Undesirable economic growth via agents\' self-protection against environmental degradation

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Journal of the Franklin Institute 344 (2007) 377–390 www.elsevier.com/locate/jfranklin

Undesirable economic growth via agents’ self-protection against environmental degradation Angelo Antocia, Marcello Galeottib,, Paolo Russua b

a DEIR, University of Sassari, Italy DiMaD, University of Florence, Italy

Received 21 February 2006; accepted 28 February 2006

Abstract Our model analyzes the effects of the interplay between environmental degradation and consumption choices on economic growth dynamics and on economic agents’ welfare. We show that if private goods can be consumed as substitutes for environmental goods, then economic growth may be fueled by environmental deterioration. In this context, undesirable economic growth may occur, characterized by an inverse correlation between aggregate capital accumulation and economic agents’ welfare. r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Self-protection choices; Environmental deterioration; Indeterminacy; Undesirable economic growth

1. Introduction In the last decade some papers (e.g., see [1–6]) have analyzed a mechanism according to which economic growth may be fueled by the deterioration of free access environmental goods. These studies have considered growth dynamics in an economy where agents can produce private goods which can be consumed as a self-protection device against environmental deterioration. In such a context, if production and consumption of private goods deteriorate the environment, economic agents’ self-protection choices may lead to a self-feeding process of economic growth. In this process environmental degradation Corresponding author. Tel.: +39 055 479 6821; fax: +39 055 4796800.

E-mail addresses: [email protected] (A. Antoci), [email protected]fi.it (M. Galeotti), [email protected] (P. Russu). 0016-0032/$30.00 r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2006.02.037

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stimulates private consumption and physical capital accumulation. In turn, a higher level of production and consumption further depletes environmental goods. Since such a growth mechanism is based on negative externalities, it may be undesirable; that is, private consumption and capital accumulation levels may be inversely correlated with economic agents’ welfare. This line of research has typically relied on the simplifying assumption of perfect substitutability between environmental goods and private consumption. While assuring analytical tractability, it is not clear to what extent such results depend on it. The present paper aims to relax such an assumption; in particular, it studies both the possibility that they are imperfect substitutes and that they are complements. The main result obtained is that the growth effects of environmental degradation crucially depends on the degree of substitutability between private consumption and environmental goods. When they are substitutes, environmental degradation stimulates both labor supply and capital accumulation, thus fostering economic growth. When they are complements, environmental degradation has the opposite effect. The paper is organized as follows. Section 2 provides some examples of private goods which are ‘‘complements’’ and ‘‘substitutes’’ for environmental goods. Section 3 presents the model. Sections 4–7 analyze the dynamics. Section 8 outlines the conclusions. 2. ‘‘Complements’’ and ‘‘substitutes’’ for environmental goods It is rather intuitive that environmental degradation may alter the utility deriving from the consumption of private goods. In particular, the increase of economic agents’ utility generated by an additional unity of consumption of a private good may be (coeteris paribus) negatively or positively affected by environmental degradation. In the former case, private goods and environmental goods are said ‘‘complements’’, while they are said ‘‘substitutes’’ in the latter case (for a formal definition see Section 3). It is obvious that the utility deriving from the consumption of private goods may be reduced by environmental deterioration: to drink a cup of coffee in front of an uncontaminated seaside is better than in front of a polluted one. Driving a car or cycling along a road across an unpolluted countryside is more pleasant than in an urban suburb. Living in a house on an uncontaminated river is obviously better than in a house placed on a polluted and smelling one. However, the relevance of substitutive consumption is also stressed by the environmental economics literature (see e.g., [1,2,7–12,15]). The consumption of substitutive goods is stimulated by environmental degradation, as individuals can consume them to protect themselves against the deterioration of the environment. We can find several examples of self-protection choices. Mineral water may substitute spring water or tap water. Medicines may mitigate the effects of respiratory diseases caused by air pollution. Individuals may react to the deterioration of the seaside near their home by going to a less deteriorated seaside by car or boat; they may build a swimming pool in their gardens; they may purchase houses in exclusive areas at the seaside or buy holiday-packages in tropical paradises. Individuals may protect themselves from external sources of noise by installing sound-proofing devices. Urban life offers other examples of this substitution mechanism. Cities are often characterized by the scarcity of free access environmental resources. At the same time, they are able to supply a considerable variety of private and expensive consumption

