the conditions for ecological sustainable development in the context of a double-limited...

Environmental and Resource Economics 9: 309–321, 1997. 309c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

The Conditions for Ecological SustainableDevelopment in the Context of a Double-LimitedSelfpurification Model of an Aggregate WaterResource

KLAUS FIEDLERUniversitat-GH-Siegen, Fachbereich 5, Holderlinstr 3, 57076 Siegen, Germany

Accepted 19 June 1996

Abstract. Based on the classical biological Monod model a so called double-limited self-purificationmodel working in an aggregate water resource is constructed. From this model we derive a stationaryselfpurification function and three conditions for ecological sustainable development.

Key words: Monod model, oxygen saturation, selfpurification

1. Introduction

In the economic literature (e.g. Barbier 1987; Pearce and Turner 1990; Serafy1992; Tisdell 1993) there are several definitions and/or interpretations of sustain-able development.1 In environmental economics one of the most commonly useddefinitions is that any economic activity has to maintain the effective resource base.This paper is interested in a ‘narrower’ interpretation of that concept concernedonly with ecological sustainability.

There is considerable agreement in the literature on environmental economicsthat water resources are renewable according to their quality which is inverselymeasured in units of respective pollutants. But in contrast very little is known aboutnatural selfpurification processes in water resources. There is not only a great vari-ety of definitions of sustainable development in the economic literature but alsoa diverse spectrum of hypotheses about stationary selfpurification processes inwater resources (Fiedler 1994). This paper aims to introduce biological selfpurifi-cation models based on the classical Monod model (Monod 1942) which are moreimportant compared to chemical and physical processes (Habeck-Tropfke 1980;Uhlmann 1988; Fiedler 1994). Monod pointed out that microorganisms use pollu-tants as food source. The Monod model is based on the enzyme kinetics of Michaelisand Menten (1913). Being well supported by experimental results (Thomas 1972;Jones 1978; Andrews 1978), it has found approval of natural science (e.g. Ohgakiand Wantawin 1989; James 1993). Since the Monod model has been used only toreflect the assimilation of pollutant stocks we have to supplement it by adding a

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310 KLAUS FIEDLER

waste flow. Further, aerobic selfpurification processes are of higher economic rel-evance, since ‘water in which wastes are being degraded anaerobically emits foulodors, looks black and bubbly, and aesthetically is altogether offensive’ (Kneese1971, p. 8). For that reason we consider aerobic microorganisms and take oxygenlimitation of selfpurification into account (Ohgaki and Wantawin 1989). Since inour selfpurification model both the stock of pollutant and the stock of oxygen workas a limiting factor2 on bacterial growth we call it double-limited following Bader(1978) and Ohgaki and Wantawin (1989). At last one characteristic of pollutantsused as food source is the so called pollutant inhibition of selfpurification process-es, i.e. a high mass of pollutants is inhibitory to microorganisms’ growth (Andrews1978; Hartmann 1988; Ohgaki and Wantawin 1989).

Starting out from the narrower concept of ecological sustainability we elaboratea set of conditions to maintain stationary assimilative services (i.e. the stabilityof a stationary biological selfpurification process) in an aggregate water resourcecontinually polluted by a single degradable pollutant:

– The first condition for sustainable ecological development our model impliesis that the continual waste flow has to be less than or equal to the assimilativecapacity of the aggregate water resource. Of course this view does not constitutea new contribution to the economic literature (e. g. Barbier and Markandya1990; Pearce and Turner 1990) but it has not yet been interpreted from theecological point of view.

– The second condition requires the stock of pollutant neither to be too low nortoo high.

– Following the third condition the oxygen saturation level has to be sufficientlyhigh to sustain life of the aerobic microorganisms.

