Hediger 2009 EDE

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Environment and Development Economics 14: 759780 C Cambridge University Press 2009doi:10.1017/S1355770X09005282First published online 31 March 2009

Sustainable development with stock pollution

WERNER HEDIGER

Bern University of Applied Sciences, Swiss College of Agriculture,Laenggasse 85, 3052 Zollikofen, Switzerland.Email: [email protected]

ABSTRACT. Optimal pollution control is an important challenge for sustainabledevelopment with three distinct cases. First, the situation where natures assimilativecapacity is completely destroyed involves normative problems that require furtherresearch. Second, environmental restoration with initial pollution above the steady-statestock requires an economy to initially allocate a relatively high share of its resourcesto cleaning-up activities. In return, this generally results in an intertemporally efficientdevelopment path that is both environmentally and economically sustainable. Third,optimal trajectories in situations with initial stocks of pollution below the long-termoptimum generally imply an increase in pollution and a decline of optimal consumption.In this case, the investment of the environmental rents accruing from natures assimilative

capacity into man-made capital is required in analogy to the famous Hartwick ruleto maintain a constant flow of instantaneous welfare. This would facilitate growth inconsumption sufficient to compensate for the rising disutility of pollution.

1. IntroductionSustainable development is a concept of concern about the well-being ofboth present and future generations. It involves trade-offs between presentand future use of environmental and natural resources. To address thisintergenerational equity concern, various concepts of sustainability havebeen proposed since the publication of the Brundtland Commission Report

(WCED, 1987). These include economic approaches that are grounded onearlier works of Solow (1974), Hartwick (1977) and Daly (1972, 1974), andencompass the integration of ecological concepts of resilience (Commonand Perrings, 1992), critical natural capital (Pearce et al., 1994) and basicneeds (Chichilnisky, 1977; Hediger, 1999) as well as the development ofnew approaches, such as Chichilniskys (1996, 1997) criterion to balance theshort run with the long run future.1

The dominant concern in most economic approaches to sustainabledevelopment is about the maintenance of current or optimal level ofconsumption or utility into the far distant future. Hence, much attention

is given to the exhaustibility of natural resources and the challenge of

1 See also Neumayer (2003) and Heal (2000) who respectively discuss recentdevelopments and limitations of traditional approaches of weak and strongsustainability, on one hand, and in valuing the present and the future, on theother.


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760 Werner Hediger

maintaining some suitably defined stocks of capital. However, most modelsof sustainability do not explicitly consider the accumulation of pollutantsin the environment, or, at least, not with the same rigour as in the

environmental economics literature on optimal pollution control. The latterevolved since the publications of Keeler et al. (1972) and Plourde (1972). Itcovers various economic aspects of pollutant accumulation and control,such as the implication of different forms of the assimilation function,economic growth and the investment in clean technologies, and theproblem of global warming. However, the relationship between trajectoriesof optimal pollution control and sustainable development has receivedlittle attention in published work, and has not been investigated in acomprehensive way.

Barbier and Markandya (1990) only identify the minimum initial level of

environmental quality required to ensure a sustainable growth path, whichis solely characterized by an increase of the total stock of environmentalassets. However, they neither investigate the effects of the optimal trajectoryupon the development of aggregate income and social welfare, norare they explicit with regard to the accumulation and assimilation ofpollution. In contrast, Cheve (2000) elucidates the importance of a declinein the natural pollution decay function upon optimal pollution controland economic growth. But, her argumentation is restricted to the caseof strong sustainability with a non-decreasing stock of natural capitalthat is simply represented by the stock of pollution. Hence, Cheves

conclusion that a better environmental quality can be associated withhigher economic growth is valid only for the case of an initial stock ofpollution above the long-term optimum. This can be referred to as a problemof environmental restoration, rather than a general problem of optimalpollution control.

Hartwick (1997), in an extension of Beckers (1982) model, also reflects onsustainable development in situations with high initial stocks of pollution.Applying a weak sustainability framework of constant utility over time,he suggests that reducing high pollution stocks can be interpreted aspaying off the current debt in environmental capital, which requires high-

cleanup activities and lowers investment in physical and human capital.But, Hartwick does not investigate the impact on the trajectory of economicdevelopment, such as Cheve (2000) that is also restricted to the case withhigh initial pollution stocks.

