Spatially distributed stocks, deterministic evolution, and ecological balance

Spatially Distributed Stocks, Deterministic Evolution, and Ecological Balance*

Dimitrios S. Dendrinos

Professor of Urban and Transportation Planning The University of Kansas Lawrence, Kansas 66045

ABSTRACT

This paper lays down a conceptual foundation for a model of a turbulent human population’s perception of and behavior in a broader environment. Violent changes in the levels of utility extracted from the environment, and abrupt shifts in the relative value that the human population places on various components of this environment and the ecology it contains, are analyzed. In presenting this foundation, two different modeling traditions are integrated: the static, behavioral utility theory, and the deterministic-descriptive Dendrinos-Sonis universal map of relative spatial population dynamics. A basic question is raised in this dynamic framework concerning the use of utility maximization.

1. INTRODUCTION

It can be argued that the human race’s attitude toward its broader environment and the ecology it contains has been subject to violent changes. The relative value placed on the various components of this ecology and the level of satisfaction extracted from it seemingly have been and still are being altered, at times drastically.

Concomitant with these shifts have been periods and places that have witnessed violent social events. Examples include large scale and sudden migrations, rapid population growth, rises in pollution levels, technological innovations, extinction of various animal and plant species and natural

*This paper is dedicated to the memory of Henry Mullally

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114 DIMITRIOS S. DENDRINOS

resources, famines, war, and the like. One may ask if these issues are connected, and assuming that they are, how.

A conceptual foundation is presented here in an attempt to provide linkages between these two elements: the changing behavior of the human race toward its environment, and the above types of social events. The basic premise is this: on the one hand, the rather ephemeral utility extracted from the ecology and the relative value placed on its components by the humans, and on the other hand various calm or violent sociospatial events, are synchronous. Both are determined by broader deterministic processes which have rather long time horizons, according to a detached observer. The behavior of humans does not affect these deterministic processes. Outcomes of the observer’s models do not interfere with the processes of evolution: the observer’s model results do not feedback upon either the deterministic or the behavioral components of the problem as stated.

The changing in discrete time short term (static) perceptions, utility, and values human populations attain from or assign to the components of the ecology contained in the broader environment are determined (according to the observer) by ecological-economic production-consumption (transforma- tion) processes. Further, spatial distribution dynamics of various stocks present in the ecology are also determined by the same processes. In turn, these sociospatial stock dynamics define at each time the human perceptions of the environment.

A central notion in the analysis is that of ecological value placed by humans on the environment. It is based on the perception of an intermediate stock’s relative worth placed on the ecology’s components. Such perceptions are referred to here as ecological prices. The intermediate stock is the human population, one of three types of stocks distributed in space-time and found in any ecology. The other two stock types are an original (pure input) and final (pure output) stock. Ecological value gives rise to the notion of ecological balance, which in turn identifies in the dynamics of regionally distributed stocks a fundamental concept: that of a complementary stock.

A descriptive and a behavioral component are found in this elementary statement of sociospatial evolution defined in sociological time. The position is advanced that within the time frame of a few human generations (defined here as “sociological time”) ecological-economic productionconsumption processes govern the perceptions and behavior of the human population. Static behavior is determined by, and does not affect, the dynamic descriptive-deterministic ecological-economic process. This process, which trans- forms stocks through productionconsumption activities, is based on the Dendrinos-Sonis discrete relative universal map. The map supplies a basis for modeling the turbulent behavior in the perceived prices that the intermediate

Spatially Distributed Stocks 115

population attaches to the other stocks in space-time, and for simulating the periodic changes in the human population’s perceived utility extracted from the ecology.

2. STOCKS, ANTISTOCKS, AND THEIR COMPLEMENTS

Recent developments in the theory of discrete time, discrete space relative stock size spatial dynamics afford the opportunity to analyze some novel and fundamental concepts in the theory of evolution of populations distributed in heterogeneous space. The main thrust of this paper is not these dynamics, which have been extensively explored elsewhere in the current literature. Instead, the focus is on certain fundamental concepts and the embedded implications they may hold for a deeper understanding of sociospatial evolution processes.

