Efficiency of conservationist measures: an optimist viewpoint

ARTICLE IN PRESS

Journal of Environmental Economics and Management 46 (2003) 310–333

Efficiency of conservationist measures: an optimist viewpoint Reyer Gerlagha,
and Michiel A. Keyzerb a

IVM/FALW/VU, Institute for Environmental Studies, Vrije Universiteit, De Boelelaan 1087, Amsterdam 1081 HV, The Netherlands b SOW-VU, Centre for World Food Studies, Vrije Universiteit, Amsterdam, The Netherlands Received 25 July 2001; revised 3 July 2002

Abstract We consider an economy with a consumer good, capital, a natural resource that provides amenity values, and heterogeneous overlapping generations. We compare a benchmark grandfathering policy that ensures efficiency through privatization with a policy of enforced resource conservation. It is shown that (i) conservationist measures do not cause any Pareto inefficiency, irrespective of whether they pass a cost– benefit test. Moreover, it is shown that (ii) there exist Pareto optimal allocations that can only be reached through resource conservation, and not through competitive markets, irrespective of compensating income transfers. Finally, (iii) equivalence is demonstrated between strict resource conservation and nondictatorship of the present generations over future generations as formalized in Chichilnisky’s ‘sustainable welfare function’. The results are shown to hold in both a first-best and a second-best setting. r 2003 Elsevier Science (USA). All rights reserved. JEL classification: H23; Q28 Keywords: Conservation; Environmental policy; Intergenerational justice; OLG models; Sustainable welfare function; Sustainability

1. Introduction Debates on environmental sustainability often confront optimistists who expect that capital accumulation and technological progress will eventually compensate for natural resource depletion [29] with pessimists who are not so sure. The optimists hope that an endless accumulation of knowledge makes it possible for economic growth to continue and to produce ever-increasing wealth, while man-made goods substitute for the higher scarcity of environmental


Corresponding author. E-mail address: [email protected] (R. Gerlagh).

0095-0696/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0095-0696(02)00037-2

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goods. The associated policy, known as weak sustainability, advocates accumulation of capital, irrespective of its composition [15,30–32]. The pessimists fear that lasting and irreversible degradation of environmental resources persists [6,23] for which there is no compensation possible. Their concerns relate among others to a loss of biodiversity and a global climate change caused by anthropogenic emissions of greenhouse gases. Some of them argue that man-made and environmental goods are complements rather than substitutes [34] and that economic growth is incompatible with sustainable environmental resource use and eventually leads to economic collapse. In this view, strict conservationist measures are the preferred policy option [7,8]. Yet, the dominant view in environmental economics rejects strict conservationism as an inefficient approach and opts for market-based policies to arrive at an efficient solution, involving Pigovian taxes and privatization of resource ownership. But efficiency is not sufficient to guarantee sustainability [2], as became clear from the theoretical literature on environmental policies. This literature largely develops within the context of the dynastic, Ramsey-type model, with a finite number of infinitely lived consumers whose additively separable utility functions apply a common and fixed, positive discount rate. In such a model the exploitation of environmental resources may lead to a persistent decline in welfare, even if competitive markets are established for all these resources [24–26]. The assumed positive and constant discount rate in the Ramsey model has been a major source of dissatisfaction, and several arguments have been brought forward that justify a discount rate decreasing over time, giving higher weights to generations living in a distant future. Weitzman [35] argues that a negative environmental externality of production lowers the social rate of return on investments, as compared to the private returns. As the relative importance of the environmental amenities is likely to increase in the future, this causes the negative externality of investments to rise and the social rate of return on investments in man-made capital to fall. In another paper, Weitzman [36] shows that differences in time preferences associated with various possible future paths along which the economy might evolve, each with a given probability of occurrence, lead to a decreasing discount rate. Li and Lo¨fgren [21] recall that, even without uncertainty, when there are heterogeneous individuals with different time preferences, those with the lowest discount rate dominate the allocations chosen in the long-run future. Thus, both Weitzman [36] and Li and Lo¨fgren [21] make clear that, from the aggregated perspective, heterogeneous time preferences imply decreasing discount rates. Gerlagh and van der Zwaan [12] use an applied model to quantify possible changes in the discount rate resulting from expected demographic changes in the coming century. They find that the continuing increase in life expectancy may cause the interest rate to decrease over time. Chichilnisky [4,5], elaborating on Radner [27], chooses a fundamentally different approach. She shows that the commonly used additively separable dynastic welfare function with welfare weights decreasing to zero implies a dictatorship of the present generation’s interests over the interests of future generations. To reconcile future and present interests, she proposes appending to the standard welfare function a separate term representing the welfare of the generation living at infinity. The ‘sustainable welfare function’ treats present and future interests as two disjoint objectives, and it successfully prevents a dictatorship of the present interests over the future. However, whereas the dynastic model itself already is a normative construct whose institutional implementation within a market economy is not easily envisaged [33], the additional term proposed by Chichilnisky only makes it more difficult to interpret the arrangement in institutional terms.

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As compared to the Ramsey-type of models, overlapping generation (OLG) models permit to formulate environmental policies more directly, e.g. as (re-)distribution of ownership. Under specific conditions, efficiency of resource use can be effectuated through privatization, e.g. by grandfathering environmental resources to the present generation. Yet, Mourmouras [22] and Krautkraemer and Batina [20] show that privatization is not always sufficient to ensure sustainability, i.e. to prevent the utilities of the generations from falling indefinitely, and that more elaborate mechanisms are called for. As such a mechanism, Gerlagh and Keyzer [11] and Gerlagh and van der Zwaan [12] propose to create a trust fund as a specialized institution that receives the initial property rights over environmental resources securing both the present and future interests. This trust fund is shown to maintain dynamic efficiency and to promote resource conservation. However, its management is demanding since it relies on competitive markets and assumes perfect foresight of private agents. Strict conservationist policies that impose explicit exploitation constraints ensure sustainability and are far simpler to implement, compared to the more complex resource management rules that aim at a careful balancing of costs and benefits. The aim of the present paper is to identify conditions under which conservationist measures are also efficient, even if the outcomes of cost– benefit ratios indicate that extraction would be profitable. Thereby, this paper finds support for the concept of stewardship, as put forward by Brown [3]. Intuitively, strict conservationism would seem to conflict with efficiency as it imposes an inflexible constraint on production space. We show that conservationist measures can be understood as the institutional implementation of the non-dictatorship principle put forward by Chichilnisky, attributing positive weight to the future generations living at infinity. Thus, our analysis links the OLG and Ramsey analysis of sustainability. The paper proceeds as follows. Section 2 introduces the OLG model with man-made consumer goods, man-made capital, and an environmental resource that also provides both extractive and non-extractive amenity values. We prove the existence of equilibrium for this model for two policies, one based on competitive markets, the other based on strict resource conservation. Section 3 studies the welfare implications of both policies. Pareto efficiency is proven for both policies in the first-best model, and it is shown that resource conservation cannot be reduced to a competitive equilibrium supported by income transfers. Section 4 links our OLG analysis with common analyses based on Ramsey-type models. It studies the relationship between present and future generations and establishes the equivalence between resource conservation and Chichilnisky’s criterion of non-dictatorship of both present and future generations. Section 5 extends the analysis to a second-best economy. We show that strict resource conservation is also ‘second-best optimal’, in the sense that no Pareto improvement can be reached through relaxing the conservationist constraint, irrespective of regulation costs, distortionary taxes, and the use of compensating income transfers. Section 6 discusses the implications for applied analyses, such as in the field of climate change. Finally, Section 7 concludes.

