Keywords: sustainability, intergenerational equity, minimal rights, viability. JEL Classification: Q01, Q32, O13, C61

This paper examines how the viability approach can be used to define sustainability goals. In an economic model with a non renewable natural resource, we define minimal rights to be guaranteed for all generations. These rights can include a minimal consumption (economic goal) and the preservation of natural resources (environmental goal). From a given economic state, it is possible to define the set of minimal rights that can be provided for all generation. To address the intergenerational equity issue, we propose to use a criterion that define the set of minimal rights that provide the maximal utility, in a Rawlsian perspective (Rawls, 1971). We describe how this criterion can be applied and computed, and discuss it with respect to usual criteria, including the maximin criterion, the Green Golden Rule, the Chichilnisky approach and the Mixed Bentham-Rawls criterion.

1 Introduction Solow (1993, p.167-168) claimed that "If the sustainability means anything more than a vague emotional commitment, it must require that something be conserved for the very long run. It is very important to understand what that thing is: I think it has to be a generalized capacity to produce economic well-being." [START_REF] Asheim | Justifying sustainability[END_REF] argue that sustainability requires the utility to be non decreasing along time, for equity concerns. [START_REF] Stavins | Interpreting sustainability in economic terms: dynamic efficiency plus intergenerational equity[END_REF] interpret sustainability as 'intergenerational equity' plus 'economic efficiency'. Intergenerational equity is defined by requiring a non decreasing utility through time, and economic efficiency is defined with respect to the neo-classical discounted utilitarian criterion. In these approaches, the social objective (sustainability of the utility) is not considered in the objective function (in the criterion to optimize) but as an added constraint to an economic criterion. This approach is criticized by [START_REF] Krautkraemer | Nonrenewable resource scarcity[END_REF], [START_REF] Pezzey | Progress and Problems in Economics of Sustainability[END_REF] and [START_REF] Cairns | Maximin, a direct approach to sustainability[END_REF] who argue that the objective function has to be defined in order to consider the sustainability issue, and especially intergenerational equity. From that point of view, the debate on sustainability becomes a debate on the social criterion to be optimized. [START_REF] Heal | Valuing the Future, Economic Theory and Sustainability[END_REF] examines the various criteria proposed to cope with the sustainability issue. Each criterion characterizes the optimal (sustainable) path and defines what is preserved for sustainability (if anything is preserved).

The most commonly used criterion is the intertemporal sum of discounted utilities max ∞ 0 ∆(t)U t dt.

(1)

This criterion, which has a solution only if the discount factor ∆(t) decreases towards zero at the infinite time 1 , is criticized because it does not take long term utility into account. According to [START_REF] Chichilnisky | An axiomatic approach to sustainable development[END_REF] this criterion is a dictatorship of the present.

Another proposed criterion [START_REF] Beltratti | The Green Golden Rule[END_REF] is inspired by the economic Golden Rule, and defines the highest indefinitely maintainable level of instantaneous utility. The criterion reads max lim

t→∞ U t (2)
This approach, called the green golden rule, does not take into account the present, and is called a "dictatorship of the future" by [START_REF] Chichilnisky | An axiomatic approach to sustainable development[END_REF]. [START_REF] Solow | Intergenerational Equity and Exhaustible Resources. Review of Economic Studies[END_REF] and [START_REF] Cairns | Maximin, a direct approach to sustainability[END_REF] address the sustainability issue by using the maximin criterion. This criterion defines the maximal sustainable level of utility, in the sense that it maximizes the utility of the poorest generation:

max min t U t (3)
This criterion has been criticized as it may lead to maintain the initial poverty by restricting the investment if the first generation is the poorest.

As mentioned by Beltratti et al. (1995, p.179), "an important task . . . lies in the analysis of criteria that combine maximization of discounted utility with elements related to the long run". In other words, sustainability issue relies on the definition of criteria that take into account both short and long run. [START_REF] Chichilnisky | An axiomatic approach to sustainable development[END_REF] develops such a study and provides the general form that the criterion must have:

max θ ∞ 0 ∆(t)U t dt + (1 -θ) lim t→∞ U t (4)
where ∆(t) is the discount factor. Nevertheless, the solution of criteria of this form is not easy to compute (See [START_REF] Heal | Valuing the Future, Economic Theory and Sustainability[END_REF]. Moreover, the criterion is not unique and depends on the choice of the parameter θ and of the discount factor ∆(t).2

In a recent contribution, [START_REF] Long | A mixed Bentham-Rawls Criterion for Intergenerational Equity[END_REF] proposes, in a complementary way of [START_REF] Chichilnisky | An axiomatic approach to sustainable development[END_REF], to mix a maximin criterion and the utilitarian discounted criterion. The criterion he proposes, called a "Mixed Bentham-Rawls criterion"

reads 3 max θ ∞ 0 ∆(t)U t dt + (1 -θ) min t U t . ( 5
)
This criterion is sensitive to the present, to the future and to the least advantaged generation.

As mentioned in the begining of this introduction, sustainability can be interpreted as the requirement to have something preserved in the long-run, in an intergenerational equity perspective. To discuss various criteria with respect to their implications 4 , the debate mainly emerges in the intertemporal resource allocation framework and aims at defining an equitable share of consumption and access to natural resources among generations. Criteria are more or less preservative with respect to the resource use and the level of consumption. The maximin criterion is the most explicit in the definition of what is preserved, as it targets the preservation of the utility along time.

In particular, the equity issue is mainly addressed with respect to the consumption of each generation, and the utility function depends on the instantaneous consumption c t , namely U (c t ). Each criterion leads to a infinite stream of consumption. The emergence of environmental issue has lead to to into account natural resource amenities, which emphasizes the preservation issue. In that perspective, the utilitarian approach can be completed: Natural resources x t can be a component of the utility function, U (c t , x t ). [START_REF] Heal | Valuing the Future, Economic Theory and Sustainability[END_REF] examines the implications of such an approach in the sustainability debate. Depending on the criteria, there is more or less consumption and thus "less or more" preservation of resource stocks.

The Brundtland report, Our Common Future (WCED, 1987), defined the sustainable development as a "development that meets the needs of the present without compromising the ability of future generations to meet their own needs". This explicit reference to the "needs" of the various generations can be a way to address the issue. An important challenging question is then the definition of the basic 3 [START_REF] Long | A mixed Bentham-Rawls Criterion for Intergenerational Equity[END_REF] develops the criterion in a discreet time framework and over a finite time horizon.

He also consider a constant discount rate. To be consistent within the presentation of the various criteria, we present the criterion a continuous time over an infinite horizon.

4 According to Dasgupta and Heal (1979, p.311), "it is legitimate to revise or criticize ethical norms in the light of their implications".

needs to be guaranteed for a sustainable development.

In the late 70's, the economic development issues where much more concerned with the intragenerational equity issue, and the development of poor countries.

It has been argued [START_REF] Chichilnisky | Economic Development and efficiency criteria in the satisfaction of basic needs[END_REF] that aggregated economic indicators like GDP were not able to encompass the development issue, as it neglects an important dimension of development: the satisfaction of basic needs. In fact, the scarcity of some basic goods can jeopardize economic development, or at least, modify the priority of the social planner: the maximization of Present Net Value of the economy may not be the primary goal of development.

From a more general point of view, one way to address the sustainability issue is to consider the satisfaction of basic needs, or equivalently, of minimal rights.

Among these basics needs, one can consider a minimal consumption and a minimal environmental quality. The issue is then to determine the way we define these minimal rights, in an equity concern. According to John Rawls' conception of justice (Rawls, 1971), the first requirement for equity is to choose the allocation of resources that provides the maximal number of minimal rights every one can enjoy. In a way similar to [START_REF] Chichilnisky | Economic Development and efficiency criteria in the satisfaction of basic needs[END_REF] that argues that economic development must be consistent with the attainment of adequate levels of per capita consumption of basic goods, a primary goal to achieve intergenerational equity can be define as a minimal per generation access to basic goods. The satisfaction of basic needs criterion aims at guaranteeing that each generation gets a minimal level of primary goods, while allowing some generation to have more than what is considered sufficient.

In this paper, we propose a methodological approach to determine minimal rights representing sustainability. Minimal rights are defined as a set of constraints (sustainability objectives) that must be respected forever, in an intergenerational equity perspective. Such constraints refer to basic needs of each generation. Chatterjee and Ravikumar (1999) study the implications of minimum consumption requirements on the rate of growth and the evolution of wealth distribution. We focus here on the implications of such a constraint on intertemporal consumption and investment paths, and on intergenerational equity, when the economy relies on the extraction of a non-renewable resource. At the same time, we consider a minimal resource constraint ensuring the preservation of a part of the natural resource.

Using the viability framework, we exhibit intertemporal paths that make it possible to satisfy a set of constraints representing minimal rights for all generations, including a minimal consumption level and a guaranteed resource stock to be preserved. It is a much more general way to address the minimal consumption and viability issue than found in [START_REF] Dawid | On sustainable growth and collapse: Optimal and adaptative paths[END_REF].

Our approach is however quite different from the sufficientarianism. We do not take a given level of minimal consumption and resource preservation, and then determine the intertemporal path that make it possible to satisfy these particular constraints in the future, like in [START_REF] Chichilnisky | Economic Development and efficiency criteria in the satisfaction of basic needs[END_REF]. On the contrary, we aim at defining the set of all the achievable goals from the initial state, i.e. the set of minimal consumption levels and preserved resource stock that can be guaranteed forever given the initial state. We then propose a criterion to choose within the set of all possible sustainability goals.