On the contrary, Stollery (1998) investigates the existence of constantutility paths in the case of irreversible greenhouse gas accumulation fromfossil fuel carbon emissions. Using a resource model where the fossil fuelstock is completely converted into the stock of atmospheric CO2 (assuming azero rate of decay) and where the associated externality (global temperatureincrease) is declining with the remaining resource stock in the ground,Stollery shows that a carbon tax is required to steer the economy alongthe sustainable path and that the constant utility level is maximizedby following the standard Hartwick rule. The latter requires the globaleconomy to invest in reproducible capital, the sum of the carbon taxrevenues and after-tax resource rents, that is, the gross energy resource rent.In essence, this is exactly the same result as Hartwick (1977). In other words,


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Environment and Development Economics 761

it does not differ from the original Hartwick rule without consideration ofpollution.2 Yet, Hediger (2006) shows that in the presence of stock pollutionthe standard Hartwick rule must be extended and include an additional

term that does not vanish even if the stock of pollution would be keptconstant over time. Hence, to comprehensively address the challenge ofsustainable development and to be consistent our models must explicitlyaddress the accumulation and decay of pollutants.3

Altogether, this indicates the need of extending our research to morecomprehensively address the problem of optimal pollution control and topay attention to both problems of environmental restoration and continuedpollution accumulation along the optimal time paths, and to evaluate theresults from a sustainable development perspective.

Building on this background, the aim of the present paper is to

provide an economic analysis of optimal pollution control and sustainabledevelopment, and to widen the above literature. To this end, we startwith the rather general model of Forster (1975) and gradually extend thisanalytical framework to gain additional insights about the relationshipbetween optimal pollution control and the sustainability of development.The basic model is introduced in section 2, while section 3 providesa first discussion of the optimal pollution control trajectories fromboth an efficiency and sustainability point of view. It shows that theexistence of constant productive capacity is not in general sufficient forsustainable development. Rather, additional investments might be required

to maintain a non-declining flow of consumption. Therefore, the modelis extended in section 4 to jointly address optimal pollution control andcapital accumulation. The extended model is also used in section 5 toinvestigate the option of a Hartwick-type investment rule as a meansto facilitate a constant flow of welfare along an intertemporally optimaltrajectory of pollution accumulation. This is also considered in brief from aperspective of sustainable preferences, as defined by Chichilnisky (1996,1997), and for the case of environmental restoration. Finally, section 6concludes.

2. The basic modelThe accumulation of pollutants is a major cause of environmentaldegradation and adverse impacts upon human health and welfare.Examples include soil and water contamination with toxic substances orsulphur and nitrogen compounds, the eutrophication of surface waters, aswell as increased concentrations of greenhouse gases and ozone depletingsubstances in the atmosphere. For these substances, damages to ecosystemsand human interests depend on the stock of pollution, which consequentlyhas an adverse impact on social welfare. This is usually represented in the

form of a concave utility function U of instantaneous consumption C and

2 This is a consequence of Stollerys simplifying assumption.3 The latter is a crucial function that cannot simply be assumed away, as in Stollery

(1998). Rather, Cesar and de Zeeuw (1995) show that slight variations in theassimilation or decay function can have dramatic effects on the steady-state values.


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762 Werner Hediger

Figure 1. Different forms of the assimilation function

the current stock of pollution S, which has the following properties:

U(C, S),

UC > 0, UCC < 0, US < 0, USS < 0, UC S = USC = 0,

limC0

UC (C , S) = , limS0

US(C , S) = 0.(1)

Following Forster (1975, 1977), the last two conditions are used for reasonsof tractability and to rule out the unrealistic boundary solutions of zeroconsumption and zero pollution. Moreover, in line with Forsters model, we

assume for the moment that the economy has a fixed productive capacityand that it produces a constant level of output Y0. This is allocated toconsumption C and anti-pollution activities Z:4

Y0 = C + Z. (2)

Assuming a constant and free flow of renewable resources, the dynamicproblem in our analysis comes from the intertemporal accumulation ofpollution

S = E(C) G(Z) A(S). (3)

This is the excess flow of net emissions E(C) G(Z), withE > 0, E > 0, G > 0 and G < 0 , above the natural capacity of pollutantassimilation, the so-called assimilative capacity A(S). According to Pearce(1976), the latter refers to the environments capability of receiving waste,degrading it, converting it to nutrients for the occupants of the ecosystemor rendering it harmless to species even if the converted product is notrequired as an input to the ecosystem. Depending on the type of pollutant

this essential natural function can take different forms, as illustrated infigure 1:5

4 This restrictive assumption is relaxed in a further step of our analysis, where wealso consider optimal capital accumulation.

5 See also Elliott and Yarrow (1977), Fiedler (1994) and Pezzey (1996).


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Case (i) is the most often used functional form in the optimal pollutioncontrol literature. It is characterized by a constant proportionate rate ofdecay , which would typically be the case for the decay of radioactive

substances. Case (ii) is rarely used in the environmental economics literature. Itdescribes the process of saturation that is typical for chemical processesof self-purification (cf. Fiedler, 1994). It could also be an importantrepresentation of the global CO2 assimilative capacity, since the uptakeof CO2 declines with the atmospheric concentration (Watson et al., 2000)and the fraction of a unit emission of CO2 remaining in the atmosphereis higher the higher the accumulated emissions are (Azar, 1995).

