Non-Catastrophic Endogenous Growth and the Environmental Kuznets Curve por J. Aznar-Márquez* J. R. Ruiz-Tamarit** DOCUMENTO DE TRABAJO 2004-15

July 2004

* Universitat Miguel Hernández d’Elx (Spain). ** Universitat de València (Spain) and IRES (Belgium).

J. Aznar-Márquez†

J. R. Ruiz-Tamarit‡

July, 2004.

Abstract

The competitive equilibrium in an endogenous growth model is not Pareto-optimal nor environmentally sustainable in presence of pollution externalities, even if costly abatement activities are allowed to be endogenously decided. In this paper we introduce the possibility of an ecological catastrophe by imposing an upper-limit to the pollutants stock. We characterize the socially optimal solution and study sustainability of the long-run balanced growth path. We find that the rate of growth depends negatively on the weight of environmental cares in utility and positively on the population growth rate. The latter efect is stronger as higher is the weight of environment in the utility function. We also identify some policies the central planner could undertake looking to guarantee sustainability. An EKC is derived in the long term using the implications of the demographic transition for the rate of population growth, and the accompanying variation in the willingness to pay for environmental quality as the economy develops.

JEL classification: C61, C62, O41, Q5.

Keywords: Environment, Optimal Growth, Ecological Catastrophe, Sustainability.

∗We acknowledge the financial support from the Spanish CICYT, Project SEC2000- 0260, and the Belgian research program ARC 03/08-302.

†Universitat Miguel Hernández d’Elx (Spain).

‡Corresponding author. Universitat de València (Spain) and IRES (Belgium). Address: Department of Economic Analysis; Av. dels Tarongers s/n; E-46022 València (Spain). Phone: (+) 34 96 3828250. Fax: (+) 34 96 3828249. e-mail: ramon.ruiz@uv.es

1 Introduction

In a recent work on endogenous growth theory and the environment [Smulders (1999)], we can read that “Many other models only incorporate a flow variable to represent the environment. Thus ignoring the accumulation of wastes and the irreversibility of environmental damage, these models are not able to examine the possible conflict between short-run and long-run consequences of economic growth on the environment, but they prove to be a useful simplification to examine, for instance, the efects of diferent environmental tax issues”. According to this, the stock of accumulated pollutants has to be explicitly incorporated into models when we study the issue of long-run growth sustainability. On the one hand, if pollution increases with economic growth, it may happen that growth ceases when the stock reaches a certain upper-bounding level. Moreover, long-run sustainability depends not simply on the level of emissions but also on the assimilative capacity of the environment. However, as López (1994) points out, the world’s capacity to absorb pollution is limited and once pollution stock approaches the absolute tolerable limit, economic growth would not become feasible anymore because the economy will be falling down into an extreme situation of catastrophic state. Consequently, it has greater importance to ask whether there are limits to growth. The global economic collapse is more probable that arises in models of endogenous growth when the economy follows a path with sustained longrun growth, if pollution emissions appear positively related to the economic activity (production and consumption) as a by-product [Gradus and Smulders (1993), Ligthart and Ploeg (1994), Michel and Rotillon (1995), Mohtadi (1996)]. In such a case, if pollution tolerance is limited it will be attained the state of ecological catastrophe that represents an efective and absolute limit to growth.

The previous problem, however, may be mitigated when it is possible for economic agents to undertake emissions abatement activities or control for the degree of pollution associated with production technologies [Gradus and Smulders (1993), Ligthart and Ploeg (1994), Byrne (1997), Stokey (1998), Andreoni and Levinson (2001), Reis (2001)]. Actually, pollution stocks can not only be diminished by increasing the regenerative capacity or by reducing the polluting activity, but also by means of pollution abatement actions that contribute to determine the degree of dirtiness associated with technology, as well as the net flow of pollutants to the environment. However, improving environmental quality requires investment expenditures that leave less resources available for growth-oriented investment activities and, hence, a trade-of between production and environmental quality has to be featured in this enlarged framework. In general, under the existence of environmental externalities, it is expected to find lower rates of growth for output and pollution when pollution is optimally controlled.¹ In this sense, the opportunity for pollution abatement generates a mechanism that may act as a limit to growth, although less strong than the previous one. According to this, the relevant question is whether economic growth and environmental protection are reconcilable. That is, whether optimal sustained growth is compatible with ecological sustainability of the economy as a whole.

Another issue arising from the previous programme is the well-known environmental Kuznets’ curve (EKC) hypothesis, which says that there is an inverted U-shaped relationship between pollution emissions and per capita income levels. Or, put in another way, that economic growth usually leads to environmental degradation in the early stages of the process, but in the end the best and probably the only way to attain a decent environment is to become rich [Beckerman (1992)]. Theoretical foundations for this hypothesis have been proposed on the ground of the short-run transitional dynamics generated into neoclassical growth models [Tahvonen and Kuuluvainen (1993), Selden and Song (1995), Kelly (2003)], as well as in models of endogenous growth where pollution is decoupled from the engine of growth under the premise that not every increase in output due to technological advances will lead to increased pollution [Byrne (1997)].² Beyond these shortrun dynamic interpretations of the EKC for an isolated country, there is a long-run lecture connected with the development process historically experienced by economies. This view, moreover, gives theoretical support to the bulk of empirical studies, because it allows for a well-defined EKC based on the variability of population growth rates and willingness to pay for cleaner environment, while it leaves any other technical and preference parameter unchanged.

¹Things could be diferent if a pollution externality on the side of production is considered [Gradus and Smulders (1993), Ewijk and Wijnbergen (1995), Mohtadi (1996), Smulders and Gradus (1996)]. In such a case, environmental quality changes production opportunities by afecting the economy’s productivity, and an increase in environmental care may boost growth.

²Alternative foundations for the EKC hypothesis may be found in Jones and Manuelli (2001) built upon a dynamic overlapping generations model, but also in the context of a static model as in Stokey (1998), Munasinghe (1999) or Andreoni and Levinson (2001).

The EKC hypothesis has lead some analysts to conclude that pollution will not be a problem in the long-run because of the beneficial efects of economic growth on the environmental quality. This proposition implicitly assumes that growth is essentially good for the environment because as levels of income go up the emissions flow will decline. Consequently, no governmental interventions are needed, being growth in itself a panacea for sustainability. However, to determine whether environmental concerns will eventually limit growth is a question that has to be answered looking at two diferent issues: first, the efects of pollution abatement on the long-run rates of growth; and second, the evolution of the stock of pollutants relative to the ecological catastrophic upper-limit.

All these questions will be analyzed more accurately here in a simple model of endogenous growth. Given that we are not directly interested on how technological change has been originated, but on conditions under which sustained endogenous growth and ecological sustainability are compatible, our model builds upon the traditional Rebelo’s (1991) one-sector AK model and then introduces pollution. Welfare depends on consumption but also on the quality of the environment where agents consume. In this model pollution arises from production and enters the consumer’s utility function playing the role of an externality. This externality may be explained by the existence of numerous agents who know which efect pollution exerts on their respective utilities, without having any influence on the generating process. We ignore here, because of empirical irrelevance, any other pollution externality which could play a role by afecting the productivity of factors via the health of workers or the quality of inputs.

One central aspect of the analysis below, which has now a long tradition in the literature, is the explicit consideration of abatement activities. These are costly because they absorb resources reducing investment and consumption possibilities. Households show environmental cares but they do not decide on abatement while firms, which take into account the cost of such activities, do not perceive the benefits. Because of this environmental externality the competitive equilibrium does not work well. On the one hand, the equilibrium path is not Pareto-optimal and, on the other, this path leads the economy to the state of ecological catastrophe. Consequently, we will study the opportunities for an eficient management of the economy with special attention to the potential risk of environmental collapse. Eficiency is not sufficient for sustainability but, as we will show, Pareto optimality is necessary to produce sustainable outcomes. Moreover, along the article we are going to answer the usual questions: (i) Do environmental externalities influence growth? (ii) What are conditions for sustained balanced growth when environment matters? (iii) Which is the efect of environmental concerns on the rate of growth? (iv) Will pollution controls and abatement reduce growth rates? (v) Under what conditions is sustainability feasible?

