Evolution of environmental concern and the dynamics of pollution


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1 Evolution of environmental concern and the dynamics of pollution Emeline Bezin March 26, 2012 Abstract Although the evolution of environmental concern has exhibited a very similar pattern in industrialized and developing nations so that it has been widely demonstrated to be now as spread in both kind of countries, many pollutions have steadily increased within the former group while it was stabilizing in the latter one. To explain those facts, we develop a dynamic model within which the evolution of pollution and the formation of environmental concern are both endogenous. We first set up an overlapping generation model based on John and Pecchenino s [1994] in which environmental concern, view as a perception of pollution, can vary within the population but is still fixed in time. We find that it is not always better for the long run environment to have a lot of environmentally concerned people and that this depends upon the available technology for the pollution production process. The technological factor matters because it determines the ease of reducing pollution in terms of its opportunity cost in consumption units. Hence, we show that a sufficient number of people who care about environment is needed in order to reach a stationary state as long as the available technology is clean. However if the technology is quite polluting a steady state will exist only if the share of environmentally concerned people is low enough. In a second step, we consider the composition of the population to be endogenous while environmental concern is transmitted through a socialization process based on Bisin and Verdier [1998, 2000] in which we further assume that pollution interacts. We show that as long as pollution and the culture sphere interact in a non negligible way, a stationary state characterized by a low (but nonzero) share of environmentally concerned people does not exist anymore. The existence of a stable steady state requires a clean technology for pollution production. This steady state is always characterized by an heterogeneous composition of the population where the fraction of people concerned by the environment is relatively high. Our model points out that it is not sufficient to be environmentally concerned to achieve pollution stabilization and that a clean technology for the pollution production process is also required. Furthermore when the share of environmentally concerned agents is endogenously determined, while having both previous conditions is still sufficient, it now turns to be also necessary. Actually with a polluting technology, individuals who are concerned with pollution cannot do anything but increase their consumption so that a rise of their number makes the environment more favorable to the transmission of their own trait. It supports the empirical evidences on the two different trends which have linked the environmental concern and the evolution of pollution. keywords: Overlapping generations, pollution, environmental concern, cultural transmission. JEL Classification: Q50, D90, J11. I am grateful to Frédéric Chantreuil and Dominique Vermersch for their advices. I would also like to thank David de la Croix and Thomas Baudin as well as François Portier for very helpful comments. UMR 1302 Smart, INRA, 4 allée BobierreCS Rennes cedex, France. Tél. (33)
2 1 Introduction Many authors have recently increase their interest toward the spread of environmental concern, a phenomenon which has begun in the early 1970 s and has never stop to intensify. An important issue is to understand the link with pollution especially because the way in which it can be involved in solving environmental problems has huge potential policy implications (mostly in terms of education). The conventional wisdom suggests that, inducing the change of agents destructive behaviour, the spread of environmental concern would consistently lead to the reduction of environmental degradations. Nevertheless some facts reveal that this relashionship does not seem so obvious. Within the developping world various kinds of pollution have risen so that many have supported the hypothesis that it was due to a lack of environmentally concerned individuals in those nations. The extansive environmental Kuznets Curve s litterature has particularly pointed in that direction and among others, McConnel [1997] for instance, explains the difference between industrialized and developping nations by a difference in environmental concern. Actually, he assumes that as income rises people are more and more concerned about the environment and this in turn lead to less environmental degradations. However number of studies which have used various measures of environmental concern have shown that the development of environemental values has not been not confined to industrialized nations but that it has been rather a worlwide phenomenon (Dunlap, Gallup and Gallup [1992], Brechin and Kempton [1994], Dunlap and Mertig [1995], Brechin [1999], Dunlap and York [2008]). In addition, simple observation of data reveals there has been two different trends linking environmental concern and the evolution of pollution. On the one hand, for a wide part of industrialized countries, the spread of environmental concern has resulted into stabilization of many kinds of pollution. On the other hand, while the same spread has occured in developping nations, most pollutions have risen. For instance, air pollutants (like PM10 and nitrogen dioxide) have exhibited upward pattern in Asia and Latin America during the last decades. As a consequence, several cities of those parts of the word range now from the highest concentration levels. Meanwhile, the same air pollutants has decreased and stabilized in Europe as well in North America (World Health Organization [2005], Baldasano and al. [2003]). In spite of those different worlwide patterns for the evolution of pollution, no statistical differences have been found between rich and poor counties, for environmental concern in general, by the studies mentionned above. It is neither the case for specific environmental issues. Precisely, concerning air pollution, 77% of mexican inhabitants and 64% of Indian were reported to be highly concerned, which is comparable with what have been reported in USA or Germany where the share is around 60 % (Kempton and Brechin [1994]). Linking those facts with what has been put forward by the EKC litterature leads us to two major observations. The first one questionned the relashonship between income and environmental concern. Indeed contrary to the EKC hypothesis, the spread of environmental concern has occured in both rich and poor nations. Hence if the incomes rise is not the engine of that change what may be at work in the evolution of environmental attitutes? Secondly, it challenges the conventional link between the evolution of environmental concern and the dynamics of pollution. What can explain that the rise of environmental concern has resulted into pollution stabilization for the industrialized world while it has been associated with an increase of many kinds of pollution in developping countries. First of all, the fact that income is not decisive for environmental concern suggests that it can not be understood in terms of monetary value. This is further emphazised by Kempton and 2
