648.7.

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 hu-man history (and especially, before CE), we do not have enough information to detect the general shape of the world population dynamics through most of hu-man 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 long-term 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 sub-sistence 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 dis-cussion. 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 sit-uation changes substantially, because this is the year of the "earliest preserved census in the world" (Bielenstein 1987: 14). Note also that this census was per-formed in China, one of the countries that is most important for us in this con-

13 Note that at that time these economies were exclusively foraging (though quite intensive in a few

areas of the world [see, e.g., Grinin 2003b]).

Introduction

12

text. This census recorded 59 million taxable inhabitants of China (e.g., Bielen-stein 1947: 126, 1986: 240; Durand 1960: 216; Loewe 1986: 206), or 57.671 million according to a later re-evaluation by Bielenstein (1987: 14).

14 Up to the

18th

century the Chinese counts tended to underestimate the population, since before this they were not real census, but rather registrations for taxation pur-poses; 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 magni-tude of the population of the Roman Empire (with available estimates in the range of 45–80 million [Durand 1977: 274]). Textual sources and archaeologi-cal data also make it possible to identify the order of magnitude of the popula-tion of the Parthian Empire (10–20 million), and of India (50–100 million) (Du-rand 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 above-mentioned 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 Re-gions" themselves. In general, then, we can be quite sure that the world popula-tion 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 im-proves by 1900

17, for which time there is not much doubt that world population

14 Or 57.671 million according to a later re-evaluation by Bielenstein (1987: 14). 15 Due to the separation of the census registration from the tax assessment conducted in the first

half of the 18th century, the Chinese population in 1800 had no substantive reason for avoiding the census registration. Therefore the Chinese census data for this time are particularly reliable (e.g., Durand 1960: 238; see also Chapter 2 of the previous part of our Introduction to Social Macrodynamics [Korotayev, Malkov, and Khaltourina 2006b]).

16 Due to the first scientific estimation of the Egyptian population performed by the members of the scientific mission that accompanied Napoleon to Egypt (Jomard 1818). Note, however, that this mission appears to have significantly underestimated the population of this country (see Chapter 1 of this book).

17 With a notable exception of China (Durand 1960; see also Chapter 2 of the previous part of our Introduction to Social Macrodynamics [Korotayev, Malkov, and Khaltourina 2006b]).


13

this year was within the range of 1600–1750 million. Finally, by 1960 popula-tion 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 Dia-gram 0.5):

Diagram 0.5

0

500

1000

1500

2000

2500

3000

-40000 -35000 -30000 -25000 -20000 -15000 -10000 -5000 0 5000

As we see the resulting pattern of world population dynamics has an unmistaka-

bly 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 long-term 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 al-

ways remain distinctly hyperbolic. Replace, for example, the estimates of

McEvedy and Jones (1978) used by us earlier for Diagram 0.4 in the range be-

tween 10000 BCE and 1900 CE with the ones of Biraben (1980) (note that gen-

Introduction

14

erally 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 pic-

ture (see Diagram 0.6):

Diagram 0.6

0

500

1000

1500

2000

2500

3000

-40000 -35000 -30000 -25000 -20000 -15000 -10000 -5000 0 5000

As we see, the overall shape of the world population dynamics remains unmis-takably 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 long-term 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 time-point within any period in question, we can be al-ready perfectly confident about the order of magnitude of the world population. Hence, it is clear that whatever discoveries are made in the future, whatever re-evaluations 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 ad-dition 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


15

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 quadratic-hyperbolic 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:

20 )(

C

ttGt

, (0.5)

where Gt is the world GDP (in billions of 1990 international dollars, in purchas-ing power parity [PPP]) in year t, С = 17355487.3 and t0 = 2005.56 (see Dia-gram 0.7):

Diagram 0.7. World GDP Dynamics, 1–1973 CE (in billions of 1990 in-ternational dollars, PPP): the fit between predictions of a quadratic-hyperbolic model and the observed data

200017501500125010007505002500

18000

16000

14000

12000

10000

8000

6000

4000

2000

0

predicted

observed

NOTE: R = .9993, R2 = .9986, p << .0001. The black markers correspond to Maddison's (2001) es-timates (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 fol-lowing equation:

Introduction

16

2)2006(

17749573.1

tG

.

Actually, as was mentioned above, the best fit is achieved with С = 17355487.3 and t0 = 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):

x

ky , (0.2)

whereas the quadratic hyperbolic equation has x2 instead of just x:

2x

ky . (0.6)

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

20 )( xx

ky

. (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 re-mains 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

long-term 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 accu-rately just by subtracting x from the "Doomsday" year and dividing some con-stant 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 hy-perbolic growth of the world population accompanied by the quadratic hyper-bolic 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

18 Whereas the answers to the questions regarding the quadratic hyperbolic growth of the world

GDP might not have been quite clear even for those readers who know the hyperbolic demo-graphic models.


17

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, tt

CNt

0

, is just the solution

for the following differential equation (see, e.g., Korotayev, Malkov, and

Khaltourina 2006a: 119–20):

C

N

dt

dN 2

. (0.8)

This equation can be also written as:

2aNdt

dN , (0.9)

where C

a1

.

What is the meaning of this mathematical expression, 2aNdt

dN ? 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 mo-

ment.

Note that by dividing both parts of equation (0.9) with N we will get the fol-lowing:

aNNdt

dN: , (0.10)

Further, note that Ndt

dN: 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

Introduction

18

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 rN, we will get a particularly simple version of the hyperbolic equation:

aNrN . (0.10')

Thus, with hyperbolic growth the relative population growth rate (rN) 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 hy-perbolic growth we should just explain why for many millennia the world popu-lation's absolute growth rate tended to be proportional to the square of the popu-lation. 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 summa-rized 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 exten-sive 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 as-sumes that overall output produced by the world economy equals

rTNG , (0.11)

19 In addition to this, the absolute growth rate is proportional to the population itself – with a given

relative growth rate a larger population will increase more in absolute numbers than a smaller one.


19

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 la-bor (1817), which in the absence of technological growth produces just Malthu-sian 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 equilib-rium 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 ap-proaches m. Note that such a dynamics was actually rather typical for agrarian societies, and its mechanisms are known very well – indeed, if per capita in-comes 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 con-stant 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 propor-

tionally more people lucky or smart enough to come up with new ideas" (Kre-mer 1993: 685), thus, "the growth rate of technology is proportional to total

20 Kremer uses the following symbols to denote respective variables: Y – output, p – population,

A – the level of technology, etc.; while describing Kremer's models we will employ the symbols (closer to the Kapitza's [1992, 1999]) used in our model, naturally without distorting the sense of Kremer's equations.

21 "This implication flows naturally from the nonrivalry of technology… The cost of inventing a new technology is independent of the number of people who use it. Thus, holding constant the share of resources devoted to research, an increase in population leads to an increase in techno-logical change" (Kremer 1993: 681).

Introduction

20

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 "Kuznet-

sian" population-technological dynamics is combined with the "Malthusian" demographic one will be denoted as "Malthusian-Kuznetsian". 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:

bNTdt

dT . (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 determina-tion equations and demonstrates that their interaction produces just the hyper-bolic 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 mil-lion, result in the growth of dN/dt 100 times? Kremer's model explains this ra-ther 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

22 Note that "the growth rate of technology" means here the relative growth rate (i.e., the level to

which technology will grow in a given unit of time in proportion to the level observed at the be-ginning of this period).

23 In Economic Anthropology it is usually denoted as "Boserupian" (see, e.g., Boserup 1965; Lee 1986).

24 Kremer did not test this hypothesis empirically in a direct way. Note, however, that our own em-pirical test of this hypothesis has supported it (Korotayev, Malkov, and Khaltourina 2006b: Ap-pendix 1).


21

times larger population). On the other hand, the growth of a population 10 times also implies a 10-fold growth of the number of potential inventors, and, hence, a 10-fold 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 technological-ly 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 posi-tive feedback mechanism that was shown long ago to generate just the hyper-bolic growth, known also as the "blow-up regime" (see, e.g., Kurdjumov 1999; Knjazeva and Kurdjumov 2005). In our case this nonlinear second order posi-tive feedback looks as follows: the more people – the more potential inventors – the faster technological growth – the faster growth of the Earth's carrying capac-ity – the faster population growth – with more people you also have more po-tential inventors – hence, faster technological growth, and so on (see Diagram 0.8):

Diagram 0.8. Block Scheme of the Nonlinear Second Order Positive Feedback between Technological Development and Demographic Growth (version 1)

In fact, this positive feedback can be graphed even more succinctly (see Dia-gram 0.9a):

Introduction

22

Diagram 0.9a. Block Scheme of the Nonlinear Second Order Positive Feedback between Technological Development and Demographic Growth (version 2)

Note that the relationship between technological development and demographic growth cannot be analyzed through any simple cause-and-effect model, as we observe a true dynamic relationship between these two processes – each of them is both the cause and the effect of the other.

It is remarkable that Kremer's model suggests ways to answer one of the main objections raised against the hyperbolic models of the world's population growth. Indeed, at present the mathematical models of world population growth as hyperbolic have not been accepted by the academic social science communi-ty [The title of the most recent article by a social scientist discussing Kapitza's model, "Demographic Adventures of a Physicist" (Shishkov 2005), is rather telling in this respect]. We believe that there are substantial reasons for such a position, and that the authors of the respective models are as much to blame for this rejection as are social scientists. Indeed, all these models are based on an assumption that world population can be treated as having been an integrated system for many centuries, if not millennia, before 1492. Already in 1960, von Foerster, Mora, and Amiot spelled out this assumption in a rather explicit way:

"However, what may be true for elements which, because of lack of adequate communi-cation among each other, have to resort to a competitive, (almost) zero-sum multiperson game may be false for elements that possess a system of communication which enables them to form coalitions until all elements are so strongly linked that the population as a whole can be considered from a game-theoretical point of view as a single person play-ing a two-person game with nature as its opponent" (von Foerster, Mora, and Amiot 1960: 1292).

However, did, e.g., in 1–1500 CE, the inhabitants of, say, Central Asia, Tasma-nia, Hawaii, Terra del Fuego, the Kalahari etc. (that is, just the world popula-tion) really have "adequate communication" to make "all elements… so strong-ly linked that the population as a whole can be considered from a game-theoretical point of view as a single person playing a two-person game with na-ture as its opponent"? For any historically minded social scientist the answer to this question is perfectly clear and, of course, it is squarely negative. Against


23

this background it is hardly surprising that those social scientists who have hap-pened to come across hyperbolic models for world population growth have tended to treat them merely as "demographic adventures of physicists" (note that indeed, nine out of eleven currently known authors of such models are physicists); none of the respective authors (von Foerster, Mora, and Amiot 1960; von Hoerner 1975; Kapitza 1992, 1999; Kremer 1993; Cohen 1995; Podlazov 2000, 2001, 2002, 2004; Johansen and Sornette 2001; Tsirel 2004), after all, has provided any convincing answer to the question above. However, it is not so difficult to provide such an answer.

The hyperbolic trend observed for the world population growth after 10000 BCE does appear to be primarily a product of the growth of quite a real system, a system that seems to have originated in West Asia around that time in direct connection with the Neolithic Revolution. With Andre Gunder Frank (1990, 1993; Frank and Gills 1994), we denote this system as "the World System" (see also, e.g., Modelski 2000, 2003; Devezas and Modelski 2003). The presence of the hyperbolic trend itself indicates that the major part of the entity in question had some systemic unity, and the evidence for this unity is readily available. In-deed, we have evidence for the systematic spread of major innovations (domes-ticated cereals, cattle, sheep, goats, horses, plow, wheel, copper, bronze, and later iron technology, and so on) throughout the whole North African – Eurasian Oikumene for a few millennia BCE (see, e.g., Chubarov 1991, or Diamond 1999 for a synthesis of such evidence). As a result, the evolution of societies of this part of the world already at this time cannot be regarded as truly independ-ent. By the end of the 1

st millennium BCE we observe a belt of cultures, stretch-

ing from the Atlantic to the Pacific, with an astonishingly similar level of cul-tural complexity characterized by agricultural production of wheat and other specific cereals, the breeding of cattle, sheep, and goats; use of the plow, iron metallurgy, and wheeled transport; development of professional armies and cavalries deploying rather similar weapons; elaborate bureaucracies, and Axial Age ideologies, and so on – this list could be extended for pages). A few mil-lennia before, we would find another belt of societies strikingly similar in level and character of cultural complexity, stretching from the Balkans up to the In-dus Valley outskirts (Peregrine and Ember 2001: vols. 4 and 8; Peregrine 2003). Note that in both cases, the respective entities included the major part of the contemporary world's population (see, e.g. McEvedy and Jones 1978; Du-rand 1977 etc.). We would interpret this as a tangible result of the World Sys-tem's functioning. The alternative explanations would involve a sort of miracu-lous scenario – that these cultures with strikingly similar levels and character of complexity somehow developed independently of one another in a very large but continuous zone, while for some reason nothing comparable to them ap-peared elsewhere in the other parts of the world, which were not parts of the World System. We find such an alternative explanation highly implausible.

Introduction

24

Thus, we would tend to treat the world population's hyperbolic growth pat-tern as reflecting the growth of quite a real entity, the World System. A few other points seem to be relevant here. Of course there would be no grounds for speaking about a World System stretching from the Atlantic to the Pacific, even at the beginning of the 1

st millennium CE, if we applied the "bulk-

good" criterion suggested by Wallerstein (1974, 1987, 2004), as there was no movement of bulk goods at all between, say, China and Europe at this time (as we have no reason to disagree with Wallerstein in his classification of the 1

st

century Chinese silk reaching Europe as a luxury rather than a bulk good). However, the 1

st century CE (and even the 1

st millennium BCE) World System

definitely qualifies as such if we apply the "softer" information-network criteri-on suggested by Chase-Dunn and Hall (1997). Note that at our level of analysis the presence of an information network covering the whole World System is a perfectly sufficient condition, which makes it possible to consider this system as a single evolving entity. Yes, in the 1

st millennium BCE any bulk goods could

hardly penetrate from the Pacific coast of Eurasia to its Atlantic coast. Howev-er, the World System had reached by that time such a level of integration that iron metallurgy could spread through the whole of the World System within a few centuries.

Yes, in the millennia preceding the European colonization of Tasmania its population dynamics – oscillating around the 4000 level (e.g., Diamond 1999) – were not influenced by World System population dynamics and did not influ-ence it at all. However, such facts just suggest that since the 10

th millennium

BCE the dynamics of the world population reflects very closely just the dynam-ics of the World System population. On the basis of Kremer's model we (Korotayev, Malkov, and Khaltourina 2006a: 34–66) have developed a mathematical model that describes not only the hyperbolic world population growth, but also the macrodynamics of the world GDP production up to 1973:

TNkG 1 , (0.11)

SNkdt

dN2 , (0.13)

NTkdt

dT3 , (0.12)

where G is the world GDP, T is the World System technological level, N is population, and S is the surplus produced, per person, over the amount (m) min-imally necessary to reproduce the population with a zero growth rate in a Mal-thusian system (thus, S = g – m, where g denotes per capita GDP); k1, k2, k3, and α (0 < α < 1) are parameters.


25

We have also shown (Korotayev, Malkov, and Khaltourina 2006a: 34–66) that this model can be further simplified to the following form:

aSNdt

dN , (0.13)

bNSdt

dS , (0.14)

while the world GDP (G) can be calculated using the following equation:

G = mN + SN . (0.15)

Note that the mathematical analysis of the basic model (0.11)-(0.13)-(0.12)

suggests that during the "Malthusian-Kuznetsian" macroperiod of human histo-

ry (that is, up to the 1960s) the amount of S (per capita surplus produced at the

given level of World System development) should be proportional, in the long

run, to the World System's population: S = kN. Our statistical analysis of avail-

able empirical data has confirmed this theoretical proportionality (Korotayev,

Malkov, and Khaltourina 2006a: 49–50). Thus, in the right-hand side of equa-

tion (0.13) S can be replaced with kN, and as a result we arrive at the following

equation:

2kaNdt

dN (0.9)

25

As we remember, the solution of this type of differential equations is

)( 0 tt

CNt

, (0.1)

and this produces simply a hyperbolic curve. As, according to our model, S can be approximated as kN, its long-term dy-namics can be approximated with the following equation:

tt

kCS

0

. (0.16)

Thus, the long-term dynamics of the most dynamic component of the world GDP, SN, "the world surplus product ", can be approximated as follows:

20

2

)( tt

kCSN

. (0.17)

25 Thus we arrive, on a theoretical basis, at the differential equation discovered empirically by von

Hoerner (1975) and Kapitza (1992, 1999).

Introduction

26

Of course, this suggests that the long-term world GDP dynamics up to the early 1970s must be approximated better by a quadratic hyperbola than by a simple one; and, as we could see above (see Diagram 0.7), this approximation works very effectively indeed.

Thus, up to the 1970s the hyperbolic growth of the world population was accompanied by the quadratic-hyperbolic growth of the world GDP, just as is suggested by our model. Note that the hyperbolic growth of the world popula-tion and the quadratic hyperbolic growth of the world GDP are very tightly connected processes, actually two sides of the same coin, two dimensions of one process propelled by the nonlinear second order positive feedback loops be-tween the technological development and demographic growth (see Diagram 0.9b):

Diagram 0.9b. Block Scheme of the Nonlinear Second Order Positive Feedback between Technological Development and Demographic Growth (version 3)

We have also demonstrated (Korotayev, Malkov, and Khaltourina 2006a: 67–80) that the World System population's literacy (l) dynamics is rather accurately described by the following differential equation:

)1( laSldt

dl ,

(0.18)

where l is the proportion of the population that is literate, S is per capita surplus, and a is a constant. In fact, this is a version of the autocatalytic model. It has the following sense: the literacy growth is proportional to the fraction of the popu-lation that is literate, l (potential teachers), to the fraction of the population that


27

is illiterate, (1 – l) (potential pupils), and to the amount of per capita surplus S, since it can be used to support educational programs (in addition to this, S re-flects the technological level T that implies, among other things, the level of de-velopment of educational technologies). Note that, from a mathematical point of view, equation (0.18) can be regarded as logistic where saturation is reached at literacy level l = 1, and S is responsible for the speed with which this level is being approached. It is important to stress that with low values of l (which would correspond to most of human history, with recent decades being the exception), the rate of in-crease in world literacy generated by this model (against the background of hy-perbolic growth of S) can be approximated rather accurately as hyperbolic (see Diagram 0.10):

Diagram 0.10. World Literacy Dynamics, 1 – 1980 CE (%%): the fit between predictions of the hyperbolic model and the observed data

0

10

20

30

40

50

60

70

0 500 1000 1500 2000

observed

predicted

NOTE: R = 0.997, R2 = 0.994, p << 0.0001. Black dots correspond to UNESCO/World Bank (2005) estimates for the period since 1970, and to Meliantsev's (1996, 2003, 2004a, 2004b) esti-mates for the earlier period. The grey solid line has been generated by the following equation:

2)2040(

3769.264

tlt

.

The best-fit values of parameters С (3769.264) and t0 (2040) have been calculated with the least

squares method.

Introduction

28

The overall number of literate people is proportional both to the literacy level and to the overall population. As both of these variables experienced hyperbolic growth until the 1960s/1970s, one has sufficient grounds to expect that until re-cently the overall number of literate people in the world (L)

26 was growing not

just hyperbolically, but rather in a quadratic-hyperbolic way (as was world GDP). Our empirical test has confirmed this – the quadratic-hyperbolic model describes the growth of the literate population of this planet with an extremely good fit indeed (see Diagram 0.11):

Diagram 0.11. World Literate Population Dynamics, 1 – 1980 CE (L, millions): the fit between predictions of the quadratic-hyperbolic mod-el and the observed data

0

200

400

600

800

1000

1200

1400

1600

1800

0 500 1000 1500 2000

observed

predicted

NOTE: R = 0.9997, R2 = 0.9994, p << 0.0001. The black dots correspond to UNESCO/World Bank (2006) estimates for the period since 1970, and to Meliantsev's (1996, 2003, 2004a, 2004b) estimates for the earlier period; we have also taken into account the changes of age structure on the basis of UN Population Division (2006) data. The grey solid line has been generated by the follow-ing equation:

2)2033(

4958551

tLt

.

The best-fit values of parameters С (4958551) and t0 (2033) have been calculated with the least

squares method.

26 Since literacy appeared, almost all of the Earth's literate population has lived within the World

System; hence, the literate population of the Earth and the literate population of the World Sys-tem have been almost perfectly synonymous.


29

Similar processes are observed with respect to world urbanization, the macro-dynamics of which appear to be described by the differential equation:

)lim

( uubSudt

du , (0.19)

where u is the proportion of the population that is urban, S is per capita surplus produced with the given level of the World System's technological develop-ment, b is a constant, and ulim is the maximum possible proportion of the popu-lation that can be urban. Note that this model implies that during the "Malthusi-an-Kuznetsian" era of the blow-up regime, the hyperbolic growth of world ur-banization must have been accompanied by a quadratic-hyperbolic growth of the urban population of the world, which is supported by our empirical tests (see Diagrams 0.12–13):

Diagram 0.12. World Megaurbanization Dynamics (% of the world pop-ulation living in cities with > 250 thousand inhabitants), 10000 BCE – 1960 CE: the fit between predictions of the hyperbolic model and empiri-cal estimates

NOTE: R = 0.987, R2 = 0.974, p << 0.0001. The black dots correspond to estimates of Chandler (1987), UN Population Division (2005), and White et al. (2006). The grey solid line has been gen-erated by the following equation:

)1990(

403.012

tut

.

The best-fit values of parameters С (403.012) and t0 (1990) have been calculated with the least

squares method. For a comparison, the best fit (R2) obtained here for the exponential model is

0.492.

Introduction

30

Diagram 0.13. Dynamics of World Urban Population Living in Cities with > 250000 Inhabitants (mlns.), 10000 BCE – 1960 CE: the fit between predictions of the quadratic-hyperbolic model and the observed data

NOTE: R = 0.998, R2 = 0.996, p << 0.0001. The black markers correspond to estimates of Chandler (1987), UN Population Division (2005), and White et al. (2006). The grey solid line has been generated by the fol-lowing equation:

2)2008(

912057.9

tU t

.

The best-fit values of pa-rameters С (912057.9) and t0 (2008) have been calculated with the least squares meth-od. For a comparison, the best fit (R2) obtained here for the exponential model is 0.637.

Within this context it is hardly surprising to find that the general macrodynam-ics of the size of the largest settlement within the World System are also quad-ratic-hyperbolic (see Diagram 0.14):

Diagram 0.14. Dynamics of Size of the Largest Settlement of the World (thousands of inhabitants), 10000 BCE – 1950 CE: the fit between pre-dictions of the quadratic-hyperbolic model and the observed data

NOTE: R = 0.992, R2 = 0.984, p << 0.0001. The black markers correspond to estimates of Modelski (2003) and Chandler (1987). The grey solid line has been gen-erated by the following equa-tion:

2max)2040(

573104020618,

tU t

.

The best-fit values of pa-rameters С (104020618,5) and t0 (2040) have been cal-culated with the least squares method. For a comparison, the best fit (R2) obtained here for the exponential model is 0.747.