It leads to define a possible mathematical formalization to define minimal rights to be provided for each generation. In that framework, we do not optimize in-tertemporal paths but level of constraints to be guaranteed forever.

This approach is applied to an intertemporal resource allocation model with a manufactured capital stock and a non renewable natural resource. It is a canonical model severally used in the literature on sustainability [START_REF] Dasgupta | The Optimal Depletion of Exhaustible Resources[END_REF][START_REF] Solow | Intergenerational Equity and Exhaustible Resources. Review of Economic Studies[END_REF][START_REF] Heal | Valuing the Future, Economic Theory and Sustainability[END_REF], which allow a comparison of results with the existing criteria, including the maximin, the green golden rule, the Chichilnisky approach and the mixed Bentham-Rawls criterion.

The paper is organized as follows. In section 2, we define the set of minimal rights that can be guaranteed to all generations in a standard economic model with an exhaustible resource. We present the framework of analysis (2.1), the model (2.2), and the results (2.3). A criterion based on the maximization of minimal rights is proposed in 3, and applied to our analytical model. The results are interpreted in section 4. We discuss the implications of this criterion with respect to the results of other criteria (maximin, green golden rule, Chichilnisky's criterion, mixed Bentham-Rawls criterion). Section 5 concludes.

2 On the possibility to achieve some minimal rights

In this section, we develop a framework to define the set of sustainability goals that can be achieved by an economy, given initial endowments. This framework is based on an extension of the viable control framework [START_REF] Aubin | Viability Theory[END_REF].

Viability framework and minimal rights

If we consider that sustainability encompasses several objectives from different nature (ecological, economic and social) it may be difficult to adopt an optimization approach and define efficiency in a multicriteria context. Another possibility is to adopt the viability approach, which studies the effectiveness of dynamic intertemporal paths, with respect to a set of objectives defined by constraints. In the viability framework, the constraint must be respected forever, inducing an intergenerational equity perspective. The constraints can thus be interpreted as minimal rights to be guaranteed to any generation.

The purpose of the viability approach [START_REF] Aubin | Viability Theory[END_REF] is to study the behavior of dynamic systems subject to constraints. The desired characteristics for the system are defined by a set of constraints, and the viability approach examines the conditions for these constraints to be satisfied forever, in a dynamic perspective.

The first step of the analysis is to define the states of the system from which at least one intertemporal path that satisfies the constraints forever starts. The set of such states is called the viability kernel of the problem. By definition, from any state in the viability kernel, there is at least one admissible decision that keeps the trajectory in the kernel. Thus, from any state outside the viability kernel, there are no decisions that make it possible to respect the constraints forever.

Whatever the decisions, the dynamics of the system will violate the constraint in a finite time.7 Viable paths are paths that stay in the kernel. The second step of the analysis is the definition of the viable decisions associated with a viable state.

Not all decisions may be viable and some of the admissible decisions may lead the system outside the viability kernel.

It means that if the initial state of the economy is not within the viability kernel, the initial endowments of the economy do not allow the sustainability goals to be achieved. The objectives (constraints level) should be diminished to be fulfilled. In that context, it is possible to extend the viability analysis by defining, for a given initial state, the set of reachable goals.

In our context, the constraints include a minimal consumption c and a minimal resource stock x to be preserved. We will examine the set of goals (c, x) that are reachable from the initial capital stocks of the economy. Doing that, we define the set of minimal rights that can be guaranteed to any generation.

As mentioned in the introduction, a particular problem one can face when trying to achieve some particular goals is that the defined goals may not be reachable from initial conditions through feasible paths of the system. In that case, the goals can be reached in some future date. This issue is discussed in section 4.3.

Model

To make comparisons with other approaches easier, we develop the proposed approach for a standard model. We consider an economy with infinitely many generations, and make the simplifying assumption that each generation is composed by an unique representative agent.8 A non renewable resource x t is extracted (r t ) and used with capital k t to produce capital. The production function is denoted f (k t , r t ). The capital can be ever consumed or invested. Such a model has been studied in [START_REF] Dasgupta | The Optimal Depletion of Exhaustible Resources[END_REF]; [START_REF] Solow | Intergenerational Equity and Exhaustible Resources. Review of Economic Studies[END_REF], and is an useful stylized model for addressing the sustainability issue: the intertemporal allocation of the exhaustible resource, and the stream of consumption through time make intertemporal comparisons possible, in an intergenerational equity perspective.

The dynamics are

k = k α t r β t -c t , (6) ẋ
= -r t . (7)
We take a Cobb-Douglas production function which is argued to be the most interesting case for studying sustainability, as mentioned by [START_REF] Dasgupta | Economic Theory and Exhaustible Resources[END_REF].9

In that model, we wonder what are the level of guaranteed consumption c and preserved resource stock x that are compatible with the initial state (k 0 , x 0 ). We are thus considering the following viability constraints

c t ≥ c, (8)
x t ≥ x.

(9)

Admissible sustainability objectives for a given initial state

In order to define the minimal rights to be guaranteed for a sustainable development, we need to define the set of all possible rights, i.e. the rights that are compatible with the initial state of the economy and with the studied dynamics.

We want to define the set of S = (c, x) there exist paths (k(.), x(.)) starting from (k 0 , x 0 ) that satisfy the constraints ( 8) and ( 9) .

(10) Martinet and Doyen (2007) describe the relationship between consumption and preservation goals in the model we consider. They examine the conditions for a minimal consumption c to be guaranteed when there is also a constraint on the preservation of the resource x. It means that they determine if there are intertemporal decisions (c(.), r(.)) that make it possible to respect the constraints c t ≥ c and x t ≥ x for all t ≥ 0, from a given initial state (k 0 , x 0 ).

As a consequences it is possible to determine which levels of guaranteed consumption c and preserved resource stock x are reachable from the initial state.

An easy way to define levels of consumption that can be guaranteed along with the preservation of a part of the resource stock is to consider the maximal level of consumption that can be sustained, under a resource preservation constraint.

c + (k 0 , x 0 , x) = max c given (k 0 , x 0 ), there exists (c(.), r(.))

such that ∀t ≥ 0, c t ≥ c and x t ≥ x . ( 11
)
This approach is linked with the maximin approach [START_REF] Solow | Intergenerational Equity and Exhaustible Resources. Review of Economic Studies[END_REF][START_REF] Cairns | Maximin, a direct approach to sustainability[END_REF]. In the classical model we study, the maximal sustainable level of consumption, given the preservation goal x, is defined as follows.

Result 1 Consider a Cobb-Douglas function with α > β. The maximal sustainable consumption from initial state (k 0 , x 0 ), associated with the guaranteed stock

x is c + (k 0 , x 0 , x) = (1 -β) (x 0 -x)(α -β) β 1-β k α-β 1-β 0 . ( 12
)
This result is proven in Martinet and Doyen (2007, Proposition 3), but an easier way to insure this result is to note that equation ( 12) leads exactly to the maximin result of Solow (1974, p.39) for x = 0. Solow's result is thus extended to take into account the resource constraint.

Fig. 1 represents equation ( 12). It represents all consumption and resource preservation levels that can be guaranteed to all generation. Eq. ( 12) is the upper bound of the set of all reachable goals. We thus know the set of all reachable goals for sustainability. Any inner pair (c, x) such that c ≤ c + (x) is feasible.10 Note that on the border, a rise of resource preservation implies a fall of sustainable consumption.

A question that arises now is to define sustainability goals in order to satisfy some intergenerational equity.

Choosing among sustainability goals

In this section we introduce an objective function to define which sustainability rights should be chosen among the possible set S.

An objective function related to minimal rights

Consider that minimal rights are based on a minimal consumption c and a minimal resource stock x. We define social preferences among the minimal rights to be guaranteed.

Definition 1 A minimal right utility function U (c, x) represents the preferences on minimal rights. We postulate

• c ∈ R + 0 ; x ∈ R + 0 • U (c, x) : R + 0 × R + 0 → R + 0 • U c ≥ 0 ; U x ≥ 0 • U c,x ≤ 0.
This utility function represents the preferences of a virtual representative agent placed under the veil of ignorance, and that should choose minimal rights to be guaranteed to any generation in a sustainability concern.

We propose to choose the pair of minimal rights (c, x) that maximizes this utility function. Hence, our approach is not to maximize some intertemporal welfare but to maximize the utility associated with some minimal rights to be guaranteed for all generations.

Definition 2 We define the Minimal Rights criterion as

max c,x U (c, x) (13)
subject to

(k 0 , x 0 ) given (14) c t ≥ c (15) x t ≥ x (16) k = k α t r β t -c t (17) ẋ = -r t (18)

Results

Using the result 1, this problem is equivalent to the maximization of this criterion among the possible pairs (c, x), i.e.

max c,x U (c, x) s.t. 0 ≤ x (19) 0 ≤ c (20) 0 ≤ x 0 -x (21) 0 ≤ c + (x) -c (22)
This problem is a classical static optimization problem under inequality constraints [START_REF] Léonard | Optimal control theory and static optimization in economics[END_REF].11

To solve this problem, we define the following functional form

φ(µ 1 , µ 2 , µ 3 , µ 4 , c, x) = U (c, x) + µ 1 x + µ 2 c + µ 3 (x 0 -x) + µ 4 (c + (x) -c) (23)
where the µ i are the dual variables of the problem.

lem are 12 φ µ 1 = x ≥ 0, µ 1 ≥ 0, µ 1 x = 0 (24) φ µ 2 = c ≥ 0, µ 2 ≥ 0, µ 2 c = 0 (25) φ µ 3 = x 0 -x ≥ 0, µ 3 ≥ 0, µ 3 (x 0 -x) = 0 (26) φ µ 4 = c + (x) -c ≥ 0, µ 4 ≥ 0, µ 4 (c + (x) -c) = 0 (27) φ x = U x + µ 1 + µ 4 dc + (x) dx ≤ 0, x ≥ 0, x U x + µ 1 + µ 4 dc + (x) dx = 0 (28) φ c = U c + µ 2 -µ 4 ≤ 0, c ≥ 0, c(U c + µ 2 -µ 4 ) = 0 (29)

Strictly positive solutions

First assume that, at the optimum, both the optimization variables x and c are strictly positives. From eq. ( 24) and ( 25), we get µ 1 = µ 2 = 0. Moreover, if consumption is positive, the preserved resource stock will be lower than the initial stock x 0 . Eq. ( 26) then leads to µ 3 = 0. We thus get a system from equations ( 27), ( 28) and ( 29), in which there are three equations and three variables.