3 Brechin [1994] who argue that surveys measuring environmental values via willingness to pay cash question, already reflect a strong bias toward economic measures of worth. The conclusion that the poor lack environmental concern is generated by the measurement instrument.. On the other hand, many authors seem to agree that perception is a good measure of the degree of concern so that it has almost always been used in the extensive related litterature. In this article, therefore, we operationalize environmental concern as a perception. More precisely, we consider agents which are not able to correctly assess the actual stock of pollution because as justified in Schumacher and Zou [2008] they are naturally more able to perceive environmental stimulus than absolute levels. This is also emphazied by many authors in environmental psychology who make use of the comparison with a frog which immediately jumps out if suddenly plunged into hot water while it would never realize the danger if water would be heated gradually to the same temperature. Thereby, in our setting agents are not affected by the level of pollution but by the variation of the level of pollution. They consider the level within which they have lived as a benchmark, against which the new level is assessed. Variations in concern for the environment are captured by the relative importance of this benchmark which stands for the amount of pollution considered as neutral. Hence, we consider two types of agents which differ in the following manner: 1. People concerned about pollution are convinced of an harmful human impact on its natural environment so that the preceeding level of pollution is thought to be already high. Consequently for those agents, the reference level is relatively close to zero so that the new level of pollution is perceived as high, and they are strongly affected by it. 2. The other type of agents strongly trust in science and innovation. They think there is no limits to growth because technological progress is able to solve environmental problems. Hence in their point of view, the preceeding level could not be harmful and their reference level is high. However they cannot denie observed changes in their natural environment and are still affected by pollution when it increases compared to the benchmark. With this formalization, environmental concern relies mostly on individual beliefs and values. This is just confirmed by an exhaustive litterature which have tried to explain variations of the concern for environmental issues (among others see Van Liere and Dunlap [1980] for a wide review). In many studies demographic variables have been shown to be slightly relevant to assess the degree of concern while cognitive ones have been demonstrated to explain a large share of observed variations. As a consequence, it is likely that, as for other beliefbased characteristics, a cultural transmission process is at work and drives the evolution. Hence rather than income rise, we think that social interactions are the driving force in the spread of environmental concern and we use the cultural transmission framework developed by Bisin and Verdier [1998, 2000] to describe this process. Furthermore, it is clear for many authors that rising pollution has also been a driving force in the spread of environmental concern (Inglehart [1995], Brechin [1999]). However the role played by objective environmental conditions still remains a puzzling question. As pointed out by Dunlap and Merting [1997] a simple stimulusresponse model of the role of objective environmental problems is much too simplistic [...] and ignores the documented complexities of environmental perception.. Hence pollution must interact in the socialization process who drives the evolution of environmental perceptions but it could occur at different stages of the transmission mechanism. We think it is realistic to assume it interacts with parental point of view (through the desire to transmit its own attitude toward environment). This is not crucial for the consistency of our results since the same conclusion can be obtained assuming pollution interacts with children s point of view. 3
4 We use an overlapping generation framework with a pollution externality based on John and Pecchenino [1994]. Pollution, arising in a accumulative way, is increased by consumption while it can be reduced by abatement spendings. The two types of agents can either use their wage for saving in order to consume latter or for the abatement of pollution through a voluntary contribution. Since we specify environmental concern as a perception, those who are concerned about pollution are actually those significantly affected by the rise of the pollution stock. This is fundamental in a dynamic framework because the direction of the changes of the decision variables, and not their absolute level, determines the evolution of the state variables and so rules out the short run and the long run behaviour of the economy. In fact, when the pollution bequeathed by preceeding generations rises, agents with environmental concern, who heavily suffer from this rise, need to undertake largest actions to offset the rising cost of past actions. To do so they can either decrease pollution or increase consumption. However, pollution is a public good so that they cannot act directly on its level. Hence is the key feature: if they must respond more to a rise of the bequeathed pollution, it does not imply that they always act for the reduction of pollution because depollution can be constrained. It is in line with a widely developed litterature capturing the existence of barriers to proenvironmental behaviour (Kollmuss and Agyeman [2002], Stern [2000]). Within this field, authors have put forward that attitudes as perception of harmful impacts of pollution do not always lead to proenvironmental behaviour and have pointed out, among others, the role played by what they called external factors. Within those factors, they mentionned the institutions as well as the technology. In our setting, such a binding technological factor is precisely at stake to link up a perception with a type of behaviour. Actually, pollution is produced according to a given process which can be more or less binding for either type of agents depending on the relative values of the efficiency of abatement and of the consumption externality. The technology interacts with environmental attitudes to determine the ease of reducing pollution in terms of its opportunity cost in consumption units (along the transistion path if not a barrier, it determines the strengh of individual actions so that it is crucially implied in stability). Thus we show that, when the production process of pollution is clean (that is the efficiency of abatement