Case (iii) is typical for biological processes of self-purification (cf.Fiedler, 1994, 1997) that are limited by some degrader populations (e.g.,

microorganisms) and general environmental conditions, such as theoxygen and other nutrients available to support the degraders. Thisfunctional form has first been used in the environmental economicsliterature by Forster (1975), and since then has gained increasingattention from other economists who are concerned about the influenceof ecological processes upon the economy and their representation ineconomic analysis.

In this study, we concentrate on the third case which is the mostinteresting from an analytical point of view, and the most relevant from

an ecological perspective. It combines the favourable case where theecosystems assimilative capacity increases with the stock of pollution forlow levels and the adverse consequences of saturation. The latter resultsfirst in a decline of the marginal rate of assimilation, followed by continuousdeterioration of the assimilative capacity for further pollution accumulationbeyond a certain turning point, say Sm. This natural process ends with thecomplete destruction of the assimilative capacity at the pollution level S,beyond which this ecosystem function is reduced to zero.6

Formally, the assimilation function can be represented as follows:

A(S) > 0, A

> 0= 0

< 0

for 0 < S < Smfor S = Sm ,

for Sm < S < S

A < 0 for 0 < S < S

A(S) = 0 for S = 0 and S S

(4)

6 In principle, this results in an irreversible pollution accumulation, such as forsimplicity assumed by Stollery (1998).However, this worst-case scenario cannot bemaintained as a rule. Rather, if it is technically feasible to reduce the accumulatedstock of pollution in the environment, then even the situation with a fully

destroyed assimilative capacity could be reversible, but implies additional costof environmental restoration. In the present paper, we do not rule out thistechnical option by assumption, since it must be taken into consideration wheninvestigating allocation problems with high levels of pollution and A(S) = 0.Accordingly, we relax the more restrictive assumption of Cheve (2000) whichimplies that, as a necessary condition, the initial stock of pollution must not beabove the irreversible level S.


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As indicated, if the net flow of emissions from the economy exceedsthis capacity, the stock of pollution accumulates in the environment andhas a negative impact upon the societys well-being. Hence, we face an

intertemporal allocation problem. This is usually formalized in terms ofmaximizing the present value of instantaneous utility by choosing theoptimal level of consumption

max{C}

0

etU(C, S)dt (5)

subject to S = f(C) A(S), S(0) = S0,

where > 0 denotes the constant social discount rate andf(C) the economys

net emissions:f(C) = E(C) G(Y0 C) (6)

with f = E + G > 0, f = E G > 0.Using Pontryagins maximum principle, this intertemporal optimization

problem can be represented by the current-value Hamiltonian

H = U(C , S) + [ f(C) A(S)] (7)

which is to be maximized at each point in time. The following differentialequations define the optimal time path of the state and costate variables, S

and , of this allocation problem:

S = f(C) A(S), (8)

= [ + A] US. (9)

Along this trajectory the allocation of the existing output to consumptionand pollution abatement must continuously be adjusted, so as to satisfythe capacity constraint (2) and the subsequent first-order condition at eachpoint in time:

UC + f = 0. (10)

Using this optimality condition, it can be shown that the shadow priceof pollution is strictly negative, which reflects that pollution generatesdisutility:

= UCf

< 0. (11)

Moreover, using equations (1) and (2), respectively, it follows that

=

UCCUC

f

f

C and C + Z = 0 (12)

and thus

sgn(C) = sgn( ) and sgn(Z) = sgn(C). (13)


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Figure 2. Phase diagram for the optimal pollution control problem with fixed productivecapacity

Consumption must develop with the same sign as the value of the costatevariable , while the share of output allocated to pollution abatement must

go in the opposite direction.This is crucial for economic growth and sustainable development, but,

so far, has not gained adequate attention in the literature and policydebate. To fill up this gap, we proceed in a straightforward manner byfirst investigating the relationship between optimal pollution control andsustainable development prospects within the framework of Forsters(1975) model with constant productive capacity, and then extend theanalysis by relaxing the latter restriction.