Two notions are incorporated into the universal map of discrete time, discrete space relative dynamics: the notions of antistock (x- ) and complementary stock (x). In an earlier paper [l] the notion of anti-stock was introduced, and it was further analyzed in [2]. The concept was discussed in reference to the U.S. regional population-income dynamics. It was derived from the trivial conservation condition governing relative spatial distributions at time t:

q(t)+x;(t)=Xi(t)+ i xj(t)=l, 0 <x,(t), x;(t) < 1, (1.1) j#i j=l

where xi(t) is the current (at time period t) relative stock size at location i. Its operational use lies in the fact that interactions or interdependencies between location i and all or part of the other locations in the environment, I, may be compressed into the interaction or interdependency between the stocks size at site i and its antistock size within the time horizon T:

Xi(t+l)=fi[Xi(t),X;(t)]. (1.2)

The second notion, that of a complementary stock x was introduced in [3]. It


is obtained from the identity (discussed later in the paper within the framework of an ecological balance) given by

x,(l) + x&) = 0, - 1 < xi(t) < 0. (2)

xi( t ) designates a quantity trivially balancing out locally the ecological system for each stock. Under this specification balancing occurs universally for all stocks and globally for all locations. Obviously, the properties of the complementary stock are the opposite of those of the actual stock. More on this in Section 4.a.ii, where the reason for dealing with anti- and complementary stocks will become apparent.

NOTE. Some readers of this paper have suggested that I reverse the assignment of the two notions. Since in physics antimatter does what the complements of stocks do in this paper, and in probability theory a complement is what I define as an antistock, there is good reason to do so. However, for consistency with previous papers, I choose to keep these assignments as they stand. The reader’s indulgence is requested.

These two quantities, x- and x, are in fact hidden in May’s [6] simple logistical prototype model

x(t+l)=Ar(t)[l-x(t)], O<x(t)<l, O<A<4. (3)

By considering the definitions for x- and x from (l), (2) and the form of the above model, one has directly the discrete time dynamics

x(t+l)=Ax(t)[x(t)+r-@)+x(t)] (4)

connecting the first iterate of x with its current size and its anti- and complementary stock sizes. Note also the symmetry

x(t+l)=Ax(t)[l-r(t)] =A[l-r-(t)]x-(t)

=A[l-x-(t)][l+x(t)].

SpatiaUy Distributed Stocks 117

Turbulent and chaotic deterministic dynamics recordable on x(t) are at- tributable, in accordance with the above conditions, to the interconnection among the first iterate of a stock and its current, anti-, and complementary stock sizes.

3. THE OBSERVERS MODEL SPECIFICATIONS

a. Definitions Three different families of stocks are considered by the observer: original,

intermediate, and final stocks. Their definition, for the purposes of this paper, depends on the time horizon incorporated into the analysis, assumed to extend over a small number of human generations, and defining the notion of sociological time.

Original stocks are those used exclusively as input factors in the production or consumption processes. They are pure inputs, in the sense that they are not produced within the time horizon considered. Examples of such stocks include all types of natural resources, geographical quantities associated with topographical, climatological, and other locational features, etc.

Intermediate stocks are responsible for setting in motion economicecological processes; they have been produced from the original stocks by actions of intermediate type populations. Economic-ecological processes are either production or consumption activities carried out by the intermediate stocks. During these production or consumption processes the original inputs are transformed into new commodities (stocks). Intermediate stocks, besides triggering these processes, are also both inputs to and outputs of these activities. That is, they are recyclable quasi input or output factors. Human, animal, and plant species are living types of intermediate stocks, which also include nonliving transformed commodities. Wholes or parts of these intermediate commodities are produced and/or consumed by other intermediate stocks. For instance, built capital (in the form of roads, buildings, nests, or houses) is an example of transformed commodities; certain animal and plant species are consumed by human or other animal populations.

Final stocks are the transformed commodities which become refuse retired from the production, but not necessarily from the consumption process within the time horizon stated. They are the waste products of the economicecological production or cwumption processes, i.e., they are pure output stocks. Examples include various types of pollutants not absorbed (diminished) by, but rather accumulated in, the environment by the economicecological productionconsumption processes at work during the time horizon considered.


b. Assumptions and the Model’s Focus For expositional simplicity, only one type in each of the three families of

stocks will be considered. An environment is conceptualized which consists of I different (heterogeneous) locations. In each location a pure (original) input stock, a quasi input/output (intermediate) population, and a pure (final) output stock are distributed at time t, within a sociological time horizon T (0 < t < T). Each time period t may be thought of as extending over a few calendar years. If Xi(t), Yi(t), and Zi(t) are correspondingly the current sizes of these three stocks at location i, then their relative size is given by

‘iCt) q(t) = -

X(t) ’ x(t) = i xi(t),

i=l

ritt) ?4iCt> = -

Y(t) ’ y(t)= i yi(t),

i=l

‘iCt) q(t) = -

Z(t) ’ z(t) = i zi(t).

i=l

(6.1)

(6.2)

(6.3)

