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a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m
w w w. e l s e v i e r. c o m / l o c a t e / e c o l e c o n
ANALYSIS
Environmental degradation as engine of undesirable economic growth via self-protection consumption choices☆ Angelo Antoci⁎ Dipartimento di Economia, Impresa e Regolamentazione Università di Sassari, Italy
AR TIC LE D ATA
ABSTR ACT
Article history:
We analyze growth dynamics in an economy where the well-being of economic agents
Received 17 April 2008
depends on three goods: leisure, a free access environmental good and a private good that
Received in revised form
can be consumed as a substitute for the environmental resource. The processes of
1 September 2008
production and consumption of the private good by each agent impose negative
Accepted 15 September 2008
externalities on other agents through the depletion of the environmental good.
Available online 22 October 2008
This paper shows that, in such context, the existence of private substitutes for environmental goods may fuel an undesirable economic growth process. This process is
Keywords:
driven by the continuous increase in agents' needs for private consumption generated by
Substitutes for environmental goods
the progressive reduction in free consumption of the environmental good.
Environmental defensive
© 2008 Elsevier B.V. All rights reserved.
expenditures Undesirable economic growth Negative externalities
Jel classification: Q56; Q01
1.
Introduction
The impact on individuals' well-being of environmental deterioration caused by the processes of economic growth in industrialized countries is evident. Less known are some undesired effects of environmental deterioration acting as a factor that conditions individuals’ choices. Environmental degradation encourages behavior perceived as rational at an individual level, that is to say capable of increasing personal well-being; however, at an aggregate level, such behavior may lead to a reduction in collective well-being. We can outline the mechanism from which these undesired effects may stem as follows. In order to defend themselves from environmental
degradation, economic agents make self-protection choices by utilizing certain private goods. The production and consumption of such goods further aggravates environmental degradation, and stimulates yet more production and consumption of goods used as a means of self-protection. The result may be a self-enforcing vicious circle that produces undesirable economic growth, in other words economic growth coupled – paradoxically – with a reduction in individuals' well-being. In this paper, we aim to analyze the possible scenarios that the above-described mechanism may generate. In particular, we study growth dynamics in an economy where only one private good is produced, a good that may be consumed as a substitute for a free access renewable environmental good or to satisfy
☆
The author would like to thank two anonymous referees of this journal for very useful suggestions and Dr. Paolo Russu for the elaboration of the numerical simulations contained in the paper. The usual caveats apply. ⁎ DEIR, University of Sassari, via Torre Tonda 34, 07100 Sassari, Italy. E-mail address:
[email protected]. 0921-8009/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2008.09.009
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needs different from those satisfied by the environmental resource. The production and consumption processes of the private good deteriorate the environmental resource; such deterioration (ceteris paribus) induces agents to increase their labor input in the production process of the private good, in order to produce and consume it in higher quantities as a substitute for the environmental good. Since economic agents take as exogenously given the aggregate negative impact of economic activity on the environmental good, production of the private good (which is assumed as non-storable) generates negative externalities. However, the production process of the private good generates also positive externalities via technical progress (which is assumed to be a pure public good) generated by a learning-by-doing mechanism of accumulation of knowledge. In this context, the negative externalities deriving from the substitution process described above may result in better exploitation of positive externalities by agents and drive the economy away from a poverty trap. However, we show also that growth paths may exist along which the (cumulative) effect of positive externalities is unable to counterbalance the effect of negative externalities. In other words, the economy may approach a stationary state characterized by relatively high consumption of private goods and technical progress, which is Pareto-dominated by other stationary states with lower private consumption and technical progress. In this case, economic growth is the consequence of a coordination failure, and the existence of private substitutes for environmental goods generates socially undesirable effects. The idea that negative externalities deriving from economic growth may fuel the growth process through the enforcement of defensive consumption has been discussed in economic literature at least since Hirsch's famous work (1976). The idea that environmental negative externalities can be an engine of undesirable economic growth was first introduced in a mathematical model by Antoci (1996) and Antoci and Bartolini (1997)1 who analyzed the selection process of labor inputs and of consumption patterns in an evolutionary game context without accumulation of assets. Similar results were obtained by Bartolini and Bonatti (2002) in a neoclassical model without capital accumulation and in Antoci et al. (2005a), where the role played by economic agents’ expectations (that can be right or wrong) on the future environmental quality in determining labor input and capital accumulation was studied via a simple two-periods static model. Finally, Bartolini and Bonatti (2003) and Antoci et al. (2005b, 2007a) have analyzed neoclassical models with perfect foresight and physical capital accumulation2. These works focus only on local stability analysis of stationary states due to analytical complexity of the proposed models; differently from these contributions in this research line, our model allows a full description of economic dynamics and an easy comparison between the dynamics with and without substitutability between the environmental and the private goods. Furthermore, it shows that environmental negative externalities can contribute to better exploit the positive externalities generated 1
See also Antoci and Bartolini (2004) for a further development of these evolutionary models. 2 In Antoci and Bartolini (1999) and in Antoci et al. (2005c, 2007b) growth models have been analyzed where economic agents can substitute ‘deteriorated’ social capital by private goods.