is large enough compared to the consumption externality), a large number of environmentally concerned agents is also necessary to the stabilization or to the sustainability when it is basically defined as reaching a stationary state. In fact, within this technological context, any rise in pollution stock incit them to increase abatement contrary to their contemporaneous who prefer to rise saving to consume more. Nevertheless, when the available technology is highly polluting, a sustainable stationary state will never be reached if the share of agents concerned is large enough because of the difficulty of reducing pollution in terms of its opportunity cost in consumption units. Indeed, they cannot act efficiently against a rise in pollution while they are strongly affected, so that they must consume more to compensate. Hence, we show that within this kind of pollution technology, a stationary state can be reached only if the share of agents concerned with pollution is low enough. If the former case can be compared with the trend, which have occured in industrialized countries, the latter one does not correspond to any observed situations. This is because the distribution of environmental concern is also a dynamical variable so that considering its evolution is essential to be able to have the whole story. Thereby, in a second time, we consider the fraction of environnementally concerned people is endogeneous. We show that if the environmental concern evolves through cultural transmission and if pollution has a non negligible impact in the socialization process, then a stable stationary state exists only when technology is clean and it is characterized by 4
5 a high share of agents concerned with the environment. When technology is polluting, however, while the share of people concerned about pollution is rising up, stabilization never occurs. The remainder of this paper is organized as follows. Section one presents and solves the model where two types of agents coexist but when the share of environmentally concerned people is fixed. In section two we endogenize the distribution of environmental concern through a cultural transmission model where pollution interacts. Section three concludes. 2 The model Let consider a perfectly competitive economy with an overlapping generation structure. Time is discrete and goes from 1 to. At each date a generation is born and lives for two periods. During the first period agents supply their labor inelastically and earn a wage which can be either saved for future consumption or spend in the abatement of pollution. Following John and Pecchenino [1994] we assume agents have preferences over consumption and the environment and can only derive utility during the second period of life (which stands for a period of retirement). Each cohort, of constant size n, consists of two types of agents who differ through their appraisal of the pollution stock. As highlighted in the introduction, agents of the first type have high perception of the pollution stock because their frame of reference is quite low. They are called environmentally concerned agents and they will be associated with the superscript G. On the other hand, the second type of agents with a relatively high frame of reference estimates pollution stock is lower than it actually is. They will be denoted by T for technologists. There are n G agents of type G and n T = n n G agents of type T. 2.1 Pollution accumulation The pollution accumulation process is described by the following equation based on John and Pecchenino [1994]: P t+1 = (1 b)p t + βn G c G t + βn T c T t γn G A G t γn T A T t, (1) where b [0, 1] is the natural rate of absorption, β > 0, is the amount of pollution per unit of consumption by agents of the old generation. Hence it stands for the intergenerational externality. On the other hand, γ > 0 is the effectivness of abatement. c i and A i, i {G, T } are respectively the typei agents consumption of the generation born at t 1, and the typei agents abatement of the generation born at t. 2.2 Agents behaviour Perception of pollution The formalisation of the pollution perception comes from Schumacher and Zou [2008]. In a model where agents live for two periods, the amount of pollution, an agent is affected by is H t+1 = P t+1 h i P t, where P t (resp. P t+1 ) is the pollution stock at time t (resp. t + 1). Individuals perceive a level between the stock and the flow from t to t + 1. h i [0, 1] is an individual specific parameter. For 5
6 a high h i, agent i almost considers as a benchmark the level where he has lived so far (P t is almost the neutral level). On the contrary agents who care a lot about about environmental problems consider the level where they have lived as already high (their frame of reference is thus closer to zero). The lower h i, the better the perception. Finally we assume h G < h T Economic choices During the first period agents have to choose the mix of abatementsaving which maximises their utility when old. We assume that, for all types of agent, U : R 2 ++ R, is twice continously differentiable and such that U c > 0, U cc < 0, U H < 0, U HH < 0 and lim c 0 U c =. An agent of type i, i {G, T } faces the following budget constraints : w t = s i t + A i t, (2) c i t+1 = (1 + r t+1 ) s i t, (3) where w t is the agent s wage (whatever his type) at time t, s i t is the amount of saving of a typei individual and A i t, his abatement expenditure. r t is the interest rate at time t. An agent of type i maxmizes his utility subject to the budget constraints (??) and (??), as well as to P t+1 = (1 b)p t + βn G c G t + βn T c T t γa i t γāt where Āt is abatement spendings of all other economic agents. It leads to the following FOC (1 + r t+1 )U i c + γu i H = 0, i {G, T }. (4) Following Zhang [1999] and Schumacher and Zou [2008], for analytical convenience we restrict to simple class of utility functions so that we assume a constant elasticity of substitution between private consumption and the environment. σ = U i H Hi U i cc i > 0, i {G, T }. (5) Note that σ does not change from one type of agent to another. As highlighted in the introduction, agents do not differ through their relative valuation of the environment but only through their perception of the pollution. Using (??), let rewrite the FOC as γσs i t = P t+1 h i P t, i {G, T } (6) At the optimum, agents choose their saving such that positive level of pollution is associated with positive amount of saving. Since h G < h T, for a given stock of pollution in t + 1 typeg agents always choose to save more than typet agents do. This is because for a given stock of pollution, typeg agents feel a higher desutility, so that all else equals, they have to consume more to offset the loss. However note that those agents are more sensitive to the intergenerational externality, 6