In the model which follows, the relative dynamics of these three types of stocks are modeled. Attention is focused on changes in relative rather than absolute sizes, because the interest is in spatially disaggregated rather than aggregate growth. If zero aggregate growth occurs, then the model is directly transferable to the aggregate growth case. If aggregate growth does not depend on the spatially disaggregated individual location stock size, then the model can be trivially extended. In the case that aggregate growth is a function of the locational distribution of the stocks, then a far more complicated model not addressed here, is needed.

c. The Deterministic-Descriptive and Behavior Models of the Obseroer According to the observer, evolution occurs within the sociological time

frame T considered, as the result of actions taken by those intermediate type populations responsible for production and consumption processes. Their actions and behavior are depicted by two distinct observer type models: a deterministic-descriptive model, and a behavioral one. These models are of the “pure observer” type because their outcome does not affect (i.e., the


observer does not interfere with) the process of evolution, including the

production-consumption activities by the intermediate stock. The descriptive model identifies the interdependencies and interactions

among the three types of stocks. It does not identify their complex structure, as it is not possible for the observer to identify them fully and in detail. It only replicates the manner in which these three types of stock have been and currently are distributed in space and their past dynamics within the time horizon T after some original perturbation (starting state). This descriptive model simulates solely the stocks’ size. It is built on the malthusian tradition, as well as on mathematical ecology of the Volterra-Lotka type. Accordingly, all factors affecting population growth are collapsed in a few growth rate and interaction parameters. Only such composite factors are used in these population dynamics models. Similar composite factors are utilized in this deterministic-descriptive type model too.

The behavioral part of the observer’s model consists of a static currently perceived “ecological value” consideration. Ecological value is an arbitrary measure of composite ecological worth as perceived by the intermediate stock, according to the observer. It implies that the human population utilizes a weighting scheme among all stocks found in the ecology (i.e., the environment of Z locations.) Ecological prices are the weights by which the intermediate stock is willing to trade off one stock for another at a given level of ecological value. Through these prices, when the perceived value of the ecology is zero, the notion of a stocks complement attains a concrete meaning. All these behavioral considerations are attributed to the human population by the observer. They do not bear (feed back) upon the deterministic processes of evolution as provided by the observer’s descriptive model. It is assumed that these perceptions are an outcome of the dynamic interdependencies and interactions among all stocks and locations, i.e., of the evolution process. To the extent that these considerations could affect the process of evolution, their existence and effects are thought to be minimal within the time horizon considered. In effect they are discounted in the descriptive model of the evolution process, as stated by the observer, in sociological time. The term “sociological time” is employed to distinguish the time frame considered here from that of the biologist’s evolutionary time and the geologist’s geological time. It spans the course of a few centuries.

d. The Deterministic-Descriptive Model Next, the observer’s descriptive-deterministic model is supplied; but it is

not elaborated upon, since it is not the main subject of this paper, which focuses instead on the behavioral model. In specifying the deterministic- descriptive model the discrete time, discrete space, relative stock dynamics

120

map of [4] is used:

DIMI’IXIOS S. DENDFUNOS

Xi( t + 1) = e(t)

Cf=rFj(t) ’ i = 1,2 ,..., I,

y,(t + 1) = Gi(t)

Cfcs ,Gj(t) ’ i=1,2 ,..., I,

q(t + 1) = Hi(t)

Cf=JZj( t) ’ i = 1,2 ,..., I,

where the functions Fi, Gi, and Hi depict locational temporal the three stocks respectively at location i, and are given by

(7.1)

(7.2)

(7.3)

advantages for

Fi(t)=Fi[xi(t),yi(t),zi(t); i=1,2 ,..., I] >O, (8.1)

Gi(t)=Gi[xi(t),yi(t)Yxi(t); i=1,2,...,Z] >O, (8.2)

Hi(t)=Hi[xi(t),y,(t),zi(t); i=1,2,...,1] >O. (8.3)

Of particular interest are the exponential specifications of these three families of functions. These specifications are broad in their interpretation and offer the opportunity to record a wide range of unidirectional, as weU as regularly and irregularly periodic, deterministic dynamics, including local and global turbulence and chaos-all widely observed events in sociospatial evolution. Under specifications given by the conditions (7.1)-(8.3), economicecological productionconsumption processes are dependent upon relative locational and temporal advantages captured by spatial ecological coefficients a, b, c:

F,(t) = Aixi(t)u”yi(t)0’2zi(t)=*3> 0, (9.1)

Gi(t) = Bixi(t)b”yi(t)b’2zi(t)bi3> 0, (9.2)

Hi(t) =Ci~i(t)Ci1yi(t)C’Zzj(t)Ci3>0. (9.3)

A,, Bj, Ci are positive bifurcation parameters, which depict prevailing environmental conditions, at location i. These parameters change at a slower pace than the state variables x, y, and z, i.e., they remain constant within


the time horizon T. The parameters [a], [b], [c] represent sets of very slowly changing interaction (elasticity) coefficients.