µ 4 (c + (x) -c) = 0 (30) x U x + µ 4 dc + (x) dx = 0 (31) c(U c -µ 4 ) = 0 (32)
As we have assumed that c = 0, eq. ( 32) leads to µ 4 = U c . We thus get from eq. ( 30) and ( 31) the conditions

c = c + (x) (33) dc + (x) dx = - U x U c (34)
It leads to the following result Corner solution x = 0

dc + (x) dx = - U x U c + (x) (35)
Assume now that x = 0. It implies that µ 3 = 0 (from eq. 26).

If c = 0, eq. ( 27) would require µ 4 = 0. But it is in contradiction with relation (29) which requires U c + µ 2 -µ 4 ≤ 0. Thus, we have µ 4 > 0 and c = c + (0), from eq. ( 27).

As c = 0, we get µ 2 = 0 from eq. ( 25). We then get µ 4 = U c from eq.( 29).

Finally, the inequality condition (28) requires

U x |(x=0) + µ 1 + U c dc + (x) dx |(x=0) ≤ 0 (36)
This equation can be expressed with respect to µ 1

µ 1 ≤ -U x |(x=0) -U c dc + (x) dx |(x=0) (37)
As

µ 1 ≥ 0, it is possible only if -U x |(x=0) -U c dc + (x) dx |(x=0) ≥ 0, or equivalently if 0 ≤ U x U c ≤ - dc + (x) dx |(x=0) (38)
It means that such a corner solution is possible if the slope of the utility function is smaller than that of the function c + (x). Such a solution is then possible only if the marginal utility of preservation for a nil resource stock is small with respect to the marginal utility of consumption. It implies that U x |(x=0) < ∞. Corner solution c = 0

We now turn toward the other case : c = 0. The inequality from eq. ( 29) implies

U c + µ 2 -µ 4 ≤ 0 =⇒ µ 4 ≥ U c |(c=0) + µ 2 > 0 (39) It is only possible if U c |(c=0) < ∞.
As µ 4 > 0, we know from eq. ( 27) that c = c + (x) which means, as c = 0, that x = x 0 . We have µ 1 = 0 from eq. ( 24). Thus, x > 0 requires from eq. ( 28) that

µ 4 = - U x |(x=x 0 ) dc + (x) dx |(x=x 0 ) (40)
Combining this condition with eq. ( 39), we get

0 ≤ µ 2 ≤ -U c |(c=0) - U x |(x=x 0 ) dc + (x) dx |(x=x 0 ) (41)
We thus have a condition on the marginal utilities in (c = 0, x = x 0 ):

0 ≤ - dc + (x) dx ≤ U x U c (42)
The slope of the utility function must be greater than that of the function c + (x).

In particular, the marginal utility of consumption, when consumption is zero, must be finite (and small with respect to the marginal utility of preservation).

Interpretation

This approach has methodological intersections with the maximin approach [START_REF] Solow | Intergenerational Equity and Exhaustible Resources. Review of Economic Studies[END_REF][START_REF] Cairns | Maximin, a direct approach to sustainability[END_REF], in that the efficient growth paths are solutions of a constrained optimization problem: maximin under constraint Whatever the case, the optimal solution always satisfies c = c + (x). The only intertemporal path that satisfies the optimal minimal rights is then a maximin under constraints and is efficient from an economic point of view. As a consequence, optimal minimal rights with respect to our criterion entail efficient resource use.

The only intertemporal path that satisfies the optimal constraint is the same path as a maximin path (on consumption) under a natural resource constraint fixed at the level x * .

Note that on the boundary of the set S, there is only one intertemporal path that satisfies the constraints. As this path is the solution of a maximin problem under constraint, it satisfies both Hartwick's and Hotelling's rules: the intertemporal use of the natural resource and the investment are optimal.

Our approach can then be interpreted as the definition of optimal constraint levels for maximin under constraint problems.

The level of the resource preservation will depend on the marginal utilities of consumption and resource stock. If we assume that the consumption marginal utility for a nil consumption U c|(c=0) is infinite, and that that of the resource for a nil resource stock U x|( x=0) is also infinite, the result will satisfy c > 0 and x > 0.

As the solution is in any case on the boundary of the set S, the solution is "pareto efficient" as it is not possible to increase one of the minimal rights without decreasing the other.

Discussion

In this section, we discuss our result with respect to existing criteria. We emphasize the links between our approach and the maximin framework. We then discuss the interest of our approach with respect to the "green golden rule" [START_REF] Beltratti | The Green Golden Rule[END_REF] and the Chichilnisky's criterion [START_REF] Chichilnisky | An axiomatic approach to sustainable development[END_REF]. We also refer to the Mixed Bentham-Rawls criterion recently proposed by [START_REF] Long | A mixed Bentham-Rawls Criterion for Intergenerational Equity[END_REF].

Minimal rights and sustainability criteria

As mentioned in the introduction, various criteria have been proposed to deal with the sustainability issue. In order to take into account both intertemporal equity issue and preservation issue, [START_REF] Heal | Valuing the Future, Economic Theory and Sustainability[END_REF] discusses the results of these criteria using an utility function that takes into account both the consumption c t and the resource stock x t ; i.e. U (c t , x t ). The natural resource thus has an amenity value.

Each criterion leads to the preservation of a part of the natural resource, and an intertemporal consumption path.

According to [START_REF] Heal | Valuing the Future, Economic Theory and Sustainability[END_REF], the criteria are more or less conservative with respect to the resource. From the less to the more conservative, we have the discounted utilitarian criterion, the Chichilnisky criterion and the green golden rule.

For the maximin criterion, it depends on the way the problem is defined and on the initial configuration.

We propose here a short comparison of the results of these criterion and ours.

Discounted utilitarian criterion

The usual economic criterion is the neoclassical sum of discounted utilities. max c(.),r(.)

∞ 0 ∆(t)U (c t , x t )dt (43)
Heal (1998) analyses the solution of this criterion in a production-consumption model with a non renewable resource, when utility depends both on the consumption and the resource stock. This analysis was started by [START_REF] Krautkraemer | Optimal Growth, Resource Amenities and the Preservation of Natural Environments[END_REF].

Unfortunately, usual mathematical tools don't make it possible to find out an explicit solution. These kind of problems are not easy to solve.

The maximin The maximin criterion maximizes the utility of the poorest generation. Within the sustainability debate, [START_REF] Cairns | Maximin, a direct approach to sustainability[END_REF] propose two ways to consider the environment in a maximin problem. The first one is to introduce the resource stock in the utility function. 13 The criterion reads max c(.),r(.)

min t U (c t , x t ) (44)
Such a criterion implies an increasing consumption like the resource stock decreases. As far as we can say, the optimal solution has not been computed in our model.

The second possible approach is to consider a maximin under natural resource constraint.14 A question then arises: at which level should we set the constraint?

Our approach proposes a framework to set the level of such sustainability constraints.

The Green Golden Rule The "green golden rule" criterion consists in maximizing the utility attained by the policy at an infinite time, when utility depends both on the consumption and the resource stock, i.e. U (c t , x t ). The criterion is of the form max lim t→∞ U (c t , x t ).

The interpretation of this criterion from a sustainability point of view is studied in [START_REF] Heal | Valuing the Future, Economic Theory and Sustainability[END_REF]. The solution of the criterion is given for two particular cases of a production-consumption model, depending on the characteristics of the production function f (k, r):

• with a non-renewable resource necessary for production, in the sense that f (k, 0) = 0, "the maximum sustainable utility level is attained by conserving the entire stock and consuming none of the produced good" (Heal, 1998, proposition 35, p.135).

• with a non-renewable resource that is not necessary to production, i.e. f (k, 0) > 0, the maximal sustainable utility level is attained without extracting any part of the stock (Heal, 1998, proposition 36, p.136).

The Cobb-Douglas case is not studied, and thus, the substitutability issue is not addressed. 15It thus seems that this criterion implies the preservation of the initial resource stock, whatever is the technological configuration. We will show that it is not always true by studying the particular case of a Cobb-Douglas production function.

In that case, we will show that it is not optimal to preserve the whole resource stock unless the marginal utility of consumption for nil consumption is small with respect to the marginal utility of the resource preservation.

Proposition 1 If U c|(c=0) ≥ - Ux |(x=x 0 ) dc + (x) dx |(x=x 0 )
, then

argmax lim t→∞ U (c t , x t ) = (c t = 0, x t = x 0 ) (45)
Proof -To prove proposition 1, we demonstrate that the stationary state (no resource extraction and no consumption of the capital good) is dominated, with respect to the GGR criterion, by the solution of our criterion.

It is straightforward, using the result of eq. ( 42) and the solution of the optimization problem 13 in the case c = 0.