by the production process. As a matter of fact, by inducing the agents to work and consume more, negative externalities can accelerate technical progress, leading the economy in some (particularly virtuous) cases on a Pareto-improving path. In our model and in the above cited literature, a low endowment of natural resources stimulates economic growth. Several recent works (the literature on the curse of natural resources) have focused on various mechanisms through which the scarcity of environmental resources may stimulate growth processes (see e.g. Matsuyama, 1992; Sachs and Warner, 1995, 1999, 2001; Gylfason et al., 1999; Gylfason, 2001; Auty, 2001a,b, 2007; Papyrakis and Gerlagh, 2007; Hodler, 2006). Most current explanations for the curse of natural resources have a crowding-out logic: natural resources crowd-out activity x; activity x drives growth; therefore, natural resources harm growth. For example, Sachs and Warner (1995, 1999) identify x with traded-manufacturing activities and the crowding-out mechanism is the following: an increase of natural resources endowment may create an increase of demand for nontraded products driving up their prices. If these non-traded goods are inputs in the production process of traded-goods (e.g. labor), the increase of non-traded goods’ prices reduces profits in the traded good sector (which sell its products on international markets at relatively fixed prices). The consequent decline of the traded activities inhibits economic growth. Matsuyama (1992) identifies x with the industrial sector; in particular, he analyses an economy with two sectors – the agricultural sector and the industrial one – in which the scarcity of natural resources is represented by low productivity in the agricultural sector. Economic agents react to the low productivity of the agricultural sector by increasing labor input within the industrial sector, where an accumulation process of knowledge driven by a learning-by-doing mechanism works3. In these studies, well resource endowed countries have been identified according to per capita land, primary export share or abundance of point resources (mining, oil)4 while in our model the environmental good is a pure public good which is a final good and not an input. Furthermore, in the literature on the curse of natural resources, economic growth is always desirable; that is, an increase in the activity level of sector x always leads to an increase in the well-being of economic agents. In our model, the development of sector x (production of private goods used as substitutes for environmental goods5) generates negative externalities which may lead to an undesirable expansion of sector x. The paper is organized as follows. Sections 1 and 2 deal with environmental self-protection choices; in sections 3, 4 and 5 we present the model; in sections 6, 7 and 8 we analyze it. Finally, Section 9 concludes the paper.
3 Matsuyama’s model is based on the open economy assumption, that is to say, economic agents can import goods not produced by the domestic agricultural sector. 4 Mineral resources and oil can be considered examples of ‘point’ resources because they are typically characterized by concentrate ownership. 5 The possibility of substituting natural resources with an increase of the activity level in sector x has essentially been contemplated in many of the theoretical works on the resource curse. For example, in Matsuyama's model the possibility of substitution is made possible by foreign trade, which – through the export of the goods produced in sector x and the import of agricultural goods – allows a reduction of the damages generated by the low productivity of the domestic agricultural sector.
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2.
