A Model of Demographic Cycles in
a Traditional Society:
the Case of Ancient
Sergey Nefedov
The Russian
and Archaeology, Ekaterinburg
It is published:
Social Evolution & History. 2004. Vol. 3. N 1. P. 69 - 80.
ABSTRACT
This publication is devoted to the theory of
demographic cycles advanced in researches of many authors. F. Braudel
named these cycles as ‘general cycles’, and R. Cameron used concept
‘logistics cycles’. The author has constructed the mathematical model of a
demographic cycle. A size of sowing areas, a population, a number of peasants and handicraftsmen are basic variables of this
model. The level of life,
reserves of grain, a sale of grounds, a growth
of large property, a transition of peasants to tenantry or handicraftsmen and
other factors of the socio-economic relations are taken into account.
Generally, the model represents an analogue of system of three
integro-differential equations.
The verification
of the model is made on a material of a history of Ancient
Population growth with limited amount
of resources and with a fixed production technology can be described by a
‘logistics equation’:
where N
(t) is population, C is capacity
of the ecological niche, and r is
rate of natural growth (
Recently economic and demographic historians such as
F. Braudel (1979), R. Cameron (1989), M. Artzrouni and J. Komlos (1985)
have employed such models, mostly for Medieval and early-modern
The purpose of
this paper is to construct a compact model describing the basic economic and
demographic processes of a pre-industrial society1. The verification of model was made by means
of data pertaining to the history of
Let us suppose that agricultural
output is characterized by production function:
Q(t) =
F[P(t)]A (1)
where P(t) is the
rural population at period t, A(t) is
a cultivated area and F(P) is some function. Q is crop output measured in kilograms. Hence, F(P) = Q/A = ksps, where ps is productivity per hectare, and ks is the multiple-cropping index (sown area divided by
cultivated area). The agrarian
technology was constant in this traditional society, therefore ps and ks do not depend on the population P2.
Thus, we have the production function
Q = kspsA.
It is necessary to take into consideration that A depends on P. We use Chinese
data to derive this relationship (Table 1).
Table 1
Population
and Cultivated Area in
Year A. D. |
Population (million) |
Cultivated area (million hectares) |
57 |
21 |
16.4 |
88 |
43.4 |
33.4 |
105 |
53.3 |
34.1 |
The graph of cultivated area as a function of the
population size is presented in Fig. 2
Fig. 2. The relationship between cultivated area and population
in Ancient
The area, A,
under cultivation in ancient
A(P)=
kP, if kP < Am (2a)
A(P)= Am, if kP
> Am (2b)
Value Am
was equal to about 34 million hectares in the period under consideration.
The society consisted of peasants, tenants and
landlords. Let Y(t) be the number of
peasants, and AF – the
area of land belonging to the farmers and AT,
the land of the tenants. We calculate the area of lands belonging to the
farmers, AF, using equations
(2a)–(2b) (but the lands of the tenants are deducted from Am). The land AT
is occupied by the tenants. The maximum land
of the peasants is
Am – AT. Then
AF(Y)=
kY, if kY < Am – AT (2c)
AF(Y)= Am – AT , if kY
> Am – AT (2d)
Let q be the
quantity of seed needed per hectare and let M
be the total grain requirements for seed. Then M = ksqAF. Let p0 be the minimal consumption per capita; in the case of
The value P0
= p0Y(t) is the minimal total consumption, and W=M+P0 is the quantity of
grain needed to cover minimum consumption and seed. Let X (t) be the quantity of grain after harvest (crop and stocks). In
case X(t) > W the peasants have
grain surpluses. The amount of grain available per capita, u, where u = (X (t) – M)/Y
(t). This is not all consumed in the current year, however. Let us assume
that half of the surplus is stored for future consumption.
Let pm
designate maximum consumption, then consumption per capita pc is:
pc
= (u +
p0)/2 if u > p0 and (u
+ p0)/2 < pm (3a)
pc = pm if (u + p0)/2 >
pm
(3b)
(If u<p0,
then X(t) < W. This case is
considered below). Total consumption is P1
= pcY(t), and total grain output is used for consumption
and seed: W1=M + P1,
so by the time of the following crop the available grain stock is Zp = X(t) – W1. Further, let l0 be the output of grain per sowing. Certainly, the productivity
was not constant, and we take it into account by adding to l0 the random variable dl0, so the real productivity becomes l = l0 + dl0 . The
production function is Q = kspsAF
= ksqlAF = lM, then the crop of the next year is
equal to lM. It is necessary to
subtract the taxes from this quantity. The taxes were equal to 1/30 of a crop
and 120 coins from each adult person (23 coins from a teenager). Each person
paid 60 coins on average. This computes the grain equivalent of this monetary
tax according to market prices, and obtains the total amount of taxes in terms
of grain, H.
After the harvest the quantity of grain is equal to
X(t+1) = lM – H + X(t) – W1
including the stocks.