Our criterion can be proposed to maximize the utility that is reached at an infinite time when the "green golden rule" does not lead to a solution. As the resource is non-renewable and its harvest irreversible, the resource stock is decreasing along time. Along the optimal path of the "minimal rights" criterion, the consumption is constant and the resource stock decreases towards the minimal value. The utility of the minimal rights (the one that has been maximized) is the utility at an infinite time. In fact, we applied some "infinite utility criterion" on a reduced set of paths: maximin under constraints. In our particular model, the criterion can be expressed as lim t→∞ U (c t , x t ) = U (c, x). In fact, as the non renewable resource stock is necessarily decreasing, and if the consumption can be smoothed along time in order to increase the level of the minimal consumption, the utility level that is sustained in the long-run is the utility associated with the minimal rights.

Doing so, one takes into account a constant consumption path, the resource preservation, and the utility of the very-long run. Thus, we combine a maximin approach under constraint and determine the constraint level with a criterion closed to the green golden rule.

The Chichilnisky's approach [START_REF] Chichilnisky | An axiomatic approach to sustainable development[END_REF] has developed a criterion that makes it possible to avoid both the "dictatorship of the present" and the "dictatorship of the future". The criterion is

W = θ ∞ 0 ∆(t)U (c t , x t )dt + (1 -θ) lim t→∞ U (c t , x t ) (46)
where ∆(t) is the discount factor that can be exponentially decreasing or hyperbolic. According to Heal (1998, p.69), "Intuitively, [the] second term [of Chichilnisky's criterion] reflects the sustainable utility level attained by a policy".

The Chichilnisky criterion is a convex combination of the utilitarian discounted criterion and the green golden rule criterion. However, as noted by [START_REF] Heal | Valuing the Future, Economic Theory and Sustainability[END_REF] and [START_REF] Long | A mixed Bentham-Rawls Criterion for Intergenerational Equity[END_REF], the Chichilnisky criterion suffers a major limitation: for many growth models (including a standard two-sector growth model with a renewable resource) there is no optimal path for the criterion.

Any path that leads to the preservation of a part of the stock will be beaten by another path that burns just an extra piece of the stock to increase the capital stock. So, an optimal path does not exist. There will be no consumption until an infinite time.

According to [START_REF] Gerlagh | Efficiency of conservationist measures: an optimistic viewpoint[END_REF], conservationist policies can be Pareto efficient, and strict resource conservation is equivalent to non-dictatorship of the present, as defined by [START_REF] Chichilnisky | An axiomatic approach to sustainable development[END_REF]. They argue that "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" (Gerlagh and Keyser, 2003, p.312).

Our approach includes such a resource conservation constraint. We thus avoid the dictatorship of the present.

The Mixed Bentham-Rawls criterion This criterion is proposed by [START_REF] Long | A mixed Bentham-Rawls Criterion for Intergenerational Equity[END_REF]. It is a convex combination of the maximin criterion and the utilitarian discounted criterion.

max θ ∞ 0 ∆(t)U (c t , x t )dt + (1 -θ) min t U (c t , x t ) . (47)
Comparing this approach to Chichilnisky's one, we have three criteria (discounted utilitarian, maximin and green golden rule), and two convex combination have been proposed. In the following section, we argue that our criterion is a particular limit case of the third combination: maximin and green golden rule.

Is our criterion a particular case of the Chichilnisky's one ? The optimal path of Chichilnisky's criterion is not easy to compute. Heal (1998) develops a particular approach combining two steps that we will detail below, in a particular case of the criterion.

Defining minimal objectives to implement sustainability

Until now, we have presented an approach to define minimal rights for sustainable development. We have compared the result of this approach in a canonical model with usual criteria, and argued that it allows avoiding some difficulties in defining optimal paths. However, any reader would argue that the use of our criterion relies on the definition of the set of all reachable goals, and then, on the viability approach. Defining such minimal rights is not an easy task, and one can argue that the approach is no more applicable than any discussed criterion.

However, there is an increasing body of literature using viability approach in complex bioeconomic systems, emphasizing its applicability is one accepts to use numerical methods to solve the problem. 17 The criterion we propose, more than a stylized criterion used to interpret the sustainability in a normative debate, can be applied to real bioeconomic problems, and help choosing between reachable sustainability objectives.

If sustainability encompasses several objectives, an optimization approach can be hard to apply. Our approach combining viability analysis and the optimization of constraints (goals) levels is an alternative solution.

In fact, when usual criteria do not have solutions, a possibility is to reduce the set of admissible path (by selecting paths that satisfy a set of constraints) and then to apply a criterion. For example, the green golden rule criterion will always postpone consumption. We propose to examine the problem when a minimal consumption is required. The problem is then to define the maximal utility level sustained by a path with a positive consumption.

The criterion is not used to define optimal paths under (already defined) constraint, but to define the set of reachable goals and to choose the sustainability target (minimal rights for sustainability). Once such goals are defined, there is no utility comparisons between generations. Everything is set with respect to the state of the system (cloosed-loop control problem) and not along a time dependent optimal path. 17 See Martinet et al. (2007) for example.

Reaching minimal right and minimal time of crisis

It is possible that none of the reachable constraints are relevant for sustainability.

For example, the level of sustainable consumption may be lower that a subsistence level. It is the purpose of the sufficientirianism approach to determine how to reach economic development paths that satisfy basic needs.

Our approach is somewhat different from the sufficientarianism, or the satisfaction of basic needs criterion, as it is presented in [START_REF] Chichilnisky | Economic Development and efficiency criteria in the satisfaction of basic needs[END_REF] and [START_REF] Long | A mixed Bentham-Rawls Criterion for Intergenerational Equity[END_REF]. In their perpectives, the criterion consists in the minimization of the time needed to reach an economic path that satisfies the basic needs. [START_REF] Long | A mixed Bentham-Rawls Criterion for Intergenerational Equity[END_REF] defines a minimal level of utility û to be reached, and [START_REF] Chichilnisky | Economic Development and efficiency criteria in the satisfaction of basic needs[END_REF] considers a multiobjective approach in which various basics needs have to be fulfilled, without considering an associated utility. [START_REF] Chichilnisky | Economic Development and efficiency criteria in the satisfaction of basic needs[END_REF] gives a mathematical formalization of economic efficiency in an economy concerned with the satisfaction of the consumption of basic goods in some near future. Efficiency is defined with respect to the minimization of the time horizon at which the minimal consumption level is reached. The criterion can be written as min u(.)

T u τ ≥ û ∀τ > T (51)

Our approach is different as the "basic needs" are not defined a priori with respect to some real level of consumption or environment quality. On the contrary, our approach aims at defining the possible sustainable development goals, interpreted as "minimal rights" to be provided to all generations. Doing this, we avoid the issue of the (normative) definition of the level at which "basic needs" are defined.

However, the viability framework can be extended by using the concept of "time of crisis" [START_REF] Doyen | Scale of viability and minimal time of crisis[END_REF]) which is the time during which the constraints are not met. The minimal time of crisis is thus a relevant approach to address the basic needs issue. This approach has been developed in Martinet et al. (2007) in a bioeconomic model.

Conclusion

Sustainability can be defined by several objectives/goals of different nature. An optimization approach is then hard to adopt. In this paper, we proposed to apply a Rawlsian approach of justice to choose among the possible sustainability goals. We describe how to apply this approach in the classical Dasgupta-Heal-Solow model.

Our approach consists in maximizing the utility associated with minimal rights to be guaranteed to all generations. It is thus a static optimization problem among all possible sustainability goals. The set of possible sustainability goals, from initial economic state, are defined using the viability approach, like in Martinet and Doyen (2007).

We discussed the interpretation of the "minimal rights criterion" with respect to existing criteria, and show that it could provide some alternative to classical criteria (Maximin, Green Golden Rule, Chichilnisky) when the solution does not exist or is not easy to compute.

We then discussed the benefits and limits of such an approach. We showed that it can be applied to real cases sustainability problem, and that adequate numerical and algorithmic methods exist for application studies. This framework can even be extended to address the issue of reaching sustainability goals that are not achievable with the initial endowment of the economy.

Figure 1 :

 1 Figure 1: Substitution between guaranteed consumption and resource conservation for a Cobb-Douglas technology.

Fig. 2

 2 Fig. 2 illustrates this result.

Figure 2 :

 2 Figure 2: Optimal minimal rights when c > 0 and x > 0.

FigFigure 3 :

 3 Fig. 3 illustrates this result.

FigFigure 4 :

 4 Fig. 4 illustrates this result.

Ackwnoledgements

  I am grateful to participants of the CIREQ seminar (June 3, 2005, Montreal) for their comments and remarks on the first draft of the paper, especially to Bob Cairns and Hassan Benchekroun. I am also indebted to the members of the French research program "Economic Modeling of Sustainable Development", especially Mabel Tidball. The research program on viability and sustainability relies on collaboration with Luc Doyen. All errors remain my own.

5

  Even if the Rawls' conception of justice was not built for intergenerational equity issues 6 , we can examine what minimal rights can be provide to all genera-

	tions. In our sustainable development issue, it is equivalent to examine what level
	of consumption, and what level of environmental quality can be provided to all
	generations.

Usually, the discount factor is decreasing at a constant rate, i.e. ∆(t) = e -δt . Other alternatives have been proposed, including hyperbolic discounting which lead to a decreasing discount rate. A general form of hyperbolic discounting is ∆(t) = (1 + αt) -γ/α . Nevertheless, the discount factor is still decreasing and the utility of the far future is not taken into account, which raises the equity issue.

The term lim can also be replaced by any function that depends only on the limiting behavior of the utility over time, such as long-run average for example.