Environmental self-protection choices
In industrialized economies, individuals can exploit a vast array of private goods and services to defend themselves from situations of environmental degradation. Textbook examples include water or air purification plants, mineral water, devices to reduce acoustic damage caused by urban traffic noise, and medicines for the treatment of respiratory diseases due to atmospheric pollution. Environmental degradation of coastal areas near town centers resulting from overbuilding and high environmental impact activities in neighboring industrial estates constitutes a strong incentive to implement a wide range of selfprotection choices. In fact, coastal areas near to the place of residence enable individuals to exploit low cost environmental goods, and the closer such areas are to major urban centers, the more significant this factor becomes. Degradation of coastal areas may motivate costly moves towards less contaminated areas by boat or automobile, or the purchase of holiday packages in some tropical paradise. Hence, individuals are obliged to pay for goods previously available free of charge. The examples cited fall unequivocally in the category of environmental self-protection choices6; this category, however, may be considered more extensively as including a very broad set of choices motivated only in part by environmental degradation. The idea that environmental degradation may make consumption patterns more dependent on the consumption of private goods rather than free access environmental goods is widely accepted in literature on environmental selfprotection choices (for example, see Hueting, 1980; Shibata and Winrich, 1983; Antoci and Bartolini, 1999, 2004; Bartolini and Bonatti, 2002, 2003; López, 2003; Escofet and Bravo-Peña, 2007; Antoci et al., 2008). Urban life is an emblematic example of such substitution process. A great deal of expenditures by inhabitants of large cities are conditioned at least in part by self-protection motivations. The use of vehicles generates pollution but at the same time may be motivated by a state of urban environmental degradation. The scarcity of parks and of areas where children may play without constant adult supervision calls for defensive spending. The widespread use of home entertainment is in part the result of urban environmental degradation; in fact, the scarcity of readily accessible environmental goods in the place of residence may induce reallocation of free time between alternative uses and increase the proportion of time spent in the home. The same is true of recourse to babysitters, who defend children from the dangers of traffic more effectively than grandparents. The use of fitness centers and swimming pools is partially motivated by self-protection needs, since rivers in large cities and waters in adjacent coastal areas are often unfit for bathing. The shortage of urban parks creates competition among the alternative uses of the park (football, jogging, walks, picnics etc.), thus increasing an individual's desire to spend defensively in order to carry out his favorite activity.
6 Several empirical contributions have shown that environmental defensive expenditures are not a negligible phenomenon and that they have been growing steadily over time, both in absolute and in relative terms. See Leipert and Simonis (1988) and Leipert (1989) for seminal contributions on the dimension of the defensive expenditures and their contribution to economic growth.
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In general, in industrialized countries the majority of selfdefensive choices are made in large cities: despite being unable to guarantee citizens an adequate supply of free access public environmental goods, these cities offer a great variety of expensive leisure opportunities.
3. Self-protection choices generating negative externalities In literature, numerous alternative classifications of environmental self-protection choices exist (see for example Leipert and Simonis, 1988; Leipert, 1989). Some authors propose a rather interesting classification (in particular Bird, 1987, Shogren and Crocker, 1991), distinguishing between choices that filter the environmental damage and those that transfer it to other subjects (public or private). The characteristic of the first type of choice is that defensive actions taken by an individual generate benefits for other individuals, while choices of the second type produce an advantage for those who make them but damage others. The second category of choice appears to be more extensive and hence forms the focus of this paper. A paradigmatic example of the second type of choice concerns the use of air conditioners (see Antoci and Bartolini, 1999). These devices cool the interior of homes but give off heat to the exterior, thereby generating a negative effect since the increase in external temperature encourages further use. This example shows how the process of deciding whether to use air conditioners may lead to an outcome in which their use is above the socially optimum level. Other choices that may generate negative externalities are those involving a house move to an area outside a degraded urban center (see Antoci et al., 2008). Generally such choices are dictated by economic motivations (the search for lower priced housing) and reasons connected with environmental degradation (irritation with traffic and urban pollution, or the desire for greater living space). Over the years, this has led to the expansion of towns into suburban areas, and the growing invasion of the countryside, a phenomenon for which experts have coined the term sprawl. Paradoxically, the spread of this process generates a series of undesired effects for individuals, who at times exacerbate the problems that induced their ‘flight from the town’: a reduction in green and public areas, or an increase in traffic congestion and consequently in pollution and journey times to work, for example. Urban sprawl thus makes travel necessary, and travel is a costly activity that aggravates the initial state of environmental degradation. The examples provided concern the negative effects deriving from the process of consuming private goods as self-protection devices. Other negative effects stem simply from the process of producing such goods, an ‘indirect’ negative effect generated by the majority of private goods, although to a different extent in each case.
4.
The model
We analyze the dynamics of an economy with an infinite number (a continuum) of identical agents. In each instant of
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time t, the representative agent's well-being depends on three goods: (1) Leisure: 1 − l(t). (2) A free access flow of a renewable environmental good: E(t). (3) A flow of a non-storable private good produced by the agent: Y(t). The flow Y(t) can be consumed by the representative agent as a substitute for the environmental good, c2(t), or to satisfy needs different from those satisfied by the environmental good, c1(t). Since Y(t) is not storable, it holds Y(t) = c1(t) + c2(t).
4.1.