3. Optimal pollution control with constant productive capacityThe solution of the above optimal control program is characterized by asteady state where = S = C = Z = 0 and a stable transition path from theinitial state S0 towards this long-term optimum. Yet, as Forster (1975) andTahvonen and Withagen (1996) already pointed out, multiple equilibria canexist for the optimal pollution control problem with a biological assimilationfunction. One of these long-term optima implies an intact assimilativecapacity A(S) > 0. This is illustrated in figure 2 by the intersection of the = 0 and S = 0 demarcation curves in the point (S, ). Here S denotesthe optimal stock of pollution in the steady state and the corresponding

shadow price of pollution. The second long-term optimum is depicted infigure 2 by the intersection of the two demarcation curves in the point(Sx,x), where the assimilative capacity is completely destroyed: A(Sx) = 0.7

7 This second long-term optimum does not exist for sufficiently low discount ratesand high marginal damage cost of pollution.


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The = 0 locus can be determined by setting the right-hand side inequation (9) equal to zero. As shown in the appendix, the resulting functionhas a pole at the pollution level S = S#, where the pollution discount rate

+ A

(S) = 0 . In contrast, the S = 0 curve is the locus of the shadow price for the stationary optimization problem with S = 0 (see the appendix).

It can further be proven that the intersections of the = 0 and S = 0curves are local saddle points (cf. Cesar and de Zeeuw, 1995; Tahvonenand Withagen, 1996; Cheve, 2000). As a consequence, each of theselong-term optima can only be achieved by proceeding on one of theoptimal trajectories that directly and consistently flow towards the relatedsaddle point. Moreover, as Tahvonen and Withagen (1996) have shown,two candidates for optimal pollution control may simultaneously exist:an irreversible policy, which finally implies zero emissions and high

pollution concentration, and a reversible policy, which maintains a positiverate of decay, strictly positive emissions, and lower pollution concentration.Thisisillustratedinfigure2bythetwopairsofstablebranchesleadingtothelong-term equilibria (S, ) and (Sx, x), respectively. All other trajectoriesturn sooner or later away from the saddle point (Chiang, 1984). In otherwords, for each initial state S0 = S, exactly one shadow price exists on theintertemporally efficient trajectory that maximizes the net present value ofinstantaneous consumption benefits and pollution damage. Environmentalpolicy is accordingly advised to internalize the damage cost of pollution bycharging emissions with a fee = ,whichisnotconstantbutmustbecon-

tinuously adjusted according to the optimal trajectory of the shadow price, in order to provide an incentive for optimal resource allocation over time.

For any current level of pollution S < S, the stock of pollution mustgrow along the optimal trajectory. As a consequence, the overall periodmarginal damage cost of pollution increases. Moreover, the absolute valueof the shadow price increases. Since the latter is negative, it follows < 0.Given the findings in equation (13), this implies that, in the case of S < S,consumption must decline over time and gradually more resources mustbe allocated to pollution abatement activities, C < 0 and Z > 0 . This is

necessary to slow down the further accumulation of pollution and stop thisprocess in the long run at the optimal level S. Altogether, this implies adecline of social welfare U(C,S) along the optimal time path, since accordingto equation (1)

U = UC C + US S

< 0 if S > 0and C < 0

= 0 if S = 0and C = 0

> 0 if S < 0and C > 0

(14)

given UC > 0 and US < 0 .The same logic with the opposite signs is valid for the upper branch of the

stable trajectory leading towards S; that is, for pollution levels within theinterval (S, S). In this case, the stock of pollution declines and consumptioncan grow along the optimal time path. As a consequence, social welfareincreases over time. In contrast, all variables would remain constant in thesteady state which is characterized by S in table 1.


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Table 1. Dynamic properties on the optimal pollution control trajectories

A(S) > 0 A(S) = 0

0 < S < S

S = S

S

< S < S S < S < Sx S = Sx S > SxS > 0 S = 0 S < 0 S > 0 S = 0 S < 0 < 0 = 0 > 0 < 0 = 0 > 0C < 0 C = 0 C > 0 C < 0 C = 0 C > 0Z > 0 Z = 0 Z < 0 Z > 0 Z = 0 Z < 0U < 0 U = 0 U > 0 U < 0 U = 0 U > 0Not

sustainableSustainable Not

sustainableSustainable (if

socially acceptable)

For the two branches of the intertemporal trajectory leading towards Sx,we get exactly the same results (see table 1), presuming that the steady statein (Sx,x) exists. This requires that the social discount rate is not too small,and that the marginal damage cost of pollution is not too high.

On the whole, table 1 shows that the development for relatively smallinitial stocks of pollution, that is on the lower branches of the twostable trajectories in figure 2, is not sustainable at all neither from anenvironmental nor economic point of view, since pollution must increaseand consumption decline over time in order to satisfy the requirementsof intertemporal efficiency and to avoid wasteful uses of scarce resources.