7 that is to a rise in the bequeathed stock of pollution P t. Hence, if pollution at time t increases, they always need to undertake largest actions. In other words, if they choose to increase abatement, they will increase it more than do typet agents. Nevertheless, if they decide to increase saving on the other hand, they will rise it more. Later we point out the specific circumstances which determinate such a choice. 2.3 The representative firm The productive sector consists in a representative firm which is supposed to be perfectly competitive and which produces using the constant returns to scale production function Y = f (K) L. We can normalize by the labor supply so that output per worker can be written as y = f (k), where k = K/n. We assume that f meets the following properties : f > 0 and f < 0. The firm maximizes its profit equalizing each marginal productivity with its market price. Namely the marginal productivity of labour is equalized to the wage rate, whereas the marginal productivity of capital minus its depreciation rate is equalized to the interest rate. Finally total savings of the young generation form the capital at time t+1, w t = f(k t) n r t+1 = f (k t+1 ) n k t n f (k t ), (7) δ, (8) n G n sg t + (1 n G n )st t = k t+1. (9) For analytical convenience we restrict to a Cobb Douglas production function f(k t ) = kt v (Zhang [1999], Schumacher and Zou [2008]) where v [0, 1] is the capital share. We further assume full depreciation of captital, that is δ = 1. In what follows, let denote by q = n G n, the share of typeg agents. 2.4 Existence of steady state Definition 1 (Intertemporal equilibrium). Given inital conditions (k 0, P 0 ), the intertemporal equilibrium is the sequence (k t, P t ) t N which satisfies the two following equations for each t. P t+1 = (bσ + h(q) σ) P t (n σ) k t+1 = where h(q) = qh G + (1 q)h T and σ n. (1 b h(q)) P t γ(n σ) (βv + γv γ)σ kt v, (10) (n σ) (βv + γv γ) kt v, (11) γ(n σ) 7
8 2.4.1 Steady states A steady state exists if some vector (k,p) solves k t+1 = k t, P t+1 = P t for all t. There are two steady states : (0,0) and a nontrivial one, which crucially depends on q, so that it will be denoted by ( P (q), k(q)) : P (q) = σγ 1 h(q) ( (1 h(q))(βv + γv γ) γ(bσ + h(q) 1) ) 1 1 v (12) Assumption 1. Preferences ( ) 1 (1 h(q))(βv + γv γ) 1 v k(q) = γ(bσ + h(q). (13) 1) (1) We assume that 1 hg b > σ, i.e. for typeg agents the cost of an additionnal unit of pollution is higher than the gain of one more unit of consumption. (2) Conversely, we assume that 1 ht b < σ, i.e. for typet agents the cost of an additionnal unit of pollution is lower than the gain of one more unit of consumption. Since a nontrivial steady state exists if bσ + h(q) 1 and βv + γv γ have the same sign, the following proposition can be deduced. Proposition 1. Existence of a nontrivial steady state (1) If the share of agents concerned about pollution is relatively high (q > q with q = bσ+ht 1 h T h G < 1), a nontrivial steady state exists if, and only if, the effectivness of abatement γ is high enough compared to the pollution externality β, that is, if the pollution production technology is clean (βv + γv γ < 0). (2) If the share of agents concerned about pollution is small enough (q < q), then a nontrivial steady state exists if, and only if, the effectivness of abatement is small enough compared to the pollution externality, that is if the technology is quite polluting (βv + γv γ > 0). For our purpose, it is interesting to investigate the impact of a change in the distribution of environmental concern for the long run state of the economy. Proposition 2. Impact of an increase of the share of typeg agents, on the long run capital and pollution stocks. (1) If the technology is clean, and if q > q, then a rise of q, the share of type G agents, induces a decrease of the steady state pollution and capital. (2) If the technology is polluting, and if q < q, then a rise of q induces an increase of the steady state pollution and capital. 8
9 Proof. Differentiating the functions P (q) and k(q) with respect to q we obtain, P (q) = P (h G h T ( ) ) bσ (1 v)(1 h(q)) (1 v) + (1 bσ h(q)) (14) k(q) kbσ(h G h T ) = (1 v)(1 h(q))(bσ + h(q) 1). (15) The sign of the above derivatives depends upon the sign of bσ + h(q) 1. If bσ + h(q) 1 < 0, which involves βv + γv γ < 0, then both dervatives are negative; while, if bσ + h(q) 1 > 0, which involves βv + γv γ > 0, then both derivatives are positive Interpretation The existence of a steady state is linked to the ease of reducing pollution in terms of its opportunity cost in consumption units. This cost depends upon the preferences, on the one hand, and upon the technology for pollution production, on the other hand. Actually, when pollution increases, typeg agents prefer to act for the reduction of pollution in increasing abatement, unless abatement is inefficient. Indeed, in this case only, the opportunity cost of consumption becomes too high so that they increase consumption. When pollution increases, typet agents prefer to rise their consumption, as long as the consumption externality is relatively low compared to the efficiency of abatement. In fact, the opportunity cost of consumption iss too high for them so that they rise consumption, unless consumption be highly polluting. More precisely, first consider the case of a clean technology. Agents with a bad perception feel little affected by pollution on the one hand, little affected by consumption externality (β low) on the other hand. Consider pollution rises. To offset this rise, typet individuals had better to save more in order to increase utility from consumption, even though this results into a further increase in pollution. In fact, the latter is felt only slightly compared to the gain provided by consumption. Therefore, if the share of technologist agents is too high, a steady state cannot exist since an increase in the long run pollution would entail further increase in pollution. Unlike typet individuals, agents of type G care a lot about environment so that when pollution rises, they are strongly affected. Hence, since abatement is efficient, slightly increasing it induces a high utility gain without inducing a too high cost, since the decrease of consumption can be moderate. Hence when q is high enough, a steady state always exists because an increase of pollution is immediately reduced. Furthermore, the higher the share of type G, the larger the number of people who want to abat more, the lower the long run pollution. Now consider the case of a polluting technology. Any rise in pollution strongly affects environmentally concerned agents. However since abatement is not efficient, they cannot sufficiently reduce pollution without significantly reduce their consumption so that they had better to increase their consumption. On the other hand, typet slightly suffer from an increase of pollution. Hence, even though abatement is not efficient, increasing it entails a sufficient reduction of pollution while a slight increase in saving would have resulted in a too high rise of pollution (because it is highly polluting). Hence when q is sufficiently low, a steady state always exists because any rise in pollution is immediately reduced. Finally the higher q, the lower the number who benefit from an increase in abatement, the higher the long run pollution. 9