The article is organized as follows. Section 2 describes the economy and introduces the assumptions featuring a general equilibrium one-sector endogenous growth model in which pollution is a by-product of economic activity, but it may be reduced by spending a fraction of the aggregate output on abatement. In Section 3 we briefly study the decentralized competitive equilibrium without regulation. In Sections 4 and 5 we study the socially optimal solution assuming suficient conditions for interior solutions. Using the non-constrained trajectories, we characterize growth in the social optimum and analyze under what conditions sustained balanced growth is feasible. Section 6 focuses on ecological sustainability and non catastrophic growth, with special attention to conditions which ensure them. There, we solve the general model allowing for corner solutions and study how, if pollution stock reaches the upper-limit, the central planner could change the value of the dirtiness index. We also characterize growth in the aftermath and compare with the previous one. Section 7 deals with the environmental Kuznets’ curve hypothesis and the implications for environmental policies. One major critique is that this relationship only describes statistically the link between income and pollution, but does not explain why it occurs. In this section we supply an alternative long term explanation for the EKC. Finally, Section 8 summarizes and concludes.

2 The economy

2.1 Production

The model economy is a one sector closed economy. Output is obtained according to an aggregate production function of the AK type where capital is the only factor needed to produce,

\[Y (t) = A K (t).\tag{1}\]

In this model is an aggregate composite of diferent sorts of capital which, in a broad sense, includes physical as well as human capital. For the sake of simplicity, we assume that this production function arises from the direct summation of the individual production functions for many identical firms.

2.2 Pollution and abatement

One feature of this model, absent from the canonical endogenous growth AK model, is the existence of a stock of pollutants that is increased by polluting activities such as production , and is reduced by abatement as well as by the corresponding natural regeneration at a constant rate Moreover, it is assumed an upper-limit for , called , which plays the role of a critical value for which if the current stock goes beyond a catastrophic state is reached in the economy. Under these assumptions, sustainable development will be characterized as a situation where the main economic variables show long-run balanced growth while, at the same time, they contribute to generate an accumulated stock of pollutants lower or equal than the critical value

The above-mentioned abatement efort , which is costly and endogenously decided by agents, will be measured in terms of output in such a way that these two variables relate to each other according to

\[\mathcal {B} (t) = Y (t) - Y _ {N} (t) = (1 - z (t)) Y (t).\tag{2}\]

Here represents, as in Stokey (1998) and the opposite to Reis (2001), a measure of the efective dirtiness of the technique used to produce. Obviously, because resources devoted to clean pollution could never pass the upper bound established by current production. Therefore, any choice for close to zero or one automatically makes the existing technique less or more polluting respectively. The above expression introduces a definition for the net output as

\[Y _ {N} (t) = z (t) Y (t).\tag{3}\]

The equation governing the motion of may be written as , where represents the emissions flow associated with the endogenously determined levels of polluting and abatement activities. This flow is increasing with respect to Y and decreasing with respect to , i.e. and . Function is assumed homogeneous of degree zero, i.e. an equal proportional increase in both output and abatement leaves the emissions flow unchanged. Consequently, the emissions flow may be rewritten as , where we assume strict concavity: lim and x 0⁺ x 1 . Actually, represents an efective upper bound for the emissions function, which is high enough to conduct the economy, if it prevails, to the state of ecological catastrophe.⁴ Now, substituting from the above variable definitions, we can specify the following diferential equation for the motion of the stock of pollutants

³An alternative to this constant exponential rate of pollution decay, which implies that natural regeneration is a linear function of the pollution stock, is the inverted U-shaped decay function modelized by Tahvonen and Withagen (1996) and Tahvonen and Salo (1996). This one implies that a pollution stock level that is suficiently high will reduce the rate of natural regeneration to zero.

\[\stackrel {\bullet} {S} (t) = E (1 - z (t)) - \delta S (t),\tag{4}\]

where , 0 < lim , lim and z 0⁺ z 1− That is, taking as reference which implies that no abatement efort is made and that emissions flow reaches the maximum level , the larger the reduction in z the more efective the reduction in emissions. Or, put in other words, as long as we produce with a cleaner technology, the efectiveness measured in terms of emissions reduction of any additional pollution abatement that reduces z, will be larger.

2.3 Investment

On the other hand, according to the aggregate resources constraint, net output may be devoted to consumption or capital accumulation. For the sake of simplicity we do not consider capital depreciation. Hence, net investment equals gross investment and the capital stock is governed by the following diferential equation

\[C (t) + \stackrel {\bullet} {K} (t) = Y (t) - \mathcal {B} (t).\tag{5}\]

⁴Tahvonen and Salo (1996) conceives this upper bound as the emissions level for which residents decide to move to other locations. On the other hand, we can interpret these emissions as a maximum level beyond which production starts to be delocalized by transferring abroad the polluting technology.

This equation also reflects the cost of the abatement activity in a very simple way. One unit of additional abatement efort is transformed automatically into a lower unit of output available for consumption or capital accumulation. This particular ‘one-to-one’ transformation, although not strictly necessary, contributes to simplify calculus.

2.4 Preferences

The economy is populated by many identical and infinitely lived agents. Population, denoted by N, is assumed to be growing at a constant rate The initial population is normalized to one. Individual preferences are assumed to be represented by a twice continuously diferentiable instantaneous utility function U (.), which depends positively on the current per capita consumption c(t) as well as negatively on the emissions flow [Gradus and Smulders (1993), Ligthart and Ploeg (1994), Selden and Song (1995), Reis (2001)]. Under this assumption, households do not take care for the stock of pollutants in the environment, but only for the current flow of polluting emissions. This may be justified on the basis that the local stock efect of pollution is assumed short-lived and the abatement activity, which reduces emissions and facilitates regeneration, makes the local stock efect negligible.⁵ As we have seen, emissions depend positively on production and negatively on abatement, two variables that appear related to each other according to (2). Moreover, the previous definition of the emissions function establishes a monotonic decreasing relationship between the emissions flow and the ratio pollution abatement to output. Hence, for the sake of simplicity, we will consider the abatement efort relative to the economy’s dimension as the second argument in the utility function. Instead of considering as a utility argument the emissions flow, which produces disutility, we assume that households derive utility directly from the aggregate efort addressed to reduce pollution emissions. That is, utility indirectly depends on environment quality, which is increased by abatement and reduced by production. In this model, however, we have the advantage that the dimension of the economy may be measured by production as well as by capital stock, given the linear form assumed for the production function. Therefore, the instantaneous utility function may be written as with and . Or, given that the relationship between abatement and production allows for the substitution , also as with . Moreover, we assume decreasing marginal utilities: and , as well as strict concavity with respect to both arguments taken together:

⁵This also implies that we ignore global stock efects in the representation of households’ preferences. An important stream of literature considers that welfare depends on the stock of pollution rather than on the current flow [Huang and Cai (1994), Mohtadi (1996), Tahvonen and Salo (1996), Byrne (1997), Kelly (2003)]. However, if the flow of pollution is increasing with production, then capital accumulation that increases future output also increases future flows of pollution. Hence, we find a general consensus in the literature [Gradus and Smulders (1993), Smulders and Gradus (1996), Aghion and Howitt (1998), Reis (2001)] according to which, in the context of this model, if we consider the stock of pollution as an argument in the utility function, we will obtain the same fundamental results but at the cost of a more complex analysis.

On the other hand, given that the structure of the model allows for the existence of a long-run balanced growth path, defined as an allocation in which per capita consumption grows at a constant rate and the dirtiness index is constant, to ensure that such a path exists for this model it is necessary to assume that the particular instantaneous utility function is multiplicatively separable and of the CIES form [King, Plosser and Rebelo (1988), Bovenberg and Smulders (1995; 1996), Smulders and Gradus (1996), Ladrón de Guevara et al. (1999)] ⁶

\[U \left(c (t), z (t)\right) = \frac {c (t) ^ {1 - \Phi}}{1 - \Phi} \left(1 - z (t)\right) ^ {\alpha (1 - \Phi)}.\tag{6}\]

In this function, the parameter that represents the relative weight of environmental cares in utility is assumed to be positive and lower than one, , and the inverse of the constant intertemporal elasticity of substitution is allowed to be lower or greater than one, . The previous function fulfill all the above mentioned assumptions concerning first and second derivatives. However, the strict concavity assumption requires as suficient condition that the determinant of the Hessian matrix be positive, which implies the additional parameter constraint:

⁶Bovenberg and Smulders (1995) also considers additional restrictions on ecological relationships and technology. With respect to the first, we anticipare that hereafter we are going to study conditions for ecologically non catastrophic states, and also analyze the problem suscited by empirical work concerning the Environmental Kuznets’ Curve. With respect to the second, we have to recall that our one sector and one accumulable factor model builds upon a linear production function which summarizes the whole set of technological requirements postulated by these authors.