2. Model specification The specification in this section adapts the usual OLG model under pure exchange with discrete time steps, tAT ¼ f1; y; Ng; to include man-made capital and a privately owned renewable

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resource that produces both amenity values and inputs for production. In addition, we allow for sustained economic growth through an increase in man-made capital and effective labor supply. The environmental resource combines features of a renewable resource, namely the potential to generate an unlimited stream of production factors and amenities valued by consumers, with those of an exhaustible resource, namely the irreversibility of resource degradation. Let the scalar variable st X0 be the environmental resource stock. It can be thought of as land that can be used for both extraction (e.g. open pit mining) and amenity use (recreation). We can also think of rain forests that can be cut for timber, or alternatively, that can be exploited in a sustainable manner, implying lower timber yields while preserving the amenities (e.g. biodiversity). Note that though we consider st as a scalar, our results carry over to an economy in which it is modeled as a vector, in which case it could stand for a rich variety of environmental goods. Extraction is considered a rival use of the resource while the amenity value, say, enjoying the beauty of the scenery, is nonrival. The distinction is important, because it introduces an additional substitutability requirement. Whereas substituting for the depletion of particular ores is a purely technological question that could possibly be resolved through capital accumulation and innovation, the substitution of natural amenities by man-made products also, and primarily, depends on consumer preferences. We assume that the resource can support a constant, sustainable flow of exploitation, rt ; proportional to the resource stock, rˆt ¼ ast ; for a constant a40; without being exhausted. However, if the use exceeds this threshold capacity rt 4ˆrt ; the resource is overexploited and degenerates irreversibly. For example, rain forests can support a modest level of sustainable timber harvesting, but they cannot recover from the clearing of land or the conversion to agricultural use, and the resulting loss of species is irreversible as well. Since resource exploitation has economic value, it will not fall short of the sustainable level, and we can write rt ¼ ast þ et ;

ð1Þ

stþ1 ¼ st  et ;

ð2Þ

for et X0; so that stþ1 pst ; damages to the resource are irreversible. The exhaustible resource has amenity value, bt X0: We follow Krautkraemer [19] and assume that the amenity levels are not affected by sustainable exploitation and can be expressed in terms of the stock level, as bt ¼ st ;

ð3Þ

where s1 is given, and without loss of generality, units are chosen such that s1 ¼ 1: Thus, bt is measured as an index, with maximum output bt ¼ 1: Resource owners receive payment for both the extractive and the amenity use. We assume that environmental services are exploited so as to maximize the value of the flow of output, X ðqt rt þ lt bt Þ; ð4Þ max t¼1;y;N

subject to (1)–(3), and given the initial resource stock s1 ; where qt is the price of rt ; and lt the price of the amenity bt :1 Let jt be the stock price of the resource at the beginning of period t; and 1

We use present prices and no separate deflator variable. A positive real interest rate is captured through a decrease in the price variables.

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qt ; jtþ1 ; lt the Lagrange multipliers (prices) associated to constraints (1), (2), and (3), respectively. We notice that, if the resource stock remains strictly positive for the infinite horizon, st 40; the resource stock price is equal to the net present value of the maximal nonexhaustive resource flow plus the amenities produced per unit of the resource stock:2 X jt ¼ ðaqt þ lt Þ; ð5Þ t¼t;y;N

provided the summation converges. Man-made capital, kt ; flow endowments (labor), lt ; measured in efficient labor units, and resource exploitation, rt ; enter the production of man-made goods, denoted by yt : ð6Þ yt ¼ Fðkt ; lt ; rt Þ: We assume linear homogeneity, so that we can write Fðkt ; lt ; rt Þ ¼ lt f ðkt =lt ; rt =lt Þ: In addition to the usual assumptions for the production function f ð:Þ; we demand that both the extracted resource and man-made capital are valuable, but not essential for production, whereas labor is essential for production. Moreover, for any extractive use of a natural resource, there exists a man-made substitute, which can be produced at finite costs, that is, the derivatives for rt are strictly positive and bounded. Conversely, we require that labor is a non-negligible production factor; we assume that the labor share in value added is bounded from below. Assumption 1. f ð:Þ is continuous, differentiable, strictly concave, and f ð:Þ satisfies: f ð0; 0Þ40; and for some 0obLB o1: kt fk ðkt =lt ; rt =lt Þ þ rt fr ðkt =lt ; rt =lt Þp 0ofr ðkt =lt ; rt =lt ÞoN; ð1  bLB Þlt f ðkt =lt ; rt =lt Þ: The assumption reflects an optimistic perspective, as there is always the possibility to rebuild the man-made capital stock, while man-made goods can substitute for the environmental resource, implying that there is no need for strict conservation from the producer’s perspective. Also, the assumption ensures a minimum level for real wages, wt =pt ; where pt is the price level for consumer goods, and wt the wage level: Lemma 1. Under Assumption 1, real wages, wt =pt ; are bounded away from zero, wt =pt 4bLB f ð0; 0Þ: Proof. We notice that wages are given by wt =pt ¼ f ðkt =lt ; rt =lt Þ  kt fk ðkt =lt ; rt =lt Þ þ rt fr ðkt =lt ; rt =lt Þ; and it follows immediately from the assumptions that & ð7Þ wt XbLB f ðkt =lt ; rt =lt ÞXbLB f ð0; 0Þ: The replacement and expansion of the capital stock require investments, it ; ktþ1 ¼ ð1  dÞkt þ it : The firms maximize the net present value of present and future profits X ðpt ðyt  it Þ  wt lt  qt rt Þ; max p ¼ t¼1;y;N 2

This follows from the first-order condition (A.2) in Appendix A.

ð8Þ ð9Þ

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subject to (6) and (8), where ðyt  it Þ is net output of the man-made good, and qt is the price of resource use. Now in equilibrium, it is possible that extraction is zero, et ¼ 0; when the exploitation price, qt ; falls short of the extraction price jtþ1 : The exploitation price, qt ¼ pt Fr ðkt ; lt ; rt Þ; represents the benefits of resource extraction in terms of the contribution of one unit of the extracted resource to the production of man-made goods,3 and jtþ1 denotes the value of future resource exploitation and amenity potentials (5). It is now possible to define the benefit–cost ðB=CÞ ratio of resource extraction as B=C ¼ qt =jtþ1 :

ð10Þ

The B=C-ratio takes the benefits of resource extraction in terms of the contribution of one unit of the extracted resource to the production of man-made goods, divided by the costs of resource extraction expressed as the total discounted resource plus amenity value, lost per unit of the extracted resource.4 If the net present value of future resource flows and amenities, jtþ1 ; is sufficiently high relative to the maximal productivity of the extracted resource qt ; B=Co1 and no extraction will take place, that is et ¼ 0:5 If the benefits and costs of resource extraction are equal, B=C ¼ 1; resource extraction et can take any positive value. Written in terms of complementarity, we have in equilibrium: qt =jtþ1 p1;

>et X0;