The representative agent's production function
We assume that, in each instant of time t, the representative agent can produce the private good by the following production function
able economic growth also in a context of inter-temporal allocation of consumption: a higher accumulation of physical capital gives economic agents the possibility to consume higher quantities of private goods in the future to protect themselves against future environmental deterioration. We assume that the dynamics of E(t) is given by h i ¯ EðtÞ gl ðtÞ½KðtÞa EðtÞ ¼ EðtÞ b½E
ð3Þ
¯ −E(t)] is the usual logistic regeneration function; the where βE(t)[E parameter βN 0 represents the regeneration rate of E(t) and the ¯ N 0 represents the value that E(t) should approach if parameter E there were no production of private goods in the economy (i.e. the ¯ may be carrying capacity of the environmental good). So E interpreted as the endowment of the environmental good. The rate of growth of E(t), EðtÞ=EðtÞ, is negatively affected by the α ̄ ; the parameter γN 0 measures the average production l (t)[K(t)] negative impact of average production on the growth rate of E(t).
Y ðtÞ ¼ lðtÞ½KðtÞa
5.
where l(t) represents the representative agent's labor input and K(t) represents technical progress; α is a parameter satisfying 0 b α b 1. We assume that K(t) is a pure public good. The environmental good is not an input in the production process of the private good.
As usual, we assume that the representative agent has to maximize the discounted flow of the values assumed by the instantaneous utility function (1) between time 0 and time ∞ Z
l
The choices of the representative agent
Uðc1 ; l; E þ bc2 Þert dt
0
4.2.
The representative agent's utility function
We assume that E enters as a final good in the utility function of agents and that E and c2 are perfect substitutes with marginal rate of substitution equal to the parameter b7; in particular, we assume that the representative agent's (instantaneous) utility function is the following Uðc1 ; l; E þ bc2 Þ ¼ lnðc1 Þ þ alnðE þ bc2 Þ þ dlnð1 lÞ
ð1Þ
with a,b,d N 0.
4.3.
Time evolution of K(t) and E(t)
As in Matsuyama (1992), we assume that K(t) evolves according to a learning-by-doing mechanism; in particular, the dynamics of K(t) is given by the differential equation
KðtÞ ¼ l ðtÞ½KðtÞa gKðtÞ
ð2Þ
where KðtÞ is the time derivative of K(t), the parameter η N 0 is the ̄ is the average labor input in the depreciation rate of K(t), l (t) α ̄ represents the average economy and consequently l (t)[K(t)] production of private goods. In our model we don't consider the inter-temporal allocation of consumption to keep the model simple from a mathematical point of view. However, in Bartolini and Bonatti (2003) and in Antoci et al. (2006, 2007a) the accumulation of physical capital is analysed; such works show that environmental degradation may be an engine of undesir-
7 The hypothesis of perfect substitutability between E(t) and c2(t) is made essentially for the sake of analytical simplicity; it could be relaxed by assuming that they are imperfect substitutes obtaining similar results (see e.g. Antoci et al. 2007a).
with respect to the control variables l, c1 and c2, subject to the dynamic constraints (2) and (3). The representative agent solves this maximization problem taking as exogenously given the average labor input l ̄; this is due to the fact that the choice of a single agent doesn't modify the average value l ̄, being economic agents a continuum As a consequence, by applying the Maximum Principle we obtain that the choices of the representative agent don't depend on the co-state variables associated to K and E (that is, the shadow prices associated to K and E) in the maximization problem under consideration. Consequently, to solve such problem, the representative agent, in each instant of time t, chooses the value of l, c1 and c2 maximizing the value of the instantaneous utility function (1). This implies that the dynamics of K and E we study don't represent the social optimum. However, since each agent plays the best response l, given the choices of the others, the trajectories followed by the economy represent Nash equilibria; that is, along these trajectories, each economic agent has no incentive to modify his choices if also the others don't revise theirs. So, in each instant of time t, the choices of the representative agent are defined by the following first order conditions AU Ka d ¼ a ¼ 08 Al lK c2 1 l AU 1 ab ¼ a þ V0; Ac2 lK c2 E þ bc2