⁷Environmental literature has long dealt with the sign of the second order cross deriv-

3 The competitive solution

First of all, we will consider the structure of this economy from the point of view of the non regulated competitive equilibrium. In this economy, assuming that there are no depreciation charges, each competitive firm faces the following stationary optimization problem, given the absence of adjustment costs and any other intertemporal element in its present value maximization problem,

\[\begin{array}{c} \max _ {\{K _ {i}, z _ {i} \}} \Pi_ {i} = Y _ {i} - r K _ {i} - \mathcal {B} _ {i} \\ \text {s.t.} 0 \leqslant z _ {i} \leqslant 1, \\ Y _ {i} = A K _ {i}, \\ \mathcal {B} _ {i} = (1 - z _ {i}) Y _ {i}, \\ K _ {i} > 0. \end{array}\]

The first order conditions are

\[r = A z _ {i},\tag{7}\]

\[0 = \left(1 - z _ {i}\right) A K _ {i}.\tag{8}\]

From (8), because of the slackness condition, we observe that it is optimal from the point of view of the individual firm to fix and then, by (7), we get . These results imply zero quasi-rents at the maximum. But this also means that individual firms have no incentive to allocate resources to pollution abatement because of the implicit externality originated in the conflict between the private nature of the cost of this activity and the social nature of its benefits, which are beyond the firm’s control. Consequently, in the competitive equilibrium we will observe that production is undertaken by firms with the most polluting of the available production techniques.

U_c2 = U_cz = α(1 Φ)c−Φ (1 z)^α(1−^Φ)−¹

ative of the instantaneous utility function [Michel and Rotillon (1995), Mohtadi (1996)]. long as Φ is greater (smaller) than one or, put in other words, as long as the intertemporal elasticity of substitution is smaller (greater) than one. Empirical evidence seems to corroborate the case of a low intertemporal elasticity of substitution and, hence, This U_c2 < 0. implies that the marginal utility of consumption decreases as the environmental quality increases, namely, consumption and pollution are complements in terms of preferences. However, the model works exactly the same in the opposite case where U_c2 > 0.

Instead, households preferences are sensitive to the pollution emissions flow as has been represented in their utility functions. They show a clear preference for reducing z below unity according to the assumption We don’t need to explicitly solve the optimization problem for households. Actually, given the form of the utility function, households will never choose such an extreme value for z and this will generate a fundamental market mismatch that exactly reflects the consequences of the above-mentioned externality. There is no incentive for agents to internalize the negative externality that they generate and, consequently, the equilibrium path is not Pareto optimal.

On the other hand, although it has not been included as an argument in the utility function, we also have to consider in our analysis the evolution of the aggregate stock of pollutants because it is crucial from the point of view of the sustainability of the long-run economy’s aggregate outcomes. However, this is not controlled by firms or households individually because both take as given the level of this stock. In fact, the competitive equilibrium that leads the firm to choose the dirtiest technique to produce, , has dreadful implications with respect to the sustainability problem. Consider the equation governing the motion of the aggregate stock of pollutants under such an extreme value: , which ofers the following solution: . Then monotonically increases converging to the value , which means that eventually the state of ecological catastrophe will be reached.

In conclusion, the presence of a welfare pollution externality in a decentralized competitive economy, which imply not much abatement and too much pollution, call for one or more kinds of interventions. Without any corrective environmental policy, the environment will be damaged up to the level of irreversible catastrophe and sustained growth, if there exists, will not be sustainable.

4 Optimality conditions

Now, we will focus on the socially optimal solution for the model economy described in previous sections, which simultaneously internalizes the costs and benefits from pollution abatement and takes into account the evolution of the aggregate stock of pollutants in the environment.⁸ In this section we will only study interior solutions. Accordingly, we resolve the problem and obtain the non-constrained optimal trajectories for which we check below whether they are ecologically sustainable or not. In fact, we derive suficient conditions on parameters that ensure such a sustainability. Although the optimization problem may be formulated introducing as an explicit constraint the no catastrophe condition, which implies that the central planner takes care of trajectories leading to catastrophic states and optimally decides to avoid them by choosing the controls appropriately, we leave such a procedure for a next section and specify for now the dynamic optimization problem without this state constraint that applies throughout the planing period.⁹

Moreover, a similar decision is made here with respect to the control constraints . At this stage, we ignore all these constraints in the formulation of the optimization problem, but later on we will check them for the optimal unconstrained trajectories. This allows us to identify two parameter conditions which make the control variable bounded. Under these premises the planner’s problem consists in choosing the sequence which, for a given positive social rate of discount , solve the optimization problem

\[\begin{array}{c} \max _ {\{K, S, c, z \}} \int_ {t _ {0}} ^ {\infty} \left[ \frac {c ^ {1 - \Phi}}{1 - \Phi} (1 - z) ^ {\alpha (1 - \Phi)} \right] e ^ {- (\rho - n) (t - t _ {0})} d t \\ s. t. \qquad (1) \text {-} (5), \\ \text {for} k (t _ {0}) = k _ {0} > 0 \text {and} s (t _ {0}) = s _ {0} > 0 \text {given}. \end{array}\]

From now on we will use lowercase letters to represent variables in per

⁸The matter of how to replicate the eficient path in a competitive economy by means of an optimal environmental policy, is beyond the scope of this paper. However, the reader may find in Mohtadi (1996), Smulders and Gradus (1996), as well as in Rubio and Aznar (2002) the study of how the government can implement pollution charges, emission standards and public abatement, which allow the competitive economy to generate eficient outcomes in an endogenous growth model with an AK technology.

⁹We do that in line with the recommendation of Chiang (1992): “Although we cannot in general expect the unconstrained solution to work, it is not a bad idea to try anyway. Should that solution turn out to satisfy the constraint, then the problem would be solved. Even if not, useful clues will usually emerge regarding the nature of the true solution.” For a complete formulation of the optimal control problem, including every relevant constraint, the reader may look at the Appendix.

capita terms. The current value Hamiltonian is

\[H _ {\{c, z, q, k, \mu , s \}} ^ {c} = \frac {c ^ {1 - \Phi} (1 - z) ^ {\alpha (1 - \Phi)}}{1 - \Phi} + q [ A k z - c - n k ] + \mu [ e (1 - z) - (\delta + n) s ],\]

where and are the co-states for k and s, respectively, and represent their corresponding shadow prices. The first order necessary conditions are

\[q = c ^ {- \Phi} (1 - z) ^ {\alpha (1 - \Phi)},\tag{9}\]

\[q + \mu \frac {e _ {z}}{A k} = \frac {\alpha c ^ {1 - \Phi} (1 - z) ^ {\alpha (1 - \Phi)}}{A k (1 - z)}.\tag{10}\]

As we have seen, gross product may be allocated to consumption, investment or abatement. On the margin, according to (9), goods must be equally valuable if they are consumed or accumulated as new physical capital. Namely, the marginal utility of consumption today must be equal to the marginal shadow value of physical capital (consumption tomorrow). According to (10), at equilibrium the implicit price of a more dirty technique, that is, the value measured in units of utility of the net marginal product from a more dirty technique plus the shadow value of the marginal emission associated with such a more dirty technique , must be equal to the marginal utility of a cleaner one. Namely, the entire valuation of a marginal reduction in resources devoted to abatement, which contributes to increase consumption (present future) as well as the stock of pollutants, must be equal to the marginal utility of those resources when devoted to abatement, which contribute to increase environmental quality. Moreover, the dynamic conditions for quantities and shadow prices are

\[\stackrel {\bullet} {k} = A k z - c - n k,\tag{11}\]

\[\stackrel {\bullet} {q} = \rho q - A z q,\tag{12}\]

\[\stackrel {\bullet} {s} = e (1 - z) - (\delta + n) s,\tag{13}\]

\[\dot {\mu} = (\rho + \delta) \mu ,\tag{14}\]

together with initial conditions and and the transversality conditions

\[\lim _ {t \to \infty} e ^ {- (\rho - n) (t - t _ {0})} q k = 0,\tag{15}\]

\[\lim _ {t \to \infty} e ^ {- (\rho - n) (t - t _ {0})} \mu s = 0.\tag{16}\]