10 Hence, an important conclusion is that a high share of agents with environmental concern is not sufficient for the existence of a steady state. A clean enough technology for the pollution production process is also required. 2.5 Stability of the nontrivial steady state Proposition 3. A necessary condition for local stability. (1) When the technology for the pollution production function is clean (βv + γv γ < 0), stability of the steady state requires that the cost of an additionnal unit of saving be relatively high (σ < n). (2) When the technology is polluting (βv + γv γ > 0), stability of the steady state requires that the cost of an additionnal unit of abatement be relatively high (σ > n). Proof. To study the local stability of the nontrivial steady state, we linearize the twodimensional map composed of equations (??) and (??) around ( P (q), k(q)). Denote by Df 2 the jacobian matrix, we have Df 2 ( k, P ) = Its characteristic polynomial can be written as v(bσ+ h(q) 1) (1 h(q))(n σ) σvγ(bσ+ h(q) 1) (1 h(q))(n σ) P (λ) = α 0 λ 2 + α 1 λ + α 2, (1 b h(q)) γ(n σ) (bσ+ h(q) σ). (n σ) where α 0 = (n σ)(1 h(q)), α 1 = σ(b 1)( h(q) 1) + vbσ (1 h(q))(v + h(q)), and α 2 = h(q)v(bσ + h(q) 1). The steady state is locally stable if each eigenvalue of the above matrix is of magnitude less than one. We denote by λ 1 = α 1+ 2α 0 and λ 2 = α 1 2α 0 the roots of P. We want to show that, at least one of the two roots of the characteristics polynomial will be of magnitude higher than 1 if, on the one hand σ > n and βv+γv γ < 0, or if, on the other hand, σ < n and βv+γv γ > 0. (1) Let consider the case βv + γv γ < 0. The existence of a steady state implies α 2 > 0 since bσ + h(q) 1 is negative. Suppose that σ > n which implies α 0 < 0. Then α 2 1 4α 0 α 2 > α 2 1 (since 4α 0 α 2 < 0) so that > α 1. Hence λ 1 = α 1+ 2α 0 < 0. Especially, we show that we always have λ 1 < 1. λ 1 < 1 α 1 + 2α 0 > which is equivalent to > ( α 1 + 2α 0 ) 2 (since α 1 + 2α 0 > λ 2 < 1 and we are done), or also to α 1 α 2 α 0 < 0. However, after simplification, we find α 1 α 2 α 0 = (v 1)( h(q) 1)(bσ + h(q) 1) which is negative since bσ + h(q) 1 is negative. Finally, when βv + γv γ < 0 and σ > n, one of the eigenvalues, is always lower than 1 and the steady state is unstable. (2) Let consider the case βv + γv γ > 0 which implies α 2 < 0 since bσ + h(q) 1 is positive. Consider σ < n which implies α 0 > 0. Then = α 2 1 4α 0 α 2 > α 2 1 so that > α 1. Hence λ 1 = α 1+ 2α 0 is always positive, but this is not the case for λ 2, which, we will we show, is always lower than 1 as long as σ < n. This is equivalent to 2α 0 α 1 <, or > ( α 1 + 2α 0 ) 2 10
11 (since λ 1 > 0 2α 0 α 1 > ). When α 0 > 0, this is equivalent to α 1 α 2 α 0 = (v 1)( h(q) 1)(bσ + h(q) 1) > 0 which is always positive since bσ + h(q) 1 > 0. Hence, when βv + γv γ > 0 and σ < n, one of the eigenvalues is always lower than 1 and the steady state is unstable An interpretation of the necessary condition for local stability On the one hand, a high relative cost of an additional unit of saving (σ < n) have to be associated with efficient abatement (βv + γv γ < 0) along the transition path. Let suppose that the cost of past actions rises, then agents (of all types but especially of type G) want to increase abatement in order to limit the rise of pollution. Since abatement is efficient, for a slight increase of abatement spendings, they reach the desired level of pollution without decreasing saving too much. Hence pollution when old undergoes a limited increase while aggregate consumption decreases so that their is a rise of the cost of past actions for the following generation but which is rather moderate. Therefore, the new generation increases abatement spendings but less than the preceeding one such that, at each iteration, the magnitude of variations decreases and the system converges toward the nontrivial steady state. On the other hand, a high relative gain of an additional unit of saving (σ > n) have to be associated with inefficient abatement (βv + γv γ > 0) along the transition path. If they prefer to increase consumption after an increase of the cost of past actions then abatement must not affect pollution too heavily. Indeed, in this case, the increase in consumption must be of large magnitude to offset the lost generated by the rising level of pollution. The large variation of consumption and pollution would lead to an even higher variation of the cost of past actions for the next generation, which thus would have to increase consumption even more. On the contrary, if the decrease of abatement does not affect pollution too heavily, then the rise of consumption can be moderate. Hence, the rise of the cost of past actions for the next generation can be limited. Finally, the magnitude of variations decreases at each iteration, so that the system converges. 3 Model with endogeneous distribution of environmental concern In the following the distribution of environmental concern is no longer constant. We allow for a cultural transmission mechanism which relies on Bisin and Verdier s works [1998, 2001]. Children are born naive and adopt a defined kind of environmental attitude through imitation and social learning. Socialization is the result of two interacting types of transmission : transmission inside the family called direct transmission and socialization outside by peers, called oblique transmission. More precisely, each child is first exposed to his parents cultural trait (the perception of pollution for us) and adopts it with probability e i, i {G, T }. If not, which occurs with probability (1 e i ), he is socialized to the cultural trait of a role model choosen randomly in the population. Thus if direct transmission failed, probabilities to pick up trait G and T are respectively q t and 1 q t. Let P ij t be the probability for a child from a family with perception i to be socialized to trait j at time t. We have the following transistion probabilities : P GG t = e G t + (1 e G t )q t, P GT t = (1 e G t )(1 q t ), P T T t = e T t + (1 e T t )(1 q t ), and P T G t = (1 e T t )(q t ). 11