ð11Þ

where the >-sign states that equality holds if extraction is strictly positive. We compare two policies, the first aims at welfare maximization through grandfathering and competitive markets and does not prevent resource exhaustion, consistent with the equations above. We assume that the natural resource is held as private property similar to man-made capital. That is, the natural resource is ‘grandfathered’ to the first generation6 and, unless an explicit conservation policy is implemented, future generations will have to pay to prevent degradation. Owners receive payments from those who enjoy the amenities as well as from those who extract the natural resource. We note that since most environmental amenities are non-rival in use and the firm supplying the environmental services is a natural monopolist, it could be difficult in practice to implement grandfathering as a competitive equilibrium. Therefore, one might think of creating a publicly supervised corporation that holds the power to compel producers and consumers to pay for resource use and environmental services at competitive prices set by this agency. Thus, we do not exclude the need for public intervention to implement first-best allocations. In Section 5, we further analyze issues related to costs of environmental programs and price distortions. Here, we consider a first-best economy with perfect markets. The second policy implements strict conservationist measures, formalized by et ¼ 0: 3

ð12Þ

See (A.7) in Appendix A. Here we follow common practice in cost–benefit analysis by considering the effect of resource extraction rather than conservation, the costs of one being the benefits of the other. 5 This follows from the first-order condition (A.1) in Appendix A. 6 In actual practice, emission permits—which can be seen as extraction permits for a clean resource—are often grandfathered to the firms that are currently polluting. Since these firms are owned by the current old generation, this amounts to awarding the property rights over the environment to the first generation. 4

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1

Zero extraction (a)

time

B/C-ratio

B/C-ratio

Positive extraction

(b)

1

time

Fig. 1. (a) Time path of benefit–cost ratio (10) under grandfathering. (b) Time path of benefit–cost ratio (10) under strict conservation.

The conservationist policy also treats the resource as public property, that is, no generation derives any income from it, but it also exempts all generations from paying for the non-rival consumption of the resource amenity.7 The first generations do not own the resource, and future generations do no have to pay for its use.8 Under a strict conservationist policy, the benefit–cost ratio (10) can have any positive value (Fig. 1b). When the benefit–cost ratio exceeds unity, a positive exhaustion level would raise profits for the owners of the environmental resources, and this understanding will lead us to the question of whether a Pareto improvement is possible when relaxing the conservation constraint. Under the grandfathering policy, the B=C ratio equals unity when extraction is strictly positive, and no extraction will take place when the ratio is less than unity as stated in (11) (Fig. 1a).9 This completes the description of producer behavior. Next, we turn to consumer behavior. The consumers in the model form an infinite sequence of overlapping generations that live two (adult) periods and that are indexed by their date of birth. The first generation has only one period to live. Within a generation, we distinguish consumer groups with different utility functions by i ¼ 1; y; I so that consumers are denoted by a pair ðt; iÞ: For ease of exposition, all groups are assumed to be of the same size, and all consumers in the same generation have the same endowments. Generations maximize their lifetime utility derived t;i from rival consumption of the man-made consumer goods, denoted by the pair ðct;i t ; ctþ1 Þ; and t;i non-rival consumption of the resource amenity ðbt;i t ; btþ1 Þ: Their utility function satisfies the common assumptions: t;i t;i t;i Assumption 2. ui ðct;i t ; ctþ1 ; bt ; btþ1 Þ is assumed to be non-negative, differentiable, concave, bounded t;i t;i t;i from above, non-satiated in both ct;i t and ctþ1 ; and non-decreasing in ðbt ; btþ1 Þ: 7 This is not the only possibility as it would also be conceivable to distribute the property rights in fixed proportions, and distribute the proceeds from Lindahl taxes in these proportions. Clearly, administration is far easier with exemption but the private incentives to protect the resource are less since there is no private owner to safeguard his property. 8 To guarantee efficiency in the conservationist equilibrium, we assume that a small share 0odo1 of the resource is grandfathered to the first generation under the management rule that the resource should not be extracted. Since we can choose d arbitrarily small, we will omit it in all following equations. 9 We have drawn figures for the general case with a non-monotonic ratio between prices. We notice, however, that if consumption of the consumer good increases monotonically, then the B=C ratio will probably be monotonically decreasing.

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In the next section, where we analyze welfare paths, we extend the assumptions on the utility function to further specify the consumer’s value of the resource amenity and the substitutability between the resource amenity and the consumption of man-made goods. Young generations have to buy the available capital stock from the old ones. For them capital only serves as store of value and they fully depend on labor for their lifetime incomes. At t ¼ 1; the first generation, that has only one period to live, receives as income the proceeds from selling the capital to the young, plus its revenue from the labor it supplies in this period. Hence, this first generation enters without any debt. It maximizes utility subject to the one period budget constraint: 0;i 0;i 0;i 0;i 0;i maxfui ðc0;i 1 ; b1 Þ j p1 c1 þ m1 b1 pw1 l1 þ ðc1 k1 þ j1 s1 Þ=Ig;

ð13Þ

while later generations, t ¼ 1; y; N; solve t;i t;i t;i t;i t;i t;i t;i t;i t;i t;i t;i maxfui ðct;i t ; ctþ1 ; bt ; btþ1 Þ j pt ct þ ptþ1 ctþ1 þ mt bt þ mtþ1 btþ1 pwt lt þ wtþ1 ltþ1 g;

ð14Þ

where pt and wt denote the given prices of the (rival) man-made consumer goods and labor, respectively, mt;i t are the given Lindahl prices for non-rival consumption of the resource amenity of t;i consumer ðt; iÞ in period t; ltt;i and ltþ1 denote the flow endowments when young and old, respectively, and the members of the first generation share the value of the man-made capital stock, c1 k1 ; and the value of the natural resource j1 s1 : Under the conservationist policy, the resource is preserved as public property, bt ¼ 1; consumers do not have to pay for the non-rival use of the resource, and all old generations receive a share of the value of the sustainable resource exploitation level, aqt st : Budget constraints are adjusted accordingly. The first generation solves 0;i 0;i maxfui ðc0;i 1 ; bt Þ j p1 c1 pw1 l1 þ ðc1 k1 þ aq1 s1 Þ=Ig;

ð15Þ

while later generations, t ¼ 1; y; N; solve t;i t;i t;i t;i t;i maxfui ðct;i t ; ctþ1 ; bt ; btþ1 Þ j pt ct þ ptþ1 ctþ1 pwt lt þ wtþ1 ltþ1 þ aqtþ1 stþ1 =Ig;

ð16Þ

for constant bt ¼ st ¼ 1: All above assumptions fit in a standard OLG model. The next assumption introduces sustained growth in the production of man-made consumer goods, which is an uncommon feature for OLG economies. In line with modern growth theory, we take the sustained increase in consumption of man-made goods to follow from a persistent rise in labor productivity that feeds on technological innovation. In our model this is directly captured via an assumed increase in the supply of efficient labor units. Assumption 3. Flow endowments are positive and non-zero, evenly distributed over consumer groups i ¼ 1; y; I; and grow uniformly and without bound over time: ðltt1;i ; ltt;i Þ ¼ gt ðl0 ; ly Þ;

ð17Þ

for g41; where l0 40 vis-a`-vis ly 40 describes the distribution of labor endowments over the old and young, respectively.