This result comes from the allocation of rights one would made under the "veil of ignorance". Rawls argues that justice should be based on two principles, with a priority order. The first principle is the definition of fundamental rights every one can enjoy ("each person is to have an equal right to the most extensive scheme of equal basic liberties compatible with a similar scheme of liberties for others"). The second principle is based on (with here again a priority order) "fair equality of opportunity" to a social position and on the "difference principle" that stipulates that inequality in the wealth distribution is justified if it is beneficial for the poorest individual, i.e. if the poorest individual in this configuration in richer than the poorest individual in all other possible allocations. This last statement leads to the maximin criterion, which is thus the less important point in Rawls' theory of justice.

According to Rawls, his theory of justice should not be applied to intergenerational problems.Nevertheless, one of the propositions of Rawls (1971) has already been applied to intertemporal equity problems: the maximin criteria[START_REF] Solow | Intergenerational Equity and Exhaustible Resources. Review of Economic Studies[END_REF][START_REF] Cairns | Maximin, a direct approach to sustainability[END_REF].

This is related to the inertia of the system. There are states that satisfy the viability constraints but from which no viable path exists. The crisis is unavoidable.

The intragenerational equity issue is not addressed. However, the perspective of growing population is discussed in section 4.3, in a discussion on the possibility to increase minimal rights along time.

If we consider Constant Elasticity of Substitution (CES) production functions, if the elasticity of substitution between the capital and the resource is greater than one, there is no sustainability issue as it is possible to produce without using the natural resource. The resource is not essential to produce. On the contrary, an elasticity of substitution lower than one implies that the resource is essential to produce, and the intertemporal production is bounded. No consumption can be sustained. The intermediate case with elasticity equals to one, i.e. the Cobb-Douglas case, makes it possible to substitute capital to the resource in the production even if the resource is essential to produce. According to[START_REF] Dasgupta | Economic Theory and Exhaustible Resources[END_REF], this case is the most interesting.

For the sake of simplicity, we will omit the initial state in the notation of function c + (k 0 , x 0 , x).

Such problems are generally easier to solve than complex dynamic optimization programs.

The variables are at optimal values. In order to get simple notations, we do not note them x * and c * but simply x and c

This utilitarian approach is related to the "weak sustainability".

This conservative approach is more related to the "strong sustainability".

Being precise, the Cobb-Douglas case satisfies f (k, 0) = 0. Nevertheless, the solution is not the one described by Heal (1998) as we will prove it. In fact, the "green golden rule" criterion has no solution in the Cobb-Douglas case.

and consider that this criterion is the first part of a "virtual" Chichilnisky's criterion. 16 It leads to the following special case of Chichilnisky's criterion:

where π(t) is the discount factor associated with the maximin path for this model.

If one applies the method developed by [START_REF] Heal | Valuing the Future, Economic Theory and Sustainability[END_REF] to exhibit the solution of such a criterion, it leads to the following: First, consider that a part x * of the resource is preserved forever, and then that the first part of the criterion is optimized. We obtain a maximin path under constraint, which is a point c + (x * ), defined by eq. ( 12). The second step consists in maximizing the level of x * in order to maximize criterion (50). If we take the limit of this criterion when θ → 0, we maximize the utility at infinite time or, equivalently, the "minimal rights" criterion we defined.

The method we propose and that consists in two steps (first defining the maximal guaranteed consumption for a level of resource preservation and then defining the level of the resource constraint) is closed from the method used to determine the optimum of the Chichilnisky criterion in [START_REF] Heal | Valuing the Future, Economic Theory and Sustainability[END_REF]. First, a part of the stock is put away and the discounted criterion is optimized. Then, the level of the resource preserved is optimized. Our approach can then be interpreted as way to avoid the difficulty induced by the resolution of Chichilnisky's criterion. 16 A maximin program can be expressed as the discounted sum of intertemporal utility (see [START_REF] Cairns | Maximin, a direct approach to sustainability[END_REF] for some particular discount factor π(t) such that a path where π(t) is the virtual discount factor of the maximin program with constant utility Ū * . Thus, a maximin program could be the first part of the chichilnisky's criterion. However, note that the virtual discount factor of a maximin problem is endogenous to that problem. Thus, all that is done in this section is a stylized fact for the following discussion.


P-A Jouvet

P Pestieau

G Ponthiere

Longevity and Environmental Quality in an OLG Model

Whereas existing OLG models with endogenous longevity neglect the impact of environmental quality on mortality, this paper studies the design of the optimal public intervention in a two-period OLG model where longevity is influenced positively by health expenditures, but negatively by pollution due to production. It is shown that if individuals, when choosing how much to spend on health, do not internalize the impact of their decision on environmental quality (i.e. the space available for each person), the decentralization of the social optimum requires a tax not only on capital income, but also, on health expenditures. The sensitivity of the optimal second-best public intervention is also explored numerically.

Introduction

The simultaneous growth of economic activity and longevity observed during the last two centuries in industrialized countries has raised the issue of the relationship between those two phenomena. In the recent years, the study of interactions between economic development and longevity has been particularly enriched by OLG models with endogenous longevity (Chakraborty, 2004; Battacharya and Qiao, 2005; Zhang et al, 2006). But those models, by making longevity depend on (private and/or public) health expenditures only, tend to neglect the influence of the natural environment on longevity. However, as this was emphasized by demographers (see Sartor, 2001), the natural environment constitutes a major determinant of longevity. Environmental quality affects longevity through various channels: the climate -i.e. temperature (Kunst et al, 1993) and solar radiations (Elwood et al, 1974)the quality of lands (Kjellström, 1986), of waters (Sartor and Rondia, 1983) and of the air (Kinney and Ozkanyak, 1991).

Given that environmental determinants of longevity are significantly influenced by economic activity, introducing the environment in the analysis can contribute to refine the study of the relationship between economic activity and longevity, which is assumed, in existing models, to be merelly 'globally positive'. Although including environmental quality does not question the positive relationship observed, this may, however, be most relevant for public policy analysis, because actual policies may be non-optimal: the observed production and longevity levels may, under actual interactions between production, environment and longevity, differ from what would maximize lifetime welfare.

The goal of this paper is to examine the issue of the optimal public intervention in a two-period OLG economy, in which the length of the second period of life is influenced not only by private health expenditures, but, also, by environmental quality. For that purpose, environmental quality shall enter our model in two distinct ways, each of these involving specific externalities, which can partially offset each others.

First, the production process is assumed to generate polluting emissions, whose negative effects on longevity are not taken into account by producers. The stock of pollution at each point in time, depending on the stock of pollution at the previous period and on current polluting emissions, tends to lower longevity, and, as such, accounts for the -widely documentednegative impact of the pollution of land, waters and the air on longevity.

Second, environmental quality is assumed to influence welfare not only indirectly, through the effects of pollution on longevity, but, also, directly, through the quantity of space available for each person. It is here postulated that individuals, when choosing their health spending, do not internalize the impact of their decisions on the number of persons. [START_REF] Boulding | Earth as a spaceship[END_REF] However, such a behaviour is not without consequences on environmental quality, because the earth is, in Boulding's (1966) terms, nothing else than a 'spaceship'. [START_REF] Bhattacharya | Public and private expenditures on health in a growth model[END_REF] Hence, individuals, by ignoring that the lengthening of their life causes -under the limiteness of the 'spaceship' -undesirable crowding effects, tend to overspend in health.

What is interesting here is that these two external effects can partially offset each others. Pollution hurts longevity, which can be desirable if longevity decisions do not internalize their effects on environmental quality. The existence of those two partially offsetting externalities raises the question of what public intervention should be: under those externalities, should a government tax or subsidize revenues from production and health expenditures?

This paper aims at casting a new light on the normative study of the growth-longevity relationship, by examining the optimal public intervention in an OLG economy where those two externalities are present. For that purpose, we shall assume that fertility is exogenous and fixed. This constitutes a non negligible restriction of the set of available instruments: undoubtedly, if the issue of 'earth as spaceship' must be tackled, it is preferable to do it through fertility than mortality. However, given that fertility decisions depend on various factors, a government has only a quite imperfect control on it, so that it makes sense, for our purpose, to abstract from fertility, and to concentrate on the interactions between production, environment and longevity under a fixed fertility.

The rest of the paper is organized as follows. Section 2 presents the model. The steady-state is characterized in Section 3. The decentralization of the social optimum is discussed in Section 4. Section 5 analyses the second best policy, whose sensitivity is studied numerically in Section 6. Section 7 concludes.

The model

We consider an overlapping generations model with endogenous life longevity and unpriced pollution.

Firms and pollution

We assume the existence of a neo-classical production sector using a quantity K of capital and L of labor. We assume that capital fully depreciates during the process of production. At each time t, firms produce a good (Y t ) with a well-behaved production function,

Y t = F (K t , L t ) (1)
Within the framework of a competitive equilibrium, each firm, at time t, chooses the quantity of capital and labor which will maximize its profit,

π t = F (K t , L t ) -R t K t -w t L t (2)
At equilibrium, the levels of return from the inputs correspond respectively to their marginal productivity:

R t = F K (K t , L t ) (3)
w t = F L (K t , L t ) (4)
where R t is the interest rate for savings and w t the wage rate. At each period, the flow of pollution emission is equal to a proportion η of current production,

E t = ηF (K t , L t ) (5)
The dynamics of the stock of pollution at time t, P t , is defined by

P t = (1 -δ)P t-1 + E t (6)
where δ the natural level of pollution absorption, 0 ≤ δ ≤ 1.