12 We can characterize the dynamic law for the share of agents with environmental concern in the population, q t+1 = q t + q t (1 q t )(e G e T ). (16) Furthermore, as Bisin and Verdier we assume socialization is an economic choice of parents which purposefully attempt at transmitting their own perception of the environment. Therefore we also consider there is cultural intolerance : the motivation to transmit comes from the fact parents feel a relatively higher gain to have a child like them. However, we will not rely on the standard Bisin and Verdier s set up. Instead, we suppose that the gain depends upon the relative advantage they judge their trait provide in given external conditions. In other words, the desire to transmit is environmentdependant and so endogeneous. As highlighted by Bisin and Verdier [2010], The preference on the part of parents for sharing their cultural trait with their children can depends on the economic and social conditions. This is because the child might feel relatively less advantaged when having his parents trait depending on its specific living conditions. In light of this reasoning, we assume the gain to be a function of perceived pollution. And so is the interpretation : for an agent of type G, the higher the perceived level of pollution, the more he thinks his child needs to be aware of what he is going to endangered in order to act accordingly, the higher the gain to have a child of his type. On the contrary, for a typet individual, the lower the perceived pollution, the more he thinks his child does not have to worry about it because technology is reliable, the higher the gain to transmit its own perception. What is interesting with this kind of specification is that evolution of perceptions is somehow governed by Natural selection because the living environment provides the conditions in favor of one particular trait. Actually we are assuming reproductibility of each trait depends upon parental effort which is given rise by the gain to transmit, the latter being environmentdependant. We denote by V ij t the gain, for a parent of type i, to have a child of type j at time t, i {G, T }. A parent of type G faces the following welfare gain functions : Vt GG (Ht+1), G dv GG dh G > 0, with Vt GT (Ht+1), G Vt GG ( ) ( ) H G t+1 > V GT t H G t+1 dv GT dh G < 0, H G t+1 > 0. The relative gain function (which stands for cultural intolerance), V G (Ht+1) G = Vt GG ( ) ( ) H G t+1 V GT t H G t+1 is positive and increasing. Furthermore, we assume that V G (0) = 0, that is typeg parents do not demonstrate cultural intolerance toward technologists if there is no pollution. Likewise, a parent of typet derives utility from : Vt T T (Ht+1), T dv T T dh T > 0, 12
13 with This results into V T T t Vt T G (Ht+1), T ( ) ( ) H T t+1 > V T G t H T t+1 V T (H T t+1) = V T T t dv T G dh T < 0, H T t+1 > 0. ( ) ( ) H T t+1 V T G t H T t+1 being positive but decreasing. Besides, let suppose that lim H T t+1 V T (Ht+1 T ) = 0, that is typet parents do not demonstrate cultural intolerance toward environmentally concerned individuals if pollution is dramatically high. Finally we assume socialization is costly. The parental effort implies a welfare loss which can be understood as time spend with the child. This effort is denoted by C(e i ) with C(e i ) = e2 i 2C, where C is a positive constant. 3.1 Agents choices Agents live for three periods but are passive when child. We assume a twostages maximisation process. During the adulthood, agents first choose their optimal combination of abatementsaving according to their concern for pollution. Then when their choices related to the economic sphere are made, they make their educational choices (to tranmsit their cultural trait). Hence, and because we assume the transmission effort enters separately in utility, we can deal with the socialization problem irrespective of the optimal savingabatement choice. The same first order conditions hold for the economic problem. The socialization problem for each agent of type i, i {G, T }, is then to choose e i to maximize ( ) θ Pt ii Vt ii (Ht+1) i + P ij (Ht+1) i C ( e i), t V ij t leading to the following FOC for agents of type G and T respectively, θ(1 q t ) V G (H G t+1) eg t C = 0, θq t V T (H T t+1) et t C = 0. We can already see that e G and e T do not necessarily respect the cultural substitution property, as defined in Bisin and Verdier [2001]. On the one hand, direct and oblique transmission are substitutes in the socialization process. Namely, parents of type i have less incentives to socialize their child whenever the cultural trait i is more widespread in the population. Thereby, effort of type i falls to zero in the limit of a perfectly homogeneous population of type i. On the other hand, parental efforts are also sensitive to the variations of q through the variations of the perceived pollution level. Hence, depending on the direction and the magnitude of this second effect, the overall impact of q on e i can be of either sign. Calculating the derivatives, e G t t = Cθ V G (H G t+1) + Cθ(1 q t ) d V G dh G P t+1 t, e T t t = Cθ V P (H T t+1) + Cθq t d V T dh T P t+1 t. 13
14 Note, that we always have e G (0) > 0, e T (0) = 0 and e G (1) = 0, e T (1) > 0 whatever the second effect. In addition, et t t (0) = Cθ V T > 0 and eg t t (1) = Cθ V G > 0. Since Ht+1 i is a function of the variables at time t, which can be written as Hi (P t, k t, q t ), let replace Ht+1 i in the intolerance functions to have an optimal parental effort depending exclusively on the variables at t, e G (k t, P t, q t ) = Cθ(1 q t ) V G (H G (P t, k t, q t )), e T (k t, P t, q t ) = Cθq t V T (H T (P t, k t, q t )). By substituting those functions in (??), we obtain the dynamic law of the share of typeg agents. Therefore, in the system we are going to study, all variables are predeterminated. Definition 2 (Intertemporal Equilibirum). Let (k 0, P 0, q 0 ) be the vector of intial values. The intertemporal equilibrium is the sequence (k t, P t, q t ) t N which satisfies for each t, the following system of equations Steady states k t+1 = (1 b h(q t )) P t γ(n σ) P t+1 = (bσ + h(q t ) σ) P t (n σ) (βv + γv γ) kt v, (17) γ(n σ) (βv + γv γ)σ kt v, (18) (n σ) q t+1 = q t + q t (1 q t )(e G (P t, k t, q t ) e T (P t, k t, q t )). (19) Proposition 4. Existence of steady sates with cultural transmission of environmental concern. (1) If the pollution production technology is clean enough (βv + γv γ < 0), the system admits two steady states. One is characterized by a perfectly homogeneous population of environnmentally concerned people : ( P (1), k(1), 1). The other one is characterized by a heterogeneous population within which, the share of agents concerned about pollution is relatively high : ( P ( q), k( q), q) with ( q > q, with q defined as in Proposition??). (2) If the technology is polluting (βv + γv γ > 0), the system will have only one steady state characterized by a perfectly homogeneous population of technologist agents : ( P (0), k(0), 0). There will have no heterogeneous steady state unless the impact of pollution on the socialization process be negligible. Proof. Existence of steady states. A steady state of the threedimensional system is a vector (k, P, q) which solves k t+1 = k t, P t+1 = P t, q t+1 = q t for all t. (As we have ever seen the point (0,0) solves the two first equations so that a vector (0,0,q) solves the whole system if q solves the third equation). 1 The variable q t can be rationnal as long as we restrcit to the case V i : R Q, i {G, T }. Indeed, it implies e i Q and hence q t Q. The behaviour of the dynamical system is decribed by a map G : R 2 Q R 2 Q. However, for the sake of simplicity, we study the same function G which maps R 3 into R 3. 14