Now population Y
(t+1) should be determined. In classical model by R. Pearl it is
Here r is
the rate of natural growth in favorable conditions, and C is the capacity of an ecological niche or the maximal population
at available food resources. In our case C=P1/p0.
We shall use a more recent model and replace the
term
by
where
n is a parameter of compensation
suggested by J. Maynard Smith and M. Slatkin (1973). The
introduction of this parameter is explained by the fact that in human societies
famine results not only in high death rate, but also in revolts and wars which
increase the death rate even more.
Let us consider now the case X(t)<W, when the peasants have grain deficits. Then the peasants
lack sufficient grain in spring sowing even if they consume p0. Then they sell a part of their land in order to compensate for
the lack of seed grain. In some cases the landowners have a limited stock of
grain and can not buy all the land sold by the peasants, then the peasants
reduce their fund of consumption P1
so, that M + P1 = X(t). In
this case u < p0 and
consumption per capita equals p(u)=P1/Y(t).
During the famine P1 <
p0 Y(t) and Y(t)/C = Y(t)/( P1/p0)
= p0Y(t)/ P1 > 1 in (4). Therefore population is reduced.
If the famine threatens destruction of a significant
part of the population, the authorities distribute grain to the peasants, increasing
consumption up to pu0 (pu0
< p0). As the peasants sell the land, the large landed
property gradually grows, and the cultivated area of the peasants decreases.
The landowners attract tenants who provide them half of the crop as land rent;
hence, a tenant should have twice more ground than a peasant, approximately 1,5
hectares per capita. In case the peasants sold land of area Da in the current year, it is
possible to locate Na = Da
/1.5 tenants on these lands and the peasant population decreases by
value Na.
Let us now consider the dynamic evolution of the
number of tenants. Let AT be
an area of grounds of the tenants, Ya(t)
is a number of the tenants in one year t,
and Xa(t) are stocks of
the grain of the tenants, except for the taxes and sowing fund. Weight of seed
grain of the tenants is equal to Ma
= ksqAT , and the minimal total consumption is Pa0 = p0Ya(t).
In case Xa(t)> Pa0
the tenants have surpluses of grain, and consumption per capita of the tenants
(pua) is calculated just
as for the peasants.
The general consumption of the tenants is Pa= pua Ya(t),
and to the next harvest the stocks Xa(t)
– Pa will be saved in
barns of the tenants. The crop of the next year is lMa, and the taxes Hà
are calculated just as for the peasants. After a deduction of taxes and seed
grain the tenant receives only half of the crop, therefore stocks of grain of the tenants will be equal to Xa(t+1) = ((l – 1)Ma –
Hà)/2 + + Xa(t) – Pa.
The maximal number of the tenants is given by C = Pa/p0 , and
the actual number is determined just as for the peasants. Each year the number
of the tenants is increased by number Na
(number of the peasants becoming the new tenants). These new tenants receive
sites of land about 1,5 hectares, sowing grain and annual norm of consumption.
They should return this grain in the future. During famine the landowners give
out to the tenants grain loans to increase consumption up to the minimum ðà1. In favorable years the
peasants return the debts.
The landowners spend a part of their income for
purchase of craft products and to maintain their servants. Let the number of
the handicraftsmen be Yr(t)
and they have stocks of a grain Xr(t).
The minimal general consumption of handicraftsmen is Pr0 = p0Yr(t). In case Xr(t)> Pr0 the
handicraftsmen have surpluses of grain, and their consumption per capita (pur) is calculated just as
for the peasants. Consumption is Pr=
pur* Yr(t), and by the next harvest the stocks of the
handicraftsmen will be equal Xr(t)
– Pr. The tax Hr
is transformed to a grain equivalent as before. In case the landowners spend
for purchase of craft products kr
% of their incomes, the next year
the stocks of the handicraftsmen will be Xr(t
+ 1) = = kr(lMa – Ha)/2 – Hr +
Xr(t) – Pr. The maximal number of the handicraftsmen
is determined by C = Pr/p0,
and the actual number is determined just as for the peasants. During famines
the handicraftsmen receive the grain loans from the landowners and try to
increase consumption at least up to the size of a minimum pr1. In favorable years they return the debts together
with interest.
The peasants, who have lost the land, engage in craft
activity. Some leave for cities, others produce craft products at times not
utilized in agriculture. It is useful to consider the peasants and
handicraftsmen separately. Assume that four peasants receiving the quarter of
their income from craft production produce an equivalent output of three
peasants and one craftsman. The handicraftsmen sell goods to buy grain. The
peasants live in a natural subsistence economy, and the handicraftsmen sell the
goods to landowners. The number of new workplaces of handicraftsmen and
servants is limited by the income of landowners received from the tenants of
the previous year. Let Da
be areas of the new tenants, and Haa
are taxes, paid by them; the income of the landowner from them will be:
G = (ksq(l
–1) Da – Haa )/2.