In contrast, development is sustainable along the optimal trajectory for aninitial stock of pollution S > S and a working assimilative capacityA(S) > 0.In this case, development is environmentally sustainable since the stock ofpollution declines along the optimal trajectory and the critical level S is notexceeded. Moreover, it is economically sustainable because consumptiongrows in line with the above process. The same conclusion can be drawnfor initial levels of pollution S > Sx, if the steady state at Sx exists and if theexistence of a totally destroyed assimilative capacity, A(S) = 0, is acceptedfor sustainability from a societal and ethical point of view.

This is a normative question that cannot be answered on purely analyticalgrounds, but requires a social discourse and an expression of values that arenot necessarily captured in our welfare function. In contrast, if S constitutesa critical level of pollution that must not be exceeded at any time, then anaccumulation of pollution above this level would be non-sustainable bydefinition. In this case, a minimum condition for sustainable developmentwould require moving above the critical level of environmental quality (cf.Hediger, 1999) and reducing the stock of pollution below S in a cost-effectiveway.8

For problems with a strictly increasing assimilation function, such as

represented by cases (i) and (ii) in figure 1, only one saddle point exists (cf.Cesar and de Zeeuw, 1995). Consequently, the above analysis reduces to thetwo branches of the optimal trajectory leading to the long-term optimum

8 This is a special problem of environmental (e.g., water quality) restoration that isnot analysed in this paper.


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768 Werner Hediger

with S. For initial levels of pollution below this steady-state level (S < S),the development along the optimal trajectory is not sustainable. In contrast,for pollution levels above the long-term optimum (S > S), an adjustment

process towards this steady state is required which is both environmentallyand economically sustainable.

4. Optimal pollution control with capital accumulationThe analysis in the previous section reveals that we must distinguishtwo different types of the pollution control problem: (a) one with furtheroptimal pollutant accumulation and (b) one of environmental restorationwith a net reduction of pollution. Moreover, it shows that the existence of aconstant productive capacity is not in general sufficient for sustainabledevelopment. Rather, an important case exists where intertemporalefficiency requires the sacrifice of consumption and the allocation of anincreasing amount of output to pollution abatement, while the stock ofpollution is allowed to further increase along the optimal trajectory. In thiscase, investments are required to further enhance the economys productioncapacity and to compensate for environmental damage. Therefore, werelax the assumption of a fixed production capacity and replace equation(2) by

Y = Y(K , L) = C + Z + I (15)

with YK > 0, YK K < 0 and limK 0 YK = .In this case, the total output Y is produced with man-made capital K and

a constant input of labour L. Total output Y is allocated to consumption C,pollution abatement Z and investment Iin the stock of man-made capital K.The latter consists of both physical assets and knowledge, and depreciatesat a constant proportionate rate :

K = I K . (16)

This results in an expanded optimization problem with two state

variables that is represented by the following current-value Hamiltonianwith the additional costate variable representing the shadow price ofcapital:

H+ = U(C , S) + [E(C) G(Z) A(S)] + [Y(K , L) C Z K ].

(17)

Using the first-order optimality conditions, we get the subsequent set ofequations that determine the long-term welfare maximizing trajectory for

given initial stocks of capital and pollution, K0 and S0, respectively:UC + E

= = G > 0, (18)

= [ + A] US, S = E(C) G(Z) A(S), (19)

= [ + YK], K = Y(K , L) C Z K . (20)


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Using equations (1) and (18) and taking into consideration that E > 0and E > 0 , we get

C = E

UCC + E

> 0= 0< 0

if

E . (21)

Thus, whether it would be optimal for consumption to grow depends onthe relationship between the optimal trajectories of the shadow prices andthe marginal rate of emissions, i.e., on whether E is positive or not.According to (20), the first term is a function of the capital stock K and ofthe discount and depreciation rates and . Thus, it is

= [ + YK ]> 0= 0

< 0

if YK

+ . (22)

The implicit price of capital must increase, if the marginal productivityof capital YK is below the long-term optimum + . This implies that thecapital stock and thus total output must shrink along the optimal trajectory(K < 0) , which is the case for initial capital endowment K0 > K

.In contrast, is negative and the capital stock must grow if YK is larger

than + . In other words, for an initial stock of capital below the steady-

state level (K0 < K), the capital stock must increase (K > 0) and the adjointshadow price decline ( < 0) .