15 Let S 1 = {{k(q), P (q)}, q [0, q[}, and S 2 = {{k(q), P (q)}, q ] q, 1]} be the sets of nontrivial steady states for the twodimensional system respectively when βv + γv γ is positive or negative. Necessary and sufficient conditions for the threedimensional system to be at steady state is thus the vector ( P, k) belongs to one either of the two sets (according to the sign of βv + γv γ) with q solving the third equation. That is q = 0, q = 1 and q solving e G t e P t = 0. Hence ( P (0), k(0), 0) is a steady state of the whole system when ( P (0), k(0)) S 1 and ( P (1), k(1), 1) is a steady state of the relevant system if ( P (1), k(1)) S 2. Furthermore the third equation admits a solution for q solving e G t e T t = 0 which is equivalent to 1 1 q = 1 + V G (Ht+1 G ) V T (Ht+1 P ). (20) As we know that the solution of the whole system ( P, k, q) must be such that ( P, k) are in either S 1 or S 2 we can fetch the solution in the restricted sets where ( P, k) are solutions of the twodimensional system. That is, in either of the two following sets : S 1 [0, q[ or S 2 ] q, 1]. The system admits a fixed point ( k, P, q) where 0 < q < 1 if the equation 1 1 q = 1 + V G (P (q)(1 h G )) V T (P (q)(1 h P )), (21) i. has at least one solution in [0, q[, when βv + γv γ > 0, ( P (q), k(q)) S 1 ; ii. has at least one solution in ] q, 1], when βv + γv γ < 0, ( P (q), k(q)) S 2. To study the existence of q let define and Ψ(q) = 1, q [0, 1] 1 q Φ 1 (q) = 1 + V G (P (q)(1 h G )) V T (P (q)(1 h P )) = 1 + V G (q) V T (q), q [0, q[, (P (q), k(q)) in S 1, Φ 2 (q) = 1 + V G (P (q)(1 h G )) V T (P (q)(1 h P )) = 1 + V G (q) V T (q), q ] q, 1], (P (q), k(q)) in S 2. We need to determine wheither Φ 1 (q) Ψ(q) = 0 admits a solution on [0, q[ on the one hand, and weither Φ 2 (q) Ψ(q) = 0 has a solution on ] q, 1] on the other hand. (1) Now let study the existence of some fixed points (different than the one where q is zero) on S 1 [0, q[. Ψ is strictly increasing on [0, q[, Ψ(0) = 1 and Ψ( q) = 1 1 q > 1. On the other hand, 15
16 q [0, q[ dφ 1 (q) dq = dp (q) dq d V G (q) dh G (1 h G ) V T (q) d V T (q) (1 h dh T T ) V G (q) ( V T (q)) 2. By assumption, d V G > 0 and d V P < 0. The second term of the product is thus positive and Φ dh G dh P 1 has the same sign as P on [0, q[. We know, from the study related to the economic sphere, that P (q) > 0 q [0, q[ (see section 2.). So that Φ 1 is strictly increasing on [0, q[. Moroever, Φ 1 (0) = 1 + V G (0) V P is strictly superior to 1 since, by assumption, the functions (0) V G and V P are striclty positive as long as for i {G, T } H i > 0, which is true, since here H i = P (0)(1 h i ) > 0. Furthermore we have, lim q q lim H G lim H P P (q) = +, V G (H G ) = δ <, V P (H P ) = 0, so that Φ( q) = lim q q Φ 1(q) = + 2. Let define the continous function g 1 (q) = Φ 1 (q) Ψ(q), we have g 1 (0) > 0, lim q q g 1 (q) =. For g 1 (q) = 0 to have solution on [0, q[, a necessary condition is g 1 to be decreasing on some interval [q 1, q 2 ], with q 1 0 and q 2 < q, or dg1 dq = dφ 1 dq dψ dq < 0 on [q 1, q 2 ]. That is, we need to find dφ 1 dq < dψ dq on the relevant interval. However, dψ dq = 1 is precisely low for low values of q, so that, the (1 q) 2 necessary condition does not hold if the derivative of Φ 1 is not small enough. More particularly, g 1 must decrease sufficiently such that it reaches the xaxis. Then for that condition to be true, dg1 dq must be negative enough, that is dφ 1 dq has to be very close to zero. Therefore, a steady state is unlikely on S 1 [0, q[. It requires that the impact of a change in pollution on the transmission process dφ 1 dq be negligible so that the effect of cultural substitution (a change in the cultural composition of the population), dψ dq, which is weak, overcomes it3. (2) Let study the existence of solutions in ] q, 1]. Ψ is strictly increasing on ] q, 1]. Ψ(1) = lim q 1 Ψ(q) = + and as we just noted Ψ( q) is a finite number which is superior to one. 2 V G (H G ) cannot be infinite, otherwise the optimal effort, which stands for the probability of direct transmission, could be higher than one as long as C, or θ, are not zero. Likewise, V T (H T ) must be finite in zero. 3 To make sure of the unlikelihood of a steady state and to confirm the conditions on its existence, we relied on a numerical exemple. This is presented in Appendix. 16
17 On the other hand, the derivative of Φ 2 on the interval ] q, 1] has the same sign as P on the same interval. From our study we know that the latter is negative on ] q, 1]. So that Φ 2 is stictly decreasing on ] q, 1]. At q=1, Φ 2 (1) = 1 + V G (P (1)(1 h G )) V P (P (1)(1 h P )) is a finite number supeior to one. Besides, lim P (q) = +, hence Φ( q) = lim Φ 1(q) = +. q q + q q + Finally, we have Φ 2 ( q) > Ψ( q) et Φ 2 (1) < Ψ(1). Φ 2 and Ψ are both continous on ] q, 1]. Moreover Ψ is strictly increasing on ] q, 1], while Φ 2 is decreasing on the same interval. According to the intermediate value theorem there is a unique q ] q, 1] such that. Ψ(q) = Φ 2 (q) Interpretation regarding existence of a steady state A steady state with a heterogeneous distribution of environmental attitudes always exists if the technology is such that abatement is quite effective compared to the strengh of consumption externality. The intuition is as follows : in that situation, as soon as pollution increases typeg agents have incentive to increase abatement (unlike typet ). The larger the share of agents of type G, the higher the number of people who benefit more from a rise in abatement, the higher the rise in total abatement spendings. Hence, an increase in q reduces the level of pollution. Both changes, in environmental conditions on the one hand, in the composition of the population on the other hand alter cultural attitudes in the same direction. Parents of type G reduce their effort not only because transmission by peers works better but also because they feel a relatively lower gain to transmit their own trait since perceived pollution is lower. The reverse is true for individual of typet so that q eventually decreases. When effectivness of abatement is low while consumption externality is quite high, a steady state associated with heterogeneous population is very unlikely. Actually, when the technology is polluting, as highlited in section 2., typet agents, as opposed to typeg, have incentive to abat more and save less after a rise in pollution (because of the opportunity cost of consumption). Contrary to the case of a clean technology, an increase in q involves a rise of the pollution stock. On the one hand, the substitution effect enables a relative increase of typet parental effort against typeg and allows a decrease of q. On the other hand, the rise in pollution alters relative parental efforts in the opposite direction and favours an increase of q. The system will admit a stationnary state if the overall effect is a decrease of q, i.e. the magnitude of the substitution effect is higher than the strengh of the impact of pollution. However when q is small, which is precisely the case when the technology is polluting (remember we are working in [0, q[), the former effect is very weak. Hence q, the fraction of typeg agents will further increases unless the effect of pollution on the socialization process is negligible. Finally, and contrary to the results we have got within our first model, when economic choices and cultural attitudes toward environment are jointly determined it is never sustainable to have a polluting technology, as long as pollution interacts in a non negligible way on the transmission of environmental concern. A clean technology and a high share of environmentally concerned people becomes the necessary and 17
