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Culture, economic structure, and the dynamics of ecological economic systems Anderies, John M. 1998

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CULTURE, ECONOMIC STRUCTURE, AND THE DYNAMICS ECOLOGICAL ECONOMIC SYSTEMS  By John M . Anderies B . S c , Colorado School of Mines, Golden, Colorado, U.S.A, 1987 M . S c , University of British Columbia, 1996  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF  PHILOSOPHY  in THE FACULTY OF GRADUATE  STUDIES  DEPARTMENT OF MATHEMATICS INSTITUTE OF APPLIED  MATHEMATICS  We accept this thesis as conforming to the required standard  T H E UNIVERSITY  OF BRITISH  COLUMBIA  July, 1998 © John M . Anderies, 1998  OF  In presenting  this  degree at the  thesis  in  partial fulfilment  of  University of  British Columbia,  I agree  freely available for reference copying  of  department  this or  publication of  and study.  his  or  her  that the  representatives.  may be It  is  this thesis for financial gain shall not be  permission.  Department The University of British Columbia Vancouver, Canada  DE-6 (2/88)  requirements  1 further agree  thesis for scholarly purposes by  the  for  an  advanced  Library shall make it  that permission for extensive granted by the understood  that  allowed without  head  of  my  copying  or  my written  Abstract  In this thesis several models are developed and analyzed i n an attempt to better understand the interaction of culture, economic structure, and the dynamics of human ecological economic systems. Specifically, how does the ability of humans to change their i n d i v i d u a l behavior quickly and easily i n response to changing environmental conditions (behavioral plasticity) alter the dynamics of human ecological economic systems? W h a t role can cultural and social institutions play i n affecting individual behavior and thus the dynamics of such systems? Finally, how do assumptions about the production and consumption of goods and services within human ecological economic systems affect their dynamics. M u c h work concerning interacting economic and natural processes has focused on technical issues and problems w i t h standard economic thought. Less attention has been paid to the role of human behavior. T h e work presented herein addresses both but emphasizes the latter. Three models are developed: a model of the Tsembaga of New Guinea which focuses on the roles of behavior, cultural practices and ritual on the dynamics of the Tsembaga ecosystem; a model of Easter Island where the linkage between economic models of utility and the resulting behavioral model is studied; and finally a model of a modern two sector economy w i t h capital accumulation where the emphasis is evenly split between behavior and economic issues. T h e m a i n results of the thesis are: behavioral plasticity exhibited by humans can destabilize ecological economic systems and culture and social organization can play a critical role i n offsetting this destabilizing force. Finally, the analysis of the two sector model indicates that there is a window of feasible investment levels that w i l l lead to a  n  sustainable economy. The size of this window depends on culture and social organization, namely the way economic growth is managed and how the associated benefits are distributed. T h e two sector model clarifies the idea of a sustainable economy, and allows the possibility of reaching one to be clearly characterized.  111  Table of Contents  Abstract  ii  List of Tables  vii  List of Figures  viii  Acknowledgement  xi  1  Introduction  1  2  T h e M o d e l i n g Framework  6  2.1  D y n a m i c a l Systems Models of Ecological Systems  7  2.2  H u m a n economic ecological systems  11  2.2.1  Background  11  2.2.2  T h e general model  15  2.3 3  A n a l y t i c a l methods  22  C u l t u r e and human agro-ecosystem dynamics: the T s e m b a g a of  Guinea  New 25  3.1  T h e ecological and cultural system of the Tsembaga  26  3.2  .The model  29  3.2.1  Definitions  29  3.2.2  Tsembaga subsistence and the population growth rate, / i  30  3.2.3  T h e ecology of slash-and-burn agriculture  32  iv  3.2.4  D y n a m i c behavior of the model  41  3.4  Behavioral plasticity  47  3.5  Modelling the ritual cycle  52  3.5.1  T h e parasitism of pigs  53  3.5.2  T h e ritual cycle  54  3.5.3  T h e behavior of the full system  60  Conclusions  61  Non-substitutibility in consumption and ecosystem stability  65  4.1  T h e Easter Island model  65  4.2  M o d e l Critique  70  4.2.1  72  4.3  4.4 5  36  3.3  3.6 4  T h e food production function  Behavioral plasticity and collapse  A d d i n g behavioral plasticity to the Easter Island model  74  4.3.1  76  M o d e l analysis  Conclusions  80  T h e dynamics of a two sector ecological economic system  81  5.1  Simple economic growth models  82  5.1.1  Basic laws of production and the theory of the firm  83  5.1.2  Consumer behavior  85  5.2  T h e ecological economic model  86  5.2.1  T h e economic system  87  5.2.2  C o m p u t i n g the general equilibrium  93  5.3  T h e ecological system model  102  5.4  Analysis of the M o d e l  104  5.4.1  104  Investment, distribution of wealth, and ecosystem stability . . . . v  5.4.2 5.5 6  Nonrenewable natural capital, efficiency, and flows between industries 115  Conclusions  130  Reflections and future Research  138  Bibliography  143  vi  List of Tables  5.1  Table of important symbols  136  5.2  Table of important symbols, continued  137  6.3  E q u i l i b r i u m consumption versus b  141  c  vii  List of Figures  2.1  Isolated predator-prey model  9  2.2  Predator-prey model embedded i n an ecosystem  10  2.3  T h e circular flow of exchange i n standard economics  12  2.4  Economic system i n the proper ecological context  15  2.5  T w o m a i n model structures: (a) attainable steady state, (b) unattainable steady state  23  3.1  G r a p h i c a l representation of nutrient cycling process i n a forest  34  3.2  Soil recovery curves  35  3.3  T h e production surface for cotton lint  38  3.4  C o m p a r i n g the Cobb Douglas and von Liebig functions  39  3.5  Bifurcation diagram for swidden agriculture  42  3.6  Bifurcation diagram for swidden agriculture  43  3.7  T w o parameter bifurcation diagram for the swidden agriculture model . .  44  3.8  Change i n dynamics accross bifurcation boundary  45  3.9  Bifurcation diagram with c™ as the bifurcation parameter i n the swidden  . .  ax  agriculture model  49  3.10 Tsembaga ecosystem l i m i t cycles  51  3.11 Work level (curve (a)) and food production (curve (b)), over time.  . . .  52  3.12 T h e influence of pigs on system dynamics  54  3.13 T h e r i t u a l cycle of the Tsembaga  55  3.14 F o r m of g(x)  57  viii  3.15 T h e dynamics of the ritual cycle  58  3.16 A n example of of the human (a), and pig (b), population trajectories under cultural outbreak dynamics  59  3.17 Sample trajectories for the full model  61  3.18 L i m i t cycle for the full model  62  4.1  Population and resource stock trajectories for Easter Island model from ([9]) 71  4.2  Per-capita growth rate from the time of initial colonization to the time of first European contact  72  4.3  Bifurcation diagram for modified Easter Island model  77  4.4  Population and sectoral labor proportion trajectories  78  4.5  Trajectories for population and total labor i n each sector over time  5.1  Schematic of two sector ecological economic model  88  5.2  Trajectories of wages, capital, and labor as the economy adjusts  99  5.3  Surface plot of utility function showing optimal combination of labor and  . . .  79  capital to agriculture  100  5.4  E x a m p l e of economic system dynamics  101  5.5  Simple economic growth model  103  5.6  State varible trajecories  110  5.7  E q u i l i b r i u m Labor, capital, and consumption trajectories  110  5.8  Bifurcation diagram for simplified model  Ill  5.9  Change i n dynamics as the bifurcation boundary is crossed  112  5.10 State varible trajectories  114  5.11 C y c l i c a l Labor, capital, and consumption trajectories  115  5.12 Resource good preference versus K  n  5.13 Effhciency curves  for different values of \k  n  117 118  ix  5.14 State variable trajectories  120  5.15 E q u i l i b r i u m states versus \k  121  5.16 C a p i t a l and investment good preferences over time  122  5.17 Bifurcation structure for full model  123  n  5.18 T w o parameter bifurcation digram for investment-good preference and k .  126  r  5.19 T w o parameter bifurcation digram for investment-good preference and (3pj. 128 5.20 T w o parameter bifurcation digram for investment good preference and  x  R .130 am  Acknowledgement  I would like to thank Dr. Colin Clark for his financial and moral support over the past 5 years; his many readings of my work and helpful ideas and comments. I would also like to thank my committee members for helpful comments and ideas as I developed the thesis, especially Leah Keshet and James Brander. Finally I am greatly indebted to my wife and friend Margaret; thanks, your turn.  xi  Chapter 1  Introduction  Since the 1970's, the impact of human activities on ecosystems has been receiving more and more attention. Through this increased awareness, 'sustainability' - the basic question of whether and how human populations can continue to live on earth indefinitely without threatening the survival of all biological populations - has become an important international issue, and the focus of much research. Unfortunately there are deep divisions between different groups of people regarding the fundamentals of the sustainability issue. Examples of such divisions are everywhere - i n the popular media and i n academic debates. For example, several authors have argued that the economic process is fundamentally influenced by entropic decay [27, 19] while others [67] argue that the entropy law is irrelevant because the earth is a thermodynamically open system. Some experts are very concerned about the degradation of agricultural ecosystems (soil erosion, etc.) [28, 49, 50] while others praise the power of technology to "liberate the environment" and give us "effectively landless agriculture" [6](p.  172) v i a "[a] cluster of innovations including  tractors, seeds, chemicals, and irrigation, joined through timely information flows and better organized markets [that will] raise yields to feed billions more without clearing new fields" [6](p. 171). T h e a i m of this thesis is to address several aspects of this division. For this purpose, different views on sustainability can be divided i n to two broad classes: A . (expansionist view) Sustainability is mainly a technical issue. T h e present paradigm 1  Chapter  1.  Introduction  2  of economic growth can continue indefinitely as long as increases i n efficiency offset increasing pressure on natural resources and ecological systems. B . (steady state view) Sustainability involves a comprehensive understanding of the place of human populations w i t h i n ecosystems. Achieving a sustainable world w i l l require a fundamental paradigm shift concerning the way humans lead their lives. There are two key points to note about these different positions. F i r s t , the existence of this difference hinders the development of effective policy to govern the relationship between human economic and ecological systems.  Second, position A is the paradigm  of choice i n present policy formation without sufficient evidence that it is the "correct" view. Clearly, the only way society can move toward a sustainable state is to extract important truths from both views and w i t h them forge some strategy to guide future human environmental interactions. This is not an easy task for two reasons. F i r s t , human agroecosystems may be too complex to understand i n enough detail to be useful i n policy formation. Second the views of people on either side of the issue may be, as Rees [54] notes, based more " [on] differing fundamental beliefs and assumptions about the nature of human-kind-environment relationships" rather than fact. A t the heart of the issue are assumptions that underly the models and arguments made i n support of either view (see the forum i n [7] for a collection of recent papers on the continuing debate). I believe there are three fundamental questions the must be addressed before real progress can be made i n resolving differences concerning the concept of sustainability. F i r s t , the expansionist view assumes that our ability to solve problems w i t h technology is necessarily a good thing. Is this so?  Second, how important are our cultural and  social institutions i n determining whether a human economic system is sustainable? F i nally, how do assumptions that underly economic growth models used to support  the  Chapter  1.  Introduction  3  expansionist position affect the dynamics of human ecological economic systems?  The  m a i n thrust of this thesis is to develop a modeling framework to help answer these three questions. M y approach is to develop dynamical systems models to study humans as ecological populations. These models focus on how human behavioral and cultural systems interact w i t h the environment, and they are deliberately stylized to avoid the trap of generating models that are too complicated w i t h too many assumptions to be of practical use, e.g. [43, 4 4 ] . O n l y the most basic features of general human economic ecological systems are included. In attempting to answer the questions posed above I develop three different models of this type, two involving simple societies of anthropological interest and one modern economic system w i t h capital accumulation, w i t h the following objectives: • T h e first model addresses the first two questions i n the context of a simple human agro-ecosystem.  T h e human ability to modify behavior quickly and over a wide  range of different activities, (defined as behavioral plasticity), is emphasized. The role that behavioral plasticity plays i n the dynamics of a human agro-ecosystem is studied i n detail.  Of special interest is the destabilizing effect of behavioral  plasticity, and the stabilizing role culture and social organization may play. • T h e second model is directed towards the third question. Here, a linkage between economic concepts and an evolving ecological economic system is developed. Economic models of behavior based on the optimization of some measure of utility are introduced. U t i l i t y measures that result i n realistic behavior i n the context of an evolving ecological economic system are identified. A g a i n , the destabilizing effect of behavioral plasticity is highlighted. • In the t h i r d model, the ideas developed i n the first two models are combined to develop the model of the modern economic system. T h i s model model addresses  Chapter 1.  Introduction  4  all three questions i n the context of economic growth i n a bounded environment. In addition to shedding light on the three fundamental questions posed above, the models developed i n this thesis provide tools to study operational aspects of sustainability. T h i s is very useful since much of the problem w i t h the sustainability concept is that it is easy to imagine what a sustainable state might be like, but few ask whether it is possible to get from our present state to a sustainable state. A s Rees [54] notes: "....sustainability w i l l require a 'paradigm shift' or a 'fundamental change' i n the way we do business, but few go on to describe just what needs to be shifted...".  Thinking  about a sustainable world is pointless unless we can find a way to get there. In a recent article, Proops et al. [52] emphasize the need to formulate a goal of sustainability, set an intermediate target, and develop feasible paths toward this goal. T h e analytical framework developed i n this thesis provides a flexible, simple, and precise means of studying (for a given set of assumptions) exactly what cultural attributes are sustainable or not, and more importantly, what key aspects affect the feasibility of potential paths to a sustainable human ecological economic system. T h e structure of the thesis is as follows. Chapter 2 outlines the background, assumptions and basic structure of the modelling framework. Next, i n Chapter 3 the modelling framework is applied to the society of the Tsembaga, a tribe that occupies the highlands of New Guinea. N e x t , the ideas developed i n Chapter 3 are extended i n Chapter 4 where a model proposed by Brander et. al [9] to explain the rise and fall of the Easter Island civilization is used to develop and study more advanced economic concepts typically used to model human consumptive and productive activities. These authors argue that the Polynesian culture that occupied Easter Island was mismatched to the ecosystem they found and thus perished. The authors also discuss the implications of their model for other societies that collapsed, and for our own society. T h e m a i n point is that more  Chapter 1. Introduction  5  complex economic models i n which agents exhibit maximizing behaviors based on a certain u t i l i t y function do not necessarily give rise to richer models behavior - indeed they can result i n very simple, not very realistic behavioral patterns. Here we emphasize how non-substitutability i n consumption fundamentally alters the behavior of the model and the nature of the approach to the sustainable state, and that realistic behavior depends on the inclusion of this aspect i n utility functions. Finally, pulling together the ideas of chapters 3 and 4, I develop a model of a two sector (a sector i n economics is a grouping of associated productive activities) economy and embed it i n a model ecosystem. T h e economy has an agricultural (bioresource) sector and a manufacturing sector. Economic agents (individuals who take part i n productive and consumptive activities within the economy) can devote the productive capacity of the economy to four different activities: the consumption of agricultural, manufactured, investment, and resource goods. This model includes a l l the components that form the basis of the current debate about human environmental interaction: we rely on flows from the environment but we can use our productive capacity to substitute for these flows, increase efficiency, reduce waste, and help regenerate the environment. Those holding the steady state view emphasize the importance of the former while expansionists emphasize the power and importance of the latter. W i t h the modelling framework developed herein, their interaction can be studied.  Chapter 2  The Modeling  Framework  In this chapter, the background and assumptions underlying the modeling framework are addressed. T h e modeling approach is outlined, and the general model that is employed throughout the thesis is developed. Next, the important features of the models that are important to the questions posed i n the introduction are discussed. Finally, the analytical techniques used to uncover these features are presented. W h e n trying to model the interaction between elements i n a system, e.g.  predators  w i t h prey, one competitor w i t h another, an organism w i t h its environment, one necessarily has to model the way each element affects how other elements change over time. T h e most common approaches are to write down differential equations, difference equations, functional differential equations (when age structure is important), or a stochastic process. Often several approaches are appropriate for a given problem so the choice of approach often depends on the intentions of the modeler. T h e models I develop i n this thesis are all deterministic dynamical systems. T h e advantage of this approach is that the models are clear and simple, allowing the underlying assumptions and concepts to be easily seen by inspecting the differential equations that constitute the model. Drawbacks are that implicit i n deterministic models is the assumption that everything is "well m i x e d " and there are no spatial or random effects allowed. T h a t is to say that each variable i n the model necessarily represents an average value of a particular quantity. Clearly no real system is well m i x e d and deviations from the average can substantially alter the dynamics of the system i n question. Fortunately, it is often  6  Chapter 2.  The Modeling Framework  7  the case that many aspects of a real system can be inferred from the structure of the "mean field" or average model given by the deterministic ordinary differential equation system. Studying the dynamics of such models is a difficult task. If the model is simple enough it can be studied by analytical methods.  T h e models i n this thesis are too complex  to study analytically. Fortunately, there are numerical techniques available that allow dynamical systems theory to be used on more complex systems. In the next section I w i l l briefly discuss the application of dynamical systems type models to ecological systems and explain how I extend them for the special case of human economic ecological systems.  2.1  D y n a m i c a l Systems M o d e l s of Ecological Systems  Ecologists have long used simple systems of differential equations to model ecosystems so as to understand how different behavioral patterns may effect the dynamics between individuals that interact i n the ecosystem. Because my interest is specifically w i t h behavior and environmental constraints, the way behavior is modeled, and the way a model is placed i n an ecological context are very important. I w i l l illustrate this by way of a simple example. Differential equation models of ecosystems often take form dx ~dt where x £  f(x,p)  describes the state of the ecosystem and p G  is a parameter vector. T h i s  type of model has been extensively studied (e.g. [65, 26, 11, 21, 42]). In such models, the behavior of organisms is often modeled by a functional response that is completely determined by the state of the system. For example the simplest Lotka-Volterra predator prey model given by  Chapter 2.  The Modeling  Framework  — dt *k  8  = =  rh — ctph -f3h + h , 7 P  (2.2a) (2.2b)  where /i(i) and p(i) are the prey and predator population densities, respectively. This model exhibits unrealistic neutral oscillations where predator and prey numbers can take on arbitrarily large values. T h i s is due to the fact that behavior is modeled too simply and there is no ecological context. Prey behavior is l i m i t e d to eating and growing. T h e y do nothing to avoid predators or carry out any other complex behavior. Predators die and eat prey; never changing their behavior whether they are hungry or full. T h e organisms are behaviorally rigid, or for our purposes, not behaviorally plastic. A l m o s t all animals have some measure of behavioral plasticity, and this is especially true of humans. Ecologists often include more complex behavior by introducing a functional response term to model the way a predator consumes prey. A t the very least, these models include some means of satiating the appetite of the predator. For example equations 2.2 could be modified by replacing the term aph i n equation 2.2a with the functional response g(h,p).  Holling [34]  proposed the functional response: g{h,p) = ^ p+ k  (2.3)  where k is the prey concentration at which the predator consumes at one-half its maxi m u m rate.  A s p increases, the rate at which prey are removed approaches ah; each  predator is consuming at a constant, m a x i m u m rate. Note that although some increased behavioral plasticity is added and the model is more realistic, the behavior or the predator is completely determined by the state of the system and not by any internal feedback. For example, if there are fewer prey and the predator becomes hungry, there is no mechanism in the model to allow the predator to change its strategy or work harder. If we attempt  Chapter 2.  The Modeling  Framework  9  to model a human ecosystem, this is a key feature to include. Indeed, i n chapter 3 we w i l l see just how important this is. To properly model a system where individual organisms are behaviorally plastic, we have to add equations that model the internal state of the organisms and how they influence behavior. I w i l l address this issue i n a moment, but first let me turn to the second point mentioned above, the ecological context. The predator prey model given by 2.2 is completely isolated from the environment. The equations model the system shown i n figure 2.1. In reality, ecological systems are not isolated but are embedded i n a physical environment and are dissipative; they continuously dissipate derivatives of solar energy.  r  *\  Predator  Prey  Figure 2.1: Isolated predator-prey model.  For a realistic model, we must include the fact that there is some abiotic component,  x, a  the m e d i u m through which this dissipative process occurs. A recent paper addressing this point [61], suggests that the equations of motion be written this way: x = f(x ,x,p,z(t),d) a  where x  a  (2.4)  are abiotic components, d describes the dissipative process, and z(t) represents  some external forcing.  This is just a general mathematical statement that instead of  modeling the system shown in figure 2.1 we must model the system shown i n figure 2.2. In such a model, the fundamental processes that make the interaction between predator and prey possible are included. In terms of equation 2.4, the abiotic components would include the soil structure of the ecosystem.  T h e forcing might be the weather  Chapter  2.  The Modeling  Framework  10  Figure 2.2: Predator-prey model embedded i n an ecosystem where the dependence on abiotic compents and the dissipative processes of nutrient generation and waste assimilation fueled by the sun is considered. patterns. The dissipative processes would include the metabolism of the plant community which generates nutrients, the animal metabolisms which convert the nutrients to energy and waste products, and the decomposer community that assimilates the waste and breaks it down for reuse. Only when these aspects are included can any ecosystem model be considered ecologically realistic. The most simple way that these important features can be included i n a model is by introducing a "carrying capacity" term. In a predator prey model the carrying capacity is often defined as the m a x i m u m number of prey that can be supported i n the given ecosystem thus lumping the dissipative process into one term. The model given by equations 2.2 could be modified to include this aspect along w i t h more complex behavior to read  Chapter 2.  The Modeling  Framework  11  aph  (2.5a)  p+ k  (2.5b) where K is the carrying capacity. This model yields a stable fixed point or a stable l i m i t cycle. T h i s is much more reasonable than the arbitrarily large fluctuations possible i n the model specified by equation 2.2. T h e key point I wish to draw out is the importance of behavior and ecological context i n ecological models. If we wish to extend this modeling framework to human ecological economic systems, these are key issues we need to address. Indeed, the issue of ecological context is fundamental i n the debate about  sustainable  development.  2.2  H u m a n economic ecological systems  2.2.1  Background  Most of the work on human economic ecological systems has been either i n the context of (optimal) economic growth, or the optimal exploitation of resources.  Unfortunately,  economic models often lack ecological context. The example above shows that modeling without proper ecological context may lead to quite absurd results, and economic models are no exception. For example, the model of Solow [58] i n the context of o p t i m a l economic growth w i t h exhaustible resources states that along an optimal growth path, constant net output can be maintained i n the face of dwindling resource inputs. Later, when further analyzing Solow's work, Hartwick [31] presented the savings rule: invest all rents from exhaustible resources (in replenishable man-made capital) to maintain constant net output indefinitely.  T h i s result is based on a model like that shown i n figure 2.3.  T h e economic  Chapter  2.  The Modeling  Framework  12  system is viewed as a circular flow of exchange between firms and households as shown on the left i n figure 2.3 interacting w i t h the physical world on the right. T h e physical world is often just viewed as a source of raw materials (to be optimally extracted as i n the case of the Solow/Hartwick model) and a sink for wastes.  Goods and Services  Factors of Production Figure 2.3: Schematic of the circular flow of exchange as perceived by standard economics. T h e connection to the real world, even as merely a source of raw materials and a waste b i n , is seldom shown.  Clearly, the underlying assumptions i n such models are critical to obtaining results such as those above. In the case above, it is assumed that the production of commodities, Y, is given by Y = K L N' a  (3  1  (2.6)  where K and L are man-made capital stocks and population respectively, N is a flow of  Chapter 2.  The Modeling Framework  13  natural.resources, and a , [3, and 7 are parameters assumed to satisfy a + /3 + 7 = 1. For the case where the population is held constant and there is no technological progress, the dynamical system for this optimal economic growth model is  (2.7a) (2.7b) where A is a constant representing the contribution to production of the fixed labor force, and C is total consumption of the population. T h e first equation states that capital, K, increases at a rate given by the total commodity production rate less what is consumed. T h e second equation states that the resource how diminishes (optimally) as resources are used up. Now, C is always less than or equal to AK N~* a  than you make) thus —— > 0. This implies that Kit)  (you can't consume more  > 0 for all t > 0 which results  in the right hand side of 2.7b being negative for all t > 0 forcing N(t)  to approach zero  asymptotically as time tends toward infinity. A glance at this model w i l l reveal its similarity to 2.2 where K is analogous to the predator and N is analogous to (in this case a finite stock of) the prey. T h e parallel I wish to draw is the similarity i n the growth function assumed for the predator and capital. T h e predator can still grow at very low prey levels if there are sufficiently many predators!  Similarly, the capital can continue to grow w i t h a very low resource flow,  as long as there are sufficient capital stocks. T h e absurdity i n the case of the predator model is obvious, and ecologists quickly modified this model as already discussed. The difficulty i n the economic growth model is more difficult to see, and economists have been slower than ecologists to modify such models. T h e Solow result depends on the assumption that the factors of production, man-made capital (a stock), and resources (a flow), are near perfect substitutes. M u c h of ecological  Chapter 2.  