31

As has been demonstrated by cross-cultural anthropologists (see, e.g., Naroll and Divale 1976; Levinson and Malone 1980: 34), for pre-agrarian, agrarian, and early industrial cultures the size of the largest settlement is a rather effective indicator of the general sociocultural complexity of a social system. This, of course, suggests that the World System's general sociocultural complexity also grew, in the "Malthusian-Kuznetsian" era, in a generally quadratic-hyperbolic way.

It is world literacy for which it is most evident that its hyperbolic growth could not continue, for any significant period, after the mid-1960s; after all, the literacy rate by definition cannot exceed 100 per cent just by definition. What is more, since the 1970s the saturation effect

27 described by our model started be-

ing felt more and more strongly and the rate of world literacy's growth began to slow (see Diagram 0.15):

Diagram 0.15. World Literacy Growth Dynamics, 1975 – 1995, the in-

crease in percentage of adult literate population, by five-year periods

4.474.15

3.64

3.13

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1975-1980 1980-1985 1985-1990 1990-1995

However, already before this, the hyperbolic growth of world literacy and of the other indicators of the human capital development had launched the process of diverging from the blow-up regime, signaling the end of the era of hyperbolic growth. As has been shown by us earlier (Korotayev, Malkov, and Khaltourina 2006a: 67–86), hyperbolic growth of population (as well as of cities, schools etc.) is only observed at relatively low (< 0.5, i.e., < 50%) levels of world liter-acy. In order to describe the World System's demographic dynamics in the last decades (as well as in the near future), it has turned out to be necessary to ex-tend the equation system (0.13)-(0.14) by adding to it equation (0.21), and by adding to equation (0.13) the multiplier (1 – l), which results in equation (0.20), and produces a mathematical model that describes not only the hyperbolic de-

27 On the ground, the saturation effect means, for example, that raising literacy from 98 to 100 per

cent of the adult population would require much more time and effort than would raising it from 50 to 52 per cent.

Introduction

32

velopment of the World System up to the 1960s/1970s, but also its withdrawal from the blow-up regime afterwards:

)1( laSNdt

dN , (0.20)

bNSdt

dS , (0.14)

)1( lcSldt

dl . (0.21)

We would like to stress that in no way are we claiming that the literacy growth is the only factor causing the demographic transition. Important roles were also played here by such factors as, for example, the development of medical care and social security subsystems. These variables, together with literacy, can be regarded as different parameters of one integrative variable, the human capital development index. These variables are connected with demographic dynamics in a way rather similar to the one described above for literacy. At the beginning of the demographic transition, the development of the social security subsystem correlates rather closely with the decline of mortality rates, as both are caused by essentially the same proximate factor – the GDP per capita growth. Howev-er, during the second phase, social security development produces quite a strong independent effect on fertility rates through the elimination of one of the main traditional incentives for the maximization of the number of children in the fam-ily.

The influence of the development of medical care on demographic dynamics shows even closer parallels with the effect produced by literacy growth. Note first of all that the development of modern medical care is connected in the most direct way with the development of the education subsystem. On the other hand, during the first phase of the demographic transition, the development of medical care acts as one of the most important factors in decreasing mortality. In the meantime, when the need to decrease fertility rates reaches critical levels, it is the medical care subsystem that develops more and more effective family planning technologies. It is remarkable that this need arises as a result of the de-crease in mortality rates, which could not reach critically low levels without the medical care subsystem being sufficiently developed. Hence, when the need to decrease fertility rates reaches critical levels, those in need, almost by defini-tion, find the medical care subsystem sufficiently developed to satisfy this need quite rapidly and effectively.

Let us recollect that the pattern of literacy's impact on demographic dynam-ics has an almost identical shape: the maximum values of population growth rates cannot be reached without a certain level of economic development, which


33

cannot be achieved without literacy rates reaching substantial levels. Hence, again almost by definition, the fact that the system reached the maximum level of population growth rates implies that literacy – especially of females – had at-tained such a level that its negative impact on fertility rates would cause popula-tion growth rates to start to decline. On the other hand, the level of development of both medical care and social security subsystems displays a very strong cor-relation with literacy (see Korotayev, Malkov, and Khaltourina 2006a: Chap-ter 7). Thus, literacy rate turns out to be a very strong predictor of the develop-ment of both medical care and social security subsystems.

Note that in reality, as well as in our model, both the decline of mortality at the beginning of the demographic transition (which caused a demographic ex-plosion) and the decline of fertility during its second phase (causing a dramatic decrease of population growth rates) were ultimately produced by essentially the same factor (human capital growth); there is therefore no need for us to in-clude mortality and fertility as separate variables in our model. On the other hand, literacy has turned out to be a rather sensitive indicator of the develop-ment level of human capital, which has made it possible to avoid including its other parameters as separate variables in extended macromodels (for more de-tail see Korotayev, Malkov, and Khaltourina 2006a: Chapter 7). Model (0.20)-(0.14)-(0.21) describes mathematically the divergence from the blow-up regime not only for world population and literacy dynamics, but al-so for world economic dynamics. However, this model does not describe the slowdown of the World System's economic growth observed after 1973. Ac-cording to the model, the relative rate of world GDP growth should have con-tinued to increase even after the World System began to diverge from the blow-up regime, though more and more slowly. In reality, however, after 1973 we observe not just a decline in the speed with which the world GDP rate grows – we observe a decline in the world GDP growth rate itself (see, e.g., Maddison 2001). It appears that model (0.20)-(0.14)-(0.21) would describe the recent world economic dynamics if the (1 – l) multiplier were added not only to its first equation, but also to the second (0.14). This multiplier might have the fol-lowing sense: the literate population is more inclined to direct a larger share of its GDP to resource restoration and to prefer resource economizing strategies than is the illiterate one, which, on the one hand, paves the way toward a sus-tainable-development trajectory, but, on the other hand, slows down the eco-nomic growth rate (cp., e.g., Liuri 2005).

Note that development, according to this scenario, does not invalidate Kre-mer's technological growth equation (0.12). Thus, the modified model does im-ply that the World System's divergence from the blow-up regime would stabi-lize the world population, the world GDP, and some other World-System de-velopment indicators (e.g., urbanization and literacy as a result of saturation, i.e., the achievement of the ultimate possible level); technological growth, how-ever, will continue, though in exponential rather than hyperbolic form.

Introduction

34

Due to the continuation of technological growth, the ending of growth in the world's GDP will not entail a cessation of growth in the standard of living of the world's population. A continuing rise in the world's standard of living is most likely to be achieved due to the so-called "Nordhaus effect" (Nordhaus 1997). The essence of this effect can be spelled out as follows: imagine that you are going to buy a new computer and plan to spend $1000 on this. Now imagine what computer you would have been able to buy with the same $1000 five years ago. Of course, the computer that you will be able to buy with $1000 now will be much better, much more effective, much more productive etc. than the com-puter that you could have bought with the same $1000 five years ago. However, open a current World Bank handbook and you will see that the present-day $1000, in terms of purchasing power parity (PPP), constitutes a significantly smaller sum than did the $1000 of five years ago. The point is that traditional measures of economic growth (above all, the GDP as measured in international PPP dollars) reflect less and less the actual growth of the standard of living (es-pecially in more developed countries). Imagine a firm that in 2001 produced 1 million computers and sold them at $1000 a piece, in 2006 the same firm pro-duced 1 million 100 thousand new, much more effective computers, but still sells them (due to increasing competition) at $1000 a piece (let us also imagine that the firm has managed to reduce production costs and thus increased both its profits and employees' salaries). How will this affect GDP, both in the country in which the firm operates, and in the world as a whole? In fact, the effect is most likely to be exactly zero. In 2006 the firm produces computers for a total price of 1100 million 2006 international PPP dollars. However, the World Bank will recalculate this sum into 2001 international PPP dollars and will find out that 1100 million 2006 international PPP dollars equal just 1000 million 2001 international PPP dollars. Thus, technological progress sufficient to raise the level of life of a significant number of people will in no way affect the World Bank GDP statistics, according to which it will appear that the above-mentioned technological advance has led to GDP increase at neither the country nor the world level. The point is that the traditional GDP measures of production growth work really well when they are connected with the growth of consumption of scarce resources (including labor resources); however, if the production growth takes place without an increase in the consumption of scarce resources, it may well go undetected. The modified macromodel predicts such a situation when the World System's divergence from the blow-up regime will have resulted in the cessation of the resource-consuming World GDP production in its traditional measures, accompanied by the transition to exponential (in place of hyperbolic) growth of technology through which an increasing standard of living will be achieved without the growth of scarce-resource consumption. Because the macrodynamics of the World System's development obey a set of rather simple laws having extremely simple mathematical descriptions, the


35

macroproportions between the main indicators of that development can be de-scribed rather accurately with the following series of approximations:

N ~ S ~ l ~ u, G ~ L ~ U ~ N

2 ~ S

2 ~ l

2 ~ u

2 ~ SN ~ etc.,

where (let us recollect) N is the world population; S is per capita surplus pro-duced, at the given level of the World System’s technological development, over the "hungry survival" level m that is necessary for simple (with zero growth) demographic reproduction; l is world literacy, the proportion of literate people among the adult (> 14 year old) population of the world; u is world ur-banization, the proportion of the world population living in cities; G is the world GDP; L is the literate population of the world; and U is the urban popula-tion of the world. Yes, for the era of hyperbolic growth the absolute rate of growth of N (but, incidentally, also of S, l and u) in the long-run is described ra-ther accurately

28 as kN

2 (Kapitza 1992, 1999); yet, with a comparable degree of

accuracy it can be described as k2SN, k3S2 or (apparently with a somehow

smaller precision) as k4G, k5L, k6U, k7l2, k8u

2, etc.

It appears important to stress that the present-day decrease of the World System's growth rates differs radically from the decreases that inhered in oscil-lations of the past. This is not merely part of a new oscillation; rather, it is a phase transition to a new development regime that differs radically from the one typical of all previous history. Note, first of all, that all previous cases of reduc-tion of world population growth took place against the background of cata-strophic declines in the standard of living, and were caused mainly by increases in mortality as a result of various cataclysms – wars, famines, epidemics; and that after the end of such calamities the population, having restored its numbers in a relatively rapid way, returned to the earlier hyperbolic trajectory. In sharp contrast, the present day decline of the world population's growth rate takes place against the background of rapid economic growth and is produced by a radically different cause – the decline of fertility rates that is occurring precisely because rising standards of living for the majority of the World System's popu-lation have meant the growth of education, health care (including various meth-ods and means of family planning), social security, etc. Decrease in the rate of growth of literacy and urbanization was not infrequent in the earlier epochs ei-ther; but in those epochs it was connected with economic decline, whereas now it takes place against the contrary background of rapid economic growth, and is connected to the closeness of the saturation level. Earlier declines, we might say, reflected a deficit of economic resources, whereas the present one reflects their abundance.

It appears necessary to stress that the models discussed above have been de-signed to describe long-term ("millennial") trends, whereas when we analyze social macrodynamics at shorter ("secular") time scales we also have to take in-

28 However, for u the fit of this description appears to be smaller than for the rest of variables.

Introduction

36

to account its cyclical (as well as stochastic) components; it is these components that will be the main task of the present part of our Introduction to Social Macrodynamics. To begin with, the actual dynamics typical for agrarian politi-cal-demographic cycles are usually the opposite of those that are theoretically described by "millennial" models and actually observed at the millennial scale. For example, as we shall see below, during agrarian political-demographic cy-cles the population normally grew much faster than technology, which naturally resulted in Malthusian dynamics: population growth was accompanied not by increase, but by decrease of per capita production, usually leading to political-demographic collapse and the start of a new cycle.

In the second part of our Introduction we reviewed available mathematical models of political-demographic cycles (Chu and Lee 1994; Nefedov 1999f, 2002a; 2004; S. Malkov, Kovalev, and A. Malkov 2000; S. Malkov and A. Malkov 2000; Malkov and Sergeev 2002, 2004a, 2004b; Malkov et al. 2002; Malkov 2002, 2003, 2004; Turchin 2003, 2005a). We considered in more detail political-demographic cycles in China, where long-term population dynamics have been recorded more thoroughly than elsewhere. We presented our own model of pre-Industrial political-demographic cycles. Finally, we provided a preliminary consideration of the interaction between long-term trends and cycli-cal dynamics.

In this part of the Introduction to Social Macrodynamics we analyze the in-terplay of trends and cyclical dynamics in Egypt and Tropical Africa.

Part I

SECULAR CYCLES

AND MILLENNIAL TRENDS

IN EGYPT

Chapter 1

General Trends of

Egyptian Demographic Dynamics

in the 1st – 18th Centuries CE

Long-term demographic dynamics of Egypt can be estimated as follows (see Diagram 1.1):

Diagram 1.1. Estimated Population Dynamics of Egypt, in thousands (10000 BCE – 2005 CE)

0

10000

20000

30000

40000

50000

60000

70000

80000

-10000 -8000 -6000 -4000 -2000 0 2000

Part I. Secular Cycles and Millennial Trends in Egypt

38

The data on Egypt population dynamics for 1950–2005 are from Maddison (2001), US Census Bureau (2006) and World Bank (2006). The data for 1897–1950 are from Craig 1917; Cleveland (1936: 7), Nāmiq (1952), McCarthy (1976: 31–3), and Vasil'ev (1990: 205). For 1800–1897 Pansac's (1987) esti-mates are used

1 (see below for more detail). For Ancient Egypt we rely on the

estimates of McEvedy and Jones (1978: 226–9), and Butzer (1976: 81–98); the general shape of the ancient Egyptian political-demographic macrocycles

2 is

deduced from Shaw (2000). For 300 BCE – 1900 CE the population dynamics can be estimated as fol-

lows (see Diagram 1.2):

Diagram 1.2. Estimated Population Dynamics of Egypt, in thousands (300 BCE – 1900 CE)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

-300 -100 100 300 500 700 900 1100 1300 1500 1700 1900

1 Taking into account McCarthy's estimates of the human losses as a result of 1835 plague epidem-

ic (1976: 15). 2 Of course, those macrocycles consisted of microcycles, whose shape for this period we leave

without reconstruction.

Chapter 1. General Trend of Egyptian Demographic Dynamics

39

For the late 1st century BCE we rely on Diodorus' (1.31.6–9) estimate that has

been shown to be one of the most reliable ancient estimates (Bagnall and Frier 1994: 53–6). We also incorporate Bagnall and Frier's (1994: 56) estimate of the Roman Egypt population at its peak as 4.75 million.

3 The general characteristics

of Ptolemaic and Roman demographic cycles are estimated after Nefedov (1999a, 2003). We estimate that the population of Ptolemaic Egypt reached a level not achieved in any of the preceding epochs on the basis of the following observation:

"The inundation was by far the most important fundamental continuity in the agricultur-al life of Egypt from Pharaonic to modern times. The ancient Egyptians devised catch-ment basins and dykes to trap the flood, and channelled water into their fields through a system of basin irrigation. However, basin irrigation only permits a single crop per year; multiple cropping requires canals. The efficient exploitation of canals was only made feasible with the introduction of water-lifting devices such as the saqia (water-wheel)4 and the Archimedean screw, which began to appear in the Ptolemaic period. It was thus under the Ptolemies that basin irrigation was much extended through a system of canals" (Bowman and Rogan 1999: 2), thus, raising significantly the overall carrying capacity of land (see also, e.g., Thompson 1999).

The population dynamics of Egypt in 170–640 CE are reproduced after Russell (1966). Ashtor's (1976: 92) estimate of the Egyptian population at the time of the Arab conquest as being between 4 and 4.5 million does not appear plausi-ble. Indeed, it is difficult to imagine that the Egyptian population could recover so rapidly almost to the level of the early Roman peak after the 6

th century cata-

clysms (and especially the Justinian plague [see, e.g., Korotayev, Klimenko, and Prussakov 1999]), the anti-Phokas rebellion,

5 and the early 7

th century Per-

sian occupation (see, e.g., Alston 2002: 361–6). What is more, the data on the cultivated area in Egypt of this time only being c. 1.5 million feddans (see, e.g., Cleveland 1936: 5) suggest that the contemporary population of Egypt could hardly exceed 2.5 million.

6 The estimate of Egyptian population as being 3 mil-

lion in 600 CE is from Kaegi (1998: 34). The general shape of the medieval Egyptian political-demographic cycles is reconstructed on the basis of Nefedov (1999b; 2003). We take into account the medieval population estimates pro-

3 Note that Bagnall and Frier themselves maintain that this figure "seems high" to them (1994: 56).

It seems high to us too, but we follow them in using this figure for the time being. 4 Note, however, that the introduction of sāqiyah might have started in the Persian period (Butzer

1976: 46; Ritner 1998: 2). 5 "Egypt experienced more fighting and consequent loss of life and property by civilians as well as

the military, in conflicts between the armies and partisans of the usurper Phokas and the rebel Heraclius, Exarch of Africa, and his son Heraclius, whose rebellion against Phokas commenced in 608, than did any other Byzantine province in the years 608–610" (Kaegi 1998: 37).

6 For example, the cultivated area of c.3.5 million feddans (e.g. Nofal 1995: 146) in the early 19th century only supported Egyptian population of c.4.5 million (see below). Note that Russell (1966: 72–3) estimates that the Egyptian population was just about 2.5 million at the time of the Arab conquest.


40

duced by Russell (1966) on the basis of the data on the amounts of land-tax rev-enues (see Diagram 1.3): Diagram 1.3. Population Dynamics of Medieval Egypt

(640 – 1422 CE) according to Russell (1966: 81, Fig. 1)

640740

805884

970 1075

1189

1346

1378

1422

0

500

1000

1500

2000

2500

3000

3500

4000

4500

625 725 825 925 1025 1125 1225 1325 1425

Though Russell's contribution to the reconstruction of the absolute levels of the medieval Egyptian population is very important, Diagram 1.3 above (made after Russell 1966: 81, Fig. 1) distorts significantly the actual picture of these dynam-ics. To start with, Russell appears to underestimate rather significantly the actu-al size of the medieval Egyptian population. This turns out to stem from his un-derestimation of the average size of the pre-Modern Egyptian household, count-ing it as 5 rather than 7 (in fact, as we shall see below, a similar mistake was made long before Russell and was a source of a very serious underestimation of the Egyptian population in the first half of the 19

th century). Thus, we had to

calibrate Russell's estimates using Pansac's correction (see below). However, even this is not the main source of distorted impression produced by Russell's diagram. The main problem is that this diagram does not reflect most of political-demographic cycles. For example, it produces an impression that between 884 and 1075 CE the Egyptian population only experienced a con-stant decline, which, we believe, oversimplifies the reality a bit too much. In fact, this could hardly be qualified as simply Russell's mistake, as Russell re-produced on his diagram only those points for which tax-survey data made


41

available; this, however, led to a significantly distorted picture of the medieval Egyptian demographic dynamics.

To start with, different Russell's data points refer to different phases of the political demographic cycles. 884 CE data point reflects the situation when the Egyptian population had experienced a considerable recovery during the Ibn Tulun rule after the demographic collapse of the 830s (Bianquis 1998; Nefedov 1999b), whereas 970 survey reflects a situation when the Egyptian population had no time at all to recover from the demographic collapse of the late 960s. The next data point refers again to the situation after the demographic collapse of 1065–1072; and, thus, Russell's diagram does not reflect the substantial growth experienced by the Egyptian population during the first decades of the Fatimid rule (Semenova 1974; Sanders 1998; Nefedov 1999b). The anchoring point for the reconstruction of the absolute levels of medieval Egyptian population is provided by the estimate of the population of Egypt in 1800. Here we rely heavily on the revision of the early 19

th century Egyptian

demographic history produced by McCarthy (1976) and Panzac (1987). Before them many scholars relied on Jomard's (1818) calculations, who estimated Egyptian population in 1800 at 2,488,950 (without nomads whose number was estimated by another member of the French Expedition as 130,000, thus pro-ducing a total 2,618,950), which no doubt affected in a very serious way esti-mates of medieval Egyptian populations. Panzac (1987: 12–13) starts with the revision of the results of Muhammad Ali's 1846 and 1847 census, showing that the actual number of Egyptian inhab-itants in 1847 is more likely to have been c.5.4 million rather than 4542 thou-sand registered by the census, accompanying this with the following commen-tary:

"This is far from surprising. As elsewhere in this part of the world at the time, a census was above all a means to establish the number of persons eligible for military service and payment of taxes. In 1840 circumstances induced Muhammad Ali to impose tax lev-ies and restrictions on the population and the memory of these policies had caused peo-ple to evade registration.7 We can reasonably assume that a fifth of the population suc-ceeded in doing so" (Panzac 1987: 12–13).

An important piece of evidence supporting his revision was found by Panzac in the French Foreign Ministry Archives. This is a report of a French diplomat to the Foreign Minister written in 1833, the relevant part of which cited the words heard by ‛Alī:

"The available data on the total population of Egypt are incomplete and unreliable. Your Eminence has seen [the report] of my discussions with the Viceroy, who insists that there are four million inhabitants in Egypt, whereas general opinion does not admit to there being more than two and a half million. You may have observed that the Viceroy

7 Note that during earlier registrations (like the ones of 1821, or 1827) people had even more

grounds (and entirely compelling ones [see, e.g., Nāmiq 1952; Rivlin 1961]) to try to avoid the registration.


42

is basing himself on the number of persons paying poll tax, who, according to him, comprise a total of 870,000 men over seventeen years of age." …The same report goes on to say: "Another basis for calculation, which I am inclined to consider more reliable, is the number of households. In 1827 there were 618,000 tax-paying households in Egypt – at five persons per household, this would mean 3,090,000 inhabitants" (Panzac 1987: 14).

At the meantime Panzac brings our attention to the fact that the first modern Egyptian census (of 1882 and 1897) recorded the mean size of an Egyptian household as respectively 7.0 and 6.9 individuals (1987: 15)

8, which leads

Panzac to the following conclusion:

"If we take this average and multiply by the number of houses that existed in 1827, we come up with a total of 4,326,000 inhabitants. The rate of growth thus obtained, even if the figure itself gives us only the illusion of precision, tends in a direction that becomes noticeable around 1830, when there were at least 4 million registered Egyptians, to which those who evaded the census should be added.9 The demographic patterns of the period and the high endemic and epidemic rate of mortality10 kept the average annual population growth rate between 1800 and 1830 down to only 3–4 per thousand. This means that in 1800, shortly before Muhammad `Ali seized power, Egypt must have had approximately 4,500,000" (Panzac 1987: 14).

8 In fact, the underestimation of the average household size is a typical cause of underestimation of

population sizes calculated on the basis of registration for the taxation purposes (on this see, e.g., Durand 1960), stemming from the fact that in such cases officials had no particular interest to register all the household members (and especially, infants and small children, a very significant part of whom in the conditions of pre-Modern Egypt had no chances to grow up to such an age when they would constitute any real interest for the tax-collectors [e.g., McCarthy 1976]). There is also some doubt that when in 1800 Jomard's team asked the village shaykhs of al-Minya prov-ince questions about the number of people living in their villages (these data were used later to estimate all the population of rural Egypt), the shaykhs told the team members actual size of re-spective populations in modern sense of this term (i.e., including infants and small children). One also wonders if the shaykhs always took into account entirely (or even partly) the female part of the respective populations. Incidentally, could not this be one of the main causes of the severe underestimation of the Egyptian population produced by Jomard?