Before the resolution of the whole dynamic system, we can study the block of equations related to the stock of pollutants and its shadow price. Looking for that, we first resolve (14): . Then, we integrate (13): −^(δ+n)(t−^t0⁾ + . Finally, if we substitute both into the transversality condition (16), the result is lim . This condition holds if, t→∞ and only if, because the integral on the r.h.s. cannot be negative. Consequently, the above-mentioned solution to (14) determines that,

\[\mu (t) = 0.\tag{17}\]

This implies that even when the central planner internalizes the costs and benefits of abatement activity, and takes into account the evolution of the aggregate stock of pollutants in the environment, he optimally assigns zero value to the social shadow price of the stock of pollutants. This is so because although the central planner ensures an eficient resource management in the economy, his main goal consists in maximizing social welfare and the stock of pollutants has been left out of that function.

If we come back to the set of first order conditions, (17), (9) and (10) imply the tangency condition

\[z = 1 - \frac {\alpha}{A} \frac {c}{k},\tag{18}\]

as well as the two control functions

\[c = c (k, q) = \left(\frac {\alpha}{A}\right) ^ {\frac {\alpha (1 - \Phi)}{\Phi - \alpha (1 - \Phi)}} q ^ {\frac {- 1}{\Phi - \alpha (1 - \Phi)}} k ^ {\frac {- \alpha (1 - \Phi)}{\Phi - \alpha (1 - \Phi)}},\tag{19}\]

\[z = z (k, q) = 1 - \left(\frac {\alpha}{A}\right) ^ {\frac {\Phi}{\Phi - \alpha (1 - \Phi)}} q ^ {\frac {- 1}{\Phi - \alpha (1 - \Phi)}} k ^ {\frac {- \Phi}{\Phi - \alpha (1 - \Phi)}}.\tag{20}\]

Now, substituting (19) and (20) into (11) and (12), we get the dynamic system

\[\stackrel {\bullet} {k} = (A - n) k - A ^ {\frac {- \alpha (1 - \Phi)}{\Phi - \alpha (1 - \Phi)}} \left[ \alpha^ {\frac {\Phi}{\Phi - \alpha (1 - \Phi)}} + \alpha^ {\frac {\alpha (1 - \Phi)}{\Phi - \alpha (1 - \Phi)}} \right] q ^ {\frac {- 1}{\Phi - \alpha (1 - \Phi)}} k ^ {\frac {- \alpha (1 - \Phi)}{\Phi - \alpha (1 - \Phi)}},\tag{21}\]

\[\stackrel {\bullet} {q} = (\rho - A) q + A ^ {\frac {- \alpha (1 - \Phi)}{\Phi - \alpha (1 - \Phi)}} \left[ \alpha^ {\frac {\Phi}{\Phi - \alpha (1 - \Phi)}} \right] q ^ {\frac {- (1 - \Phi + \alpha (1 - \Phi))}{\Phi - \alpha (1 - \Phi)}} k ^ {\frac {- \Phi}{\Phi - \alpha (1 - \Phi)}},\tag{22}\]

with the initial condition and the transversality condition (15). These two diferential equations conform a non-linear dynamic system, which has a particular structure that makes it susceptible of being solved in closed form. Following the method developed in Ruiz-Tamarit and Ventura-Marco (2004), from which Propositions 1 and 2 lead us to conclude that it does exist a unique optimal solution trajectory for and , we get the closed form representation

\[k (t) = k _ {0} \exp \left\{\frac {A - \rho - \alpha (\rho - n)}{\Phi - \alpha (1 - \Phi)} (t - t _ {0}) \right\},\tag{23}\]

\[q (t) = q (t _ {0}) \exp \left\{- \Phi \frac {A - \rho - \alpha (\rho - n)}{\Phi - \alpha (1 - \Phi)} (t - t _ {0}) \right\},\tag{24}\]

\[q (t _ {0}) ^ {\frac {1}{\Phi - \alpha (1 - \Phi)}} k _ {0} ^ {\frac {\Phi}{\Phi - \alpha (1 - \Phi)}} = \frac {\Phi - \alpha (1 - \Phi)}{\rho - A + \Phi (A - n)} \left(\frac {\alpha}{A}\right) ^ {\frac {\alpha (1 - \Phi)}{\Phi - \alpha (1 - \Phi)}}.\tag{25}\]

Given the initial capital stock , equation (25), which arises directly from the transversality condition, gives the initial value for the shadow price Once the two initial values are known, equations (23) and (24) determine unequivocally the complete trajectories for these two variables. For any other than the one given by (25) the economy places on an explosive trajectory which does not satisfy optimality conditions, in particular the transversality condition. Moreover, given , the transversality condition holds if, and only if, . This parameter constraint must be satisfied for any positive intertemporal elasticity of substitution, i.e. , which is not obvious. However, the strict concavity assumption on the utility function imposes the additional parameter constraint

\[\Phi > \alpha (1 - \Phi).\tag{26}\]

Then, the transversality condition (15) holds if, and only if,

\[\rho > A (1 - \Phi) + \Phi n.\tag{27}\]

5 Sustained optimal balanced growth

Given (23) and the production function in per capita terms that arises from (1), we obtain

\[y (t) = A k _ {0} \exp \left\{\frac {A - \rho - \alpha (\rho - n)}{\Phi - \alpha (1 - \Phi)} (t - t _ {0}) \right\},\tag{28}\]

\[\gamma_ {y} (t) = \gamma_ {k} (t) = \gamma^ {*} = \frac {A - \rho - \alpha (\rho - n)}{\Phi - \alpha (1 - \Phi)}.\tag{29}\]

The growth rates of per capita capital stock and output are equal to each other and constant over time along their respective optimal solution trajectories. On the other hand, using the control functions as given in (19) and (20) for consumption and the degree of dirtiness associated with the technique, we get the optimal solution trajectories for these two variables

\[\begin{array}{r l} & c (t) = \frac {\rho - A + \Phi (A - n)}{\Phi - \alpha (1 - \Phi)} k _ {0} \exp \left\{\frac {A - \rho - \alpha (\rho - n)}{\Phi - \alpha (1 - \Phi)} (t - t _ {0}) \right\}, \\ & \frac {c (t)}{k (t)} = \left(\frac {c}{k}\right) ^ {*} = \frac {\rho - A + \Phi (A - n)}{\Phi - \alpha (1 - \Phi)}, \\ & \gamma_ {c} (t) = - \frac {\gamma_ {q} (t)}{\Phi} = \gamma^ {*}, \\ & z (t) = z ^ {*} = \frac {A \Phi - \alpha \rho + \alpha \Phi n}{A (\Phi - \alpha (1 - \Phi))}, \\ & \gamma_ {z} ^ {*} = 0. \end{array}\tag{30}\]

(31)

The second one is expected to be bounded, i.e. . However, for this to be ensured we need additional parameter constraints. In particular,

\[\Phi (A - n) + n (\Phi - \alpha (1 - \Phi)) \geqslant \alpha (\rho - n),\tag{32}\]

\[\Phi (A - n) \geqslant A - \rho ,\tag{33}\]

where it may be easily checked that (33) encompasses (27).

These results completely characterize the dynamic system corresponding to the socially optimal solution. k, c and y grow at the same constant rate; the ratio consumption to capital stock is constant and positive; and the dirtiness index remains fixed forever at a constant value between zero and one. Therefore, the model does not predict transitional dynamics and all the endogenous variables conform a balanced growth path from the beginning. 10

¹⁰The intuition for why the rate of growth is constant is related to the social rate of return on capital. In this model:

α = 0,

∂Y_N AΦ αρ + αΦn r∗ 三 = Az∗ = ∂K Φ α(1 Φ)

the real return to capital is constant, although endogenously determined by preferences and technology parameters. The previous expression also shows that only in the absence of environmental concerns, the interest rate is equal to A, as in the standard model. Otherwise, it is lower because of the transversality condition (27).