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opportunities. The scarcity of areas where individuals can meet away from the dangers of city traffic brings on additional expenses for childcare (baby-sitters, playgrounds, etc.), as well as for the leisure of adults. In the cities, individuals can do almost nothing without spending money. To a degree, the reason for the constant increase in the consumption of ‘‘home entertainments’’ in the industrial societies can indeed be found in this substitution mechanism. The lack of parklands can induce individuals to pay for an entrance ticket to a national park or to spend money on a day out away from home on leisure activities. Another possible expenditure may be joining a gym, substituting physical activity in the open air with exercise carried out in a sports centre. The self-feeding process we consider in our model is well described by the metaphor of the ‘‘air conditioner syndrome’’: We refer to the example of Tokyo, which is very hot in the summer. The temperature of the city is considerably increased by the air conditioners in general and constant use. These machines cool the interior of buildings but emit heat to the exterior. Hence people are forced to buy air-conditioners by their widespread use: that is, the increase in output rises the demand for output... [1, p. 19]. Another paradigmatic example concerns the choice of using a car as a means of transport. This choice can partially be caused by air pollution: individuals who would have preferred to go by bicycle or on foot are forced to use their car because the air is unbreathable. Thus air pollution reinforces the decision to use private cars, which in turn increases air pollution (for other examples see [1,2,7,12]). 3. The model We assume that in the economy there exists a continuum of identical economic agents. Therefore we can analyze the choices of a representative agent. We assume that, at each instant of time t, the representative agent can produce a private good by his own labor input lðtÞ and the accumulated physical capital KðtÞ. The output Y ðtÞ he produces can be consumed ðCðtÞÞ or invested ðIðtÞÞ to increase KðtÞ. Consequently, Y ðtÞ ¼ CðtÞ þ IðtÞ and  K ðtÞ ¼ IðtÞ, where K denotes the time derivative of K. The instantaneous utility function of the representative agent depends on the consumption of the private good CðtÞ, on labor input lðtÞ and on a free access renewable environmental good EðtÞ. Furthermore, we assume that the representative agent has to solve the following intertemporal maximization problem:  Z 1 ðCEÞ1a l 2 rt max  (1) e dt C;l 1a 2 0 with the constraints 

d

K ¼ al b K 1b l  C,

ð2Þ



E ¼ E½ðE 1  EÞ  gC,

ð3Þ

where r, a, d, g and E 1 are strictly positive parameters. The parameter b belongs to the interval ð0; 1Þ and the parameter a satisfies the conditions a40, aa1. l and C indicate the economy-wide average labor input and consumption. The function ðCEÞ1a =ð1  aÞ  l 2 =2 represents the utility UðC; E; lÞ of the representative agent, who has to choose CðtÞ and lðtÞ in order to maximize the integral in (1). Such a

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quantity measures the discounted flow of the values assumed by the utility function in the time interval ½0; þ1Þ, i.e., the representative agent’s welfare along the growth trajectories. Note that, if a41, C and E are substitutes, that is, U CE o0. Vice versa, if 0oao1, then C and E are complements, that is, U CE 40.1 At such an aggregate level, environmental degradation may reduce or increase the utility of private consumption. Which effect dominates is clearly an empirical question and, to the best of our knowledge, no research has so far addressed this problem. Therefore we shall analyze the economy dynamics under both assumptions. When a ! 1, the utility function becomes logarithmic in C and E ; this implies that the utility deriving from the consumption C does not depend on the value of E. At each instant of time, the representative agent produces the output Y by the d production function Y ¼ al b K 1b l and, according to Eq. (2), the difference between the output Y and the consumption C is invested and accumulated as productive capital. For the sake of analytical simplicity, we omit to consider physical capital depreciation. However, we conjecture that relaxing this assumption has no effect on the qualitative results of our model (e.g., [6]). The production process of each agent benefits from a positive externality due to the economy-wide labor input l. Coeteris paribus, the productivity of l and K increases as l increases. Furthermore, the assumption bo1 implies that the production function exhibits a constant return-to-scale technology in l and K (i.e., it is a homogeneous function of degree 1). The parameter a measures the level of exogenous technical progress. We assume that the average values l and C are considered as exogenously given by the representative agent when optimizing. Since economic agents are a continuum, the impact of each agent’s choices on l and C is negligible. This assumption implies that the growth dynamics we analyze are not optimal. However, each orbit under the dynamics of our model is a Nash equilibrium; that is, no agent has an incentive to modify his choices, given the choices of the others. Since all agents are identical, they make the same choices; consequently, the average values l and C coincide, ex post, with the values of l and C. Note that, by posing l ¼ l, we obtain the production function Y ¼ al bþd K 1b , which would be considered by a social planner who had the possibility of coordinating agents’ choices. Such a function exhibits decreasing (respectively, constant and increasing) marginal productivity of the labor input if b þ do1 (respectively, b þ d ¼ 1 and b þ d41). We consider a positive externality in the model to show that, according to the mechanism we analyze, negative externalities may play the role of a coordination device among agents’ choices. That is, the increase in labor input generated by the increase of substitutive consumption can lead economic agents to better exploit positive externalities. In particular, we shall show that negative externalities generate utility growth if the value of d is low, while the opposite holds if d is high enough. Eq. (3) describes the dynamics of EðtÞ. The parameter E 1 represents the carrying capacity of the environmental good and can be interpreted as the endowment of the environmental good in the economy. That is, the value of E would reach E 1 without the negative effect of the consumption C. The expression EðE 1  EÞ is the usual logistic regeneration function. We assume, for the sake of analytical simplicity, that the renewable natural resource is depleted only by the consumption and not by the production of the