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Having described producers’ and consumers’ behavior, we come to the equilibrium conditions. For every period t; there is a commodity balance for flow endowments, X ðltt1;i þ ltt;i Þ; ð18Þ lt ¼ i¼1;y;I

and for the man-made consumer goods, X ðct1;i þ ct;i t t Þ þ it ¼ yt :

ð19Þ

i¼1;y;I

Non-rivalry of the demand for the resource amenity is expressed through a pair of commodity balances in every period: ¼ bt;i bt1;i t t ¼ bt ;

ð20Þ

which indicates that in equilibrium contemporary consumers should agree about the amenity level. Associated to these balances are Lindahl prices paid for the amenity use, which should add up to the production price: X ðmt1;i þ mt;i ð21Þ lt ¼ t t Þ: i¼1;y;I

This completes the model description for the grandfathering and conservation scenario.10 We can now define equilibrium: Definition 1. A competitive equilibrium of model (1)–(21) is an intertemporal allocation supported by prices pt ; qt ; lt ; wt ; mt1;i ; mt;i t t ; ct ; jt ; for t ¼ 1; y; N; and for pt ; qt ; lt ; wt normalized on the simplex, in which producers choose yt ; lt ; rt ; bt that maximize profits subject to the technology constraints (1)–(3), (6)–(8), and to the policy constraint (12) in case a conservation t1;i t;i ; ct;i ; bt that maximize utility subject to policy is implemented, consumers choose ct1;i t t ; bt lifetime budget constraints (13) and (14), or subject to (15) and (16) in case of a resource conservation policy, and markets clear as in (18)–(20), while Lindahl prices satisfy (21). The economy satisfies all usual assumptions for existence of equilibrium, except that the consumption space becomes unbounded over time. We can now state and prove: Proposition 1. Under Assumptions 1–3, the grandfathering and conservation equilibrium of Definition 1 exist. The proof of the proposition is given in Appendix B; it is based on [1]. 10

ctt



In the remainder we omit, whenever convenient, the superscripts i when denoting the sums over i ¼ 1; y; I; e.g. P t;i i¼1;y;I ct :

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3. Welfare paths In an OLG economy, there are an infinite number of consumers and markets, and the first welfare theorem does not apply. The competitive markets will ensure efficient use of resources within a finite number of periods, but not over the infinite time horizon [1,28]. The problem can be understood as follows. If prices do not decrease sufficiently over time,11 then the net present value of the future consumption flow becomes unbounded and there is no welfare program that produces the competitive equilibrium as its solution. We can reinforce the first welfare theorem when we make additional assumptions that keep the value of future consumption bounded. And since the consumer’s budgets imply that the value of consumption is equal to the value of endowments, we can do so when we ensure that the value of future endowments decreases to zero. Alternatively, we may assure that the value of endowments of the first generation constitute a non-negligible part of total endowments [9]. For the economy described in this paper, this can be established if the first generation owns the natural resource, and if for every period, expenditures on the environmental resource make up a strictly positive share of total expenditures [10,11]. This condition is satisfied if there is at least one consumer group i in every generation, for which the resource amenity constitutes a non-negligible fraction of total expenditures: t;i i t;i t;i bt ui3 ðct;i t ; ctþ1 ; bt ; btþ1 Þ þ btþ1 u4 ðct ; ctþ1 ; bt ; btþ1 Þ t;i i t;i t;i i t;i t;i Xect;i t u1 ðct ; ctþ1 ; bt ; btþ1 Þ þ ectþ1 u2 ðct ; ctþ1 ; bt ; btþ1 Þ;

ð22Þ

for some e40; where u1 ; y; u4 denote the first till fourth derivative of the utility function, that is the prices for the man-made goods in periods t and t þ 1; and the Lindahl prices for the i environmental goods in periods t and t þ 1; respectively. The values for ct;i t u1 ð:Þ; t;i ctþ1 ui2 ð:Þ; bt ui3 ð:Þ; and btþ1 ui4 ð:Þ give the corresponding consumer expenditures. We actually prove efficiency under the slightly weaker requirement: Assumption 4. There is a consumer group i to whose utility the resource amenity makes a nonnegligible contribution: t;i i t;i t;i ui3 ðct;i t ; ctþ1 ; bt ; btþ1 Þ þ u4 ðct ; ctþ1 ; bt ; btþ1 Þ t;i i t;i t;i i t;i t;i Xect;i t u1 ðct ; ctþ1 ; bt ; btþ1 Þ þ ectþ1 u2 ðct ; ctþ1 ; bt ; btþ1 Þ;

ð23Þ

t;i for all 0oðct;i t ; ctþ1 Þ; 0oðbt ; btþ1 Þp1; for some fixed e40:

This assumption is weaker than condition (22), since it allows for a value share of the resource amenity that decreases to zero if the resource amenity level itself decreases to zero. The relevant property is that this assumption prevents the value of the natural resource stock to decrease to zero, when the consumption of man-made goods increases without bound. We can now state and prove:

11

Stated in other terms: if the real interest rate is too low or negative.

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Proposition 2. Under Assumptions 1–4, the grandfathering equilibrium is Pareto efficient. The proof of the proposition is given in Appendix B. Whereas, in a Ramsey-type model, the first welfare theorem applies and competitive equilibria are obviously Pareto efficient, in an OLG economy, inefficiency may occur. Assumption 4 on the non-negligibility of the contribution of the resource to utility is made to avoid this [9]. We show that under sustained economic growth, as ensured by Assumption 1, the conservation policy also results in a Pareto efficient allocation. This is noteworthy, since the conservation rule operates like a quota, and the common understanding is that quota result in inefficiency. Specifically, if the benefits of resource extraction exceed the costs, qt 4jtþ1 ;

ð24Þ

i.e. if extraction passes a cost–benefit test, B=C41 so that Eq. (11) is violated, profits could be made and the net present value of aggregated welfare could be increased by relaxing the strict conservation constraint (12). In this case, one would expect that positive extraction, if needed accompanied by compensating transfers, would increase the utility of all consumers. However, Proposition 3 below indicates that in our economy, this cost–benefit test is not decisive because a strict conservationist policy supports a dynamically efficient allocation, irrespective of the outcome of this cost–benefit test, basically because at infinity for any loss in amenity there are no man-made goods consumers could be compensated with. In fact, a similar problem can arise in a finite horizon context, as soon as one nature loving individual becomes satiated in man-made goods, and non-satiated in the amenity value, since there is no way to compensate his losses from resource extraction. Yet, within our optimistic perspective, we assume that man-made good can compensate for loss of natural amenities. All consumers are non-satiated in the man-made good (Assumption 2), but it is when time approaches infinity that consumers are confronted with a potential decrease in utility due to resource exhaustion. To analyze this mechanism, we follow [13] and define vi ðbÞ; the (long-term) environmental value function, as the supremum utility level that can be reached for given resource amenity b (recall that, without loss of generality, e.g. through a monotone transformation, uð:Þ is assumed to be bounded from above):12 t;i t;i t;i vi ðbÞ ¼ supfui ðct;i t ; ctþ1 ; b; bÞ j ct ; ctþ1 X0g:

ð25Þ

The supremum preserves non-decreasingness and concavity of the underlying utility function. Because of concavity, we have that vi ð:Þ is continuous on the interior 0obp1: Moreover, because of concavity and non-decreasingness, if vi ð:Þ is strictly increasing at b
; then it is strictly increasing for all 0obpb
: On the other hand, if vi ð:Þ is constant at b
; then it is maximal at b
and vi ðbÞ ¼ vi ðb
Þ for all bXb
: We can thus distinguish three cases: (i) constant: vi ðbÞ is constant for all 0obp1; (ii) increasing: vi ðbÞ is increasing and non-constant for all 0obp1; and (iii) increasing-to-constant: vi ðbÞ is non-constant for 0obob
; and constant for b
ob; for some b
o1: 12

Our analysis of the environmental value function vð:Þ extends the analysis by Gerlagh and van der Zwaan [13] as we have heterogeneous consumers.