As shown in the previous section, the shadow price of pollution increasesaccording to equation (19) with the accumulation of pollution and declinesif the stock of pollution decreases along the optimal trajectory. In table 2,this information on and S is combined with the above results aboutthe relationship between and K . It shows the same pattern for thedomain with intact assimilative capacity, A(S) > 0 (i.e., for 0 < S < S), andthe one where this ecosystem function is fully destroyed (A(S) = 0, i.e.,

for S>

S).Given the relationship between and S , on the one hand, and between and K , on the other, table 2 indicates that along the optimal trajectoryconsumption should grow for low initial levels of capital (K0 < K

) andhigh levels of pollution (S < S0 < S and S0 > S

x, respectively). In contrast,consumption should decline if the initial level of capital is above the long-term optimum but pollution below the steady-state level S and between S

and Sx, respectively. The economy is obviously in a steady state if K = S =C = = = 0 , which is the case in the long-term optimum (S, K).

To further investigate the optimal trajectories for economic development,

we employ equations (20) and (21) and get the modified condition foreconomic growth

C

>=

=

0 = 0, K = 0

0 < S0 < S < 0, S > 0 ? > E C < 0

S0 = S

= 0, S = 0 < E

C > 0 = E

C = 0 S < S0 < S > 0, S < 0 < E C > 0 < E C > 0

S < S0 < Sx < 0, S > 0 ? > E C < 0

S0 = Sx = 0, S = 0 < E C > 0 = E C = 0

S0 > Sx > 0, S < 0 < E C > 0 < E C > 0


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with

= E

=

[ + A] + US + UC /E

. (24)

Since E > 0 and > 0 , it follows furthermore that

sgn() = sgn( ). (25)

Thus, is positive for low initial pollution and capital endowment(0 < S0 < S

and 0 < K0 < K). This implies that the equality condition

YK = + + > + which determines that the limit for optimalconsumption growth is higher than the standard Ramsey rule (without

pollution) would suggest. Given the declining marginal productivity ofcapital, it further follows that the steady-state level of capital K, whereYK(K

, L) = + + , is below the reference level K for the case withoutpollution, where YK(K

, L) = + . This conclusion can also be drawn forthe initial situation with S < S0 < S and K0 > K

, where < 0 and thereforeYK(K

, L) = + + < + = YK(K, L).9 Thus, when taking welfare losses

due to pollutant accumulation into account, we get a lower equilibriumstock of capital (K < K), below which the economy (consumption andcapital) would optimally grow, and above which it would have to shrinkalong the optimal trajectory.

However, as table 2 indicates, the solution is ambiguous for the fields thatshow a question mark because the sign of E is not clearly determined.For a further analysis, we define

YK (K , L) g(K ). (26)

This allows us to represent K as an inverse function of its marginalproductivity YK

K = g1(YK |L). (27)

Under consideration of equations (23) and (24), it follows

C

>=

S but S0 < S, the investment

guideline (29) should not be applied as a rule, because the optimal trajectoryis generally characterized by an increase in environmental quality (S < 0)and instantaneous consumption (C > 0) , or at least in social welfare(U > 0).13 One exception, however, could be the restoration of water quality

in a eutrophicated lake or the restoration of a contaminated soil that requires

13 This contrasts with the result of Hartwick (1997), who propagated a zero netinvestment rule to maintain utility constant over time for the case with high initiallevels of pollution.


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costly investments in the beginning, irrespective of whether S0 < S or S0 > S.In both cases, a Hartwick-type rule could be used to make the polluterspaying a fee equivalent to the environmental rent and to amortize the related

debt that results from an initial investment required for environmentalquality improvement in the long run.

6. ConclusionThe control of accumulative pollution in the environment is an importanttask that must be included in any sustainable development policy. From awelfare economic and efficiency oriented point of view, this will not imply astrong sustainability constraint of a constant or declining stock of pollution,such as propagated by Daly (1991) and investigated by Cheve (2000).

Rather, to avoid wasteful uses of scarce resources, sustainable developmentrequires an efficient resource allocation. This theoretically goes along withthe optimal control of accumulative pollutants. As shown in this paper,the latter involves three distinct types of problem that must be differentlyanalysed with regard to the implementation of environmental policy withina sustainable development framework.

The first one is related to the potential long-term optimum (saddlepoint) that can exist at a level of accumulated pollution where theecosystems assimilative capacity is completely destroyed. From a purelyefficiency-oriented perspective, it can be optimal to let the stock of

pollution further increase if the initial level is already in the domain wherethe assimilative capacity is zero. From a societal perspective, however,the normative question remains whether it can be desirable to let anecosystem die, and how the environmental quality could be restoredwith technological measures. This issue is not specifically investigated inthe present paper, and remains subject to further theoretical and appliedresearch.