18 sufficient conditions for the existence of a steady state, that is for sustainability. 3.3 Stability of steady states Stability of the steady states with homogeneous population Proposition 5. Unstability of the homogeneous stationnary population. (1) The steady state characterized by an homogeneous population of agents with environmental concern ( P (1), k(1), 1) (existing if (and only if) the technology is clean (βv + γv γ < 0)) is unstable. (2) The steady state characterized by an homogeneous population of technologist agents ( P (0), k(0), 0), (existing if (and only if) the technology is polluting (βv + γv γ > 0)) is unstable. Proof. To study the local stability of each steady state let linearize the system consisting of equations (16), (17) and (18) around ( P (0), k(0), 0) on the one hand (βv + γv γ > 0), and around ( P (1), k(1), 1), on the other hand (βv + γv γ > 0). (1) The Jacobian matrix evaluated at ( P (0), k(0), 0) is v(bσ+ h(0) 1) (1 h(0))(n σ) σγv(bσ+ h(0) 1) (1 h(0))(n σ) (1 b h(0)) γ(n σ) (bσ+ h(0) σ) (n σ) We can express its characteristic polynomial as γ(h G h P ) (n σ) (h G h P ) (n σ) P e G (0) e T (0) Q 1 (λ) = ([1 + e G (0) e T (0)] λ)(α 0 λ 2 + α 1 λ + α 2 ), where α 0, α 1 and α 2 are defined as in section 2.5. The second term of this product is precisely P (λ), the characteristic polynomial of Df 2 defined in section 2.5, evaluated at q = 0. Hence, two of the eigenvalues can be of magnitude less than one as long as the twodimensional system of section 2 is stable. That is Df 2 has two eigenvalues of magnitude less than one (which requires especially σ > n). However, the third eignevalue is equal to 1 + e T (0) e G (0). This is always higher than one since the socialization mechanism implies e G (0) > 0 and e T (0) = 0. Hence, the system consisting of (16), (17) and (18) is never stable around ( P (0), k(0), 0). (2) The Jacobian matrix evaluated at ( P (1), k(1), 1) is P v(bσ+ h(1) 1) (1 h(1))(n σ) σγv(bσ+ h(1) 1) (1 h(1))(n σ) (1 b h(1)) γ(n σ) (bσ+ h(1) σ) (n σ) γ(h G h P ) (n σ) (h G h P ) (n σ) P e T (1) e G (1) P 18
19 Its characteristic polynomial can be express as Q 2 (λ) = ([1 + e T (1) e G (1)] λ)(α 0 λ 2 + α 1 λ + α 2 ). Again if stability holds for the twodimensional system studied in section 2.5. (especially, σ < n), then two of the eigeinvalues are of magitude less than one. Then, the threedimensional system is stable if and only if λ 3 = 1 + e T (1) e G (1), has absolute value less than one. However within our set up, e T (1) > 0 while e G (1) = 0, so that we always have λ 3 > 1 and the system consisting of (16), (17) and (18) is never stable around ( P (1), k(1), 1) Stability of the steady state with heterogeneous population When linearizing the system at ( P ( q), k( q), q ), we obtain the following Jacobian matrix v(bσ+ h( q) 1) (1 h( q))(n σ) σγv(bσ+ h( q) 1) (1 h( q))(n σ) G( q, P )σγv(bσ+ h( q) 1) (1 h( q))(n σ) where (1 b h( q)) γ(n σ) (bσ+ h( q) σ) (n σ) G( q, P )(bσ+ h( q) σ) (n σ) G( q, P ) = q(1 q)βc ((1 q) V G H T γ(h G h P ) (n σ) (h G h P ) (n σ) 1 + G( q, P )(h G h T ) P (n σ) + q(1 q)βc( V T V G ) P P ) V T q H T. Expressions of the eigenvalues are far more complicated so that we rely on a numerical exemple to study the local stability. Contrary to the other steady states, we find that this one could be stable for large combinations of the parameters. We also observe that σ < n is still necessary for stability Interpreting stability The steady state characterized by an homogeneous population, either in the clean technology case (q = 1) or when technology is polluting (q = 0), is always unstable whatever the interaction between the pollution generated by the economic sphere and the cultural composition of the population. Actually, what matters is the effect of economic choices on capital and pollution on the one hand, and the effect of cultural choices on the cultural composition of the population on the other hand. Hence no matter if the former effect enables the economy to converge to a stationnary state, the system will never be stable because cultural choices lead to move away from full homogeneity of the population. Actually, vertical and oblique transmission are cultural substitutes, so that, all else equal, parents have less incentives to socialize their children, the more widely dominant are their environmental values in the population. In the limit of a perfectly homogeneous population of type G or T, parents of type G or T, respectively, do not directly socialize their child. The cultural substitution property makes homogeneous distributions of environmental concern unstable. Conversely, the stability of the steady state characterized by an heterogeneous population crucially depends upon the interaction between the economic sphere, and the cultural sphere which is linked to the value of σ. First, as long as σ < n, economic as well as cultural choices enable the convergence of capital and pollution stocks. Actually, the cultural composition of the population affects the economic sphere in strengthening the effect of economic choices on the stocks of capital and pollution. On the one hand, as more detailed in section 2, as long as σ < n a rise in the, 19
20 bequeathed stock of pollution encourages agents to increase abatement and enables convergence toward a stationnary state (since abatement is efficient). On the other hand, agents of type G are precisely those who increase more abatement since they are sensitive to pollution. Hence an increase of the share of typeg agents allows for stability because it lowers the increase of capital and pollution in strenghtening the stabilizing effect of aggregated economic actions. Secondly, the cultural choices enable the convergence toward the heterogeneous composition of the population. Importantly, when σ < n the cultural substitution property holds, that is parental effort of typei decreases with the fraction of typei. On the one hand, the technology of direct socialization allows for cultural substitution property : parents have less incentives to socialize their children whenever their cultural trait is more widely dominant because direct and oblique transmission act as substitutes. Nevertheless a sufficient condition for cultural substitution to hold, involves the impact of the distribution of environmental concern, at some t, on the next pollution stock. In fact, since pollution favours the transmission of environmental concern (through the intolerance functions), an increase of the fraction of environmentally concerned have to imply the reduction of pollution in order to reduce the desire to transmit. As we just point out in the first paragraph, it is precisely the case when σ < n. Hence the system can converge to an heterogeneous steady state but σ < n is a necessary condition. 20
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