As is the case with the Volterra-Lotka mathematical ecology framework, in the signs of the pairwise combinations among these coefficients all possible ecological associations among stocks and locations can be captured. These coefficients change at a pace even slower than A, B, and C, or randomly as a result of large scale technological innovations or other shocks, some occurring in geological time. In the universe of their space, the behavior of the fast-moving state variables (x, y, z) is recorded. Together with A, B, and C, these parameters remain constant in the sociological time horizon T; evolutionary processes extending over longer time spans determine them.

The above universal map (7.1-3), (9.1-3) produces calm and turbulent dynamics in the stock size, involving regular and nonregular periodic motion [4, 31. Nothing will be added to this model here. The reader will be assumed to be familiar with the behavior of these dynamical equations. This variety in behavior is used next to derive similar dynamics in the human population’s perception of ecological prices and utility.

4. THE BEHAVIORAL MODEL: PREFERENCES AND ECOLOGICAL CONSERVATION CONDITIONS

In this section, the observer’s behavioral model of human population perceptions is constructed. It is based on the notions of ecological value, perceived utility, and ecological price of stocks. The behavioral model contains two key elements: a perceived ecological constraint and a preference function. In its analytical structure it resembles the economist’s microeconomic model of consumer behavior.

a. Strong Local Ecological Conditions without Spatial lnteractim To set the stage for presenting different ecological conditions it is neces-

sary that at the outset the basic components of the human behavior be discussed. They include the notions of preferaces, ecological value, and ecological prices, which are presented in turn next.

i. Prefmences. Individuals at (i, t ) belonging to the homogeneous intermediate stock found there and then are assumed (by the observer) to have intralocationally (but not necessarily interlocationally) identical preference levels y(t). These functions V are defined by the relative sizes of only the local stocks, which enter these functions as arguments:


in this strongly local ecological condition where the individuals’ utility at (i, t) is not affected by stock sizes at other locations and/or time periods. The components (arguments) of V are such that

ax(t) axi(t) ’ O7

aw) <o

axi ’

w(t) ayi(t) co,

a2v,(t) ____ < 0, aYi(tJ2

w(t) azi(t) < O3

a%(t) <o

az,(t)2 *

(11.1)

(11.2)

(11.3)

These conditions indicate that the human population’s utility level y(t) is an increasing function of the relative abundance of the input stock at a decreasing rate [Equation (ll.l)], it is a decreasing function of the intermediate stock concentration at (i, t) at a decreasing rate, i.e., intraspecies aversion is assumed [Equation (11.2)], and it is a decreasing function of the final stock accumulation at a decreasing rate [Equation (11.3)]. Thus, the principle of diminishing marginal utility/disutility is present.

As the relative sizes of the state variables {x, y, z } change in time, so does the level of V at (i, t). Violent/calm changes in these stocks’ size result in violent/calm changes in V.

Since the levels of the state variables are deterministically determined at (i, t ) as described in Section 3.d, the utility level enjoyed by the intermediate stock is also deterministically determined there and then. According to the observer, what is behaviorally determined at (i, t) by the human population is the relative worth assigned by it to these three stocks as found there and then, and the implied ecological value. These behavioral determinations are carried out through the use of indirect utility function theory as presented next.

ii. Ecological Value, Prices, and Balunce. To the observer, the intermediate stock attaches a total value to the local ecology, and relative values to each of the stocks found in (i, t). The observer notes that an increase/de- crease by one unit of the relative concentration in any stock at (i, t ) affects differently the total perceived value of the ecology and the relative value placed on the three stocks by the human population. Expressed in units of any of the three stocks (x, y, z) in question, the total perceived value (denoted as p, r, s, respectively) of the local ecology at (i, t) is

xi(t)+ai(t)Yi(t)+Pi(t)zi(t)=Pi(t)~o (12.1)

Spatially Distributf3d Stocks 123

expressed in unit shares of x, or

Yi(t)xi(t)+ yi(t)+ Gi(t)Zi(t) =ri(t) ><O (12.2)

expressed in unit shares of y, or

Ei(t)lCi(t)+17j(t)yi(t)+zi(t)=si(t)><o (12.3)

expressed in unit shares of z, where the parameters (Y, /3, y, 6, E, 17 are any real numbers. Since they convert one stock to its respective numeraire stock for each type of total perceived ecological value in any of the above conditions (12.1-3), they play the role of “ prices.”