1

C and E are said ‘‘perfect substitutes’’ if U C =U E is constant, whatever the values of C and E are.

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private good. In fact, in many real life examples, the negative impact of consumption largely overcomes that of production. Think of the use of cars. The parameter g measures the negative impact of the consumption of one unity of private good. An increase of g can be produced by the diffusion of consumption technologies with high environmental impact, for example packaging, or by changes in the composition of the aggregate private consumption C; for example, when the use of cars increases and the use of bicycles reduces. 4. Dynamics The current value Hamiltonian function for our problem is HðE; K; C; l; l; yÞ ¼

ðCEÞ1a l 2 d  þ lðal b K 1b l  CÞ þ yE½ðE 1  EÞ  gC, 1a 2

where l and y, the co-state variables associated with K and E, can be interpreted as the (subjective) prices associated to K and E. By applying the maximum principle conditions and identifying, a posteriori, C and l with C and l, we are reduced to study a threedimensional system (as y does not appear in the expressions of K, l, E), that is, 

K ¼ al bþd K 1b  C, 

l ¼ l½r  að1  bÞl bþd K b , E_ ¼ E ½E 1  E  gC 

ð4Þ

with the conditions (equivalent to qH=ql ¼ qH=qC ¼ 0) l 2 þ abll bþd K 1b ¼ 0, C a E 1a  l ¼ 0.

(5)

Furthermore, the transversality condition lim KðtÞlðtÞert ¼ 0

t!1

requires that the discounted value of K approaches zero ‘‘at the end’’ of the maximization horizon. By Eq. (5), system (4) can be written as follows: 

K ¼ al bþd K 1b  C,     1a 1 ðE 1  E  gC Þ þ að1  bÞl bþd K b  r , C¼C a a _ E ¼ E½E 1  E  gC, where l is defined by l

2bd

¼ abK 1b C

a

ð6Þ

E 1a .

Remark. It can be proved that the three-dimensional system reduces to a two-dimensional one when a ¼ 1. Therefore, in the following, we shall assume aa1.

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5. Fixed points We shall analyze only the empirically more relevant case b4d (i.e., in the production function Y the exponent of l is greater than the exponent of l). We are going to seek fixed points S ¼ ðK n ; C n ; E n Þ with E n 40. In fact, under our assumptions, system (6) is defined, when E ¼ 0, only if ða  1Þð2  b  dÞo0. In this case, as l ¼ 0, it is easily computed that along any trajectory on the invariant plane E ¼ 0 either KðtÞ or CðtÞ tends to zero, according that ð1  aÞE 1  r_0. This justifies our research of fixed points S with E n 40. It is easily computed that the coordinates of S must satisfy the following equalities: r K n, Cn ¼ 1b gr K n, En ¼ E1  1b gr ðð1aÞdðaþ1ÞbÞ=ða1ÞðbþdÞ K n þ mK n , ð7Þ E 1 ¼ cðK n Þ :¼ 1b where m is a positive quantity depending on a, b, g, d, a, r. Hence (1) if a41, the fixed points are generically zero or two (two if, coeteris paribus, E 1 is sufficiently high); see Fig. 1; (2) if 0oao1, the fixed point is unique.

Ψ(K)

E∞

__ E∞ 0

__ K

K*

Fig. 1. Fixed points for a41.

