A Compact Model of Pre-Industrial Population Growth

 

John Komlos, Department of Economics, University of Munich

Sergey Nefedov, The Russian Academy of Sciences, Institute of history and archaeology, Ural department

 

It is published:  Historical Methods. 2002. Vol. 35. ¹ 2. P. 92-94.

 

Abstract: We propose a compact macro-model of pre-industrial population growth between the Middle Ages and the demographic revolution. We attempt to capture two salient features of the demographic history of this epoch, i.e., that population growth was slow on average and cyclical, but there were phases during which growth was relatively fast. Our model synthesizes Malthusian notions with endogenous technical progress. The latter continually shifts the constraints on population growth. The simulation based on the model is able to reproduce well the estimated size of the European population in this half a millennium.

 

        We propose a compact macro-model of pre-industrial population growth conceived to capture its two salient features: a) that growth over the long run was extremely slow on average, in spite of the fact that there were phases during which population expanded relatively rapidly; and b) that growth was cyclical (Coale 1974; Durand 1967, 1977; Biraben 1979, 23; Schacht 1981).[1] Our focus is Europe between 1200 and 1750, i.e., the last half millennium prior to the demographic revolution; i.e., the demographic and industrial revolutions are not part of our model.[2] Some reasonable estimates do exist for the European population during this epoch (McEvedy and Jones 1978), and we use them as the basis of  our simulation model.[3] We assume, in keeping with the current consensus, that the epoch under consideration was essentially Malthusian (Malthus 1798),[4] that is, constrained by the availability of nutrients, and that there was an incessant contest between population growth and society's resource base (Cameron 1989; Braudel 1973; Artzrouni and Komlos 1985; Dokkum 1999). Although the secular trend of population growth was ever upward, the cycles testify to the continued existence of the Malthusian population trap: population could not grow beyond an upper bound imposed by the resource, technology, and capital constraints of the economic structure in which it was imbedded (Habakkuk 1953; Postan 1972; Abel 1974, 1980).[5]

We assume that population growth can be described by a logistic-type function, such as:[6]

(1)                                   

Where P(t) is population at time (t), r is 1 plus the rate of population growth without any food constraint (assumed constant, i.e., the limit as K). K(t) is the maximum carrying capacity of an ecological niche, given the current level of technology, measured in terms of  the number of people that can be supported. K(t) is the maximum population size that can be sustained with the maximum exertion of labor, given the level of technology. As P(t)K(t) the denominator reduces to r, and therefore cancels out the r in the numerator. Hence, as the size of the population approaches carrying capacity, P(t+1) = P(t), and population growth approaches zero.

K(t) corresponds to a certain amount of available food, assumed to be made up of, K1, current food output, and K2, inventories of grain and of animal stock:  K(t) = K1(t) + K2(t). K1 is produced by a Cobb-Douglass production function:

(2)                                           K1(t) =  [T(t)1/3 P(t)2/3]

where T(t) is the current state of technology. Technology is assumed to be produced by people, and its level is proportional to the number of people who ever lived, with c being a constant factor.

(3)                                                                                          T(t+1) = T(t)+cP(t).

We assume that people produce new ideas, and these ideas, in turn, are converted into technology (Eq. 3). This formulation resembles recent endogenous growth models (Aghion and Howitt 1998). Simon was among the first to conceive of “endogenous knowledge in connection with population growth” (Simon 1977, 2000, 39).[7] Kremer builds on this supposition as follows: “[He] adopts Kuznets’ and Simon’s view that high population spurs technological change because it increases the number of potential inventors.... in a larger population there will be proportionally more people lucky or smart enough to come up with new ideas” (1993, 685). This can be seen as a complement to the Boserupian notion that population pressure forces people to adopt new technologies (Boserup 1965; Lee 1986). The evolution of K2 is given by:  

(4)                                                  K2(t+1)=K2(t)+q[K1(t)-P(t)]

where q is a constant. Eq (4) implies that when current output is above conventional norms [K1(t) -P(t)]>0 then inventory levels are increased by a factor q of the excess, and the remainder (1-q)[K1(t) -P(t)] is consumed by the population. In contrast, whenever current output is insufficient to meet the current needs, [K1(t) -P(t)]<0 , a share q of the shortfall is made up from inventories. If this situation persists then the inventories are all used up and, in fact, K2(t) can become negative. This is our conceptualization of famine and wars, when even a part of the current output is destroyed, and these disasters reduce the ecological niche K(t) in Eq. (1). In such cases the death rate increases, and if K1(t) continues to be less than P, the famine persists, and the wars and epidemics intensify until the population decreases sufficiently so that K1(t) >P.