Agents

We suppose that the population is constant, and that at each time t, N identical agents are born. Each agent lives through two life periods. He or she works during the first period of life and is a pensioner for the second period of life. The first period is of unitary length; the second period lasts a period h, inferior to 1, which can be increased through health spending (like primary prevention in first period), x t , in the second period and decreased by industrial pollution, P t+1 (see Evans and Smith, 2005). The longevity function, h(x t , P t+1 ), is strictly concave with h x > 0, h xx < 0, h P < 0, h P P > 0.

Any agent born in period t derives utility from consumption, c t and the amount of space or land per person, q t , in her/his first period of life and from consumption, d t+1 , and the amount of space or land per person, q t+1 , in her/his second period of life. The preferences of the agents are represented by a utility function, U(c t , q t ) +h(x t , P t+1 )U (d t+1 , q t+1 ) where q t = Q/N(1 + h(x t-1 , P t )) with Q the given total quantity of space. U(.) is supposed to be strictly concave with U i > 0

(for i = c, d), U q > 0 and the cross derivative is assumed to be non negative, U iq ≥ 0. [START_REF] Browning | Micro Data and General Equilibrium Models[END_REF] During the first period of life, each agent supplies one inelastic unit of labor and receives the wage, w t , which he or she consumes, c t and saves s t in the form of capital and spend x t for health. Let us introduce a tax ξ on health spending and a tax τ on capital income, along with a lump-sum subsidy a. The budget constraint in the first period of life is:

w t + a = c t + s t + (1 + ξ)x t (7)
In the second period of life, the agents receive the return of their saving, R t+1 s t and consume d t+1 during h(x t , P t+1 ). The budget constraint in the second period of life of an agent born in t is therefore:

h(x t , P t+1 )d t+1 = (R t+1 -τ ) s t . (8)
The problem of each individual is thus to maximize:

U (c t , q t ) + h(x t , P t+1 )U(d t+1 , q t+1 )
subject to (7) and ( 8) by choosing x t and s t . What is crucial is that, when choosing x t and s t , individuals do not perceive the effect of their decisions on the environmental variables: pollution and space. Since the individual does not see the effect of savings on pollution, nor the effect of health spending on total population N(1 + h(x t , P t+1 )), one obtains the first order conditions for savings

-U c t + (R t+1 -τ ) U d t+1 = 0, (9)
and for health spending

-(1 + ξ)U c t + h x t U(d t+1 , q t+1 ) -d t+1 h x t U d t+1 = 0 (10)
3 Equilibrium and steady state

Intertemporal equilibrium

The intertemporal equilibrium is defined, by a sequence of prices, individual variables and aggregate variables satisfying all the equilibrium conditions.

Firms maximize their profit (conditions ( 19) and (20) hold) and consumers their utility (conditions ( 9) and ( 10) hold).

The capital stock is equal to savings,

K t = Ns t-1 (11)
The market of labor clears,

L t = N, (12)
as well the market of goods,

Y t = F (K t , N) = Nc t + Nx t + Nh(x t-1 , P t )d t + K t+1 (13)
In addition, the dynamic equation of pollution holds.

P t = (1 -δ)P t-1 + ηF (K t , N) (14)
At time 0, consumption of the retirees satisfies:

h(x -1 , P 0 )d 0 = R 0 s -1 . (15)
Further we take for given the initial capital stock K 0 = Ns -1 , the pollution stock P -1 and health expenditure x -1 .

Stationary equilibrium

At the steady state with a given policy, the stock of capital is given by the sum of saving, K = Ns and the economy's resource constraint per young is,

f (k) = c + x + h(x, P )d + k (16)
where k = K/N is the capital per young and f (k) = F (k, 1). Long-run equilibrium pollution is defined by,

P = η δ Nf(k) (17)
The amount of space or land per person is defined by,

q = Q N [1 + h(x, P )] (18)
Using relation (2), the interest factor (with tax) and the wage rate are,

R = f 0 (k) (19)
and

w = f (k) -kf 0 (k) (20)
The optimal consumers' decisions are given by,

-U c + (R -τ ) U d = 0 (21)
and for health spending

-(1 + ξ)U c + h x U(d, q) -dh x U d = 0 (22)
4 Social Optimum and optimal policy

We now turn to the analysis of the social optimum and optimal policy. At the long-run equilibrium, we are looking for the maximum possible utility in the economy.

Social optimum

In a centralized economy, the objective of the central planner is to maximize the welfare of the agents by choosing the level of consumptions (c, d), health spending (x) and the level of capital (k), under constraints of resources, pollution and space per person.

max c,d,x,k U(c, q) + h(x, P )U(d, q) s.t. :    f (k) = c + x + h(x, P )d + k P = η δ Nf(k) q = Q N[1+h(x,P )]
Denoting by λ the Lagrangian multiplier of resource constraint (16), the Lagrangian L is defined by

L(c, d, x, k) = U µ c, Q N[1+h(x, η δ Nf(k))] ¶ +h ¡ x, η δ Nf(k) ¢ U µ d, Q N[1+h(x, η δ Nf(k))] ¶ +λ £ f (k) -c -x -h(x, η δ Nf(k))d -k ¤ (23)
One obtains thereby the first order conditions,

U c = λ (24) U d = λ (25) h x U(d, q) - NQh x (N [1 + h(x, P )]) 2 µ = λ(1 + h x d) (26)
and

h P η δ Nf 0 (k)U (d, q) - N Qh P η δ Nf 0 (k) (N [1 + h(x, P )]) 2 µ (27) = λ(1 + h P η δ Nf 0 (k)d -f 0 (k))
where µ ≡ U q (c, q) + h(x, P )U q (d, q) is the marginal lifetime utility of an increase in q. [START_REF] Chakraborty | Endogenous lifetime and economic growth[END_REF] From ( 24) and ( 25) we obtain,

U c = U d (28)
Using ( 28), ( 26) and ( 27), we have

U c (c, q) (1 + h x (x, P )d) = h x U (d, q) -h x QN [N [1 + h(x, P )]] 2 µ (29)
and

f 0 (k) = 1 + f 0 (k) h p η δ N • Q (N(1 + h)) 2 µ U d - µ U(d, q) U d -d ¶¸( 30)
Note that without the environmental variables, these two optimal conditions would reduce to:

U c (c, q) = h x [U (d, q) -U d (d, q)d] and f 0 (k) = 1.
The first equation is equivalent to (22) with ξ = 0. As to the second, it is the Golden rule (population growth is here 0). With the environmental variable, we have some externalities. Starting with (29) associated with health spending, the externality comes from the fact that individuals do not internalize in their decisions the effect of increased longevity on the number of inhabitants of a finite earth.

The choice of investment (eq. ( 30)) entails two externalities. The first one is positive. Keeping in mind that investment, production and pollution are closely related, more capital means more pollution and thus less population, which is good for the quality of the environment. The second externality is negative: more pollution means lower longevity and thus lower utility in the second period of life. However, by substituting (29) in (30), we obtain:

f 0 (k) = 1 -f 0 (k) h p h x η δ N (30')
This implies that the optimum level of capital accumulation should be lower than that corresponding to the standard golden rule level. In other words, the negative externality more than offsets the positive one.

Optimal policy

Contrasting the market solution with the first-best optimum, one sees that the social optimum can be decentralized with appropriate choices of "Pigouvian taxes" ξ and τ for the environmental damage:

ξ = h x Q N [1 + h(x, P )] 2 µ U c > 0 (31)
and for saving

τ = f 0 (k) -1 = - h P h x η δ N 1 + h P h x η δ N > 0 (32)
To these two taxes one should add an intergenerational transfer device that leads to Rτ = 1 from (28). With our environmental externality, we thus have f 0 (k) = R > 1. In other words, the optimal level of capital stock is inferior to that consistent with the standard Golden Rule (f 0 (k) = 1).

Second-best policy

We now turn to the second-best setting. We assume that both τ and ξ are available but that we do not have an instrument to achieve the optimal level of capital accumulation. We conduct this second-best analysis in a steadystate framework.

With a zero population growth, we have s = k. We can easily show that the resource constraint implies the revenue constraint

f (k) = c + x + hd + k = f 0 (k) + c + s (1 + ξ) + s -a and thus, a = τs + ξx,
where a is a lump-sum transfer given in the first period of life. Note that if, besides a, we had also a transfer in the second period of life, one would get the first-best, with ξ and τ having the values (31) and (32) and the tax transfers leading to f 0 (k)τ = 1.

We will express the problem of the social planner using the revenue and not the resource constraint. To keep the notation simple, we assume N = 1 without loss of generosity. The Lagrangian expression can now be written as:

L(τ, ξ, a) = U (c, q) + h ³ x, η δ f (s) ´U (d, q) +γ [τs + ξx -a] -µ • q - Q 1+h ³ x, η δ f (s) ´¸.
where γ and µ are the multipliers associated with the revenue constraint and with the definition of q.

We maximize L with respect to τ, ξ and a, our tax parameters and with respect to q, an adjustment variable.