The Modeling  Framework  14  economics is concerned with exposing the underlying physical problems associated w i t h such models and developing more realistic models (for recent examples see [60, 12]). T h e emphasis of this work is the non-substitutability among different stocks and between stocks and flows. Even if these modifications were made to the Solow model, there is still no clear ecological context; the only connection to the physical world is through a finite stock of resources to optimally use up. H e r m a n D a l y [18] and Nicholas Georgescu - Roegen [27] were among the first (ecologically minded) economists to recognize the need to study the system shown i n figure 2.4 and to emphasize that i n addition to the issue of finite resource stocks, there is the issue of ecological context: we are embedded i n a natural world that is important to our survival regardless of its connection to the economic process. T h i s is the type of model which is developed and analyzed i n the rest of this thesis. T h e other key component that governs the evolution of an ecological economic system, namely human behavior, has received much less attention i n the literature than technical issues related to economic models and ideas. For example m a x i m i z a t i o n of u t i l i t y over the next twenty years is most often assumed as the primary goal driving behavior. This has two important consequences:  this assumption has become ingrained i n standard  economics, encouraging this behavior within society whether natural or not; i n policy formation the model implies that only the next few years are important. In defense of his model, Solow [59] makes this very point. He indicates that the m a i n purpose of these models is for planning over the next 60 years. How feasible is this planning strategy? Before turning our attention to the mathematical model, note two m a i n points: • A n y realistic model of the interaction of organisms w i t h their environment must address the role of individual behavior. • M a i n t a i n i n g realism i n the way that different inputs interact i n the productive  Chapter 2.  The Modeling  Framework  15  Figure 2:4: Schematic of the circular flow of exchange as perceived by standard economics embedded i n the proper ecological context. process is important, but ecological context may be more so. E x p l i c i t modelling of the influence of organisms on the abiotic components and dissipative processes upon which they rely is crucial to capturing the dynamics of the system.  T h e topic of the next section is the mathematical expression of these ideas.  2.2.2  T h e general model  It is difficult to define a model that would be suitable to study a wide variety of ecological economic systems because of the variability of human cultural and social systems. Thus,  Chapter  2.  The Modeling  Framework  16  the following is a general description of the model intended to emphasize basic structures common to human ecological economic systems. The general model w i l l then be made specific i n later chapters. State variables will be defined, a behavioral model is developed and the dynamics of the physical system are specified. Consistency w i t h these definitions is maintained where possible, but there are slight notational differences between different models.  State variable definitions T h e m i n i m u m ecological contextual variables are the productivity of the biophysical processes and the stock of low entropy material i n the ecosystem. T h e only organisms explicitly modeled are humans.  Unique to economic systems is the ability of humans  to create capital which greatly enhances their ability to carry out productive activities. Thus, the following (stock) variables are necessary to track the state of the system: h  =  H u m a n population density,  k  =  Stock of renewable natural capital,  k  =  Stock of nonrenewable natural capital,  kh  — Stock of man-made capital.  r  n  T h e precise definitions of the state variables and their units are as follows: • H u m a n population density. Units are people per cultivable hectare. These units were chosen because organisms are inextricably linked to some energy conversion process. A population of 100 people occupying 1,000,000 hectares would seem a low population density - but not if only 100 hectares of the total land were productive. Thus we are explicit about population per cultivable hectare.  For comparison, this number might  typically be 0.0001 for hunter-gatherers [51], 0.5 for swidden agriculturalists i n New Guinea [51], and about 4 for the industrialized world [6].  Chapter 2.  The Modeling  Framework  17  • Renewable natural capital. It is difficult to assign units to capital, natural or manmade.  Consider an example of man-made capital, the common passenger car. Should  we measure the capital by a physical quantity? Should it be measured i n tons of rubber, steel, or glass?? T h e entire heap of physical objects that comprise the car is totally useless without one quart of transmission fluid or some fuel. Clearly, we must define capital i n terms of the service it provides per unit of input. Car engine capital could be defined as horsepower output per fuel input. Now an engine that has been used for 80,000 miles can be compared to a new one. The objects are almost physically indistinguishable, but the service they provide per unit of input is discernibly different. T h e case is similar for renewable natural capital. Renewable natural capital can be measured as the potential of natural systems to generate streams of biophysical processes that stabilize the biosphere's structure and function (natural income streams). T h e capital value of agricultural land, for example, is measured as its productivity per unit of input. • Nonrenewable natural capital. A g a i n there are difficulties w i t h units but I simply define nonrenewable natural capital as any low entropy material such as iron ore, petroleum, etc. for which human society can find a use. • H u m a n made capital. A s w i t h natural capital, the units of human made capital are related to productivity, or ability to do work. In our model, capital is related to how much work can be accomplished per capita. In a community w i t h no human made capital, the per-capita work potential is somewhere between 200 kcal/hour for light activity to 1000 k c a l / h o u r for extremely hard work. For a highly capitalized society, the per-capita work potential would be 100-1000 times these values. I would like to stress the idea of work potential - for without fuel, the work potential provided by the capital stock is not realizable.  Chapter  2.  The Modeling  Framework  18  T h e behavioral model T h e behavioral model consists of two components:  a description of the  population's  allocation of available time and energy to different tasks, and a description of how a particular allocation would change i n response to a change i n the state of the system. T h e model is based on neo-classical theories of production and consumer behavior [32, 14, 64]. A s already mentioned, these models often have no ecological context. To remedy this, these models are modified to reflect thermodynamic considerations and l i m i t s to substitutability that many economists and scientists stress [60, 13, 16, 17, 30, 28, 55, 18]. T h e basic model of behavior assumes that people act to m a x i m i z e their utility, i.e. they solve the optimization problem:  max U(y ,y -....,y ]c)  (2.8)  E ? = i ViPi = w  (2.9)  1  s-t. where U(yi, y i , y  n  )  2  n  is the utility associated w i t h the consumption of c o m m o d i t y y,-  whose prices are pi, c is a vector of parameters that describe the preferences (or culture) of the society being modeled, and w is the wage rate. The solution of this problem generates an expenditure system which specifies how much of each good w i l l be purchased, and thus how many resources should be devoted to the production of each of these goods for any given set of prices. Prices are determined by firms trying to m a x i m i z e profits i n the face of a given demand with a certain technology specified by a production function of the form Vi = fi(xi, :,x ) m  where  is the output of the i  th  (2.10)  commodity and the Xj are inputs, or i n the language  of economics, factors of production.  In economics, the "classic" factors of production  were labor, land, and man-made capital. In my models, factors of production include  Chapter 2.  The Modeling Framework  19  labor, man-made capital, renewable natural capital, and nonrenewable natural capital. T h e inclusion of these latter two inputs links the productivity of the economy to the physical state of the system. Thus human preferences influence the nature of economic activity which i n turn influences the ecosystem. This two step linkage connects human culture to the physical environment.  T h e other component of the cultural model is  to specify a decision process to cope w i t h the situation when the o p t i m a l solution to the consumer problem is not feasible for the state of the physical system and current technology.  Mathematically, this amounts to parameters that define the utility and  production functions changing over time. T h e nature of the utility function plays a very important role i n the dynamics of the system as does the way the population changes its preferences over time. These issues are explored i n detail i n chapters 3, 4, and 5. The final element we must address i n developing the model is the set of rules that govern the dynamics of the system. Before describing the dynamics of the system, I would like to make clear the usage of the term "behavioral plasticity". A s used i n this thesis, behavioral plasticity refers i n d i v i d u a l behavior. Each individual can change their behavior i n response to changing environmental conditions. T h e group behavior is then the result of the aggregation of i n d i v i d u a l behaviors. This is to be contrasted w i t h behavioral plasticity at the group, or cultural level, i.e. cultural or social institutions changing w i t h changing environmental conditions. T h i s assumes that cultural process form w i t h some purpose, an assumption w i t h which I disagree. I view cultural processes as outgrowths of individual interactions, or "emergent variables".  Whether or not a particular set of cultural processes (e.g.  the r i t u a l cycle of the Tsembaga) are adaptive is, to a large extent, accidental. Social institutions, on the other hand, can and do form i n response to particular problems. T h e y can be viewed as behaviorally plastic at the group level. I do not address this issue directly i n the thesis, but propose some directions for further research i n chapter 6.  Chapter 2.  The Modeling  Framework  20  System dynamics T h e dynamics of the system are based on the following basic assumptions:  • A l l human activities require materials and energy and create waste flows - there are no 'free lunches'. Statements about feeding billions w i t h clusters of innovations while sparing land are really about shifting our reliance from one resource to another and this must be recognized. • Ecosystems provide flows of critical services - climate stabilization, waste assimilation, food production, etc. • M a n can, through capital creation, innovation and technical advances increase the efficiency w i t h which both renewable and non-renewable resources are used. • There are limits to substitution i n both production and consumption. • H u m a n economic activity can degrade natural capital (e.g. pollution, soil erosion, etc.). Humans can offset this degradation to some extent by directing a portion of the economy's productive capacity toward this end. • T h e dissipative nature of the system requires the constant input flow of energy to m a i n t a i n a certain level of organization at a given level of technology (i.e. things wear out). • A s materials become more scarce, more work w i l l be required to collect and transform them into useful objects. In order to simplify notation, I represent the state of the system w i t h a vector, i.e. let s = (h, k , k , kh) -the human population density, the stock of renewable natural capital, r  n  Chapter 2.  The Modeling  Framework  21  nonrenewable natural capital, and man-made capital, respectively, at an instant i n time. T h e n , a general model that embodies the assumptions listed above has the form:  dh —  = gh{s,c)h  -jjj-  = 9k (s,c) - d (s,c)  (2.11b)  -jf  =  9k {s,c) - d (s,c)  (2.11c)  dkh -jf  =  9k {s,c) - d {s,c).  (2.lid)  (2.11a)  r  kr  dk n  kn  h  kh  A l l of the functions above depend on the state of the system, s, and the preferences (culture) of the population as represented by c. In equation 2.11a, gh(s,c) represents the per-capita growth rate of the population. It w i l l depend on, among other things, per-capita consumption of commodities, and percapita b i r t h rates. Similarly i n equation 2.11b, g (s,c) kr  defines the natural regeneration  of bioresources. A common form for g (s, c) might be the logistic function, or G o m p e r t z kr  function commonly used i n fisheries [15]. T h e growth of nonrenewable natural capital modeled by g  is associated with the continued discovery of new reserves, new materials,  kn  and new and better ways to use materials. stocks, g  kh  Finally, the growth i n man-made capital  is the result of new investment.  T h e t e r m d (s, c) models decreasing quality of renewable natural capital as nutrients kr  are removed and soil structure is damaged through agricultural activities. T h e function d (s,c) kn  represents the simple fact that flows of resources are required to produce  economic output, while d (s,c) kh  captures the simple fact that machines wear out.  Associated w i t h each dynamical system for the physical state space outlined by equations 2.11a through 2 . l i d is one for the cultural state space. T h e cultural dynamics are very specific to a particular model realization and are impossible to state i n general. In a  Chapter 2.  The Modeling  Framework  22  pure labor economy for example, the cultural dynamics might simply consist of how the population changes its work effort over time. In an economy w i t h capital accumulation, work effort, desired capital to output ratio, and savings rate might constitute the cultural state space. In each of the models discussed i n chapters 3, 4, and 5 the cultural models are slightly different.  2.3  A n a l y t i c a l methods  A given family of models specified by equations 2.11 can be cataloged by a parameter space i n which each point represents a realization of the model. T h e m a i n objective of studying this family of models is to divide this parameter space into regions where the model has the same qualitative behavior. W h e n a "boundary between these regions is crossed, the behavior of the model fundamentally changes-i.e. a bifurcation occurs. A n example is a parameter space divided into two regions, one where the model exhibits a stable e q u i l i b r i u m (sustainable economy), and one where the model exhibits only large amplitude cyclical behavior (unsustainable economy). The nature of these regions generally depends on key parameters or ratios of parameters.  For example, i n the specific  application of the model i n chapter 3, the nature of the model behavior depends on three parameters, the work level of the population and the marginal rates of technical substit u t i o n of land and labor. Parameter combinations where the model exhibits a sudden change of behavior generate the boundaries between regions i n parameter space. T h e two basic model features of stable equilibrium and cyclical behavior relate to whether an economy can  attain a  sustainable state. In both cases, one can describe a  stationary point where each of the state variables remains constant. Such a description would correspond to one for a sustainable economy where human population, natural, and man-made capital stocks are constant.  This says nothing of whether the system  Chapter 2.  can  The Modeling  Framework  23  sustain the flows of materials necessary to maintain this state.  T h i s is directly  related to the difficult question of the meaningfulness of assessing sustainability using the idea of natural capital versus flows of materials [33]. T h e analysis applied herein illustrates the importance of both measures.  If the steady state is stable, then the  flows of materials necessary to maintain it are feasible. If it is not, the steady state is unattainable.  T h e bifurcation from a steady state to limit cycle marks the boundary  between these possibilities. Figure 2.5 illustrates this point.  Natural Capital (a) Figure 2.5: T w o m a i n model structures: steady state.  Natural Capital (b) (a) attainable steady state, (b) unattainable  In graph (a), any reasonable initial condition w i t h high renewable natural capital and low population w i l l evolve to a sustainable state. In graph (b), on the other hand, no reasonable i n i t i a l condition w i t h high renewable natural capital and low population w i l l evolve to a sustainable state. In this case, the difference between e q u i l i b r i u m natural capital stocks might not provide enough information to discriminate between the two cases as [33] points out. The modelling framework developed herein does. Unfortunately, computing the boundary between the behavior exhibited i n graph (a) from that shown i n graph (b) is a difficult task i n general.  If the system is of  low dimension, standard analytic methods of dynamical systems theory can be applied  Chapter 2.  The Modeling  Framework  24  reasonably easily [39]. For large dimensional systems, such analysis becomes impractical. The main tool I employ is a numerical technique known as pseudo arclength continuation available in the software package Auto [20]. The analysis amounts to starting at a known fixed point of the system and tracking its behavior in very small steps. By locating points where the stability of the fixed point changes, we can detect local bifurcations and use these to divide the parameter space as mentioned above. The main transition we encounter in the models presented in this thesis is called a Hopf bifurcation. Hopf bifurcations occur when a stable fixed point changes to an unstable fixed point surrounded by a stable limit cycle. In mathematical terms, two eigenvalues of the Jacobian of the system in question occur as complex conjugates, and all other eigenvalues have negative real parts. When a parameter is varied, if the real parts of the eigenvalues that occur as complex conjugates change from negative to positive, then the steady state changes from being locally stable to locally unstable, and a periodic orbit develops around the steady state. It is the detection of these Hopf bifurcation and the tracking of their dependence on parameter values using the software package Auto that helps us to study the underlying structure of the models presented herein.  Chapter 3  C u l t u r e and human agro-ecosystem dynamics: the Tsembaga of New  In his classic ethnography of the Tsembaga of New Guinea, Pigs for the  Guinea  Ancestors,  R o y Rappaport [53] proposed that the cultural practices and elaborate r i t u a l cycle of these tribal people was a mechanism to regulate human population growth and prevent the degradation of the Tsembaga ecosystem. This is probably the best known work i n applying ecological ideas, especially systems ecology [45], i n anthropology.  Rappaport  treated the Tsembaga ecosystem as an integrated whole i n which the the r i t u a l cycle was a finely tuned mechanism to maintain ecosystem integrity. A l t h o u g h Rappaport provided detailed ethnographic and ecological information to support his claim, many aspects of his model were subsequently criticized. T h e m a i n points of criticism were that his work ignored historical factors and the role of the individual, relied on the controversial concept of group selection, and focused too much on the idea of equilibrium. Several simulation models of the Tsembaga ecosystem were constructed to test Rappaport's hypothesis [57, 23] and evaluate possible alternatives, e.g. [24]. T h e basic conclusions were that it was possible to develop models supporting Rappaport's hypothesis but they were extremely sensitive to parameter choices, and other simpler population control mechanisms might be more likely [10, 24]. Rappaport's original work and associated modeling work by others provide an excellent context i n which to apply the modeling framework outlined i n chapter 2. T h e Tsembaga system is a perfect example by which to address the first two questions proposed i n the introduction: W h a t role does behavioral plasticity play i n this ecosystem?  25  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  26  Does it cause problems or solve them? Do cultural processes play as important a role as Rappaport suggested, and if so how? To answer these questions, the model is developed in three stages. After summarizing the relevant information for the model in the next section, a physical model for a simple human agro-ecosystem is developed and calibrated based on quantitative information provided by Rappaport [53]. Behavior (in terms of the effort devoted to agriculture) is fixed, and the focus is on the importance of the food production function and associated feedbacks on the dynamics of the physical system. Next, the model is extend to allow for changing levels of work effort in agriculture based on the needs of the human and pig populations (i.e the behavioral plasticity of the population is increased). Finally, more complex behavioral dynamics representing the ritual cycle of the Tsembaga are added.  3.1  T h e ecological and cultural system of the Tsembaga  The Tsembaga occupy a rugged mountainous region in the Simbai and Jimi River Valleys of New Guinea along with several other Maring speaking groups with whom they engage in some material and personnel exchanges through marriages and ritual activity. These groups each occupy semi-fixed territories that intersperse in times of plenty and become more rigidly separated in times of hardship. Outside these interactions, the Tsembaga act as a unit in ritual performance, material relations with the environment, and in warfare. The Tsembaga rely on a simple swidden (slash-and-burn) agricultural system as a means of subsistence. At the time of Rappaport's [53] field work they occupied about 830 ha, 364 of which were cultivable. The Tsembaga also practice animal husbandry (the most prominent domesticated animal being pigs) but derive little energetic value from this activity. Pork probably serves as a concentrated source of protein for particular segments of the population as it is rarely eaten other than on ceremonial occasions, and  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  27  several taboos surround its consumption that seem to direct it to women and children who need it most. M u c h of the activity of the Tsembaga is related to the observance of rituals tied up w i t h spirits of the low ground and the red spirits. T h e spirits of the low ground are associated w i t h fertility and growth while the red spirits which occupy the high forest forbid the felling of trees. The ritual activity that is the focus here is the K a i k o .  The  K a i k o is a year long pig festival where a host group entertains other groups which are allies to the host group i n times of war. T h e K a i k o serves to end a 5 to 25 year long r i t u a l cycle that is coupled w i t h pig husbandry and warfare. It is this r i t u a l cycle that Rappaport hypothesized acted as self-regulatory mechanism for the Tsembaga population preventing the degradation of their ecosystem. T h e three m a i n ingredients of the ritual cycle, pig husbandry, the K a i k o itself, and the subsequent warfare, are intricately interwoven w i t h the political relationships between the Tsembaga and the neighboring groups. T h e Tsembaga maintain perpetual hostilities w i t h some groups and are allied w i t h other groups without whose support they w i l l not go to war.  There are two important aspects of pig husbandry:  raising pigs requires  more energy than is derived from their consumption; pigs are the m a i n source of conflict between neighboring groups because they invade gardens. keeping of pigs is completely nonsensical.  F r o m this perspective the  However, the effort required to raise pigs  is a strong information source about pressure on the ecosystem.  T h e greater the pig  population, the greater the chance an accidental invasion of neighboring gardens w i l l occur. E a c h time a garden is invaded, there is a chance that the person whose garden was invaded w i l l k i l l the owner of the invading pig. Records are kept of such deaths which must be avenged during the next ritually sanctioned bout of warfare. F r o m this perspective, pigs provide a meter of ecological and human population pressure and help "measure" the right amount of human population reduction required to prevent  the  Chapter 3. Culture and the dynamics of the Tsembaga  ecosystem  28  degradation of their ecosystem. T h e K a i k o , when a l l but a few of the pigs i n the herd of the host group are slaughtered, helps facilitate material transfers w i t h other groups, allows the host group to assess the support of its allies, and resets the pig population. The ritual cycle as the homeostatic mechanism proposed by Rappaport operates as follows: h u m a n and pig populations grow until the work required to raise pigs is too great. A K a i k o is called and most of the pig herd is slaughtered for gifts to allies and to meet ritual requirements. T h e Tsembaga then uproot the r u m b i m plant i n an elaborate ritual and thus release themselves from taboos prohibiting conflict w i t h neighbors. Warfare, motivated by the requirement of each tribe to exact blood revenge for a l l past deaths caused by the enemy tribe, begins w i t h a series of minor "nothing fights" where casualties are unlikely then escalates to the "true fight" where axes are the weapons of choice and casualties are much more likely. Periods of active hostilities seldom end i n decisive victories but rather when both sides have agreed on "enough k i l l i n g " related to blood revenge from past injustices. The ritual cycle then begins anew w i t h both the pig and human populations reduced to (hopefully) levels that w i l l not cause ecological degradation. A s the model is developed I w i l l fill i n the relevant details of each of the components summarized here. A n obvious question is if the ritual cycle does play such and important role i n the Tsembaga ecosystem, how did it come about?  It is this point that has received much  attention i n subsequent literature regarding Rappaport's hypothesis. In this thesis, the focus is not how the Tsembaga cultural system evolved, but rather on the more general question of how behavioral plasticity (i.e. the very presence of humans) and associated cultural practices affect the structure and dynamics of agroecosystems. For more on the issue of the evolution of group behavior (culture) versus individual behavior, and how a cultural system such as the Tsembaga might come about, see Anderies [4, 3] and A l d e n S m i t h [2].  Chapter 3. Culture and the dynamics  3.2  of the Tsembaga  ecosystem  29  The model  3.2.1  Definitions  Following the framework set out i n chapter 2, the following physical state variables apply to the Tsembaga: h(t): Tsembaga population density i n persons per cultivable hectares.  A t the time of  Rappaport's [53] study the Tsembaga numbered 204 and occupied 364 cultivable 204 hectares, thus h — = 0.56. 364 k (t): Renewable natural capital i n the Tsembaga ecosystem. Here, renewable natural T  capital is related to the productive potential of the 364 hectares upon which the Tsembaga rely for their survival. T h e variable k should be thought of as an index r  of productivity, i.e. productivity per unit of land per unit of effort directed to agriculture. Similarly, the appropriate cultural state variables are: ci(t): Tsembaga per capita birthrate. c (t): Fraction of population devoting 1 man year of energy (2000 hours at 350 k c a l / h r ) 2  to horticulture each year. Thus the total energy devoted to horticulture at time t is given by c?,(t) • h(t) • A man years of energy per year, where A c  c  is the total  number of cultivable hectares available to the population. We then specify the dynamics for each of the variables based on the interaction of human activities and the energy flows through the system. We define the function that governs human population growth as fi(h, k , c\, c ) - the formal statement that popular  2  tion growth depends on the human population, land productivity, per capita birthrate, and work effort directed to cultivating the land. Similarly, the biophysical regenerative  Chapter 3. Culture and the dynamics  of the Tsembaga  process of forest recovery is defined as f (h,  ecosystem  30  k , c\, C2). The functions fx and f  2  r  2  represent  the change i n the human population and renewable natural capital over time which leads to the two dimensional dynamical system:  ^  =  fi(h,k ,c ,c )  (3.1a)  =  f {h,k ,c ,c ).  (3.1b)  r  1  2  dk 2  r  x  2  In the next two sections, we explicitly define the forms of fx and f  2  based on the ecology  of the Tsembaga system. Major considerations are: the nutritional requirements of the Tsembaga population, soil properties and the food production process of the Tsembaga that couples them to the land.  3.2.2  T s e m b a g a subsistence and the population growth rate, fx  T h e canonical way to represent fx is fx = (b-d)h  (3.2)  where b and d are the per capita b i r t h and death rates respectively. We are specifically interested i n how these rates depend on food production and nutrition, so we separate influences on b i r t h and mortality into a constant component not associated w i t h food intake and a component that does depend on food intake. production of the population as e(h,k ), r  First we define the food  then fx can be written as:  fi = (b (ci) - d (e(h, k , c )))h. n  n  r  (3.3)  2  T h e term b is the "net birth rate" which is the natural (culturally dependent) b i r t h rate n  less the natural death rate and does not depend on food intake. T h e term d (e(h, k ,c) is n  T  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  31  the "net death rate" which is the difference between the portions of fertility and mortality that d o depend on food intake. T h e form of d  n  is inferred from the subsistence pattern of the Tsembaga who rely  almost completely on fruits and vegetables (99% by weight) for their usual daily intake, the greatest portion of which come from their gardens. Of this non-animal intake, taro, sweet potato, and fruits and stems constitute the largest part (over 60%) of the diet. These starchy staples combined with a wide variety of leafy vegetables and grains, including protein rich hibiscus leaves, combine to provide adequate calories for the entire population and adequate protein for all but the young children. A t low levels of production, below a m i n i m u m requirement of around 2500 kcal/day, the net per capita death rate increases quickly due to malnutrition. Buchbinder [10] proposed that the mechanism linking malnutrition and mortality could be increased malaria infection due to reduced immunity. Above this m i n i m u m , the net death rate of the population can be decreased through the improved nutrition associated with better quality animal protein that i m proves characteristics such as sexual development, immunity, etc. T h i s decrease i n net death rate is, however, small compared w i t h the increase i n net death rate associated w i t h malnutrition. T h e simplest way to represent d (-)  mathematically is to assume that once the per  capita food requirements are met, d (-)  approaches 0 asymptotically. Below this m i n i -  n  n  m u m requirement, d (-) n  rises quickly. If we choose the units of e(h, k , c ) to be energy r  requirements per person per year then the quantity e(h,k ,c )/h r  2  2  represents the relative  level of nutrition of the population. If this ratio is one, the nutritional needs of the population are just being met. If this ratio is larger than one, the population is producing more than it needs. It devotes the excess to pig husbandry and receives the benefits i n terms of increased intake of concentrated protein and fat. The ratio being less than one has the obvious implications. A convenient function w i t h the desired properties is the  Chapter 3. Culture and the dynamics of the Tsembaga  ecosystem  32  exponential, and we can represent the mortality,rf (-) as n  d (e(-)) = a exp —a—— n  )  (3.4)  where the parameter a characterizes the speed at which people die due to malnutrition and a indicates the response to nutrients. For example if a = 3 and there is no nutrient intake, 40% percent of the population would be dead within two months, and 78% would perish by 6 months. In the model, I have chosen a and a in the interval [1,10]. There are many reasonable choices but the behavior of the model is qualitatively unchanged by any reasonable combination of these parameters. We can now define fi(h, k , c ) completely r  2  as fi(h, k ,c ,c ) r  3.2.3  1  2  = (6 (cx) - a exp n  T h e ecology of slash-and-burn  -a  e(h,  fc ,ci,c ) r  h  2  (3.5)  agriculture  The Tsembaga agricultural system amounts to a piece of land being cleared, cultivated for one year and then left fallow for 15 to 25 years. The gardens are cut in the wetter season in May and early June, allowed to dry, then burned in the dryer season between June and September, and planted immediately thereafter. Because the Tsembaga live on a fixed amount of land, the fallow period and amount of land in production at any one time are directly related. For the Tsembaga, the 15 to 25 year fallow period correlates to about 19 hectares or a little over five percent of the available land being cultivated at any one time. The dynamics of slash and burn agriculture can be viewed as a cycle with two phases: the cultivation phase and the fallow recovery phase. During the cultivation phase, nutrients contained in the biomass of the forest are released into the soil through burning, a portion of which are subsequently removed through cultivation. In addition to direct nutrient removal, gardening has other negative effects on soil quality, especially on soil  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  33  structure. Juo et al. [37] have cataloged some of these indirect effects: - The removal of ground cover exacerbates erosion. - Increased frequency of clearing and cultivation causes the gradual destruction of soil macropore system due to increased foot traffic and tilling. - Burning and cultivation lead to the gradual destruction of the root mat, the decomposition of humidified organic matter, and the reduction of the contribution of organic and microbial processes to nutrient cycling. Frequency and intensity of cultivation probably both effect recovery times (Szott et al. [62]) and the negative effects of agriculture on soil productivity probably increases nonlinearly with food production. I assume, probably conservatively, that these effects increase linearly with food production. During the subsequent fallow phase, the nutrient cycling process shown schematically in Figure 3.1 is reestablished through forest succession. The rate of the cycling process and the associated rate at which nutrients are recycled and fixed in the soil depends on the four processes depicted in Figure 3.1: litter fall, decomposition, mineralization, and uptake [47]. Uptake and litter fall are related to standing biomass which, of course, depends on soil nutrients. Thus, the rate of change of soil nutrients depends on the level of nutrients in the soil. Finally, the nutrient cycling process is governed by the characteristics of the community of decomposing and mineralizing organisms in the soil which set an upper limit on the amount of nutrients in the soil.  