The second problem area is characterized by a relatively high initialstock of pollution that is above the long-term optimum. In this case whichhas also been investigated by Hartwick (1997) and Cheve (2000) optimal

resource allocation results in a decline of the accumulated stock of pollutionand consumption growth. Apparently, this does not pose a problem forsustainable development since the optimal trajectory satisfies both criteriaof economic and environmental sustainability. Hence, a policy of optimalpollution control will in general contribute to sustainable development insituations with relatively high initial pollution and a working assimilationcapacity of the environment. But, it will require a relatively high share ofcapacity allocated to pollution abatement, which can subsequently declineover time and allow for an increase in the rate of consumption. Yet, thisdoes not imply a recommendation to first overpollute the environment

and then to start a sustainable development program, as this would notbe compatible with the minimum requirement of intertemporal efficiencyand would result in avoidable welfare losses. Rather, an optimal pollutioncontrol strategy may avoid such high stocks of pollution. This brings us tothe next case of investigation.


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776 Werner Hediger

The third domain of the optimal pollution control problem is the onethat, apart from global problems like the depletion of the ozone layer andthe enhanced greenhouse effect, might gain the least attention in the public,

since it refers to the situation with a relatively low initial stock of pollutionbelow the long-term optimum and with still low or even neglectable damagecosts. But, as shown in this paper, this situation refers to the most importantchallenge with regard to sustainable development because the optimalpollution control trajectory in this domain generally involves a continuedaccumulation of pollution and a decline of consumption. This is particularlythe case in wealthy economies with a high initial stock of capital. Incontrast, poorer economies might have the option of following a trajectoryof optimal capital accumulation that generates sufficient additional outputto increase anti-pollution efforts and, at the same time, facilitate growth in

consumption. Thus, despite continued pollution accumulation, the poor-country case can be sustainable from a weak sustainability perspectiveof non-declining welfare, whereas the one of the wealthy economy maybe environmentally and economically unsustainable. In the latter case,additional investment in man-made capital will be required to achieve asustainable development path. To this end, the economy might invest theenvironmental rents accruing from the use of natures assimilative capacityand increase its total productive capacity which is required for growthin consumption and pollution abatement activities. With the applicationof such a modified Hartwick-type investment rule, the economy could

maintain a constant flow of welfare, even in the case of continued pollutionaccumulation along an intertemporally optimal trajectory.

The determination of the above rent is a particular research challenge. Itrequires adequate knowledge of the assimilative capacity and the dynamicsof pollution accumulation, and it calls for the solution of the entireoptimal control problem in order to provide policy information aboutthe shadow prices of pollution and capital. Without this information, thesustainability of economic development and environmental policy cannotbe judged in the presence of continued accumulation of pollution, such asthe enhanced greenhouse effect. Moreover, the choice of a particular policy

and investment of environmental rents into reproducible capital involvesthe selection of different levels of instantaneous consumption and welfare.This is a matter of societal preferences for short-term and long-term benefits,respectively. For wealthier economies with a relatively high initial stock ofcapital, this selection problem can be formally solved with Chichilniskyssustainability criterion. The latter captures a priority given to immediatebut non-sustainable consumption, on one side, a priority for a constant andsustainable flow of welfare, on the other, and a set of mixed cases betweenthese two extremes. In the first case, net present value maximization ofsocial welfare is the decision criterion without investment of resource rents.In the other extreme, the above Hartwick rule must be applied and allenvironmental rents accruing from natures assimilative capacity mustbe invested into man-made capital in order to maintain a constant flowof instantaneous welfare. This would facilitate growth in consumptionsufficient to compensate for the rising disutility of pollution. However, theultimate choice is about the weighting factors between the two extremes,


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or in other words between traditional net present value maximization andweak sustainability, while the efficient use of scarce resources and thus theneed for optimal pollution control is a prerequisite.

References

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Barbier, E.B. and A. Markandya (1990), The conditions for achievingenvironmentally sustainable development, European Economic Review 34: 659669.

Becker, R.A. (1982), Intergenerational equity: the capital environment trade-off,Journal of Environmental Economics and Management 9: 165185.

Cesar, H. and A. de Zeeuw (1995), Sustainability and the greenhouse effect:

robustness analysis of the assimilation function, in C. Carraro and J. Filar (eds),Control and Game Theoretical Models of the Environment, Boston, Basel, Berlin:Birkhauser, pp. 2545.

Cheve, M. (2000), Irreversibility of pollution accumulation: new implications forsustainable endogenous growth, Environmental and Resource Economics 16: 93104.

Chiang, A.C. (1984), Fundamental Methods of Mathematical Economics, 3rd ed.,Singapore: McGraw-Hill.

Chichilnisky, G. (1977), Economic development and efficiency criteria in thesatisfaction of basic needs, Applied Mathematical Modelling 1: 290298.

Chichilnisky, G. (1996), An axiomatic approach to sustainable development, SocialChoice and Welfare 13: 231257.

Chichilnisky, G. (1997), What is sustainable development?, Land Economics 73: 467491.