If we examine a little more the socially optimal balanced growth path, we find that there could be either positive or negative growth, as well as stationarity. Given (29) and the strict concavity assumption on the utility function, a positive rate of growth arises when . This condition is compatible, and may be combined, with the parameter constraint corresponding to the transversality condition as well as those representing the lower and upper bounds for , giving

\[\Phi (A - n) + n (\Phi - \alpha (1 - \Phi)) > \Phi (A - n) \geqslant A - \rho > \alpha (\rho - n) > 0.\tag{34}\]

On the other hand, stationarity arises when ， which combined with the remaining parameter constraints leads to

\[\Phi (A - n) + n (\Phi - \alpha (1 - \Phi)) > \Phi (A - n) \geqslant A - \rho = \alpha (\rho - n) > 0.\tag{35}\]

In this case all the variables conform a steady state, which is ‘chosen’ among a multiplicity by the predetermined initial value of per capita capital stock. Finally, although less economically relevant, we could also find negative growth, , when . This is the only case among the previous ones that allows for

The absence of transitional dynamics that makes the short-run identical to the long-run, allows us to simplify the comparative analysis for the socially optimal rate of growth and dirtiness index. In fact, we find the following parameter dependences for these two endogenous variables of the model

\[\gamma^ {*} = \gamma \left( \begin{array}{c} + \\ A, \bar {\rho}, \bar {\Phi}, \bar {\alpha}, \bar {n} \end{array} \right),\tag{36}\]

\[z ^ {*} = z \left( \begin{array}{c} +, - \\ A, \bar {\rho}, \bar {\Phi}, \bar {\alpha}, n \end{array} \right).\tag{37}\]

The signs associated with A, and Φ are the usual in the canonical AK model: the larger the capital productivity and the higher the patience of agents, the greater the rate of growth. A newer but very intuitive result is found here: the higher the weight of environmental cares in utility the smaller the rate of growth. That is, for higher values of that imply a higher marginal utility of abatement and a lower rate of return on capital, the central planner optimally decides to devote more resources to abatement and less to capital accumulation and, hence, to growth. However, an striking result arises in this model because of the apparent positive relationship between the rate of growth and the population growth rate. This result is absolutely dependent on the presence of environmental cares in the model, because only in such cases a higher population growth rate leads the central planner to divert resources from abatement and consumption towards capital accumulation. This investment flow is strong enough to compensate for the new capital requirements associated with a greater population, and so it also contributes to generate a greater rate of growth. Moreover, this efect is stronger as higher is the weight of environmental cares in the utility function.¹²

¹¹More technical details about these cases, as well as a complete geometric characterization, may be found in Ruiz-Tamarit and Ventura-Marco (2004).

The dirtiness index, in turn, depends positively on the productivity parameter when the intertemporal elasticity of substitution is greater than one, but the sign of this relationship cannot be analytically decided for values of such elasticity lower than one. Moreover, for a positive balanced growth path, the higher the patience of agents the higher the value of z. This happens because when consumers show a high level of patience, the central planner optimally decides to reallocate resources towards capital accumulation, which enhance growth. This is done so intensively that even diverts some of the resources previously devoted to pollution abatement, which leads to produce with a more dirty technique. On the other hand, because of the crowding out efect, the higher the weight of environmental cares in the utility function the smaller the dirtiness index. And finally, the greater the population growth rate the higher the dirtiness associated with the efective production technique, which occurs because for higher population growth rates the central planner decides to divert more resources from abatement efort.¹³

These two variables are closely related to each other. Actually, we can make this relationship evident from the first order conditions (11) and (18). If we take the first one and divide by k, and then substitute for the ratio c

¹²A similar result, although there it is based on a diferent explanation, may be found in Bartolini and Bonatti (2003).

¹³The results concerning the population growth rate are consistent with propositions dicussed and tested in Cropper and Grifiths (1994). In that paper, the environment is not a factor that limits productivity as population expands, but a good which quality is degraded by a growing population.

from the second, we get for any

\[\gamma^ {*} = - \left(\frac {A + \alpha n}{\alpha}\right) + \left(\frac {A + \alpha A}{\alpha}\right) z ^ {*},\tag{38}\]

from which we can deduce the following pairs of values of reference: and The positive relationship between and suggests that tighter pollution controls and increased abatement, which reduce the dirtiness index, will have negative efects on the optimal rate of growth. This fact only reflects the previous crowding out result according to which greener preferences associated with a shift in preferences towards more environmental concern, i.e. a rise in afects negatively both the dirtiness index and the rate of growth.

6 Environmental catastrophe and optimal growth sustainability

One major problem considered in previous models of endogenous growth is the sustainability of the long-run socially optimal and competitive balanced growth paths. This problem has been largely studied in environmental literature where several definitions of sustainability have been proposed [Pezzey (1997), Chichilnisky (1997)]. The most usual concepts of sustainability rely on the intertemporal evolution of utility and consumption, and impose a nondeclining trajectory for any of these two variables. In general, sustainability has been conceived as a situation where the needs of the present are satisfied without compromising the needs of the future. This suggests a related problem associated with the valuation of the well-being of present and future generations, and points to the equity condition according to which it has to be avoided the underestimation of future utilities. From the perspective of our own model, the previous requirements for sustainability are always met given that the socially optimal choices guarantee a monotonically increasing profile for consumption and utility. However, although conditions for a positive long-run rate of growth are satisfied, there is a trade-of between growth and environmental quality. Environmental quality is measured by the absence of pollution and increases with abatement but diminishes with production. This trade-of, which results from agent decisions, involve the level of abatement expenditures and, consequently, the value of the dirtiness index.

Sustainability may also be inspected into the model looking at the positive relationship between the dirtiness index and the rate of growth that was shown in (38), which represents in other words the above-mentioned tradeof. As such relationship shows, the more clean the used technology is, the lower the rate of growth. In particular, our model built upon the assumption that people show environmental awareness predicts a lower, or at most an equal, rate of growth relative to the rate of growth arising from the standard model where no environmental concerns do exist . Moreover, as has been shown in the previous section, may be zero provided that , which is in accordance with the premise that more environmental concern comes in detriment of growth. But this is an extreme case and, for any other value of α smaller than the previous one, we are still allowed to conclude that positive sustained growth and environment preservation may be compatible in the long-run. That is, as long as

Nevertheless, the main question we are going to consider now is whether the socially optimal results from the previous section are compatible with the general condition of no environmental catastrophe, which imposes the constraint that must be lower or equal than . In particular, we have to study the current level of at any moment in time associated with the positive balanced growth path for the variables of the model. This approach becomes necessary because along the previous formulation of the central planner’s optimization problem we did not take explicitly into account the inequality constraint Actually, none of the previously studied results give rise to explicit constraints on the dynamics of

The equation governing the dynamics of the stock of pollutants is , with and . Solving backward for we get

\[S (t) = S _ {0} e ^ {- \delta (t - t _ {0})} + \int_ {t _ {0}} ^ {t} E (1 - z (\tau)) e ^ {- \delta (t - \tau)} d \tau .\tag{39}\]

The first term on the r.h.s. is a finite value that approaches zero as tends , and the integral of the second term is a convergent one as long as the function grows at most at a positive exponential rate lower than In fact, the assumed more restrictive condition sufices to guarantee such a convergence of the integral, given the presence of an exponential discount term. Beyond this, since the model predicts that is always chosen as a constant value, the above expression for may be

simplified to

\[S (t) = \left(S _ {0} - \frac {E (1 - z)}{\delta}\right) e ^ {- \delta (t - t _ {0})} + \frac {E (1 - z)}{\delta},\tag{40}\]

where lim Given and a constant value for t→∞ the stock of pollutants always converges monotonically to a constant finite value, which is determined by emissions corresponding to such a value of the dirtiness index and the constant rate of natural regeneration This result, however, does not sufices to exclude the threat of an environmental catastrophe. This situation may arise in our model for a suficiently high value of given that and depends positively on

In particular, z may be chosen at a level for which emissions flow is exactly balanced out by the regeneration corresponding to the natural capacity of the environment to absorb pollution. In this case, a steady state arises from the beginnin and then t

\[S (t) = \bar {S} _ {\infty} = S _ {0} = \frac {E (1 - \bar {z})}{\delta}.\tag{41}\]