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6. Local stability Utilizing Eqs. (6) and (7), we can write the Jacobian matrix J at some fixed point S ¼ ðK n ; C n ; E n Þ as follows: 0 1 2r 2  ða  1Þðb þ dÞ ð1  aÞðb þ dÞrK n B C ð2  b  dÞð1  bÞE n 2bd 2bd B C B C 2 2 ðd  bÞr ðb þ dÞr a  1 ð1  aÞðb þ dÞr K n J¼B C. a1 þ a gC n C þ gC n B @ að1  bÞð2  b  dÞ 2  b  d A að2  b  dÞð1  bÞE n a 0 gE n E n (8) Hence det J can be easily computed. It turns out that (a) when two fixed points (generically) exist, then b  K n Þ, sign det J ¼ signð2  b  dÞða  1ÞðK where b :¼ E 1 ð1  bÞ½ða  1Þd þ ða þ 1Þb ; K 2gr½aðb þ dÞ  d (b) when the fixed point is unique, then sign det J ¼ signð2  b  dÞ. b corresponds to a fixed point if Let us further discuss case (a). It is easily checked that K 0 b b and only if c ðKÞ ¼ 0, implying K ¼ K and E 1 ¼ E 1 (see Fig. 1). Consider the function b 1 ÞÞ jðE 1 Þ :¼ cðKðE and the equation jðE 1 Þ ¼ E 1 .

(9)

Hence the only solution of Eq. (9) is E 1 ¼ E 1 , whereas two fixed points exist if and b only if E 1 4E 1 , that is, if E 1 4cðKÞ. In other words, assume two fixed points exist and denote them by S1 ¼ ðK 1 ; C 1 ; E 1 Þ and b S 2 ¼ ðK 2 ; C 2 ; E 2 Þ, with K 1 oK 2 . Then K 1 oKoK 2 , i.e., signðdet JðSi ÞÞ ¼ ð1Þi1 signð2  b  dÞða  1Þ;

i ¼ 1; 2.

The local stability analysis is summed up by the following proposition. Proposition 1. Let b4d. Then (i) if 0oao1, the unique equilibrium is a saddle with a two-dimensional stable manifold; (ii) if a41 and two equilibria S 1 and S2 exist (using the above notations), then S 2 is a saddle with one-dimensional stable manifold, while S 1 either is always a saddle with a twodimensional stable manifold or turns from a saddle with a two-dimensional stable manifold into a repeller.

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Proof. Denote by l1 ; l2 ; l3 the eigenvalues of J and write the characteristic polynomial PðlÞ ¼ l3 þ trðJÞl2  gðJÞl þ detðJÞ, where gðJÞ ¼ J 11 þ J 22 þ J 33 ¼ l1 l2 þ l1 l3 þ l2 l3 . (i) If 0oao1, then, at the unique equilibrium S, detðJðSÞÞ404gðJðSÞÞ, which implies the above statement (see the proof of Proposition 2). (ii) If a41 and two equilibria exist, when K n crosses the value K (defined by c0 ðKÞ ¼ 0), one eigenvalue of the matrix J changes sign, passing through 0.

Denote by S the equilibrium corresponding to K. Hence, if gðJðSÞÞo0; two eigenvalues at S 2 , when K n is close to K, say l1 and l2 , have, respectively, positive and negative real part, so that, being detðJðS 2 ÞÞo0, l3 is necessarily positive. Vice versa, as it will be seen, gðJðSÞÞ40 implies trðJðSÞÞ40 as well, so that, in such a case, there exist again two eigenvalues with positive real part at S 2 . Moreover no Hopf bifurcation can occur for K n 4K (see Proposition 2), implying our statement. & In our model, K and E are the state variables and C is a jumping variable. Given the initial values K 0 , E 0 , the representative agent chooses C 0 so that the trajectory passing through the point ðK 0 ; C 0 ; E 0 Þ approaches a fixed point or a periodic trajectory. Consequently, if at a fixed point S ¼ ðK n ; C n ; E n Þ the Jacobian matrix has one positive and two negative real part eigenvalues, there exists generically a unique value of C 0 that the representative agent can choose to approach S. On the contrary, if the fixed point S has one or no eigenvalue with negative real part, then the fixed point is generically non-reachable by the economy. In particular, when two fixed points S1 and S2 exist (a41), then the only reachable fixed point is S1 . This implies (see Fig. 1) that an exogenous reduction of the endowment E 1 of the environmental resource leads to an increase of the aggregate product Y and of the capital accumulation K in the economy (evaluated at the reachable fixed point). It is easily checked that the same effect is produced by an increase of the parameter g, measuring the negative impact on the environment due to C. In such a context, economic growth is fueled by the increase of ‘‘work motivation’’ of economic agents. The gradual deterioration of the environmental resource induces agents to alter their consumption patterns, concentrating more and more on the consumption of expensive private goods, rather than on the consumption of free access environmental goods. An important example of exogenous reduction of the endowment E 1 is given by British parliamentary enclosures of the 18th century, which contributed to the proletarianization of rural population (see [1,13]). The exclusion from common land access reduced considerably non-wage sources of subsistence available to rural classes and left them more dependent on wages and on private goods bought in the market. Another cause producing a reduction of E 1 is the construction of dams. The population living in the areas involved in the projects suffer forced displacement and loss of open access to natural resources (for a deeper discussion see [1,2,14]).