ð4Þ
c2 z0; c2
AU ¼ 0: Ac2
ð5Þ
8 With the utility function (1), the conditions 0 b l(t) b 1 and c1(t) N 0 always hold.
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From Eqs. (4) and (5) we obtain the following choice functions c 2 ðK; EÞu0 c2 ¼~ c2 ¼~ c 2 ðK; EÞu
if
Ez
ab a K 1þd
1 ½abKa ð1 þ dÞE bð1 þ a þ dÞ
~ l ¼ l ðK; EÞu
1 1þd
~ l ¼ l ðK; EÞu
1 E bð1 þ aÞ d a bð1 þ a þ dÞ K
if
Ez
if
Eb
ab a K 1þd
Eb
ab a K : 1þd
ab a K 1þd if
The choice of c1 is obtained by subtracting the substitutive consumption c̃2 from the output l ̃(K, E)Kα. Note that the curve Ω, defined by the equation E¼
ab a K 1þd
separates the positive orthant of the plane (K, E) into two regions; in the region above Ω, the environmental good is not perceived as ‘scarce’ relative to the value of K and the representative agent has no incentive to produce and consume output as a substitute for the environmental good (that is, c̃2 = 0);note that, above Ω, the labor input l ̃(K, E) is constant and doesn't depend on the values of K and E. Below Ω, the representative agent chooses c̃2 N 0 and the labor input becomes strictly increasing in K and strictly decreasing in E.
6.
Growth dynamics
Since all agents are identical, the average labor input l ̄ coincides (ex-post) with the representative agent's choice l ̃(K, E). So, the dynamical system (2)–(3) becomes ~ a K ¼ l ðK; EÞK gK h i ~ E ¼ E b E E g lðK; EÞKa :
ð6Þ
¯ ) and corresponding Fig. 1 – Threshold values in the plane (g g,E dynamics.
Note that K is a decreasing function of E (given K). More precisely, above Ω, the evolution of K does not depend on E (see Eq. (7)); below Ω, K is a strictly decreasing function of E (see Eq. (8)). Therefore, the accumulation of K is fuelled by the depletion of the environmental good. A reduction of E increases agents' need of private substitutes for the environmental good and consequently their labor supply. The consequent increase of aggregate output has a positive effect on the accumulation of K (positive externalities) but causes a further depletion of the environmental good (negative externalities). Therefore, system (6) describes a self-enforcing mechanism according to which negative externalities are an engine of economic growth. By such mechanism, environmental degradation is not only a consequence of economic growth, but it plays a key role as a push factor in the growth process.
The former equation of system (6) can be explicitly written as
K¼
7.
1 Ka gK 1þd
ab Ka (that is, above the curve Ω), and for Ez 1þd
K¼
1þa d Ka E gK 1þaþd bð1 þ a þ dÞ
ð8Þ
ab for Eb 1þd Ka (below the curve Ω). The latter equation of system (6) can be explicitly written as
E¼E b EE
g Ka 1þd
ð9Þ
ab for Ez 1þd Ka , and
E¼
Classification of dynamics
ð7Þ
E bbE ð1 þ a þ dÞ gbð1 þ aÞKa þ ½gd bbð1 þ a þ dÞE bð1 þ a þ dÞ
In this section we give a classification of the dynamic regimes in our model. The mathematical details are straightforward but tedious and so they are omitted. Our classification does not consider ‘non-robust’ cases, that is those dynamic regimes which hold only for particular values of the parameters of the model9. In the classification there are some threshold values of the ¯ and γ; when such values are crossed, the parameters E dynamics of the economy pass from one dynamic regime to ¯ may be interpreted another. Remember that the parameter E as the endowment of the environmental good while the parameter γ represents the negative impact of economic activity on the environment.
ð10Þ ab Ka . for Eb 1þd Systems (7), (9) and (8), (10) describe dynamics above the curve Ω and below the curve Ω, respectively.
9
More precisely, we define as ‘non-robust’ the dynamic regimes that are observed only if an equality condition on the values of parameters is satisfied.
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Fig. 2 – Dynamics in which the stationary state D is globally attracting. These thresholds values are the following abb þ g
T
E u
b½ga ð1 þ dÞ
TT
1 1a
E u
1 1a g 1þa a b g ð1 þ a þ dÞ
a !1a 1 1a abðbð1 þ aÞÞa E u d gðbbð1 þ a þ dÞ gdÞ
T
gLu
bb ½1 þ a þ d að1 þ aÞð1 þ dÞ d
gUu
bb ð1 aÞð1 þ a þ dÞ d
L U ¯ T N max(E ¯ ⁎,E ¯ ⁎⁎), while E ¯ ⁎bE ¯ ⁎⁎ if and It always holds γ ¯ bγ ¯ and E only if
gNgu
bab 1
1a
ð1þaÞð1þdÞ 1þaþd
1
L U where γ ¯bγ ¯ always. See Fig. 1 for a graphical representation ¯ bγ ¯ ). of these threshold values in the plane (γ,E Figs. 2–6 show all the dynamic regimes that may occur under system (6); attractive stationary states of dynamics are represented by full dots (•), repulsive ones by open dots (◦) and saddle points by drawing their inset and outset10. In Fig. 1, the symbol Fi is used to indicate the region of the ¯ ) where dynamics are represented by Figure i, i = 2, …, 6. plane (γ,E In Fig. 2 there exist four stationary states: the repulsive stationary state O, the saddles C and F, and the (globally) attracting stationary state D. In D, it holds E N abKα/(1 + d) (that
10 The inset (outset) of a saddle type point is the subset of the positive orthant of the plane (K, E) constituted by the union between the two trajectories approaching (respectively, diverging from) such point and the point itself.