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Three examples may provide some illustration of cases (i)–(iii). First, consider the class of utility functions with constant elasticity of substitution.13 Functions with elasticity greater or equal to unity belong the first category, whereas functions with elasticity less than unity belong to the second. Next, consider the class of additively separable utility functions of the form t;i c t;i t;i b c b ui ðct;i t ; ctþ1 ; bt ; btþ1 Þ ¼ Hðw ðct ; ctþ1 Þ þ w ðbt ; btþ1 ÞÞ; where w ð:Þ and w ð:Þ return scalars, and Hð:Þ is a monotonic transformation to keep utility uniformly bounded. This form is often used in integrated assessment models (IAMs) that study the effect of reductions in carbon emissions (e.g. [17]). The utility function falls in the first or in the second category depending on whether wc ð:Þ is t;i unbounded or bounded. More generally, the class of nested utility functions ui ðct;i t ; ctþ1 ; bt ; btþ1 Þ ¼ t;i c u˜ i ðwc ðct;i t ; ctþ1 Þ; bt ; btþ1 Þ for vector-valued and uniformly bounded w ð:Þ also belongs to the second category. Finally, to construct utility functions in the third category, we can take any utility function of category (i), and add a function wb ðbt ; btþ1 Þ that is constant at ðbt ; btþ1 Þ ¼ ð1; 1Þ and strictly increasing for low levels of ðbt ; btþ1 Þ: The following lemma makes use of the environmental value function vi ð:Þ and shows that both the grandfathering and the strict conservationist scenario define a long-term welfare level. Furthermore, this lemma enables us to characterize the long-term welfare effects of present day resource use in a straightforward and rigorous manner. Lemma 2. Under Assumptions 1–3, in both the grandfathering and strict conservationist equilibria, the amenity level converges to some limit point bN ; which depends on the scenario, and in case of t;i i multiple equilibria, on the equilibrium selected; utility levels ui ðct;i t ; ctþ1 ; bt ; btþ1 Þ converge to v ðbN Þ; as t-N: Proof. Convergence of bt follows immediately from the fact that in both the grandfathering and conservationist scenario, bt is a non-negative non-increasing sequence. We notice that, in general, the convergent level will differ between the scenarios as well as on the equilibrium selected, in case of multiplicity. The upper bound for utility levels follows from the definition of vð:Þ in (25). Furthermore, by Lemma 1, income increases without bound and any consumption vector of mant;i i made goods ðct;i t ; ctþ1 Þ becomes feasible for t sufficiently large. Hence, utility converges to v ðbN Þ; as t-N: & Whether consumption of man-made consumer goods can substitute for the amenity loss, so that resource extraction causes no irreversible loss, depends on the location of b1 ¼ 1 relative to b
: In case (i), vi ð:Þ is constant and a drop from level b1 has no effect. In case (ii), vi ð:Þ is increasing and non-constant, 1ob
; and there is a welfare loss for future generations. In case (iii), 0ob
o1 and there is no welfare loss if bt remains above b
; but there is a welfare loss if at certain period t; bt falls below b
: Using the terminology of [13], we may say that perfect long-term substitutability between man-made and environmental goods prevails if vi ð:Þ is constant, and poor long-term substitutability prevails if vi ð:Þ is strictly increasing. 13

Note that a CES function may need a bounded monotonic transformation to meet the boundedness requirement for uð:Þ:

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The value for b
can differ between consumer groups. The next assumption states that there is at least one group of consumers for whom an amenity loss can eventually not be compensated by growth of the man-made consumer good: Assumption 5. There is a consumer group i whose maximal potential welfare vi ðbÞ is strictly increasing in the resource amenity level for all 1ob: The assumption says that, in the long term, there is some consumer group for which perfect substitutability between man-made goods and environmental resources does not hold. We are now in a position to state and prove: Proposition 3. Under Assumptions 1–5, the strict conservationist equilibrium of Definition 2 is Pareto efficient. Though the formal proof is somewhat technical and given in Appendix B, the intuition is readily conveyed. The strict resource conservation constraint (12) is the only possible cause for inefficiency. Therefore, the question to answer is whether a Pareto improvement can be produced if we relax the constraint. This is not the case, since Lemma 2 says that future generations can reach the utility level vi ðb1 Þ; only if the resource constraint is maintained. In other words, the resource constraint is essential for the utility levels of future generations and its relaxation cannot produce a Pareto improvement. We remark that in the literature an intergenerational policy of income transfers is sometimes proposed as a means to realize resource conservation in a competitive equilibrium [18]. This would seem preferable since it avoids stringent environmental policies. Indeed, as stated in the Second Welfare Theorem, in a finite horizon economy, under standard convexity assumptions and in the absence of all market failures, any Pareto efficient allocation can be reached through income transfers. However, when the time horizon becomes infinite, the conditions of this theorem no longer apply. In the OLG model, income transfers can lead to dynamically inefficient outcomes [9].14 Moreover, the following proposition shows that there exist Pareto optimal allocations effectuated through conservationist measures that cannot be implemented in a competitive equilibrium without conservation measures, irrespective of the use of intergenerational income transfers.15 Also, reinvestments in man-made capital of revenues from resource extraction [15] cannot compensate future generations for the loss of amenities. Proposition 4. If, under the conservationist policy, the strict conservation constraint (12) has a positive shadow price, that is, if the benefit/cost ratio (11) exceeds unity for some period t; then there exists no competitive equilibrium without the conservation constraint, supported by income transfers, that decentralizes the same allocation.

14

An extensive discussion on dynamic efficiency, intergenerational transfers, and natural resources is given in Gerlagh [10, Chapter 3]. 15 See Appendix B for the proof.

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4. Avoiding dictatorship of present and future generations In this section, we will interpret our results in terms of weights given to the welfare of present and future generations. Resource exhaustion is often regarded as an indication of a preferential treatment of the early generations at the expense of their successors. This uneven treatment has been characterized as a dictatorship of the present over the future [5]. In this section, we show that a strict conservationist policy allows for an equal treatment of present and future generations, without a dictatorship of the present over the future, or conversely. Indeed, resource conservation can be used to balance the interests of the present and the future. Our analysis relates to Radner’s [27] and Chichilnisky’s [5] approach, based on welfare maximization in a dynastic (Ramsey) framework. Chichilnisky defines a ‘sustainable welfare function’ in which there is no dictatorship of the present over the future. The welfare function appends to a (uniformly bounded) additively separable dynastic welfare function a separate term representing the welfare function of the generation living at infinity. For our economy, the welfare objective in a Negishi format is given by16 N X I X t;i max W ¼ ait ui ðct;i ð26Þ t ; ctþ1 ; bt ; btþ1 Þ; t¼0

i¼1

ait ;

are decreasing to zero for t-N; and thereby these weights mainly represent the where weights interests of present generations. To construct a ‘sustainable welfare function’, the objective function has to become N X I I X X t;i t;i ait ui ðct;i ; c ; b ; b Þ þ bi lim ui ðct;i ð27Þ max W ¼ t t ; ctþ1 ; bt ; btþ1 Þ; tþ1 t tþ1 t¼0

i¼1

i¼1

t-N

where the weights bi represent the interests of future generations. We show below that adding the weights for the generations at infinity to the welfare objective as in (27) is equivalent to inserting the strict conservation constraint (12) within the welfare program as in (26). Through the equivalence, we demonstrate that conservationist measures can be understood as the institutional implementation of the additional term in the ‘sustainable welfare function’, representing nondictatorship of both the present and future generations. This links our OLG model to the Ramsey analysis of sustainability. Since the strict conservation constraint implies that bt ¼ b1 for all t (recall that b1 is given), we can rewrite (12) as b1 pbt :