2. The Monod Model

Fundamental models considering bacterial growth and pollutant assimilation areoften based on the classical Monod model (Monod 1942; James 1993) expressedby the following two coupled homogeneous differential equations:

_s :=dsdt

= �M0(s) � x (2.1)

_x :=dxdt

= �M0(s) � x� kd � x: (2.2)

In the Monod model the specific growth rate M0(s) [1 / period] of the microor-ganism stock4 x [mass] is defined as M0(s) := �0�s

km+s; �0 [1 / period] symbolizes

the maximum specific growth rate, km [mass] is the Michaelis-Menten constantand s [mass] represents the stock of a single degradable pollutant (Jones 1978, p.265; Ohgaki and Wantawin 1989, p. 250).5 The net rate of change of microbialmass in equation (2.2) is equal to the growth rate �M0(s) � x [mass / period]

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THE CONDITIONS FOR ECOLOGICAL SUSTAINABLE DEVELOPMENT 311

minus the decay rate kd � x [mass / period], where kd [1 / period] represents adecay parameter. The decay rate represents altogether death, cell decompositionand consumption as food source by higher terrestrial organisms (Thomas 1972).The consumption by higher land organisms causes the biomass x (included themass of the dead microorganisms) probably to get out of the water resource. Thisfact will be analysed later in the context of water resource eutrophication. In thefollowing we set for simplicity the yield coefficient [biomass / pollutant mass]equal to one ( = 1).

Moreover, following Jorgenson (1988, p. 205) and James (1993, pp. 94 sq.) thepollutant is assumed to be completely mixed up with all the other contents beinginvolved in the selfpurification process occurring within the water resource. Thusour watersource has to be regarded as a ‘homogeneous selfpurification reactor’concerning the single degradable pollutant s. Further, the temperature and thepressure insight the aggregate watersource is assumed to be constant over time.Finally observe that the pollutant s works as a limiting factor on the microbialgrowth and on the assimilation rate. We precise the limiting factor by the followingexpressions: M0(s) � x = _s = 0� _x < 0 , s = 0.

3. Construction of a Double-Limited Selfpurification Model Based on theMonod Model

Since water pollution is a continual pollution problem we have to supplementthe Monod model by considering a waste emission rate e entering the aggregatewater resource directly. e is measured in units of the single degradable pollutant perperiod. Further, we focus our attention on aerobic selfpurification processes being ofhigher interest for economic analysis than the anaerobic ones and assume the stockof microorganisms to be aerobic. Therefore we have to take into consideration thataerobic selfpurification processes are limited by the oxygen stock. For that reasonwe couple up an additional differential equation describing the time evolution of theoxygen stock. Most of the pollutants result in a so-called pollutant inhibition whenbeing used as food source by microorganisms in water resources (e.g. Andrews1978; Ohgaki and Wantawin 1989; Bever et al. 1993). This fact requires to substitutethe specific growth rate M0(s) by the term M(s) := ��s

km+s+kps2 [1 / mass �

period] where kp [1 / mass] symbolizes an inhibition constant6 and the parameter� := K� � �0 is of the dimension [1 / mass � period].7 Taking these additionalcircumstances into account our selfpurification model has the following form:

_s = e�M(s) � x � o (2.3)

_x = M(s) � x � o� kd � x (2.4)

_o :=dodt

= k0(os � o)�M(s) � x � o: (2.5)

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312 KLAUS FIEDLER

In equation (2.5) k0 [1 / period] represents the reaeration constant and os [mass]is the oxygen saturation constant.8 The net rate of change of the oxygen stock o isequal to the reaeration rate ko(os � o) > 0 minus the rate of oxygen consumptionM(s) � x � o. The former term should also be interpreted as the selfpurification rateand as the microbial growth rate starting out from the equations (2.3) and (2.4).Observe that both the stock of pollutant and the stock of oxygen work as limitingfactors on bacterial growth. Therefore we call our selfpurification model followingBader (1978) and Ohgaki and Wantawin (1989) double-limited. The double-limitedselfpurification model can be even better understood by introducing the followinginvestigations.

A double limit on emissions is determined by

_s>0 0 , e>0 min[k0(os � o); kd � x]: (2.6)

The stock of pollutant increases / remains constant / decreases over time iff thecontinual waste flow is greater than / equal to / less than the smaller amount bothof the reaeration rate k0(os � o) and the decay rate kd.

_x>0 0 , min[e; k0(os � o)]>0 kd � x: (2.7)

The biomass increases / remains constant / decreases over time if the smalleramount of both the pollutant flow and the reaeration is higher than / equal to /less than the decay rate. The case _x > 0 indicates an eutrophication. That is theaccumulation of the biomass x in the water resource over time (James 1993).Observe that the relation (2.7) expresses the double-limit on microbial growth.