is, D lies above the curve Ω); therefore, in such a point, there is no scarcity of the environmental good (relative to the value of K) and, consequently, agents don't consume output as a substitute for the environmental good. In the dynamics showed in Fig. 3 there exist six stationary states: A, B, C, D, F, O; among these, B and D are locally attractive; almost all trajectories approach either B or D and their attraction basins are separated by the inset Γ of the saddle A. The remaining stationary states are repulsive or saddles. In Fig. 4 dynamics are characterized by a bi-stable regime in which there exist two (locally) attracting stationary states, C and D. Almost all trajectories approach either C or D and their attraction basins are separated by the inset Γ of the saddle point A. The stationary state O is repulsive and F is a saddle. In C, the environmental resource is completely depleted and agents have to rely completely on the consumption of the private good as a substitute for the environmental good. In Figs. 5 and 6, the dynamics are characterized by the existence of a globally attracting stationary state. In Fig. 5, in the globally attracting stationary state B it holds E N 0 and the relatively low level of E induces agents to consume a share of output as a substitute; in Fig. 6, in the globally attracting stationary state C the environmental good is completely depleted.
8.
Interpretation of results
In order to interpret the results of the classification showed above, it is useful to compare the dynamics we have described with those under the assumption b = 0. If b = 0, there is no possibility of substitution between the private good and the environmental good and the dynamics are described by systems (7)–(9) only, which in such case holds for every K and E. Under the assumption b = 0, it holds K ¼ 0 for K = 0 and
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Fig. 3 – Bi-stable dynamics: the stationary states B and D are locally attracting.
Fig. 4 – Bi-stable dynamics: the stationary states C and D are locally attracting.
1
along the vertical line K ¼ K 1 u1=½gð1 þ dÞ1a . This means that environmental degradation doesn't affect the evolution of K11 and, consequently, the dynamics becomes very simple. In 11
This occurs as we are analyzing an economy made up of a continuum of agents; as a consequence, the economic activity of each agent does not have a relevant impact on the environmental good and in the absence of coordination, none of them will enforce activities for the protection of the environmental good.
particular, there always exists a globally attracting stationary ¯ 1. state in which K = K In the case b N 0, above the curve Ω, the locus K ¼ 0 coincides with the locus K ¼ 0 in the case b = 0; however, below the curve Ω, ¯ 1. This implies that the it lies entirely on the right of the line K =K region where K N0 is wider in case b N 0 than in case b = 0 (see Figs. 2–6). Therefore, the evolution of K is sensibly conditioned by the value of E and this gives rise to the variety of possible dynamic regimes showed in the previous classification.
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Fig. 5 – Dynamics in which the stationary state B is globally attracting.
¯ 1 may Note that, in the case b N 0, the accumulation level K still be reached by the economy; in particular, it is the value assumed by K at the stationary state D (when it exists), where private goods are not consumed as substitutes (see Figs. 2–4). ¯ 1 is the minimum value that the economy However, for b N 0, K may reach starting from a strictly positive value of K. If the economy doesn't approach D, then it follows a trajectory converging to a stationary state with a higher value of K.