ð28Þ

Since by Assumption 5, vi ðb1 Þ is strictly increasing for at least one consumer group i; these bounds can also be expressed as vi ðb1 Þpvi ðbt Þ;

ð29Þ

for all i; all t: In program (26) with fixed welfare weights a; let constraint (29) replace (12), and associate to this restriction a Lagrange multiplier Zit ; we note that this multiplier is set valued in view of the complementarity of the resource output across periods. Next, constraint (29) can be 16

See the proof of Proposition 2 in Appendix B.

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exchanged with a Lagrangean term Si;t Zit vi ðbt Þ in the welfare objective (26). The resulting program is N X I N X I X X t;i max W ¼ ait ui ðct;i ; c ; b ; b Þ þ Zit vi ðbt Þ; ð30Þ t tþ1 t tþ1 t¼0

i¼1

t¼2

i¼1

subject to (1)–(3), (6), (8), (18)–(20), where consumption of the first generation ‘0’ in period t ¼ 0 is given. Furthermore, in view of the output complementarity across periods, the second term could be added over any set of periods f2; y; Tg: Instead of (30), we may write N X I I N I X X X X i i i i t;i t;i at u ðct ; ctþ1 ; bt ; btþ1 Þ þ b v ðbT Þ þ Zit vi ðbt Þ; ð31Þ max W ¼ t¼0

i¼1

i¼1

t¼Tþ1

i¼1

where bi ¼

T X

Zit :

ð32Þ

t¼2

In a finite horizon model with last period T; it would suffice to impose constraint (29) for t ¼ T only. In our setting with an infinite time horizon, we can consider a sequence of programs restricting constraint (29) to periods t ¼ T; y; N; and letting T-N: In the limit, the strict conservation condition can be understood as an expression of attributing a positive welfare weight to the generation at infinity. N X I I X X t;i ait ui ðct;i ; c ; b ; b Þ þ bi lim vi ðbt Þ; ð33Þ max W ¼ t tþ1 t tþ1 t¼0

i¼1

i¼1

t-N

where bi ¼

N X

Zit :

ð34Þ

t¼2 t;i i Now, because bt is non-increasing and since utility ui ðct;i t ; ctþ1 ; bt ; btþ1 Þ converges to v ðbN Þ; as t-N (Lemma 2), the welfare programs (27) and (33) are equivalent. The fact that we can represent strict conservationism in terms of Chichilnisky’s sustainable welfare function raises the question as to whether it is preferable to deal with sustainability by explicit resource exploitation constraints, or by a proper objective function. Within the context of a first-best Ramsey model, one could argue that the inclusion in the objective function of a sustainability requirement through the second term in (27) is preferable, since this adheres more closely to the Ramsey framework. If the modeler takes interest in future welfare levels, then obviously Chichilnisky’s sustainable welfare function is the most direct route to take account of this interest.17 But, Proposition 4 suggests a counter argument. Since in an OLG economy income transfers between consumers are equivalent to adjusting welfare weights in a Ramsey framework, this proposition implies that the sustainable welfare function (27) specifies a program that cannot be reduced to a welfare program of type (26) without a conservation constraint, and thus program 17

The argument was brought forward by Geoffrey Heal when the first author asked for his comment on the equivalence between the resource conservation constraint and the sustainable welfare function.

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(27) is not decentralizable as a competitive equilibrium if not supported by explicit conservation constraints. Moreover, there is a second argument for relying on the explicit resource exploitation constraint. Despite the suggestion of allowing for an evenhanded treatment of the present and future interests in the sustainable welfare function, both terms in (27) have a different basis. The first term, with welfare weights decreasing to zero, fits in a descriptive perspective, reflecting a competitive equilibrium. The second term, however, with the welfare function for the generations living at infinity, is of an explicitly normative character and there is no objective way to decide on the form of the preferences, and the welfare weight attributed to it. Even though individual consumers may appreciate a sustainable welfare function, in a competitive market, they have no opportunities to signal their priorities in this respect. This illustrates that the use of a Ramsey framework does not preclude the need for government intervention.

5. Resource policies in a second-best world Clearly, the economy considered so far is perfectly undistorted. It has complete markets, perfect competition, no indirect taxes, and resource conservation measures are at no cost. Pessimists often argue that it is precisely because these causes of market failure that the economy is unable to deal with environmental problems. It is therefore natural to ask whether our results carry over to an economy with distortionary taxes and costly environmental programs. More specifically, we have shown that, in a first-best economy, cost–benefit analysis is not decisive as a project selection mechanism since it may incorrectly reject strictly conservationist measures that can be Pareto efficient. Is it possible to uphold the conclusion in a second-best economy, where protectionist measures can be costly, and which need distortionary taxes to finance resource conservation? For this, we modify the model by allowing for taxes on consumer goods, resource use, and labor, tct ; trt ; tlt ; all supposed to be less than unity. These taxes drive a wedge between prices observed by consumers and by producers. We use a superscript c to denote prices observed by consumers, and a superscript y to denote prices observed by producers. pct ¼ ð1 þ tyt Þpyt ;

qct ¼ ðt þ trt Þqyt

and wyt ¼ ð1 þ tlt Þwct :

ð35Þ

Profit maximization (4) and (9) and utility maximization (13)–(16) are adjusted accordingly by using consumer and producer-specific prices. Tax revenues are used for public consumption of man-made goods, denoted gt ; and government maintains budget neutrality tct pyt yt þ trt qt rt þ tlt wt lt ¼ pct gt : The commodity balance for the man-made goods (19) is also modified X ðct1;i þ ct;i t t Þ þ it þ gt ¼ yt :

ð36Þ ð37Þ

i¼1;y;I

Furthermore, we assume that public consumption is proportional to total output gt ¼ yyt ;

ð38Þ

for constant 0oyo1; and does not provide any utility. Hence, public consumption purely reflects the cost of regulation, which will differ between scenarios. Efficient resource use under a