_o>0 0 , k0(os � o)>0 min[e; kd � x]: (2.8)

The stock of oxygen increases / remains constant / decreases over time iff thereaeration rate k0(os � o) is greater than / equal to / less than the smaller amountof both the pollutant flow and the decay rate.9

Our major interest is focussed on stationary self-purification, i.e. _s = _x = _o = 0.The stationary reaeration rate e = k0(os � o) > 0, the specific growth rate M(s)and the stationary oxygen demand function

o = O(s) :=kd

M(s)6 os (2.9)

determine the stationary selfpurification function of the model:

e = S(s) := k0 � os � k0 �kd

M(s)> 0: (2.10)

In equation (2.10) the relation S(s) > 0 results from equation (2.9). The eco-logical interpretation of the stationary selfpurification function can be given as

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Figure 1. Geometrical derivation of the stationary selfpurification function.

follows: The continual pollution rate e is assimilated into biomass in each pointof time by the aerobic microorganisms within the water resource. This stationaryselfpurification process runs in such a way that all the stock of pollutant, the stockof microorganisms and the stock of oxygen simultaneously remains constant overtime. Figure 1 illustrates the geometrical derivation of the function S(s).

The selfpurification function depicted in the first quadrant of Figure 1 has thefollowing analytical properties: S(s) is concave shaped, Sss = �

2kdk0km��s3 < 0,

hits the zero locus at the points smin := �2 �

h�2

4 �kmkp

i1=2> 0 and smax :=

�2 +

h�2

4 �kmkp

i> smin with� defined as � := �os�kd

kdkp, reaches its global maximum

at s0 :=hkmkp

i1=2with s0 � [smin; smax] and has the assimilative capacity e0 =

S(s0) := k0 os � k0kd

M(s0)= max

�S(s)

�. The term o0 := kd

M(s0)= O(s0) =

min�O(s0)

�> 0 has to be interpreted as the absolute minimum of the stationary

oxygen stock (point B in Figure 1) enough to sustain a stationary microbial life.Further, the following properties of our stationary selfpurification function have tobe proved:

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314 KLAUS FIEDLER

I. e0 > 0 , os > o0: This property results immediately by employing thedefinition o0 := kd

M(s0)in e0 = k0 os�k0

kdM(s0)

. Thus we obtain e0 = k0(os)�

o0) > 0 , os > o0. �II. �

4 �kmkp> 0 , os > o0: With the help of the definition of � and of o0 this

assertion results after some rearrangements of terms.10 �

III. smin > 0: Using the definition of smin and of � the relation km=kp > 0 results.This is true since km > 0 and kp > 0. �

IV. smin 6 s0 6 smax , os > o0: Employing the definitions of smin; s0; smax and� we obtain the following two relations after some transformations: smin 6

s0 , os > o0 and s0 6 smax , os > o0. �

4. Conditions for Sustainable Ecological Development

Based on our selfpurification model we are now in the position to give a definitionof ecological sustainable development.

DEFINITION: In the context of our selfpurification model a sustainable ecologicaldevelopment is achieved if the stationary selfpurification process e = S(s) going offwithin the watersource is stable.

This definition requires to examine the stability of stationary selfpurification andcan easily be related to the following conditions of sustainable ecological devel-opment given by Pearce and Turner (1990) and Pearce, Barbier and Markandya(1990): Maximizing the net benefits of economic development

– must not decrease the natural capital stock over time, and– have to maintain the assimilative services and the quality of natural resources

over time.

Our definition of sustainability is equivalent to these two criteria if we identifythe stationary selfpurification process with a natural capital stock giving perpet-ual assimilative services to any economic sector. The stability of the stationaryselfpurification, i.e. ecological sustainability, is bound to the following three con-ditions which have to be satisfied simultaneously. These conditions are determinedheuristically.

Condition 1: e 6 e0

The pollution rate must not exceed the assimilative capacity within the waterresource. This condition has been given by Pearce and Turner (1990) but it hasnot yet been interpreted from the ecological point of view. To interpret the firstcondition in the context of our selfpurification model we assume that this conditionis violated (e.g. by e1 > e0 in Figure 1). In this case the stationary stock of oxygenis not sufficient to sustain the life of the aerobic microorganisms in the long run.