9 In fact, the registration of 1821 produced rather similar results, it "was based on a census of hous-es with an average of eight persons per dwelling in the metropolis and four persons per dwelling in the provinces" and estimated the population of Egypt to be 2,536,400 (Cleveland 1936: 7; see also ‘Azmī 1937: 9 and Nāmiq 1952: 12). Indeed, if we calibrate this number along the lines suggested by Panzac (that is, taking 7 instead of 4 as an average number of persons per house-hold, and assuming that a fifth of the population avoided the registration), we will arrive at a fig-ure, which will be only slightly less than the one arrived at by Pansac, and which could be easily explained by the population growth between 1821 and 1827 just at 3–4‰ per year rate suggested by Panzac.

10 For additional evidence of a really "high endemic and epidemic rate of mortality", attested in the period between 1800 and 1830 see, e.g., McCarthy 1976, who also points out that the fertility rate in this period should have been somehow depressed due to the draft to the army of more than 200,000 reproductive age males, that is almost a quarter of all the males of reproductive age (McCarthy 1976: 15); Edward Lane (1966 [1836]:23) even believed that this factor coupled with high mortality rates attested at this period led to the population decline, which has been shown, however, to be a definite exaggeration (McCarthy 1976; Pansac 1987).


43

Note, however, that in the late 18th

century Egypt experienced a demographic collapse caused by state breakdown, famines and epidemics (see, e.g., al-Sayyid Marsot 1985: 49–50; Atsamba and Kirillina 1996: 17–8; Zelenev 1999; Cre-celius 1998: 83–6; Raymond 2001: 225; Sāmī Bāshā 2002–2004, 2: 95–116; Harīdī 2005: 280–2). McCarthy (1976: 15) estimates that the 1835 plague led to 500,000 deaths (adding that "the actual death rate could have been considerably higher, but not lower") which corresponds to an approximately 10% population decrease. In the late 18

th century Egypt encountered two such epidemics, in

1784–1785 and 1791 (in addition to many other hardships experienced by the Egyptian population in this period

11). Note that in both cases the plagues oc-

curred against the background of state breakdown and catastrophic famines, whereas 1835 plague took place within a state, which was better organized than any of its predecessors during the past few centuries, when any symptoms of state breakdown were absent and no famine was observed. Hence, there is every reason to believe that each of the 1784–1785 and 1791 plagues removed a sig-nificantly higher proportion of the Egyptian population than the 1835 one. In general, total decline of population as a result of the late 18

th century demo-

graphic collapse could hardly be less than 25%12

, which suggests that the popu-lation of Egypt around 1775 could hardly be less than 6 million (and most likely was substantially higher). Of course, in conjunction with Bagnall and Frier's (1994) estimates, accord-ing to which the population of Roman Egypt peaked to 4.75 million, this sug-gests that contrary to a wide-spread belief (see, e.g., Lane 1966: 23; McEvedy and Jones 1978: 226–9; Bowman and Rogan 1999: 6; Maddison 2001: 239) the overall population trend in the 1

st–18

th centuries CE in Egypt was upward, but

not downward. To us the conclusion that the population of Egypt in the Islamic period (but before the 19

th century) reached levels that were higher than those

achieved in any of the pre-Islamic epochs does not appear absurd. There is evi-dence for a considerable number of very important technological innovations that were introduced in Egypt during the Islamic period of its history and this, in turn, raised significantly the carrying capacity of land in Egypt. The first half of the Islamic period saw the introduction of a very considera-ble number of domestic plants

13, including such important calorie sources as

11 To cite just a few lines from a description of this period of the Egyptian history: "Murad Bey

[one of the leaders of al-Muhammadiyyah mamluks] made annual rounds through Lower Egypt, demanding extraordinary taxes from the already overtaxed farmers. Many fled the land or re-belled; Murad… leveled many villages as examples to others" (Crecelius 1998: 83).

12 For example, Altson (2002: 364) estimates that "general and severe plague visitations removed around 20 per cent of the Egyptian population" (see also Alston 2001).

13 Eggplant (Solanum melongena), artichoke (Cynara cardunculus), spinach (Spinacia oleracea), watermelon (Citrullus lanatus), plantain (Musa paradisiaca), sour orange (Citrus aurantium), lemon (Citrus limon), lime (Citrus aurantifolia), shaddock (Citrus grandis), Old World cotton (Gossypium arboreum and Gossypium herbaceum), banana (Musa sapientium), taro/colocasia


44

sorghum, taro, Asiatic rice, sugar cane, hard wheat14

, and banana, which raised the carrying capacity of land in a very significant way. Thus, for example, "sor-ghum has tended to displace millets and other grains of antiquity because of its higher yields and the ability of some cultivars to withstand drought and to toler-ate poor soils" (Watson 1983: 9); by the thirteenth century "many writers saw it growing in Upper Egypt…, where it had clearly become the principal food" (Watson 1983: 13). "Though it may be grown in all parts of the Mediterranean basin, colocosia [taro] has become a particularly important crop in tropical and semi-tropical regions

15 because of various advantages: its exceptionally high

per-acre yield of starch, which it produces in an unusually digestible form; its ability to withstand slightly colder weather than, for instance, the greater yam; its short growing season, which allows two or three crops of it to be grown on the same land in a single year; and the easy preservation over many months of the fresh and dried tubers, which made it useful for tiding people over barren seasons…" (Watson 1983: 66); and al-Maqrīzī (1364–1442) notes that by his time taro (qulqās) became one of the most important components of the diet of the inhabitants of Lower Egypt (al-Maqrīzī 1959, 1: 78).

16 Sugar cane is "capa-

ble of producing more human food per acre than any other crop" (Harrison, Masefield, and Wallis 1969: 14), and al-Maqrīzī (1364–1442) notes that by his time the produced from the sugar cane became a very important component of the diet of the inhabitants of Upper Egypt (al-Maqrīzī 1959, 1: 78). Sugar cane was in general a rather important source of calories for the commoner population mainly not in the form of sugar, but rather in the form of the so-called "black honey" (‛asl iswid), a by-product of sugar production that was consumed in combination with tahini (sesame) paste (a by-product of the sesame oil production).

17

In the Ottoman period the carrying capacity of the Egyptian land was raised further through the introduction of some New World crops, especially maize. Note that in this period "the crops of the Late Summer Cultivation [the 3

rd crop

of the year] were planted in the months of May, June, July, and August…, when the Nile first began its rise in Upper Egypt. They consisted principally of maize" (Shaw 1962: 51). The introduction of maize increased significantly the food security of Egypt (and, hence, its carrying capacity). For example, in 1809 when the wheat crops failed, this was "only the supply of maize [that] saved the country from starvation" (Rivlin 1961: 50).

(Colocasia antiquorum), sugar cane (Saccharum officinarum), hard wheat (Triticum durum), Asiatic rice (Oryza sativa), and sorghum (Sorghum bicolor) (Watson 1983: 9–75).

14 However, the introduction of the hard wheat in Egypt appears to have started in the Byzantine period of its history (Watson 1983: 20).

15 Hence, Egypt is that very part of the Mediterranean region where taro can be grown in the most effective way.

16 See also, e.g., al-Baghdādī 2004: 60–8 (the early 13th century). 17 We would like to express our gratitude to Elizabeth Sartain and Eleonora Fernandes of the

American University in Cairo for bringing our attention to this point.

Chapter 2

Some Features of Medieval Egyptian

Political-Demographic Cycles

Though the presence of a general upward demographic trend in the population dynamics of the 1

st – 18

th century Egypt is very likely, it is also quite clear that

this trend was very weak, especially when compared with the one observed in other parts of the World System (outside the Middle East) (see Diagrams 2.1 and 2.2):

Diagram 2.1. Population Growth between the Early 1st millennium CE

and the 18th century (in millions)

0

50

100

150

200

250

300

Early

1st

mil.

18th

cent.

Early

1st

mil.

18th

cent.

Early

1st

mil.

18th

cent.

Early

1st

mil.

18th

cent.

Egypt Europe India China

NOTES: the comparison is made between the maximum population levels achieved in the respec-tive areas in the early 1st millennium and the 18th century. Data sources: Egypt – see above; Eu-rope, India and China – Korotayev, Malkov, and Khaltourina 2006b; Durand 1960; McEvedy and Jones 1978: 19–39, 170–4, 182–9.


46

Diagram 2.2. Population Growth between the Early 1st millennium CE

and the 18th century (maximum population level reached

in the early 1st millennium = 100)

100

147

100

500

100

543

100

508

0

100

200

300

400

500

600

Early

1st

mil.

18th

cent.

Early

1st

mil.

18th

cent.

Early

1st

mil.

18th

cent.

Early

1st

mil.

18th

cent.

Egypt Europe India China

Note that we have employed here the maximum estimate of Egyptian population in 1775 (7 million) rather than the minimum one (6 million) arrived at in Chap-ter 1. Yet, the contrast between upward demographic dynamics of Egypt, on the one hand, and all the major Old World regions outside the Middle East remains extremely sharp. Within the period in question the population of Egypt grew not more than 50% (and it may well have grown less than 35% if the minimum es-timate is correct), whereas in the major regions of Eurasia (outside the Middle East) the population grew 5 times or more. We believe this phenomenon could be at least partly accounted for by the mechanisms of political-demographic cy-cles typical for Egypt (and apparently the other countries of the Islamic Middle East) in comparison with the ones typical for the other major regions of Eurasia.

Chapter 2. Medieval Egyptian Political-Demographic Cycles

47

The first feature that cannot avoid the attention of a student of medieval Egyptian political-demographic cycles is that those cycles are very short. Let us compare their length, for example, with the one of the country whose political-demographic dynamics are known best of all (see, e.g., Korotayev, Malkov, and Khaltourina 2006b), those of China (see Table 2.1 and Diagrams 2.3–4):

Table 2.1. Egyptian and Chinese Political-Demographic Cycles Compared

EGYPT Cycle Name Beginning of

growth phase (CE)

Beginning of demo-graphic collapse (CE)

Growth phase length (years)

`Umayyid Cycle 642 737 95

`Abbasid Cycle 740 830 90

Tulunid-Ikhshidid Cycle 868 968 100

1st Fatimid Cycle 970 1065 95

2nd Fatimid (al-Jamali) Cy-cle

1073 1143 70

1st Ayyubid "Cycle" 1171 1195 24

2nd Ayyubid "Cycle" 1218 1243 25

Bahri Cycle 1250 1347 97

1st Ottoman Cycle 1525 1601 76

2nd Ottoman Cycle 1737 1784 47

Average 72

Average, excluding "Ayyubid" cycles 84

CHINA Cycle Name Beginning of

growth phase (CE)

Beginning of demo-graphic collapse (CE)

Growth phase length (years)

East Han Cycle 57 188 131

Early T'ang Cycle 650 754 104

Late T'ang "Cycle"1 786 870 84

Sung Cycle 960 1120 160

Yüan Cycle 1280 1355 75

Ming Cycle 1368 1626 258

Qing Cycle 1680 1852 172

Average 141

Average, excluding the Late T'ang "cycle" 151.5

DATA SOURCES: China – see Korotayev, Malkov, and Khaltourina 2005: 177–227; 2006b; Egypt – Nefedov 1999b, 2003; Atsamba and Kirillina 1996; Zelenev 1999; Semenova 1974;

1 It is not entirely clear if the Late T'ang period should not be regarded as a part of the T'ang – Sung

intercycle rather than a separate cycle (e.g., Fairbank 1992: 86; Korotayev, Malkov, and Khaltou-rina 2005: 185; 2006b). The Ayyubid "cycles" might be also regarded as parts of the Fatimid-Bahri intercycle rather than separate cycles.


48

Ayyūb 1997; Crecelius 1998; Ibrāhīm 1998; Sāmī Bāshā 2002–2004, 2: 8–112; Harīdī 2005: 243–84.

Diagram 2.3. Egyptian and Chinese Political-Demographic Cycles Compared (years)


49

Diagram 2.4. Average Length of Egyptian and Chinese Political-Demographic Cycles Compared (years)

84

151.5

0

20

40

60

80

100

120

140

160

Egypt China

Thus, as we see, the average length of the Chinese political-demographic cycles was almost twice as long as the one of the medieval Egyptian cycles. Note that the medieval European political-demographic cycles were even longer than those of China (see, e.g., Turchin 2003, 2005b and Diagram 2.5 below).

Diagram 2.5. Population Dynamics of Italy, 1000–1750 CE (in millions)

0

2

4

6

8

10

12

14

16

1000 1100 1200 1300 1400 1500 1600 1700

SOURCE: McEvedy and Jones 1978: 106–9.


50

However, to our mind, there is an even more striking characteristic feature of the medieval Egyptian political-demographic cycles. As was mentioned earlier (Korotayev, Malkov, and Khaltourina 2006b), the main study of the political-demographic cycles in pre-Modern Asia and North Africa was done by Nefedov. We have practically no long-term population data outside China (and, to some extent, Europe), and this made it difficult to detect demographic cycles outside Europe and China. However, we can sometimes find long-term data on some other variables whose dynamics is predicted by Nefedov's model (espe-cially per capita consumption rates), and quite regularly they have the form pre-dicted by Nefedov's model of political-demographic cycles. Using such indirect data, as well as his system of qualitative indicators of various phases of demo-graphic cycles Nefedov (1999a, 1999b, 1999c, 1999d, 1999e, 2000a, 2000b, 2001a, 2001b, 2002a, 2002b, 2003, 2004, 2005 etc.) has managed to detect more than 40 demographic cycles in the history of various ancient and medieval societies of Eurasia and North Africa, thus demonstrating that the demographic cycles are not just specific for Chinese and European history, but should be re-garded as a general feature of complex agrarian system dynamics (see Korota-yev, Malkov, and Khaltourina 2006b: Chapter 1 for more detail). The main quantitative data that were used by Nefedov in order to detect the political-demographic cycles are those that deal with real daily wages of un-skilled workers. Here he did find a very interesting pattern quite consistent with the political-demographic cycle models (see Diagrams 2.6–7):

Diagram 2.6. Consumption Level Dynamics in Babylonia, the 6th –

early 5th Centuries BCE. Figures refer to amounts of

barley in liters that an unskilled worker could buy with his daily wage (Nefedov 2003: Fig. 4)


51

Diagram 2.7. Consumption Level Dynamics in Northern India, the late 16

th – 17

th Centuries CE. Figures refer to amounts of

wheat in liters that an unskilled worker could buy with his daily wage (Nefedov 2003: Fig. 12)

As we see, at the beginning of political-demographic cycles, when population is well-provided with resources, the real wages of unskilled workers tend to be relatively high. However, with increasing population essential resources become more and more scarce, and per capita consumption shrinks, eventually launch-ing a political-demographic collapse. Thus, real daily wages are used quite ap-propriately by Nefedov as an indicator of a relative overpopulation (the lower the real wages are the higher is the relative overpopulation level). Against this background it is rather interesting to compare the real wages found at the brink of demographic collapses in medieval Egypt, on the one hand, and outside it (both in space and time), on the other (see Table 2.2 and Diagram 2.8):


52

Table 2.2. Real Wages at the Beginning and at the End of Medieval Egyptian and other Political-Demographic Cycles

Medieval Egypt

Cycle Name Real daily wages of an unskilled worker (liters of wheat)

At the begin-ning of demo-graphic cycle

Years At the brink of a demo-graphic col-

lapse

Years

`Abbasid Cycle 15 743 4.3 early 9th cent

Tulunid-Ikhshidid Cycle 40 after 868 4 954–62

1st Fatimid Cycle 14 970s 4.9 1053

2nd Fatimid (al-Jamali) Cy-cle

22 1097 8 1st half of 12th cent.

1st Ayyubid "Cycle" 18 1172 12.5 1180s

2nd Ayyubid "Cycle" 30 1203–18 16 1232

Bahri Cycle 29 1293–96 5.1 1341–43

Average 24 7.8

Average, excluding "Ayyu-bid" cycles

24 5.25

OTHER

Cycle Name, Country Real daily wages of an unskilled worker Unit of measurement At the begin-

ning of demo-graphic cycle

Years At the brink of a demo-graphic col-

lapse

Years

Qing Cycle, China 3.2 1730–50 1.7 1800–20 liters of rice/day

Moghol Cycle, India 4.6 1595 2.5 1670–90 liters of wheat/day

New Babylonian Cycle, Babylonia

7.7 605–562 BCE

2.0 522–486 BCE

liters of bar-ley/day

1st Ptolemaic Cycle, Egypt 5.3 c.270 BCE

1.7 c.210 BCE

liters of wheat/day

2nd Ptolemaic Cycle, Egypt 7.2 c.140 BCE

2.6 c.60 BCE liters of wheat/day

Roman Cycle, Egypt 6.0 c.1 CE 1.7 2nd half of the 2nd

cent.

liters of wheat/day

Late Abbasid Cycle, Iraq 12.2 1132–66 5.7 1243 liters of wheat/day

Average 6.6 2.6 liters of grain/day

Average, excluding the Ab-basid Iraqi cycle

5.7 2.0 liters of grain/day

DATA SOURCES: Nefedov 1999a, 1999b, 1999d, 1999e, 2000a, 2001b, 2003.


53

Diagram 2.8. Real wages at the brink of demographic collapses (liters of standard grain)


54

As we see, at the brink of demographic collapses in medieval Egypt the real wages of the unskilled workers were on average more than twice those in other places and times. The only exception here is one that only confirms the rule, medieval Iraq. This suggests that here we might be dealing with some feature that was characteristic for medieval Middle Eastern political-demographic dy-namics. The same striking contrast is observed if we compare real wages at the beginning of political-demographic cycles (see Diagram 2.9):

Diagram 2.9. Real wages at the brink of demographic collapses (liters of standard grain)

However, we shall get especially telling results if we compare the medieval Egyptian pre-collapse real wages with the "beginning-of-cycle" ones in other places and times (see Diagram 2.10):


55

Diagram 2.10. Comparison of the Medieval Egyptian pre-collapse real wages with the "beginning-of-cycle" ones in other places and times (liters of standard grain per day)


56

As we see at the brink of demographic collapses medieval Egyptian real wages were not so different from the ones observed in other places and times at the ini-tial, the most prosperous phases of the demographic cycles, when the highest consumption levels (and real wages) were usually observed.

After Nefedov, if we consider medieval real wages as an indicator of under-population, the results obtained suggest that medieval Egypt suffered from un-derpopulation rather than overpopulation, and that the medieval Egyptian popu-lation fluctuated well below the carrying capacity level. Incidentally, this seems to account (at least partly) for an apparent contradiction between the agricultur-al history data indicating a significant increase in the carrying capacity of land between the 2

nd and 18

th centuries and a relatively insignificant demographic

growth. The point is that when comparing the 2nd

century population with that of the 18

th century, we are comparing a population that had reached its carrying

capacity ceiling with the one that was significantly below it. One has the strong-est possible doubts that in the 60 years, preceding 1784, that were relatively free from great plagues and famines, the Egyptian population had enough time to fill the ecological niche up to saturation after the 17

th century demographic collapse

(Ibrāhīm 1998; Crecelius 1998; Hathaway 1998; Sāmī Bāshā 2002–2004, 2: 32–112; Harīdī 2005: 243–84). In general the relatively high real wages attested to the end of medieval Egyptian political demographic cycles and their general-ly short duration seem to be well connected; during medieval political-demographic cycles the Egyptian population simply had not enough time to reach the carrying capacity of land ceiling (which would lead to the real wages dropping to really low levels). In general, the analyzed data suggest that in medieval Egypt political-demographic collapses took place well before the population reached the carry-ing capacity ceiling. Thus the political-demographic cycle models that connect demographic collapses with the ecological niche saturation and that describe ra-ther well political-demographic dynamics of pre-Modern China (e.g., Usher 1989; Chu and Lee 1994; Nefedov 2004; Korotayev, Malkov, and Khaltourina 2005: 221–7; 2006b), do not appear appropriate for medieval Egypt. This might not be a coincidence that the mathematical model that appears to describe the medieval Egyptian political-demographic best of all, is the one (Turchin 2003: 131–7) that was developed in an attempt to formulate in a math-ematical form some part of the theory of political-demographic dynamics pro-posed by a person who spent a substantial part of his life specifically in medie-val Egypt, ‛Abd al-Rahman ibn Khaldun (1332–1406) (see, e.g., Ibn Khaldun 1958; 2004; ‛Inān 1933; Mahdi 1937; Batsieva 1965; Ignatenko 1980; Alekseev and Khaltourina 2004; Anderson and Chase-Dunn 2005).

Chapter 3

‛Abd al-Raḥmān ibn Khaldūn's

Sociological Theory as a Methodological

Basis for Mathematical Modeling of Medieval

Egyptian Political-Demographic Dynamics

We believe it makes sense to start this chapter with a summary of those points of Ibn Khaldūn's sociopolitical theory, on which Turchin's (2003) model is based (and on which, to a considerable extent, our own versions of this model will also be based). 0. A central notion of Ibn Khaldūn's theory is ‛aṣabiyyah (عصبية), which will be interpreted below after Turchin (2003: 38–9) as "collective solidarity". 1. New dynasties can be established only by groups with a very high ‛aṣabiyyah (Ibn Khaldūn 1958, 1: 284–5, 313; 2: 119; 2004: 183–4, 201, 360). 2. After a new dynasty is established the ruling elite's ‛aṣabiyyah tends to progressively decrease (Ibn Khaldūn 1958, 1: 339–46; 2: 119–20; 2004: 217–22, 360–1). 3. This process is accounted for up to a considerable extent by the growing prestige consumption (taraf/ ترف , "luxury") of the ruling elite (Ibn Khaldūn 1958, 1: 338–341, 343–347, 353–355; 2: 90, 123, 125–126; 2004: 216–23, 226–8, 343, 360–3). 4. Within four generations the ruling elite's ‛aṣabiyyah decreases to such a critical level that it leads to the dynasty (and, hence, political system) break-down and the establishment of a new dynasty by a new high-‛aṣabiyyah group (Ibn Khaldūn 1958, 1: 278–82, 343–6; 2004: 180–1, 220–2). Note that this suggests that a typical length of a political-demographic cycle should be be-tween 80 and 100 years

1, which is indeed, extremely close to what we could see

above with respect to medieval Egyptian political-demographic cycles. Ibn Khaldūn also makes a number of other relevant observations and gener-alizations: 5. Growing prestige consumption of ruling elite leads to increasing burden of taxation on the commoners (Ibn Khaldūn 1958, 2: 90–1, 103–4, 109–11, 136–7; 2004: 343–4, 351–2, 354–5, 370).

1 If we estimate a generation length as 20–25 years. Note that Ibn Khaldūn himself estimated it as

40 years (1959, 1: 344, 346; 2004: 220–2).