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grandfathering policy requires that resource management includes the value of non-rival environmental amenities, and in practice, management and regulation costs can be substantial when we aim to exploit environmental resources in agreement with a cost–benefit evaluation. Since no regulation costs are involved if government follows a laissez-faire policy, an obvious question is whether the benefits of a more efficient resource use can outweigh the cost of resource regulation. Yet, if conservationist measures are to be effectuated by government it is costly as well. In short, it is not evident in which of both scenarios the regulation costs will be higher. Therefore, we allow y being scenario dependent. We can now define a second-best equilibrium. Definition 2. A second-best equilibrium of model (1)–(38) is an intertemporal allocation ; mt;i supported by prices pct ; pyt ; qct ; qyt ; wct ; wyt ; mt1;i t t ; ct ; jt ; for t ¼ 1; y; N; in which yt ; lt ; rt ; bt maximize profits subject to the technology constraints (1)–(3), (6)–(8), and to the policy constraint t1;i t;i (12) in case a conservation policy is implemented, ct1;i ; ct;i ; bt maximize utility subject to t t ; bt lifetime budget constraints (13) and (14) or subject to (15) and (16) in case of a resource conservation policy, markets clear as in (18), (19), (37), Lindahl prices satisfy (21), and prices are distorted as in (35), and government chooses its consumption in accordance with (36) and (38). Clearly, in the second-best economy, the notion of Pareto efficiency has to be replaced by another concept. Here we consider second-best efficient allocations, which are those equilibrium allocations that cannot be improved upon by means of changes in tax rates tct ; trt ; tlt or changes in the mechanism affecting the regulatory costs y: Thus, every equilibrium is characterized by the ; ct;i level of its flow variables: yt ; lt ; rt ; bt ; ct1;i t t : Let E denote the set of equilibria that are attainable, given the tax structure tct ; trt ; tlt and the regulatory cost y: Now we can prove that, within the set of attainable allocations, no Pareto improvement of the strict conservationist policy is possible through a relaxation of the conservation constraint, under any change in tax structure and regulatory costs. Proposition 5. Consider a second-best equilibrium with enforced strict resource conservation (12). Under Assumptions 1–3, and 5, there exists no second-best equilibrium with positive resource extraction, and possibly different tax structure and regulatory costs, that Pareto improves on the conservationist equilibrium. The proof is given in the appendix.

6. Application We also briefly investigate possible implications of our formal analysis for applied studies with so-called integrated assessment models (IAMs) on global environmental issues. In these applied studies, often, the irreversible deterioration of environmental resources comes out as the most efficient strategy of resource exploitation. Climate change is a case in point. The majority of the economic studies on climate change is directed towards determining the costs and benefits of

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different levels of emission reduction efforts, and concludes that no aggressive carbon dioxide mitigation regulations are called for, at least in the short run. These studies allow for a continued increase in carbon dioxide emissions since the net present value of the emission reduction costs appears to exceed the net present value of the benefits from prevention of damages caused by future changes in global average atmospheric temperature. This is not to say that damages due to climate change are unsubstantial in these studies. When it comes to biodiversity, vector-borne diseases, and other impacts on ecological systems irreversible and significant damages cannot be ruled out. However, because of discounting at a fixed rate the future benefits of resource conservation, their net present value cannot outweigh the present costs of conservation measures. The favoring of present benefits of resource extraction over future benefits of conservation has led many authors to characterize standard cost–benefit calculations as exponents of a myopic perspective on resource use. As an alternative they propose remedies that attribute higher weight to future generations, often in a Ramsey-type framework. Indeed, various examples have been given to illustrate that a ‘low’ or zero discount rate for future generations tends to promote natural resource conservation. However, as pointed out by Nordhaus [23], such high weights also appear to imply unrealistically high investments in both environmental resources and man-made capital. Another approach (e.g. [16]) suggests applying ‘normal’ or ‘high’ discount rates for investments in man-made capital, and low discount rates for decisions on resource conservation, but this results in an inefficient composition of the overall capital stock. Instead, Nordhaus [23] argues that if resource conservation is an objective in its own right, this should be recognized as such, and an explicit resource exploitation constraint should be added to the welfare program. Nordhaus’ support for a resource conservation constraint is noteworthy, since he maintains the standpoint that such a constraint is inefficient, that is, a Pareto superior solution exists. This paper has formulated conditions under which the common understanding, which says that conservationism for its own sake implies inefficiency, does not hold. This might be due to the typical assumption of most integrated assessment models, that environmental damages are reversible or, alternatively, that man-made goods constitute a proper substitute. Such assumptions imply that there is no need to worry for natural resource losses, since welfare will continue to increase, irrespective of present environmental damages. Consequently, the rejection of explicit resource conservation constraints as an inefficiency is justified. However, if we suppose that some environmental damages are irreversible and, for some consumers, no long-term perfect man-made substitutes exist, resource conservation may prove an appropriate objective in its own right. While inclusion of this insight in existing IAMs would require the specification of heterogeneous consumers and utility and production functions that are different from those used at present—and this may be too demanding—our results establish that conservation is not always inefficient, and would not necessarily come out as such in suitably adapted IAMs.

7. Conclusion We have described an economy with natural resources whose capacity to produce amenity values can be irreversibly damaged by over-exploitation. The economy has an equilibrium (Proposition 1) and supports sustained income growth (Lemma 1, Assumption 3). Efficient

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resource use that partly exhausts the resource will lower welfare levels for some consumers over an indefinite future period, as compared to a situation with an unspoiled environment (Lemma 2). Both a grandfathering policy and conservationist measures are compatible with Pareto efficiency (Propositions 2 and 3). In contrast with the Second Welfare Theorem, there exist Pareto optimal allocations that can be reached as an equilibrium through strict conservationist measures, but that cannot be reached by means of income transfers without strict conservationist measures (Proposition 4). It follows that a cost–benefit analysis is insufficient to decide on the rejection or acceptance of resource conservation. Even if a policy of competitive resource markets produces higher net present value then a policy of resource conservation, the latter may still produce a Pareto efficient allocation. This conclusion also holds in a second-best setting with distortionary taxes and costly resource regulations (Proposition 5). In other words, independently of whether the costs of resource conservation consist of resource rents foregone or of resource regulation costs, a cost–benefit analysis is indecisive for efficiency. We also investigated the relation between resource conservation and sustainability issues in Ramsey-type of models. Strict resource conservationist measures can be represented in a welfare program by appending a separate welfare function for the generation living at infinity. We thereby offer a practical consequence, strict resource conservation, of the non-dictatorship of the present generations, as proposed by Chichilnisky. The relationship between resource conservation and non-dictatorship of the present may be understood as follows. If present generations exploit environmental resources up to a level that these resources degenerate irreversibly, and if some future consumers cannot be compensated for the environmental loss, the current exploitation levels reduce the possibility set of welfare levels future generations can attain. In other words, by overexploiting the environmental resources, the present generations dictate the possible paths of future welfare levels. If non-dictatorship of the present requires that the present generations do not constrain the set of attainable future welfare levels, this requires the implementation of strict resource conservation. Our results depend on two essential properties of the model. First, environmental resources do not regenerate if overexploited, and second, in the far future there is some consumer group alive of nature lovers for whom there is no man-made substitute for any loss in these amenities. The properties remain valid if we consider market imperfections or introduce more pessimistic assumptions on technology whereby natural resource degradation reduces the capacity to produce man-made goods. The special feature emphasized here is that even if substitution is relatively strong, but not perfect for some consumers, technical progress eventually makes for these consumers a further increase in man-made goods insignificant relative to the resource amenity.

Acknowledgments The authors are grateful to Larry Karp, Richard Howarth, and two anonymous referees for their careful reading of and comments on two previous versions of this paper. R. Gerlagh also thanks the Dutch National Science Foundation (NWO) that funded part of the research under contract nr. 016.005.040.