The simplest way  to capture this behavior is by the well known logistic function. This is obviously an oversimplification for a very complex process. However, if compared to a detailed, much more complex model for this process [35], the qualitative behavior is captured reasonably well by the logistic. Combining the effects of biophysical regeneration and degradation  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  34  due to agriculture, the rate of change of renewable natural capital is f (h, k ,c ) 2  r  2  = n k (l r  r  - k /k™ ) ax  r  ~ Pe(h, k , c ) r  2  (3.6)  where n is the m a x i m u m regeneration rate, k™ * is the m a x i m u m soil nutrient level for 11  r  the ecosystem, and j3 is the appropriate conversion factor relating food production to productivity.  Figure 3.1: Graphical representation of nutrient cycling process i n a forest. from ([47])  Adapted  There is some difficulty associated w i t h the determination of the intrinsic regeneration rate, n , for the forests the Tsembaga occupy. It is possible, however, to get an idea of the r  order of magnitude n from other studies. T h e time of successional recovery from slash r  and burn to stable litter falls ranges from seven years i n the plains of the U n i t e d States [56] to 14-20 years i n the tropics [22]. T h e numbers for Guatemala closely m a t c h the  Chapter 3. Culture and the dynamics  of the Tsembaga ecosystem  35  fallow periods for the Tsembaga i n New Guinea, so we can scale n  for a characteristic  r  recovery time of 15 to 25 years if the forest is left undisturbed. Figure 3.2 shows recovery curves for different values of n and different initial conditions for k (0). r  know k (0)  r  we can only bracket reasonable values of n  r  Since we do not  i n the following way. If enough  r  nutrients are removed to reduce k to 20% of its m a x i m u m value, we examine recovery r  curves from this value (graph (a) i n Figure 3.2) to see that if n = 0.3 or 0.5, the system r  recovers too fast. T h e recovery time for this initial condition and n = 0.2 is reasonable r  so we take 0.2 to be the upper bound for n . r  If cropping does not reduce soil nutrients  so drastically, say to a level of 50%, lower values of n  r  Figure 3.2 shows the results for n  r  are reasonable.  G r a p h (b) i n  = 0.05, 0.1, and 0.15 respectively; suggesting that  0.05 might be taken as a lower bound for n . r  Thus we assume that n £ [0.05,0.2]. This r  range could be significantly narrowed from a quantitative measurement of soil parameters before and after cropping. Unfortunately, it seems that when these measurements have been attempted, the range of error of measurement exceeds the magnitude of the variables themselves.  postcrop interval (years)  postcrop interval (years)  (a)- post crop nutrient levels: 20% of original  (b)- post crop nutrient levels: 50% of original  Figure 3.2: Recovery curves for different values of the condition of the soil after cropping and recovery rate n . In figure (a), the values of n coresponding to curves of increasing steepness are 0.2, 0.3, and 0.5. Likewise, i n figure (b), these values are 0.05, 0.1, and 0.15. r  r  Chapter 3. Culture and the dynamics  W i t h fi and f  2  of the Tsembaga  ecosystem  36  now completely defined, we can rewrite the dynamical system repre-  sentation of the Tsembaga ecosystem defined by Equations 3.1a and 3.1b as  (3.7a)  G i v e n the problems w i t h associating units to renewable natural capital, it is convenient to rescale the model by k™  ax  by letting k = k • k™ , w i t h k G [0,1]. Now, k represents ax  r  r  r  r  the mean productivity index per hectare of the land the population is occupying, one being m a x i m u m productivity, zero being barren. We also drop the explicit dependence of b on C i by assuming b is a linear function of c and treating b as a parameter. T h e n  n  x  n  rescaled equations are (dropping the tilde notation):  (3.8a) (3.8b) Our final task is the specification of e(-).  3.2.4  T h e food production function  For E q u a t i o n 3.8b of the model, we need an explicit form of the food production function, e(/i,fc ,C2). Unfortunately, although several simple causal relationships are understood, r  there is no fundamental scientific understanding of how nutrients, soil processes, and crop output are related. Examples of work on this problem include France and Thornley's [25] development of plant growth models and Keulen and Heemst's [38] empirical work on  Chapter 3. Culture and the dynamics of the Tsembaga  crop response to the supply of macronutrients.  ecosystem  37  Economic approaches that focus on  energy inputs and resource degradation can be found i n work by Cleveland [16, 17] and Giampietro et al. [28]. Econometric work on determining the form of production functions has been carried out by many authors, see for example [1, 48]. Several functional forms have been suggested for modeling crop output i n the work just mentioned, but two are of interest for the model: the von Liebig and the CobbDouglas. T h e von Liebig function is based on von Liebig's law which states that crop output is a function of the most limiting resource. T h e functional form is y= A  sw  where y is output, A  sw  availability of the i  th  mm iei  [/.-(s,-)]  (3.9)  is the yield plateau set by the soil and weather, x,- is the total  nutrient, and each /,• is a concave function from 1Z to [0,1]. Lanzer  and Paris [40] proposed to use this functional form i n place of the commonly used polyn o m i a l forms and i n a later paper, Ackello-Ogutu, Paris, et al. [1] tested the von Liebig crop response against polynomial specifications and were able to reject the hypothesis that crop response is polynomial. Further, they could not reject that crop response was of the m i n i m u m or von Liebig type. Paris et al. [48] estimated the von Liebig function for cotton lint response to the input of water and nitrogen. They assumed that /jv and fw were linear and l u m p e d all other scarcities into one variable m , to get y=  min  [Q>N  +  PNN,(XW  + PwW,m\.  (3.10)  N,W  Note that a j v and aw represent nutrients already present, while the other terms represent applied nutrients. T h e production surface for this production function is shown i n Figure 3.3. T h e key point to note is that the variable m places a constraint on production due to a l l the other variables not accounted for.  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  38  Nitrogen Input  Figure 3.3: T h e production surface for cotton lint as modeled by the von Liebig production function. A , B , and C are the Nitrogen, Water, and " m " l i m i t i n g planes respectively. A l t h o u g h the von Liebig function may be the best representation of reality, the fact that it is not smooth w i l l cause difficulties when analyzing the dynamical system. Instead, a commonly used production function from economics, the Cobb-Douglas given by y = kf[x?  (3.ii)  i=l  where X{ is the i  th  input and a; are constants is used as an approximation. T h e problem  w i t h this function is that it allows infinite substitutability. T h a t is, if the inputs were land and water, this function says that productivity can be maintained i n the face of a drought by bringing more land under cultivation. This is clearly absurd. If on the other hand, the inputs of interest are not physical quantities, for example energy input, the situation is different. If the general form of the von Liebig function given by E q u a t i o n 3.9 is used to model  Chapter  3. Culture and the dynamics of the Tsembaga  ecosystem  39  output where the input variable is human work energy, the physical inputs / (energy in) 2  may be nonlinear. T h i s is definitely the case for the Tsembaga w i t h regard to the amount of land brought into cultivation for a given amount of labor. Here, the Cobb-Douglas is not such a bad approximation to the von Liebig as shown in Figure 3.4.  Figure 3.4: T h e Cobb Douglas production function overlayed on the von Liebig function. Case (a) - inputs are physical quanities. Case (b)-one input is a nonphysical quantity, work, upon which the physical input, land depends i n a non-linear way. T h e two inputs to agriculture accounted for i n my model are human energy and renewable natural capital. Other inputs such as rainfall and solar energy input are assumed to be fairly constant, which based on the indications of the Tsembaga, is accurate. They indicate that the weather never fluctuates significantly enough to influence crop output, at least not i n their lifetimes. Under these assumptions, the food energy production function is of the form: e(h,k ,c ) r  where w(h,c ) 2  2  = k{w(h,c )) k * 2  ai  a  r  (3.12)  is the amount of energy the population directs towards agriculture, a and t  a are the output elasticity of energy and renewable natural capital respectively, and k 2  is a proportionality constant. Fortunately Rappaport [53] made detailed measurements of the energy input per unit area of land cultivated along w i t h the associated output.  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  40  Using this information we can calibrate the food energy production function, i.e., for a given choice of a\ and a , Rappaport's data can be used to compute an estimate of k as 2  follows. Rappaport indicates that when the human population was 204 and the pig population was 169 animals weighing between 120- and 150 pounds, the amount of land cultivated was about 18 hectares or 6% of the total cultivable land, leaving 94% fallow. T h e trophic requirements of pigs are similar to those of humans, and their population can thus be converted into equivalent Tsembaga numbers. T h e average Tsembaga weighs 94 pounds so their 169 pigs would have the same trophic demands as 240 Tsembaga. Thus, the 18 hectares supported approximately 444 Tsembaga equivalents. Based on his energetic analysis, one person year ( 2000 hours at 350 k c a l / h r ) of energy input is sufficient to clear, burn, cultivate, and harvest one hectare of land. Using energy units i n human annual energy requirements, 18 man years of energy input produced 444 units of total energy output or 1.22 energy units per hectare. Now, m a k i n g a guess at the stage of recovery the secondary forest when brought into cultivation, we can estimate k. Supposing the nutrient level is 80% that of a mature forest, we have 1.22 = A;(18) 0.8 ai  v  ;  a2  1 22 k = . .' . (18) i0.8 * a  n  (3.13) '  a  v  T h e n , given the definition of c , the work devoted to agriculture is w(h, c ) = h c A . For 2  2  2  c  the situation described above, c = 0.09, and A — 364. 2  c  Assuming that the villagers do not waste labor, a certain work effort is roughly correlated to the amount of land being cultivated. If the relationship were linear, increased effort would increase land under cultivation proportionately. If an additional proportional amount of land of equal quality is is brought under cultivation, one would expect that output would increase proportionately.  This situation would be modeled by choosing  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  41  cti = 1. G i v e n the terrain of the Tsembaga, however, increased work input w i l l not increase the amount of land cultivated proportionately. Each marginal unit of land brought into cultivation requires further travel distances which may require substantial elevation gains, and the passage of natural barriers such as ridges and rivers. T h i s suggests that ai < 1 but not substantially. E s t i m a t i n g a reasonable value for a is more difficult and 2  w i l l be discussed later. T h e model is now fully specified:  (3.14b) and we can now study its behavior.  3.3  D y n a m i c behavior of the model  Equations 3.8a and 3.8b represent a family of models parameterized by c , a i , and 2  a. 2  A p p l y i n g the techniques described i n chapter 2 to our model system allows us to assess its sensitivity to the structure of the food production function and the work level of the population. Over a wide range of physically meaningful values for b , a, a, rc , and n  r  the model exhibits a (locally) asymptotically stable equilibrium population density of around 0.6 when c = 0.09 which agrees w i t h the demographic data previously discussed. 2  T h e corresponding equilibrium renewable natural capital value is around 0.75; quite reasonable given that cultivated land is rotated so at any one time at least 10% of the land has just been cultivated and other land is i n various stages of recovery. T h e model's qualitative behavior is sensitive to c , a i , a n d a . 2  2  If we fix a\ = 0.7  and a = 0.3 representing the case where bringing more land under cultivation is more 2  marginally productive than increasing renewable natural capital (soil quality), the model  Chapter 3. Culture and the dynamics of the Tsembaga  ecosystem  42  exhibits a Hopf bifurcation when c is varied as shown i n the bifurcation diagram i n 2  Figure 3.5. Points on the solid line represent stable equilibria while those on the dotted line represent unstable equilibria. The large solid circles represent stable l i m i t cycles. For c less than approximately 0.1354 the system w i l l exhibit a stable equilibrium. For 2  c greater than 0.1354, the equilibrium becomes unstable, and a stable l i m i t cycle w i t h a 2  period of about 300 years appears i n which population builds and reaches its m a x i m u m after about 250 years then declines over the next 20 to 30 years. W h e n the population density is extremely low, the land recovers over the next 20 to 30 years and the process begins again.  0.06  0.08  0.1  0.12  Work level, c  0.14  0.16  0.18  2  Figure 3.5: Bifurcation diagram for swidden agriculture w i t h a i = 0.7 and a, = 0.3. The heavy solid line represents stable equilibria points while the t h i n line represents unstable equilibrium points. T h e dark circles represent the m a x i m u m and m i n i m u m values taken on by x\ on the stable limit cycle, i.e. as the system goes through one cycle, a;, varies from 0.1 to 0.8 people/cultivable hectare.  T h e key point is that if the population works at a level c = 0.09 as it was during 2  Chapter  3. Culture and the dynamics of the Tsembaga  ecosystem  43  Rappaport's field work, the ecosystem is very stable. M o r e interesting is the model's dependence on the relative marginal productivities of soil and labor. If we make the common assumption that a i + o-i = 1 (the economic implications of which w i l l be discussed later), then the effect of the output elasticity of soil and labor on the dynamics of the model can be studied by varying one parameter, either a\ or a . It turns out that there is a relationship between the output elasticity of 2  energy input versus renewable natural capital as is made clear by comparing Figure 3.6 w i t h Figure 3.5.  0 8  o  r  I 0  I 0.25  I 0.5  I 0.75  Work level, c  I  I  1  2  Figure 3.6: Bifurcation diagram for swidden agriculture w i t h a\ = 0.4 and a i n figure (3.5) the solid line represents stable fixed points.  W h e n ai — 0.7 a bifurcation occurs near c  2  2  = 0.6. As  = 0.1354 as previously noted but when  cii = 0.4, no bifurcation occurs for any value of c as indicated by Figure 3.6. 2  In order to understand this behavior, we create a two parameter bifurcation diagram, Figure 3.7, that shows all the combinations of c  2  and a\ for which a Hopf bifurcation  Chapter  3. Culture and the dynamics of the Tsembaga  ecosystem  occurs.  T h e curve generated by these points separates c  2  44  - a\ parameter space into  regions w i t h qualitatively different behaviors shown i n Figure 3.8. Curves for two different cases are shown, one where the population is more and less susceptible to death due to malnutrition as indicated on the diagram. In each case there is a threshold value of a\ below which no bifurcation occurs, i.e. the system remains stable for any level of work. T h i s phenomenon has an interesting ecological interpretation.  more sensitive to malnutrition  o c3  O  0.6  J$  0.4  <L>  t less sensitive to malnutrition  3  O  0.2  0.2  0.3  0.4  Work level, c  0.5  0.6  0.7  2  Figure 3.7: T w o parameter bifurcation diagram for the swidden agriculture model. The curves represent parameter combinations at which a Hopf bifurcation occurs.  In any ecological model, the relative strengths and t i m i n g of feedbacks between state variables governs model stability. In our case, the agriculturalists receive feedback from the land i n terms of productivity per unit effort and the land receives feedback from the agriculturalists i n the form of population density. G i v e n that e(h, k , c ) = / c ( c / i A ) fc^ , the marginal productivity of each input is ai  r  2  2  c  2  Chapter  3. Culture and the dynamics of the Tsembaga  ecosystem  45  SH  o  £ 0.8 h  PH  o  0.6 h 0  0.2  0.1 Work level, c  2  Figure 3.8: Change i n dynamics as the bifurcation boundary is crossed. T h e system goes to a stable equilibrium for parameter values to the left and below the curve while for those above and to the right, the system exhibits stable, cyclic behavior. defined as de(h k , c )  • a\e(h, k , c )  dh  c /iA  :  r  2  r  2  2  (3.15)  c  and de(h,k ,c ) r  a e(h,fc ,c )  2  2  dk  r  2  (3.16)  r  respectively.  T h e parameters a\ and a , called output elasticities i n economics, are 2  measures of proportional increase i n productivity associated w i t h increasing work effort and renewable natural capital respectively. If the output elasticity of labor is higher than the output elasticity of natural capital, it w i l l pay to bring more lower quality land into production (shorter fallow periods) as opposed to preserving soil quality. T h e declining  Chapter 3. Culture and the dynamics of the Tsembaga ecosystem  46  natural capital feedback is weakened by the stronger feedback of increased yields due to increased cultivation effort. Under these circumstances, the ecological system exhibits a bifurcation from a stable to an unstable system if the work level becomes too high. If on the other hand, the output elasticity of labor is lower and that of renewable natural capital correspondingly higher, the possibility of bifurcating from a stable to an unstable system is reduced. The feedback from decreased renewable natural capital is now stronger and exerts more pressure on the population. This pressure keeps the population i n check before natural capital is degraded to the point below which the population can not be supported.  F r o m the agriculturalists' point of view, the gains from cultivating  more land are more than offset by the productivity losses associated w i t h reduced soil quality and nutrient levels resulting from the shorter fallowing periods, a strong feedback to avoid working the land too hard. Notice that the curve for the case where the population is less sensitive to m a l n u t r i t i o n and disease extends to lower values of a\ for which a bifurcation occurs. M a l n u t r i t i o n and disease is the mechanism through which reduced agricultural productivity affects the population. If this mechanism is weakened, the stabilizing influence of reduced natural capital is also weakened. This has the effect of making the model unstable for wider range of values of a . T h e critical point to take away from this analysis is that as output elastic2  ity of labor is increased and the relationship of malnutrition and disease to m o r t a l i t y i n the population is weakened, the potential for ecosystem instability increases. Whether or not that potential is realized depends on how behaviorally plastic the population is, the issue to which we now turn our attention.  Chapter 3. Culture and the dynamics  3.4  of the Tsembaga  ecosystem  47  Behavioral plasticity  In general, i n models of animal population dynamics, behavior, although state dependent, is relatively inflexible. Dynamics and stability characteristics are determined by physical aspects of the ecosystem coupled w i t h the fixed behaviors of organisms that occupy it. Mechanisms that might cause a change i n the qualitative behavior of such a system might be changes i n the external environment (e.g. [8]) , or evolutionary dynamics (e.g. [29]). In an ecological model involving humans, the situation is quite different. T h e system can move i n and out of regimes of stability and instability very quickly w i t h changing behavior. For example, the amount of land that the Tsembaga put into cultivation (the value of c ) is not constant-it depends on the human and pig population. To investigate 2  the effect this has on the model, we now treat c not as an exogenously set parameter, 2  but rather, as an endogenously determined quantity by allowing the population to adjust c to attempt to meet nutritional requirements. T h e work level is governed by the differ2  ence between actual food production and desired food production and the availability of additional labor. A simple expression for the dynamics of c is: 2  (3.17) where dj is the food demand, c™  a:c  is the upper limit on the fraction of the population  working full time cultivating the land, and A  C2  is the speed of response of the population  to changes i n demand. T h e food demand is culturally set, and I define it as follows: if the m i n i m u m food requirements of the population are being met on average (about 3000 calories per day), then df = 1. Significant deviations away from one are possible, as human populations exist on a daily caloric intake ranging from around 2000 up to 6000 calories. T h e parameter c™  ax  could be culturally set or set by physical limitations. T h e parameter A  C2  is  a measure of the behavioral plasticity of the population, setting the time scale on which  Chapter 3. Culture and the dynamics  behavioral change can occur. A s A  C2  of the Tsembaga  ecosystem  48  increases, the population can change its behavior  on shorter time scales. If we append Equations 3.8a and 3.8b w i t h E q u a t i o n 3.17 we have a three dimensional dynamical system that describes the human agroecosystem. T h i s system exhibits a steady state if either food demand is met (£(Mp£2l — \^ population is working at the m a x i m u m permissible level (c = 2  B y treating c™  ax  o  r  t  n e  c ). nax  2  as a bifurcation parameter, we can explore the behavior of the  system defined by Equations 3.8a, 3.8b, and 3.17. T h e results are shown i n Figure 3.9. If max  c  <  0.1354 the model exhibits a stable equilibrium. The stable equilibrium vanishes  when c™  ax  > 0.1354 and a stable l i m i t cycle develops.  If the population is somehow limited i n the m a x i m u m effort it devotes to agriculture, the n u t r i t i o n and disease population control mechanism proposed by Buchbinder [10] would effectively stabilize the system. From the description of their computer simulation model, it seems that Foin and Davis [24] set an upper l i m i t on "cultivation intensity" which would explain their conclusion supporting Buchbinder's hypothesis. If, on the other hand, the m a x i m u m effort the Tsembaga could devote to agriculture if necessary is above the critical level, (which"is reasonable to believe since, for example, this would only require that 15% of the population be willing to work i n agriculture if necessary) the stabilizing mechanism proposed by Buchbinder would not be sufficient to stabilize the system. Thus, if there is any hope of the ecological system being stable, some other mechanism, perhaps cultural, must come into play. If we let c!^  ax  = 0.25, meaning one fourth of the population could devote a person  year of energy to agriculture if necessary, the population could work hard enough to meet food demand and then c is dynamically set by the relation 2  1 =  h  T h e n from E q u a t i o n 3.8a and 3.8b, for equilibrium we must have  (3.18)  Chapter  3. Culture and the dynamics of the Tsembaga  0.05  0.15  0.2  0.25  0.3  ecosystem  ,max  M a x i m u m work level, d2  0.35  0.4  49  0.45  0.5  Figure 3.9: Bifurcation diagram w i t h c™ as the bifurcation parameter i n the swidden agriculture model. T h e upper inset is an exploded view of the boxed region i n the m a i n bifurcation diagram showing the increase i n complexity of the dynamics when culture is added to the system. These dynamics occur over an extremely narrow parameter range, thus having a low probability of being observed i n the physical system. ax  b — a exp (—a • 1)  =  0  (3.19a)  - k ) - (3h =  0.  (3.19b)  n  k n (l r  r  r  If the parameters b , a, and a are such that Equation 3.19a is satisfied, the nonlinear n  system defined by Equation 3.18 and 3.19b defines a one dimensional manifold of fixed points i n 3ft . T h e equilibrium population, natural capital level, and work level depend on 3  i n i t i a l conditions. O f interest to us is how the net birthrate must be exactly balanced by  Chapter  3. Culture and the dynamics of the Tsembaga  ecosystem  50  the net death rate associated w i t h the nutritional level achieved when food demand is met. If the population could, through some cultural mechanism such as infanticide or some other type of b i r t h control, match these rates, the system would be (neutrally) stable. Here, we see how extreme behavioral plasticity can destabilize a system by nullifying the feedback control of resource l i m i t a t i o n and transferring the responsibility of ecosystem regulation from environmental to cultural mechanisms. It is probable that the net growth rate of the population is positive when food demand is met which violates the stability condition given by 3.19a. In this case the ecosystem exhibits cyclic behavior.  It is very interesting to compare the l i m i t cycle behavior of  the cases w i t h and without behavioral plasticity. Figure 3.10 shows the l i m i t cycles that develop i n the system where the work level is treated as a parameter (inner cycle) set constant at c = 0.14 and those that develop when the work level is dynamically set w i t h 2  max _ o 25 (outer cycle). Figure 3.11 shows the work level and food production over  c  time. Several interesting points are worth making about these figures. F i r s t , the period of the outer cycle where the work level is dynamically set is about twice that of the case were the work level is constant. T h e reason for this can be seen in Figure 3.11. T h e initial work level is very low, around 0.05, because if the population is low and renewable natural capital is high at t = 0 little effort is required to meet food demands. T h e population does not over exploit its environment just because it can, and just meets food demand. W i t h the case where the work level is constant at 0.14, the population exploits the environment at a constant rate. W h e n renewable natural capital is high, the population can produce an abundance of food which increases the growth rate of the population. Thus, when the level of renewable natural capital is high, a population that just meets food demand grows more slowly than a population w i t h a constant work level. The difference is indicated i n Figure 3.10 by the difference i n time required for the population to reach a m a x i m u m : 240 versus 720 years for the constant  Chapter  3. Culture and the dynamics of the Tsembaga  ecosystem  51  t=720 t=735 1  o  /  t=741 o  0.2  0.4  0.6  0.8  Natural Capital, k  r  Figure 3.10: L i m i t cycles that develop as the the system becomes unstable. T h e inner cycle is for the case where the work level is constant at 0.14. T h e outer cycle represents the case where the work level is set by demand. and dynamic work level cases, respectively. Next, notice that i n the constant work level case, after the population reaches a maxi m u m , it begins to decline immediately. T h i s decline to the lowest population level takes about 60 years. In the dynamic work level case, by increasing work level dramatically as shown i n Figure 3.11 around t = 720, the population is able to delay the precipitous decline i n population for about another 15 years. In doing so, however, the population puts itself into a more precarious position of very high population density i n a very degraded environment. T h e precipitous decline now takes 6 years instead of 60! Since the Tsembaga do adjust their work level, the model suggests that unless some mechanism intervenes, their ecosystem is doomed to crash. T h i s could be avoided by maintaining the knife edge set of parameters required for stability i n 3.19a by controlling  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  52  0.25  .2 o  >  o  o  o 0.05  200  400  600  800  1000  Time Figure 3.11: Work level (curve (a)) and food production (curve (b)), over time. b i r t h and death rates w i t h i n the population, or possibly by the r i t u a l cycle. It seems that the former is not the case; the Tsembaga actively seek to be as "fertile" as possible as evidenced by their rituals to improve fertility. In the next section, we add the dynamics of the r i t u a l cycle and determine the conditions under which it could m a i n t a i n a balance i n and prevent the degradation of the Tsembaga ecosystem.  