Therefore, if then monotonically decreases below converging to a certain , while then monotonically increases converging to some . The no-catastrophe condition requires that for every and, in particular, that , being the socially optimal path for the stock of pollutants. This one emerges from the optimal balanced growth path, and is determined by substituting the value into (40). Given monotonicity, the no-catastrophe condition is ensured when , where is the limiting value for the stock of pollutants

Let be the value of that eventually makes the stock of pollutants to catch up with the catastrophic level S^max, and which satisfies . Then, looking for a socially optimal positive balanced growth path that be compatible with a non-catastrophic ecological state of the economy, we additionally need to impose the parameter constraint

¹⁴Sometimes, the condition for stationarity has been taken as a condition for sustainability of the balanced growth path [Chevé (2000)]. This one may be considered as a very strong, near the consevationists, position where the stock of pollutants and other environmental variables remain constant, while the rest of economic variables are still allowed to grow at a constant positive rate. However, as we will show immediately, it becomes excessive unless the upper ecological limit had been reached.

\[A \Phi - \alpha \rho + \alpha \Phi n \leqslant z ^ {\max} (A \Phi - \alpha A + \alpha \Phi A).\tag{42}\]

This condition, which corresponds to , is necessary and suficient and also establishes the margins for sustainable optimal growth, given that parameter depends positively on the ecologically determined parameters and δ. Thus, if parameters determining the values of the variables along the balanced growth path satisfy condition (42), then sustained socially optimal growth is also ecologically sustainable in the long-run.

Now, to complement the previous study of the sustainability of the longrun balanced growth path, we will derive the socially optimal behavior of the main variables in the model once the limits of ecological sustainable growth have been reached. Namely, the behavior of the economy when catch up with the upper-limit . Here we look at the general optimal control solution when the non-negativity constraint, the control variable constraint and the state variable constraint are explicitly introduced in the dynamic optimization problem. This has been done in the Appendix and, from now on, we will work with the first order conditions appearing there.

Consider that for some . This happens when the socially optimal value of the dirtiness index may be found on the interval . According to what has been previously shown in this section then , whereas if then . Actually, only the latter becomes economically interesting at this point because in such a case the ecological limit to growth appears as a binding constraint in finite time.¹⁵ Hence, . Then, given that θ is not allowed to be negative, we conclude that and . The latter implies that at the dirtiness index, a variable that admits jumps, will take the value

\[z \left(t ^ {c}\right) \equiv z ^ {c} = 1 - E ^ {- 1} \left(\delta S ^ {\mathrm{max}}\right) = z ^ {\mathrm{max}},\tag{43}\]

\[t ^ {c} = t _ {0} + \ln \left(\frac {\frac {E (1 - z ^ {*})}{\delta} - S _ {0}}{\frac {E (1 - z ^ {*})}{\delta} - S ^ {\mathrm{max}}}\right) ^ {\frac {1}{\delta}},\]

¹⁵ It is easy to show that

tc − t0

T =

T µ −z∗, −S0, ⁺Smax, −^,+δ ¶. 十 -,+ , δ

which implies that the span of time elapsed before we reach the ecological limit, tc − ^t0^, depends on the involved parameters according to

irrespective of its previous value. This means that at the central planner optimally decides a discrete and instantaneous change in the value of the dirtiness index from to . Moreover, the strict inequality , which is required for a non-zero marginal utility of consumption, also implies

Therefore, the dynamic equation for s becomes , from which we deduce that and , which in turn implies that and for all . Thus, the dirtiness index will remain stuck to the value , and then , for all . On the other hand, the solution to the dynamic equation , starting from t^c, is . The dynamics for this shadow price couples to such of , which is caused by population growth. Moreover, the transversality condition lim imt→∞ plies , where the integral on the r.h.s. is bounded because the integrand converges to zero, given that θ is positive but decreasing and

The particular dynamics for and k arise from the solution to the dynamic system

\[q = (1 - z ^ {c}) ^ {\alpha (1 - \Phi)} c ^ {- \Phi},\tag{44}\]

\[\stackrel {\bullet} {c} = \left(\frac {A z ^ {c} - \rho}{\Phi}\right) c,\tag{45}\]

\[\stackrel {\bullet} {k} = (A z ^ {c} - n) k - c,\tag{46}\]

with the initial condition k (t^c) and the transversality condition

\[\lim _ {t \to \infty} e ^ {- (\rho - n) (t - t ^ {c})} c ^ {- \Phi} k = 0.\tag{47}\]

This dynamic system is similar in its structure to the standard AK model, but the constant marginal productivity of capital is now . Consequently, the particular solution may be represented by

\[q (t) = q (t ^ {c}) \exp \left\{- (A z ^ {c} - \rho) (t - t ^ {c}) \right\},\tag{48}\]

\[c (t) = \frac {\rho - A z ^ {c} + \Phi (A z ^ {c} - n)}{\Phi} k (t ^ {c}) \exp \left\{\frac {A z ^ {c} - \rho}{\Phi} (t - t ^ {c}) \right\},\tag{49}\]

\[q (t ^ {c}) = \left(\frac {\Phi (1 - z ^ {c}) ^ {\frac {\alpha (1 - \Phi)}{\Phi}}}{\rho - A z ^ {c} + \Phi (A z ^ {c} - n)}\right) ^ {\Phi} \frac {1}{k (t ^ {c}) ^ {\Phi}},\tag{50}\]

\[k (t) = k \left(t ^ {c}\right) \exp \left\{\frac {A z ^ {c} - \rho}{\Phi} (t - t ^ {c}) \right\},\tag{51}\]

\[\frac {- \rho + A z ^ {c} - \Phi (A z ^ {c} - n)}{\Phi} < 0,\tag{52}\]

\[y (t) = A k \left(t ^ {c}\right) \exp \left\{\frac {A z ^ {c} - \rho}{\Phi} (t - t ^ {c}) \right\},\tag{53}\]

\[\gamma_ {y} (t) = \gamma_ {c} (t) = \gamma_ {k} (t) = \gamma^ {c} = \frac {A z ^ {c} - \rho}{\Phi}.\tag{54}\]

These results completely characterize the economic system after period , just on the limits of ecological catastrophe with an accumulated stock of pollutants equal to and the dirtiness index fixed at the level ， which depends exclusively on the parameters of the emissions function, the rate of natural regeneration and the previous maximum stock according to (43). Moreover, variables c and y grow at a common constant rate, which is positive under the assumption of a constant return to capital greater than the social discount rate, . In this case, however, the socially optimal rate of growth and dirtiness index do not depend on the weight of environmental cares in utility, or the population growth rate, . Finally, according to the parameter constraint (52), which comes from the transversality condition (47), the ratio consumption to capital stock is constant and positive. Therefore, the model does not show transitional dynamics beyond and all the endogenous variables conform a unique socially optimal balanced growth path.

Along this balanced growth path, the rate of growth may be greater, equal or smaller than depending on the sign of the relationship between and . That is if, and only if, . However, given that we have been considering initially a higher value of the dirtiness index, which lead the economy to the limits of ecological catastrophe in finite time, and then it was optimally changed to a lower value that ensures a constant aggregate stock of pollutants in the environment, we conclude that the rate of growth along this new path is smaller than the previous one.

7 Long term environmental Kuznets’ curve and environmental protection policies

Beyond the problem of sustainability of the optimal balanced growth path we have to deal with the environmental Kuznets’ curve (EKC) hypothesis, which suggests an inverted U-shaped relationship between pollution emissions and per capita income levels. Recent empirical work on this subject have documented cases for which the previous pattern holds. That is, economic growth leads to higher emissions until income reaches a critical turning point, and thereafter emissions decrease. Some analysts recognize in this hypothesis the justification for the classical proposition which asserts that pollution will not be a problem in the long term because of the beneficial efects of economic growth for the environmental quality.¹⁶ In our opinion, these two problems (long-run growth sustainability and the environmental Kuznets’ curve) require a diferentiated and particular analysis each of them. Along the previous section we have studied the first one. Now, we will show that an inverted U-shaped function connecting emissions and production may also be deduced from our framework. Overall, we conclude that growth is not a definitive solution for the environmental pollution problem and admit that environmental active policies are still needed.