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Therefore the selfpurification process will come to a standstill and the stock ofpollutant will increase over time. This mechanism calls for closer inspection: Thestationary reaeration rate is given by e = k0(os � o) implying

o = os �e

k0: (2.11)

For the case e = e0 equation (2.11) leads to (point B in Figure 1)

o = os �e0

k0=

kd

M(s0)=: o0 > 0: (2.12)

Employing the assumption e1 > e0 in equation (2.12) exhibits the followingrelation:

o = os �e1

k0=: o1 < o0 = os �

e0

k0: (2.13)

Further, o1 < o0 = kdM(s0)

implies _x < 0 (point C in Figure 1) by consideringequation (2.7). A decreasing stock of microorganisms leads to x = 0 in the long run,consequentlyM(s) � x� = 0 is valid and therefore using equation (2.3) _s = e1 > 0results. �

Condition 2: smin < s < smax

This condition is immediately obtained by solving the relation e = S(s) > 0for the variable s. If the second condition is violated the microorganisms will nothave enough food to sustain life for s < smin and will be poisoned if s > smax

is valid. Consequently the selfpurification process stops and the stock of pollutantincreases over time. This fact should be proved: The relations s 6 smin and s > smax

imply M(s) 6M(smin) = M(smax).11 From the system (2.3)–(2.5) we obtain thestationary stock of microorganisms

x =e

kd=

k0

kd� (os � o) > 0: (2.14)

Further, equation (2.9) implies:

o =kd

M(s)(2.15a)

os =kd

M(smin)=

kd

M(smax)8x > 0: (2.15b)

Employing these equations into equation (2.14) yields

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316 KLAUS FIEDLER

x = k0 �M(s)�M(smin)

M(smin) �M(smin)> 0

,M(s) >M(smin), smin 6 s 6 smax (2.16)

According to equation (2.16) the stationary stock of microorganisms x is greaterthan / equal to zero if and only if the specific growth rate M(s) is greater than /equal to the minimal specific growth rate M(smin). Consequently in the cases ofs 6 smin and s > smax the microorganisms become extinct. Therefore in the longrun x = 0 is valid and thus due to

_s = e > 0: � (2.3)

A similar condition is stated by Barbier and Markandya (1990). However, accordingto these authors there exists only an upper limit smax to the pollutant stock in such away that environmental degradation destroys the natural selfpurification processesif the pollutant stock exceeds smax.

Condition 3: os > o0

This condition requires the oxygen saturation limit to be sufficiently high to sustainthe microbial life, i.e. to exceed the threshold o0. This should be proved: In thesteady-state the relation 0 6 S(s) = e = k0(os � o) , os > o > o0 is valid.Consequently our third sustainability condition e = S(s) > 0 , os > o0 is alsovalid.12 �

In the case of os = o0 the oxygen saturation level is too low to sustain themicrobial life in the long run. According to x = k0

kd� (os � o) > 0 , os > o > o0

the case of os = o0 results in x = k0kd� (os � o) = M(s) � o � x = 0 and thus

_s = e > 0. In geometrical terms the line representing the stationary reaeration in thesecond quadrant of Figure 1 is shifted parallelly from the position os es0(os > o0)to the position o0 e

s0. Hence the function S(s) degenerates to one single point

s = s0 = smin = smax on the s-axis. In this point e = 0 is valid.For a better understanding of our simple selfpurification model it should be

mentioned that there is only one single possibility for man to influence the massof the microorganisms13 and the level of oxygen: This is, as can be seen fromequation (2.11) and (2.14), the variation of the waste flow e.

5. Some Basic Implications of the Stationary Double-LimitedSelfpurification Process on Economic Modelling

To study the economic impact of the stationary selfprurification process (2.10),as a first step, it is convenient according to Maler (1974, p. 64), Pethig (1976, p.