3.5  M o d e l l i n g the ritual cycle  T h e r i t u a l cycle dynamics are added i n two parts. First we address pig husbandry to find that even without the ritual cycle, pig husbandry alone can help stabilize the system. Next we add the r i t u a l cycle to show that under certain assumptions the r i t u a l cycle can stabilize the system, and that stability is not as sensitive to parameter choices as it is to how the number of people who ought to be killed during warfare is related to pig  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  53  invasions.  3.5.1  T h e parasitism o f pigs  T h e bulk of the responsibility of keeping pigs falls on Tsembaga women. T h e y do most of the work i n planting, harvesting and carrying the crops used to feed the pigs. In this sense, the pigs can be viewed as parasitizing Tsembaga women. T h e y benefit from energy derived from the ecosystem but do not contribute to obtaining that energy. It turns out that this relationship, i n and of itself, is enough to help stabilize the ecosystem.  The  mechanism is related to the fact that working too hard is a major factor i n destabilizing the ecosystem. If the human population is the sole benefactor of its agricultural effort, it grows i n number, produces a larger labor pool, and the per-capita work level remains constant. If, on the other hand, the population keeps pigs, as the pig population grows relative to the human population, the per-capita work level increases.  In this way, the  pigs act as an ecosystem monitoring device. This is clearly illustrated by the model. In all the previous investigations, it was assumed that the Tsembaga devoted a constant 55% of their harvest (based on demographic information at a point i n time) to pigs maintained a constant pig to person ratio (no ritual cycle). B y treating this ratio as a parameter, r , we can generate a figure similar to p  Figure 3.8 where the parameters of interest are the percentage of food being consumed by humans and c™ . Figure 3.12 is the result. T h e curve i n graph (a) separates regions aa;  i n parameter space of stability and instability. Notice that the more food the humans eat themselves, i.e. r unstable.  —> 1, the lower the level of c^  at which the system becomes  ax  p  Recall that w i t h r  = 0.45, the system goes unstable when c^  ax  p  = 0.1354.  T h i s represents only a 50% increase i n work effort which is plausible. Now consider the case where r  = 0.3, the system remains stable until c™  ax  p  reaches approximately 0.22.  T h i s represents a more than doubling of work effort which may be intolerable to the  Chapter 3. Culture and the dynamics of the Tsembaga ecosystem  ,max  54  V  Figure 3.12: T h e influence of pigs on system dynamics. Figure (a) shows the bifurcation boundary i n c™ -r parameter space. Figure (b) shows the equilibrium natural capital level function of r„. ax  v  population. Thus, just by being there, the pigs help stabilize the system. Note that this stability comes at the expense of human nutrition. In this model, food is first fed to the pigs and the remainder is fed to the population. This is not what happens; the Tsembaga eat the best food first and give the rest to their pigs. This difference requires the more elaborate ritual cycle mechanism to stabilize the system.  3.5.2  T h e ritual cycle  T h e ritual cycle consists of periods of ritually sanctioned truces separated by warfare. T h e rituals that mark the transitions between the phases are the K a i k o that marks the end of the truce period and the planting of a plant called r u m b i m {cordyline fruticosa) that marks the beginning of the next truce. Figure 3.13 is a representation of the cycle. T h e length of the arcs on the circle is loosely representative of the times between events.  T h e K a i k o itself lasts one year.  Warfare lasts for a matter of months.  The  time between planting the r u m b i m that signifies truce and the K a i k o (typically about  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  55  Figure 3.13: T h e ritual cycle of the Tsembaga 12 years) depends on the demographics of the pigs. In this period enough pigs must be grown to satisfy r i t u a l requirements, but the staging of the K a i k o also depends on when women get tired of being parasitized by pigs. T h e question mark between the uprooting of the r u m b i m and the beginning of warfare indicates uncertainty about the t i m i n g of the onset of warfare, although Rappaport indicated that fighting had usually resumed w i t h i n 3 months of the uprooting of the r u m b i m . After a truce, the populations return to tending gardens and pigs. A s the pig population increases, work load on the women also increases. Rappaport computed that there were an average of 2.4 pigs of the 120- to 150-pound size to each mature female at the outset of the 1962 K a i k o . This translates into a pig to person ratio (in terms of biomass) of about 1.2.  T h e range of the number of pigs kept was 0 to 6. Rappaport observed  only one woman keeping 6, and four keeping 5 and postulates that these figures may represent the m a x i m u m physically possible. W h e n females are burdened w i t h this many pigs, their complaints to their husbands become more frequent. T h e husbands then call  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  56  for the K a i k o to be staged during which the pig herd is drastically reduced v i a ritual sacrifice. To model this we add variables for the pig population (p) and the "harvest" (g) level of pigs. W h e n p is less than the level tolerable by the Tsembaga women, q is very low. W h e n p reaches a critical level of about 2-3 pigs per woman, the K a i k o "breaks out" and q increases very rapidly. T h e dynamics of this type of system can be modeled by a dynamical system of the form:  dq dt dp  r(p/h  - g(q))  (r - q)p  (3.20b)  where r is the intrinsic growth rate of the pig population and the function g(q) has the form i n Figure 3.14, and r , which is relatively large, is the relaxation time. T h e trajectory i n the phase plane generated by the dynamics i n 3.20a and 3.20b is superimposed on g(q)- W h e n the quantity p/h is between 0.2 and 1.2, Equation 3.20a forces q to track the function g(x) very closely. Once outside these limits, the difference between p/h  and  g(q) grows causing q to change very quickly, as shown i n Figure 3.15. After the staging of the K a i k o , the ritually sanctioned truce between hostile groups is ended by the uprooting of the r u m b i m plant. Hostilities are then allowed to, but do not necessarily, resume. If hostilities can be avoided through two ritual cycles, lasting peace between the two hostile groups can be established. Rappaport notes, however, that hostilities are generally resumed by three months after the K a i k o and can last up to six months. D u r i n g actively hostile periods, actual combat is frequently halted for the performance of rituals associated w i t h casualties and for pigs and gardens to be tended. Warfare comes to a halt w i t h another ritual truce when both sides feel that enough k i l l i n g has taken  Chapter 3. Culture and the dynamics of the Tsembaga  0 I -0.2  J  1  0  0.2  1  ecosystem  1  0.4  0.6  57  1  1  0.8  I  1 1.2  Harvest rate, q Figure 3.14: F o r m of g(x) i n equation (3.20a) and the associated l i m i t cycle. place or combatants simply tire of fighting.  Since the fighting forces are composed of  principal combatants and their allies, as time goes on, the support of allies becomes more difficult to maintain which increases pressure to bring hostilities to an end. To model this we use the fact that after several casualties have occurred, the people to pig ratio begins to decrease. A s this happens, the per-person work level begins to increase and daily living activities become more pressing. T h e pig to person ratio acts as a proxy for this increased work effort and the warfare outbreak dynamics can be expressed by:  ^  = r{h/p - < 7 H + S) 7  (3.21)  where w is the per-capita death rate due to war and 7 and 6 merely scale and shift the ratio of people to pigs where the outbreak of war and ritual truce occur. T h e human and pig population dynamics under this scenario are shown i n Figure 3.16. T h e most critical aspect of the model for the ritual cycle and its effect on the human population is the set assumptions made about the effect of warfare on the population.  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  58  Kaiko  J time  1  time  Figure 3.15: T h e dynamics of the ritual cycle. These represent the time plots of the l i m i t cycle shown i n figure (3.14). Between Kaikos, the harvest rate is very low. W h e n the pig to person ratio exceeds the tolerable level, the harvest rate increases dramatically representing the pig slaughter associated w i t h the K a i k o as shown i n the graph on the right. Unfortunately, data on warfare-related mortality are not rich - estimates range from two to eight percent of the population [23]. This is not an important issue w i t h regard to stability, however. T h e key point is the assumption that the number of deaths due to warfare is a constant fraction of the population. If we make this assumption then the human population dynamics would be given by (3.22) If the system is to evolve to a stable limit cycle, the parameters that govern the dynamics of w must be chosen such that the average value over one cycle of the quantity (3.23) vanishes. Since the cultural dynamics act to drive e(h, k , c i , c ) toward 1, the growth r  2  rate of the human population is nearly constant and only very weakly dependent on the physical state of the system over most of a cycle. T h e average war mortality over a cycle must be balanced against essentially a constant growth rate, and there is no mechanism by which the model can "seek" an equilibrium population level. In this case the ability  Chapter 3. Culture and the dynamics  0  of the Tsembaga  5  10  ecosystem  15  20  59  25  time Figure 3.16: A n example of of the human (a), and pig (b), population trajectories under cultural outbreak dynamics. After the K a i k o when the pig population drops drastically (curve (b)) warfare resumes and the the human population drops (curve (a)). A s people are killed, the human pig ratio drops until a cutoff is reached and a truce is called. of the K a i k o to stabilize the system is very sensitive to parameter choices. T h i s may help explain why the model due to Shantzis and Behrens [57] was neutrally stable and, of course, why when Foin and Davis [23] used different parameters (making the counterpart of expression 3.23 i n their model positive i n mean over one cycle) found that the K a i k o would not stabilize the system. Here, there is no mechanism by which the model can "seek" an equilibrium population level. If, on the other hand, we assume that mortality due to warfare increases nonlinearly w i t h the population size, the K a i k o can stabilize the system. Rappaport actually indicated that this was the case. A s there are more pigs, people, and gardens there are more ways for pigs to invade gardens and cause conflict, increasing the number of required blood revenge deaths during an active period of warfare.  T h e number of ways a pig  might invade an enemy's garden rises much faster than linearly w i t h increases i n pig and  Chapter 3. Culture and the dynamics  garden numbers.  of the Tsembaga  ecosystem  60  If we assume that number of war moralities behaves roughly as the  square of the population size, the human population dynamics are given by dh / e(7i, k , c i , c ) \ — = (b - aexp f -a ) - wh)h. r  n  2  ,  , (3.24)  h  We then define the full ecological system by the physical component defined by Equations 3.24, 3.14b, and 3.20b and the cultural component defined by Equations 3.20a, 3.17, and 3.21 to arrive at the following dynamical system:  —j-^ =  {b — a e x p I— a  J—wh)h  n  (3.25a)  dk  J  ^  dq  dt dw ~dt  3.5.3  =  k n (l-k )-  =  (r-q)p  r  r  r  /3k(c hA ) k^ c  (3.25c)  = ( ^t^hflSL) ._ K  d  (cr  C2)  (  3.  2 5 d )  (3.25e)  T ( p / h - g ( q ) )  T(h/p-<yg(w)  (3.25b)  ai  2  + 6).  (3.25f)  T h e b e h a v i o r of the full s y s t e m  T h e dynamics of the ritual variables are confined to stable l i m i t cycles and the work level follows food demand forcing the overall system behavior to be cyclic. W i t h the human population dynamics defined by 3.24, the ritual warfare acts to drive the system to e q u i l i b r i u m keeping the human population i n check. Figure 3.17 illustrates the behavior of several trajectories beginning from different reasonable initial conditions. They a l l collapse onto a very small amplitude stable limit cycle. Projections of this cycle onto the h — p and k — h planes are shown i n Figure 3.18. r  J  Chapter 3. Culture and the dynamics  of the Tsembaga  ecosystem  61  Biophysical Capital, k  r  Figure 3.17: Sample trajectories for the full model. A n y time that the human population is large compared to biophysical capital, the pig to people ratio w i l l be high and warfare w i l l break out. T h i s drives the population to a more stable (or sustainable) region i n the state space whence the system collapses onto the very low amplitude l i m i t cycle shown in flgure( 3.18). T h e r i t u a l cycle effectively keeps the human population density i n the interval (0.41,0.49) and the natural capital i n the interval (0.86,0.89).  Compare these fluctuations to the  case without the ritual cycle (see Figure 3.10). T h e model predicts that if the Tsembaga attempt to meet food demand, it is possible that the ritual cycle could play a critical role i n stabilizing the ecosystem.  3.6  Conclusions  T h e dynamical system model for the Tsembaga ecosystem based on the ethnographic work of Rappaport [53] developed i n this paper suggests that behavioral plasticity, feedback from the land, and the relationship between people and pigs are the m a i n factors affecting ecosystem stability.  Behavioral plasticity, i n the form of the ability of the  Chapter  3. Culture  and the dynamics of the Tsembaga  H u m a n population density, h  ecosystem  62  Biophysical capital, k  r  Figure 3.18: L i m i t cycle for the full model projected into the x\ — a; 3 and x — xi planes respectively. 2  Tsembaga to adjust food production based on demand, is strongly destabilizing because it allows people to attempt to overcome nutritional deficiencies that would otherwise help stabilize the system. C r i t i c a l to the effect behavioral plasticity has on the model is the relative productivity of labor. If the increased nutritional intake generated by increased effort more than offsets the soil productivity loses due to the associated shorter fallow periods, the model stability structure is sensitive to changes i n effort directed to agriculture. Increased output elasticity of the soil (sensitivity of soil productivity to increased effort) has a stabilizing influence, reducing the importance of behavioral plasticity i n determining the stability of the system. If the output elasticity of labor (in the short run) is higher than that of soil (probably reasonable) then the destabilizing effect of behavioral plasticity can be so strong as to nullify the stabilizing effect of malnutrition and disease proposed by Buchbinder  [10]  opening up the possibility of temporally violent oscillations i n population numbers. B y extending the model, it was shown that pig husbandry, i n and of itself, helped stabilize the  Chapter 3. Culture and the dynamics of the Tsembaga  ecosystem  63  system. Finally, pig husbandry combined with the ritual cycle can act as a homeostatic mechanism to stabilize ecosystem as proposed by Rappaport if war mortality  is density  dependent. This runs contrary to earlier results [23, 24, 57] that emphasized sensitivity to parameters. T h e model presented here is fairly robust to changes i n parameters and suggests that the key factors are the structure of the food production function and density dependence of war related mortality. M a n y of the original criticisms of Rappaport's work centered on the problem of explaining how the Tsembaga cultural system might have come about, and the appropriateness of the ecosystem concept as he applied it. Of course, no model can explain the evolution of behavior, at best it can only shed light on how certain behavior could be adaptive.  T h e focus of this paper was to study the effects humans and their cultural  practices can have on an ecosystem. We found that culture can be both destabilizing (how hard a population decides to work) and stabilizing (the r i t u a l cycle). T h e model presented here supports the claim that a cultural mechanism such as the Tsembaga r i t u a l cycle can operate to prevent ecosystem degradation. If an individual can do better by participating i n the existing cultural "environment" rather than going against it, any cultural construct that prevents ecosystem destruction could have adaptive value for the i n d i v i d u a l . In this sense the ritual cycle of the Tsembaga could have adaptive value as Rappaport originally proposed. T h e model also highlights the destructiveness of a society that directs ever increasing quantities of energy to agriculture i n the face of continually degrading soil quality, and the importance of the role "sustainable culture" might play in both past and present sustainable human agroecosystems. T h e m a i n point to take away from this model is that the human ability to modify behavior to overcome short term resource shortages does not, as many economists believe, help the society reach a sustainable state.  It has the opposite effect:  it makes the  sustainable state harder to achieve. The model suggests that collective social action is  Chapter  3. Culture and the dynamics of the Tsembaga  ecosystem  64  more critical making a sustainable world a reality. Also, it must be emphasized that this social action can not be "soft" by which I mean actions that focus on trying to continue what we are doing w i t h less. The social action has to be an emergent of i n d i v i d u a l beliefs.  property  T h i n k , for example, if excessive individual wealth accumulation  and greed were viewed w i t h as much indignation and disgust as say incest or rape, we might be faced w i t h a quite different present and future world. Simple economic and technological fixes that are not accompanied by cultural change might do nothing more than help paint us into a corner. This will be illustrated i n chapter 5 w i t h regard to investment and wealth distribution practices.  Chapter 4  Non-substitutibility in consumption and ecosystem stability  If we wish to extend the modelling framework to more complex economic systems w i t h a wider range of possible activities and more state variables, defining how the linkage between them operates becomes the main challenge. The m a i n question is how do people decide to allocate energy to the different activities and how do feedbacks from the environment influence this allocation. Economists have dealt w i t h this problem in great detail through the use of the market, where the m a i n feedbacks from the environment are prices, and u t i l i t y functions determine how income is allocated among available activities. The a i m of this section is to examine i n detail the implications of assuming a standard economic model for the interaction between behavior and environment, i.e. how certain assumptions about utility generate very specific cultural structures. We accomplish this by studying and extending a model of the economic system of Easter Island developed by Brander et al. [9]. In this model the authors develop the hypothesis that the culture and economic system of the invading Polynesians were incompatible w i t h the physical properties of Easter Island. This mismatch between cultural and ecological systems lead to the eventual collapse of the system. This is an excellent example of the importance of studying culture and economic systems within an ecological context.  4.1  T h e Easter Island model  Brander et a l . [9] developed a simple general equilibrium model to characterize the collapse of the society on Easter Island that created the stone monuments for which the 65  Chapter 4. Non-substitutibility  in consumption  and ecosystem  stability  66  Island is so well known. The model has two state variables: S(t):  Renewable resource stock (= k i n m y notation)  L(t):  Available labour i n the population (= h i n m y notation)  r  T h e renewable resource stock would include agricultural output and fish catch potential. As is traditional w i t h economic models, the population is modeled as a labor pool that is proportional to the physical population. T h e dynamics of the Easter Island ecosystem according to Brander et al. [9] are then given by 7  where G(s)  C  —  =  G(S)-H(S,L)  (4.1a)  M  =  (b-d  (4.1b)  + F(H,L))L  is the intrinsic growth rate of the renewable resource (food and wood),  H(S, L) is the harvest rate of the resource, b and d are the constant b i r t h and death rates for the labor force (population) and F(H, L) is the variable growth rate of the population that depends on resource use. the determination of H(S,L)  and F(H,L).  The cultural subsystem is associated w i t h T h e cultural system is modeled by treat-  ing the inhabitants of Easter Island rational economic agents attempting to m a x i m i z e u t i l i t y through the consumption of material goods. This cultural structure, of course, determines a large part of the model's behavior, just as it did i n the Tsembaga case. T h i s provides an example of how cultures can be compared. Tsembaga r i t u a l culture (non economic behavior) stabilized the system while if the culture commonly ascribed to modern industrial man prevailed on Easter Island, they would be doomed to "overshoot and collapse". W i t h i n this cultural model, the population consumes two goods - bioresource goods (agricultural output and fish), H, and manufactured goods (tools, housing, and artistic output), M.  T h e cultural dynamics, i.e. the way the population decides to partition  Chapter 4.  Non-substitutibility  in consumption  and ecosystem  stability  67  available energy among possible activities of producing and consuming goods are then determined by solving a constrained m a x i m i z a t i o n problem. Brander et al. use a CobbDouglas utility function, u(h,m) = / i m / 3  (4.2)  1 _ / 3  where h and m are per capita consumption rates of the bioresource and manufactured goods respectively, and f3 defines the preferences for these goods. If w is the wage rate, the budget constraint is Phh + p m  = w,  m  Ph and p  m  (4.3)  being the respective prices of the two goods. B y the choice of units Brander  et al. set p  m  — 1 ( M is defined as the numeraire good whose price is the benchmark  by which all prices are measured).  Solving this m a x i m i z a t i o n problem results i n the  following per-capita demand functions: h= — Ph  and  m = w(l-  f3).  (4.4)  E q u a t i o n 4.4 thus defines the demand side of the economy. To model the supply side, we must employ production functions to link demands w i t h physical possibilities. T h e production functions chosen by Brander et al. are H  =  cxSL  H  (4.5a)  M  =  L  .  (4.5b)  M  E q u a t i o n 4.5a asserts that the quantity of H produced is proportional to the product of the size of the resource stock and the quantity of labor devoted to obtaining it, Lff. Such production functions are commonly used i n fisheries [15]. E q u a t i o n 4.5b states that M depends on labor alone, LM and by choice of units, one unit of labor produces one unit of Af.  Chapter 4.  Non-substitutibility  in consumption  and ecosystem  stability  68  The link between the supply and demand side is, of course, the market. T h e market w i l l equilibrate when the supply prices equal the demand prices.  A s s u m i n g that the  economic processes are much faster than natural processes, Brander et al. assume that the market is always i n equilibrium so that linking the supply and demand sides of the economy reduces to solving a set of algebraic equations. Assuming that the only costs of production are due to labor, the per-unit supply prices are given by Ph = —jjPm =  w  L  (4.6a) I A  m  (4.6b)  F r o m equation 4.5b we see that jff- = 1 and since p  m  = 1 we must have that the wage  rate is also 1. Combining this fact with equations 4.5a and 4.6a we see that Ph = -^  (4-7)  which merely says as the resource stock decreases, its supply price increases. Substituting the supply prices and wage rate into equation 4.4 yields the actual per-capita amounts of H and M produced: h  =  a/3S  (4.8a)  m  =  1- P  (4.8b)  In order to extend this model and illustrate how the choice of u t i l i t y functions relates to the level of behavioral plasticity exhibited by the populations we express culture as the amount of energy devoted to each available activity. This requires relating the per-capita consumption to the energy required to produce it. We w i l l accomplish this i n the same manner as w i t h the Tsembaga model. Let us assume that the available labor is a fraction of the total population, i.e. L  = -/N  (4.9)  Chapter 4.  Non-substitutibility  in consumption  and ecosystem  stability  69  where N is the total population at time t. Brander et al. assume that 7 is equal to 1 (again by choice of units) thus N — L. B y definition, the total demand for H and M is the per-capita demand multiplied by the total population: H = Nh = Lh = La(3S  and  M = Nm = Lm = L(l — f3)  (4.10)  Now, using the production functions once again, we can determine the energy (or labor) required to meet these demands, i.e. we set the total production equations equal to the total demand equations: Laf3S = L aS  => L  H  L{1 -f3)  H  = (3L  = L  M  (4.11a) (4.11b)  Thus, the Easter Island Culture as characterized by this economic model is one i n which a constant proportion, (3, of the labor force is directed towards producing bioresource goods, while the remaining portion of the labor force, 1 — (3, directs its energy towards the production of manufactured goods. T h e final aspect of the model to be specified is how the fertility function F depends on the per-capita intake of bioresource goods (nourishment). Here Brander et al. make the assumption that net fertility increases linearly w i t h per-capita consumption of bioresources, i.e. the better life is the higher the propensity to reproduce. Thus they let F = tj  (4.12)  where <> / is a positive constant and the ratio of H to L represents the actual per-capita intake of bioresource goods. Thus the culture of Easter Island can be completely specified by two parameters:  /3, its taste for bioresource goods and <f>, its fertility response  coefficient. W i t h the cultural sub-model specification complete, we are left to quantify the physical aspect of the model; the growth rate of the bioresource, G(S).  Here Brander et  Chapter 4. Non-substitutibility  in consumption  and ecosystem  al. assume the common logistic function: G(S) = rS(l  — S/K)  stability  70  where r is the intrinsic  growth rate and K is the carrying capacity. The planar dynamical system we wish to study is then given by: 7  4.2  C  —  =  rS(l  - S/K)  - a/3SL  ^  =  (b-  d + <f>apS)L.  (4.13a) (4.13b)  M o d e l Critique  A glance at equations 4.13a reveals that they are equivalent to a Lotka-Volterra predatorprey system w i t h density-dependent prey growth rate. T h e behavior of such systems is well known and I w i l l not discuss it here (see [11]).  Rather, I w i l l focus on how  assumptions about culture affect the model - especially focusing on the role of behavioral plasticity. The model specified by equations 4.13a has one non-trivial equilibrium point (S*, L*) that satisfies S* > 0, L* > 0 and dS(S*,L*) dt dL(S*,L*) dt  =  0  (4.14a)  =  0.  (4.14b)  T h i s equilibrium point is globally asymptotically stable, the proof of which relies on a simple application of a theorem due to Kolmogorov relating to planar systems of this type (see [42] or [21]).  Beginning from any interior initial condition, the system w i l l  converge to the steady state.  Depending on parameter values, the steady state will  either be a node or a spiral which w i l l force the system to converge to the equilibrium either monotonically or through a series of damped oscillations. O f interest to Brander et al. is that for certain parameter values representative of the situation on Easter Island, the system w i l l exhibit transitory oscillatory behavior which manifests itself i n overshoot  Chapter 4. Non-substitutibility  in consumption  and ecosystem  stability  71  and collapse. Figure 4.1 shows the human population and resource stock trajectories for an initial condition of 40 humans landing on Easter Island w i t h the resource stock at carrying capacity (The units for the resource are a matter of scaling. Brander et a l . [9] choose a carrying capacity of 12,000 units for convenience.)  12(100 |  400  f  1  1  1  1  1  1  600  800  1000  1200  1400  1600  1800  1  Time Figure 4.1: Population and resource stock trajectories for Easter Island model from ([9]) T h e archaeological record indicates the first presence of humans at around 400 A D . T h e population increases which is accompanied by a decrease i n resource stock.  The  population (and available labor) peaks at around 1250 A D corresponding to the period of intense carving i n the archaeological record. T h e population subsequently declines due to resource depletion. T h e model predicts a population of about 3800 i n 1722, close to the estimated value of 3000. T h e model thus gives a reasonable qualitative picture of what may have happened to the culture on Easter Island. T h e culture became very productive and able to undertake the construction of major monuments, i.e. the labor force increased thus making LM large enough to complete such a large scale project. T h e population subsequently declined due to resource degradation which left the small  Chapter  4.  Non-substitutibility  in consumption  and ecosystem  stability  72  population who knew nothing of the origin of the great monuments to meet the D u t c h ships i n the eighteenth century. The discussion i n Brander et al. [9] is very interesting and I refer the reader there for more detail.  4.2.1  Behavioral plasticity and collapse  In this section we examine how the nature of the population collapse depends on the level of behavioral plasticity exhibited by the population. The nature of the collapse can be more clearly understood by examining the per-capita growth rate over a time scale meaningful to a member of the population. Figure 4.2 shows the annual per-capita net growth rate of the population from the time of initial colonization to the time of the D u t c h ships arrived i n the eighteenth century.  4(xi  am  800  IOOO  1200  1400  i«xi  isoo  Time Figure 4.2: Per-capita growth rate from the time of initial colonization to the time of first European contact.  T h e population exhibits positive growth up to approximately 1200 A D when it peaks at around 10,000 individuals. The m a x i m u m per-capita annual growth rate is around  Chapter 4.  