If there is no interlocational uniformity in the utility level among populations, i.e., the condition

y(t) #v(t) 03)

holds, then the level of total ecological value at (i, t ) is arbitrary. On the other hand, if there is a uniform level of utility at all locations during each time period t, then the total perceived ecological value is endogenously determined at each t, under conditions of interlocational equilibrium. In both cases, these ecological prices (and the total ecological value in the event that there is interlocational equilibrium) are behaviorally derived at (i, t ) through the use of indirect utility theory, as it will be demonstrated later.

Before doing so, however, a special case of the above specifications will be addressed: that in which the ecological values (pi or ri or si) are zero. This case will be referred to as the perceived ecological balance condition, whereas the case of positive (or negative) ecological value will be referred to as the perceived ecological profit condition. Under a perceived ecological balance, a state of “nihilism” prevails. It implies that a positive perceived value by the humans of one stock type carries with it an appropriate negative value for another stock.

Under perceived ecological balance, the notion of a complementary stock { x, #, 3 } obtains a concrete meaning through the conditions

Xi(t)=CYi(t)yi(t)+Pi(t)Zi(t)~ -_i(t)<O, (14.1)

+bi(t)=yi(t)xi(t)+Gi(t)zi(t)= -yi(t)<O, (14.2)

Si(t)=Ei(t)Xj(t)+qi(t)yi(t) ~ -Zi(t)<O. (14.3)

124 DIMITFUOS S. DENDFUNOS

Thus, a complement of a stock is the perceived ecological value of sustaining the actual stock at its current level at (i, t) in the absence of ecological profit. It is expressed in terms of the other (two in this case) stocks’ sizes locally at t.

Since the contribution of y and z is negative (as stated in Section 4.a.i), both ecological prices {a, p } must be negative. Put differently, from (12.1) one now has

(15.1)

If {(Y, p} are known, then y, 6, E, 1 are immediately derived, since the conditions (14.1-3) hold as identities. Next, the absolute values for (Y, p are computed, having dropped the absolute value signs for simplicity. The condition (15.1) can also be written as a budget constraint:

aj(t)Yi(t> + PiCt)'iCt) = xj(t)* (15.2)

iii. Zndirect Utility under Ecological Balance Conditions. Perceptions, as expressed by these ecological price parameters, enter as arguments in the intermediate stock preference function defined at (i, t) and to be referred to as an indirect utility j%nctiun [S, p. 1721. An indirect utility function, by definition, is determined by prices and income but not quantity levels. This function is strictly convex, separable, and nonlinear in its arguments. Its form is determined by factors operating in a time frame longer than T, and thus outside the scope of this paper. The form is assumed here to be

maxU(i,t)=U{ai(t),&(t)}. (16.1)

Rises in any price diminish U at a decreasing rate:

(16.2)

(16.3)

The optimization problem is subject to the conservation condition, from (15.2):

Ai( -Xi(t)+~j(t)yj(t)+jOi(t)Zj(t) <Oo. 07)

Spatially Distributed Stacks 125

The parameters in this indirect utility function (i.e., the ecological prices) are any real positive numbers. As the three stocks are deterministically defined from the universal map (discussed in Section 3.d) at each time period t, and consequently their complements are also deterministically defined at t, the unknowns in this optimization problem are the prices (weights) of the perceived ecological (value) balance equations at (i, t).

Prices may oscillate in time either periodically or chaotically, following the oscillations in their corresponding stocks. These oscillations cause the value of U(i, t) also to oscillate in time. Nonperiodic, violent oscillations in prices, quantities, and utility levels raise a basic question: if the time that it takes for an economic agent to optimize utility is longer than the time it takes for the parameters of the utility function to change under the deterministic changes of (7.1-3), is it then desirable for anyone to attempt maximizing utility? Further, if conditions change faster than the time to determine the prices { cy*, p*} which optimize U, is it then efficient for the humans to derive an optimum appreciation of this ecology’s components? If, then, utility maximization and derivation of optimum prices is not an efficient strategy, what is? The answer to this basic question will be left to future research.

The price parameters contain two distinct pieces of information each: first, they identify an ecological value equivalence between each pair of stocks, translating one stock to another through a perceived willingness to substitute one for another at some level of U and V; second, they depict appropriate ecological prices that render the perceived aggregate spatiotemporal value in the community of three stocks nil for their current stock sizes in (i, t).

