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Журнал: Social Evolution & History. Volume 13, Number 1 / March 2014 - подписаться на статьи журнала

**ABSTRACT**

*The discussion about the Malthusian character of pre-industrial economies that has arisen in the recent years extensively uses simple mathematical models. This article analyzes some of these models to determine their conformity with Malthusian postulates. The author suggests two models that are more adequate for the description of the Malthusian patterns.*

Until recently, most economic historians were inclined to think that the medieval economies in Eurasia had a Malthusian nature (Allen 2008: 951). However, following the publication of Lee and Anderson's work (Lee and Anderson 2002), many came to dispute this opinion. A discussion has arisen about how the available data confirm the Malthusian relationship between demographic dynamics and consumption (*i.e.*, real wages). This discussion has largely involved simple mathematical models of the Malthusian economy.

In 1980, Lee published the first and most popular of these models. This model describes the relationship between the real wage, *w _{t}* (consumption), and labor resources,

Or, in the logarithmic form:

Here, *t *is time;* μ, ρ, η *are some non-negative constants; and *ε _{t} *is a variable that takes into account the climatic effect and other exogenous parameters. The factor

Wood (1998) suggests a feedback option for which he derives an equation basing on the same equation (1) as Lee, but formulates it as follows:

Here, *θ* is the minimum per capita consumption rate, and *S _{t }*is the maximum population that can subsist in the given territory when the consumption equals

where *β _{0} , β_{1} , β_{2}*,

Deriving *b _{t}* and

where *с _{0} *and

where* с _{2} *and

However, Malthus did write about population loss and depopulation:

The power of population is so superior to the power of the earth to produce subsistence for man that premature death must in some shape or other visit the human race. The vices of mankind are active and able ministers of depopulation. They are the precursors in the great army of destruction, and often finish the dreadful work themselves. But should they fail in this war of extermination, sickly seasons, epidemics, pestilence, and plague advance in terrific array, and sweep off their thousands and tens of thousands. Should success be still incomplete, gigantic inevitable famine stalks in the rear, and with one mighty blow levels the population with the food of the world (Malthus 1798: 61).

Wood's model does not, therefore, describe the population dynamics envisioned by Malthus himself. Nevertheless, it is used in many studies devoted to the analysis of the Malthusian economy in traditional societies.

Sometimes an iterative version of this model is used, implying calculations on an annual basis. Equation (1a) in the version put forth by Møller and Sharp (2009) has logarithmic form:

Birth and death rates are calculated from the simplified equations:

The population *N _{t} *is related to the population

here, *A, c _{0}, c_{1} , a_{0} … a_{3} *are certain constants

where *u *and *v *are certain constants. The resulting equation is:

This equation generates a series of population values. If the population at the initial moment equals a million (*i.e*.*, N _{1 }=1*), then

Another iterative version of the model is the one developed by Ashraf and Galor (2011). Beginning with equation (1a), the authors of this model take into consideration the number of adults and children, and optimize expenses. They, nevertheless, ultimately come to the same equation (8).

One more version of Wood's model is that of Voigtlander and Voth (2009). They use equation (1a), but replace equations (3) and (4) with (3a) and (4a):

where *b _{0 }*and

where *p *and *q *are certain constants. The differential equation (6a) has the time-independent solution *N _{t} = N_{0 }= p^{ 1/η(m – n)}*, which represents a horizontal line. As with the above model, the solution curves that are beneath this line monotonically increase and those lying above the line – monotonically decrease. Thus, this model has the same limitation as Wood's model and its other derivatives: it does not offer oscillating solutions.

Brander and Taylor (1998) have suggested another popular model. This model analyzes some abstract renewable resource consumed in the course of human activities. For example, it might be forest resources or soil yield. *S _{t}* is the available amount of this resource (in year

where *r *and *u *are certain constants. The first term on the right side describes the process of natural resource renewal; the second term describes resource depletion owing to economic activity. The population number is given by the following equation:

where *d *and* v *are constants, and *d < *0 in this case. This equation shows that natural population growth depends on the availability of resource *S _{t}*.

Brander and Taylor have shown that the system of equations (8)–(9) has oscillating solutions: when the resource is abundant the population grows, when it is exhausted the population decreases until the resource is renewed. Brander and Taylor refer to their model as ‘Malthusian-Ricardian’. Initially, the model was intended to describe the economy of Easter Island, but afterwards it got wider application as a sufficiently general model of the Malthusian economy (*e.g*., Maxwell and Reuveny 2000; D'Alessandro 2007). It is essential to note, however, that resource *S _{t}* in the Brander–Taylor model is not the harvest gathered by farmers. According to Brander and Taylor, the crop is denoted by the term

Up to this point, I have confined my discussion to the analysis of simple models of the Malthusian economy that consist of no more than two differential equations. Naturally, there exist more complicated models (*e.g.*, Usher 1989; Komlos and Artzrouni 1990; Chu and Lee 1994; Galor and Weil 2000; Lee and Tuljapurkar 2008) that allow for better behavioral freedom and offer oscillating solutions, as well. Many such models have been constructed within the framework of cliodynamic research actively performed in Russia and the USA (*e.g.*, Tsirel 2004; Korotayev, Malkov, and Khaltourina 2005, 2006; Korotayev, Malkov, and Grinin 2007; Turchin 2007, 2009; Malkov 2009). However, almost all models described in literature have the same drawback: they contain uncertain coefficients whose values are unknown and cannot be determined in principle. The more complicated is a model, the more uncertain coefficients it contains. Meanwhile, these coefficients determine the behavior of the model, and different coefficient values result in different population dynamics. Thus, an uncertainty originates: as coefficient values are unknown, it is also uncertain which of the possible behavioral variants corresponds to historical reality and which of them could not possibly be realized.