Non-substitutibility  in consumption  and ecosystem  stability  73  0.92%-very low by today's standards. Similarly, the m i n i m u m net growth rate is -0.262% which implies that even under the most extreme resource shortage conditions the population is decreasing very slowly. It takes 600 years for population to drop from 10,000 to 3800. Compare this to populations doubling every 40 years at present. Next consider the perceived change i n an individual's standard of living over a life span of say 70 years from the year 1000 A D to 1070 A D when it is decreasing most rapidly . In this period one would experience a 12% decrease i n bioresource intake over an entire lifetime. A l t h o u g h the quality of life is going down, it is not changing catastrophically. F r o m our present clay point of view the manner i n which the population adjusts to the environment depicted by the model might not be that bad. We can now investigate the role behavioral plasticity has to play i n the nature of the collapse. Recall from equations 4.11 we deduced that the population directs a constant proportion (3 of the labor force towards the bioresource sector while what is left is directed to the manufacturing sector. Further, equations 4.8 indicate that the per-capita rate of consumption of the manufactured goods is constant, no matter what quantity of bioresources are being consumed. This implies that as the bioresource stock is depleted and becomes more expensive to produce, individuals continue to consume the same amount of manufactured goods and consume less and less bioresources. T h e population could be starving, yet the utility maximizing strategy is to keep the proportion of labor directed to each activity constant. T h e problem here is substitutability. Cobb-Douglass utility functions allow for one input to be substituted for another without affecting utility. Based on this model, the o p t i m a l strategy i n the face of a resource good shortage is to increase consumption of cheaper manufactured goods. This is reasonable i n some cases, but not where bioresource goods that sustain one's very life are concerned. In short, the standard Cobb-Douglass u t i l i t y function cannot capture the possibility that labor could be shifted from one sector  Chapter  4.  Non-substitutibility  in consumption  and ecosystem  stability  74  to the other-the structure of the economic system is fixed over time. T h e only aspect of the model that allows for behavioral flexibility is the fertility function, and this depends on how it is interpreted. If the change i n per-capita growth is due to active choices on the part of individuals depending on "quality of life" as measured as per-capita intake of bioresource goods then these changes would be considered the result of behavioral plasticity. If on the other hand, these changes are due to indirect effects and not active choice, then there is no behavioral plasticity built into the model.  4.3  A d d i n g behavioral plasticity to the Easter Island m o d e l  There are two aspects of the Easter Island model where behavioral plasticity might manifest itself, either i n the structure of the economy, or i n the overall effort expended by each i n d i v i d u a l i n the population. One way to introduce the possibility for structural change i n the economy is to modify the utility function. I do so by utilizing a Stone-Geary type utility function which assumes that there is a m i n i m u m amount of bioresource goods (subsistence level) at which utility is zero, i.e.: (4.15) where h > h { . m  n  Modifying the model so that overall work effort can change is accom-  plished by changing 7 from equation 4.9 from a constant to a state variable. A s before, we can determine the o p t i m a l consumption of resources by m a x i m i z i n g U(h,m)  as defined  by 4.15 subject to the income constraint Phh + p m  < 71/j  m  (4.16)  where w is the wage paid per unit of labor. T h e resulting o p t i m a l consumption levels are: h  =  (1 - (3)h  min  +  Ph  (4.17a)  Chapter 4. Non-substitutibility  in consumption  m  =  and ecosystem  stability  )  Phkmm  (1-(3)C  W  75  (4.17b)  Now we have that the optimal consumption level of h consists of a price dependent and a price independent portion. This is more realistic as it says to spend excess income on certain proportions of h and m only after meeting m i n i m u m nutritional requirements. Equations 4.17 only make physical sense when  Ph < 7 T ^  (4-18)  but this condition w i l l always be satisfied if h > h i .  Substituting equation 4.7 for  m n  Ph into equation 4.18 and assuming as before that w = 1 and p  m  = 1, we see that the  condition for the system to make physical sense reduces to hmin < ICiS which simply says that if the demand h i  m n  (4.19)  can be met at the present work level, use the  optimality conditions given by 4.17 to divide excess capacity to the tasks of producing m and h. If 4.18 is not met, the optimality conditions do not say what to do. C o m m o n sense suggests that if people are tying to meet m i n i m u m nutritional requirements, they would produce all the bioresource goods possible, i.e. h = jaS.  (4.20)  Finally, we can, by combining the above equations with the production functions given by 4.5a and 4.5b, compute the amount of labor (available work) the population should devote to producing bioresource goods and manufactured goods: + L  h  =  N  r  f  0  i f  h  m  m  <  i  <(  a  S  (4.21a) /V7  otherwise  Chapter 4. Non-substitutibility  in consumption  and ecosystem  ( l - / W 7 - ^ f ) Lm.  76  ifh < aS min  7  (4.21b)  —  0 The  stability  otherwise  "culture" defined by 4.15 combined w i t h the physical system defined by 4.5a  and 4.5b generates the decision process defined by 4.21. Notice that i n contrast to the original model, the division of labor is no longer fixed.  A s the price of bioresource  goods increases, labor is shifted out of the production of manufactured goods into the bioresource sector - i.e. there is structural change i n the economy. Finally, the population has the option to increase the work level 7 i n an effort to meet its needs, just as i n the Tsembaga model. I assume that the population w i l l increase its work level only after all labor is shifted into producing bioresource goods. This leads to the new system we wish to analyze:  —  = rS(l - S/K) - aSL  (4.22a)  =  (b-d  + (/>aSL )N  (4.22b)  =  \{h  - h )(-f  h  dN — —  opt  h  prod  max  - 7).  (4.22c)  where h d = ^aS is the quantity of bioresource goods actually produced. W h e n conpro  dition 4.18 is met, h  opt  < h d and the amount of bioresource goods the population is pro  capable of making w i l l exceed the amount it wishes to make so work levels w i l l decrease to the o p t i m a l level. If, on the other hand, condition 4.18 is not met, the population w i l l try to increase its work level to meet optimal demand. We can now analyze how the dynamics of the model change under these conditions. 4.3.1  M o d e l analysis  We begin the analysis by first letting A = 0 and focusing our attention on the effect that h i  m n  has on the model. If we take w(0) = 1 and and h i  m n  = 0 , we retrieve the  Chapter  4. Non-substitutibility  in consumption  and ecosystem  stability  77  original model. For the parameters chosen by Brander et al., we know there is globally stable equilibrium point at N = 4791.7 and S = 6250.  W e can again use pseudo-  arclength continuation to investigate the nature of this equilibrium point as h i  m n  is varied.  Figure 4.3 is the result of this exercise.  10000  0  I 0  i  i  0.005  0.01  i  i  0.015  0.02  i  i  i  0.025  0.03  0.035  I 0.04  hmin  Figure 4.3: Bifurcation diagram for modified Easter Island model. As w i t h the Tsembaga model, the way i n which the population partitions its energy profoundly affects the dynamics of the model ecosystem. We see from figure 4.3 that a stable equilibrium point persists up to a value of h  mm  occurs. For values of h i  near 0.017 where a Hopf-bifurcation  beyond the bifurcation point, not only does the system lose  m n  stability, but the nature of the dynamics far from the singular point change as well. Figure 4.4 shows the change i n the dynamics as well as the role behavioral plasticity has to play. T h e figure to the left shows the population trajectories for the original model and for the modified model w i t h h  min  — .02 T h e figure to the right shows how the structure of  the economy evolves over time, initially, the two trajectories are roughly the same. For  Figure 4.4: Trajectories for population and proportions of labor i n each sector over time. In the leftmost graph, curve (1) is for the original model as proposed by Brander while (2) is from the modifed model. the first 400 years the structure of the economy remains fairly stable w i t h approximately 48% of the labor force working i n the bioresource sector and the remainder i n the manufacturing sector.  A s bioresources become more scarce, the economic structure begins  to change and labor is shifted into the bioresource sector until all of the population is working i n this sector by between 1100 and 1200 A D . This shifting of available work into the bioresource sector enables the population to grow about 100 years longer than i n the original model up to a peak of around 14,000 as compared to 10,000. Also evident is the much more rapid decline that the more behaviorally plastic population must endure after it has pushed its ecosystem too far. Here, behavioral plasticity enabled the population to m a i n t a i n its positive growth trajectory longer resulting i n a more dramatic decline. T h e final aspect of this model to be discussed is the effect of allowing the population to decide to work harder, i.e. set A > 0. Figure 4.5 shows the results for w  max  = 3, i.e.  the population is willing to triple its work effort if necessary. T h e graph on the right i n figure 4.5 shows the structure of the economy changing over  Chapter 4.  Non-substitutibility  in consumption  and ecosystem  stability  79  Figure 4.5: Trajectories for population and total labor i n each sector over time for the case A =fi 0. In the leftmost graph, curve (1) is for the original model as proposed by Brander, (2) is from the modifed model w i t h A = 0, and (3) is the case for the modified model w i t h A ^ 0. time as bioresources become more scarce. In this case, when all the labor force has shifted into the bioresource sector the population begins to increase its work effort. Trajectory (3) i n the figure to the left shows the case where the population increases its work effort. B y doing so, the population averts a further decrease i n the intake of bioresources (and thus quality of life) for about 80 years. Unfortunately, this decision u l t i m a t e l y increases the price the population has to pay i n the rate of decrease of the population when it finally does collapse. T h e rate of decrease is four times that of the original model and two times that of the modified model w i t h a fixed work level. This type of scenario is very reminiscent of our situation today. We are increasing the amount of work we do as we attempt to maintain our standard of living. Obviously, we may be simply buying ourselves a little time and increasing the ultimate price we w i l l have to pay.  Chapter 4.  4.4  Non-substitutibility  in consumption  and ecosystem  stability  80  Conclusions  In this section we have studied the interaction between culture and ecosystems i n the context of a model where the economy is more complex. The model I proposed where both the structure and the overall work level of the economy were allowed to change experienced a bifurcation from a stable steady state to a l i m i t cycle which produced more dramatic changes i n population dynamics. T h e key point to observe is that, as w i t h the Tsembaga model, increased behavioral plasticity decreased the stability of the system. In this light, the ability of modern economies to change their structure quickly i n response to changing environmental conditions so frequently lauded by the expansionist view, might not be such a positive asset i n achieving sustainability. Obviously one can argue that this model is not rich enough to capture our ability to become more efficient, to utilize different goods to perform certain tasks, to generate capital, and to try to improve natural capital before it degrades, thus averting the collapse experienced by the simple model and enabling a transition to sustainability. E x a m i n i n g such a model is the focus of the next chapter of this thesis.  Chapter 5  T h e d y n a m i c s of a two sector ecological economic s y s t e m  In this chapter, I w i l l extend the concepts I have developed so far to study the dynamics of a model of a two sector economy w i t h capital accumulation. T h i s is a much harder problem than we have addressed so far. The Tsembaga and Easter Island models were both pure labor economies.  T h e only decisions taking place i n these economies were  how hard to work and what portion of available labor to devote to each activity. In an economy w i t h labor and capital, the decisions are more complex.  Here we have  firms that are t r y i n g to utilize resources efficiently while consumers are simultaneously trying to maximize utility. In order to tackle this problem, we w i l l have to develop more sophisticated economic concepts for modeling economic growth. To this end, this chapter is organized as follows.  In the first section, I summarize  important concepts from the theory of economic growth that are important for this model. N e x t , I outline the relevant concepts from production and utility theory and related issues such as non-substitutability of consumer goods that we investigated i n chapter 4 and the importance of the nature of the production function that we encountered i n chapter 3 that are used to construct the model economic growth system. Finally, I develop the ecological system i n which the economic growth system is embedded. T h e final step is then to analyze the dynamics of the resulting system.  81  Chapter  5.1  5.  The dynamics of a two sector ecological economic system  82  Simple economic growth models  Jensen [36] gives an exhaustive treatment of simple economic growth models w i t h two state variables: labor and capital.  Such simple models have received much attention  i n the economic literature, often focusing on the steady state growth trajectory of an economy. T h i s steady state trajectory corresponds to a constant capital-labor ratio w i t h economic output growing w i t h capital and labor growth. necessarily consists of three components:  A n economic growth model  relationships that describe the dynamics of  labor and capital over time, a relationship between economic output and a given level of capital and labor (factors of production), and information specifying what society does w i t h economic output. Mathematically, the model consists of a dynamical system coupled w i t h algebraic equations governing production and consumption. A common example of a simple economic growth model w i t h a single production sector would be: dL/dt  =  nL  dK/dt  =  sY  .  (5.1) (5.2)  where L is labor (generally viewed as the number of workers i n a population), K is the quantity of capital, n is the per-capita growth rate of the population, Y is the physical output of the economy and s is the proportion of output that is saved. T h e output of the economy is typically given by a function of the form Y = f(L,K) assumed to satisfy the following conditions: / ( L , 0 ) = f(0,K) > 0,  < 0, J^4  where f(L,K)  is  = 0, V i i ' and L, | £ > 0,  < 0. T h e behavioral dynamics of the population modeled here  are obviously quite simple - a constant proportion s of output is devoted to savings and (1 — s)Y units of output are consumed. Clearly, the behavior of such a system hinges on the assumptions about the production function and the behavior of the population.  Chapter  5.  The dynamics of a two sector ecological economic  system  83  It is easy to see that for the conditions normally placed on / , the behavior of the above system is very simple.  Using simple differential inequalities one can see that  any trajectory beginning i n the first quadrant (both capital and labor are positive) w i l l remain there for all time and both state variables w i l l grow without bound. Thus, the population, capital stocks, and productivity all grow exponentially. To address economic growth i n a bounded ecosystem the dynamical system has to be extended to include d y n a m i c resource constraints and economic model must be extended to more complex behavior.  accommodate  In order to develop such a model, some additional concepts  from production and u t i l i t y theory must be employed, which I w i l l briefly review i n the next section.  5.1.1  Basic laws of production and the theory of the  firm  Very basic to an economic growth model is the specification of the laws of production or the production technology of the economy.  Some specific examples of production  functions were discussed i n the model for agricultural output i n the Tsembaga ecosystem (Chapter 3). T h e production technology is represented by a production function, Y = f(xi,X2,...,x ), n  that characterizes technological alternatives for the inputs  maximal output Y obtainable for a given choice of these inputs.  and the  T h e characteristic of  the production function most important for this model is the possibility of technical substitution between inputs. T h e technical substitution possibilities specified by a particular production function refers to what extent one input may be substituted for another to m a i n t a i n a fixed level of output. A s we already saw, the Cobb-Douglas allows infinite substitutability between inputs, an assumption that may be completely unrealistic.  Problems associated with  such assumptions have received much attention i n the ecological economics literature (e.g. see [60] for a review). A t the opposite end of the spectrum is the Leontief production  Chapter  5.  The dynamics of a two sector ecological economic system  84  function usually written as Y =  min  {—}  (5.3)  i=l,..,n Pi where /?,• is the requirement of input i per unit of output.  T h i s is the analogue of the von-Liebig function used to describe agricultural production that we have already met. Here, there is absolutely no possibility for substitution between inputs. Clearly, neither extreme is entirely realistic, and different levels of substitutability are to be found for different types of inputs and outputs. For example, land can't be substituted for water to maintain productivity during a drought. A sewing machine and electrical energy can be substituted for a person w i t h needle and thread i n the construction of a garment. In m y model, I assume that the overall production technology is of the Leontief form for physical inputs but capital and labor are substitutable to carry out productive activity i n the production process. That is, let X{ be the i t h physical and let £(L,K)  input  represent productive activity where L is labor i n hours and K represents  services provided by capital, then (5.4) I represent £(L,K)  w i t h a Cobb-Douglas production function i.e. £(L,K)  = L K^. a  The  resulting production function given by equation 5.4 allows infinite substitution between capital and labor, but no substitution between labor and capital (stocks), and raw materials (flows).  T h i s production function would not allow labor to be substituted for  a l u m i n u m i n the production of a bicycle, but it does allow a frame j i g to be substituted for a human hand to hold the frame i n place as it is welded. Recall from Chapter 3 that a and (3 measure the marginal productivities of labor and capital respectively. It is commonly assumed that a + f3 = 1 or that the production function has constant returns to scale (or the elasticity of scale is 1). E l a s t i c i t y of scale (e ) s  is a measure of the proportionate change i n output associated w i t h a proportionate change  Chapter  5.  The dynamics of a two sector ecological economic system  85  of all inputs. If e = 1, doubling all inputs exactly doubles output. If e > 1, doubling of s  s  all inputs more than doubles output, etc. In m y model I assume that productive activity exhibits constant returns to scale. Next, I assume perfect competition (individual firms cannot affect prices by their choices of output levels) and that firms are making decisions i n the "short r u n " . In the economics literature, time scales are resolved to the "short run" and the "long r u n " . T h i s distinction is related to what managers are able to change as they make decisions. It is assumed that i n the short run, managers can't change capital stocks. Thus for short run decisions, managers are faced w i t h a fixed capital stock and w i l l select the o p t i m a l labor input. In the long run, managers can adjust both capital and labor stocks i n response to the conditions i n the labor and capital markets.  In m y model, there is no explicit  modeling of investment supply and demand, managers make only short run decisions and capital growth is determined completely by savings rates. F i n a l l y I assume that firms w i l l make full and efficient utilization of available factors of production. T h e y w i l l attempt to fully utilize capital stocks and select the o p t i m a l labor and output levels to minimize cost (or maximize profit).  For an economy w i t h  m u l t i p l e firms, full and efficient utilization means the total capital is divided optimally among the firms and then optimal labor is selected w i t h i n each industry. T h e final aspect of firm behavior important to this model is the labor market. T h e o p t i m a l labor input for a given industry depends on the relationship of the cost of labor (wage) to the cost of capital. Thus given the cost of capital as fixed, the availability and cost of labor will determine the o p t i m a l combination of labor and capital.  5.1.2  C o n s u m e r behavior  T h e behavior of consumers is modeled using the standard approach from neo-classical economics: consumers maximize utility subject to an income constraint. We have already  Chapter 5.  The dynamics  of a two sector ecological economic  system  86  seen the importance the. form of the utility function plays i n ecosystem dynamics i n Chapter 4. We saw w i t h the Easter Island model that restricted substitutability between bioresources and manufactured goods was destabilizing. T h e Stone-Geary u t i l i t y function is given by  logu = J2log(q - r ) n  t  q  (5.5)  t=i  where u is utility, qi are commodities, and q™ are the m i n i m u m amounts of a commodity %n  required. T h i s function is intuitively appealing. If the economy is capable of production levels above m i n i m u m requirements, people w i l l substitute among favorite goods, trading off nightly fillet mignon for a better quality compact disc player. However, starving people won't try to ease their suffering by making bead necklaces, simply because there is no food and there are beads. The Stone-Geary utility function nicely captures this behavior as demonstrated i n chapter 4.  5.2  The ecological economic model  T h e model that is the focus of the rest of this thesis is a two sector economic model coupled w i t h an ecological model. T h e economy has an agricultural and non farm business sector (manufacturing). T h i s choice of division for economic activities is motivated by the fact that we wish to model the effects of economic activity on two basic stocks: renewable natural capital and nonrenewable natural capital. A more common division of economic activity is between the agricultural, manufacturing, and service sectors. In m y model I have vertically integrated the manufacturing and service sectors w i t h the idea that the provision of services relies heavily on manufactured goods (insurance agents use cars, cell phones, computers, fuel, paper, etc. to do their jobs) and that the impact of these activities tend to be more focused on nonrenewable natural capital. T h e economic ecological system model is shown schematically i n figure 5.1. There  Chapter 5.  The dynamics  of a two sector ecological economic  system  87  are two basic flows i n the model: the flow of raw materials and services from the state variables into the economic system and the flow of goods and services out of the economic system.  T h e economic system represented by the non-farm business and agricultural  sectors draw flows of low entropy materials from the stock of nonrenewable natural capital and services from labor, man-made capital, and renewable natural capital converts them to a flow of goods and services. T h e arrows between the two sectors represent the interindustry transfer of goods and services. T h e human population, based on its preferences, can decide to consume goods and services, direct them towards investment, or increasing nonrenewable natural capital stocks through research and development for new materials, recycling, more efficient use of materials, or more efficient extraction techniques. The model attempts to capture as simply as possible the fundamental aspects of both sides of the argument about sustainable development. A l l of the processes by which many believe we w i l l continue to avert environmental degradation are included: ever-increasing efficiency, better material use, etc., but the achievement of these ends a l l require flows of economic goods and services and generate their own impact on the ecosystem. A perfect example is recycling. Recycling reduces the environmental impact of some production processes but requires capital, labor, energy input, and generates a waste stream, i.e. it merely transfers ecological stress from one form to another.  5.2.1  T h e economic system  In this section I w i l l solve the simultaneous consumer and firm o p t i m i z a t i o n problems in order to specify how labor and capital are allocated to each sector.  We begin by  specifying the technology i n each of the sectors. Should the need arise, please refer to the table provided at the end of the chapter for an easy reference for the definitions of symbols. A s we have seen before, agriculture is best modeled w i t h the von-Liebig or Leontief  Chapter 5.  The dynamics  of a two sector ecological economic  system  88  Economic System NonFarm Business -— { Sector J  —"  Agricultural { Sector  J  (Social Organization Preferences, etc)  Nonrenewable Natural Capital, k n  Renewable Natural Capital, k  Man-made Capital, k  r  Human Population, h (Labor)  A  Flow of economic outputs (goods and services) Flow of economic inputs (raw materials and services from capital stocks)  Figure 5.1: Schematic of two sector ecological economic model, function. I assume that Y  a  = E (k )min{-£-, a  r  PU  where Y is annual agricultural output, E (k ) a  a  r  —, Pi  (5.6)  —} PN  is a measure of efficiency related to soil and  weather and is a function of the stock of natural capital, k . r  T h e inputs are productive  activity £ , land /, and nutrients N (phosphorus, nitrogen, potassium, etc.). T h e /?'s are a  the per unit input requirements per unit of output. Efficient utilization implies that  (5.7) thus for a given amount of land, there is a set nutrient requirement and a physically  Chapter 5.  The dynamics  of a two sector ecological economic  system  89  determined amount of work required to carry out the production process. T h e population w i l l decide how much productive activity ( £ ) to direct to agricultural production v i a the a  o p t i m a l combination of capital (K ) a  and labor (L ) a  based on the production function  (a = Ll«K °.  (5.8)  b  a  In the model, natural capital provides several free services and could be called an economic sector i n a sense. A m o n g other things, it generates soil and soil nutrients, assimilates waste, and irrigates v i a the solar water pump. In equation 5.6 this is reflected by the fact that efficiency is a function of the stock of natural capital, but also through the nutrient input required for a given level of output.  T h e required nutrients can be  supplied by the "natural sector" as is the case i n the Tsembaga ecosystem, or by the manufacturing sector (fertilizer, etc.). Thus at low levels of agricultural output, natural nutrient production is sufficient to meet demand.  A s output increases, nutrients in the form of fertilizer, pesticides, and  genetically engineered seed must be provided from the manufacturing sector. Let R  ma  be  the manufactured goods required per unit of agricultural output. A s agricultural production increases R  ma  increases from zero up to some m a x i m u m where most of the nutrients  for agriculture are supplied by the manufacturing sector. It is a messy bookkeeping and computational problem to try to relate R  directly to agricultural output.  ma  Instead,  the ratio of population density to renewable natural capital, — is used as an indirect measure of agricultural output. T h e higher this ratio, the more pressure is being put on k and more nutrients must be injected into the system from the manufacturing sector. r  T h e functional relationship is  Rma(x) =  3  f!  (5-9)  3 3  where 0N is the nutrient requirement per unit of agricultural output, and  fihaif  is the  Chapter 5. The dynamics  level of — at which R  ma  of a two sector ecological economic  system  90  is one-half the m a x i m u m . This function has the property that  below a certain threshold value of x, R (x)  is very small (nutrients are being provided  ma  by natural capital). A s x increases above the threshold, R (x) ma  begins to increase rapidly  up to a m a x i m u m where all nutrient inputs come from the manufacturing industry. Choosing the units so that /3^ = 1, and assuming efficient factor utilization we have a  Y = E (k )L «K \ a  a  a  r  (5.10)  b  a  a  w i t h nutrient demand from the manufacturing sector, Y  given by  ma  Y  = R {^)Y .  ma  ma  (5.11)  a  fXirf  T h e story is similar for manufacturing (= non farm business sector) except that here, the manufacturing industry includes the production of inputs and the finished product. T h i s is necessary to avoid including a t h i r d sector i n the model for the production of raw materials. Thus we can write manufacturing production i n terms of the productive activity directed towards the process of extracting raw materials and using them to deliver goods and services: Y  = E (k )£  m  where Y  m  is manufacturing output.  m  n  (5.12)  m  T h e efficiency of the manufacturing process,  E, m  depends on the stock of nonrenewable natural capital, k , because as stocks of low entropy n  materials go down (e.g. metal per ton of ore, reservoir petroleum saturation, etc.), more and more work is required to extract raw materials. A s i n the agricultural sector £  = L^K^  1  m  thus we have Y  = E {k )L^K -.  (5.13)  b  m  m  n  If we define the capital-labor ratio r\i = -y-, and assume constant returns to scale, Ei equations 5.10 and 5.13 can be rewritten i n the form Y  a  =  E {k )L rfc a  r  a  = E {k ) ; "K a  a  r  V  a  (5.14a)  Chapter 5.  The dynamics  of a two sector ecological economic  Y  E (k )L T] ™  =  m  which we w i l l employ later.  m  n  = E (k ) - ™K m  n v  91  (5.14b)  a  b  m  system  m  Equations 5.10 and 5.13 determine how agricultural and  manufacturing outputs are related to labor and capital devoted to them. T h e question remains: how does society decide how much to consume of each product and how m u c h labor and capital should be devoted to each activity? To answer the first question, we assume that society directs energy to producing agricultural, manufactured, investment, and resource goods. T h e first three require no explanation.  