From the above classification, note that (ceteris paribus) D ¯ is high enough and the exists if and only if the endowment E negative impact γ of economic activity is low enough (see Fig. 1). In such context (see Figs. 2–4), D may be globally ¯ is ‘very’ high) or a bi-stable regime may attracting (when E occur where the economy approaches D only if it starts sufficiently near to it. Observe that, in the bi-stable regimes, the inset Γ of the saddle A can be considered as the graph of a strictly increasing function E = Γ̃(K) defined for every K ∈ (0,+∞). Given any initial value K0 N 0 of K, the economy reaches D if the initial value E0 of E is such that E0 N Γ̃(K0) while it approaches the other attracting stationary state (B in Fig. 3, C in Fig. 4) if E0 b Γ̃(K0). Therefore, in bi-stable regimes, whatever the initial value K0 N 0 of K is, the economy can always follow a trajectory leading it to B or C if the initial value E0 of E is low enough. Being E = Γ̃(K) a strictly increasing function, the lower K0 is, the lower E0 must be for the economy to approach B or C. In bi-stable regimes, from the point of view of technical progress K, the stationary state D is a ‘poverty trap’ with respect to B and C. However, as we shall see in the following section, agents’ well-being evaluated at D may be higher than that at B and C. ¯ is low enough and γ is high enough, the stationary state If E D doesn't exist and dynamics always reach a globally ¯ 1 and output is attracting point where K is greater than K consumed as a substitute for the environmental good (cases represented in Figs. 5 and 6).
9.
Numerical examples
Note that all the stationary states with K N 0 are characterized by an inverse relation between K and E: the lower the
Fig. 6 – Dynamics in which the stationary state C is globally attracting.
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value of K is at such points, the higher the value of E is. Since representative agent's labor supply l ̃(K, E) is increasing in K and decreasing in E, this means that going from the left to the right of plane (K, E) we encounter stationary states with higher levels of technical progress K, of work effort, of private consumption and of environmental degradation. Therefore, in an economy where agents do not internalize negative and positive externalities, it is possible that a positive correlation between the value of K (and of the aggregate output) and the agents’ well-being doesn't exist. This paragraph provides some numerical examples showing how dynamics are affected by varia¯ tions of the most significant parameters of the model: γ, E and b. In the following Figures, for simplicity, only a portion of curves K ¼ 0 and E ¼ 0 has been traced, specifically that which lays under curve Ω. In Fig. 7 we consider an example that can help to show how economic agents' well-being and the dynamics of the economy are influenced by variations of γ, the parameter measuring the impact on the environment due to economic activity. With γ = 0.04, D is globally attracting and the dynamics are those described in Fig. 2 (F2). The same applies with γ = 0.07; however, in this case, the curves K ¼ 0 and E ¼ 0 are closer to each other. With γ = 0.1 and γ = 0.12, said curves meet, producing the bi-stable regimes defined in Figs. 3 and 4 (i.e.
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F3 and F4) respectively. It is worth mentioning that an increase in γ gives rise to an attracting stationary state where the value of K is higher than in D. Therefore an exogenous increase of γ may lead to an increase of K and of the aggregate output. We observe that in both cases of bi-stable dynamics, in D the value assumed by the utility function U is higher than in the other attracting stationary states (B in the case γ = 0.1, C in the case γ = 0.12). Fig. 8 shows the effects produced by the variation of ¯ , which measures the carrying capacity of the parameter E ¯ = 2.8, the point C is globally environmental good. With E attracting and dynamics are of the type F6, described in ¯ = 5.6 the stationary states A and D emerge, very Fig. 6. For E near to each other, and dynamics are those described in Fig. 4 ¯ = 8.4, the (F4). As the carrying capacity increases, for E stationary state B also emerges and it becomes attracting in ¯ = 11.2, place of C (dynamics F3, described in Fig. 3). Finally, for E the point D becomes globally attracting (dynamics F 2, described in Fig. 2). In all the considered cases, the value of the utility function U evaluated in the stationary state D (when it exists) is higher than in the other attracting stationary states. The numerical exercise represented in Fig. 9 shows the variations of economic dynamics and of the values of the utility function in relation to the variation of parameter b, the parameter representing the marginal rate of substitution
¯ = 8.7. Fig. 7 – Numerical simulation with parameters’ values: α = 0.5, b = 0.1, g = 0.05, a = 8, b = 1, d = 5, E
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Fig. 8 – Numerical simulation with parameters' values: α = 0.5, b = 0.1, g = 0.1, g = 0.05, a = 8, b = 1, d = 5.