The EKC hypothesis has had a traditional intertemporal dynamic interpretation for an isolated country [Borghesi (2001)], built upon growth models that show short-run transitional dynamics. Our model, instead, because of its particular nature cannot produce transitional dynamics. Consequently, we regain here an alternative long-run lecture of this hypothesis, which connects with the concept of development and relates to some parameter changes experienced by economies along such a process [Arrow et al. (1995), Bruyn (1997), Vincent (1997)].

Our construction relies on two cornerstones. On the one hand, beyond the three most conventionally assumed channels whereby income growth afects environmental quality (scale, composition and technique efects), Grossman and Krueger (1995) considers that the state of the environment may deteriorate or improve along time if consumer tastes shift toward less or more environmental awareness, causing an autonomous shift in demands for environmental safeguards. Hence, after the initial deterioration, an eventual improvement of the environment may arise from the increased demand for environmental protection, based on the increased willingness to pay for environmental cares at higher levels of income per capita. On the other hand, there is an empirically well-documented demographic relationship between per capita income levels and population growth rates: the demographic transition phenomenon, which happens along the development process. This transition has very clear implications for the rates of population growth in agricultural, or subsistence, industrializing and services-oriented economies respectively [Kremer (1993), Mincer (1995), Barro and Sala-i-Martín (1995), Dahan and Tsiddom (1998), Perman et al. (1999), Tabata (2003)]. Combining the two elements, we can first postulate for low rates of population growth and high environmental concerns at the initial stages of the development process, when economies are essentially agricultural and experience a limited impact from economic activities on the environment. Then, at the intermediate stages, when economies become fundamentally industrial, the rates of population growth are higher and the environmental concern lower. Finally, for high developed and basically services-oriented economies the rates of population growth are again low and the environmental concern high.

¹⁶According to this, if the EKC hypothesis is satisfied, instead of being a threat to the environment, economic growth that moves the economy from lower to higher levels of income per capita improves it. This conclusion, however, is not generally accepted in the literature because the EKC seems to be only a valid description for a subset of all possible pollutants and countries [Grossman and Krueger (1996), Bimonte (2001), Borghesi (2001)]. Despite this, many authors have recommended a policy of wait-and-see, based on an absolute trust in such a naive interpretation of the the EKC hypothesis.

Therefore, we can modelize a long term EKC on the basis of the evolution and changes experienced by two structural parameters of the model alone. In the long term the economy moves from the less-developed state with a low level of income per capita, towards the more developed one with a higher level of income per capita. According to what has been said in the previous paragraph, this economy may be characterized with the corresponding low or high values of the rate of population growth, and the environmental concern, for any ven set of invariant parameters , and . Taking into account the comparative results for the long-run rate of growth as summarized in (36), , we can hypothesize the following relationship between the level and the rate of growth of the per capita income

\[\gamma = \phi y (2 \omega - y),\tag{55}\]

where the constant and positive parameter represents the transformation coeficient from the level to the rate of growth, while ω stands for the level of income per capita for which the maximum rate of growth is attained. Moreover, the result previously shown in (38) allows us to transform from the rate of growth to the value of the dirtiness index z, which in combination with (55) gives us

\[z = \left(\frac {A + \alpha n}{A + \alpha A}\right) + \left(\frac {\alpha \phi}{A (1 + \alpha)}\right) y (2 \omega - y).\tag{56}\]

Finally, we may connect with the emissions flow, using the function , which has been characterized before as a function satisfying 0, lim lim 2 and x 0⁺ x 1− Then, substituting for from (56) we get the Environmental Kuznets’ Curve

\[E = E \left(\left(\frac {\alpha (A - n)}{A (1 + \alpha)}\right) - \left(\frac {2 \alpha \phi \omega}{A (1 + \alpha)}\right) y + \left(\frac {\alpha \phi}{A (1 + \alpha)}\right) y ^ {2}\right).\tag{57}\]

This function shows the properties: (i) being positive for and negative for Consequently, the relationship between emissions flow and income per capita is strictly concave, increasing for low levels of income per capita and decreasing for higher levels of this variable beyond the critical value ω. This pattern just replicates the observed hump-shaped relationship between pollution and income¹⁷, and emerges as a direct consequence of the inverted U-shaped relationship between z, the index of dirtiness associated with the technique, and the level of activity y, which was obtained in (56).

Moreover, one variable that has played an important role in the discussion of the EKC hypothesis is the income elasticity for environmental quality. The value of this elasticity is placed among the main factors causing the downturn of polluting emissions, but there is not a general consensus about the exact definition of this ‘good’ with respect to income [Magnani (2000)]. In spite of the fact that many authors have claimed that environment is a luxury good and the income elasticity is above unity, others manifest serious doubts about this assumption and even prove that this is neither a necessary nor suficient condition for the EKC hypothesis to be satisfied. In our framework, the abatement efort is an indirect indicator of the environmental quality. If we rewrite (2) in per capita terms as , then the income elasticity may be easily computed using the previous expressions as

¹⁷Recent empirical studies have shown that, for some countries and pollutants, the best functional form is cubic, implying that for very high levels of income per capita environmental degradation starts to increase again [Torras and Boyce (1998)].

\[\epsilon_ {\mathfrak {h}, y} = 1 - \frac {2 \alpha \phi y (\omega - y)}{A (1 + \alpha) \left(\left(\frac {\alpha (A - n)}{A (1 + \alpha)}\right) - \left(\frac {2 \alpha \phi \omega}{A (1 + \alpha)}\right) y + \left(\frac {\alpha \phi}{A (1 + \alpha)}\right) y ^ {2}\right)}.\tag{58}\]

The last term on the r.h.s. is positive for and negative for Consequently, the income elasticity for environmental quality is less than one for but bigger than one for . That is, along the initial stages of development the elasticity of abatement efort to income remains below unity, but for higher development levels this elasticity becomes greater than one. Accordingly, the environmental quality appears as a luxury good only for high levels of income per capita: as countries get richer abatement expenditures will increase, but only when a certain level of income per capita has been surpassed will they increase more than proportionally. This feature is also shown by many other goods, as for example education, with which environmental quality shares the important property that they generate positive externalities over the economy.

This view of the EKC may be supported by a vast empirical literature which, analyzing cross-sectional or panel data, finds that economic growth and development bring an initial phase of environmental deterioration followed by a subsequent phase of improvement [World Bank (1992), Selden and Song (1994), Holtz-Eakin and Selden (1995), Grossman and Krueger (1995), Bruyn et al. (1998), List and Gallet (1999), Harbaugh et al. (2000)]. The picture has been perfectly summarized by Panayotou (1993) in the following sentence: “At low levels of development both the quantity and intensity of environmental degradation is limited to the impacts of subsistence economic activity on the resource base and to limited quantities of biodegradable wastes. As economic development accelerates with the intensification of agriculture and other resource extraction and the take of of industrialization, the rates of resource depletion begin to exceed the rates ofresource regeneration, and waste generation increases in quantity and toxicity. At higher levels of development, structural change towards information-intensive industries and services, coupled with increased environmental awareness, enforcement of environmental regulations, better technology and higher environmental expenditures, result in leveling of and gradual decline of environmental degradation”. Therefore, the EKC hypothesis accounts for an evolutionary progression associated with the diferent stages of the development process historically followed by many nations, from clean agricultural economies to clean services economies, going through polluting industrial economies with high detrimental efects on the environmental quality.

Despite the previous considerations about the classical hypothesis of an inverted U-shaped relationship between pollution emissions and per capita income levels, only a very superficial interpretation of its meaning could lead the analysts to believe that the best thing the policy-makers can do is to keep out of active environmental protection policies. Actually, growth is not a panacea for the environment [Arrow et al. (1995)]. As we have seen, the externality associated with pollution emissions and abatement makes the environmental problem very dificult, if not impossible, to resolve in a competitive decentralized economy. Consequently, in no one case can be expected that the sustainability problem will automatically be solved as a result of economic growth without government interventions and environmental policies.