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Appendix A. First-order conditions for equilibrium The first-order conditions for the owner of the environmental resource, maximizing the value of P output, t¼1;y;N ðqt rt þ lt bt Þ; subject to (1)–(3), read jtþ1 Xqt ;

>et X0;

jt pjtþ1 þ aqt þ lt ;

ðA:1Þ >st X0;

ðA:2Þ

where the >-sign refers to complementarity conditions: the constraint on the left is binding if the right-hand side is a strict inequality. Notice that the first-order condition for rt defines the resource exploitation price qt as the shadow price of the first constraint. Because of constant returns to scale in (2) and (3), for every period, the zero-profit condition holds jt st ¼ qt rt þ lt st þ jtþ1 stþ1 :

ðA:3Þ

This equation states that the value of the resource, jt st ; is equal to the value of its output, which also can be written out for the entire time horizon X ðqt rt þ lt bt Þ: ðA:4Þ jt st ¼ t¼t;y;N

The first-order conditions for the producer of man-made goods are pt Xctþ1 ;

>it X0;

ðA:5Þ

wt Xpt Fl ðkt ; lt ; rt Þ;

>lt X0;

ðA:6Þ

qt Xpt Fr ðkt ; lt ; rt Þ;

>rt X0;

ðA:7Þ

ct Xð1  dÞctþ1 þ pt Fk ðkt ; lt ; rt Þ;

>kt X0:

The associated zero-profit condition reads X ðpt ðyt  it Þ  wt lt  qt rt Þ ct kt ¼

ðA:8Þ

ðA:9Þ

t¼t;y;N

or recursively, ct kt ¼ pt ðyt  it Þ  wt lt  qt rt þ ctþ1 ktþ1 :

ðA:10Þ

Appendix B. Proof of propositions Proposition 1. Under Assumptions 1–3, the grandfathering and conservation equilibrium of Definition 1 exist.

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Proof. The existence proof of Balasko et al. [1] imposes two requirements. First, endowments should be bounded for every toN (they do not have to be bounded at t ¼ N). This property holds in our model. Second, all generations should be connected through positive payments to ensure irreducibility of the economy. We show that this condition is also satisfied. Non-satiation of utility in man-made goods ensures that in every period, there is at least one good with a strictly positive price. Then, the strictly positive slopes of F ð:Þ guarantees that all endowments have strictly positive price. Since endowments are non-zero for every generation in both periods of its life, income is strictly positive, and the endowments of generation t can be used to increase welfare of generations t  1 and t þ 1: Thus, the economy is irreducible. & Proposition 2. Under Assumptions 1–4, the grandfathering equilibrium is Pareto efficient. Proof. To prove efficiency, we show that the equilibrium solves a so-called Negishi welfare maximization program. For this, we take welfare weights ait equal to the inverse marginal utility of expenditure of consumer ðt; iÞ at the equilibrium solution: t;i i i i i i t;i t;i ðpct ; pctþ1 ; mt;i t ; mtþ1 ÞXat ðu1 ; u2 ; u3 ; u4 Þ>ðct ; ctþ1 ; bt ; btþ1 ÞX0:

ðB:1Þ

All welfare weights are strictly positive because of the irreducibility of the economy, which can be established along the same lines as in Proposition 1. We can now rewrite the model as a Negishi welfare program (see e.g. [14], Chapter 8) N X I X t;i ait ui ðct;i ðB:2Þ max W ¼ t ; ctþ1 ; bt ; btþ1 Þ; t¼0

i¼1

subject to constraints (1)–(3), (6), (8), (18)–(20), for given s1 ; and for given consumption of the first generation ‘0’ before the first period t ¼ 1: We start by noting that the equilibrium satisfies the first-order conditions of this program. Moreover, because the first generation owns the nonnegligible resource and future generations pay a positive part of their income for the resource amenity (Assumption 4), the first generation owns a non-negligible part of the total value of all endowments, and the cumulated value of consumption and endowments is bounded. Since the program is convex, the value W cannot exceed to total value of income, it is bounded as well, that is the welfare program is well defined and has a unique solution, that is fully characterized by the first-order conditions. From the existence of an OLG equilibrium follows that the program’s firstorder conditions are also well defined, with OLG equilibrium prices as (bounded) Lagrange multipliers, and the OLG equilibrium verifies these first-order conditions. Hence, the OLG equilibrium also solves the welfare program. & Proposition 3. Under Assumptions 1–5, the strict conservationist equilibrium of Definition 2 is Pareto efficient. Proof. We can follow the same Negishi procedure as for the previous proposition. Furthermore, to prove efficiency, we show that the strict conservation constraint (12) does not cause inefficiency, even though the cost–benefit criterion might warrant extraction, in the sense that (24) holds. We take welfare weights ait equal to the inverse marginal utility of expenditure of consumer ðt; iÞ at the equilibrium solution (B.1), and we rewrite the model as a Negishi welfare program

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(B.2) with (12) as additional constraint. Because the first generation owns a small, positive share d of the non-negligible resource (see footnote 8), the welfare program is bounded and has a unique solution, that is fully characterized by the first-order conditions. From the existence of an OLG equilibrium follows that the program’s first-order conditions are also well defined, with OLG equilibrium prices as (bounded) Lagrange multipliers, and the OLG equilibrium verifies these first-order conditions. Hence, the OLG equilibrium also solves the welfare program. It is now easy to show that no Pareto improvement is possible. We first check that any allocation that is a Pareto improvement must satisfy the strict conservation condition (12). Let an alternative allocation have positive extraction, et 40 for some t; which implies that there is a consumer group i for which utility converges to a strictly lower value vi ðbN Þ (Lemma 2) and there must be a period T and a consumer group i such that consumers ðt; iÞ has strictly lower utility in the alternative allocation for all t ¼ T; y; N: Such a lower utility level for some consumers contradicts a Pareto improvement, and thus, any Pareto improvement must have strict conservation. Finally, from the optimality of Program (26), the positiveness of welfare coefficients and the non-satiation of consumers follows that of all allocations with strict conservation, there is none that Pareto improves on the welfare maximizing allocation in program (26). & Proposition 4. If, under the conservationist policy, the strict conservation constraint (12) has a positive shadow price, that is, if the benefit/cost ratio (11) exceeds unity for some period t; then there exists no competitive equilibrium without the conservation constraint, supported by income transfers, that decentralizes the same allocation. Proof. We proceed by indirect demonstration. Suppose that the allocation of Proposition 3 can be implemented as an equilibrium supported by income transfers, such that the resource constraint (12) receives no shadow price. Using the same construction as for the proof of Proposition 3, we find the same welfare weights ait such that (B.1) holds. Now, it should be the case that the allocation satisfies all first-order conditions for a Negishi welfare program as in (B.2) subject to the same constraints but without (12), for given s1 ; and for given consumption of the first generation ‘0’ before the first period t ¼ 1: However, since constraint (12) has a positive shadow price in the proof of Proposition 3, we cannot omit this constraint without changing the welfare maximizing allocation. Thus, the welfare weights ait cannot support an equilibrium with strict conservation but with zero shadow price for the resource constraint (12). & Proposition 5. Consider a second-best equilibrium with enforced strict resource conservation (12). Under Assumptions 1–3, and 5, there exists no second-best equilibrium with positive resource extraction, and possibly different tax structure and regulatory costs, that Pareto improves on the conservationist equilibrium. Proof. Since the tax rates, the labor tax in particular, are less than unity, consumer income eventually rises to infinity and Lemma 2 applies. Hence, welfare of future generations will converge to the utility level vi ðb1 Þ; if and only if the resource is conserved, and any other equilibrium yields a lower utility for some consumer group i; from some period T onwards (see the second part of the proof of Proposition 3). &

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