58

6. However, over-taxation undermines the economy; as a result, the elites turn out to be unable to increase their revenues to a degree that would satisfy their growing demands (Ibn Khaldūn 1958, 1: 340–1; 2: 90–1, 103–4, 109–11, 136–7; 2004: 218–9, 343–4, 351–2, 354–5, 370). Ibn Khaldūn connects this process directly to the dynastic decline, explaining that when the elite consump-tion grows faster than the state revenues, this would undermine its military force (Ibn Khaldūn 1958, 1: 340–1; 2004: 218–9). Turchin describes another way in which this process would contribute to the dynastic decline: it would lead to in-creased competition within the ruling elite, thus destroying the elite's collective solidarity (‛aṣabiyyah) (Turchin 2003: 132–6). 7. During early phases of political cycles a very high rate of elite reproduc-tion is observed: "…A tribe that obtained royal authority and luxury is prolific and produces many children, and the (elite) community grows" (Ibn Khaldūn 1958, 1: 351; 2004: 225). Ibn Khaldūn did not study a link between this phe-nomenon and the process of dynastic decline; however, Turchin argues in a very convincing way that it should have been a very important factor, as a very high rate of elite reproduction would lead to a further increase of amount of re-sources needed to satisfy the growing prestige consumption demands of the elite, thus increasing both the competition within the ruling elite and the over-taxation. These points describe what could be called a theory of political cycles, but it could hardly be denoted as a theory of political-demographic cycles. However, Ibn Khaldūn suggests a number of other observations and generalizations, which in conjunction with points 1–7 produce a veritable theory of political-demographic cycles: 8. Low taxation and political order observed at the early phases of dynastic cycles result in overall economic and demographic growth leading to a signifi-cant increase in numbers of not only elite members, but commoners as well (Ibn Khaldūn 1958, 2: 135; 2004: 369). 9. Overtaxation and rebellions observed during the late phases of dynastic cycles lead to the destruction of economy, famines and epidemics (directly con-nected with famine, state breakdown, as well as relative overpopulation) and, thus, to the demographic collapse affecting not only elites, but commoners as well (Ibn Khaldūn 1958, 2: 103–4, 109–11, 136–137; 2004: 351–2, 354–5, 370). Turchin (2003) developed two models that he denoted by himself as "Ibn Khaldūn models", within which political-demographic collapses are produced not by actual overpopulation, but rather by elite overpopulation, elite overpro-duction that can well take place in a generally underpopulated country (or at least in a country whose population is still significantly below the saturation level), and thus suggests direction within which the political-demographic dy-namics of medieval Egypt could be adequately described. The first "Ibn Khaldun model" is described by Turchin in the following way:

Chapter 3. Ibn Khaldun's Sociological Theory

59

"The dynamics of the Ibn Khaldunian “world-system” are determined by the interaction between the civilized society and the desert tribes. The civilized region is the site of re-current state building/collapse episodes. It is inhabited by an indigenous commoner population, who provide the productive basis of the society. The desert is inhabited by stateless tribes, who periodically conquer the civilized region and establish a ruling dyn-asty there. Desert tribes, thus, supply the elites (nobility) for the civilized state. In the model I assume that the dynamics of the commoner population are largely disconnected from the elite dynamics. Dynasties come and go, but peasants and merchants continue to grow food, trade, and pay taxes to whichever government is currently in power. Thus, the amount of resources extracted from commoners is a constant, R. During the early years of the dynasty, the extracted resources are divided in two parts: taxes to support the government, γR, and rents to support the elites, (1 − γ)R. The parameter γ, the pro-portion of resources going to the state, is assumed to be a constant. The income per no-ble is therefore μ = (1 − γ) R/E, where E is the current number of nobles. I now intro-duce two other parameters. Let μ0 be the per capita income that is necessary to maintain and replace exactly one noble. In other words, when per capita income falls below μ0, the elite numbers will decline, while if μ > μ0, then the elite numbers will increase. The second parameter is μmin, the per capita income that nobles consider to be the minimum that accords with their station. This “minimal acceptable income” is determined socially and can vary between societies. In general, however, μmin ≥ μ0 since it is unlikely that nobles would consider acceptable an income on which they cannot afford to perpetuate their family line to the next generation. Ibn Khaldun argued that with time former tribesmen forget the rude ways of the desert, and subsequent generations grow accus-tomed to ever-increasing luxury. Thus, μmin is a variable that starts at some low level (e.g., μ0) at the beginning of the dynasty and then increases at a certain rate (e.g., δμ per year). As I stated above, elite numbers increase when per capita income μ is greater than μ0. The rate of elite population growth is proportional to the difference μ − μ0, subject to not exceeding the maximum rate of increase, rmax. Because the model is tailored to a specific society, in which the elites practiced polygamy, I set rmax = 0.08 yr−1 at four times the intrinsic rate of population increase typical for preindustrial populations. My justification for this assumption is that the legal maximum of wives that a Muslim man could have is four. Of course, in Islamic societies many high-rank individuals would have large numbers of concubines and would usually acknowledge their sons as legiti-mate heirs. This practice would lead to a much higher rate of population growth. On the other hand, poorer members of the elite might be unable to afford the full complement of legal wives. On balance, I believe that a multiplier of four is a reasonable one to choose as the reference value for Islamic societies (as is usual in theory building, we will need to determine how variation in this and other parameters affect the model predic-tions). The next key assumption of the model is that as long as per capita income gener-ated from rents exceeds the minimum acceptable income, (1 − γ) R/E > μmin, the state and elites live in harmony. However, if elite numbers grow to the point where their per capita incomes fall below μmin, then nobles become dissatisfied, and will use a variety of usual techniques to divert some of the taxes into their pockets… The model makes the assumption that at this point μ = μmin, that is, the elites steal just enough from the state to be able to maintain what they perceive as the minimal standard of living appropriate to their station. If elite numbers grow to the point where R/E < μmin, that is, there is not enough extracted resource to satisfy all the nobles, even if the state gets no taxes, then the model assumes that μ = R/E. In other words, at this point the nobles divide all ex-


60

tracted resource among themselves, and the state gets nothing. The state fiscal dynamics are modeled as usual, with revenues as described above, and the expenditures propor-tional to the elite numbers. Thus, the dynamics of S, the accumulated state resources, follows the typical trajectory, in which S grows during the early period of the dynasty, because elite numbers are few and their appetites are modest. At some point, however, the revenues drop to the point where they cannot match expenditures, and S declines and eventually become 0. At this point, the model assumes that the dynasty failed. The state becomes vulnerable to conquest, which (at least in the model) happens immediately, be-cause the desert tribes provide a ready and spatially adjacent source of the next dynasty. A typical trajectory predicted by the model is illustrated in Figure 7.4a.:

[Diagram 3.1. Dynamics of the Ibn Khaldun – Turchin model: elite numbers, E (solid line) and accumulated state surplus S (broken line) (=Turchin 2003:134, Fig. 7.4a) ]


61

Note that I chose to frame this model in economic (or fiscal) terms. However, we can easily cast the model in terms of collective solidarity (asabiya). For example, we can re-interpret S as the asabiya of the elites supporting the current dynasty (the conquering tribesmen and their descendants). Then, at the beginning of the dynasty, asabiya is high,2 and it stays high until elite numbers reach the threshold where per capita income falls below the minimal acceptable level. At that point, intraelite competition intensifies, and the asabiya begins to decline. When asabiya reaches a certain threshold, the dynasty (and the state) collapses, and is replaced by a new tribal group from the desert. The dy-namics of such a model are essentially identical to those shown in Figure 7.4a. A numer-ical investigation of the effect of parameter values on the dynamics of the Ibn Khaldun model indicates that the main parameters that affect the period of the cycle are the max-imum rate of elite population increase (rmax) and the rate at which the minimal accepta-ble income grows with time. Rather rapid cycles of about one century in period, shown in Figure 7.4a, obtain for high values of rmax that should be typical for societies where elite polygamy is widespread. By contrast, reducing rmax to 0.02 leads to longer cycles, of around 1.5 centuries" (Turchin 2003: 132–4).

Note that the cycle length generated by the first Turchin model turns out to be astonishingly close to the observed average length of the medieval Egyptian po-litical-demographic cycles. However, the first model does not describe ade-quately the commoner (and, hence, overall) population, counterfactually assum-ing it to be constant. Turchin acknowledges this and develops the second "Ibn Khaldun" model aimed at the description of demographic dynamics of both the elite and commoner population (and, hence, not only political, but also veritable political-demographic cycles): "Now that we have some understanding of the dynamics predicted by the Ibn Khaldun

model, it would be a good idea to investigate the effects of some of the simplifying as-

sumptions we employed to derive the model. Probably the most drastic simplification

was the assumption that we can neglect the commoner dynamics. To check on the validi-

ty of this approximation, I developed a more complex Ibn Khaldun model with class

structure. This model predicts essentially the same dynamics for E and S as the simple

Ibn Khaldun model (Figure 7.5a):

2 One still has to note that according to Ibn Khaldūn a dynasty's ‛aṣabiyyah has the maximum val-

ue at the moment of its founding (Ibn Khaldūn 1958, 1:284–5, 313; 2:119; 2004:183–4, 201, 360), whereas within the reinterpreted Turchin model it has the minimum (0) value at this mo-ment.


62

[Diagram 3.2. Dynamics of the Ibn Khaldun – Turchin model with class structure. E: elite numbers, S: accumulated state surplus, and P: com-moner numbers. (a) Dynamics with high elite reproduction rate and lim-ited extraction ability. (b) Dynamics with low elite reproduction rate and high extraction ability (=Turchin 2003: 135, Fig. 7.5)]

Additionally, it reveals the commoner dynamics during the elite cycle. We see that the growth of the commoner population is slowed by periodic state breakdown, but this ef-fect is slight. Numerical investigation of the model parameters indicates that commoner population declines are more pronounced when we increase the elite extraction ability (by increasing the parameter , the maximum proportion of commoner production that can be extracted by elites). Additionally, the cycle period is lengthened and the ampli-tude of commoner oscillation is increased when we decrease the per capita rate of elite reproduction. This happens because longer cycles mean longer periods of very intense exploitation of commoners by elites, resulting in a greater decline of commoner popula-


63

tions by the end of the cycle. A sample trajectory for = 1 and rmax = 0.02 is shown in Figure 7.5b" (Turchin 2003: 134–6).3

The dynamics produced by the second Turchin – "Ibn Khaldun" model are quite close to those actually observed, and we believe this model should be regarded as a very important step towards the adequate explanation of the medieval Egyptian political-demographic pattern. A very significant achievement of Turchin is that basing himself on some of Ibn Khaldun's suggestions he man-aged to describe mathematically how the population could experience 90-year fluctuations below the carrying capacity of land level. The whole idea that this should be connected with the elite dynamics and elite (rather than commoner) overpopulation appears very productive and promising.

However, there are still some problems with this model.4 According to this

model the commoner population5 starts declining about the middle of the politi-

cal-demographic cycle and declines more or less gradually up to the end of the cycle. The actual situation is quite different.

The data analyzed by Nefedov suggest that the commoner population tended to grow (though with slowing speed) almost up to the very end of a cycle and tended to decline in a rather precipitous way at the very end of the cycle (see Diagrams 3.3 and 3.4):

Diagram 3.3. Unqualified Worker Real Wage Dynamics (in liters of wheat per day) in Egypt, the 8

th – 9

th centuries (Nefedov 2003: Fig. 8)

NOTE: triangles denote demographic collapses.

3 For the mathematical description of the model see Turchin 2003: 212–3. 4 Note that they could be hardly considered as its true defects, as the model was designed as the

most basic one, and was not aimed to describe the actual political-demographic dynamics of me-dieval Middle Eastern polities.

5 And taking into consideration the fact that the elite constituted a tiny fraction of the overall popu-lation, the same can be said about the overall population as well.


64

Diagram 3.4. Unqualified Worker Real Wage Dynamics (in liters of wheat per day) in Egypt, the 11

th – 14

th centuries (Nefedov 1999b: Fig. 5)

NOTE: triangles mark catastrophic famines.

Thus, for example, by the end of the Ikhshidid rule, in 954–962, the average re-al daily wages of the unskilled workers dropped to an equivalent of 4 liters of wheat. After the political-demographic collapse of 968–969, the real daily wag-es rose to an equivalent of 14 liters, which appears to indicate an extreme de-gree of labor shortage and, hence, a very rapid depopulation. On the other hand, Nefedov's data suggest that the population tended to grow up to the end of the political-demographic cycles, though its growth tended to slow down signifi-cantly at the last (pre-collapse) phases of those cycles. According to the second Turchin's model, commoners depopulation started well before the elite demo-graphic collapse. Historical data suggest that the political system collapses (and the old elite "depopulation") tended rather to occur very shortly after the com-moner depopulation (see, e.g.: Bianquis 1998; Crecelius 1998; Sanders 1998). Finally, in Turchin's model which generates political-demographic cycles with the length that is typical for medieval Egypt we do not see actually observed characteristic demographic collapses, whereas the version generating such col-lapses is characterized by the cycle length that is not typical for medieval Egypt.

Chapter 4

Basic Model of Medieval Egyptian

Political-Demographic Dynamics

Although Turchin's models were inspired by Ibn Khaldūn's treatise, in the pro-cess of the model development Turchin moved rather far from the original theo-ry of Ibn Khaldūn and less of it survived in the final versions of his models. We believe that in order to produce a mathematical model describing medieval po-litical-demographic dynamics more accurately we should try to follow Ibn Khaldūn's theory more closely. Note in particular that Turchin abstained from modeling the part of this theory described in the previous chapter under number 9.

We think it makes sense to reproduce at this point a full quotation of that part of al-Muqqadimah where the mechanisms of the political-demographic col-lapses are described by Ibn Khaldūn in a most clear and succinct way:

"In the later (years) of dynasties, famines and pestilences become numerous. As far as famines are concerned, the reason is that most people at that time refrain from cultivat-ing the soil. For, in the later (years) of dynasties, there occur attacks on property and tax revenue and, through customs duties, on trading. Or, trouble occurs as the result of the unrest of the subjects and the great number of rebels (who are provoked) by the senility of the dynasty to rebel. Therefore, as a rule, little grain is stored. The grain and harvest situation is not always good and stable from year to year. The amount of rainfall in the world differs by nature. The rainfall may be strong or weak, little or much. Grain, fruits, and (the amount of) milk given by animals varies correspondingly. Still, for their food requirements, people put their trust in what it is possible to store. If nothing is stored, people must expect famines. The price of grain rises. Indigent people are unable to buy any and perish. If for some years nothing is stored, hunger will be general. The large number of pestilences has its reason in the large number of famines just mentioned. Or, it has its reason in the many disturbances that result from the disintegration of the dynas-ty. There is much unrest and bloodshed, and plagues occur" (Ibn Khaldūn 1958, 2: 136; 2004: 370).

This text gives us very clear and convincing advice on how the model under consideration should be amended so that it could describe medieval Egyptian political-demographic dynamics more accurately. In fact Ibn Khaldūn suggests that we should take into account the effect of climatic fluctuations. Needless to say, this factor is especially relevant for the "pre-Aswan Dam" Egypt, where ag-ricultural yields depended largely on the levels of the Nile inundations which


66

are totally unpredictable, and vary with an extremely significant magnitude (see, e.g., Park 1992). Fundamentally, this means that people have no way to predict the forthcoming year's yield, but they know that sooner or later the Nile will not rise to the necessary level, resulting in a catastrophic crop failure. Within such a context, the storage of sufficient amounts of food turns out to be essential to prevent demographic collapse, and indeed the shrinking of food reserves under the influence of over-taxation (caused in its turn by elite over-population and over-consumption) could be an extremely important cause of such collapses.

In fact, the influence of this factor on the demographic cycle dynamics has already been modeled by Nefedov (2002a, 2004; for the analysis of this model see Korotayev, Malkov, and Khaltourina 2006b). However, this model does not describe the actual political-demographic dynamics of medieval Egypt, as Nefedov does not take into consideration the effects of elite overpopulation that were modeled so convincingly by Turchin.

Thus, in order to produce a "more Khaldunian" model of the medieval Egyptian political-demographic dynamics it appears necessary to combine re-spective elements of Turchin's and Nefedov's models into one model.

In our model the annual production (Y) is measured in minimum annual ra-tions (MARs)

1. Thus, Yi, the production in year i, can be also regarded as Ki, the

carrying capacity of land for this year, the maximum number of people that this annual product can support.

In our first model we assume that technological base (the technologically de-termined production per unit of cultivated land) and cultivated area remain con-stant, thus, Yi, the production in year i, is determined entirely by climatic condi-tions in this year.

In this model the annual production in most years fluctuates in the range that can potentially support 6.0–8.0 million people. However, once in every 10 years catastrophic crop failures are randomly observed, when the annual product turns out to be sufficient to support (even at the bare survival level) only 2.5–3.0 mil-lion people.

2

In this model the problem of population survival in the "lean" years is solved in the following way. In a "good" year i the state collects as taxes (Ti) half of "surplus", Yi – Pсi-1, that is, the yield produced over the annual amount that is absolutely necessary for bare survival of commoner population found at the beginning of year i (Pсi-1) at zero reproduction level (in the "lean" years the commoners are assumed not to pay taxes). At the beginning of a cycle a quarter of the remaining surplus is stored by the population in order to procure its sur-vival through forthcoming "lean" years.

1 Minimum annual ration (MAR) is an amount of product that is barely sufficient to support a sur-

vival of one person for one year. 2 Technically this was done in the following way: a random number generator generated yield val-

ues in the range 6–8 million. The minimum value within a given decade was automatically trans-formed in a random number in 2.5–3 million range.

Chapter 4. Basic Model of Medieval Egyptian Dynamics

67

Thus, the amount of these counter-famine reserves (R) at the end of year i (if this year is not "lean") will be equal to

Ri = Ri-1 + dRi,

where Ri-1 denotes food reserves by the end of the previous year, and dRi de-notes resources added to these reserves in year i. In good years of the cycle be-ginning this variable will equal

dRi = 0.25(Yi – Pсi-1 – Ti),

whereas the amount of collected taxes will be

Ti = 0.5(Yi – Pсi-1).

Thus, in good years of the cycle beginning

dRi = 0.25(Yi – Pсi-1 – Ti) = 0.25(Yi – Pсi-1 – 0.5(Yi – Pсi-1)) = = 0.25(Yi – Pсi-1) – 0.125(Yi – Pсi-1) = 0.125(Yi – Pсi-1),

thus, the population stores 12.5% (i.e., one eighth) of all the surplus. This slows down the commoner population growth in good years. In general,

the growth of commoner population is described by the Verhulst population dy-namics equation (Verhulst 1838; Riznichenko 2002; Korotayev, Malkov, and Khaltourina 2005; 2006a):

dP/dt = r(1 – P/K)P,

where dP/dt is the absolute population growth rate, K is carrying capacity, and r is relative population growth rate when the resource limitations are totally ab-sent. With Turchin (2003) we assume this rate to be equal to 2% a year (i.e., 0.02 year

-1).

In order to determine the value of the commoner population growth rate in year i the main importance for us is associated not with the potential carrying capacity of land, К, but with the actual carrying capacity in this particular year, Каi. Thus, we are interested not in how many people this year's yield could gen-erally support, but in how many commoners it could support within the concrete sociopolitical context of this year. We should know the amount of resources that is actually available for the commoners' consumption, that is this year's yield minus taxes and those resources moved to food reserves. Thus the actual carry-ing capacity in year i will be equal (at least in good years in the beginning of the cycle) to

Каi.= Yi – dRi – Ti,

whereas the commoner population by the end of this year (Pсi) will be equal to

Pсi = Pсi-1 + 0.02(1 – Pсi-1/Kai)Pсi-1.


68

On the other hand, the food storage in good years helps to prevent demographic collapses in lean years. For this reason in lean years (if sufficient food reserves are available) a part of them, Pсi-1 – Yi (note that in lean years this difference has a positive value), is used to prevent the decrease of commoner population. Thus, when (in the first phase of a cycle) adequate reserves are available, the actual carrying capacity in lean years will be equal to

Каi.= Yi + (Pсi-1 – Yi) = Pсi-1.

As a result, by the end of lean year i the commoner population will be equal to

Pсi = Pсi-1 + 0.02(1 – Pсi-1/Kai)Pсi-1 = Pсi-1 + 0.02(1 – Pсi-1/Pсi-1)Pсi-1 = = Pсi-1 + 0.02(1 – 1)Pсi-1 = Pсi-1 + 0 = Pсi-1,

that is, in a lean year within the first phase of a cycle the commoner population does not grow, but it does not decrease either (due to the presence of adequate food reserves).

However, this is not the only purpose for which the reserve resources are used in the lean years, as a part of elite's demands could be also covered from this source.

In good years of the first phase of a cycle the collected taxed are divided in-to two parts. One part (Сei) is spent to cover the elite consumption demands, whereas the other part (dSi) is accumulated in the state treasury (S).

The elite consumption in year i (Cei) is equal to

Cei = PeiBi,

where Pei is the elite population in year i, and Вi is an average per capita elite consumption in year i.

Basically, the dynamics of this variable follow the assumptions of Ibn Khal-dūn and Turchin. Thus, the average per capita elite consumption is assumed to grow linearly throughout a dynastic cycle:

Bi = Bi-1 + dB, dB = const.

With Turchin we assume that the elite population grows with a faster speed due to general polygyny practiced by the elites, though in our basic "medieval Egyp-tian" model the elite population growth rate is more modest (4% a year) than in Turchin's model (8% a year). Thus,

Pei = Pei-1 + 0.04Pei-1.

At the first phase of a cycle the elite consumption is provided with resources collected as taxes; as a result, the total amount of the tax revenue is divided into two parts:


69

Ti = Cei + dSi,

where dSi is a part of taxes moved to the state treasury. Thus, the amount of re-sources accumulated in the state treasury by the end of year i in good years of the first phase of a cycle will be equal to

Si = Si-1 + dSi = Si-1 + (Ti – Cei).

If in the given year the amount of resources needed to cover the constantly growing elite demands (Cei) exceeds the tax revenues of this year (Ti), no transfers to the state treasury are made (dSi = 0, that is, all the collected taxes are consumed by the elites), whereas the deficient resources (Сei – Ti) are extracted (in the form of various extortions, requisitions, and "illegitimate taxes"/mukūs [مكوس], as so vividly described by Ibn Khaldūn [1958, 2: 103–4, 109–11, 136–7; 2004: 351–2, 354–5, 370]) from the surplus produced by the commoners. Hence, at the second phase of the cycle (when the elite demands start to exceed systematically the amount of tax revenue) in good years (with Yi > Pi-1) the actual carrying capacity of land will be equal to

Каi.= Yi – dRi – Ti – (Сеi – Ti) = Yi – dRi – Сеi.

Respectively the amount of resources accumulated in counter-famine reserves (R) in good (Yi – Pсi-1 – Cei > 0) years of this phase of the cycle will be equal to

dRi = 0.25(Yi – Pi-1 – Ti – (Сеi – Ti)) = 0.25(Yi – Pi-1 – Сеi).

During the third phase of a cycle the growth of the prestige consumption of the elites leads to the situation when even in good years all the produced surplus turns out to be insufficient to cover the demands of numerous elites addicted to luxury. Now even in good years the elites extract the entire surplus produced by the commoners, the resources stop being accumulated in the counter-famine re-serves; what is more, if the entire surplus extracted by the elites from the com-moner turns out to be insufficient to cover the elite demands, the deficient re-sources are taken from the anti-famine reserves. Thus, with Cei > Yi – Pсi-1 even in relatively good years (Yi > Pci-1) the counter-famine reserves will not grow; what is more they will decrease by the amount that is equal to Cei – (Yi – Pci-1), thus the variable dRi will have a negative value

dRi = Yi – Pi-1 – Cei.

As in such relatively good years (with Yi > Pi-1) of the third phase all the surplus (Yi – Pi-1) will be extracted from the commoners, consequently, the actual carry-ing capacity of land in these years will be equal to

Каi.= Yi – (Yi – Pci-1) = Pci-1.