In this paper, while studying the socially optimal solution to the environmental problem as opposed to the decentralized one, we have identified diferent opportunities for government interventions. First of all, an institutional one, which involves the government correcting the externality associated with pollution, by setting the usual pollution standards and taxation that make the competitive economy to work eficiently. Alternatively, the government may develop an allocative function, which implies a direct participation that will lead him to a gradual public abatement provision, greater than the one decided in a decentralized economy. Moreover, when given the socially optimal choices for economic variables the stock of pollutants catch up with the ecological upper-limit, then the government must impose abruptly a cleaner technology increasing abatement suddenly. This will just make stationary the stock of pollutants, which is needed to avoid the catastrophic state and simultaneously to guarantee sustained growth. Finally, from a long term perspective, the government has an important role to play implementing indirect environmental policies such as information or awareness campaigns [Chevé (2000)]. These policies influence social preferences for environmental conservation and, hence, the demand for environmental quality. In other words, participation makes people more environmentally conscious and prevents that the environment is felt as an obstacle to growth [Bimonte (2001)]. Additionally, there are also the population control policies and other development encouraging actions that accelerate the demographic transition, which is important because of the strong impacts of population growth on the environment. Taken together, the above-mentioned policies may afect in the long term both the population growth rate and the environmental willingness to pay. Namely, the two parameters that contribute to generate an environmental Kuznets curve. In conclusion, what is important to avoid irreversible damages is to implement policies that involve people in the growth and decision making processes.

8 Conclusions

In this paper, we have built a general equilibrium one-sector endogenous growth model in which pollution is a by-product of economic activity but it may be reduced by spending a fraction of the aggregate output on abatement We consider the existence of a welfare pollution externality associated with the emissions flow, and introduce an absolute upper-limit to the accumulated stock of pollutants beyond which we fall in an ecological catastrophe. First of all, we studied the decentralized competitive economy. The equilibrium path is not Pareto-optimal and sustained growth is not sustainable, leading the economy to an environmental catastrophe. Consequently, we gone to study the socially optimal equilibrium. We have proved that the optimal path does exists, it is unique, and does not show transitional dynamics. We found that the rate of growth depends negatively on the weight of environmental cares in utility and positively on the population growth rate. Moreover, the latter efect is stronger as higher is the weight of environment in the utility function. We also found a trade-of between growth and environmental quality, which results from agent decisions, because increased abatement efort crowds out resources from capital accumulation and growth. Put in other words, the higher the rate of growth the higher the dirtiness index associated with production. However, this is not a problem at least until the stock of pollutants reaches the upper-limit.

Here, we have analyzed the opportunities for an eficient management of the economy with special attention to the potential risk of environmental collapse. In the context of our model, eficiency is not suficient for sustainability but, as we have shown, Pareto optimality is necessary to produce sustainable outcomes. We have identified conditions for sustainability of the optimal balanced growth path. However, if for the optimal path pollution stock reaches the upper-limit, the central planner still could change the value of the dirtiness index by increasing drastically abatement activity. Obviously, this means that even in the aftermath sustainability may be guaranteed but at the cost of a lower rate of growth. In this sense, sustainability and optimality seem to be in conflict because strong optimal growth is detrimental for the environment due to higher levels of pollutant emissions.

Based on the EKC hypothesis some authors claim that growth alone is necessary to improve environment and that the above-mentioned trade-of is only a temporary phenomenon. However, there is no reason to believe that the positive relationship linking growth and environmental quality is inevitable. Even though economic growth directly fosters higher abatement expenditures, it also increases pollution. Policy has a very important role to play because, on one hand, growth and development are not a substitute for environmental policy and, on other hand, a permanent conflict between economic policies encouraging growth and environmental quality seem to be omnipresent. In this context, policy-makers, specially in developing countries, should not assume that economic growth will automatically solve pollution problems. Instead, some policies can help to promote both sustainable growth and the environment. Since environmental preferences and population growth rates are central in this framework, government may implement indirect policies such as information and awareness campaigns that make people more environmentally conscious, enhance education levels, improve health and promote population control actions that precipitate the demographic transition process. These long term policies should be complemented with the more direct ones which focus on incentives to adopt cleaner technologies using environmental corrective taxes and subsidies.

9 Appendix

Consider the more general optimal control problem involving either nonnegativity constraints, pure control variable constraints and pure state-space constraints, applied to our endogenous growth model with environmental concern and awareness. In particular, such constraints are: , (ii) and (iii) , which add to the usual dynamic and boundary constraints for k and S.

The third category consists of one constraint in which no control variables are present. This constraint places a restriction on the state space, delimitating the permissible area for the accumulated stock of pollutants. Writing the current stock in per capita terms we get . However, given that is not allowed to exceed , when the constraint is binding, , we impose the new condition

\[\frac {d \left(e ^ {n t} s (t)\right)}{d t} = e ^ {n t} \left[ e (1 - z (t)) - \delta s (t) \right] \leqslant 0 \quad \left(\mathrm{whenever} e ^ {n t} s (t) = S ^ {\max}\right),\]

where we have made use of the equation representing the motion of the stock of pollutants in per capita terms . Then, the current value Hamiltonian after writing all the variables in per capita terms is

\[\begin{array}{r} H _ {\{c, z, q, k, \mu , s, \eta , \theta \}} ^ {c} = \frac {c ^ {1 - \Phi} (1 - z) ^ {\alpha (1 - \Phi)}}{1 - \Phi} + q [ A k z - c - n k ] + \mu [ e (1 - z) - (\delta + n) s ] + \\ + \eta [ 1 - z ] - \theta e ^ {n t} [ e (1 - z) - \delta s ]. \end{array}\]

Here, and are the co-states for k and s respectively, and and are Lagrangian multipliers associated with the control variable constraint and the state variable constraint respectively. Both and are dynamic multipliers because their corresponding constraints must be satisfied at every period t. Given that the control inequality constraint is linear, the first order necessary conditions arising from the Pontryagin’s Principle and the Kuhn-Tucker theorem are

\[\begin{array}{c} q = c ^ {- \Phi} (1 - z) ^ {\alpha (1 - \Phi)}, \\ q A k + \mu e _ {z} - \eta - \theta e ^ {n t} e _ {z} - \frac {\alpha c ^ {1 - \Phi} (1 - z) ^ {\alpha (1 - \Phi)}}{1 - z} \leqslant 0, \\ z \geqslant 0, \quad z \left[ q A k + \mu e _ {z} - \eta - \theta e ^ {n t} e _ {z} - \frac {\alpha c ^ {1 - \Phi} (1 - z) ^ {\alpha (1 - \Phi)}}{1 - z} \right] = 0, \\ \stackrel {{\bullet}} {{k}} = A k z - c - n k, \\ \stackrel {{\bullet}} {{q}} = \rho q - A z q, \\ \stackrel {{\bullet}} {{s}} = e (1 - z) - (\delta + n) s, \\ \stackrel {{\bullet}} {{\mu}} = (\rho + \delta) \mu - \delta \theta e ^ {n t}, \\ 1 - z \geqslant 0, \quad \eta \geqslant 0, \quad \eta (1 - z) = 0, \\ e (1 - z) - \delta s \leqslant 0, \quad \theta \geqslant 0, \quad \theta [ e (1 - z) - \delta s ] = 0. \end{array}\]

To make clear that the three latter rows of conditions apply only when , we append the complementary-slackness condition and the restriction on the way θ changes over time

\[S \leqslant S ^ {\max}, \theta [ S - S ^ {\max} ] = 0, \stackrel {{\bullet}} {{\theta}} \leqslant 0 (= 0 \text {when} S < S ^ {\max}).\]

Finally, we also need the initial conditions and and the transversality conditions

\[\begin{array}{l} \lim _ {t \to \infty} e ^ {- (\rho - n) (t - t _ {0})} q k = 0, \\ \lim _ {t \to \infty} e ^ {- (\rho - n) (t - t _ {0})} \mu s = 0. \end{array}\]

These necessary conditions are also suficient for a maximum because the Hamiltonian function satisfies the required concavity conditions. It is easy to see that, when the constraints mentioned at the beginning of this appendix are nonbinding, the previous first order conditions reduce to the ones studied in Section 4, giving only interior solutions. However, if anyone of such constraints changes its status from nonbinding to binding, then these first order conditions become fully operative, and corner solutions are also feasible. This is the case analyzed in Section 6 with special regard to the pure state-space constraint.

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