Resource goods would consist of any effort to find more raw materials,  improve material efficiency or develop new materials. Consumers then solve the following constrained m a x i m i z a t i o n problem: max  U(q ,q ,q q ) a  m  1:  = (q - q* ) {q - q*J qf Ca  r  a  subject to: P q a  where q ,q ,(lii a  m  +Pq  a  CM  a  m  m  m  + Piqi + P q r  r  q/  (5.15)  < I  (5.16)  c  and q are the per-capita consumption rates of agricultural, manufacr  turing, investment, and resource goods, P , P , a  m  Pi, and P  r  are their respective prices,  / is per-capita income, and c through c are the cultural parameters that characterize a  r  the preference for each good. A s i n the Easter Island model, there are m i n i m u m intake levels of certain commodities below which the population w i l l alter its behavior. Here we assume that there is a m i n i m u m level of agricultural goods q* set by h u m a n nutritional requirements and a m i n i m u m quantity of manufactured goods, q* necessary to meet m  housing, clothing, and m i n i m a l capital requirements such as very simple tools. There is no m i n i m u m investment or resource-good levels - when faced w i t h merely surviving, the population concentrates on the bare essentials. B y applying the technique of Lagrange multipliers, we can solve the problem specified by 5.16. Define supernumery income, I by s  I = I-P q: s  a  + P q* m  m  (5.17)  Chapter 5. The dynamics of a two sector ecological economic system  92  then we obtain the following first order conditions for the optimal per-capita consumption levels :  la  c I o + -7^  =  (5.18a)  a  a  I  C  9m  =  q + -=—  (5.18b)  ^ ^  (5.18c) (5.18d)  m  ft = <?r =  Equations 5.18 are interpreted as follows. After meeting m i n i m u m demands of agricultural and manufactured goods, a proportion of the income left over, the supernumery income I is devoted to each of the four activities. T h i s defines the demand side of the s  economy. T h e supply side of the economy is characterized by firms m a x i m i z i n g profits. T h e profit functions for the agricultural and manufacturing sectors (=non-farm business) are U (L ,K ) a  a  a  ^-m{L ,  K)  m  m  =  P Y -wL -rK -Y R P  =  PY  a  a  m  a  a  a  ma  — wL  m  (5.19a)  m  rK  m  m  —YR P m  am  (5.19b)  a  where w and r are the per-unit costs of labor and capital respectively, R  ma  is the rate  at which manufacturing goods are utilized by the agricultural industry, and R  am  is the  rate at which agricultural goods are utilized by the manufacturing industry. I assume that labor and capital decisions made i n one industry w i l l not affect prices i n the other so firms w i l l maximize profits by finding the optimal labor-capital inputs v i a first order conditions given by (for example i n agriculture) dIl (L ,K ) a  a  a  dL dU (L ,K ) a  a  a  a  dK  a  aY a  a  L bY  { P a - R  m  a  P m ) - w  =  0  (5.20a)  =  0  (5.20b)  a  a  a  K  B  -(Pa  -  RmaPm)  - r  Chapter 5.  The dynamics  of a two sector ecological economic  system  93  w i t h an analogous set of equations for the manufacturing industry. These two equations determine the o p t i m a l capital labor ratio:  which says that the o p t i m u m factor inputs depend on the labor to capital cost ratio and the factor productivities. Next, by adding equations 5.20a and 5.20b we arrive at the zero profit condition: PY a  a  = wLf  + rKT  + YR P , a  ma  m  (5.22)  which says that, at o p t i m u m , the revenue generated by the production and sale of agricultural goods exactly covers the production costs. This relationship is true for any C R S technology. U n t i l further notice, all the quantities I w i l l be referring to are the o p t i m a l quantities (where this makes sense), and I w i l l drop the superscript.  Equations 5.18  characterize the demand for goods while 5.21, and 5.22 along w i t h their counterparts for the manufacturing industry characterize the demand.  5.2.2  C o m p u t i n g the general equilibrium  C o m p u t i n g the general equilibrium reduces to setting the aggregate demand equations equal to the aggregate supply equations. T h e demand for agricultural goods is composed of the per-capita consumption multiplied by the population level plus the agricultural goods used i n the manufacturing industry, i.e. Y  D a  = hq + Y°R a  am  (5.23)  where h is the human population, and the superscript indicates "demanded". T h e demand for manufactured goods is composed of the demands of consumption, investment, and resource goods all of which are produced by the manufacturing sector, plus the  Chapter 5.  The dynamics  of a two sector ecological economic  system  94  manufactured goods consumed by the agricultural sector. Thus Y£ = Hm + hq + hq + Y R .  (5.24)  D  {  r  a  ma  T h e demands for agricultural and manufactured goods are easily computed by dividing equation 5.22 and the counterpart for manufacturing through by the appropriate prices. Setting the results equal to the right hand sides of equation 5.23 and 5.24 yields the general e q u i l i b r i u m equations: P hq  + PY R  + P hq  + P YR  a  P hq m  m  + P hq m  l  m  a  r  a  m  m  a  am  ma  =  wL  =  wL  a  m  + rK  + YR P  a  + rK  a  Equations 5.25 specify the equilibrium w i t h efficient factor  m  ma  (5.25a)  m  + YR P m  utilization.  am  (5.25b)  a  Recall that i n  the model full factor utilization is enforced. T h i s requires that L  a  + L  m  Ka + Km  =  L  (5.26a)  =  K  (5.26b)  where L and K are the total labor and capital available, respectively. Equations 5.21, 5.25 and 5.26 constitute a system of five equations (of which three are nonlinear because prices and output are nonlinear functions of capital and labor) and six unknowns: L , L , a  K, m  m  K, a  r , and w. Thus given any one variable, all other equations could be solved for the  other variables. Since i n this model money acts only as a numeraire, the system is closed by fixing r (the factor cost of a unit of capital) as the numeraire good and measuring prices i n terms of r. There are several problems w i t h this approach. First and most obvious is the problem of existence and uniqueness of solutions to systems of nonlinear equations.  Then,  supposing there is a unique solution, there is the difficulty of locating it. T h e algebraic system of equations that characterize the economic system is coupled w i t h a dynamical  Chapter 5.  The dynamics  of a two sector ecological economic  system  95  system that characterizes the ecosystem - i.e. the human population, capital stocks, natural capital stocks, and so on. Thus, the economic system equations must be solved continuously as the physical system evolves. If there is no explicit solution to the economic model as was the case for the models i n Chapter 4, the ecological economic system model is a set of differential algebraic equations ( D A E ) A l t h o u g h there are techniques to solve D A E ' s (i.e. collocation, [5]), dynamical system and bifurcation analysis tools such as X P P a u t and A u t o are not set up to handle this situation. Thus, i n order to study the structure of the model, we must reformulate the general equilibrium problem. I reformulate the problem by adding a labor market and writing the five equation system as one explicit algebraic equation and one differential equation. F i r s t , we substitute the values of Y , Y , a  and q given by 5.14a, 5.14b, and 5.18a respectively into 5.25a  m  a  to get P hq* + c hl - c hP q* a  a  a  a  a  - c hP q*  a  a  m  + P R E {k )r]- ™  K  a  m  a  am  wL  m  n  + rK  a  =  m  + P R E {k )r,- *K .  (5.27)  a  a  m  ma  a  r  a  T h e n , from equation 5.21 and its counterpart for the manufacturing industry, we get a set of coupled equations for the optimal prices:  ~ = £Z]£L - "  p  -  +R P  (5 2Sb)  We can again use equation 5.21 to eliminate capital and labor from equations 5.28, i.e., at o p t i m u m we have: Lw =  (5.29)  a  thus  K ra a  L iu + K r E (k )ri-^K a  a  a  r  a  a  X E (k ) -^K +  a  r  V  K  a  a  T  _  + E (k )rj-^  r  {  l  a  r  rtf E (k )b ' a  r  a  (5.30)  Chapter 5.  The dynamics  of a two sector ecological economic  system  96  A similar relation holds for the manufacturing sector, enabling us to write equations 5.28 as  Pa  —  n  /  ~\~ RraaPra  \1  7  (5.31a)  E (k )b a  Pm  r  a  f  =  -C'm ( "'TI ) "m  +RamPa.  '  (5.31b)  Solving these coupled equations for the prices yields: =  p  a  j ^° Rma Ram \E (k )b  1  a  _  p  _j_ RmaV E (k )b  r  /  _  r  1  r  m  m  a  m  VnT  n  j J  m  ^ Ram )q r  (5.32a)  a  j  ^  32b)  Rma Ram V E  m  (k )b n  E (k )b )  m  a  r  a  Notice the upward effect decreasing efficiencies and increasing inter-industry transfers have on prices. It is important to include this aspect i n the model to capture the important fact of the heavy reliance of modern agriculture on manufacturing inputs. Notice that the prices i n 5.32 depend only on physical constants, the per unit capital cost, and the capital-labor ratios rj and r\ . A t o p t i m u m , the capital labor ratio can be replaced ai  m  by the factor cost ratio v i a 5.21. Thus, given the factor cost ratio, o p t i m a l prices are determined up to the constant r. Thus equations 5.32 can be rewritten as Pa = rf (co)  andP  a  m  = rf (u)  (5.33)  m  where LO = — and r M  w  )  =  \ J-  f (co) m  B—T7~ —  =  iX lXam ma  1 1  —  j? \  (  u  u  —+  rp (  ^ -(b /a ya  RmaRam \  m  m  -^m (  u  U  :—  ^m\^n)Om  ^ay^rjOa |  ) ^ro  R mu (ba/a y"\ a  aa  a  E [k )b a  r  (5.34a) J  a  _  (  5  3  4  b  )  J  B y writing the prices this way, we w i l l see that r cancels and the e q u i l i b r i u m labor and capital devoted to agriculture and manufacturing depend only on the factor cost ratio LO. Finally, if / , the per-capita income of the economy could be written i n terms of cu,  Chapter  5.  The dynamics of a two sector ecological economic system  equations 5.34 and 5.27 can be combined to write K  a  97  as an explicit function of CJ. Since  hi, the total income of the economy, is equal to the sum of the total income generated by labor and capital, respectively, (factor rewards) we have, hi = Lw + Kr = (L  + L )w  a  + (K  w  + K )r  a  (5.35)  m  and using 5.29 we can eliminate the labor terms arriving at hI= -^ +  4^-  r  (5-36)  r  0  Oa  m  Here we see that income and prices both depend on r. F i x i n g r is equivalent to choosing units for the money i n the system - i.e. r is a numeraire. Since we are only including the dynamics of the labor market, we fix r = 1, then CJ = w.  Finally, combining  equations 5.25, 5.36, 5.26b, and 5.21 we arrive at at explicit formula for K  i n terms of  a  K, h, and w: K (K,h,w)  =  a  C K  (1 - c )hf (w)q* a  l _ —:  C  C  a  a  + -j— + RamE (k ) m  f a  o  hi  b  f  h RamE  o  m  \  a  n  \  QJ  0,171  I —j- ]  f (w)K a  fa(w)-\-R E (k ) ma  m  hc f q* a m  ( a  m  )\~r~) \wb J  -  a  r  \  a  [—-) \wb J  m  ' '  a  '  f (w) m  a  Thus given the total capital endowment of the economy, the human population, and the wage rate (= factor cost ratio), the optimal amount of capital to devote to agriculture is easily computed by 5.37.  T h e n using 5.26b, and 5.21, the o p t i m a l levels of capital  and labor to manufacturing and labor to agriculture can be computed.  T h e problem  is that the o p t i m a l labor quantities computed this way may not be equal to the labor endowment of the economy, that is: L + L a  m  ^ L i n general and the economy is out of  equilibrium. T h i s is where the role of the labor market comes into play. T h e labor market w i l l link wages to available labor and force the economy to tend towards equilibrium. Before discussing the labor market, however, I would like to make a critical point about  Chapter 5.  The dynamics  of a two sector ecological economic  system  98  equation 5.35. This equation says that a certain portion of the revenue generated by the productive process is paid to workers i n the form of wages while the remainder is paid to ' c a p i t a l ' i n the form of interest, dividends, etc. It says nothing however about the distribution of income. I w i l l address this point i n more detail later. Several (nonlinear) algebraic relationships have been proposed to relate labor supply, demand, and wages, e.g. [66], but I w i l l employ a simple linear (in labor supply and demand) differential equation to model wage dynamics. The assumptions are basic: an oversupply of labor w i l l put downward pressure on wages while and under-supply will have the opposite effect. This simple-minded model does nothing to address important labor market issues such as union activity and so on, but is sufficient for a start. Thus we have (5.38) where \  w  is the speed of response of wages to disparities between labor supply and de-  mand.  E q u a t i o n 5.38 coupled w i t h 5.37 comprise a fast efficient method forcing the  economy to seek equilibrium i n a dynamically evolving system. T h e alternative of solving a set of coupled nonlinear equations for the equilibrium is not only slower and more difficult, but also artificial. Economies are never i n equilibrium, and equation 5.38 captures this fact. Further, we can actually adjust "out of equilibriumness" v i a the factor X  W  and study its effect on the dynamics of the system. In order to illustrate the operation of the economic system, I have computed the  e q u i l i b r i u m w i t h arbitrary initial capital and labor endowments of 100 units each. Parameters are: a = 0.3, a a  E  a  = E  m  m  = 0.8, c]* = 0.5, q* = 0.1, c = 0.2, c m  a  m  = 0.8, c,- = c — 0, and r  are constant and set equal to 1. Figure 5.2 shows the results of this exercise.  The i n i t i a l guess at the wage rate is 0.5 so each unit of labor is half as costly as a corresponding unit of capital. W i t h such cheap labor, it is o p t i m a l to use well over 200  Chapter 5.  0.2  0  The dynamics  0.4  0.6  0.1 I  1.2 1.4  1.6  of a two sector ecological economic  1.1  0  2  0.2  0.4  0.6  0.1  1  1.2  1.4  1.6  1.1 2  0  Time  Time  system  0.2  0.4  0.6  99  0.1  1  1.2  1.4  1.6  I.I  2  Time  Figure 5.2: Trajectories of wages, capital, and labor as the economy adjusts. units which far exceeds labor availability. Upward pressure on wages drives the system very quickly to the equilibrium state w i t h w = 1.121, K  a  30.767, L  m  = 80.574, K  m  — 19.426, L  a  =  = 62.233. T h e question is, is this solution unique and optimal? Figure 5.3  helps put this question i n perspective; it shows the utility function and the o p t i m u m solution above. Note that the utility function is strictly convex inside the region where the economy can exceed its m i n i m u m demands of q* = 0.5 and a  = 0.1. T h e inset figure on the  upper right is a contour plot of the surface on the lower left showing the o p t i m u m w i t h a white dot, the region where m i n i m u m demands can't be met w i t h available labor and capital endowments (white area), and where they can (grey scale area).  For values of  labor and capital i n the grey scale region, it is tedious but not difficult to show that' the necessary condition for optimality given by 5.18 is sufficient and the solution is unique. In the region i n the L  a  —K  a  plane where m i n i m u m needs cannot be met, the util-  ity function is defined to be identically 0. In this case there is no o p t i m u m solution so some other mechanism must be defined to allocate available resources to different activities. I accomplish this by assuming that if m i n i m u m needs cannot be met, the economy  Chapter  5.  The dynamics of a two sector ecological economic  system  100  Figure 5.3: Surface plot of utility function showing optimal combination of labor and capital to agriculture. w i l l first attempt to meet food needs and devote what is left over to other activities. Mathematically, this translates to:  h> 0  a* + — <  Ha  p  a  <?*  / , < 0 and / - P q* > 0  a  a  E (k )L K aa  a  otherwise  ba  r  -ft  q* +  c  m  q  m  =  I < 0 and / - P q* > 0 s  0  0  a  (5.39b)  a  otherwise  z± J  (5.39a)  a  I  s  >0  m  otherwise  (5.39c)  Chapter 5.  The dynamics  of a two sector ecological economic  7^  /, >0  0  otherwise  system  101  (5.39d)  Before turning our attention to the physical system, I would like to emphasize two important aspects of the economic system: the effect of inter-industry transfers, and the (sensible) way the economy evolves when it becomes more difficult to meet m i n i m u m demands (i.e.  how equations 5.39 work) .  I do this by examining the evolution of  the economy as the amount of manufactured goods purchased by the agricultural sector increases. Figure 5.4 shows how the consumption and expenditure patterns change under these conditions.  Figure 5.4: G r a p h (a) shows q versus q . Notice that consumption evolves toward (9o>4m)- G r a p h (b) shows q (dotted) and q (solid) over time. G r a p h (c) shows the proportion of income devoted to purchasing manufacturing and agricultural goods, I and I respectively. m  a  m  a  m  a  Figure 5.4(a) plots q  m  (QmiQa)-  versus q and illustrates how the economy moves to the point a  Beyond this point, the economy first meets agricultural needs and uses what  is left for manufactured goods as illustrated by the vertical line.  Figure 5.4(b) shows  consumption over time - large sacrifices i n the consumption of manufactured goods are necessary to m a i n t a i n agricultural production. Finally, figure 5.4(c) shows how increased reliance on manufactured inputs i n agriculture w i l l cause relative price increases for  Chapter 5.  The dynamics  agricultural goods.  of a two sector ecological economic system  102  W i t h the economic system model complete, we now t u r n to the  final task of specifying the physical system.  5.3  T h e ecological system model  T h e cultural (distributional) component of the model is contained i n the economic system i n the four parameters:  c , c , c;, and c that govern how the productive capacity of a  m  r  the economy is portioned to the different activities of consuming food, manufactured goods, investment goods, and resource goods respectively. W e are left to specify how these activities interact w i t h the state variables h, kh, k , and k as defined in chapter 2. n  r  T h e dynamical system that we will analyze for the remainder of this chapter is: r£ dt  ^dt  =  m  (5.40a)  a  = e^foi-MH  ±H = dt  d  *k  (b(q ) - d(q ))h  =  -e , Y kn m  n (l  kr  r  m  (- ) 5  + e , hqr  (5-40c)  kn r  - k) - e ,Y r  kr a  40b  a  (5.40d)  where b(q ) is the per capita b i r t h rate as a function of per capita consumption of m  manufactured goods which incorporates the idea of "demographic transition", d(q )) is a  the nutrition dependent death rate function just as i n the Tsembaga model, the e ; j are (conversion) factors measuring the effect of the j t h process on the iih state variable, i.e. e ,a measures the effect of agriculture on renewable natural capital, 8 is the rate of kr  depreciation of man-made capital, and n is the (possibly dependent on economic output r  or the state of the system) regeneration rate of renewable natural capital. T h e model specified by 5.40 is perhaps the simplest possible that incorporates a l l the key features that are debated i n the literature. For example, equation 5.40a taken w i t h equation 5.40b w i t h 8 = 0 and b — d held constant is a typical example of an  Chapter  5.  The dynamics of a two sector ecological economic system  103  economic growth model w i t h no connection to the physical world. This would correspond to the model i n figure 2.3. Figure 5.5 shows the evolution of a model economy under these circumstances.  G r a p h (a) shows the trajectory of the economy i n phase space  from different initial capital and labor endowments.  In this case, capital and labor  grow without bound, converging to a fixed capital labor ratio determined by the level of investment of the economy, c,- as shown i n graph (b). W h i l e the capital labor ratio is below the long run equilibrium level, standard of living increases up to a m a x i m u m as indicated i n graph (c). After the long r u n equilibrium is reached, economic output grows exponentially, w i t h per capita consumption constant.  od  fflO  oi  4(x>  O  200  4 0 0  6 0 0  B O O  1 0 0 0  Labor (a) Figure 5.5: G r a p h (a) shows capital versus labor for the simple economic growth model corresponding to figure (2.3) and equations (5.2). Notice each trajectory has the same slope. G r a p h (b) shows the capital-labor ratio. G r a p h (c) shows the per capita consumption of manufactured (dotted) and agricultural goods (solid) over time.  E x p o n e n t i a l economic growth is unrealistic i n the long run, and the model incorporates important implications of entropic considerations called for by authors such as [27, 18] by allowing things to wear out - i.e. S ^ 0 i n equation 5.40b, and including the physical reality that producing goods can degrade both renewable and nonrenewable natural capital i n equations 5.40c and 5.40d. Now, if one sets the right hand sides of equations 5.40 to zero to find the steady  Chapter 5.  The dynamics  of a two sector ecological economic  system  104  state(s), this would correspond to locating a steady state economy i n phase space. Indeed, setting the equations above to zero and reading off the conditions for this to be true matches our intuitive idea about what a sustainable human agro- ecosystem is, i.e. at a steady state, b i r t h rates w i l l be depressed by changing economic structure (improved living standards and the increased marginal cost of children); investment rates w i l l just offset depreciation (entropic decay) keeping capital stocks constant; and recycling, more efficient resource use, and reduced waste streams w i l l offset degradation of natural capital. So what can be gained studying a complicated dynamical system? T h e verbal description does not say anything about the magnitudes of the state variables at equilibrium, nor does it say anything about whether the equilibrium is attainable, i.e. under what conditions can a system arrive at a sustainable state. It is one thing to characterize a sustainable state, but another to study its structure, the task to which we now t u r n our attention.  5.4  A n a l y s i s of the M o d e l  Because the model structure is very rich, it w i l l be explored a piece at a time.  The  first issue we w i l l explore w i t h the model is the interaction of investment, evenness of economic growth, and the distribution of wealth i n an economy that relies on renewable natural capital - i.e. one step up from the most basic economic growth model involving only labor and capital. C o m p l e x i t y w i l l then be added step by step, finishing w i t h the analysis of the full model.  5.4.1  Investment, d i s t r i b u t i o n of wealth, and ecosystem s t a b i l i t y  Intuitively, the process of investment by which productive capacity is increased should make everyone's life better off. It is possible however to invest too m u c h whereby, for example, the capital stock may grow to such a point that its maintenance puts such  Chapter  5.  The dynamics of a two sector ecological economic system  105  a drain on the economy that the standard of living is reduced. Another problem w i t h too much investment is associated w i t h overexploitation of resources due to being too efficient.  In our model, investment helps productivity not only i n the  goods sector, but also i n agriculture.  manufactured  This increased productivity i n agriculture may  destabilize the system by allowing the population to grow far beyond the level that an ecosystem could bear without degradation. One mechanism that might halt this process is behavioral changes associated w i t h changing economic structure sometimes referred to as the "demographic transition". A s the structure of the economy changes, the roles children play i n the economy change which i n turn suppresses b i r t h rates. We investigate the interplay between these two process by analyzing the dynamics of the model while two parameters are varied: Cj - the investment level, and b - a parameter that relates how c  sensitive the b i r t h rate is to per capita consumption of manufactured goods which I w i l l explain i n a moment. In this analysis, we assume that the efficiency i n the manufacturing sector is constant and does not depend on the availability of low entropy materials. T h i s leaves only three physical state variables: h, kh, and k . r  T h e function b(x) relates the b i r t h rate to per capita consumption of manufactured goods. A s economic structure changes, there are several factors that might influence birth rates. F i r s t , the marginal cost of children increases as economic complexity increases. In simple rural economies, children can produce more than they consume at a young age (below 10 years). In a complex industrial economy, children are a financial burden to their parents for a much longer time. Values might also shift - the enjoyment of having children and of family life might be replaced w i t h other leisure activities aided by having fewer children . W h a t ever the mechanism, changing economic structure and the associated increased economic productivity seem to depress b i r t h rates. It is this rationale that leads to the idea that continued economic development is the best policy if we wish to guide the global economy to a sustainable state. A g a i n , although this argument is very  Chapter  5.  The dynamics of a two sector ecological economic  system  106  attractive, there is the question of under what circumstances this goal is attainable. To capture this, I assume that b(x) has the form  where b is the per-capita birth rate when no manufactured goods are consumed and b 0  c  measures the sensitivity of b i r t h rates to the level of consumption. For large values of b , births decrease very rapidly w i t h increased per capita consumption of manufactured c  goods and vice versa. The physical interpretation of b could be either that each indic  vidual i n the population has a certain response to consumption or it could measure the distribution of income, or more precisely, the evenness of economic development. latter is of most interest to us.  Notice that the argument of b(x) is q  m  The  which is the  average per capita consumption of manufactured goods. If economic development is not even, some individuals might enjoy certain benefits that reduce mortality w i t h out experiencing other aspects of the development process that might suppress b i r t h rates. In this case the response of the birth rate to consumption levels would be weak. T h i s situation is modeled by a low value of b . If, on the other hand, economic growth is more even c  and income is distributed evenly, b i r t h rates would fall off more quickly as consumption increased because more individuals i n the population would reduce births for the same level of per capita intake. It turns out that for an economy that decides to invest, how evenly the the economy develops and distributes income is an important factor for its survival. To illustrate, we examine the structure of the model as the parameters c,- and b are c  varied. To set the stage, suppose that economic growth is even and income is distributed very well w i t h i n the economy. T h e system is then integrated w i t h the following parameter values: • E c o n o m i c parameters: for the marginal productivities of labor i n each industry  Chapter 5.  The dynamics  we take a  a  of a two sector ecological economic  — 0.3 and a  m  system  107  = 0.8. T h e value for manufacturing.is based on some  empirical work that suggests that values i n the range of 0.7 to 0.8 are reasonable [32]. T h e value for agriculture is more speculative and is based on the heavy reliance on capital i n modern agriculture.  We take q* = 0.5 and q* = 0.1 which are m  arbitrary and depend on scaling and choice of units i n the rest of the model. T h e only important thing is that agricultural goods become relatively more important i n times of scarcity. The cultural parameters are c = 0.05, c a  m  = 0.9, c; = 0.05, c = 0. r  I selected these values based on consumer data from the 1994 Statistical Abstract of the U n i t e d States [46]. I simply adjusted the parameters u n t i l the proportion of income spent i n each category generated by the model roughly matched those for the U . S . , roughly 11 percent to food, 13 percent to investment, and the rest to personal consumption (manufactured goods). Next I set E  a  = 10k  r  and E  = 1.  m  T h e efficiency i n agriculture is based on energy data for agricultural production [51]. In this case, I assume that the efficiency of manufacturing is constant and unity and that there are no interindustry transfers - assumptions that w i l l be relaxed later. • Ecological parameters:  8 = 0.03,  = 0.35, e^  n)m  = 0. T h e parameter ek , is  irrelevant because no income is directed toward resource goods. 0.005, and n  r  n r  F i n a l l y , c ,a — kT  = 0.1. These parameters merely scale time i n the model (i.e. just  specify the units of measurement). The key physical parameters are bo and b . For c  example if b = 0.05, at low levels of consumption, a couple (on average) would 0  have around 6 births over a lifetime. affects the model.  Now we can study how the parameter b  c  Chapter 5. The dynamics of a two sector ecological economic system  108  W i t h these assumptions, we are left to analyze the following dynamical system: — dt dk  =  h  dt  ^ dt  (0.05exp(-6 <? )-7exp(-10q ))/i c  -  m  (5.42b)  =  0.35%  =  0.1fc (l -  =  0.l(h - L {w) - L {w))  r  (5.42a)  a  O.OSkh  kr) -  (5.42c)  0.005K  a  (5.42d)  m  where the following set of algebraic constraints apply. T h e o p t i m a l capital levels to devote to agriculture and manufacturing are 0.054f w0  K  + 0.156*;* - 0 . 0 0 5 W -  3  K  8  a  <  k  a  a  kh K  h -  m  (5.43a)  otherwise (5.43b)  K. a  T h e n equations 5.21, 5.14a, 5.14b, and 5.32 allow the o p t i m a l labor, output, and price levels to be computed: L  a  = 0.429 — w  L  m  Y = 7.76kw - 0 . 3 a  Y  r  m  3;.-l P = 0.184™ 0-. fc, U  P  J  a  Recall that L = L + L a  m  =4  K„  (5.44)  w  = 3Mw- 0 . 8 = 1.649w; - . 0  m  (5.45) 8  (5.46)  so per capita income and supernumery income can be computed:  I =  k + wL h  I = I - 0.5P - 0.1P„ s  o  (5.47)  Chapter  5.  The dynamics of a two sector ecological economic system  109  Finally, the per capita consumption levels are given by 0.5 + ^ J  la  =  0.5 aa  m  =  I  < 0 and I - 0.5P > 0  (5.48a)  a  otherwise  b a  r  q  >  s  10k h k  0  I, a  h  0.1 + ^  Is >  ± = § ^  I  0  otherwise  s  0 (5.48b)  < 0 and I - 0.5P > 0  0.05/.A  -7, > 0  0  otherwise  a  (5.48c)  and the model is fully specified. Figure 5.6 shows the trajectories of the model i n phase space for b = 3 (relatively c  even economic development and wealth distribution). G r a p h (a) shows the population versus natural capital. As population grows, natural capital is reduced, but the system comes to stable equilibrium, i.e. a sustainable state. G r a p h (b) shows the population versus man-made capital. Notice that when the population is low, capital and labor grow maintaining a constant ratio (i.e. the labor versus capital curve is a straight line) as is common for simple economic growth models. However, as the system grows, it encounters limitations i n natural capital which restricts human population and, i n turn, capital growth. The capital-labor trajectory tends away from the linear growth trajectory (that would continue on indefinitely i n a simple economic growth model including just labor and capital) and comes to equilibrium. Here we see the distinct difference embedding the economic growth model i n a physical environment makes - population and capital cannot grow indefinitely. Nonetheless, the outcome of the model under these conditions is very positive. If economic growth is even and wealth is reasonably distributed, the economy settles down  Chapter 5.  The dynamics  of a two sector ecological economic  Natural Capital, k (a)  system  110  Man-made C a p i t a l , kh (b)  r  Figure 5.6: G r a p h (a) shows h versus k . G r a p h (b) shows h versus khr  to a steady state w i t h each individual enjoying a high standard of living. T h e population equilibrates at a little over 6 people per (cultivated) hectare, w i t h natural capital at about 65 % of the m a x i m u m . Figure 5.7 shows the evolution of capital, labor, and consumption over time.  fl  200  400  600  Time (a)  SOD  1000  1200  0  200  400  600  Time (b)  800  1000  I2O0  0  200  400  600  800  1000  I20O  Time (c)  Figure 5.7: Graphs (a) and (b) show the distribution of labor and capital to agriculture and manufactuing respectively. G r a p h (c) shows the per capita consumption of manufactured and agricultural goods over time.  