Common, M. and C. Perrings (1992), Toward an ecological economics ofsustainability, Ecological Economics 6: 734.

Daly, H.E. (1972), In defense of a steady-state economy, American Journal ofAgricultural Economics 54: 945954.

Daly, H.E. (1974), The economics of the steady state, American Economic Review 64:1521.

Daly, H.E. (1991) Steady-State Economics, 2nd ed. (with new essays), Washington, DC:Island Press.

Elliott, D. and G. Yarrow (1977), The limits of cost-benefit analysis as a guide toenvironmental policy: a comment, Kyklos 30: 300309.Fiedler, K. (1994), Naturwissenschaftliche Grundlagen naturlicher Selbstreinigung-

sprozesse in Wasserressourcen, Zeitschrift f ur Umweltpolitik und Umweltrecht 17:323355 (in German).

Fiedler, K. (1997), The conditions for ecological sustainable development in thecontext of a double-limited selfpurification model of an aggregate water resource,Environmental and Resource Economics 9: 309321.

Forster, B.A. (1975), Optimal pollution control with a nonconstant exponential rateof decay, Journal of Environmental Economics and Management 2: 16.

Forster, B.A. (1977), On a one state variable optimal control problem: consumption-

pollution trade-offs, in J.D. Pitchford and S.J. Turnovsky (eds), Applications ofControl Theory to Economic Analysis, Amsterdam, New York, Oxford: North-Holland, pp. 3556.

Hartwick, J.M. (1977), Intergenerational equity and the investing of rents fromexhaustible resources, American Economic Review 67: 972974.

Hartwick, J.M. (1997), Paying down the environmental debt, Land Economics 73:508515.


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Heal, G. (2000), Valuing the Future, New York: Columbia University Press.Hediger, W. (1999), Reconciling weak and strong sustainability, International

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AppendixAs illustrated in figure 2, the two long-term optima (S, ) and (Sx, x) aredefined by the intersection of the = 0 and S = 0 demarcation curves.The former can be determined by setting = 0 in equation (9)

|=0 =

US + A

< 0 for + A > 0

for + A

0> 0 for + A < 0

(A1)

This partial equilibrium locus is zero for S = 0 and has a pole at thepollution level S = S#, where the pollution discount rate + A

(S) = 0.Moreover, it is discontinuous at the level S = S where the assimilative


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capacity is destroyed

(S)=0

=US(C , S)

+ A(S)> 0 and (S)

=0=

US(C , S)

< 0. (A2)

For S S#, the gradient of the = 0 curve is strictly negative and, forS# S S, it can change sign if + A

< AUS/USS

S

=0

=USS( + A

) USA

( + A)2

< 0= 0> 0

for + A >for + A =for + A 0= 0< 0

for 0 < S < Smfor S = Sm and S Sfor Sm < S S

. (A6)

The shape of the S = 0 curve is characterized by the A(S) curve. Starting

with S = 0 in the point

x = Uc( f

1(0), 0)

f( f1(0)), (A7)

the conditional shadow price |S=0 increases for low levels of pollution,reaches the maximum at S = Sm, declines afterwards and remains constantfor very high levels of pollution where the assimilative capacity isdestroyed.

The dynamics of state and costate variables are caused by any deviation

from the = 0 and S = 0 isoclines. For different conditions, the directionof the resulting adjustment processes is depicted by the arrows in fig-ure 2. Their orientation can be determined on the basis of economicreasoning, taking as price charged on pollutant emissions. For eachpoint above the S = 0 curve, the incentive from the hypothetical effluentcharge would not be sufficient to maintain a constant level of pollution.


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780 Werner Hediger

The latter would increase, which is indicated by arrows pointing to theright. The opposite is the case for any point below the S = 0 curve.

To determine the orientation of the adjustment process from points

deviating from the = 0 locus, we consider equations (9) and (A1), andget

>=

=

0 and + A > 0 , i.e., 0 < S < S#

For points below the = 0 locus i.e., < |=0 we get < 0 .This is indicated by the downward pointing arrows in figure 2. Theopposite applies for points above the = 0 curve.

(b) A> 0 and + A < 0 , i.e., S# < S < SThe orientation of the arrows changes because + A is negative.

(c) A = 0 , A = 0 and > 0 , i.e., S > SThe same logic applies as for case (a) after replacing + A by .

Additional remarks

The solution is ambiguous in S due to the discontinuity of A(S).

The optimal trajectory for S > Sx only exists if E(C) < G(Z), i.e., ifa substantial part of the economys productive capacity is devotedto pollution abatement and cleaning activities Z, and therefore notavailable for consumption.

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Hediger 2009 EDE