Hence, the commoner population stops growing even in relatively good years. In the lean years the anti-famine reserves are used both to prevent the decrease


70

of commoner population and to cover the elite demands. These reserves are used more and more to cover elite demands even in good years. As a result, these reserves are depleted rather quickly. After this the elites start robbing the state treasury, and the fourth and final phase of dynastic (= political-demographic) cycle begins.

During this phase even in the best years the demands of more and more nu-merous and luxury-addicted elites are not covered by surplus produced by the commoners. At this phase the counter-famine reserves are totally depleted (and in full accordance with Ibn Khaldūn's theory they are not replenished). The de-ficient resources are found by the elites in the state treasury that is also depleted in a rather rapid way. As in the "Khaldunian" models of Peter Turchin when the state treasury is entirely depleted, the state breaks down and (in the following year) the country falls under the control of a new relatively small, "high-as ", not corrupted by luxury elite, which is modeled by ascribing to variables Pe and B their initial values. Thus,

if Si-1 = 0, Pei = Pe1 and Bi = B1.

However, within the model even before the dynastic collapse a demographic collapse takes place. Indeed, during the 4

th phase the elites systematically ex-

tract the entire surplus produced by the commoners, the counter-famine reserves are absent, and the commoner population turns out to be totally defenseless in the face of any forthcoming catastrophic crop failure. Thus, if the given annual yield is only sufficient to support, say, 2.5 million commoners, while the actual commoner population is 4 million, the commoner population will decrease pre-cisely to 2.5 million.

In general, our basic mathematical model of medieval Egyptian political-demographic cycles can be presented in the following form (see Table 4.1):

Table 4.1. Basic Mathematical Model of Medieval Egyptian Political-Demographic Cycles

Varia-

ble

symbol

This symbol denotes = Value in year i If Equation #

Yi

Yield produced in year i, measured as a number of people that can be potentially supported by it at zero reproduction level, or minimum an-nual rations (MARs)

=

random number, mostly fluctuating in the range of 6.0–8.0 mln., but once every 10 years randomly taking values be-tween 2.5 and 3 mln.

(4.1)

Pсi Commoner population at the end of year i

={

Pсi-1 + 0.02(1 – Pсi-1/Kai)Pсi-1

Kai ≥ Pсi-1 (4.2)

Kai Kai < Pсi-1 (4.3)


71

Varia-

ble

symbol

This symbol denotes = Value in year i If Equation #

Kai

Actual carrying capacity of land in year i, that is commoner population that this year pure yield (i.e., total yield minus transfers to counter-crisis reserves, as well as legal and illegal taxes and requisitions) could support at zero repro-duction level

={

Yi – dRi – Ti Yi ≥ Pсi-1 and Cei ≤ Ti (4.4)

Yi – dRi – Cei Yi ≥ Pсi-1 and Cei > Ti and Cei < Yi – Pсi-1

(4.5)

Pсi-1 Yi ≥ Pсi-1 and Cei ≥ Yi – Pсi-1

(4.6)

Yi + (Pсi-1 – Yi) = Pсi-

1 Yi < Pсi-1 and R > Pсi-1 – Yi

(4.7)

Yi Yi < Pсi-1 and Ri = 0

(4.8)

Ri Counter-famine reserves at the end of year i

={

Ri-1 + 0,25(Yi – Pi-1 – Ti)

Yi ≥ Pi-1 and Cei < Ti

(4.9)

Ri-1 + 0.25(Yi – Pсi-1 – Cei)

Yi ≥ Pi-1 and Cei ≥ Ti

(4.10)

Ri-1 – (Pсi-1 – Yi) – Cei

Yi < Pi-1 and Ri-1 > Pi-1 – Yi + Cei

(4.11)

0 Ri-1 < Pi-1 – Yi + Cei (4.12)

Cei Resources consumed by the elite population in year i

= PeiBi (4.13)

Pei Elite population in year i

={ Pei-1 + 0.04Pei-1 Si-1 > 0 (4.14)

Pe1 Si-1 = 0 (4.15)

Pi Overall population in year i

= Pci + Pei (4.16)

Bi Average per capita elite consumption in year i

={

Bi-1 + dB, dB = const

Si-1 > 0

(4.17)

B0 Si-1 = 0

(4.18)

Ti Amount of legitimate taxes collected in year i

={ 0.5(Yi – Pci-1) Pci-1 – Yi > 0 (4.19)

0 Pci-1 – Yi ≤ 0 (4.20)

Si Resources accumulated in the state treasury by the end of year i

={

Si-1 + Ti – Cei Cei ≤ Ti (4.21)

Si-1 Cei > Ti and Cei ≤ Ri-1 (4.22)

Si-1 – (Cei – Ri-1) Cei > Ti and Cei > Ri-1 and Cei < Si-1

(4.23)

0 Cei > Ti and Cei < Ri-1 and Cei ≥ Si-1

(4.24)

Typical dynamics generated by this model is represented at Diagram 4.1:


72

Diagram 4.1. Typical Dynamics Generated by the Basic Model of Medieval Egyptian Political-Demographic Cycles

(a) Dynamics of Overall Population (black curve, thousands) and Counter-Famine Reserves (grey curve, tens of thousands of MARs)

(b) Dynamics of Elite Population (black curve, thousands) and Re-

sources Accumulated by the State (grey curve, hundreds of thou-sands of MAR equivalents)

NOTE: this diagram reproduces results of a computer simulation with the following values of ini-tial conditions and parameters not mentioned in Table 4.1: Pс0 = 2500 (thousands); R0 = 0 (MARs);


73

S0 = 0 (MARs); Pe1 = 10 (thousands); В1 = 10 (MARs); dB = const = 0.2 (MAR/year). The initial value for the commoner population is given for year 0, whereas the one for the elite population is given for year 1, as our computer simulation starts at the very beginning of a cycle when, in ac-cordance to Ibn Khaldūn's theory the control over the country is seized by a new relatively small "high- " not corrupted by luxury elite that finds a commoner population decimated by the previous social-demographic collapse, absent counter-famine reserves and an empty state treasury. Thus, a new dynasty is founded in the country by the new elite in the year that is consid-ered within our model as the first year of the respective dynasty rule, whereas the decimated com-moner population, empty treasury and totally depleted counter-famine reserves are inherited by the new dynasty from the previous one, thus they come over to it from year 0 of the computer simula-tion.

This model provides a relatively adequate description of the political-demographic dynamics of medieval Egypt in all its basic characteristics (espe-cially up to 1347): political-demographic cycles of approximately 90-year length, the reduction of population growth during the last phases of a cycle, precipitous depopulations in the course of political-demographic collapses, and the absence of pronounced intercycles whereby a rather steady recovery growth starts immediately (or almost immediately) after the demographic collapse.

Special attention is deserved by the fact that, as in Turchin's extended mod-el, in our basic model the increase of the natural elite growth rate

3 leads to the

decline of the political-demographic cycle length, whereas its decrease results in the lengthening of those cycles. Thus, it turns out to be possible to produce the model version describing rather adequately the basic features of medieval Euro-pean political-demographic dynamics through the decrease by 4 times of the "Egyptian" natural elite growth rate coefficient (which would correspond to the strictly monogamous reproduction context typical for all the medieval European Christians, including the elites) (see Diagram 4.2):

3 In Table 4.1 it corresponds to coefficient 0.04 in equation (4.14).


74

Diagram 4.2. Typical Dynamics Generated by the Basic Model of Medieval European Political-Demographic Cycles

(a) Dynamics of Overall Population (black curve, thousands) and Coun-ter-Famine Reserves (grey curve, tens of thousands of MARs)

(b) Dynamics of Elite Population (black curve, thousands) and Re-

sources Accumulated by the State (grey curve, hundreds of thou-sands of MAR equivalents)

NOTE: this diagram reproduces results of a computer simulation with the values of initial condi-tions and parameters (including the ones for climatic fluctuations) that are entirely identical with the ones used above for the simulation whose results are reproduced at Diagram 4.1. The ONLY difference between the two simulations is that the value of natural annual elite population growth coefficient within the present simulation is equal not to 0.04 (as within the "Egyptian" version of the model), but is 4 times as small and equals 0.01.


75

As we see, this version of the model describes adequately the medieval European po-litical-demographic macrodynamics in its basic characteristics: c.200-year-long polit-ical-demographic cycles and pronounced intercycles (see, e.g., McEvedy and Jones 1978; Turchin 2005b). Special attention should be paid to the fact that within the "European" version of the model population systematically approaches the carrying capacity of the land much closer than in the "Egyptian" version. Thus, with respect to two computer simulations whose results have been preliminary discussed above, the situation looks as follows (see Table 4.2):

Table 4.2. Ecological Niche Saturation within "Egyptian" and "European" Versions of the Model

"Egyptian" ("polygynous"4) version "European" ("monogamous"5) version

Cy

cle

#

Sim

ula

tio

n y

ears

Max

imu

m p

op

ula

tio

n l

evel

rea

ched

du

rin

g

the

cycl

e (t

ho

usa

nd

s)

Av

erag

e ca

rry

ing

cap

aci

ty

(th

ou

san

ds

of

peo

ple

)

Max

imu

m l

evel

of

eco

log

ical

nic

he

satu

ra-

tio

n r

each

ed d

uri

ng

th

e cy

cle

Cy

cle

#

Sim

ula

tio

n y

ears

Max

imu

m p

op

ula

tio

n l

evel

rea

ched

du

rin

g

the

cycl

e (t

ho

usa

nd

s)

Av

erag

e ca

rry

ing

cap

acit

y

(th

ou

san

ds

of

peo

ple

)

Max

imu

m l

evel

of

eco

log

ical

nic

he

satu

ra-

tio

n r

each

ed d

uri

ng

th

e cy

cle

1. 1–94 3940 6663 59.1% 1. 1–228 4983 6663 74.8%

2. 95–186 3958 6663 59.4% 2. 229–452 5197 6663 78%

3. 187–278 4403 6663 66.1% 3. 453–679 5120 6663 76.8%

4. 279–374 3993 6663 59.9% 4. 680–910 5066 6663 76%

5. 375–465 4001 6663 60% 6663

6. 466–557 4292 6663 64.4% 6663

7. 558–650 4111 6663 61.7% 6663

8. 651–739 4225 6663 63.4% 6663

9. 740–830 4258 6663 63.9% 6663

10. 831–925 4063 6663 61% 6663

Average 4124 6663 61.9% Average 5091.5 6663 76.4%

4 The "Egyptian" version of our model of medieval political-demographic cycles is also denoted by

us as "polygynous", as the characteristic feature of this version consists in the extremely high rate of the natural growth of the elite population due to the extensive practice of polygyny by the elites.

5 The "Europian" version of our model of medieval political-demographic cycles is also denoted by us as "monogamous", as the characteristic feature of this version consists in the extremely low rate of the natural growth of the elite population due to the strict monogamy imposed by the Christian Church on all the population (including the elites).

Chapter 5

Extended Model of Medieval Egyptian

Political-Demographic Dynamics

One of the main simplifying assumptions of our basic model is that the techno-logically determined carrying capacity is considered to be constant.

1 In reality,

of course, it was not a constant, but a variable with a pronounced long-term up-ward trend dynamics. This trend is conditioned by technological innovations whose intensity also tends to grow (see, e.g., Grinin 2000, 2003b, 2006; Koro-tayev, Malkov, and Khaltourina 2006a, 2006b). This way we treat this variable in our extended model, and it allows us to investigate numerically the influence of the "secular cycle" structure on the "millennial" economic and demographic trends.

2 Our extended model of medieval political-demographic and economic-

technological dynamics differs from the basic model with the two following points:

1) Instead of the climatically determined annual yield (Yi) in the extended model we take into account the annual production (Ai), determined not only by the climatic conditions of the given year, but also by technological development level (index) achieved by the beginning of the given year (Ii-1):

Ai = Ii-1Yi. (5.1)

2) With Kremer (1993), Cohen (1995), Podlazov (2000, 2001, 2002, 2004), Komlos, Nefedov (2002), and Tsirel (2004) we use the following technological growth equation

3:

1 Note that this is relevant for most of the other models of pre-industrial political-demographic cy-

cles (Usher 1989; Chu and Lee 1994; Nefedov 1999f, 2002а, 2004; Malkov 2002, 2003, 2004; Malkov et al. 2002; Malkov and Sergeev 2004; Malkov, Selunskaja, and Sergeev 2005; Turchin 2003, 2005a; Korotayev and Komarova 2004; Turchin and Korotayev 2006), whereas the excep-tions from this rule are still rather rare (Komlos and Nefedov 2002; Korotayev, Malkov, and Khaltourina 2006b).

2 For a preliminary investigation of this influence see the previous issue of our Introduction to So-

cial Macrodynamics (Korotayev, Malkov, and Khaltourina 2006b). 3 See also the previous issues of our Introduction to Social Macrodynamics (Korotayev, Malkov,

and Khaltourina 2006a, 2006b). Note that the first model where the Verhulst population dynam-ics equation and Kremer technological growth equation were united within one system was pro-posed by Tsirel (2004).

Chapter 5. Extended Model of Medieval Egyptian Dynamics

77

dI/dt = aPI,

where I denotes technological development index, and P is population. Within our model this differential equation is used in its following difference version:

Ii = Ii-1 + aPci-1Ii-1. (5.2) Our extended model also takes into account the "Boserupian" effect. Ester Boserup (1965) has shown that the relative overpopulation creates powerful stimuli to generate and introduce innovations that raise the carrying capacity. Indeed, if there is no land shortage, the population has no strong incentives to generate and/or apply such innovations, whereas under conditions of relative overpopulation such incentives become really strong, as the introduction of such innovations becomes literally a question of "life or death" for a substantial part of the population, and the intensity of generation and introduction of the carry-ing capacity increasing innovations significantly grows. In our model the Boserupian assumption is modeled by a two-fold increase of the coefficient val-ue in equation (5.2) when the ecological niche gets filled more than 50%. As a result, the technological growth is modeled in the following way (see Table 5.1):

Table 5.1. Technological Growth Equations in the Extended Model of Medieval Political-Demographic Cycles

Variable sym-

bol This symbol denotes =

Value in year

i If Equation #

Ii Technological level reached by the end of year i

={

Ii-1 + aIi-1Pci-1 Pci-1/Ai ≤ 0,5

(5.3)

Ii-1 + 2aIi-1Pci-1 Pci-1/Ai > 0,5

(5.4)

Typical dynamics generated by this model are represented at Diagram 5.1 and Table 5.2:


78

Diagram 5.1. and Table 5.2. Typical Dynamics Generated

by the Extended Mathematical Model of Medieval Political-Demographic Cycles (a 700-year computer simulation)

0

2000

4000

6000

8000

10000

12000

14000

0 100 200 300 400 500 600 700

NOTES to Diagram 5.1: thick grey line – overall population dynamics within the "Egyptian" (po-lygynous) version of the model (in thousands); thin grey line – average carrying capacity dynamics within the "Egyptian" (polygynous) version of the model (in thousands of people); thick black line – overall population dynamics within the "European" (monogamous) version of the model (in thousands); thin black line – average carrying capacity dynamics within the "European" (monoga-mous) version of the model (in thousands of people). This diagram presents the results of a com-puter simulation with the values of initial conditions and parameters (including the ones for climat-ic fluctuations) that are entirely identical with the ones used above for the simulation whose results are reproduced at Diagrams 4.1 and 4.2. The only differences are constituted by the extension of the model with equations (5.1, 5.3–4) and, hence, the addition of one more initial condition, the in-itial level of subsistence technology (I0), which within the present series of computer simulations


79

was assumed to equal 1. The value of coefficient a in equations (5.3–4) in this series of simulations equals 1×10-7. The average value of carrying capacity for year i was calculated as the average value of climatically determined yield at the initial level (1.0) of subsistence technology (Y = 6663 thou-sand MARs) multiplied by the subsistence technology development index achieved in year i (Ii).

"Egyptian" ("polygynous") version "European" ("monogamous") version

Cy

cle

#

Sim

ula

tio

n y

ears

Max

imu

m p

op

ula

tio

n l

evel

rea

ched

d

uri

ng

th

e cy

cle

(th

ou

san

ds)

Av

erag

e ca

rry

ing

cap

acit

y (

tho

usa

nd

s o

f p

eop

le)

at t

he

cy

cle

pea

k

Lev

el o

f ec

olo

gic

al n

ich

e sa

tura

tio

n a

t th

e cy

cle

pea

k

Cy

cle

#

Sim

ula

tio

n y

ears

Max

imu

m p

op

ula

tio

n l

evel

rea

ched

d

uri

ng

th

e cy

cle

(th

ou

san

ds)

Av

erag

e ca

rry

ing

cap

acit

y (

tho

usa

nd

s o

f p

eop

le)

at t

he

cycl

e p

eak

Lev

el o

f ec

olo

gic

al n

ich

e sa

tura

tio

n a

t th

e cy

cle

pea

k

1. 1-95 3968 6934 57.2% 1. 1-240 5348 8126 65.8%

2. 96-189 4139 7223 57.3% 2. 241-492 6939 10309 67.3%

3. 190-286 4785 7668 62.4% 3. 493-778 9359 14578 64.2%

4. 287-385 4714 8065 58.5%

5. 386-485 4912 8501 57.8%

6. 486-582 5672 9134 62.1%

7. 583-688 5434 9675 56.2%

Thus, our numerical investigation of this model suggests that within the "Euro-pean" ("monogamous") version of the model subsistence technologies do tend to develop faster than those within the "Egyptian" ("polygynous") version. Thus, in a 700-year simulation the technological development index grew by more than 63% within the first version and constituted less than 47% in the sec-ond. At the meantime a comparison of population at cycle peaks indicates that within the "Egyptian" model population tends to approach the ceiling of the car-rying capacity of land to a much smaller degree than is observed within the "Eu-ropean" version of the model. It is highly remarkable that within the "Egyptian" model a significant growth of the carrying capacity could exist without a paral-lel demographic growth. Moreover, for considerable periods of time it can be accompanied by a certain population decline (cp. cycles 3 and 4, or 6 and 7), which, as we have seen this above, appears to have been actually observed for certain parts of the medieval Egyptian history. The contrast between trajectories of the countries developing according to these two models will look especially salient if we continue the period of com-puter simulation twice, up to 1400 years (see Diagram 5.2):


80

Diagram 5.2. Typical Dynamics Generated by the Extended Mathematical Model of Medieval Political-Demographic Cycles (a 1400-year computer simulation)

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

0 200 400 600 800 1000 1200 1400

NOTE to Diagram 5.3: thick grey line – overall population dynamics within the "Egyptian" (po-lygynous) version of the model (in thousands); thin grey line – average carrying capacity dynamics within the "Egyptian" (polygynous) version of the model (in thousands of people); thick black line – overall population dynamics within the "European" (monogamous) version of the model (in thousands); thin black line – average carrying capacity dynamics within the "European" (monoga-mous) version of the model (in thousands of people).

By the end of this period the technological development in a country evolving according to the "European" model exceeds by almost 5 times the technological level of a country that develops according to the "Egyptian" model. What is more, within the first model by the end of period in question demographic col-lapses disappear, the rate of economic growth starts to systematically outstrip the population growth rate. Thus, the per capita production begins to grow with accelerating speed and the system begins in a rather confident way its escape


81

from the "Malthusian trap"4, within which the population of the country evolv-

ing according to the "polygynous" model continues to remain (see Diagram 5.3):

Diagram 5.3. Typical Dynamics of Per Capita Production Generated by the Extended Mathematical Model of Medieval Political-Demographic Cycles (years 1350–1650 of the computer simulation)

NOTE: the diagram reproduce the data on average annual per capita production (for different dec-ades) measured in MARs.

The closest fit with the actually observed long-term political-demographic dy-namics of Egypt in the 1

st – 18

th centuries CE is observed when we bring our

model closer to the Egyptian history realities and take into consideration the

4 On the models of the "Malthusian (low-equilibrium) trap" and escape from this trap see, e.g.,

Artzrouni and Komlos 1985; Steinmann and Komlos 1988; Komlos and Artzrouni 1990; Steinmann, Prskawetz, and Feichtinger 1998; Wood 1998; Kögel, and Prskawetz 2001.


82

fact that during the period covered by the present study we observe the transi-tion from strictly monogamous elites to those practicing polygyny in a rather ex-tensive way. In the last version of our extended model (the "monogamous-polygynous" one) this is modeled in the following way: the initial value of the elite natural growth coefficient in equation (4.14) is taken to be equal to 0.01, and after one of the cycles, at the moment corresponding to the power in the country being seized by a new, polygynous, elite its value increase 4-fold up to 0.04, which corresponds to the transfer of power from monogamous elites to polygynous ones. Typical dynamics generated by the "monogamous-polygynous" version of the extended model of medieval political-demographic cycles is represented at Diagram 5.4 and Table 5.3:

Diagram 5.4. and Table 5.3. Typical Dynamics Generated

by the "Monogamous-Polygynous" Version of Extended Mathematical Model of Medieval Political-Demographic Cycles

0

2000

4000

6000

8000

10000

12000

14000

0 100 200 300 400 500 600 700 800

NOTES to Diagram 5.4: thick black line – overall population dynamics (in thousands); thin black line – average carrying capacity dynamics (in thousands of people). This diagram presents the re-


83

sults of a computer simulation with the values of initial conditions and parameters (including the ones for climatic fluctuations) that are entirely identical with the ones used above for the simula-tion of the "monogamous" version of the extended model whose results are reproduced at Diagram 5.2. The only difference is constituted by the following point: the value of the elite natural growth coefficient in equation (4.14) is taken to be equal to 0.01 only for the first cycle, whereas at the very beginning of the second cycle, at the moment corresponding to the power in the country being seized by a new, polygynous, elite its value increases 4-fold up to 0.04, which corresponds to the transfer of power from monogamous elites to polygynous ones. It is assumed that afterwards the country continues to be controlled by polygynous elites, hence, the value of the elite natural growth coefficient remains 0.04 throughout the rest of the simulation.

Cycle # Simulation

years

Maximum population

level reached during

the cycle (thousands)

Average carrying capaci-

ty (thousands of people)

at the cycle peak

Level of ecologi-

cal niche satura-

tion at the cycle

peak

1. 1-240 5349 8126 65.9%

2. 241-337 5161 8741 59.0%

3. 338-436 5384 9285 58.0%

4. 437-539 5565 9885 56.3%

5. 540-638 6463 10803 59.8

6. 639-741 6621 11688 56.6%

Maximum population level reached during the first cycle (thousands) 5349

Maximum population level reached during the last cycle of the simulation (thousands) 6621

Average carrying capacity (thousands of people) at the peak of the first cycle 8126

Average carrying capacity (thousands of people) at the peak of the last cycle of the simulation 11688

Population growth between the first and the last cycle peaks 23.8%

Carrying capacity of land growth between the first and the last cycle peaks 43.8%

As we see, the respective model provides a rather adequate mathematical de-scription of the phenomenon that has been detected through the analysis of the economic-demographic dynamics of Egypt in the 1

st – 18

th centuries, when a ra-

ther significant increase in the carrying capacity of land was accompanied by a comparatively insignificant population growth; thus, our models suggests possi-ble ways to account for this phenomenon.

