Model structure and analytical methods

action and the associated institutions. The need for collective action emerges from increases in population numbers, ability to harvest the re- source, or both. That collective action is an emergent property of an evolving system suggests that successful institutions would be very tightly woven into the lives of the participating agents. Work concerning culture, social structure, and ecology from this perspective has roots in anthropology. In contrast to the view that social structure was largely a product of unpredictable historical processes, in the 1950s Julian Steward suggested that social structure and modes of subsistence were causally linked Moran, 1990. Subsequently, researchers, e.g. Geertz 1963, Vayda and Rappaport 1968 and Vayda 1969, took the ecosystem rather than culture as the primary unit of analysis in attempts to explain culture as an evolved adaptation to the environment. Pigs for the Ancestors, in which Rappaport 1968 proposed that cultural pro- cesses, such as the ritual cycle of the Tsembaga play critical regulatory functions in their ecosys- tem, is probably the best known work in applying the ecosystem concept in anthropology. Rappaport was subsequently criticized for his focus on a ‘functionalist’ explanation of Tsem- baga culture. The argument was that the possible adaptive value of cultural processes cannot ex- plain how a particular set of institutions evolved. This is true, of course, for any evolutionary argu- ment: they can never be used to explain how a particular structure evolved. The best we can do is better understand how particular structures might be adaptive. This is valuable in itself — it can help us understand what characteristics different sets of institutions capable of preventing popula- tions from destroying their resource base and thus going extinct might share. This is the spirit of the analysis of the Tsembaga model presented here. In another recent work on traditional societies, Brander and Taylor 1998 focused on Easter Island. The authors combined a simple produc- tion structure with open access exploitation of a renewable resource to shed light on why the cul- ture of Easter Island vanished while other islands occupied by Polynesians did not experience the same fate. The main point made by Brander and Taylor 1998 is whether or not a population would experience a boom and bust cycle depends on an ecological parameter: the intrinsic growth rate of the palm species on which the Polynesians depend for food. The authors question why insti- tutional changes did not emerge to prevent the collapse. In this case, perhaps, social capital did not evolve sufficiently fast for the population to adjust to its new environment. Indeed, some so- cial change did occur: the great statues were top- pled, but this was probably not helpful in coping with their degraded resource base. On the other hand, the Tsembaga may have been more lucky.

3. Model structure and analytical methods

The models of Easter Island and the Tsembaga of New Guinea developed in Brander and Taylor 1998 and Anderies 1998a, respectively, have the same general form. They are simple dynamic models of renewable resource use where resource consumption affects human fertility. The most basic form of such a model must incorporate at least two state variables: the stock of renewable resources upon which the population depends de- noted as St, and the human population or Fig. 1. Caricture of the Tsembaga and Easter Island models. Table 1 Summary of model attributes Production structure Model Demographics Ecosystem Tsembaga Preferences: meet minimum food Exponentially declining with Logistic regeneration of forest after burn requirements increased nutrition Production technology: Cobb–Douglas in labor and Natural Capital reasonable for slash and burn agriculture on fixed area Social capital: ritual cycle Preferences: Cobb–Douglas in food Logistic regeneration of BT Easter Island Linearly declining with increased and manufactures nutrition palm forest Production technology: mass action in labor and Natural Capital reasonable for harvestable resource Social capital: none Same as BT Easter Island Preferences: Stone-Geary in food and Extended Easter Linearly declining with increased manufactures nutrition Island Production technology: mass action in labor and Natural Capital reasonable for harvestable resource Social capital: none available labor pool as is the custom in econom- ics denoted by Lt. The basic model structure is depicted in Fig. 1. The figure depicts three interacting compo- nents; the two state variables and the socio-eco- nomic system that governs their interaction. The nature of the socio-economic system depicted in the center is characterized by individual prefer- ences, production technology, and social capital. The interaction of these components governs the dynamics of the model. The model can be stated in a general as dS dt = RS − HS, L 1a dL dt = GH, LL 1b where RS is the renewal rate of the renewable resource, HS, L is the harvest rate of the re- Fig. 2. Two main model structures: a attainable steady state, b unattainable steady state and limit cycle. source, and GH, L is the per capita growth rate of the human population. The roles these func- tions play in the model are caricatured in Fig. 1. In the context here, RS and GH, L are taken as biophysically determined. The form of HS, L, on the other hand, is based on the nature of the socio-economic system and will be our focus. Table 1 summarizes the specific assumptions for each of the models discussed here. This model given by Eqs. 1a and 1b is a simple dynamical system in two dimensions, and its range of behaviors is well known. Trajectories will either approach an equilibrium monotoni- cally, after a series of damped oscillations, or not at all. If trajectories do not approach an equi- librium, they will approach a limit cycle and undergo persistent oscillations. Fig. 2 illustrates these possibilities. In graph a, any reasonable initial condition with high biophysical capital and low population will evolve to a stable steady state perhaps through a series of damped oscillations. This corresponds to the case where either individual behavior, collective behavior, or both are such that the resource is not degraded to the point where it can no longer sustain the population. In graph b, on the other hand, no reasonable initial condition with high biophysical capital and low population will evolve to a steady state, and will instead converge to a limit cycle. The fact that a model evolves to a limit cycle as opposed to a steady state does not necessarily mean that it is less preferable. Whether such behavior is less preferable depends on nature of the limit cycle, namely, the amplitude and characteristic time scales involved. Consider graph b in Fig. 2 and the points labeled A, B, and C. Important to whether a limit cycle is less preferable is its amplitude and period, i.e. the distance in time and state space between the above points. For example, the case in which the time between points A and B is 300 years, the time between points B and C is 10 years, the time between points C and A is 100 years, and the population difference between points B and C is large, is undesirable. This corresponds to the growth and development of a society or civiliza- tion depending on the scale causing the slow degradation of the resource base A “ B, which was followed by a rapid decline in the society B “ C. The collapse of the society is followed by a slow recovery of the resource base C “ A, after which the cycle may and probably has many times in the past repeat itself. The key point is that the large fluctuation from B to C occurs on a human time scale — 10 years, and could be devastating. The slow degradation and recovery phases occur over much longer time scales, i.e. the resource base will recover, but over what period of time? On the other hand, a small amplitude, low period limit cycle may be a desirable outcome. It turns out that the ritual cycle of the Tsembaga serves to move the system away from the large amplitude fluctuations in behavior b to a limit cycle with very small amplitude fluctuations driven by the periodicity of the ritual cycle. Nonetheless, the ritual cycle prevents the in- evitable degradation of the resource base associ- ated with model behavior b. The point, graphs a and b only serve as illustrations of the ex- tremes — there are many possibilities in between, as we shall see. A successful institution need not return the system to behavior a, it need only move it away from behavior b. The interesting questions are what can cause the model to shift from behavior a towards behavior b and what institutional response can move the system away from behavior b. Unfor- tunately, finding the boundary in parameter space between the behavior exhibited in graph a from that shown in graph b is a difficult task in general. One of the advantages of the use of stylized models such as given by Eqs. 1a and 1b is the relative ease with which parameter space can be explored for different classes of behavior. The behavior of more complex mod- els, such as agent based models or detailed simu- lation models, can be extremely difficult if not impossible to accurately characterize. The closer a model gets to capturing reality, the more difficult it is to interpret. Of course, the more stylized the model, the more stylized the facts one can derive from it. Nonetheless, stylized facts can be useful in their own right, and help guide more detailed enquiry. Even with simple models, the analysis is not easy. The two model behaviors a and b are