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162) and Fiedler (1994, p. 346) to define the following normalized index of waterquality q measured in units of the pollutant stock s:

q = V (s) := qnat � s [Index function]14 (2.17)

The parameter qnat > q denotes the natural water quality. Moreover, the zero level

of water quality is defined with the help of the parameter smax := �2 +

h�2

4 �kmkp

i1=2

from the double-limited selfpurification model as

q = V (smax) = qnat � smax =: 0 (2.18)

We carry out the transformation of the measuring concept by substituting the waterquality index (qnat � q) > 0 for the pollutant stock variable s in the stationaryselfpurification function. By considering S(qnat � q) =: E0(q);M(qnat � q) =:M 0(q) and es0 := k0 � os we obtain the corresponding stationary water qualityregeneration function:

e = E0(q) := es0 � k0 � ka �qnat � q + km + kp(qnat � q)2

�(qnat � q)(2.19)

Since the parameters e0; o0 and os are invariant with respect to the transformationof the pollutant concept into the measuring concept of water quality the process ofquality regeneration is bound to the conditions e < e0 and os > o0 as well as theunderlying selfpurification process. Due to equation (2.17) and (2.18) the condition2 reads 0 < q < qmax := qnat � smin.

The water quality regeneration function, which is shaped like the yield-effortcurve implied by the fishing and forest models (Clark 1976), has the followingproperties:

E0q :=

kd � k0 � kp(qnat � q)2� kd � k0 � km

�(qnat � q)2>0 0 , q

61 q0 := qnat � s0

q0 = arg maxhE0(q)

i

e0 = E0(q0) := es0 � k0 �kd

M0(q0)= max

hE0(q)

i

E0qq = �

2 � k0 � ka � km

�(qnat � q)3 < 0

e = E0(qmin) = E0(qmax) = 0: (2.20)

In a second step based on equation (2.19) and (2.20) a simple economic model isdeveloped: A representative firm (industrial sector) produces a consumption goodY according to the function.15

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318 KLAUS FIEDLER

y = Y (ay; e) (2.21)+ +

where y denotes the quantity of the consumption good, ay represents the labourinput and e stands for the waste flow directly discharged into the aggregate water-source. In this model the industrial sector has to be regarded as a direct discharger.Even though e represents an undesired joint output it is formally treated as aninput, namely as the industry’s demand for selfpurification processes (assimilationservices) occurring in the watersource. The production function encompasses intra-industrial waste treatment activities by means of labour input (Pethig 1979).16 Forsimplicity we suppose that labour is fully employed and constant, i.e. ay = a0

(Pethig 1994, p. 7). Further, it is plausible to assume that the maximum emis-sion emax exceeds the assimilative capacity e0 of the aggregate watersource, i.e.emax > e0. In this case ymax := Y (a0; emax) denotes the maximum output withoutany intra-industrial waste reduction. Observe that for sustainability the firm has toreduce the amount (emax � e0) > 0 of waste emission until the condition e = e0

is valid. Therefore the maximum sustainable output reads y0 := Y (a0; e0) < ymax

(Pethig 1994, p. 10).Assume that a representative consumer maximizes his (or her) utility depending

on the quantity of the consumption good y and on the water quality q:17

maxe;q;y

U(q; y) (2.22)+ +

s.t.

e = E0(q)

y = Y (a0; e)

The associated Kuhn-Tucker conditions yield an interior solution

+

Tq := Ye(e�) �E0

q (q�) = �

Uq(q�; y�)

Uy(q�; y�)< 0 (2.23)

– + –+

whereE0q is defined byE0

q := kd�k0�kp(qnat�q)2�kd�k0�km

�(qnat�q)2 . The simple condition (2.23)for a Pareto efficient allocation has the following important implication: Since theterms Uq; Uy and Ye are positive, E0

q has to be negative. Therefore in the long runa welfare maximizing equilibrium is selected from the subset [q0; qmax] with E0

q <0 of the function E0. Moreover the condition (2.23) states that a Pareto efficientallocation is reached if the marginal willingness to pay Uq=Uy for water quality isequal to the marginal opportunity costs Tq of producing water quality by reducingwaste emissions.18

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Further models making use of the function E0 have been developed by Pethig(1989, 1994), Pethig and Fiedler (1989, 1992) and Fiedler (1996).

6. Concluding Remarks

A double-limited selfpurification model based on the classical Monod modelimplies the following three conditions, which have to be fulfilled simultaneouslyfor a sustainable ecological development of an aggregate water resource:

Condition 1: The continual waste flow has to be less than or equal to the assimila-tive capacity of the aggregate water resource.

Condition 2: The stock of pollutant has to be neither too low nor too high, i.e. wehave to pollute the water to a certain degree to prevent the microorganismsfrom starving on the one hand and from poisoning on the other hand.