The condition (17) expresses an identity balancing the perceived value of all stocks expressed in units (the numeraire price) of the original stock at (i, t) in a manner similar to a budget constraint in the microeconomic theory of consumer/producer behavior. It is noted that all coefficients (prices) in this condition are subscripted by location and time. The problem as stated in this case does not contain flows (i.e., interactions) among locations or stocks; it only addresses interdependencies among stocks expressed in the form of the universal algorithm of Section 3.d, the utility function and its associated conditions (16.1-3), and the ecological balance equation (17).

The Lagrange formulation of the above problem, to the extent that the behavioral component is concerned, is as follows:

LP(i,t) = U(i,t) -hi(t) 08)

where Xi is the corresponding Lagrange multiplier associated with the ecological balance equation. It can be interpreted as the current and local marginal utility of the original stock to the intermediate population in the


community of three stocks. First order conditions require that

kqi,t) al(i,t)

= aa, -hj(t)Yi(t)=O, 1 xJ(i,t)

aai(t>

hi(t) = - YiW %(t> ’

(19.1)

&qi, t) aqi, t) - X,(t)z,(t) = 0,

1 aqi,t>

aPi(t) = abi(t)

A,(t) = -

zi(t) aPi ’

(19.2)

which, in turn, directly imply that

r3u(i,t) ix@, t) y,(t)

affi(t) I aBi(t) =zi(t)* (20)

Thus, I separable nonlinear systems of three equations each containing three unknowns are produced, which may or may not have a solution. If they do have a solution, it may or may not be unique. These possibilities present important implications for this time-location and stock specific intermediate stock’s perception of an ecological balance. The possibility that the solution may not be unique (and in general it is not) implies that different individuals may have different perceived prices of the ecology’s components.

It is also noted that V*(i, t) = V{ a,*(t), /3:(t)} is not necessarily identical at all locations, and consequently it does not represent an individual’s interlocational equilibrium under the deterministic and behavioral framework adopted here. Thus, a dynamic stock equilibrium state does not necessarily imply a static interlocational individual equilibrium-a more realistic condition than that found in static spatial economics. It does, however, represent an intralocational individual equilibrium.

Since the utility at equilibrium, V*(i, t) = V{ x:(t), y,*(t), z:(t)}, is equal to the indirect utility V*( i, t) = U( a:(t), Pi*(t)} by definition of an indirect utility function, then at each point in (i, t ) the equilibrium ecological prices produce the optimum utility level for the deterministically obtained stock quantities. Further, these stock quantities are, for the human population, optimal under the price schedules a(*, /3* at (i, t), and under a utility function which has as its indirect equivalent the form U( i, t) = V{ ai( t ), P,(t)}. According to indirect utility theory, the stock sizes at (i, t) are then viewed

Spatially Lhtibuted Stocks 127

by the human population as optimum subject to the derived ecological prices and the implied ecological value of the community.

An example is provided next for the case where the indirect utility function is given by a polynomial expression containing separable components without cross-impact effects. This type of utility is often used in microeconomic analysis. It is used here for demonstrational simplicity only, and it produces results which seem reasonable:

u(i, t) = - [ “i(t)2+ p,(t)“]. (21.1)

The corresponding Lagrangian functions are given by

(21.2)

and the first order conditions are

2ol,(t)+ X,(t)y,(t) = 0,

2&(t) + &(t)q(t) = 0.

(21.3)

(21.4)

The ecological balance equation

-~i(t)+ai(t)y,(t)+j3i(t)zi(t)=0 (21.5)

closes the system of three equations in three unknowns. The system supplies the following solution:

Al(t) = 2xi(t)

yi(t)2+ Zi(t)2 ‘O,

a?(t) = xi(t)Yi(t) , o

yi(t)2+ Zi(t)2 ’

P,*(t) = xi(t>zi(t) , o

yi(t)2+ Zi(Q2 .

(22.1)

(22.2)

(22.3)

128 DIMMXIOS S. DENDRINOS

This solution indicates that the marginal utility of the original stock at all points in (i, t) is positive. Further, the ecological balance equation

a:(t)Yi(t)+Pi*(t)zi(t) d xi(t> (23.1)

is indeed a budget, with positive prices { cui(t), Pi(t)} and an upper bound equal to the original stock’s current and local relative size. The ecological price of the relative share of the intermediate population at (i, t), expressed in units of the original stock, is positively affected by increases in the local relative share of the original stock:

aq( t) YiCt> 40 -= =->o. ax,(t) yi(t)2+ Zi(t)2 'iCt)

(23.2)

Further, the ecological price of the relative share of the final stock at (i, t) is also positively affected by rises in the local share of the original stock:

a&*(t) zi(t) -= _ Pi*(t) >. ax,(t) yi(t)2+Zi(t)2 xi(t) .