T h e bulk of the labor and capital are directed towards non farm business, consistent w i t h what would be observed i n a modern economy. The population consumes around 0.7 units of agricultural goods and manufactured goods respectively, both above their  Chapter 5. The dynamics of a two sector ecological economic system  111  m i n i m u m values -i.e. life is quite good. Now suppose we reduce b . Figure 5.8 is a bifurcation diagram showing the effect this c  has on the model. A s b is reduced, a sub-critical Hopf bifurcation occurs at b (=s 1.5 c  c  Below this point the steady state is unstable, and the system undergoes large amplitude oscillations.  This is to say that if the system begins from an i n i t i a l condition w i t h a  value of b below 1.5, there is a barrier that precludes the system from arriving at a c  "sustainable state".  12  I  1  1  1  1  1  2  3  4  r  10 h  a .2  v= S PH  o  PH  0  5  6  Figure 5.8: Bifurcation diagram for simplified model.  It turns out that there is an explicit relationship between investment, evenness of economic growth and distribution of wealth, and system stability that we can elucidate by performing a two-parameter continuation with b and c;. Figure 5.9 is the result. c  For combinations of c; and b i n the region below the bifurcation boundary (more even c  Chapter 5.  The dynamics  of a two sector ecological economic  system  112  development and wealth distribution for a given level of investment) there is always an a t t a i n a b l e sustainable state.  For combinations of c- and b i n the region above 8  c  the bifurcation boundary (less even development wealth distribution for a given level of investment) the steady state is u n a t t a i n a b l e . T h e steady state is surrounded by a stable l i m i t cycle which forms a boundary between any initial state outside the l i m i t cycle and a sustainable economy.  o  2  4  6  8  10  Income Distribution, b  c  Figure 5.9: Change i n dynamics as the bifurcation boundary is crossed. T h e system goes to a stable equilibrium (sustainable economy for parameter values to the right and below the curve (lower investment and better income distribution). For parameter combinations above and to the left , (higher investment and less even economic development and wealth distribution) the system undergoes stable, large amplitude fluctuations.  Figure 5.10 shows the trajectories for the model i n phase space for b  c  = 1, and  Ci = 0.1 G r a p h (a) shows the population versus natural capital. A s population grows, natural capital is reduced but i n this case the population does not come to a steady state.  Chapter  5.  The dynamics of a two sector ecological economic system  113  Instead, after the human population density reaches a m a x i m u m , continued increase i n capital stocks and efficiency i n agricultural production allows the population to be maintained for a short time while natural capital continues to decline. Figure 5.11 shows the evolution of labor, capital and consumption over time. T h e n we see both labor and capital being shifted out of manufacturing into agriculture i n an attempt to m a i n t a i n agricultural output. This corresponds to the flat portion of the curve i n k — h phase r  space on the left i n figure 5.10. Increased productivity that accompanies capital growth masks the degradation of natural capital allowing the population to grow far beyond the capacity of the environment to support it. Finally, the population cannot maintain either agricultural or manufacturing output and capital stocks fall as shown i n figure 5.10. Notice that i n graph (c) i n figure 5.11, per capita output of agricultural and manufactured goods are maintained up to the point when the system collapses suggesting that the signals to consumers about environmental degradation through the market system would not be strong enough to cause them to change their habits. Thus the first prediction of the model is that investment must be accompanied by efforts to insure that economic growth is even and and its associated benefits are evenly distributed to have any hope of reaching a "sustainable economy". There are several other points that could be addressed here. For example how does changing the productivities of labor i n agriculture and manufacturing change the structure of the model? One might also argue that the model does not really correctly characterize the nature of the the agricultural sector because it does not take into consideration measures that might preserve natural capital. O n the other hand, both sectors are perfectly non-polluting. Also the manufacturing sector has a constant efficiency which does not capture the negative effects of dwindling resource supplies or the positive effects of innovation. A r e the model predictions of any value then? I believe so. T h e model predictions relate to a general phenomenon that transcends  Chapter  5.  The dynamics of a two sector ecological economic system  Natural Capital, k (a)  114  Man-made C a p i t a l , k (b)  r  h  Figure 5.10: G r a p h (a) shows h versus K . r  G r a p h (b) shows h versus  K. m  the actual assumptions about the organization of a particular social system. T h a t phenomenon is when the society can no longer bear increased complexity and must necessarily collapse. A s Joseph Tainter [63] puts it, the marginal benefits of increased complexity approach zero. In our simplified model, as the society increases i n complexity (manufactured capital increases) it receives positive benefits i n terms of improved standard of living. If, however, the society moves into a position where it can no longer maintain the complex structure it has created, it becomes a burden and may cause the society to collapse. In our simple model, this occurs when all capital and labor is shifted into agriculture i n an attempt to feed the population. W h e n this occurs, capital stocks are neglected and decay - i.e. the society can no longer maintain its complex structure. T h e point is, i n one case increasing complexity leads to a sustainable economic ecological system and i n the other case, increasing complexity leads to collapse. T h i s emphasizes the important role that evenness of economic development and the management of the benefits of increased complexity play i n the evolution of an economy.  In Collapse of  Complex Societies [63], Joseph Tainter describes several societies that he believes went  Chapter 5.  The dynamics  /  o  Lm/  /  /  -  of a two sector ecological economic  o o  'EH  I  7  115  PI  !  J La y  /  system  O SH CD  PH  Time (a)  Time (b)  Time (c)  Figure 5.11: Graphs (a) and (b) show the distribution of labor and capital to agriculture and manufactuing respectively. G r a p h (c) shows the per capita consumption of manufactured and agricultural goods over time. through a process of increasing societal complexity reaching a point where this increasing complexity became a burden and forced the society to collapse. Perhaps how well these societies managed the benefits of increased complexity is related to their subsequent collapse. T h e full model given by equations 5.40 can help explore this idea further.  5.4.2  Nonrenewable natural capital, efficiency, and flows between industries  In the previous example, it was assumed that the depletion of the nonrenewable natural capital had no effect on manufacturing efficiency which was assumed constant. It was also assumed i n the previous example that neither industry relied on output from the other, i.e. there were no interindustry transfers of goods and services. Finally, the efficiency of agricultural output was modeled as a linear function of the renewable natural capital stock. In this section these unrealistic assumptions are relaxed. F i r s t , resource scarcity is explicitly modeled by making the parameters ek , , n m  a  n  d  ekn,mr  nonzero. T h e dynamics  of the model are then explored under different assumptions about how society responds to resource shortages. N e x t , the effect of the relationship between natural capital stocks and the efficiency of production i n the two sectors on the model is explored i n more  Chapter 5.  The dynamics  of a two sector ecological economic  system  116  detail. Finally, the role of interindustry transfers (i.e. the dependence of agriculture on a flow of manufactured goods and services) on the model is investigated. F i r s t , consider the role of nonrenewable natural capital depletion as modeled by equation 5.40d. A t equilibrium, we must have  Hr =  -^Y .  (5.49)  e  m  &kn,r Since the amount of manufacturing output devoted to maintaining nonrenewable natural capital stocks (through such activities as exploration and technological development) is Cfc  n  m  a fraction of the total output Y , the ratio — m u s t be less than 1. This simply means m  that the output used to find new nonrenewable resources has to more than replace those used i n producing that output. The next question is how society allocates output to the activity of generating new nonrenewable natural capital stocks. A simple way to model this process is to let the preference for resource goods increase as these stocks become more scarce. A reasonable function representing this relationship is c = \ ' r  C  ; ~  C  ; .  (5.50)  As resources become more scarce, society shifts its preference for consumption of goods and services to replacing sources of raw materials. Since the preferences must add up to one, the m a x i m u m value of c is 1 — c — c;, the preference "remainder" after food r  a  and investment needs are met. Afc is a measure of how responsive society is to resource n  shortages. Figure 5.12 depicts the relationship between k  n  and c for different values of r  \kn- T h e lower Afc , the more responsive the society is to raw material shortages. If \k n  n  is large, society w i l l not devote output to replacing raw material stocks u n t i l the actual stock is quite low. F i n a l l y , before exploring the implications of resource scarcity on the model, the dependence of the efficiency of the manufacturing and agricultural sectors on resource stocks  Chapter 5. The dynamics of a two sector ecological economic system  117  0.8  CD CJ  a  CU SH  0.6 0.5  ,v —I  CD OH  O  o  bO  0.2  I  CD (J SH  O  0.1  co CD  Figure 5.12: Resource good preference versus K for different values of A ^ . F r o m top to b o t t o m , the values for Afc are 10, 30, and 50. n  n  n  must be modeled. Above a certain level, the relative abundance of raw materials has little effect on manufacturing efficiency because only a small portion of total economic output must be directed towards .their procurement. A s they become more scarce, more economic output must be directed towards obtaining raw materials which reduces the overall efficiency of the production process. A simple function that captures this effect is Em{k ) — n  (5.51)  kn ~f" k  n  where k is the resource level at which efficiency is half the m a x i m u m . A similar functional n  form is used for productivity i n agriculture, but is scaled so that when k = 1, E {k ) = r  r  r  10. T h e result is E (K) a  =  10£v(l  +7c ) r  (5.52)  Figure 5.13 illustrates the form of these relationships. G r a p h (a) shows the manufacturing efficiency for k  n  = 0.1. Efficiency is m i l d l y reduced until k  n  original endowment) after which it falls off rapidly.  G r a p h (b) shows the analogous  relationship between E and k for different values of k . r  r  = 0.5 (one-half of the  r  In the following example,  Chapter  5.  k = 1, k r  n  The dynamics of a two sector ecological economic system  118  = 0.1. This choice is arbitrary, w i t h the only motivation being to capture the  effects of nonlinearities i n efficiency that are consistent w i t h common sense. T h e effects of these parameters on the structure of the model are addressed i n the next section where the full model is analyzed.  tf  0  0.2  0.4  0.6  0  1  O.B  0.2  0.4  0.6  0.6  Renewable N a t u r a l C a p i t a l , k (b)  Nonrenewable N a t u r a l C a p i t a l , k. (a)  r  Figure 5.13: G r a p h (a) shows E versus k w i t h k = 0.1. G r a p h (b) shows E versus k for three different values of k : 10, 1, 0.1 w i t h decreasing values corresponding to increased curvature. m  r  Nonrenewable N a t u r a l  n  n  r  r  Capital  Here it is assumed that ek ,m = 0.01, ek ,r — 0.1, and b — 3. n  n  c  In this analysis, the  assumption of no interindustry transfers is maintained. T h e dynamical system analyzed in this section is given by equations 5.42 appended w i t h the expression for nonrenewable natural capital,  = -0.0ir  m  + 0.1%.  (5.53)  1  Chapter 5. The dynamics of a two sector ecological economic system  119  A l s o , now that c ^ 0, the per capita consumption equations given by 5.48 must be r  appended w i t h an expression for q : r  ^  I  s  > 0  (5.54)  = <  0  otherwise,  where 0.9  Finally, using the definitions of  E  m  (k ), n  (5.55)  and E (k ) a  r  given by equations 5.51 and 5.52,.  equations 5.43, 5.45 and 5.46 are replaced by  Kn  0.296hP  +  a  I  =  0.156** -  0.0031/tP  K  m  =  m  k  h  - K  k  a  (5.56a)  kh K  <  a  otherwise a  (5.56b)  ,  and Y  08  3  n  r  ° =  - =^TT^T  \+kr  _ 0.092(1 + Pa -  3mk w-  15.52k w-°-  ^  7  (5  Y  fc>°-  _ 1.649(0.1 + k )w -  3  0  7  •  Figure 5.14 shows the state variable trajectories for the case for \k  n  57)  .  8  n  Rm -  -  (5.58)  — 10. This cor-  responds to the society being relatively responsive to resource shortages and the raw material replacement process being able to generate ten times the raw materials it consumes. A s long as society devotes economic output to replacing raw material stocks, the economic system can reach a sustainable steady state  (h,k ,kh,k ) r  n  (8,0.6,1.6,0.68).  T h e economic system is still subject to the problem of over-exploiting renewable natural capital and collapsing. T h e problem introduced by nonrenewable natural capital occurs when investment is too low, or stocks are allowed to dwindle to a low level before efforts are made to replace them (high value for A/^„).  Chapter  5.  The dynamics of a two sector ecological economic system  Renewable N a t u r a l C a p i t a l (a)  120  Nonrenewable N a t u r a l C a p i t a l (b)  Figure 5.14: G r a p h (a) shows human population versus renewable natural capital. G r a p h (b) shows man-made capital versus nonrenewable natural capital. Notice i n figure (b) how nonrenewable natural capital is transformed into man-made capital as the economy develops. Once the economy is sufficiently developed, new sources of raw materials are being found (via improvements i n efficiency, using new materials, using materials i n new ways, etc) as fast as they are used i n the production of goods and services. After this point, nonrenewable natural capital remains constant as the economy continues to develop towards its final state.  If \k  n  is large, the situation is different.  Figure 5.15 shows the equilibrium human population and man-made capital levels for different values of  \k n  As long as Xk is below about 45, the economy w i l l reach a sustainable stable equilibn  r i u m state. A s Xk is increased, equilibrium values of man-made capital decreases because n  society waits too long before addressing resource scarcity. W h e n it finally does, manufacturing efficiency is low, more economic output must be directed towards maintaining raw material flows, and less can be directed to increasing man-made capital stocks. In this case the economy begins to develop just as w i t h low levels of \k  n  but reaches a level  Chapter 5.  The dynamics  of a two sector ecological economic  system  121  Figure 5.15: G r a p h (a) shows the stable equilibrium human population versus Afc . G r a p h (b) shows the stable equilibrium man-made capital versus Afc . n  n  of complexity where it can no longer maintain agricultural and manufacturing output as well as look for new sources of raw materials. Figure 5.16 shows the transient dynamics for Afc = 60, and c,- = 0.07. n  G r a p h (a) shows the evolution of man-made and nonrenewable natural capital over time. A s w i t h the previous example, nonrenewable natural capital is depleted as it is transformed into man-made capital. Here however, nonrenewable natural capital stocks are quite low (around 0.1 versus 0.7 i n the example w i t h Xk = 10) before society responds n  and begins to replace these stocks (around t = 100). Between t = 100 and t = 200 nonrenewable natural capital stocks are maintained by directing more economic output towards their replacement at the expense of new investment (as well as consumption but to a lesser degree) as shown i n graph (b). T h e problem is that the effort to find replacements for nonrenewable natural capital stocks comes too late. A t around t = 225, the cost of maintaining economic infrastructure, feeding the population, and replacing nonrenewable natural capital becomes to high for society to bear. A l l remaining factors  Chapter 5.  The dynamics  of a two sector ecological economic  system  122  a o o P, 0.06 o f-l P-1 0.04  <P  1  fi 0.02  o  u ° time fa)  Time (b)  Figure 5.16: G r a p h (a) shows man-made and nonrenewable natural capital over time. G r a p h (b) shows resource and investment-good preferences over time. G r a p h (c) shows the human population density over time. of production are then directed to feeding the population which is maintained for another 50 years and then the populations crashes as shown i n graph (c). As w i t h the model where overexploitation of renewable natural capital was the cause of collapse, here we have a period of economic development by which the economic-ecological system reaches a bottleneck. Society attempts to negotiate the bottleneck by changing economic structure, but subsequently collapses. In the first case, economic development proceeds to a point where flows from renewable natural capital are insufficient to maintain the structure of the system. This "road to collapse" sets an upper bound on investment. In the second case, it is lack of flows from man-made capital that ultimately causes collapse. T h i s "road to collapse" sets a lower bound on investment. T h e higher Xkn-, the higher the level of investment required to develop economic infrastructure to cope w i t h resource scarcity before it is too late. This increased investment, on the other hand, might cause collapse due to natural capital overexploitation. These facts pose an interesting problem for a developing economy: there is a safe window of investment below which non-renewable natural scarcity poses the greatest threat to achieving sustainability and above which, overexploitation of renewable natural capital is the l i m i t i n g factor.  Chapter 5. The dynamics  The  of a two sector ecological economic  system  123  problem of finding the appropriate window to grow fast enough to overcome  limitations i n m a n made capital yet slow enough to avoid destroying natural capital is illustrated i n figure 5.17.  1  i  non feasible: nonrenewable natural capital scarcity /  feasible non feasible: overexploitation of" renewable natural capital  0.045  0.05  0.055  i  0.06  Investment-good preference, c, (a)  i  0.06  0.08  Investment-good preference, Cj (b)  Figure 5.17: G r a p h (a) shows the bifurcation structure for \k = 10. G r a p h (b) is the two parameter bifurcation diagram for \k versus investment good preference. n  n  G r a p h (a) shows the bifurcation structure for \k  n  = 1, i.e. society is relatively  responsive to resource shortages. T h e window of feasible investment-good preference is quite narrow. T h e economy w i l l evolve to a sustainable steady state if investment good preference is between 0.028 and 0.053. Investment good preferences outside this range will give rise to an economic development path that leads to collapse due to resource shortages or overexploitation of natural capital respectively. G r a p h (b) shows the dependence of this result on the responsiveness of society to resource shortages. right depicts a l l the combinations of \k  T h e curve on the  and investment-good preference for which a  n  Hopf bifurcation occurs. For a given Xk the corresponding value for investment-good n  preference is an upper bound for the feasible level of investment-good preference that w i l l lead to a sustainable steady state economic ecological system. T h e curve on the  Chapter 5.  The dynamics  of a two sector ecological economic  system  124  left is the corresponding lower bound for investment-good preference to prevent resource shortages. T h e region between these two curves defines the feasible region of investment-good preferences that w i l l lead to a sustainable economy. Given that the range of possible values for investment-good preferences is from 0 to 1 — c (=0.95 i n the example above), a  the w i d t h of the feasible region (about 0.025 i n the example above) is quite narrow. O f course, these numbers should not be taken as representative of those a modern economy might face, but i n the context of the model, they do indicate that the possibility of attaining a sustainable economic ecological system may be very sensitive to investment patterns.  Efficiency and feasible investment patterns T h e nature of the relationship between investment patterns and feasible paths can depend on many things.  T w o key aspects of the model that affect this relationship are the  relationships between efficiency and capital stocks and the transfer of goods between industries. In the above example, recall that k = 0.1, and k = 1. A low value like this n  r  for k corresponds to the fact that if an economy has a stock of raw materials available n  for productive activities, the size of that stock does not affect these activities u n t i l it is reduced to a level where some portion of productive capacity must be diverted to maintaining the stock. The lower k , the more dramatic this transition. T h e significance n  of the relative nonlinearity i n the relationship between k and E r  a  is more difficult to  imagine. It could correspond roughly to the idea of ecosystem resilience. If an ecosystem is not resilient, productivity would decline rapidly due to agricultural disturbances (high value for k ). If an ecosystem is resilient, it might remain fairly productive even with r  a high level of disturbance, but break down more rapidly after some threshold level of disturbance is surpassed.  The question is, how do different values for k and k affect n  r  Chapter 5.  The dynamics  of a two sector ecological economic  system  125  the results shown i n figure 5.17? To investigate this, the model is analyzed by fixing \k  n  = 10 and varying /c , and  k,  n  r  leaving the rest of the model assumptions unchanged from the previous section. Thus, we now have 0.9  (5.59)  I0k + 1' n  and 7.76fc (l + k )w-°-  3 . 0 3 f ew- - -  3  ^  r  =  r\j  T  ~Y~  =  (  ri  3  r  p  =  ~j~  rv  5  6  o  )  n  k )w 0  n  8  n  k  r  n  n  1.649(fe +  A:,. (1 ~\~ k ) It turns out that increasing k  8  =  r\jf  0.184(1,. + k )w 0  p  0  ^  r  •  n  shifts the feasible region to the right but does not sig-  nificantly affect the w i d t h of the region. This is consistent with intuition: increasing k  n  makes manufacturing efficiency more sensitive to resource shortages requiring more  investment to avoid them.  Also, reduced efficiency associated with increased k  n  a drag on the economy slowing the growth process.  puts  This allows for a higher level of  investment without overexploiting renewable natural capital. Thus both the m i n i m u m and m a x i m u m feasible values for investment-good preference are increased, shifting the feasible region to the right. T h e model is much more sensitive to k . r  This sensitivity is illustrated i n figure 5.18  which shows a two parameter bifurcation diagram for investment-good preference versus k . A s ecosystems become less resilient (higher k ) , the system can tolerate more investr  r  ment. T h i s seems a bit counter intuitive, but is similar i n nature to the Tsembaga model where increased productivity of renewable natural capital had a stabilizing tendency. The key is that the feedback from ecosystems is stronger if they are less resilient. U n l i k e k , increasing k widens the feasible range. For k — 10 and k n  r  r  n  = 0.1 the feasible  values for investment-good preference lie between 0.028 and 0.082, about double the range for the case with k = 1. A s k is reduced, ecosystems remain productive at higher levels r  r  Chapter 5.  The dynamics  of a two sector ecological economic  1  ••••  T  non feasible: overexploitation of renewable natural capital  CD O  V <V  1  0.08  i  r  system  i  126  T—  "  —•  CD  o.o7  feasible: no overexploitation of renewable natural capital  o Sb ^  ( )  "  6  0.05  -  /  CO £  0.04  (=1 <  •  •  i  i  i  i  Figure 5.18: T w o parameter bifurcation digram for investment-good preference and  k. r  of agricultural disturbance. T h i s weakens the feedback from natural systems and allows the human economic system to develop beyond the capacity of ecosystem to support it. Thus the more resilient ecosystems are, the more likely it is for human economic systems develop into situations from which they cannot extricate themselves. Thus the human propensity to try to fix things through attempting to increase productivity may be the worst development strategy possible.  T h e effect of interindustry transfers T h e final aspect of the model that we address i n this section is the role of interindustry transfers. In the previous examples, each industry was assumed to operate independently of the other. Neither sector relied on the other for raw material inputs. T h i s is unrealistic for modern agriculture which relies heavily on manufactured pi'oducts, most notably chemicals. Similarly, the manufacturing sector relies on fibers from the agricultural sector. In order to study the effects of interindustry transfers, we examine the effect that the  Chapter 5.  The dynamics  parameters  fa,  of a two sector ecological economic  and R  f3haif,  system  127  have on the model. A l l other parameters are fixed and the  am  model assumptions remain unchanged from previous sections, i.e. the d y n a m i c a l system is given by equations 5.42 and equation 5.53, optimal consumption by equations 5.48 and 5.54, output by 5.57, labor by 5.44, income by 5.47, and resource-good by 5.59. Because R  am  and R  ma  preference  are not zero, no simplifications occur for the o p t i m a l  capital and price levels. T h e full equations for the optimal capital and price levels given by 5.37 and 5.34, respectively, must be used. Recall that fa measures the quantity of nutrient inputs required per unit of agricultural output (a unit conversion factor) whileflhaifmeasures the productivity of natural capital. A s  Pkaif is increased,  the higher the ratio of — can be before nutrients produced  by biological processes are no longer sufficient to meet demand. It turns out that the effect of material transfers from manufacturing to agriculture has a stabilizing effect. This is illustrated by the two parameter bifurcation diagram i n figure 5.19 w i t h  = 6  (meaning as population density per hectare approaches a typical value for a modern industrial economy, depending on the level of degradation of natural capital, a substantial amount of manufactured inputs would be required to meet food demand).  A s fa in-  creases, there is more pressure on the manufacturing sector which allows for increased investment without overexploiting renewable natural capital. A g a i n , the harder natural capital is to exploit, the more stable the model. Interestingly, changing fa does not affect the m i n i m u m investment level necessary to avoid raw material shortages i n the manufacturing sector. For example for c,- = 0.05 and fa = 0.1, the feasible window for investment-good preference is 0.02838 to 0.1034. For fa = 0.2, the feasible window for investment good preference is 0.02838 to 0.2462. T h i s result is slightly counterintuitive.  One would think that increased demand for  manufactured goods i n the agricultural sector would divert productive capacity away from investment and nonrenewable natural capital replacement. A v o i d i n g resource shortages  Chapter 5.  The dynamics  of a two sector ecological economic  system  128  0.5 0.45 0.4 0.35 0.3  feasible: no overexploitation of renewable natural capital  0.25 0.2  so.  0.15  non feasible: overexploitation of renewable natural capital  0.1  Investment-good preference, c; Figure 5.19: T w o parameter bifurcation digram for investment-good preference and /3/v. would then require a higher investment-good preference. The reason why this is not the case is related to the pattern of economic growth associated w i t h different values of /3^. For each of the cases above, the equilibrium levels of per capita output of goods and services are very similar w i t h q  a  = 0.64, q  m  = 0.48, qi = 0.024, and qi = 0.06 which  translates into 16.6, 71, 3.6, and 8 percent of income spent on food, consumption, investment, and nonrenewable resource replacement respectively. W h a t does change is the e q u i l i b r i u m levels of the state variables w i t h (h,kk,k ,k ) n  for /3  N  = 0.1 and (h,k ,k ,k ) h  n  r  T  = (5.199,1.484,0.632,0.789)  = (3.93,1.119,0.627,0.852) for /3  N  = 0.2.  For larger  values of /3N, equilibrium population and m a n made capital levels are lower, the renewable natural capital level is higher, and the non renewable natural capital level is almost unchanged. D u r i n g the i n i t i a l growth period of the economy, the increased price of food due to inputs from the manufacturing sector causes consumers to shift spending away from food. T h e lower food intake slows population growth slightly which, i n turn, slows man-made capital growth. The overall growth of the economy is slowed so it equilibrates  Chapter 5.  The dynamics  of a two sector ecological economic  system  129  w i t h a smaller human population and man-made capital stock. T h e result is that the scale of the final economy is smaller, putting less pressure on both non-renewable and renewable natural capital stocks. Thus the lower bound for feasible investment remains unchanged while the upper bound increases. It is interesting how the two cases above which differ only very slightly i n terms of their development over time and equilibrium economic output differ much more significantly i n the equilibrium scale of the economy and levels of state variables. A drag on the economy that slows economic growth, which is often considered bad, may i n the long run produce the same economic outcome as faster growth. The only difference is that the final scale of the slower growing economy is smaller, and the quality of renewable natural capital higher. If the state of the natural environment is related to quality of life, then the slower growing economy produces the better end result. This should be a major concern when considering how policy affects economic growth. Next, we t u r n our attention to the role that transfers from the agricultural to the manufacturing sector have on the model. These transfers simply put more pressure on renewable natural capital for a given level of economic output. Figure 5.20 illustrates the relationship between the m i n i m u m and m a x i m u m feasible investment-good and R  A  M  preference  .  T h e m a x i m u m feasible investment-good preference is more sensitive to increases i n R  AM  than is the m i n i m u m . This causes the feasible region to narrow as R  AM  is increased.  Thus the more taxing the manufacturing sector is on the agricultural sector, the smaller the feasible investment region and the more difficult achieving sustainability is.  