Note that our analysis suggests that pre-industrial Christian Europe was characterized by rates of demographic and technological growth that were sig-nificantly higher than in the Islamic Middle East to some extent because of the strict monogamy imposed and maintained by the Christian Church in an effec-tive manner throughout the whole population, including the elites,

5 as well as

5 Note that even in the Islamic world the Christian Church imposed the monogamy within the

Christian communities in the strictest possible way: "The Moslems were astonished mainly by


84

some other norms and practices imposed by the Church, whose application led to a significant decrease of the natural elite population growth (Korotayev and Bondarenko 2000, 2001; Korotayev and Tsereteli 2001; Korotayev 2003a, 2003b, 2004; Korotayev and de Munck 2003; Bondarenko and Korotayev 2004). Of course, Greeks and Romans were monogamous prior to their Chris-tianization. However, pre-Christian Germans, Celts, Slavs, and Hungarians practiced polygyny, especially among the elites (see, e.g., Herlihy 1993: 41), which is of particular importance for us within the context of our discussion. Thus, the total absence of the polygyny in the Christian part of the Circum-Mediterranean region (but not in its Moslem part

6) could be hardly explained by

anything other than by the strict prohibition of the polygyny by the Christian Church, which it managed to impose quite effectively as a result of a few centu-ries of systematic efforts. Though some regulations which established the mo-nogamy as the norm were imposed by the Church still in the Roman times, even in the 12

th century, when marriage was declared a sacrament, the Church had to

struggle severely against rudiments of polygyny among both the elite and com-mon people, for example in France. And the struggle for the observation of the Christian marital norms among the elite strata of the knighthood went on even in the 13

th century (Bessmertnyj 1989).

In addition to this, already in the 4th

century the Christian Church imposed the regulations which prohibited close marriages, discouraged adoption, con-demned concubinage, divorce and remarriage. As has been suggested by Goody (1983: 44–6), in this way the Church appears to have striven towards obtaining the property left by couples lacking legitimate male heirs. Note, however, that among other things those norms contributed in a rather significant way to the bringing down of the natural elite population growth rates in the Christian world.

Last but not least, we wish to refer to an important medieval Christian insti-tution of clergy celibacy. This institution contributed to the blocking of the elite

the fact that the female slaves in the Christian and Jewish houses were not at the sexual disposal of the houses' heads... The cause of this was that the Christian regulation in the East considered the liaison of a man with his female slave as lechery which should have been expiated by the formal penance... The Khalif al- r once sent to his physician Georgios three beautiful Greek female slaves and 3,000 golden coins. The physician accepted the money, but returned the girls back saying to the Khalif: 'I cannot live with them in one house, because for us, the Chris-tians, it is permitted to have one wife only, whereas I already have a wife'..." (Mez 1996/1922: 159). However, in the Islamic world Christians did not constitute anything more than a confes-sional minority that did not affect the medieval Near Eastern elite population dynamics in any significant way. Of course, within the Christian states the Church had much more opportunities to impose the strictest monogamy among the whole population including the uppermost strata.

6 It appears remarkable that we would find the total absence of polygyny in Christian societies neighboring the Moslem societies living under entirely similar economic and ecological condi-tions and practicing (at least occasionally) polygyny (e.g. the Montenegrans [Jelavic 1983: 81–97; Fine 1987: 529–536] vs. the Highland Albanians [Pisko 1896; Durham 1909; 1928; Coon 1950; Hasluck 1954; Jelavic 1983: 78–86; Fine 1987: 49–54, 599–604, etc.]).


85

overproduction in two important ways. On the one hand, the absence of legiti-mate natural heirs of the members of the Church elite meant that for every gen-eration a very large number of elite positions became open within the Church system, which alleviated significantly the problem of the elite overproduction, as very large numbers of excessive elites members could be canalized to the Church system, as they had an opportunity to make a socially approved and ac-ceptable carrier within the Church hierarchy. On the other hand, these excessive elite members having found themselves within the Church hierarchy were effec-tively taken out of the process of the natural reproduction of the elite that con-tributed in a rather significant way to the decrease of the rate of the natural elite population growth.

It is difficult not to notice that the political-demographic dynamics generated by the "European" version of the extended model is quite close to the one that is observed not only in the pre-industrial Europe, but also in pre-industrial China (see Diagram 5.5):

Diagram 5.5. Population Dynamics of China (1300–1900 CE, millions)

0

50

100

150

200

250

300

350

400

450

500

1300 1400 1500 1600 1700 1800 1900

NOTE: for the justification of the estimates used for this diagram see Korotayev, Malkov, and Khaltourina 2006b.


86

This similarity might not be entirely coincidental. Though the polygyny was quite wide-spread among the highest echelons of the Chinese elite, the number of children in the Chinese polygynous families was significantly smaller than in the Islamic Middle East, apparently due to some birth control measures prac-ticed by the Chinese elites (Lee and Wang 1999). In addition to this, the growth of the upper strata of the Chinese elites (which had a real access to the state re-sources) was quite effectively restricted through the examination system. It appears necessary to add that the proposed "Egyptian" version of the ex-tended model of medieval political-demographic cycles describes more ade-quately the political-demographic dynamics of Egypt before 1347. Since 1347–1348 a new factor appears that changes the overall picture of the political-demographic dynamics in a rather dramatic way. In the mid 14

th century we ob-

serve the beginning of a "pathogenic attack at the World System" (Korotayev, Malkov, and Khaltourina 2005: 108–13), and Egypt was one of the countries that suffered from this attack to an extremely high degree.

It appears necessary to stress that this attack itself was an almost inevitable and natural product of the World System development. By the mid 14

th century

within the World System a huge interconnected human population of about 300 million people had been reproducing itself and constantly growing for a few centuries, which made the appearance of a new generation of particularly lethal pathogens (that cannot reproduce themselves in the scale of smaller popula-tions) almost inevitable. At the meantime a rather low level of health-care sub-system that was totally inadequate to the new level of pathogenic threat made the diffusion of the new generation of pathogenic waves throughout the whole World System almost as inevitable (Diamond 1999: 202–5; Korotayev, Malkov, and Khaltourina 2005: 106–14). Egypt was one of the countries that suffered most of all from this attack, or rather a whole series of such attacks which con-tinued just with two substantial breaks up to the mid 19

th century (Dols 1977;

Crecelius 1998; Garcin 1998; Harīdī 2005; Hathaway 1998; Ibrāhīm 1998; McCarthy 1976; Northrup 1998; Raymond 2001; Sāmī Bāshā 2002–2004, vols. 1–2; al-Sayyid Marsot 1985); for example, only during the period of the Circas-sian (Burjī) Mamluks (1382–1517) the plague epidemics attacked Egypt in 1388–1389, 1397–1398, 1403–1407, 1410–1411, 1415–1419, 1429–1430, 1438–1439, 1444–1449, 1455, 1459–1460, 1468–1469, 1476–1477, 1492, 1498, 1504–1505 and 1513–1514 (Garcin 1998: 308).

In general, the Islamic World suffered from the pathogenic attack on the World System more than any other part of it. It appears that a considerable role here was played by the system of Islamic pilgrimage (al-Ḥajj). Of course, this system constituted a very important mechanism securing the integration of a huge intersocietal network covering some most important central areas of the World System (and many peripheral areas as well), a mechanism which secured the unity of some significant patterns, values and practices throughout all this territory, guaranteeing the annual meeting of the representatives of all the socie-


87

ties covered by the respective network in one place, the exchange of infor-mation between them, the constant reintegration of the network, etc. However, after 1347 this system started performing a "function" that nobody designed for it, the function of the pathogen exchange between the participants of this com-municative network, as a result of which the appearance of new pathogens in one of the network zones led to a very high probability of this pathogens' diffu-sion within 1–2 years throughout the whole (or almost whole) Islamic commu-nicative network. As was noted by McCarthy (1976), epidemics typically pene-trated to Egypt with the pilgrims returning from al-Ḥajj

7. Note that Egypt

played an especially important role in the organization of al-Ḥajj and it played a role as an important terminal for the pilgrims making their way to Mecca from most of the African Continent (who naturally also returned back through Egypt) (see, e.g., Crecelius 1998). The situation only changed in the 19

th century, after

the plague epidemic of 1835, with the introduction of the quarantine systems (McCarthy 1976). The radical growth of the pathogenic pressure on the Egyptian population led to significant changes in the political-demographic dynamics of this country. Note first of all the appearance of pronounced intercycles, when after demo-graphic collapses a steady recovery of the population growth could not start for many decades, as the population increase achieved for a few years after an epi-demic was "eaten" (frequently with a significant excess) by new epidemic waves.

8 Our mathematical model does not describe such intercycles, as it does

7 Or (more rarely) with pilgrims moving through Egypt to Mecca from the other parts of the Afri-

can continent. 8 Note that, as has been shown by Borsch (2004), the demographic collapse (or rather a series of

demographic collapses) of the second half of the 14th – 15th centuries led not to the growth of per capita consumption of commoner population (as was observed during the previous demographic collapses), but to its decrease. The point is that in contrast to the previous demographic collapses of the Islamic epoch that were followed by a rather rapid stabilization and a start of a steady re-covery growth, after the mid 14th century Black Death the recovery population growth was con-stantly interrupted by new and new epidemic waves that led to further and further decrease of the country population. As a result we observe an increasingly acute shortage of labor that was nec-essary to maintain the Egyptian irrigation systems. As a result, a substantial part of those irriga-tion systems was ruined, which led to the decrease of the cultivated area (and, hence, food pro-duction) that was even sharper than the population decrease. This led to the decrease of per capita production, and, hence, per capita consumption. Note that similar consequences were observed in Egypt as a result of prolonged demographic collapse caused by the 2nd century CE Antonine pan-demic (cp. Bagnall and Frier 1994; Nefedov 1999а). One does not see grounds not to suggest that a similar consequences were also produced by a prolonged demographic collapse caused by the 6th century Justinian's Plague, or by a prolonged series of epidemics observed in the 17th century (see, e.g., Ibrāhīm 1998). On the other hand, Borsch's model suggests that in pre-industrial Egypt stabilization after a series of epidemic waves could have led to the situation when for a consider-able period of time (of an order of 20–30–40 years) the population growth could have been ac-companied not by the decrease (which was typical for most of agrarian systems) but by the in-crease in per capita production, per capita consumption, and, hence, particularly fast rates of re-covery population growth (which seems to have been the case in the first decades of the Ottoman rule in Egypt).


88

not take the epidemic factor into account. Thus, to achieve a more accurate de-scription of the political-demographic dynamics of medieval Egypt it appears necessary to undertake a further extension of the model taking the above men-tioned factor into consideration. This, however, goes out of the scope of the present monograph.

Chapter 6

Secular Cycles

and Millennial Trends in Egypt:

Preliminary Conclusions

1. The overall population trend in the 1

st – 18

th centuries CE in Egypt is sug-

gested to be upward, rather than downward. 2. On the other hand, the carrying capacity of land appears to have grown in

medieval Egypt considerably higher than population, whereas the population growth was significantly slower than throughout the World System (outside the Middle East).

3. We believe this phenomenon could be at least partly accounted for by the mechanisms of political-demographic cycles typical for Egypt (and appar-ently for the other countries of the Islamic Middle East), on the one hand, and by the very different mechanisms in the other major regions of Eurasia, on the other.

4. Medieval Egyptian political-demographic cycles had a rather short length (approximately 90 years).

5. During the relatively short medieval Egyptian political-demographic cycles, population simply had not enough time to reach the carrying capacity of land. Political-demographic collapses took place well before the population reached the carrying capacity level, and medieval Egypt suffered from un-derpopulation rather than overpopulation. The population of medieval Egypt fluctuated well below the carrying capacity.

6. Thus the political-demographic cycle models that connect demographic col-lapses with the ecological niche saturation and that describe rather well po-litical-demographic dynamics of pre-Modern China do not appear appropri-ate for medieval Egypt.

7. Hence, it might not be a coincidence that the mathematical model that ap-pears to be more appropriate for describing the political-demographic dy-namics of medieval Egypt than the rest of the models is the one (Turchin 2003: 131–7) that was developed in an attempt to formulate in a mathemati-cal form some part of the theory of `Abd al-Raḥmān Ibn Khaldūn (1332–1406) who spent a substantial part of his life specifically in medieval Egypt.

8. Turchin (2003) developed two "Ibn Khaldūn models", within which politi-cal-demographic collapses are produced not by actual overpopulation, but


90

rather by elite overpopulation. Elite overproduction can take place in a gen-erally underpopulated country (or at least in a country whose population is still significantly below the saturation level). Hence, these models suggest a direction within which the political-demographic dynamics of medieval Egypt could be adequately described.

9. However, in a few points these models fail to describe adequately the politi-cal-demographic dynamics of medieval Egyptian.

10. Though Turchin's models were no doubt inspired by Ibn Khaldūn's treatise, Turchin moved rather far from the original Ibn Khaldūn's theory in the pro-cess of the model development, and thus, not so much of it survived in the final versions of the models. We believe that in order to produce a mathe-matical model describing the medieval political-demographic dynamics in a more accurate way it makes sense to try to follow Ibn Khaldūn's theory more closely.

11. Ibn Khaldūn's observations on the role of climatic fluctuations as an im-portant factor of political-demographic dynamics appear of special interest. By taking them into consideration it is possible to develop the basic mathe-matical model that describes the medieval Egyptian political-demographic dynamics more accurately.

12. As in Turchin's extended model, in our basic model the increase of the natu-ral elite growth rate leads to a decline in the length of the political-demographic cycles, whereas its decrease results in the lengthening of those cycles. Thus, it turns out to be possible to produce the model that describes rather adequately the basic features of medieval European political-demographic dynamics through the decrease by 4 times of the "Egyptian" natural elite growth rate coefficient (which would correspond to the strictly monogamous reproduction context typical for all the medieval European Christians, including the elites).

13. One of the main simplifying assumptions of our basic model is that the tech-nologically determined carrying capacity of land is assumed to be constant. In reality, of course, it was not a constant, but a variable with a pronounced long-term upward trend dynamics. This trend is conditioned by technologi-cal innovations whose intensity also tends to grow. This way this variable is treated in our extended model, which makes it possible to investigate numer-ically the influence of the "secular cycle" structure on the "millennial" eco-nomic and demographic trends.

14. Our extended model also takes into account the "Boserupian" effect – such that relative overpopulation creates powerful stimuli for generating and in-troducing innovations that raise the carrying capacity of land (Boserup 1965).

15. Our numerical investigation of this model suggests that within the "Europe-an" ("monogamous") versions of the model subsistence technologies do tend to develop faster than they do within the "Egyptian" ("polygynous") ver-

Chapter 6. Preliminary Conclusions

91

sions. The comparison of population at cycle peaks indicates that within the "Egyptian" model population tends to approach the ceiling of the carrying capacity of land to a much smaller degree than is observed within the "Eu-ropean" version of the model. It is highly remarkable that within the "Egyp-tian" model a significant increase in the carrying capacity could take place without a parallel demographic growth; what is more, for considerable peri-ods of time the growth of carrying capacity can be accompanied by a certain population decline, which appears to have been actually observed for certain parts of the medieval Egyptian history.

16. The closest fit with the actually observed long-term political-demographic dynamics of Egypt in the 1

st – 18

th centuries CE is observed when we bring

our model closer to the Egyptian history realities and take into consideration the fact that during the period covered by the present part of our monograph we observe the transition from strictly monogamous elites to elites who practiced polygyny in a rather extensive way. This model provides a mathe-matical description of the phenomenon that has been detected through the analysis of the economic-demographic dynamics of Egypt in the 1

st – 18

th

centuries: significant increases in the carrying capacity of land were accom-panied by a comparatively insignificant population growth. Thus, our mod-els suggest possible ways to account for this phenomenon.

Part II

CYCLICAL AND TREND DYNAMICS

IN POSTCOLONIAL TROPICAL AFRICA

Chapter 7

Postcolonial Tropical Africa:

Trends and Cycles

Modern demographic development is radically different from that of previous historical epochs. The emergence and the mass spread of effective contracep-tion practices have drastically changed patterns of historical demographic pro-cesses. Nowadays, population size has been stabilized, or is well on the way to-wards stabilization in most countries of the world. But the development of his-torical societies, especially the complex agrarian ones, had regularly been inter-rupted by demographic collapses that were accompanied by millions of human tragedies (see, for example: Abel 1974, 1980; Postan 1973; Kuhn 1978; Mugruzin 1986, 1994; Usher 1989; Kul'pin 1990; Chu and Lee 1994; Huang 2002; Nefedov 2003, 2004, 2005; Turchin 2003b, 2005a, 2005b; Nepomnin 2005; Turchin and Korotayev 2006; Korotayev, Malkov, and Khaltourina 2006b).

People have produced innovations that increased the carrying capacity of the environment in terms of its ability to maintain a certain number of people (see, e.g., Grinin 2000, 2003b). In the recent centuries this process has accelerated and per capita GDP has increasingly begun to exceed the minimum subsistence level. Unprecedented modernization processes have greatly changed the shape of demographic cycles. As we suggested in the previous volume of our Intro-duction to Social Macrodynamics (Korotayev, Malkov, and Khaltourina 2006a) the length of demographic cycles tended to grow in the pre-industrial era. Dur-ing the early stages of modernization, demographic cycles increased in frequen-cy, compared with those of agrarian empires. At the same time, depopulation began affecting a progressively smaller percentage of population, and gradually tended to disappear (although the absolute figures of human losses might have increased at certain stage). The smoothing of demographic cycles during the

Chapter 7. Trends and Cycles

93

course of modernization is, to our opinion, described quite neatly by the trend-cyclical models presented both in Chapter 5 above, and in Chapter 4 of the pre-vious volume of our Introduction to Social Macrodynamics (Korotayev, Malkov, and Khaltourina 2006b). The Diagram 7.1 presents the population dy-namic of the world and China between 200 BC and 2003 AD:

Diagram 7.1. The World and China Population Dynamics (millions) between 200 BCE and 2003 CE

0

1000

2000

3000

4000

5000

6000

7000

-200 300 800 1300 1800

NOTE: black curve – the world's population dynamics; grey curve – population dynamics of Chi-na. For sources see the previous volumes of our Introduction to Social Macrodynamics (Korota-yev, Malkov, and Khaltourina 2006a, 2006b).

The major transformations of the relationships between trend and cyclical com-ponents of demographic dynamics during the last two centuries are especially

Part II. Postcolonial Tropical Africa

94

evident when comparing the dynamics between 700 BCE and 1800 CE on the one hand, and that of 1800–2003 on the other (see Diagrams 7.2 and 7.3):

1

Diagram 7.2. The World and China Population Dynamics (millions) between 200 BCE and 1800 CE

0

100

200

300

400

500

600

700

-200 300 800 1300 1800

Diagram 7.3. The World and China Population Dynamics (millions) between 1870 and 2000 CE

0

1000

2000

3000

4000

5000

6000

1870 1890 1910 1930 1950 1970 1990

1 At the same time, the comparison of the two types of demographic dynamics also demonstrates

that radical transformation of evolution pattern during the two recent centuries is a logical conse-quence of the previous World-System development processes that emerged long before the mod-ernization (Korotayev, Malkov, and Khaltourina 2006b).


95

Note that during the modernization period we observe a certain shortening of political-demographic cycles. While the Qing cycle in China (1650–1851) last-ed more than two hundred years, the succeeding "Republican" cycle (the 1880s – 1930s) took less than seventy years, and the "Maoist" cycle (1940-s – early 1960-s) took about twenty years. Depopulation reached 30% at the end of the Qing cycle, less than 10% in the end of the "Republican" cycle, and less than 5% of population in the end of the Maoist cycle

2 (see the previous volume of

our Introduction to Social Macrodynamics [Korotayev, Malkov, and Khaltouri-na 2006b]).

Demographic transition and the increase in agricultural productivity due to major technological advances in the recent centuries allowed most states to es-cape the Malthusian trap. The first stage of the demographic transition is char-acterized by a decline in mortality due to the advancement and spread of mod-ern medical technologies. This leads to the acceleration of population growth. In the second stage of demographic transition, the development of medicine in combination with other processes (especially, with mass education among women) leads to a widespread use of contraceptives and, as a result, to a de-crease of population growth rates (see, e.g., Chesnais 1992, or the first volume of our Introduction to Social Macrodynamics [Korotayev, Malkov, and Khalto-urina 2006a]).

Modernization processes started in Subsaharan Africa, in general, later than in the other regions of the world. The populations of the developed counties have been stabilized, and in some cases populations are declining. At the same time, many states of Subsaharan Africa have just recently begun to move from the first phase of demographic transition, which is the demographic explosive phase, to the phase of declining population growth rates. As a result, the popula-tion of Subsaharan Africa is still increasing in a rather rapid way (see Diagram 7.4):

2 If we consider as a demographic collapse the depopulation in connection with the "Great Leap

Forward" events.


96

Diagram 7.4. Population of Subsaharan Africa in 1960–2003

DATA SOURCE: World Bank 2006.

Despite the fact that the African population is increasing, its growth rate has been declining in the last decades. This happens, first of all, due to fertility de-cline, but secondly, because of the spread HIV/AIDS epidemics

3 (World Bank

2006; see also Polikanov 2000). Relative population growth rates stabilized in the early 1980s and have gradually declined since the early 1990s.

4 The dynam-

ics of birth rate, death rate and population growth rate in Subsaharan Africa are represented on Diagram 7.5:

3 It may be said that the AIDS epidemics in contemporary Tropical Africa has a certain Malthusian

component that is similar to the one typical for the epidemics which were one of the main mech-anisms of demographic collapses in the complex agrarian systems. Indeed, if the African popula-tion had not grown so rapidly, the African countries would have had much more resources that could be used to fight this lethal disease. Another important factor of the diffusion of AIDS and other infectious diseases (like tuberculosis) throughout Africa is the formation of a pan-African communication network, and in this respect the situation in contemporary Africa might have some resemblance with the "pathogenic attack on the World System" in the 14th century (see Chapter 5 above).

4 Note that the world population relative growth rates stopped increasing in the 1960s, and began to gradually decline in the 1970s (see, e.g., Korotayev, Malkov, and Khaltourina 2006a).


97

Diagram 7.5. Dynamics of Birth Rate, Death Rate and Population Growth Rate in Subsaharan Africa in 1960–2003

Data source: World Bank 2006.

If this tendency continues, one could predict stabilization of African population figures within a few decades (Akimov 2004).

Population growth has been accommodated by a growth in agricultural productivity. Average cereals yield increased in Subsaharan Africa approxi-mately 50%, from 8 to 12 centner/hectare in 1960–2003

5 (see Diagram 7.6):

Diagram 7.6. Dynamics of Cereal Yields (rice equivalent, centners per hectare) in Subsaharan Africa in 1960 – 2003

DATA SOURCE: FAO 2006.

5 However, cereal yields in Subsaharan Africa are still among the lowest in the world (FAO 2006).


98

This productivity growth has been due to the implementation of modern agricultural technologies, such as fertilizers, more productive varieties of agricultural plants, scientifically based crop rotation practices and so on.

The increased yield per hectare is not the only factor contributing to the green revolution in Africa. The cultivated area has also increased (see Diagram 7.7):

Diagram 7.7. Cultivated Land Area Growth in Subsaharan Africa (thousands of hectars) in 1961–2001

DATA SOURCE: FAO 2006.

In most cases growth is not only extensive, but also intensive. A great amount of uncultivated lands in Africa are fallow lands. Here the extension of cultivated area has been accompanied by a decrease of fallow periods, which means an intensification of agriculture. This process, however, is accompanied by soil degradation (Potemkin and Ksenofontova 2001; Chernjaev 2002).