The FOC's are given by:

∂L ∂a = U c + γ • τ ∂s ∂a + ξ ∂x ∂a -1 ¸-• µ Q (1 + h) 2 -U(d, q) ¸hp η δ f 0 (s) ∂s ∂a - µ Q (1 + h) 2 h x ∂x ∂a = 0 ∂L ∂τ = -U c s R -τ + γ • τ ∂s ∂τ + ξ ∂x ∂τ + s ¸-• µ Q (1 + h) 2 -U (d, q) ¸hp η δ f 0 (s) ∂s ∂τ - µ Q (1 + h) 2 h x ∂x ∂τ = 0 ∂L ∂ξ = -U c x + γ • τ ∂s ∂ξ + ξ ∂x ∂ξ + x ¸-• µ Q (1 + h) 2 -U(d, q) ¸hp η δ f 0 (s) ∂s ∂ξ - µ Q (1 + h) 2 h x ∂x ∂ξ = 0 ∂L ∂q = U q (c, q) + hU q (d, q) -µ = 0 (33)
As usual with this type of problem, we express the tax formula in compensated terms. In other words, an increase in either τ or ξ is compensated by an increase in a and we want to know the effect of such a compensated increase on social welfare. We use the superscript c to denote the compensated effects. The multiplier µ in (33) expresses the social value of environmental quality.

∂L c ∂τ = [1] γ • τ ∂s c ∂τ + ξ ∂x c ∂τ ¸+ [2] sU c µ R -τ -1 R -τ ¶ - [3] µ Q (1 + h) 2 h x ∂x c ∂τ -    [4] µ Q (1 + h) 2 [5] -U (d, q)    h p η δ f 0 (s) ∂s c ∂τ (34)
∂L c ∂ξ = γ [1] • τ ∂s c ∂ξ + ξ ∂x c ∂ξ ¸- [3] µ Q (1 + h) 2 h x ∂x c ∂ξ -    [4] µ Q (1 + h) 2 [5] -U (d, q)    h p η δ f 0 (s) ∂s c ∂ξ (35)
We distinguish among five terms in those two conditions:

1. the traditional Ramsey formula, 2. the gap between the (net) rate of interest and the population growth rate, 3. the crowding effect induced by longevity-enhancing investment, 4. the decrowding effect induced by pollution, 5. the utility loss arising from shorter lifetime induced by pollution.

To get further insight, we make a quite extreme assumption: the crossderivatives are negligible. In other words, τ has no influence over x and ξ over s. Then we have:

∂L c ∂τ = γ [1] τ ∂s c ∂τ + s [2] U c µ R -τ -1 R -τ ¶ -    [4] µ Q (1 + h) 2 - [5] U (d, q)    h p η δ f 0 (s) ∂s c ∂τ ∂L c ∂ξ = γζ [1]
∂x c ∂ξ -

[3] µ Q (1 + h) 2 h x ∂x c ∂ξ
Starting with τ , we have two positive terms: [2] and [5] and two negative ones [1] and [4]. The terms [2] and [4] are standard. The term [5] reflects the fact that by increasing τ , there is less capital accumulation and thus less pollution, which leads to an increased longevity and thus to an increased utility in period two. The term [4] shows also that a tax on saving has a positive effect on longevity, but longevity has a crowding effect on the fixed environmental quality Q.

Turning to ξ, we have one negative effects [1] and a positive one [3]. The first one [1] is standard and negative; the second one, positive effect [3] says that by taxing health care, people do not live as long as without such a tax and this has a relief effect on the fixed quality of environment.

Admittedly, the result -i.e. health care ought to be taxed -obtained both in the first-best and in the second-best (under particular conditions) is a bit surprising and, as such, has to be qualified. Subsidizing health care is often recommended on the grounds of other considerations: redistribution, externality, etc. The negative effect underlined in this paper is likely to be dominated by these other considerations.

The two external effects of production and pollution are quite interesting. On prior grounds, one cannot say whether saving ought to be taxed or subsidized. We know that in the first-best it has to be taxed.

A numerical application

Let us now consider, in the light of numerical examples, the implications of the present model for public policy. For that purpose, we shall first introduce and calibrate functional forms for preferences, production and longevity. Then, we shall explore the sensitivity of the optimal (second-best) taxes on capital income and health expenditures -τ * and ξ * -to various parameters of the model.

Functional forms

Lifetime welfare is assumed to take the following, additive form:5

U t = £ (c t ) γ (q t ) 1-γ ¤ 1-σ 1 -σ + βh t+1 £ (d t+1 ) γ (q t+1 ) 1-γ ¤ 1-σ 1 -σ ( 36
)
where β is a time preference factor, whereas γ reflects the importance of consumption with respect to environmental quality (0 ≤ γ ≤ 1). γ is assumed to be constant across periods, which is a simplification, as old agents may be more or less sensitive to environmental quality than young ones. [START_REF] Elwood | Relationship of melanoma andd other skin cancer mortality to latitude and ultraviolet radiation in the United States and Canada[END_REF] The parameter σ is the inverse of the intertemporal elasticity of substitution for consumption. Moreover, we shall assume that production takes a Cobb-Douglas form:

Y t = AK α t L 1-α t ( 37
)
with 0 < α < 1 and A > 0.

Longevity is determined by health expenditures and pollution as follows:7

h t+1 = Bx φ t p ψ t ( 38
)
where B denotes the 'natural' longevity level, that is, its level in the hypothetical case where health expenditures and pollution have no influence on longevity (B > 0). φ is the elasticity of h t+1 with respect to health expenditures (φ > 0), whereas ψ is the elasticity of h t+1 with respect to the stock of pollution at time t (ψ < 0), denoted here in intensive terms (i.e. p t = P t /N is the stock of pollution per worker). [START_REF]Population, densité et part de la population urbaine des principaux pays du monde[END_REF] In order to capture the intergenerational dimension of pollution, longevity is here affected by the stock of pollution faced when being young: individuals, even if they were non-myopic, could choose their longevity only within a range allowed by previous generations, because some of its causes -here p t -result from past decisions on which they have no control.9

Calibration

Regarding preferences, the discount factor β is assumed to be equal to 0.30, which corresponds, given that the length of a full period is 40 years, to a quarterly subjective discount factor of 0.99. In the light of empirical studies showing that the intertemporal elasticity of substitution for consumption is slightly above unity (see Browning et al, 1999), the parameter σ is fixed to 0.83, which implies an elasticity of intertemporal substitution of 1.25.

Regarding the parameter γ, we shall first consider the benchmark case where the available space does not affect welfare (i.e. γ = 1), and, in a second stage, use lower values for γ.

As far as production parameters are concerned, α is fixed to 0.36, in conformity with the literature (see de la Croix and Michel, 2002), while the scale parameter A is fixed to 10.

The calibration of pollution parameters δ and η depends on the particular pollution process under study. [START_REF] Kjellström | Itai-Itai disease[END_REF] Given that each period is of length 40 years, it makes sense to assume that δ is relatively high, but its level depends on what pollution consists of. We shall, as a benchmark, assume that δ is equal to 0.9 (i.e. 9/10th of the pollution have vanished after a time lag of 40 years). Regarding η, we shall take 0.10 as a benchmark, and consider also higher values.

Regarding the space available per active person Q/N, we shall assume that it is equal to about 3000 square-meters (i.e. equal approximately to the available space in countries such as the Netherlands). But, in order to explore how the optimal policy is affected by the 'number problem', we shall also compute (τ * , ξ * ) under a higher population density (i.e. a lower Q/N).

Finally, as far as the calibration of the longevity function is concerned, we shall assume that the elasticities of longevity with respect to health spending φ and pollution ψ are equal respectively to 0.15 and -0.05. Under those values, assuming that B is equal to 0.30 implies, under k 0 equal to 0.1, and under τ = 0 and ξ = 0, an initial longevity equal to about 65 + 0.25(40) = 75 years.

Results

Let us now consider the policy to be implemented by a government maximizing steady-state lifetime welfare subject to the budget constraint. The government collects a tax τ on capital income, taxes health expenditures at a rate ξ, and uses the fiscal revenues to fund a transfer a, which is remitted to individuals during their first period of life. Individuals cannot anticipate the impact of their health spending on a, and take it as given. Given the absence of a second-period transfer, the derived optimal policies are second-best policies.

When choosing their savings and health expenditures, agents do not internalize the impact of those decisions on pollution and on the available space. Moreover, we shall assume, for simplicity, that, when making their decisions, individuals form static expectations for the space available in the next period. [START_REF] Kunst | Outdoor air temperature and mortality in the Netherlands: a time series analysis[END_REF] To study the sensitivity of the optimal second-best policy (τ * , ξ * ), we shall first focus on parameters describing pollution (i.e. η and δ), and, then, on preference parameters (i.e. β, γ and σ). For simplicity, we shall, in a first stage, assume that the available space does not affect welfare (i.e. γ = 1).

As illustrated by Figure 1, lifetime welfare at the steady-state is a nonmonotonic function of the tax on capital income τ . [START_REF] Michel | Disutility of pollution and endogenous growth[END_REF] For low taxation levels, a higher τ raises lifetime welfare, whereas the opposite holds when τ is high. Although the relationship between steady-state lifetime welfare and taxation on capital income has the same inverted-U shape for the three values of [START_REF] Kunst | Outdoor air temperature and mortality in the Netherlands: a time series analysis[END_REF] Under those assumptions, the optimal saving and health expenditures are equal to:

s * = (w-(1+ξ)x * +a) • βht+1 ³ (R-τ ) h t+1 ´γ(1-σ) ¸1 γσ-γ+1 1+ • βh t+1 ³ (R-τ ) h t+1 ´γ(1-σ) ¸1 γσ-γ+1 and x * = φs * [ γσ-γ+1 1-σ ] γ (1+ξ)
.

ξ, the level of lifetime welfare is not insensitive to the tax on health spending. Actually, the computation of lifetime welfare under different values of ξ suggests that subsidizing health spending is here socially desirable: the highest inverted-U curve is obtained under ξ equal to -38 %, so that the optimal second-best policy involves τ * equal to about 4.9, and ξ * equal to -0.38.