For  example, the model predicts that our reliance on paper products and wood fiber for use in the manufacturing sector may significantly reduce the range of feasible investment for our economy. Another important aspect of the manufacturing industry is the pollution it generates.  Chapter 5.  The dynamics  of a two sector ecological economic  system  130  non feasible: overexploitation of renewable natural capital  0  o  o.i  0.05  0.15  0.2  0.25  0.3  R•am Figure 5.20: T w o parameter bifurcation digram for investment good preference and  R am  A l t h o u g h I have not addressed pollution directly (eg. as a state variable) its effect on the dynamics of the system can be studied indirectly. One key aspect of pollution i n an ecological system is its negative effect on the operation of ecosystems. T h i s can be modeled as a reduction i n renewable natural capital associated w i t h economic activity. T h i s is similar to the effect R  am  has on the economy - when manufacturing puts in-  creased pressure on renewable natural capital, whether by compromising its operation through contamination or direct removal of nutrients, attaining sustainability is made more difficult.  5.5  Conclusions  In this chapter we have developed and studied the dynamics of a model for a two sector ecological economic system. T h e m a i n results of this modelling exercise are that increases in efficiency (or more generally, productivity) do not necessarily increase the likelihood  Chapter 5.  The dynamics  of a two sector ecological economic  system  131  that a human ecological economic system can attain a sustainable state.  Increasing  productivity through capital growth (increased investment), and increasing the efficiency of the utilization of nonrenewable resources both make achieving a sustainable state less likely. T h i s and the similar result i n the Tsembaga model are mounting evidence that the answer to the first question posed i n the introduction is " N o " . Our ability to solve problems is not necessarily a good thing. N e x t , cultural parameters, like i n the case of the Tsembaga, do play a key role i n achieving sustainability. Here, key cultural parameters are investment good preference and how society manages economic growth and distributes its benefits.  These results  suggest that the answer to the second question posed in the introduction is " V e r y " . C u l ture is very important i n determining whether a human economic system is sustainable. These two points taken together suggest that the requirement for a sustainable ecological system are the right k i n d of values and cultural institutions, not the right technological fixes. Finally, recall that nonsubstitutability i n consumption is very destabilizing as demonstrated i n chapter 4 . T h e two sector model suggests that nonsubstitutability i n production, on the other hand, can have both positive and negative impacts on the possibility of achieving a sustainable ecological economic system. Difficulty i n finding substitutes for agricultural goods used i n manufacturing dramatically reduces the possibility of achieving a sustainable ecological economic system. The possibility of substituting manufactured products for nutrients generated by renewable natural capital can have a stabilizing effect. T h e mechanism is the fact that diverting output from the manufacturing sector to agriculture can slow overall economic growth. Several specific points that came to light through the analysis of the two sector model are:  Chapter 5.  The dynamics  of a two sector ecological economic  system  132  • There is a critical relationship between the level of investment (speed of economic growth) an ecological economic system can tolerate and the evenness of economic growth.  If an ecological economic system is to attain a sustainable state, for a  certain level of investment, there is a m i n i m u m evenness of growth and distribution of wealth that must be maintained. If not, the system w i l l grow beyond a point where the renewable natural capital can renew itself while providing sufficient flows of goods and services to maintain economic complexity, and the system w i l l crash. Thus for a given value of b (which measures evenness of economic growth), the posc  sibility of overexploiting renewable natural capital sets an upper bound on feasible levels of investment. • If an economic system relies on flows of raw materials from non renewable natural capital stocks, there is a m i n i m u m level of investment and willingness to address resource shortages i n a timely manner to attain a sustainable state.  If not, the  system w i l l collapse because economic output is insufficient to maintain man-made capital and simultaneously maintain raw material flows.  T h i s possibility sets a  lower bound on feasible levels of investment. • T h e window of feasible levels of investment set by natural capital constraints is affected by the nature or the dependence of efficiency of production on natural capital stocks. If this relationship is highly nonlinear, and efficiency remains relatively high as stocks decline but then declines rapidly when stocks are below a certain threshold level, the window for feasible investment significantly narrows. • T h e window of feasible levels of investment set by natural capital constraints is affected by the structure of the economic system. If the agricultural sector relies heavily on inputs from the manufacturing sector, the upper bound for feasible investment increases while the lower bound remains unchanged and the feasible  Chapter 5.  The dynamics  of a two sector ecological economic  system  133  window is widened. If the manufacturing sector relies on the agricultural sector for inputs, pressure on renewable natural capital increases and the feasible investment window is narrowed. These aspects of the model structure have several interesting policy implications: • A n y policy that affects the rate of economic growth should be assessed as to its affect on the evenness of growth and the distribution of the benefits of that growth. How w i l l the benefits of economic growth affect different segments of the population? A n y economic activity that provides benefits from economic growth without the associated societal context associated w i t h that economic growth should be viewed as highly suspect and fundamentally destabilizing. A n example might be the green revolution which provides products to enhance agricultural production to groups who live outside the technologically based social structure that produces those goods. T h e result: potentially improved nutrition and increased b i r t h rates without the increased marginal cost of children or other factors that might reduce b i r t h rates. • How much can we rely on market signals for resource scarcity? T h e market may signal shortages, but depending on the relationship between efficiency and resource stocks, the market signal may come too late. This is not due to a failure of the market, but rather to fundamental "unknowability" i n the behavior of complex systems. • Feedback generated by economic activity regarding the health of renewable natural capital stocks may be very weak and this fact must be built i n to management policies.  Such a scenario corresponds to graph (b) i n figure 5.13 for k  r  =  0.1  (highest curvature), which recall was highly destabilizing and narrowed the range  Chapter 5.  The dynamics  of a two sector ecological economic  of feasible investment-good preference.  system  134  This type of situation has been receiving  more attention with respect to the specific renewable natural capital stock of marine fisheries [41]. A l t h o u g h terrestrial ecosystems are more easily observed than marine ecosystems, they are no less complex. Their artificially maintained productivity masks the continued degradation of agricultural resources due to erosion, loss of soil structure, and contamination, which may eventually cause a crash i n productivity similar to what has been witnessed i n marine fisheries. • A n y process that puts a drag on economic growth should not be viewed as necessarily bad i n terms of the big picture of reaching a sustainable ecological economic system. Indeed, the model predicts that the propensity of humans to view these drags negatively and attempt to remove them through improvements i n efficiency is fundamentally destabilizing and may severely reduce our chances of ever achieving a sustainable ecological economic system. This runs directly counter to the argument that increased efficiency will rescue us from ecological disaster. Further, any manufacturing process that puts pressure on renewable natural capital severely restricts the amount of economic growth an ecological system can endure. Thus any argument that proposes increased economic productivity as i m p r o v i n g chances for achieving a sustainable ecological economic system without specifically addressing the pressure this economic activity places on ecosystems is flawed. In this chapter we have studied not sustainable economic growth, but rather, feasible economic growth paths that w i l l lead to a sustainable ecological economic system. T h e first implies that there is some way to grow sustainably (such as through environmentally friendly consumption). A d m i t t e d l y , it seems economic growth is a necessary part of the particular evolutionary trajectory the human race is presently on, but we need economic growth of a very special kind. We need economic growth where the benefits  Chapter  5.  The dynamics of a two sector ecological economic system  135  and responsibilities of growth are evenly distributed among the participants i n the economic system. Thus the concept of sustainable growth is not very useful. T h e concept of feasible economic growth paths generated by the two sector model we have studied i n this chapter is. Such models help clarify critical relationships that may help i n the design of policy to direct future development down such paths. Granted, the work presented here is speculative, but I believe that it is an important step i n the right direction. I have only begun to explore the basic structure of the model. There are many directions to go from here to gain more understanding about economic growth i n a bounded environment. I outline some directions for future research i n the final chapter.  Chapter  5.  The dynamics of a two sector ecological economic system  SYMBOL  a  a  b b b(-) a  m  bo b c  C  a  Ci Cm C  r  d(.) Ea(-)  INTERPRETATION  M a r g i n a l productivity of labor i n agriculture Marginal productivity of labor i n manufacturing Marginal productivity of capital i n agriculture Marginal productivity of capital i n manufacturing Per-capita birth rate. Depends on per-capita consumption of manufactured goods. M a x i m u m per-capita b i r t h rate Response of b i r t h rate to per-capita consumption of manufactured goods. Agricultural good consumption preference Investment good consumption preference Manufactured good consumption preference Resource good consumption preference Per-capita death rate. Depends on per-capita consumption of agricultural goods. Agricultural sector production efficiency. Depends on renewable natural capital stock, k . Manufacturing sector production efficiency. Depends on nonrenewable natural capital stock, k . Effect (conversion factor) of j - t h process on i-th state variable H u m a n population density Per-capita income Supernumery per-capita income. (Income left over after basic needs have been met.) Man-made capital stock Man-made capital devoted to agriculture Man-made capital devoted to manufacturing Nonrenewable natural capital Renewable natural capital Nonrenewable natural capital level at which efficiency is half of the m a x i m u m Measure of the nonlinearity i n the relationship between renewable natural capital and efficiency i n the agricultural sector. Intrinsic regeneration rate of renewable natural capital r  E (-) m  r  h I Is  h Ka Km  kn k r  n  r  Table 5.1: Table of important symbols  136  Chapter  5.  The dynamics of a two sector ecological economic system  SYMBOL  Pa P p Pr <la qi q  m  q  r  *  q q  a  m  •^ma(')  Ram (') r [/(•) w Y Y m Va Vm a  1  LO 8  Xyj Xkn  INTERPRETATION  Per-unit price of agricultural goods Per-unit price of investment goods Per-unit price of manufactured goods Per-unit price of resource goods Per-capita consumption of agricultural goods Per-capita consumption of investment goods Per-capita consumption of manufactured goods Per-capita consumption of resource goods M i n i m u m tolerable per-capita consumption of agricultural goods M i n i m u m tolerable per-capita consumption of manufactured goods Manufactured goods required per unit of agricultural goods produced Agricultural goods required per unit of manufactured goods produced Per-unit cost of man-made capital Utility Per-unit cost of labor (wage rate) Output of agricultural goods Output of manufactured goods Man-made capital to labor ratio i n agriculture Man-made capital to labor ratio i n manufacturing Factor cost'ratio Depreciation rate of Man-made capital Speed of response of wages to differences between labor supply and demand Speed of response of resource-good preference to resource scarcity  Table 5.2: Table of important symbols, continued  Chapter 6  Reflections and future Research  In this thesis I have tried to develop the fundamental idea that the extreme behavioral plasticity of humans can be a fundamentally destabilizing force i n the ecosystems they inhabit.  It seems that the most stabilizing force is also related to this plasticity; our  ability to generate culture and social organizations. ritual cycle.  For the Tsembaga, this was the  W h a t stabilizing forces are available for modern industrial economies is  unclear. W h a t does modern industrial society and its associated culture have to offer to counter its own destabilizing tendencies? I also tried to put the idea of behavioral plasticity and social structure i n the context of neoclassical economic theory by addressing the affects that different assumptions about u t i l i t y and production have on the evolution of ecological economic systems. I addressed non substitutability i n consumption i n the Easter Island model and non substitutability in b o t h consumption and production i n the two sector model. F i n a l l y I attempted to address the relative importance that cultural versus physical parameters play i n the evolution of ecological economic systems. T h e analysis of these models seem to point i n the direction that social organization and cultural practices may be more influential than technical prowess i n attaining a sustainable ecological economic system. Recall that if society directs enough economic output to replacing non renewable resources, the system w i l l reach a sustainable equil i b r i u m . T h i s result is i n a similar vein as that of Solow [58] and Hartwick [31] i n the context of the theory of economic growth. M y result is conservative; it assumes that  138  Chapter 6. Reflections  and future  Research  139  efforts directed towards finding new resources or substitutes and improving efficiency are always successful. T h e problem i n my model of a two sector economy it not too little investment, but rather too much investment and too much efficiency. In this case, social organization and cultural practices must play a role i n reaching a sustainable state. T h e y must offset destabilizing forces of investment and increasing efficiency. Critics would argue that the model did not include the possibility of substituting man-made capital for renewable natural capital, the possibility of investing i n natural capital, or intergenerational equity. Future research should focus on three m a i n areas:  Simplifying the model Based on the results of the analysis of the two sector model, we have a good idea of what the most important aspects of the model are, namely the over exploitation of natural capital. If we assume that society invests enough to avoid non renewable natural capital scarcity we can simplify the model considerably. We can drop equation 5.40c. If interindustry transfers could be neglected, this would simplify the model considerably, but we saw the significant effect that transfers from the agricultural sector to the manufacturing sector had on the model. We could retain this aspect of the model by including the negative effects of manufacturing processes on the environment directly rather than through the economic system. The simplification of the economic system would allow the temporary equilibrium wage rate to be computed directly, eliminating the need for equation 5.38. T h e model would then consist of only three differential equations for which it might be possible to obtain closed form analytical results for feasible investment paths.  Investing in natural capital W h a t if society set aside a reserve of renewable natural capital? B y adding the possibility of society directing some portion of economic output to maintaining such a reserve  Chapter 6. Reflections  and future  Research  140  or enhancing the quality of renewable natural capital being exploited we can explore this question. T h e idea of maintaining such reserves i n fisheries has recently been addressed [41].  C u l t u r e versus Social Institutions Recall that throughout the thesis, behavioral plasticity referred to individuals. A t this level, I concluded that behavioral plasticity could be a very destabilizing force. Whether or not the culture of a particular group offsets this destabilizing force is accidental. O n the other hand, behavioral plasticity can operate at the group level when a group decides to set up an institution i n response to changing environmental conditions w i t h a particular purpose i n m i n d . A very important question is whether social institutions be set up to mediate human environmental interactions even though the underlying culture is destabilizing. For example, can social institutions stop the degradation of an ecosystem inhabited by a group where cultural practices attach social status to hoarding?  This  question could be addressed by extending the model to include both i n d i v i d u a l behavior and the behavior modifications induced by institutions.  O p t i m a l economic growth G i v e n the possibility of investing i n renewable natural capital (resource good preference), society would now have the following problem: W h a t is the best set of preferences for consumption, investment, and resource goods and evenness of economic development? T h i s depends on the definition of best. One definition might be a path that would provide the highest per-capita consumption levels over time w i t h the least degraded environment possible.  Table 6.3 shows some equilibrium levels of consumption of agricultural and  manufactured goods and renewable natural capital for the model w i t h no interindustry transfers.  T h e first line of the table shows that lower levels of b , low levels of investc  Chapter  6. Reflections  and future  Research  K 7 3 0.04 6 0.06 6 . 0.08 8 0.01  141  q 0.472 0.562 0.593 0.624 m  0.636 0.832 0.849 0.990  K 0.612 0.955 0.638 0.731 r  Table 6.3: E q u i l i b r i u m consumption and renewable natural capital levels versus b . c  ment seriously degrade renewable natural capital resulting i n low e q u i l i b r i u m levels of consumption and natural capital. In this case, people would have low standards of living and to add insult to injury would be living i n a degraded environment. W i t h more even economic growth, increased investment is possible resulting i n higher standards of living w i t h much better environmental quality as shown on line 2. More is not necessarily better i n the case of investment. For b = 6 increasing investment good preference from 0.06 to c  0.08 increases consumption levels but significantly degrades the environment. Thus for a given level of b there is i n some sense an optimal level of investment. c  B y increasing both b and 7 consumption levels can be increased still further and c  shown on line 4 of the table but to make the model realistic, there would have to some negative aspect of high b . T h i s is not difficult to envision looking back on the different c  economic experiments of this century. It is often argued that the possibility of making it big fosters entrepreneurship which i n turn drives improvements i n efficiency. If wealth is distributed very equally, there may be no incentive for entrepreneurship.  Thus if b  c  increased too much and efficiency began to decline, there would be reason to tolerate a certain amount of distributional inequity that would make everyone better off. In the model, these cultural parameters are constant over the evolution of the system. Certainly, culture changes over time, and an interesting optimal control problem would be to determine the optimal time paths of b (t), j(t) and v(t). c  E a r l y i n the evolution of an  Chapter 6. Reflections  and future  Research  142  ecological economic system investment i n man-made capital may be the most important activity while later, evenness of growth and wealth distribution along w i t h investment i n natural capital might be more important to utility maximization. If it were possible to obtain a feedback control for this system, then it could be used to develop o p t i m a l future policies given the present state of our system. Given the incredible challenges that lie ahead for the world ecological economic system, I am hopeful that future work i n this area might provide some insight into possible means of dealing w i t h them.  Bibliography  [1] C . A C K E L L O - O G U T U , Q . P A R I S , A N D W . W I L L I A M S , Testing a von liebig crop re-  sponse function against polynomial specifications, Economics, 67 (1985), pp. 873-80.  A m e r i c a n Journal of A g r i c u l t u r a l  [2] E . A L D E N S M I T H , Anthropology, evolutionary ecology, and the explanatory limitations of the ecosystem concept, i n T h e Ecosystem Concept i n Anthropology, Westview Press, Boulder, C O , 1982. [3] J . A N D E R I E S , Culture and human agro-ecosystem dynamics: guinea., Journal of Theoretical Biology, 192 (1998).  the tsembaga of new  [4] J . M . A N D E R I E S , An adaptive model for predicting !kung reproductive performance: A stochastic dynamic programming approach, Ethology and Sociobiology, 17 (1996), pp. 221-245. [5] U . M . A S H C E R A N D R . J . S P E T E R I , Collocation software for boundary value differential-algebraic equations, S I A M J . Sci. Comput., 15 (1994), pp. 938-952. [6] J . H . A U S U B E L , The liberation of the environment, A m e r i c a n Scientist, 84 (1996). [7] S. A U T H O R S , Forum section on resources and economic growth, Ecological Economics, 22 (1997). [8] W . B A L T E N S W E I L E R , The relevance of changes in the composition of larch bud moth popxdations for the dynamics of its numbers, i n T h e Dynamics of Populatons, P . V . den Boer and G . R . Gradwell, eds., Wageningen: Center for agricultural publishing and documentation, 1970. [9] J . A . B R A N D E R A N D M . S. T A Y L O R , The simple economics of easter island: a ricardo-malthus model of renewable resource use, A m e r i c a n Economic Review, (forthcoming, 1997). [10] G . B U C H B I N C E R , Nutritional stress and post-contact population decline among the maring of new guinea, i n M a l n u t r i t i o n , Behavior and Social organization, C . S. Greene, ed., Academic Press, 1977. [11] M . B U L M E R ,  Theoretical  Evolutionary  Mass., 1994.  143  Ecology, Sinauer Associates, Sunderland,  Bibliography  144  M . C A B E Z A G U T E S , The concept of weak sustainability,  Ecological Economics, 1 7  (1996).  P . F . C H A P M A N A N D F . R O B E R T S , Metal Resources London, 1983. J . W . C H U N G , Utility and production functions  and Energy,  Butterworths,  : theory and applications,  Blackwell,  1994.  C . W . C L A R K , Mathematical Bioeconomics: able Resources, J . Wiley, New York, 1990.  The Optimal Management  of Renew-  C . C L E V E L A N D , The direct and indirect use of fossil fuels and electricity in us a agriculture, 1910-1990, Agriculture, Ecosystems & Environment, 55 (1995), pp. I l l — 121.  , Resource degradation, technical change, and productivity culture, Ecological Economics, 1 3 (1995), pp. 1 8 5 - 2 0 1 .  of energy in u.s. agri-  H . E . D A L Y , Steady-State Economics : the Economics of Biophysical and Moral Growth, San Francisco, W . H . Freeman, 1977.  Equilibrium  H . E . D A L Y A N D K . N . T O W N S E N D , Is the entropy law relevant to the economics of natural resource scarcity? - yes, of course it is!; comment, Journal of E n v i r o n m e n t a l Economics and Management, 2 3 ( 1 9 9 2 ) , pp. 9 1 - 1 0 0 .  E . D O E D E L , A program for the automatic  bifurcation analysis of autonomous sys-  tems, Cong. N u m . , 3 0 ( 1 9 8 1 ) , pp. 2 6 5 - 2 8 4 .  L . E D E L S T E I N - K E S H E T , Mathematical  models in biology, R a n d o m House, New York,  1988.  J . E W E L ; Litter fall and leaf decompostion in a tropical forest succession guatmala, Guatamalan Journal of Ecology, 64 (1986), pp. 2 9 3 - 3 0 8 .  in eastern  T . C . F O I N A N D W . G . D A V I S , Ritual and self-regulation of the tsembaga maring ecosystem in the new guinea highlands, H u m a n Ecology, 1 2 (1984), pp. 3 8 5 - 4 1 2 . , Equilibrium and nonequilibrium models in ecological anthropology: an evaluation of 'stability' in maring ecosystems in new guinea, A m e r i c a n Anthropology, 89 ( 1 9 8 7 ) , pp. 9 - 3 1 .  J . F R A N C E A N D J . H . M . T H O R N L E Y , Mathematical terworths, London, 1984.  Models in Agriculture,  But-  Bibliography  145  [26] H . I. F R E E D M A N , Deterministic Dekker, New York, 1980.  Mathematical  [27] N . GEORGESCU R O E G E N , Entropy Press, Cambridge, Mass., 1971.  Models in Population  Law and Economic  Process.,  Ecology, M .  Harvard U n i v .  [28] M . G l A M P I E T R O , G . C E R R A T E L L I , AND D . P l M E N T E L , Energy analysis of agricultural ecosystem management, Agriculture, Ecosystems & Environment, 38 (1992), pp. 219-244. [29] M . E . G l L P I N , Group Selection in Predator-Prey sity Press, Princeton, N . J . , 1975.  Communities,  Princeton Univer-  [30] H . GROSCURTH, R . K U M M E L , AND V . G . W . , Thermodynamic optimization, Energy, 14 (1989), pp. 241-258.  limits to energy  [31] J . M . H A R T W I C K , Intergenerational equity and the investing of rents from haustible resources, A m e r i c a n Economic Review, 67 (1977), pp. 972-974. [32] D . F . HEATHFIELD AND S. W l B E , An Introduction tions, M a c m i l l a n Education, 1987.  to Cost and Production  ex-  Func-  [33] F . H l N T E R B E R G E R , F . L U K S , AND F . S C H M I D T - B L E E K , Material flows vs. 'natural capital :what makes an economy sustainable?, Ecological Economics, 23 (1997), pp. 1-14. 7  [34] C . S. H O L L I N G , The functional response of predators to prey density and its role in mimicry and populations regulation., M e m . E n t o m o l . Soc. C a n . , 45 (1965), pp. 1-60. [35] R . J . H U G G E T T , Modelling the Human Impact on Nature : Systems Analysis Environmental Problems, Oxford University Press, Oxford ; New Y o r k , 1993. [36] B . S. JENSEN, The Dynamic Academic, 1994.  Systems of Basic Economic  of  Growth Models, Kluwer  [37] A . J U O AND A . M A N U , Chemical dynamics in slash-and-burn ture, Ecosystems & Environment, 58 (1996), pp. 49-60.  agriculture,  Agricul-  [38] H . V . K E U L E N AND H . D . J . V . H E E M S T , Crop response to the supply of trients, tech. report, Pudoc, Wageningen, 1982.  macronu-  [39] I. A . K U Z N E T S O V , Elements of applied bifurcation theory, Springer-Verlag, 1995. [40] E . L A N Z E R AND Q . PARIS, A new analytical framework for the fertilization A m e r i c a n Journal of Agricultural Economics, 63 (1981), pp. 93-103.  problem,  Bibliography  [41]  T.  146  LAUCK,  C . C L A R K , M . M A N G E L , A N D G . M T J N R O , Implementing  cautionary pronciple in fisheries management through marine Applications, 8 (1998), pp. S72-S78. Supplement. [42] R . M . M A Y , Stability and Complexity Press, Princeton, N . J , 1973.  in Model Ecosystems,  the  pre-  reserves, Ecological  Princeton University  [43] D . H . M E A D O W S E T A L . , The Limits to growth; a report for the Club of Rome's project on the predicament of mankind, Universe Books, New York, 1972. [44]  D.  H. MEADOWS,  fronting 1992. [45] [46]  E.  D. MEADOWS,  Global Collapse, Envisioning  ODTJM,  Fundamentals  AND J.  R A N D E R S , Beyond  a Sustainable  the  limits:  Con-  Future, Chelsea Green P u b . C o . ,  of Ecology, Saunders, Philidelphia, 1953.  U . D. OF COMMERCE, ECONOMICS, S. ADMINISTRATION, A N D B . OF T H E C E N -  SUS, Statistical  Abstract of the United States, U . S . Department of Commerce, 1994.  [47] C . P A L M , M . S W I F T , A N D P . W O O M E R , Soil biological dynamics in slash-and-burn agriculture, Agriculture, Ecosystems & Environment, 58 (1996), pp. 61-74. [48] Q . P A R I S A N D K . K N A P P , Estimation of von liebig response functions, Journal of A g r i c u l t u r a l Economics, 71 (1989), pp. 178-86.  American  [49] D . P l M E N T E L , Ecological systems, natural resources, and food supplies, i n Food and N a t r u a l Resources, D . P i m e n t e l and C . H a l l , eds., Academic Press, 1989. [50] D . P l M E N T E L , World Soil Erosion and Conservation, 1993.  Cambridge University Press,  [51] D . P l M E N T E L A N D M . P l M E N T E L , Food, Energy and Society, Colorado University Press, Niwot Colorado, 1996. [52]  J . P R O O P S , M . F A B E R , R . M A N S T E T T E N , A N D F . J O S T , Acheiving  a  sustainable  world, Ecological Economics, 17 (1996). [53] R . A . R A P P A P O R T , Pigs for the ancestors : ritual in the ecology of a New people, Yale University Press, 1968. [54] W . E . R E E S , Achieving  sustainability,  [55] M . R U T H , Integrating Economics, demic, Dordrecht ; Boston, 1993.  Guinea  Journal of Planning Literature, 9 (1995). Ecology,  and Thermodynamics,  Kluwer Aca-  Bibliography  147  [56] M . S C O T T , The importance of old field succession on biomass increments agriculture, R o c k y M o u n t a i n Geological Journal, 6 (1976), pp. 318-327.  to shifting  [57] S. B . S H A N T Z I S A N D W . W . B E H R E N S , Population control mechanisms in a primitive agricultural society, i n Towards global equilibrium, D . L . Meadows and D . H . Meadows, eds., W r i g h t - A l l e n Press, 1973. [58] R . S O L O W , Intergenerational equity and exhaustible resources, i n S y m p o s i u m on the Economics of Exhaustible Resources, 1974, pp. 29-45. [59]  , Georgescu-roegen versus solow/stiglitz:  Reply, Ecological Economics, (1997).  [60] D . . S T E R N , Limits to substitution and irreversibility in production and consumption: A neoclassical interpretation of ecological economics, Ecological Economics, 21 (1997). [61] M . S T R A S K R A B A , Cybernetic theory of complex ecosystems, i n C o m p l e x Ecology : The P a r t - W h o l e Relation in Ecosystems, B . C . Patten and S. E . Jorgensen, eds., Prentice H a l l , Englewood Cliffs, N . J . , 1995. [62] L . S Z O T T , C . P A L M , A N D C . D A V E Y , Biomass and litter accumulation under managed and natural tropical fallows, Forest Ecology and Management, 67 (1994), pp. 177-190. [63] J . A . T A I N T E R , The collapse of complex societies, 1988. [64] H . R . V A R I A N , Microeconomic [65] R . W A L T M A N , Competition  Cambridge University Press,  analysis, Norton, 1992.  Models in Population  Biology, S I A M , 1983.  [66] A . L . W R I G H T , An introduction to the theory of dynamic economics : a theoretical study in long run disequilibrium, Oxford University Press, 1984. [67] J . T . Y O U N G , IS the entropy law relevant to the economics of natural resource scarcity?, Journal of Environmental Economics and Management, 21 (1991), pp. 169-179. -  

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