The rate of growth in agricultural production in Tropical Africa has been, on average, 1.4–1.8% in the 1990s, which is significantly lower than the population growth rate, 2.5–3% (Roshchina 1999: 55). However, this is not only agricultural growth that leads to the rise of the carrying capacity. The development of other spheres of economy and international aid also contribute to the increase in the actual carrying capacity, as they make it possible for African countries to import deficient food.

The threat of Malthusian trap for African countries is not the possibility of population reaching a stable ceiling of the carrying capacity. It is the threat of population growing faster than the carrying capacity of territory. Despite


99

considerable advances of Africa in development (see, e.g., Pavlova 2001), the food crisis is the sword of Damocles that hangs above a number of African states.

Modernization processes developed slower in Tropical Africa than in other regions in the 20

th century (see, e. g., Vasil'ev 1999). This is why agriculture is

still playing a major role in many African economies. Although the food prob-lem is a critical one for many African countries, the high share of agriculture in their GDP is an alarming indicator. In the absence of complementary industrial resources, agricultural growth is limited by finite land resources, as well as by limitations of crop growth capacity (see, e. g., Brown 2001: 51). A large share of agriculture in GDP, as a rule, coexists with high fertility rates and indicates the persistence of an agrarian society for which the Malthusian political-demographic cyclical dynamics should be relevant.

Our cross-national analysis indicates that the share of agriculture in a na-tional economy in 2003 correlates positively with the number of internal mili-tary conflicts in a country within the previous 30 years (see Diagram 7.8):

Diagram 7.8. Correlation of Share of Agriculture in GDP (%) in 2003 (X-axis) and the Number of Internal Military Conflicts within the Previous 30 years (1973–2003, Y-axis)

NOTE: Rho = 0.43; p = 10-11. Data source for share of agriculture in GDP: World Bank 2006; data sources for the number of military conflicts: Sarkees 1997; Vasiljev et al. 2002; White 2005.

The Diagram 7.8 shows that the decline of the share of agriculture in GDP (due to the diversification of economy) leads to the decreasing probability of internal military conflicts, with 35–40% being the threshold.


100

The situation in some African countries appears to correspond to the pre-collapse phases of political-demographic cycles (see, e.g., Nefedov 2002a, 2002b, 2003, 2004, 2005; Korotayev, Malkov, and Khaltourina 2006b): malnu-trition, frequent famines; shortage of arable land; mass migration to towns and cities; declining real incomes; cheap labor cost; relatively high staple food pric-es; relatively high land prices; large number of unemployed people and beggars; hunger riots and rebellions; rise of popular movements under the slogans of so-cial justice etc. (see, e.g., Morozov 2002).

Large scale demographic collapses are fortunately no longer present in con-temporary Subsaharan Africa. In fact, they are not supposed to take place there, because Africa has moved onto the path of modernization and development; and the carrying capacity of land is constantly growing.

Population data show that modern type demographic cycles, accompanied with considerable population decline, took place in Somalia in the last few decades (see Diagram 7.9):

Diagram 7.9. Somalian Political-Demographic Cycles (1960 – 1990)

YEAR

19921991

19901989

19881987

19861985

19841983

19821981

19801979

19781977

19761975

19741973

19721971

19701969

19681967

19661965

19641963

19621961

1960

SO

MA

LIA

po

pu

latio

n

8000000

7000000

6000000

5000000

4000000

3000000

2000000

DATA SOURCE: World Bank 2006.

In fact, the recent Somalian political-demographic dynamics are similar to that of many currently developed countries when they were at their early


101

modernization stages (see, e.g., Nefedov 2005; Korotayev, Malkov, and Khaltourina 2006b).

Most African countries are more developed than Somalia. This is why humanitarian catastrophes in the hot spots of Africa, rather than decreasing the population, in most cases only reduce the rate of population growth, due to increase in mortality and immigration.

6 A cyclical dynamics can be traced with

respect to the population growth rates of some Tropical African countries (see Diagram 7.10):

Diagram 7.10. Population Growth Rates in Burundi, Sierra Leone and Liberia in 1960–2003

Data source: World Bank 2006.

Some mechanisms of such cyclical dynamics will be considered in the next

chapter.

6 Similarly, World War II resulted not in the decline of the world population but in a temporary

slowdown of its growth rate, which was compensated by a sharp increase in the world population growth rates in the post-war years (see, e.g., the first book of our Introduction to Social Macro-dynamics [Korotayev, Malkov, and Khaltourina 2006a]).

Appendix

Cyclical Dynamics and

Mechanisms of Hyperbolic Growth

The compact macromodels, which we discussed in the Introduction to this vol-ume, specify the most general mechanisms of the world population growth. However, we also need to know how these macrotrends are produced on a more specific level.

In the first part of our Introduction to Social Macrodynamics (Korotayev, Malkov, and Khaltourina 2006a: 92–104) we have discussed the most evident specific mechanism accounting for both the hyperbolic population growth

1 in

1850–1962/3 and inverse-hyperbolic (logistic) trend afterwards – the one asso-ciated with the demographic transition (e.g., Chesnais 1992; Kapitza 1999). As is well known, during the first phase of demographic transition a rather sharp decline in mortality rates is observed. This is followed by decline in fertility rates (through the introduction of family planning practices and technologies as a proximate cause), but with a substantial time lag. As a result for considerable periods of time we observe pronounced trends towards the rise of the popula-tion growth rates against the background of growing population. This, of course, exactly produces a hyperbolic effect – the higher is the population (N), the higher is the population growth rate (r). Since the 19

th century more and

more populations of the world entered the demographic transition. Up until the 1960s the number of populations which entered the 2

nd phase of demographic

1 Let us recollect that hyperbolic population growth implies that the absolute population growth

rate is proportional to the square of population (unlike exponential growth, for which the absolute growth rate is lineally proportional to population). Thus, with exponential growth if, at a world population level of 100 million, the absolute annual growth rate was 100 thousand people a year, at a level of 1 billion level it will be 1 million people a year (a 10-fold growth of population leads to an equivalent 10-fold increase in the absolute population growth rate). For hyperbolic growth, if, at the world population level of 100 million, the absolute annual growth rate was 100 thousand people a year, at a level of 1 billion it will be 10 million people a year (the 10-fold growth of population leads to a 100-fold increase in the absolute population growth rate). Note that the rel-ative population growth rate will remain constant with exponential growth (0.1% in our exam-ple), whereas it will be lineally proportional to the absolute population level with hyperbolic growth (in our example, population growth by a factor of 10 leads to an increase in the relative annual growth rate also by a factor of 10, from 0.1% to 1%). Such a growth is hyperbolic, just because it implies that the relative population growth rate is proportional to population size: dN/dt N = kN. If we multiply both sides of this equation by N, we will get dN/dt = kN 2, where-as the solution of this differential equation is just the hyperbolic formula Nt = C/(t0 – t), where С = 1/k (see the Introduction).

Cyclical Dynamics and Hyperbolic Growth

117

transition did not compensate for the hyperbolic growth of the 1st phase popula-

tions, hence, the hyperbolic growth trend was characteristic not only for indi-vidual populations, but also for the world population as a whole.

The only problem with the mechanism of demographic transition is that it is impossible to use it to account for the hyperbolic growth trend in the pre-19

th

century history of the humankind. In fact, against the background of our earlier discussion of the pre-industrial

cyclical dynamics it should not appear strange that the presence of hyperbolic population growth trend in the pre-industrial period of human history looks quite counterintuitive for those specialists who deal with particular historical demographies. Indeed, whenever we manage to acquire any quantitative data on pre-Modern population dynamics for any particular countries on a century time scale (and this happens infrequently), we tend to observe just the contrary trend – the higher the population, the lower its relative growth rate. Let us re-turn

2 to the historical population dynamics in China.

World population growth

and historical demographic dynamics of China

For example, the data of the Chinese census of the Late (Eastern) Han period (in fact, the first period and place in human history, for which we have any di-rect and systematic population data) looks as follows (see Diagrams A1–2, and Table A1):

Diagram A1. Population Dynamics of China during the Eastern Han Pe-riod according to Contemporary Census Data (millions)

15

25

35

45

55

65

50 75 100 125 150 175

t, years CE

N, p

op

ula

tio

n (

millio

ns)

SOURCES: Bielenstein 1947: 126; 1986: 240–2; Durand 1960: 216; Loewe 1986c: 485; Zhao and Xie 1988: 536.

2 For our earlier detailed treatment of the historical population dynamics of China see the previous

volume of our Introduction to Social Macrodynamics (Korotayev, Malkov, and Khaltourina 2006b: Chapter 2).

Appendix

118

Table A1. Population Numbers and Growth Rates in China during the Eastern Han Period according to Contemporary Census Data

Census year Population registered by census Population growth rate for subse-quent intercensus period (%%)

57 21008000 6.25

75 34125000 1.86

88 43356000 1.22

105 53256000 –0.34

125 49691000 –0.07

140 49150000 0.29

144 49731000 –0.41

145 49524000 –4.1

146 47567000 1.73

156 56487000

Diagram A2. Correlation between Population Numbers and Population Growth Rates for Eastern Han China

Population

6000000050000000400000003000000020000000

Po

pu

latio

n G

row

th R

ate

(%

%)

8

6

4

2

0

-2

-4

-6

NOTE: r = – 0.82, p = 0.007.


119

The extremely high population growth rate for 57–75 CE is, no doubt, a result of underregistration in 57 CE. Otherwise, the overall picture is quite clear, logi-cal, and precisely contrary to the one implied by the hyperbolic growth mod-els – the higher the population, the lower the population growth rates.

As we could see above, a rather convincing explanation for this pattern re-currently found in pre-industrial populations is provided by demographic cycle models – at the initial phases of such cycles resources are abundant, consump-tion levels are high, and thus, the population growth rates are also high; with population growth per capita acreage decreases, which against a context (typi-cally observed within pre-industrial agrarian systems) of stable, or very slowly growing subsistence technology levels would normally lead to decreasing per capita food production and food consumption, and to decreasing population growth rates, which after reaching the ceiling of the carrying capacity could eventually drop to zero, or even negative values (see, e.g., Abel 1974, 1980; Postan 1973; Kuhn 1978; Mugruzin 1986, 1994; Usher 1989; Kul'pin 1990; Chu and Lee 1994; Huang 2002; Nefedov 2003, 2004, 2005; Turchin 2003b, 2005a, 2005b; Nepomnin 2005; Turchin and Korotayev 2006; Korotayev, Malkov, and Khaltourina 2006b).

Within the East Han cycle the population growth slowed down dramatically after the population approached 60 million. Note that after the population had approached this level very closely, the system experienced demographic col-lapse (starting in 186 CE). It is remarkable that demographic collapses also oc-curred at the same level during the West Han (in the early 1

st century CE), Sui

(in the early 7th

century) and Early Tang (mid 8th

century) cycles (see, e.g., Bielenstein 1947: 126, 1986: 240; Durand 1960: 216, 223; Loewe 1986b: 206; Nefedov 1999e: 5; 2003: Fig. 10; Lee Mabel Ping-hua 1921: 436; Wechsler 1979a; Wright 1979: 128–49; Zhao and Xie 1988: 536–7).

Note also that Sung China experienced all the pre-collapse symptoms after its population approached the same level in the early 11

th century – famines, ris-

ing rebellions etc. (e.g., Lee Mabel Ping-hua 1921: 281–2; Smolin 1974: 311–57; Nefedov 1999e: 9, etc.).

3 All this, of course, suggests 60 million as an effec-

tive ceiling of the carrying capacity of land for 1st millennium CE China. This

ceiling was radically raised only in the 11th

century through the Sung "green revolution" (e.g., Ho 1956, 1959: 169–70, 177–8; Shiba 1970: 50; Bray 1984: 79, 113–4, 294–5, 491–4, 597–600; Mote 1999: 165).

Another interesting observation on population dynamics during the East Han cycle is that, according to Chinese census in 57–105 CE the average annual population growth rate was c.2%. Against the background of data on fairly high life expectancies in China during the phases of high population growth (e.g., Harrell and Pullum 1995: 148; Liu 1995: 118–9; Heijdra 1994, 1998: 437) we

3 However, the Sung mid-phase demographic crisis resulted not in a demographic collapse, but in

the non-catastrophic solution of the crisis through the radical raising of the carrying capacity of land ceiling (see below).

Appendix

120

do not see why the possibility of 2% annual population growth rates during ini-tial phases of Chinese demographic cycles could be completely excluded.

However, even if we take a much more conservative estimate of the Chinese population growth rates in the second half of the 1

st century CE as being c.1.5%

(e.g., Durand 1960: 216–21), we will still get a value of the world population growth rate far exceeding the one attested for the last 50 years (1750–1800) of the pre-industrial period, < 0.45% (Kremer 1993: 683). This is accounted for by the fact that at the end of the 1

st century the population of China constituted

around one third of the world population and the point that the Roman Empire (encompassing by that moment almost another third of the world population) al-so experienced a significant demographic growth during the 1

st century (see,

e.g., Turchin 2003: 162). This, of course, suggests that 0.45% attested as the average annual world

population growth rate for the second half of the 18th

century (see, e.g., Kremer 1993) was not only significantly lower than the one achieved by particular (and rather significant) regional populations long before that time, but also that long before the 18

th century the world population growth rates could be equal or

higher than the ones attested at the end of the pre-industrial epoch. This also suggests that the hyperbolic effect might have been created not by the absolute increase in the population growth rates during the pre-industrial demographic expansion periods, but rather by the changes in lengths and spacing of those pe-riods.

Indeed, the actual increase in annual population growth rates in pre-industrial era would imply the growth of life expectancies at birth, whereas the evidence does not indicate any significant growth of life expectancies between the Neolithic and Industrial revolutions, suggesting rather a general trend to-wards its decline in the Neolithic and Post-Neolithic epoch up to the Modern Age when life expectancies started to grow significantly marking the beginning of demographic transition (e.g., Lee and de Vore 1968; Mel'jantsev 1996; Kozintsev 1980; Storey 1985; Fedosova 1994; Cohen 1977, 1987, 1989, 1995, 1998; Cohen and Armelagos 1984; Ember and Ember 1999: 152–3, etc.) . In the rest of this appendix we shall demonstrate how the hyperbolic trend of world population growth in the pre-Modern Age could co-exist with the absence of an increased annual growth rate during the pre-industrial demographic ex-pansion (as well as, by definition, stagnation) phases.

Let us consider now the overall of demographic dynamics in China. Against the background of what has been mentioned above it might be somehow coun-terintuitive to find that we do observe a hyperbolic growth trend for this popula-tion. Naturally this trend turns out to be more pronounced if we take into ac-count the last 150 years of Chinese demographic history (see Diagrams A3–5):

4

4 We use the estimates of historical Chinese population surveyed in the previous issue of our Intro-

duction to Social Macrodynamics (Korotayev, Malkov, and Khaltourina 2006b: Chapter 2).


121

Diagram A3. Population Dynamics of China, millions (700 BCE – 2003 CE)

0

200

400

600

800

1000

1200

1400

-700 -500 -300 -100 100 300 500 700 900 1100 1300 1500 1700 1900

It is easy to see here a pattern of interplay of cyclical and trend dynamics. How-ever, what kind of trend do we observe here? Linear regression suggests a sta-tistically significant (p < 0.001) relationship with R

2 = 0.398.

5 Exponential re-

gression produces an even stronger result with R2 = 0.685 (p < 0.001), see Dia-

gram A4:

5 All regressions for pre-industrial and industrial periods combined were calculated for years 57–

2003.

Appendix

122

Diagram A4. Curve Estimations for Chinese Population Dynamics, millions, 57 – 2003 CE (linear and exponential models)

YEAR

2100

2000

1900

1800

1700

1600

1500

1400

1300

1200

1100

1000900

800700

600500

400300

2001000

1400

1200

1000

800

600

400

200

0

Observed

Linear

Exponential

NOTES: the thin black line corresponds to the observed population dynamics surveyed in Chap-ter 2 of the previous issue of our Introduction to Social Macrodynamics (Korotayev, Malkov, and Khaltourina 2006b). Lineal regression: R = 0.631, R2 = 0.398, p < 0.001. The respective best-fit thin light grey line has been generated by the following equation: Nt = 0.2436t – 124.25. Exponen-tial regression: R = 0.828, R2 = 0.685, p < 0.001. The respective best-fit thick dark grey line has been generated by the following equation: 13.3575 × e 0,0015t. The best-fit values of parameters have been calculated with the least squares method.

However, a simple hyperbolic growth model produces a much better fit with the observed data (R

2 = 0.968, p << 0.001

6), see Diagram A5:

6 In fact, to be exact, statistical significance of the fit in this case reaches an astronomical level of

1.67 10-19.


123

Diagram A5. Population Dynamics of China (57 – 2003 CE), millions, correlation between the observed values and the ones predicted by a hyperbolic growth model

t, years

21001800150012009006003000

1400

1200

1000

800

600

400

200

0

predicted

observed

NOTE: R = 0.984, R2 = 0.968, p = 1.7 × 10-19. The black markers cor-respond to empirical es-timates surveyed in Chapter 2. The grey sol-id line has been generat-ed by the following equation:

tN t

2050

63150.376.

The best-fit values of parameters С (63150.376) and t0 (2050) have been calcu-lated with the least squares method.

Yet, even if we consider only the pre-Modern history of China (up to 1850), we will still find the hyperbolic growth trend for this part of Chinese history too (see Diagrams A6–8):

Diagram A6. Population Dynamics of Pre-Modern China (700 BCE – 1850 CE)

0

50

100

150

200

250

300

350

400

450

500

-700 -400 -100 200 500 800 1100 1400 1700

What kind of trend do we observe here? Linear regression again suggests a sta-tistically significant (p < 0.001) relationship with R

2 = 0.469. Exponential re-

gression again produces an even stronger result with R2 = 0.593 (p < 0.001), see

Diagram A7:

Appendix

124

Diagram A7. Curve Estimations for Pre-Modern Chinese Population Dynamics, millions, 57 – 1850 CE (linear and exponential models)

YEAR

1900

1800

1700

1600

1500

1400

1300

1200

1100

1000

900

800

700

600

500

400

300

200

100

0

500

400

300

200

100

0

Observed

Linear

Exponential

NOTES: the thin black line corresponds to the observed population dynamics surveyed in Chap-ter 2 of the previous issue of our Introduction to Social Macrodynamics (Korotayev, Malkov, and Khaltourina 2006b). Lineal regression: R = 0.689, R2 = 0.469, p < 0.001. The respective best-fit thin light grey line has been generated by the following equation: Nt = 0.1098t – 27.97. Exponen-tial regression: R = 0.770, R2 = 0.593, p < 0.001. The respective best-fit thick dark grey line has been generated by the following equation: 16.9785 × e 0,0012t. The best-fit values of parameters have been calculated with the least squares method.

However, a simple hyperbolic growth model once more produces a much better

fit with the observed data (R2 = 0.884, p << 0.001

7), see Diagram A8:

7 To be exact, statistical significance of the fit in this case again reaches an astronomical level

(2.8 × 10-19).


125

Diagram A8. Population Dynamics of Pre-Modern China (57 – 1850 CE), millions, correlation between the observed values and the ones predicted by a hyperbolic growth model

1850165014501250105085065045025050

450

400

350

300

250

200

150

100

50

0

observed

predicted

NOTE: R = 0.94, R2 = 0.884, p = 2.8 × 10-19. The black markers cor-respond to empirical estimates surveyed in Chapter 2. The grey solid line has been generated by the fol-lowing equation:

tN t

1915

33430.518.

The best-fit values of parameters С (33430) and t0 (1915) have been calculated with the least squares meth-od.

Note that the population dynamics of China for the pre-Modern period correlate rather well with the population dynamics of the world (see Diagram A9):

Diagram A9. Population Dynamics of the World and China, millions, 700 BCE – 1850 CE

0

200

400

600

800

1000

1200

1400

-750 -250 250 750 1250 1750

China

World

NOTE: The world population data here and elsewhere are from Kremer 1993: 683 (the other sources consulted are: Thomlinson 1975; Durand 1977; McEvedy and Jones 1978: 342–51; Haub 1995: 5; Biraben 1980; U.S. Bureau of the Census 2004; UN Population Division 2004). The Chi-na population data are from Bielenstein 1947, 1986; Durand 1960; Ho 1959; Lee 1921;

Appendix

126

Mel'jantsev 1996; Nefedov 2003, 2004; Zhao and Xie 1988 (surveyed in the previous issue of our Introduction to Social Macrodynamics [Korotayev, Malkov, and Khaltourina 2006b]).

As could be expected from the graph above, the population of China for this pe-riod correlates very well (R = 0.907, R

2 = 0.822, p < 0.001) with the world pop-

ulation. There is, of course, a very significant autocorrelation component here, which can be easily controlled, if we compare the dynamics of Chinese popula-tion with the one of the rest of the world. It is highly remarkable that in this case the correlation remains very strong and significant (see Diagram A10):

Diagram A10. Population of China vs. Population of the Rest of the World (700 BCE – 1850 CE)

Population of China (millions)

4003002001000

Po

pu

latio

n o

f th

e R

est

of

the

Wo

rld

(m

illio

ns)

600

500

400

300

200

100

0

NOTE: R = 0.793, R2 = 0.628, p < 0.001.

This, of course, suggests that the structure of Chinese pre-industrial population dynamics (known in more detail than for any other part of the world) could re-veal a lot with respect to the world population dynamics in the pre-Modern pe-riod.

As can be easily seen in Diagram A6, the upward hyperbolic trend is created by just 4 relatively long periods of population growth accompanied by series of carrying capacity increasing innovations – during West Han, Sung, Ming and Qing dynasties. As the census data for China are only available since 2 CE, more or lest reliable dynamics of Chinese population during West Han (206


127

BCE – 9 CE) remains unknown, the shape of the West Han population curve in diagrams above is based on estimates by Zhao and Xie (1988: 536), which could hardly be used for any exact analysis. The only thing, which seems to be clear is that during this period the Chinese population did not only manage to restore its numbers to the level preceding the demographic collapse of the Qin – Han transition, but grew substantially (due to a series of the carrying capacity enhancing innovations [see, e.g., Bray 1984]) over the pre-Han level (which is not known exactly either, e.g., according to Zhao and Xie's [1988: 536] esti-mates, it was around 20–32 million) up to c. 60 million, whereas afterwards (up to the 11

th century) the Chinese population oscillated below this level. There do

not seem to be any factual grounds to estimate the population growth rates dur-ing the restoration and "pure growth" phases

8 of the West Han cycle.

During the Sung cycle a relatively high population growth was observed for about a century, which could be roughly split into two parts, or phases – a resto-ration phase, and a pure growth phase (see Diagram A11):

Diagram A11. Sung China Population Dynamics (millions)

0

20

40

60

80

100

120

950 1000 1050 1100

8 The following terminology is used throughout this appendix. The demographic cycle phase when

the population is restored to the pre-collapse (usually close to the carrying capacity of land ceil-ing) level is denoted as a "restoration phase", the phase when population grows over this level (through the introduction of the carrying capacity enhancing innovations) is denoted as a "pure growth phase". Though restoration phases are observed in all Chinese demographic cycles, the pure growth phases are only found within the West Han, Sung, Ming, and Qing cycles, and not observed in the East Han, Sui, Early and Late Tang, as well as Yüan cycles. During the Late Tang and Yüan cycles demographic collapses took place before the population reached the carrying capacity of land ceiling.