With this four-equation model we start the simulation in the year 1200 and do annual iterations. We choose the following values of the constants and initial conditions:[8] r = 1.022; T(0) = 48; c=0.0019; K2(0) = 30; P(0) = 26; C = 0.0019; q = 0.12.

A typical outcome of the simulation, presented in Figure 1, indicates that the model is actually capable of replicating quite reasonably the population estimates of McEvedy and Jones not only in most of their cyclical characteristics (amplitude and period), but also in terms of the important turning points.[9] The demographic crisis associated with the Black Death is reproduced very well without an external shock to the population.[10] Contrary to the notion that the population downturn was exogenous, the model emphasizes their endogenous nature, i.e., that the cycles can be conceptualized as being an inherent part of the European demographic system. Our interpretation is that the European population has reached a Malthusian ceiling by 1300, so that a prolonged downturn would have occurred in any event. To be sure, the Black Death exacerbated the process and was perhaps even its proximate cause, but not, it seems to us, its fundamental determinant.[11]

The proposed model is essentially a food-constrained quasi-homeostatic conceptualisation of population growth between the Middle Ages and the demographic revolution. The model does not have stochastic elements, exogenous shocks, or climatic change which could alter fundamentally the nature of the ecological niche. Rather, the trend and cycles are determined by the model’s functional forms, and by the initial conditions. We have assumed that technology is produced endogenously by people, and that population growth can be described by a logistic-type function. With such a compact model we are able to reproduce rather well the cycles and the long-run growth of population between the Middle Ages and the demographic revolution.

 

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Notes

We appreciate the comments received from George Alter, Marc Artzrouni, Jack Goldstone, Eugene Hammel, Peter Turchin, and an anonymous referee on an earlier version of the paper. Any remaining errors are the sole responsibility of the authors.


[1] Goldstone (1991) conceives of these as ecological cycles. On the application of other types of mathematical models to historical processes see Turchin (2002).

[2] For such models see, for example, Steinmann et al. ( 1998), Kögel and Prskawetz (2001), Galor and Weil (1999, 2000), and (Komlos and Artzrouni 1990).

[3] For the theory, practice and limitations of simulation models see, Pindyck and Rubinfeld 1998) and Hodder (1978).

[4] See also Leibenstein (1954, 8).

[5] On population cycles see Galloway (1986, 1988).

[6] This bears some similarity to Pearl (1926).

[7] See also Simon and Steinmann (1981) and Hammel and Howell (1987).

[8] We adopted the following procedure: we chose the initial values of the variables P(0), and then by trial and error suitable values of K2(0) and T(0). K2(0), and T(0) do not have any substantive implications – since we do not have historical estimates available, and therefore can be chosen freely. We believe that the initial value of P(0) given in McEvedy and Jones is overestimated. This is also the implication of the fact that their population growth between 1200 and 1300 is slower than that found for England (Wrigley et al. 1997). As a consequence, we prefer a lower starting value than that given by McEvedy and Jones. Given the initial values for P(0), K2(0), and T(0) we chose the constants in such a way as to minimise the sum of squared residuals between McEvedy and Jones’s values and those predicted by our model. We were able to accomplish this by regressing McEvedy and Jones’s values on the population estimates implied by our model in order to find the optimal values of the constants, given the initial guess of the constants. K2(0) determines the extent of the ecological niche, and includes such factors as include unexploited labor intensity. As a consequence, the larger is K2(0) the larger would be the oscillations in population. An increase in T(0), the initial technological base, would induce a once-and-for-all upward shift in the curve, but would not affect either the amplitude or the periodicity of the fluctuations. The consequence of a larger c would be faster population growth. An increase of q or r would decrease the period of oscillations.

[9] The population value for 1200 is an exception. We interpret this discrepancy as an indication that McEvedy and Jones’s estimate is inaccurate for 1200. Turchin comes to a similar conclusion. Referring to McEvedy and Jones’s “conservatism”, he states, “This in many ways excellent compilation of long-term population dynamics,... suffers, in my opinion, for one failing, the tendency of the authors to underemphasize the degree of fluctuation” (2002, 151). Hence, we do not think that the more pronounced upswing of our estimates in the pre-plague era is worrisome. See also the higher population growth rates in the 13th century in (Wrigley et al. 1997).

[10] This pattern is not directly the outcome of the starting values, even if these do determine the amplitude and periodicity of the cycles. Rather, the model’s conceptualisation is the fundamental cause of our views on the Black Death.

[11] There are indications that population actually started to decline even before the plague. This was the case, for instance, even in Greenland, where the plague did not appear at all (Lynnerup  1998, 126).