A share (kr1)
of this income is paid to the handicraftsmen, and the number of the new
handicraftsmen and servants can be krG/p0.
Anther part (kr2) of the
income of the landowners is stored in terms of grain stocks, and the third part
(kr3) is spent for
consumption3. Let kr1 = 50%, kr3
=25% and the consumption per capita of the landowners is five times more
than consumption of the handicraftsmen. Then the number of the landowners will
be ten times less than number of handicraftsmen. The maximum number of the
landowners is nearly half million.
The state also stores grain. Half of the grain received by the state as
the ground tax is accumulated in state barns according to the recommendation of
the treatise ‘Kuan-tzu’. The remaining portion was
about 2,5 millions tons according to estimates. There were about 150 thousand
officials in
The values of many of the parameters used are taken
from the historical documents. But mathematical models usually contain some
arbitrary parameters, which are selected by numerical experiment. In our case
the parameter of compensation (n) is
the most important one, because it describes the death rate of the population after consumption p(u) falls below the critical level p0. In traditional models (n = 1) the reduction of output below half of p0 results in a reduction
of the population by only 3 %. Such a reduction after a major crisis is
obviously implausible. This is also the case for n = 2 and n = 3. Therefore
we shall consider cases n = 4 – 6.
The results of simulating
the above described model are presented in fig. 3 (with n = 6). The presence of casual fluctuations of productivity causes
the variations in the curves. However, the fluctuations of productivity do not
influence the population until about the year 100. The calculations show that
at in that period the peasants had long-term stocks of grain, and the poor
harvest did not result in famines. The curve of growth of population is smooth
and steady in this period. The calculated population differs from actual
population levels estimated on the basis of historical documents only very
slightly. The data on the population and sowing areas are taken from (Lee 1921: 436; Krukov, Perelomov, Sofronov and
Cheboksarov 1983: 41).
The general tendencies are also correlated. In the
years between 57 and 85 the peasants intensively brought virgin soil into cultivation
and possessed large stocks of grain (see fig. 3). During this period the
consumption was large and the population grew quickly. After the year 85 the
internal colonization was slowed down owing to gradual exhaustion of reserves
of free lands, but the population continued to grow. Hence, per capita
consumption began to exceed the currently produced crops and inventory of grain
began to decrease.
About the year 102 the stocks were depleted and
famines began (Malijvin 1983: 80). The peasants began to sell land owing to famine; this
enabled many of them to avoid death through malnutrition, however population
decreased a little. The historical records document the first large revolts of
peasants. This crisis is of great importance because after it the stability of
economic processes was disturbed. Thereafter peasants had no inventories and
poor harvests could result in terrible famine and demographic catastrophes (fig.
4). However, the state tried to maintain stability
and during famines officials distributed grain from the state inventories.
Catastrophes were thereby avoided for an extended period of time. According to
historical evidence after the year 157 the state barns became empty and the
distribution of grain stopped (Malijvin 1983: 77).
Then the catastrophe
became inevitable. At this time the ‘Yellow Turbans’ revolt took place and then
the long internecine wars began.
Fig.
4. Economic dynamics in the Later Han period. Variant of calculations in the case of the lack of the state support. Such short
demographic cycle took place in
Thus, the main reason of
the catastrophe was the instability of an economy with lack of virgin land and
absence of grain inventories. This instability increased when the peasants
could freely sell the ground to the landowners. During years 102–160 many
peasants were ruined: the poor peasants lived in conditions of incessant
famines and sold much of their lands to landowners. The ruined farmers became
tenants, handicraftsmen and servants. The number of the farmers was decreasing,
but the area of their arable land was decreasing faster. Thus, the
disproportion between their number and the area they cultivated grew, bringing
about a shortage of cultivated land (Fig. 3). Eventually there remained so
little land in their possession, that its sale could no longer rescue the
peasants, and a terrible period of famines accompanied by epidemics and revolts
began. These catastrophes appear very quickly in the graph. However, in reality
the revolts resulted in disintegration of the state and long civil wars. The
real losses in lives were more dramatic, than the trends in the graph show4. We have demonstrated that the above model
can reproduce the salient features of Chinese demographic experience in the 1st
and 2nd centuries of our era.
ACKNOWLEDGMENTS
The author thanks Professor John Komlos for
discussion of the paper and valuable advice.
NOTES
1
A more detailed exposition of algorithm is available
at
http://hist1.narod.ru/Science/Cekl/index.htm
2
But the situation is different in some other cases, for example at the Sung
epoch.
3
The dynamic evolution of the number of landowners is outside of the scope of
this study.
4
The calculations show, that if the peasants had no right to sell the land,
their acreage in their possession is more stable. Probably, this was
the reason why Oriental monarchies often forbade the peasants to sell their
land.
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