Appendix

128

The Sung restoration phase occupied the late 10th

– early 11th

centuries and last-

ed until the 1030s, when the population grew over 50 million and approached

the carrying capacity limit, which resulted in a political-demographic crisis with

all the pre-collapse symptoms. However, Sung China reacted to this crisis in a

rather adequate way, through the well-coordinated introduction of a series of the

carrying capacity increasing innovations (e.g., Ho 1956, 1959: 169–70, 177–8;

Shiba 1970: 50; Bray 1984: 79, 113–4, 294–5, 491–4, 597–600; Mote 1999:

165). As a result, the carrying capacity in China was raised about twice, and af-

terwards the Chinese population oscillated well above the 1st millennium CE

level.

During the pure growth phase (c.1050–1110) the average annual growth rate was c.1.1%, which was somehow lower than the growth rate achieved during the East Han restoration phase (57–105 CE), but was still much higher than the highest pre-industrial world population growth rate evidenced for 1750–1800 CE.

The official Ming census records give rather lower figures, indicating that the population grew up to 60.5 million by 1393 and then fluctuated between slightly more than 50 million (1431–1435, 1487–1504) up to 63–65 million (1486, 1513, 1542–1562); in 1602 it was 56.3 million, in 1620–1626 it was 51.7 million (e.g., Durand 1960: 231–2). There is a consensus that the actual population of Ming China was much higher.

What is more, this appears to have been clear for the Ming Chinese them-selves:

"The official census records were hopelessly out of touch with demographic reality. The compiler of a Zhejiang gazetteer of 1575 insisted that the number of people off the offi-cial census registers in his county was three times the number on. A Fujian gazetteer of 1613 similarly dismissed the impression of demographic stagnation conveyed by the of-ficial statistics: 'The realm has enjoyed, for some two hundred years, an unbroken peace which is unparalleled in history,' the editor pointed out. 'During this period of recupera-tion and economic development the population should have multiplied several times since the beginning of the dynasty. It is impossible that the population should have re-mained stationary.' A Fujian contemporary agreed: 'During a period of 240 years when peace and plenty in general have reigned [and] people no longer know what war is like, population has grown so much that it is entirely without parallel in history.' Another of-ficial in 1614 guessed that the increase since 1368 had been fivefold. China's population did not grow between 1368 and 1614 by a factor of five, but it certainly more than dou-bled" (Brook 1998: 162).

Thus, nobody appears to doubt that the actual population of Ming China was much higher than is indicated by the Ming census (what is more, many Ming Chinese do not seem to have had doubts about this either); however, there is no consensus at all as regards how much higher it was.

The lowest estimate is 100 million (Zhao and Xie 1988: 540). Most experts suggest for the end of the Ming much higher figures: 150 million (Ho 1959:


129

264), 120–200 million (Perkins 1969: 16), 175 million (Brook 1998: 162), 200 million (Chao 1986: 89), or even 230–290 million (Heijdra 1998: 438–40; Mote 1999: 745), though the last figures appear to be overestimations (see, e.g., Marks 2002). In any case, as we have argued in the previous volume of our In-troduction to Social Macrodynamics (Korotayev, Malkov, and Khaltourina 2006b), there does not appear to be much doubt that the Ming cycle included both restoration and pure growth phases, but there does not seem to be suffi-cient grounds to estimate population growth rates during the Ming pure growth phase. As the highest population level achieved during the Ming period remains unknown, it is difficult to demarcate the boundary between the restoration and pure growth phases within the Qing cycle.

If we accept as such the figure of 150 million, then the Qing pure growth phase would start in c. 1740. The pure growth phase would then last for 110 years, during which the average growth would be c. 1%.

9

Thus, there are no grounds to believe that the population growth rates during later phases of pure growth were higher than during pure grow phases of earlier cycles. The available evidence rather suggests that as soon as free resources be-came available (due either to previous depopulation or to series of carrying ca-pacity increasing innovations); the Chinese population grew with fairly similar rates up to the point when the available resources were exhausted.

How could this condition coexist with the hyperbolic growth trend? We shall try to start answering this question below.

Pure growth phase lengthening mechanism

Let us now try to apply the compact macromodels of hyperbolic growth (see the Introduction, and Korotayev, Malkov, and Khaltourina 2006a) to Chinese data. Of course, the main problem with the direct application of these models to the Chinese data (and, as we shall see soon, to the world data, in general, as well) is that we do not have evidence for the actual systematic rise of population growth rate during later stagnation and expansion periods of Chinese demographic his-tory in comparison with earlier ones. Thus, this model has to be translated into more specific mechanisms explaining how hyperbolic growth could appear against a background of absence of increase in population growth rates during pure growth phases.

9 Note that this rate was significantly higher than the one attested for the world population growth

in the contemporary period. Naturally, as a result, the proportion of Chinese population to the whole population of the world grew very significantly from 16.5% in 1700 to > 36% in 1850. This proportion experienced a sharp decline in 1850–1870, as a result of the "Taiping" demo-graphic collapse with the total human life losses as high, as 118 million (see, e.g., Huang 2002: 528) against the background of a rather high population growth rates attested in most other parts of contemporary world.

Appendix

130

One possible prediction which could be made on the basis of the compact macromodels is as follows: the higher the population at the beginning of the given pure growth phase, the greater the number of innovations that will be made during this phase, the higher the level, to which the carrying capacity (and, thus, the population) will grow. Note that even if we assume that the population growth rate during the pure growth phase is constant, this will still produce a certain hyperbolic effect.

The immediate logic of the compact macromodels suggests that the rate of carrying capacity increase during the pure growth phase would be proportional to the square of population at the beginning of the phase. This will result in the following dynamics.

In our first auxiliary model we assume that the pure growth phases are sepa-rated by equal 50-year interphases.

10 During each pure growth phase the popu-

lation growth is assumed to be 1.5 %. After the first 50-year interphase the first pure growth phase starts (at 100 million level), during which the carrying capac-ity is raised twice, whereas the population will grow with 1.5 % rate for about 50 years, after which the new limit of the carrying capacity is reached, and the population stabilizes at the 200 million level. A new pure growth phase starts after a 50-year interphase. This time the population growth starts from 200 mil-lion, so the logic of compact macromodels would suggest that during this phase the carrying capacity would be raised 4 times, which would make it possible for the population to grow up to 800 million level, thus securing c. 100 years of population growth with 1.5% annual rate. During the next growth phase we would expect the carrying capacity to be raised 64 times, which would secure 1.5% a year growth for 300 hundred years. The population dynamics for the 450 years just described starting with zero interphase would look as follows (see Diagrams A12–13 and Table A2):

10 We denote periods of innovations leading to the absolute increases in carrying capacity and pure

growth of population followed by stagnation periods as "developmental cycles"; we subdivide developmental cycle into "growth phase" and "stagnation phase" (= "interphase").


131

Diagram A12. Dynamics Produced by the Pure Growth Phase Lengthening Mechanism

0

500

1000

1500

2000

2500

3000

3500

4000

0 100 200 300 400

The pre-industrial population growth is usually measured for 100–200 year pe-riods. If we split 450 years in 3 equal intervals, we will get the following picture (see Table 5.2):

Table 5.2. Dynamics Produced by the Pure Growth Phase Lengthening Mechanism

Period Years Population at the begin-ning of cycle (millions)

Average population growth rate during 150-year period

1 0 – 150 50 0.5

2 151 – 300 100 0.9

3 301 – 450 400 1.5

Appendix

132

Diagram A13. Relationship between Population Size and Growth Rate Produced by the Pure Growth Phase Lengthening Mechanism

Population (millions)

5004003002001000

Po

pu

latio

n G

row

th R

ate

(%

%)

1.6

1.4

1.2

1.0

.8

.6

.4

NOTE: R = 0.962, R2 = 0.926, p = 0.088 (1-tailed).

As we see, if higher populations raise the carrying capacity to higher levels, this creates a certain hyperbolic effect, even if the annual growth rates during pure growth phases do not increase, and even if the interphases do not become short-er. There is some evidence that during the last pre-industrial pure growth phase the carrying capacity was raised in greater proportion than during the previous phase (Table A3):

Table A3. Two Last Pre-Industrial World Population Growth Phases Compared

Growth phases Population at the beginning of

the phase

Population at the end of the

phase

Population growth achieved during the phase

1400–1600 350 545 55%

1650–1800 545 900 65%

1650–1850 545 1200 120%


133

Note that this difference becomes more pronounced if we consider as the end of the last pre-industrial pure growth phase 1850 rather than 1800.

11 In any case,

the mechanism under consideration accounts to a very considerable extent for the fact that the last period of uninterrupted world population growth (combin-ing pre-industrial and industrial phases) was longer than the previous period, which in its turn contributed to the hyperbolic population growth trend. Two 2

nd millennium pre-industrial Chinese growth phases about which we

know some detail are the Sung and Qing ones12

(see Table A4): Table A4. Sung and Qing Pure Growth Phases Compared

Growth phases Population at the beginning of the phase (millions)

Population at the end of the phase

(millions)

Population growth achieved during the phase

1050–1110 54 104.5 93.5%

1740–1850 150 436.3 190.8%

An interesting thing about Sung – Qing comparison is that during the Qing pure growth phase the annual growth rates were even a bit lower (c. 1.0 %) than dur-ing the Sung one (c. 1.1 %), and the hyperbolic effect here was created specifi-cally by the pure-growth-phase-lengthening mechanism.

As we see, for the last pre-Modern pure-growth phases for both China and the world we do observe a certain positive correlation between the population at the beginning of the phase and the increase in carrying capacity achieved during a respective phase. Thus the hyperbolic trend during the last centuries of pre-industrial population growth turns out to be accounted for to some extent by the mechanism under consideration. Note, however, that the respective proportion does not appear to be quadratic, but is rather linear. Thus, though the respective mechanism appears to contribute to the appearance of hyperbolic trend in the population growth in the last centuries of the pre-industrial history, this contri-bution does not appear to be very high (unlike the one of the mechanism, which we will discuss next). Interphase Shortening Mechanism

Let us now further re-formulate compact macromodel logic in the following way: the higher the population at the beginning of an interphase, the less time it will take it to start a new series of innovations resulting in a new pure growth

11 This makes sense, as the world population growth in 1800–1850 resulted mostly from the re-

gions (first of all East Asia), where industrial revolution and demographic transition had not started yet.

12 As has been mentioned above, due to the defectiveness of the Ming statistics, no such detail is known for the Ming phase.

Appendix

134

phase. The empirical evidence appears to support this hypothesis (see Ta-ble A5):

Table A5. Interphase Characteristics

China The World

Interphase Population at the begin-ning of in-terphase, millions

Interphase length

Interphase Population at the be-ginning of interphase,

millions

Interphase length

13 – 1050 CE

59.85 1037 100 – 1000 180 900

1110 – 1500 (?)

104.5 390 1200 – 1400 360 200

1580 – 1740

150 160 1600 – 1650 545 50

R 0.963 R 0.934

R2 0.92 R

2 0.87

p (1-tailed) 0.087 p (1-tailed) 0.088

As we see, we seem to observe a relationship between the population at the be-ginning of interphase and the interphase length which is close to the inverse quadratic, i.e., the increase in population during the pure growth phase by factor of X leads to the decrease of the subsequent interphase X

2 times (in fact, in

most cases even more). Actually, this is just what the compact macromodels' logic suggests: the innovation rate is assumed to be proportional to the popula-tion size and technology level. Thus, a population twice as large at the begin-ning of interphase B (tB) as compared to the one at the beginning of interphase A (tA) implies that the technology level at tB was twice as high as at tA. Thus, we have grounds to predict that it will take twice as large population having twice as high technology a 4 times smaller period (2

2 = 4) to accumulate the same

amount of innovations necessary to initiate a new pure growth phase. We will now model what the contribution of this mechanism to the hyper-bolic population growth trend will be. In our model the first pure growth phase starts in 300 BCE from a 100 mil-lion level. During every pure growth phase the population grows with 0.7% an-nual rate for 100 years, thus doubling within a century. The first interphase is assumed to be 800 years. The length of each subsequent interphase is inversely related to the square of the population growth during the preceding pure growth phase. During each growth phase as the population increases twice, each subse-


135

quent interphase becomes shorter than the preceding one by a factor of four. This results in the following dynamics (see Diagrams A14–15 and Table A6):

Diagram A14. Dynamics Produced by the Interphase Shortening Mechanism

0

200

400

600

800

1000

1200

1400

1600

1800

-500 0 500 1000 1500 2000

Table A6. Dynamics Produced by the Interphase Shortening Mechanism

Cycle Years Population at the beginning of cycle

(millions)

Average population growth rate during subsequent cycle (growth phase + subsequent inter-

phase) (%%)

Interphase Length

1 –300–1000 100 0.054 1200

2 1001–1400 200 0.175 300

3 1401–1575 400 0.4 75

4 1576–1695 800 0.59 18.75

Appendix

136

Diagram A15. Relationship between Population Size and Growth Rate Produced by the Interphase Shortening Mechanism

Population (millions)

10008006004002000

Ave

rag

e A

nn

ua

l Gro

wth

Ra

te (

%%

)

.5

.4

.3

.2

.1

0.0

NOTE: R = 0.992, R2 = 0.984, p = 0.001.

As we see, the interphase shortening mechanism produces a rather strong hy-perbolic effect, and we believe that it had the most important contribution to the creating of the pre-industrial world population hyperbolic growth trend. How-ever, it accounts for the increase in the population growth rates during each sub-sequent developmental cycle (whereas each subsequent developmental cycle occurred on a significantly higher population level); however, it cannot account for the increase in population growth rate during each subsequent pure growth phase, whereas such a trend is also observed (see Table A7): Table A7. Trend towards the Increase in the World Population Growth

Rate during Each Subsequent Pure Growth Phase

Pure growth phase Average population growth rate (%%)

–500 – 100 0.1

950 – 1200 0.15

1400 – 1600 0.24

1650 – 1850 0.4


137

This trend appears to be augmented by the effects of the "increasing synchroni-zation of pure growth phase" mechanism, which we shall discuss in the next section of this appendix.

"Increasing synchronization of pure growth phases" mechanism

An important feature of the World System history is the increasing synchroniza-tion of the growth and decline phases in the various World System centers demonstrated recently by Chase-Dunn et al. (2003), as well as by Hall and Turchin (2003) (see Diagram A16):

Diagram A16. Growing Phase Synchronization of the World System (population rate of change dynamics in 3 World System regions, 800 – 1800 CE)

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

800 1000 1200 1400 1600 1800

East Asia Europe West Asia

NOTE: Adapted from Hall and Turchin 2003: 15.

There are grounds to believe that this increasing synchronization of pure growth phases was caused to a considerable extent just by the World System population

Appendix

138

growth. The higher is the population of the World System regions, the more contacts they will have, the faster the innovations will spread, and the higher will be the growth phase synchronization. What will be the impact of this increasing growth phase synchronization on the population dynamics? It will result in exactly a hyperbolic effect – the aver-age population growth rate within every subsequent growth phase, occurring at the population level higher than during the previous one, will be also higher. Let us model this effect. We assume that the World System population at the beginning is 100 million and it consists of 4 regions comprising 25 million each. During each growth phase the carrying capacity grows twice; as a result the population of each region grows twice at constant annual rate of 0.5%. Dur-ing the first phase the synchronization takes 800 years. During each subsequent developmental cycle the synchronization period is assumed to be inversely pro-portional to the population level reached during the previous cycle. As a result, if the population grows twice during the given cycle, the synchronization period during the subsequent period will shorten twice. This will result in the following dynamics (see Table A8):

Table A8. Dynamics Produced by the "Increasing Synchronization of Pure Growth Phase" Mechanism

Synchronization period (years)

Population at the beginning of growth phase (millions)

Average annual growth rate (%%)

800 100 0.0625

400 200 0.125

200 400 0.25

100 800 0.5

As we see, up to the complete synchronization this gives a very considerable hyperbolic effect.

Let us model the combined impact of the three above described hyperbolic growth mechanisms (different from the one of the demographic transition first phase).

In this model the World System is assumed to consist of four regions with equal population. The first growth phase accounted for by the model starts in 650 BCE from 120 million level. During the growth phases regional popula-tions grow at 0.4% rate. The first interphase is assumed to be 600 years. Inter-phase lengths are inversely proportional to the square of population; increase in the pure growth phases in each region is lineally proportional to the population increase during the previous phase. The first synchronization period is 800 years. The length of a synchronization period is inversely proportional to the population growth during the previous cycle. This model generates the follow-ing dynamics (see Diagram A17 and Table A8):


139

Diagram A17. Dynamics Produced by the Combined Action of Mecha-nisms of "Pure Growth Phase Lengthening", "Interphase Shortening", and "Increasing Pure Growth Phase Synchronization

0

200

400

600

800

1000

1200

-1000 -500 0 500 1000 1500 2000

Table A8.

Period Average world

population growth rate

World population

at the beginning of the period

World population at the end

of the period

Regional pure growth

phase lengths

–650 – –550 0.1% 120 135 100

–550 – –417 0 135 135

–417 – –317 0.1% 135 150 100

–317 – –184 0 150 150

–184 – –84 0.1% 150 165 100

–84 – 50 0 165 165

50 – 150 0.1% 165 180 100

150–750 0 180 180

750–1150 0.11% 180 280 110

1150–1400 0 280 280

1401–1600 0.29% 280 500 145

1601–1635 0 500 500

1636–1835 0.4% 500 1110 200

This model demonstrates an especially close fit with the observed data. The main discrepancy is produced by the fact that this model does not account for the hyperbolic trends within the pure growth phases. This trend is especially pronounced within the last pre-industrial pure growth phase (1650–1850, for which we, incidentally, have the most accurate data within the pre-industrial epoch) (see Table A9):

Appendix

140

Table A9. World Population Dynamics, 1650–1850 (according to McEvedy and Jones [1978] and Kremer [1993])

Period Population at the beginning

of the period (millions)

Average world population growth

rate during the period (%%)

1650–1700 545 0.225

1700–1750 610 0.332

1750–1800 720 0.446

1800–1850 900 0.575

Is it possible to account for such a trend without dropping the assumption that the average population growth rate within any pre-industrial pure growth phase cannot exceed 0.4%? Yes, this is possible if we take into account two more mechanisms of hyperbolic growth. We start with the Innovation Diffusion Mechanism.

Innovation diffusion mechanism

Its logic can be formulated as follows: The diffusion of a carrying capacity increasing technology within a world

system with a stagnant population will result in a quasi-hyperbolic demographic growth trend (even if the annual population growth rate after this technology in-troduction remains constant) due to the rise of the proportion of the growing population.

Let us model the impact of this mechanism on population dynamics using the following model.

In this model the World System consists of 4 regions, each of which consists of 4 zones. At the beginning all the zones have equal populations. A new tech-nology starts to be introduced in all the 4 regions. It raises the carrying capacity so that it allows the population growth at 0.6% rate for 200 years. Assume that the innovation is not introduced immediately in all the zones; it is implemented during each phase in one more zone of each World System region. This will re-sult in the following dynamics (see Table A10):

Table A10. Dynamics Produced by the Innovation Diffusion Mechanism

Sub- Phase

Years Zones where

innovations are implemented

Annual population growth rate

at the beginning of sub-phase (%%)

Population at the beginning and at the end of

sub-phase (millions)

1 1–50 25% 0.15 545–587

2 50–100 50% >0.3 587–681

3 100–150 75% >0.45 681–852

4 150–200 100% 0.6 852–1150

NOTE: average annual growth rate during the phase = 0.4%.


141

Note that within this model we arrive at the hyperbolic growth within the pure growth phase, though the average population growth rate during the whole phase remains 0.4%.

Differential growth mechanism

Note that within the model above a quasi-hyperbolic growth starts immediately after the carrying capacity increasing technology is introduced in 25% of the world system zones and is observed within 50 years of Phase 1, even though this technology is assumed to only spread to the next belt zones in Phase 2. Thus, though the number of zones where the new technology is introduced re-mains constant, and the population in these zones increases at constant rate, the World System population during 50 years of sub-phase 1 experiences a quasi-hyperbolic growth simply due to the differential growth mechanism.

Its logic can be formulated as follows: If a new carrying capacity increasing technology is introduced only in one of

the World System zone (A) (with all the other zones having stagnant popula-tion), it does not diffuse to the other zones, and the annual population growth in Zone A remains constant, the World System population growth will be charac-terized by a quasi-hyperbolic trend, due to the increase in the portion of the population that is growing.

Let us model this mechanism impact on population dynamics using the fol-lowing model:

In this model zones comprising 25% of the world introduce innovations and their population starts increasing at 2.05% growth rate (thus, growing c. 50% every 20 years). The population of the rest of the world does not grow. This re-sults in the following dynamics (Table A11):

Table A11. Dynamics Produced by the Differential Growth Mechanism

Year Population of innovation

zones (millions)

Population of the world

(millions)

World population

growth rate (%%)

1 100 400 0.5125

20 150 450 0.683

40 225 525 0.879

60 337.5 637.5 1.085

80 506.25 806.25 1.287

100 759.375 1059.375 1.47

Introducing the effect of the last two mechanisms for the last phase (note that the average annual growth rate in all the regions during the last pure growth phase still remains 0.4%) we arrive at the following dynamics showing the clos-est fit with the observed data (see Diagram A18):

Appendix

142

Diagram A18.13

Dynamics Produced by the Combined Effect of the Considered Mechanisms

0

200

400

600

800

1000

1200

1400

-1000 -500 0 500 1000 1500 2000

Predicted

Observed

Conclusion

We believe that the 5 mechanisms of hyperbolic growth suggested in this part of our book (in addition to the one of the first phase of demographic transition) ac-count quite satisfactorily for the hyperbolic trend observed for the pre-industrial world population without making a counter-factual assumption that the growth rate of world populations tended to increase with each subsequent cycle. Hence, this model does not contradict the available data suggesting the absence of any significant world trend toward the growth of life expectancies in the pre-industrial era (as we remember, these data rather suggest a weak opposite trend).

Of course, the hyperbolic growth generated by our model is somewhat im-perfect in that it is rather different from the one generated be simple hyperbolic growth models; but in this it is similar to the one observed in the historical rec-ord of pre-industrial world – all the main deviations from hyperbolic growth turn out to be totally regular phenomena predicted by the last model.

13 Note that our model predicts a short (c.12.5 years) intercycle at the end of the last pre-industrial

pure growth phase around 1850 (not reflected in this diagram). There are some reasons to expect that this interphase in the world population growth actually existed, i.e., in 1863 the world popu-lation was not higher than in 1851 (due to enormous population losses in China during the Tai-ping rebellion and accompanying episodes of internal [as well as external] warfare [see, e.g., Nepomnin 2005; Korotayev, Malkov, and Khaltourina 2006b]).


143

However this does not deny the value of compact mathematical models of the World System growth that describe rather accurately the overall shape of millennial trends and account for their most basic "Kuznetsian" mechanism and logic: more people – more potential inventors – faster technological growth – the carrying capacity of the Earth grows faster – faster population growth – more people – more potential inventors – faster technological growth, and so on, whereas, as we could see at the very beginning of this monograph, such a positive nonlinear feedback finally just produces hyperbolic growth dynamics.

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