XIVth International Economic History Congress
Helsinki, Finland, 21-25 August 2006
Session 95
Evolutionary Theories of Long-Run World Economic History:
The Theory/History Interconnection Re-Examined
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A Mathematical Model of the World System Demographic,
Economic, Technological and Cultural Growth
Artemy Malkov
Keldysh Institute for Applied Mathematics, Russian Academy of Sciences
Andrey Korotayev
Russian State University for the Humanities, Moscow
Daria Khaltourina
Centre for Civilizational and Regional Studies, Russian Academy of Sciences
This is a draft conference paper, not to be quoted without permission of the authors
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Abstract
We study macro-historical processes concerning the period from 1 CE to 2000 CE and taking demographic,
economical, technological and cultural factors into account. Our main research purpose consists in construction
of a mathematical model describing the dynamics of all mentioned characteristics.
1. Introduction
In 1960 Heinz von Foerster and his co-authors found that human population follows a hyperbolic curve from the
extreme antiquity up to the modern period (Fig. 1). This means that population must reach infinity at some finite
moment of time. They counted up the precise date of singularity – Friday, 13th November 2026. However a few years
after this publication growth rate stopped its sharp increase and started falling down abruptly (Fig. 2).
In 1960 von Foerster proposed a formula
, which describes hyperbolic growth of population but
doesn’t involves the sudden decrease of growth rate which started a decade later.
Sergey Kapitza (1992) published a paper on global demographic dynamics and proposed a formula or hyperbolic
growth and its modification for global demographic transition.
Fig. 1 Hyperbolic growth of population
Fig. 2. Growth rate dynamics
As a result Kapitza’s model perfectly describes both the hyperbolic growth and demographic transition (Fig.3), but
doesn’t explain them as a phenomenological model.
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Fig.3. Kapitza’s model.
In 1993 Michael Kremer considered not only demography but also economics, and proposed a model of simultaneous
growth of population and technology. He assumed that technology growth is proportional to current technological level
T and to the number of potential inventors N (population):
. His model gives the same hyperbolic growth
of population, but also describes the growth of technology. Kremer also tries to describe the demographic transition
assuming that high income leads to birth rate decline. However his description is not enough clear and, in general, is not
quite adequate. Moreover Kremer does not define the sense of terms yield and technology level in his model.
2. Model
We propose to use term of technology level for equilibrium world GDP per capita:
.
Our analysis gives an unexpected result that GDP also follows a hyperbolic curve (Fig. 4), and moreover there is perfect
relationship between GDP G and population N:
,
where w =
$; m = 221.15 $ as a minimum for over last 2000 years (Fig. 5). Today when N is high
enough this equation is very close to just G = wN2. In our model we also include additional value S = T – m (resource
surplus per capita), which represents average surplus gained by people thanks to technological advances comparatively
with wild state when G = mN. It is clear that empirical relation between N and S is just S = wN.
Fig. 4 Hyperbolic growth of world GDP.
As for demographic transition, our analysis shows that income itself does not change birth rate, the main reason of birth
rate decline lies in the field of culture, or more precisely of education.
It is rather difficult to measure cultural variables, but we take one aspect of culture, related with all other aspects –
education. Measuring of education is also problematic, and in general, the only disposable historical data is literacy
level. Fortunately, literacy level is actually very significant variable. Our analysis shows that female literacy is the main
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factor of birth rate decline. Other important factors – such as medicine and urbanization are considerably less significant
according to statistical multifactor analysis.
So we propose a model of simultaneous growth of population, GDP, technology and literacy level:
where N is population, S – surplus, L – literacy level, a, b and c are constants. Technology level T is calculated as T =
m + S, and GDP as G = TN = (m + S)N.
Fig. 5. Relation between GDP and population.
The first equation describes the increase of population which depends on population itself N, technology S, and suffers
from literacy L.
The second equation describes the growth of technology level
which depends on current technological level and on the number of potential inventors, proportional to N.
The third equation describes the growth of literacy level which is proportional to technological surplus S, on the
percentage of literate people and number of illiterate people (it is clear that if there is no illiterate people – there will be
zero growth – as well as if there is no literate people to teach the others).
3. Conclusions
We should once again recall the difference between T and S. T is never close to zero. Per capita output of human
ancestors was equal to their minimum needs m. While S is addition, caused by technological advances. So we consider
that just advances S but not output itself T, increase growth of N, S and L in our model. That’s why S but not T is
included into equations.
Results of simulation gives excellent fit with reality (Fig.6). Parameters of the model and initial conditions at 1 CE
are
a = 1.085·10-5 b = 6.51 ·10-12
= 8.2·10-6 ($
-1
($ · year)-1 ; (person · year)-1 ; · year) ;
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N0 =
S0 = 17.47 $ ;
L0 = 0.052 ;
170000000;
m (minimal needs per person) = 420 $
Fig.6. Results of simulation (normal and double logarithmic scales)
In conclusion, we would like to note a curious corollary of our analysis. Empirical data shows that N, S and L are
related (with considerable precision):
But there are no apparent limits of N and S, while L is limited by unity. What could mean this limit of growth for
humankind?
References
Foerster, H. von, P. Mora, and L. Amiot. 1960. Doomsday: Friday, 13 November, A.D. 2026. Science 132:1291–5.
1960.
Kapitza S. 1992 Mathematical model of world population growth, Matematicheskoe 4/6: 65–79. In Russian
Kapitza S. 1999 How many people did, do and will live on Earth, Nauka. 1999. In Russian
Kremer, M. 1993. Population Growth and Technological Change: One Million B.C. to 1990. The Quarterly Journal of
Economics 108: 681–716.
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