Let us now explore how this optimal second-best policy (τ * , ξ * ) varies with the various parameters of this model. For that purpose, we shall present steady-state lifetime welfare as a function of τ under different parameters's values, while assuming that the tax on health expenditures takes its optimal level ξ * for each parametrization.

A first set of parameters concerns the pollution process. As illustrated on Figure 2, which shows steady-state lifetime welfare as a function of τ under low emissions (i.e. η = 0.1), medium emissions (i.e. η = 0.3) and large emissions (i.e. η = 0.5), lifetime welfare is, as expected, lower when emissions are larger, that is, when η is higher. [START_REF] Sartor | Hardness of municipal waters and cardiovascular mortality in four small Belgian towns[END_REF] But another important thing to observe is that the optimal second-best policy is not insensitive to η: (τ * , ξ * ) is equal to (4.9, -0.38) when η equals 0.1, and to respectively (5, -0.39) and (5.3, -0.41) when η equals 0.3 and 0.5. Thus, higher emissions lead here to a higher optimal tax on capital income, and to a higher optimal subsidy on health spending. One should notice that assuming a lower δ (i.e. a lower natural absorption of pollution) has the same effects as a rise of η: it reduces, ceteris paribus, steady-state lifetime welfare, and implies a higher optimal tax on capital income, and a higher optimal subsidy on health expenditures. [START_REF] Sartor | Les déterminants environnementaux de la mortalité[END_REF] But the optimal second-best policy is also sensitive to preference parameters β, σ and γ, as we shall now discuss.

The influence of parameters β and σ on the optimal second-best policy is shown on Table 1. [START_REF] Zhang | Health investment, saving, and public policy[END_REF] Regarding the impact of β, the first three rows of Table 1 illustrate that a lower impatience tends to raise steady-state capital, longevity and welfare. From the point of view of public intervention, a higher β reduces the optimal tax on capital income and the optimal subsidy on health spending. Regarding the influence of σ on the optimal second-best public intervention, Table 1 suggests that a higher σ (i.e. a lower elasticity of intertemporal substitution) tends to reduce the size of the optimal subsity on health expenditures, and may turn it into a tax. However, its impact on τ * is more ambiguous.

Turning now to the parameter γ, which captures the importance, in welfare terms, of consumption with respect to the available space per person, Table 2 suggests that reducing γ leaves the optimal tax on capital income τ * unchanged. 16 However, assigning a higher weight to environmental quality tends, ceteris paribus, to reduce the subsidy on health expenditures, and turns it into a tax (i.e. ξ * > 0). That result is not surprising: the lower γ [START_REF] Sartor | Les déterminants environnementaux de la mortalité[END_REF] For instance, under the above calibration and η = 0.1, (τ * , ξ * ) is equal to (4.9, -0.38) when δ equals 0.9, and to (5.1, -0.39) when δ equals 0.1.

15 Table 1 is based on A = 10, α = 0.36, δ = 0.9, η = 0.1, B = 0.3, φ = 0.15 and ψ = -0.05. Initial capital is 0.1. Q/N = 3000.

16 Table 2 is based on A = 10, α = 0.36, δ = 0.9, η = 0.1, B = 0.3, φ = 0.15 and ψ = -0.05. Initial capital is 0.1. is, the larger is the welfare loss due to the non-internalization by agents of the influence of health spending on the available space (through the rise in longevity). Such a larger welfare loss must necessarily lead, ceteris paribus, to a higher tax rate ξ * .

The second part of Table 2 shows the optimal policy (τ * , ξ * ) when the available space per active person Q/N is reduced to a level equal to 1000 square-meters, which corresponds approximately to the space availability in a highly densified country like Bangladesh (where the population density is about 1018 persons per square kilometer). 17 As shown by the last column of Table 2, postulating a smaller available space Q/N tends, ceteris paribus, to lower steady-state lifetime welfare, except in the special case where γ equals 1 (i.e. environmental quality does not influence welfare). However, reducing the available space does not, under the postulated functional forms, affect the optimal second-best policy (τ * , ξ * ), which is the same under Q/N equal to 3000 and 1000. Thus, although a smaller space reduces welfare, it does not affect the optimal second-best public intervention.

Conclusions

The goal of this paper was to study the design of the optimal long-run public intervention in an economy where longevity, which is influenced negatively by pollution due to production, tends, by raising population density, to affect environmental quality. For that purpose, we developed a two-period OLG model, where agents do not, when choosing their savings and health expenditures, internalize the impact of their decisions on the natural environment.

As this was shown with general functional forms for production, preferences and longevity, the first-best public intervention in that economy involves, besides adequate transfers leading to the Golden rule, a positive taxation of capital income, as well as a positive taxation of health expenditures. Whereas the former pigouvian tax allows the internalization of the pollution externality, the latter corrects agents's tendency to overspend in health.

Regarding the optimal public policy under a limited set of instruments (including the two taxes and a first-period transfer), it was shown that the optimal second-best levels of the tax on capital income and on health spending can hardly be signed with certainty, but, rather, depend on various factors, such as the intensity of preferences for environmental quality and the pollution process.

The sensitivity of the optimal second-best policy to those factors was also illustrated by means of numerical examples based on a time-additive CES utility function, a Cobb-Douglas production technology, and a longevity exhibiting constant elasticities with respect to health spending and pollution. As one may expect, pollution processes involving higher emissions tend, ceteris paribus, to raise the optimal second-best tax on capital income. Moreover, increasing the importance of environmental quality as a determinant of human welfare tends to turn health care subsidies into a tax.

While this paper does not allow us to draw precise conclusions about what public policy should be in the actual world, this allows us, however, to highlight that public intervention should take into account the multiple relationships between production, longevity and the natural environment, because these determine the corrections to be made by governments. Given the complexity of those relations, it cannot be overemphasized that this paper is only a first step in the study of optimal policy under endogenous longevity and environment.

Fig. 1 :

 1 Fig. 1: Steady-state lifetime welfare as a function of τ.

Fig. 2 :

 2 Fig. 2: Relation between U and τ for different η.

Table 1 :

 1 Optimal policy (τ * , ξ * ) and preference parameters

	β	σ	γ	τ *	ξ *	k *	x *	p *	h *	U *
	0.30 0.83 1.00 4.90 -0.38 0.396 0.468 0.796 0.271 8.452
	0.40 0.83 1.00 3.50 -0.25 0.628 0.613 0.940 0.280 8.853
	0.50 0.83 1.00 2.70 -0.15 0.887 0.764 1.064 0.287 9.217
	0.30 0.70 1.00 4.40 -0.59 0.362 0.309 0.771 0.255 5.889
	0.30 0.83 1.00 4.90 -0.38 0.396 0.468 0.796 0.271 8.452
	0.30 0.95 1.00 4.90 0.16 0.487 1.198 0.857 0.311 23.693

Table 2 :

 2 Optimal policy (τ * , ξ * ) for different γ and Q/N.

	β Q/N = 3000	σ	γ	τ *	ξ *	k *	x *	p *	h *	U *
	0.30	0.83 1.00 4.90 -0.38 0.396 0.468 0.796 0.271 8.452
	0.30	0.83 0.75 4.90 -0.19 0.421 0.533 0.814 0.276 10.979
	0.30	0.83 0.50 4.90 0.14 0.455 0.644 0.836 0.283 14.246
	Q/N = 1000
	0.30	0.83 1.00 4.90 -0.38 0.396 0.468 0.796 0.271 8.452
	0.30	0.83 0.75 4.90 -0.19 0.421 0.533 0.814 0.276 10.478
	0.30	0.83 0.50 4.90 0.14 0.455 0.644 0.836 0.283 12.976

Actually, the same externalities appear in fertility decision, which involves the wellknown tragedy of the commons: with fixed natural resources, free riding agents adopt a too high fertility rate. In the present context, free riding agents tend to live too long.

[START_REF] Bhattacharya | Public and private expenditures on health in a growth model[END_REF] Boulding's (1966) expression means that the earth is becoming a single spaceship without illimited reservoirs of anything, either for extraction or for pollution, and in which therefore man must find his place in a cyclical ecological system.

This assumption converts a complementary effect of space and consumption. An alternative assumption is that space and consumption are substitutable, a negative crossed derivative. For an in-depth discussion of these assumptions we refer the reader to[START_REF] Michel | Disutility of pollution and endogenous growth[END_REF].

Note that if there is separability between consumption and q, N Q(N (1 + h)) 2 µ = v 0 (q) q with U (c, q) = u(c) + v(q).

Under that functional form, the cross derivatives U cq and U dq are non-negative.

Another non-negligible assumption is that pollution does not enter individual utility directly, but, only indirectly, through its influence on longevity h t+1 .

Under φ > 0 and ψ < 0, that functional form satisfies the properties mentioned in Section 2: h x > 0, h xx < 0, h p < 0 and h pp > 0.

Given that each cohort is of constant size, this does not have any influence on the results.

This assumption does not affect the conclusions drawn in previous sections, because these concerned the steady-state, where, by definition, p t+1 and p t are equal.

It should be stressed here that the assumed dynamic expression for pollution may not cover many existing pollution processes, whose dynamics is far more complex.

Figure 1 is based on A = 10, α = 0.36, β = 0.3, γ = 1, σ = 0.83, δ = 0.9, η = 0.1, B = 0.3, φ = 0.15 and ψ = -0.05. Initial capital is 0.1. Q/N = 3000.

Figure 2 is based on A = 10, α = 0.36, β = 0.3, γ = 1, σ = 0.83, δ = 0.9, B = 0.3, φ = 0.15 and ψ = -0.05. Initial capital is 0.1. Q/N = 3000.

Sources: INSEE (2006).