Condition 3: The oxygen saturation limit has to be sufficiently high to sustainmicrobial life.

By a simple linear transformation we obtain a stationary regeneration function ofwater quality from the underlying selfpurification function based on the double-limited model. This regeneration function is shaped just as the yield-effort curvebelonging to the classical fishing model. In conclusion a simple economic model isdeveloped. Finally, as pointed out by Cesar and de Zeeuw (1994), the implicationsof the environmental economics’ models (e.g. the shape of the transformationfrontier with pollutant versus waste flow or water quality versus waste flow) aredepending significantly on the shape of the stationary selfpurification function S.For that reason the knowledge of natural selfpurification processes is indispensible.

Acknowledgments

Earlier versions of this paper were presented at the 5th EAERE Annual Confer-ence, University College Dublin, held on June 22–24, 1994 and at the conference,‘Governing Our Environment’, in Copenhagen, held on November 17–18, 1994. Iwould like to thank Rudiger Pethig (Universitat-GH-Siegen, VolkswirtschaftslehreIV (Department of Public Economics) and Naomi Zeitouni (University of Haifain Israel, Natural Resources & Environmental Research Centre), as well as twoanonymous referees, for very useful comments. At last but not least I have to thankmy wife, Petra Ullmann-Fiedler, for critically reading this text. Remaining errorsare my own responsibility.

Notes

1. Pezzey (1989) has cited about 60 definitions of sustainable development.2. A factor works as a limiting factor on any process if and only if this process does not go off

without this factor.3. For simplicity we do not distinguish between various types of watersources e.g. rivers, estuaries

or lakes.

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320 KLAUS FIEDLER

4. The term for M0(s) with M0(s = km) = 1=2 � �0 has been derived from the enzyme kinetics ofMichaelis and Menten (1913) by Morris (1972), Thomas (1972) and Murray (1989).

5. Note that water quality is negatively correlated to the ambient stock of the pollutant s (Maler1974, p. 64; Pethig 1976, p. 162 and Fiedler 1994, p. 346). Further, by assumption, the stock ofmicroorganisms does not damage the water quality.

6. The term M(s) has been derived from the generalised enzyme kinetics of Michaelis and Mentenby Morris (1972). Observe that so := (km=kp)

1=2 = arg max [M(s)] and M(so) = � s02 km+s0

.7. The constant k� is of the dimension [1 / mass]. As one can verify from the subsequent system

(2.3) – (2.5) the dimension of the term M(s) is [1 / mass � period] because M(s) is multipliedby the variables x [mass] and o [mass].

8. Typical numerical values of kinetics constants, e.g. for nitrifying microorganisms are listed inTassin and Thevenot (1989 p. 231) and in Ohgaki and Wantawin (1989 p. 251).

9. Observe that in the biochemical system (2.3) – (2.5) the selfpurification rate is triple-limited, i.e.M(s) = min [e; k0(os � o); kd � x].

10. In geometrical terms the case �2

4 �kmkp

= 0 , os = o0 means that the function S(s) degeneratesto one single point in which the three parameters smin; s0 and smax coincide.

11. This equation results form substituting smin and smax into the specific growth function after somerearrangements. Observe that the effect of pollutant scarcity and abundance is symmetrical in thesense that M(smin) = M(smax) is valid.

12. We have proved above that e0 > 0 , os > o0. Observe that this assertion is a special case ofthe condition 3, since e = e0 is a functual value of S(s) according to e0 = max [S(s)].

13. Moverover observe that our selfpurification model does not take into account any possibility toget microorganisms to clean water from other sources.

14. Further index functions are cited for example by Ott (1978, pp. 52 sq.).15. For details compare Pethig (1979) and Siebert et al. (1980).16. Labour is both employed for the production and for waste treatment.17. Observe that water quality is a public good. However, this publicness does not cause any analytic

difficulties, because our aggregate model assumes one single consumer, who reports his (or her)true marginal willingness to pay for water quality to a social planner. A disaggregated model isintroduced in Pethig (1994, pp. 7 sq.).

18. In a disaggregated model with many consumers the condition (2.23) has to be replaced with theSamuelson condition, which states that Pareto efficiency is reached if the sum of all individuals’marginal willingness to pay is equal to the marginal opportunity costs (Samuelson 1954).

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