between the environmental good and the private one. We can observe that, with b = 0.4, the stationary state D is globally attracting (dynamics F2). With b = 0.8, we experience the same dynamic regime; however, curves K ¼ 0 and E ¼ 0 are closer to each other. With b = 1.2 we achieve a bi-stable regime (dynamics F3), in which the stationary state D Pareto-dominates B, the other attracting stationary state. Finally, with b = 1.6, the point D is very near to the point A and the point B is very close to the point C; if b further increases, the bi-stable regime ceases to exist and C becomes globally attracting (dynamics F6). In this example, it is interesting to notice that in the cases with b = 0.4 and b = 0.8, the value of the utility function assumed in D (which is globally attracting) is higher than that of B in the cases with b = 1.2 and b = 1.6. This suggests that an improvement in the substitution possibilities between the private consumption good and the environmental good does not always produce desirable effects. In fact, in this example, as b increases, the attracting stationary state B ‘emerges’, Pareto-dominated by D, which does not exist with lower values of b. In the above examples, the stationary state D always Pareto-dominates the other attracting stationary states (B or C), characterized by a higher value of K and a lower value of E. The last numerical example we present (Fig. 10) shows that if the marginal rate of substitution (represented by the value of
the parameter b) between the private consumption good and the environmental good is high enough, then bi-stable dynamics regimes may be observed where D is Paretodominated by an attracting steady state with a higher value of K; in particular, Fig. 10 shows the values assumed by the utility function in each of the stationary states A, B, C, D of the bi-stable regime F3 represented in Fig. 3, in relation to the variation of parameter b. It is worth remembering that in D the value of K is at its lowest possible level; then A, B and C follow, in this order. We observe that the value of the utility function in D does not depend on the value of the parameter b, given that in D output is not consumed as a substitute for the environmental good. In the other stationary states, utility increases as b increases. Notice that as b increases, at first D Pareto-dominates all the other stationary states; in particular, it dominates B, the other attracting point of the bi-stable regime. Then, in the case of considerably high values of b, B dominates D. This numerical example suggests that, in the analyzed economy, two different ‘undesirable’ outcomes are possible, both a consequence of a coordination failure among economic agents; the economy may reach a ‘poverty trap’, characterized by low levels of technical progress and of private consumption – from which economic agents could step out by increasing their private consumption and their labor input within the industrial
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¯ = 8.7. Fig. 9 – Numerical simulation with parameters' values: α = 0.5, b = 0.1, g = 0.1, η = 0.05, a = 8, d = 5, E
sector. In this context – characterized by a ‘high enough’ value of the parameter b – environmental negative externalities can be an engine of desirable growth in that the deterioration of the
environmental good can play the role of a coordination device leading economic agents to a wider exploitation of positive externalities.
¯ = 8.7. Fig. 10 – Numerical simulation with parameters' values: α = 0.5, b = 0.01, g = 0.1, η = 0.05, a = 8, d = 5, E
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If however the parameter b is not high enough, the economy could converge to a ‘private consumption trap’, characterized by an excessive consumption of private goods, from which economic agents could step by reducing private consumption.
10.
Conclusion
The general prediction of the model is that the higher the ¯ of the environmental impact γ and the lower the endowment E environmental good are, the higher the economy's technical progress and consumption level will be. An exogenous reduc¯ or an exogenous increase in γ may generate an increase tion in E of the aggregate product and of the level of K. Economic growth is fueled by the increase in the work motivation of economic agents, as a consequence of the gradual deterioration of the environmental resource. Such deterioration induces agents to modify their consumption patterns, and concentrate more and more on the consumption of private and expensive goods, rather than free access environmental goods. The analysis of the model has shown that the outcome of this economic growth mechanism may be undesirable, and call for coordination by the Public Administration. What guidelines should the Public Administration follow in planning environmental policies? Public administrators should ask themselves the following question: how many free access leisure opportunities are available to individuals? The question becomes particularly relevant when we consider the issue of managing large urban centers. The negative environmental effects of interaction between individuals are particularly evident in environments with high human density and high production density such as urban areas, the very places in which the majority of self-protection choices are made. Cities have the advantage of offering a great variety of leisure opportunities; nevertheless, they also have the disadvantage that almost nothing on offer is free of charge. This damages weaker categories such as the elderly, children, and lowincome families. Traffic bans at weekends designed to reduce urban pollution are one example of an extremely effective public measure that induces citizens to change their consumption patterns. By reducing atmospheric and noise pollution, such measures considerably increase the supply of free access places available to citizens for their leisure time and hence help to reduce undesired self-protection consumption. Another important area of intervention is management of coastal areas, the objective of which should be not only a reduction in the degradation of ‘nearby’ bathing areas but also a reduction in the cost of access to such areas. In fact, it is appropriate not only to protect these areas, but also to guarantee free access to beaches made private by the granting of concessions, in order to avoid the effective transformation of free access public goods into costly private goods. Generally, it would be desirable for the Public Administration to identify and classify all activities which may give rise to self-protection choices, with reference to the type of subject involved (individuals, firms, Public Administration), the places where such activities are undertaken, and possible responsive actions to be undertaken by the Public Administration.
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