(23.3)

Thus, if at some (i, t) the rate of depletion of natural resources exceeds the global average rate, the perceived prices of the intermediate and final stocks then and there will fall. Looking at the effect of rises in the relative share of the final stock at some (i, t) upon the two prices, one obtains from (22.2,3)

aa: 2xi(t)Yi(t) -= -

adt)

2aF(t)2zi(t) <o

Yi(t)2+ Zi(t)2 = - Yj(t)xi(t) ’ Pw

w,*(t) --pi*(t)

a,%(t)

xi(t> -2zi(t) , o

x,(t)q(t) < * (23.5)

The above indicate that rises in the final relative stock size result in lower perceived ecological prices for the intermediate stock, whereas the effect of rises in z upon its perceived price is ambivalent. Similar results are obtained for the effects of y upon (Y and p, as one observes that

4w zi(t) -=-

Pi*tt> Yitt> ’ C-=6)


b. Strong Ecological Balance Conservation Condition with Znterspatial Interaction

Interspatial interaction, in the framework of this model, implies that the complements of stocks are now given by the following conditions:

xi(t)+Xi(t)=Xi(t)+ f: [LYij(t)yj(t)+~ij(t)Zj(t)] ~0, (24.1) j=l

yi(t)+J/i(t)=yi(t)+ i [U,j(t)~j(t)+Sij(t)zj(t)] ‘0, (24.2) ,j = 1

Zi(t)+<i(t)=Zi(t)+ i [Eij(t)ri(t)+Ilij(t)Yj(t)] ‘0, (24.3) j=l

so that the complement of a stock depends on the relative size of the stocks at all locations, and not simply at i as was the case in section 4.a. Now there is interspatial interaction, so that the spatial distribution of the other stocks at time t are also included in the value of a stock’s complement. The ecological balance is maintained for each stock, at each point in (i, t).

Parameters, which can be any real numbers, under this formulation contain three information components: ecological value, equivalence, and interaction. Similarly to the case in Section 4.a.ii, the matrices [y], [S], [E], and [q] are immediately obtained from the matrices [a] and [ fi]. The last two matrices are components of an argument vector in an indirect utility function U( i, t ) with similar properties to those defined in Section 4.a.iii, so that

m~U(i,t)=U{aij(t),Bij<t)}~ (25.1)

aU(i,t) aU(i,t) <o

&xij( t) ’ api, ’ (25.2)

i?“U(i,t) a2U(i,t) <o

aail( t)2 ’ asij(t)2 ’

(25.3)

The process of optimization (but not the analytical difficulties) is similar to that defined in Section 4.a.


c. Weak Ecological Balance Condition Under this specification, the locationally weak ecological value identity is

given by

A(t)= C [cui(t)xj(t)+Pi(t)yi(t)+Yi(t)zi(t)l =O, (26.1) i=l

so that the ecology is perceived by the intermediate (buman) stock as being balanced only globally and collectively for all stocks at any time period t. It is not perceived to be balanced for every stock at every location. One might make the condition (26) even weaker by adding the time dimension. This addition implies that the ecological value is perceived as being balanced, through appropriate (real) values attached to the existing stocks (and their complements), so that

A= 5 i [a,(t)x,(t)+&(t)y,(t)+y,(t)z,(t)] ~00. (26.2) t-0 i=l

Under this specification, the broader community of stocks, extending over a time horizon T and spatial horizon I, is perceived as being very weakly balanced. In both conditions (26.1,2), the ecological prices (i.e., the coefficients of the ecological balance equations) may be given by one of the following two indirect utility functions:

(27.1)

in the case of ecological balance at each time period t, or

maxU(T)= ; i - [~i(t)“+P,(t)“+u,(t)‘]e-“, (27.2) t=O i=l

where r is a discount rate, greater than zero. Of interest is the form of the first order condition under (26.2), (27.2):

9 = U(T) -AA, (28.1)

a3 aw? -= ---Xx,(t) + aa, &Y,(t)

- 2e-” ai( t) = Xx,(t), (28.2)


so that at all time periods and locations, and for all three stock types,

2e-‘“q(t) -

'iCt>

= h = const, (28.3)

a result indicating that the marginal utility of the original stock to the intermediate population is independent of location and space. Further, as t increases (and e-*” declines), either a higher ecological price ai or a lower relative stock accumulation results. In the case of (28.3), lower xi(t) implies relative original stock depletion and thus higher currently self-perceived worth of the intermediate stock, ai( as t approaches T.