In the remainder of this article, I would like to discuss two simple models that contain no uncertain parameters and, in my opinion, are sufficiently adequate for the description of the Malthusian population dynamics. Within the first model, *N _{t}* is the population in the year

*K _{t}* in this logistic equation denotes the carrying capacity (

Thus, we have the simplest system of two differential equations (10)–(11). This system has an equilibrium state, when the population and stock remain constant, namely, when *K _{0 }= = N_{0 }= a – d.*

If *N* in the equation for *dP/dN* tends to 0, we will obtain the harvest *a/d* (number of consumption rates) gathered by one farmer in favorable conditions (when the population is small and he or she is able to cultivate the maximum area). Thus, the value *q = a/d *shows how many households one farming family can support. The history of agricultural societies shows that *q* usually varies within the limits *1.2 < q < 2. *We can express *a *and* d* in terms of *q* and *N _{0}:*

*d = N _{0 }/ *(

*N _{0} *can be conventionally set equal to 1 and thus, there are two constants in this model,

*T = *2*π / √*(*r – r/q – r ^{2}/*4)

The period *T* decreases when *r* and *q* increase, and increases, respectively, when these values decrease (Table 1 and Fig. 1).

*Table 1*

**Period of oscillations with various r and q (in years)**

Thus, the period of oscillations in this model is comparable to the duration of secular demographic cycles observed in the history of many states (Turchin and Nefedov 2009).

**Fig. 1. Example of calculations using the model ( r = 0.01; p = 1.2)**

The dynamics of the agricultural population according to this model have an oscillating nature. In theory these oscillations die out and the system tends to the equilibrium state, but various random impacts and influences neglected herein (*e.g*., catastrophic crop failure) disturb the system equilibrium, after which there starts a new series of dying oscillations. The peculiar feature of the agricultural society is that its economic dynamics substantially depend on such a random value as the crop yield. The random factors that impact such systems are generally assumed to be exogenous; however, the dependence on crop yield variations is an intrinsic, endogenous feature of agricultural production. Therefore, one arrives at the conclusion that a special random value describing crop yield must be incorporated into the ideal model of the Malthusian cycle. This can be conveniently done within the iterative model where the calculations are made from year to year.

For convenience, I consider production years to start with the harvest, not a specific calendar date. The population number *N _{t}* at the beginning of year

Let *a _{t} *represent the annual crop yield

*Y _{t }= a_{t}N_{t} *if

If there is a grain surplus in the year *t*, that is per-capita production *y _{t}* =

The population growth rate *r _{t}* is determined as the ratio of the population

Considering the typical case from the Middle East or Russia from the sixteenth to the eighteenth centuries, when every family could obtain two minimal consumption rations from one standard parcel, one can assume *a _{0} = *2 for the numerical experiment. The scatter of crop yield (ratio

**Fig. 2. Example of calculation using this model for r^{0} =1.02, p^{0} = 2, a_{0} = 2, а_{1}=1.2, p_{1}=1.05**

Naturally, this model describes just the basic mechanism of the demographic cycle omitting many details (*e.g*., the existence of the state and military elite, the emergence of large landowners). Such factors are taken into account in other models (*e.g.*, Nefedov and Turchin 2007) and the calculations made using these models show that the qualitative cyclic pattern changes insignificantly compared to the suggested model. On the whole, it seems quite certain that the availability of grain stock in farms contributes to a long-term economic stabilization. However, the population growth results in the stock depletion, and, sooner or later, major harvest failures provoke catastrophic starvations followed by events like epidemics, uprisings of starving people, and/or invasions by external enemies seeking to take advantage. As a result, the population size can decrease even by half and a new demographic cycle starts. While the model calculations suggest that this new cycle might start immediately after the catastrophe, in real life such crises as wars and uprisings have some inertia and impede economic revival. In such cases, stabilization is delayed.

Finally, it is worth noting that after the publication of Wood's model economic historians came to consider the Malthusian economy as a system where the population size cannot exceed the carrying capacity and, consequently, the ‘Malthusian crisis’ is impossible. For example, Read and LeBlanc (2003: 59) suggest that

… there is a standard model for the pattern of human population growth and its relationship to carrying capacity (*K*), namely, that most of the time human populations have low to nonexistent rates of growth… The model is often implicit and may simply assert that, until recently, population sizes have always been well below *K *and growth rates very low.

But Le Roy Ladurie, Postan, Hatcher, and many other economic historians insist that ‘the Malthusian crises’ were quite common phenomena in lived history, a fact acknowledged by Wood himself. The models described in this article show that the inevitability of similar crises arises from the simple laws that rule the functioning of agrarian economies.

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