d. Ecological Profit Instead of assuming an ecological balance condition according to which

the sum of a stock and its complement equals zero, a perceived ecological profit may be considered, so that

xitt)-aitt)Yi(t)-Pitt)zitt)=~i(t). (29.1)

In the above formulation all quantities are real and positive. The implicit assumption is still that the original stock carries a positive perception by the human population, presumably because of positive net relative external effects of agglomeration. However, the intermediate and final stocks have a negative perceived effect: negative net relative externalities associated with a higher concentration of the intermediate stock, and a net negative effects of pollutants and other similar final stocks. The perceived ecological profit then is a net perceived effect upon the intermediate stock from the current and local state of the community. It could be either negative or positive; in the example which follows it is assumed to be positive.

The indirect utility is stated as

m~~(i,t)=“{ai(t),Pi(t)~pi(t)}~ (29.2)

so that

aqi, t) aqi, t) <o

&x,(t) ’ a&(t) ’

aqi, t) api >O,

(29.3)

(29.4)

a2qi, t) a2qi, t) a2u(i,t) ~ o

a”i(t)2 ’ api(t)2 ’ api . (29.5)


The problem is solved as in the previous sections. All specifications of the current Section 4 are special cases of the conditions (29.1,2), resulting when pi(t) is equal to zero. It is noted that in the absence of interlocational equilibrium [V*(i, t) = V*(i, t) f v(t)], the value of p,(t) is arbitrary, whereas in the case of inter-locational equilibrium v(t), the additional condition defines the ecological profit perceived at (i. t) due to the existence of the equilibrium.

All results from indirect utility theory hold at (i, t), no matter what the dynamics of stocks there within T, be they cahn or chaotic. This statement becomes of interest in the case of period doubling cycles in stock sizes and consequently in ecological prices. Otherwise statically stable ecological prices may commence regular oscillations and bifurcate according to welldefined period doubling sequences as stock sizes locally cross critical thresholds, pulling along with them the perceived level of utility negotiated from the environment by the humans. Sharp changes in the environment (picked up by the deterministic model’s parameters, the slow-moving variables) and either regularly or irregularly oscillating levels in stocks cause similar oscillations in the humans’ attitude toward the ecology. In such a violent environment the observer may expect changes in the form of the utility function so that some past averages may enter it as arguments, a possible extension of this work.

5. CONCLUSIONS

An elementary model of fundamental spatial evolutionary processes has been stated, involving three types of stocks (original, intermediate, and final). The notion of a complementary stock was defined and its role in the evolution process outlined. A descriptive and a behavioral component were identified in a relative dynamics, “pure observer” type model simulating a process extending over a (sociological) time horizon T.

Ecological value, prices, and balance were specified under a number of scenarios in which a preference function and its indirect utility formulation were stated. The assumption was made that the conditions determining production and consumption, i.e., economic-ecological processes, obey longer time constants (sociological time) than those affecting the human population’s shorter term (a few calendar years or one generation) perceptions of the ecology and its components.

The central objective of the paper is to tie deterministic nonlinear dynamics to the traditional static utility theory of consumer behavior. This link produces turbulent behavior in the human population’s perception and evaluation of the broader environment and its components. Periodic move-


ment in stock sizes results in oscillatory behavior in the perceived ecological prices and level of utility extracted from the environment by the intermediate stock, an ellusive component in static utility theory. These oscillations raise a basic question: whether, in a dynamic framework, utility maximization is always an efficient strategy.

REFERENCES

D. S. Dendrinos, Regions, antiregions, and their dynamic stability: the U.S. case (1929-1979), ]ournaZ of Regional Sci. 24(1):65-83 (1984). D. S. Dendrinos, The structural stability of the U.S. regions: Evidence and theoretical underpinnings, Enuimrment and Plunn. A 16:1433-1443 (1984). D. S. Dendrinos, Theoretical developments in discrete maps of relative spatial population dynamics, presented at XXVII European Congress of the Regional Science Association, Athens, Aug. 1987; Sistemi Urbani 2/3 (1988). D. S. Dendrinos and M. Sonis, The onset of turbulence in discrete multiple spatial dynamics, Appl. Math. Cum@. 22:25-44 (1987). M. D. Intrilligator, Mathematical Optimization and Economic Theory, Prentice- Hall, 1971. R. May, Simple mathematical models with very complicated dynamics, Nature 261:459-467 (1976).

Spatially distributed stocks, deterministic evolution, and ecological balance

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