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Observe that, if ao1, the curve E 1 ¼ cðK n Þ is strictly increasing in the plane ðK n ; E 1 Þ (see Eq. (7)). Hence, a reduction of E 1 causes a reduction of the capital accumulation K n at the fixed point. The same result is produced by an increase of g. Therefore, when private and environmental goods are complements, environmental degradation inhibits economic growth. This result is due to the fact that the utility deriving from private consumption is reduced by environmental degradation; consequently, economic agents are not stimulated to work hard to produce private goods. Fig. 2 considers the case a41 and shows, by a numerical exercise (a ¼ 0:6, b ¼ 0:2, d ¼ 0:01, r ¼ 0:03, a ¼ 1:5, E 1 ¼ 800), that the value of K n increases with the value of g. A continuous (respectively, a dotted) line indicates that the fixed point is a saddle (respectively, a repeller). The point H corresponds to the occurrence of a Hopf bifurcation. By using the same parameter values, Fig. 3 illustrates a somewhat paradoxical result: the representative agent’s utility, evaluated at the reachable fixed point, is inversely correlated with the value of K n , when g increases. This implies that welfare is inversely correlated with gross national product (GNP). In this case, we can say that GNP is not an appropriate welfare index. In another numerical exercise, obtained by taking the same parameter values of the example in Fig. 2, except for a higher value of d (d ¼ 0:18), an increase of g produces an increase of the representative agent’s utility. This result is depending on the interplay between positive and negative externalities. In fact, negative externalities increase with g, while positive externalities increase with d. So the effect of positive externalities can more

2.15

x 104

2.1

2.05

K*

2 H

1.95

1.9

1.85

1.8 0.75

0.76

0.77

0.78

0.79 γ

0.8

0.81

0.82

0.83

Fig. 2. Numerical example: capital accumulation increasing with environmental degradation.

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−4.6

x 10−3

−4.7 −4.8 −4.9

U

−5 −5.1 −5.2 −5.3 −5.4 −5.5 1.84

1.86

1.88

1.9

1.92

1.94

1.96

K*

1.98 x 104

Fig. 3. Numerical example: agents’ utility decreasing with capital accumulation.

than compensate the undesirable effects of negative externalities, which become a coordination device leading to a better exploitation of positive externalities. Finally, Fig. 4 illustrates another exercise (b ¼ 0:2, g ¼ 0:75, d ¼ 0:01, r ¼ 0:03, a ¼ 1:5, E 1 ¼ 800) giving a counter intuitive result: when the economy experiments exogenous technical progress (i.e., the value of the parameter a rises), the representative agents’ welfare reduces. In other words, without suitable environmental protection policies, even technical progress may be an ‘‘evil’’ for the economy. For a discussion on the implications of the above results for economic policy see [1,2,14].

7. Hopf bifurcations We recall that, in the case of two equilibria, when K n crosses the value K, defined by c0 ðKÞ ¼ 0 (see Fig. 1), one eigenvalue of the matrix J changes sign. Vice versa, in the intervals where c0 ðK n Þa0, only Hopf bifurcations can generically occur. In fact, as signðdet JÞ does not change, the only possible change in local stability takes place when the real part of two complex conjugate eigenvalues becomes positive from negative or vice versa.

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−3

−4.5

x 10

−4.55

U

−4.6

−4.65

−4.7

−4.75

−4.8 0.57

0.575

0.58

0.585

0.59

0.595

0.6

0.605

α Fig. 4. Numerical example: agents’ utility decreasing with productivity.

If S is such a bifurcation point, the characteristic polynomial of JðSÞ is ðl2 þ b2 Þðl þ l0 Þ ¼  l3 þ l0 l2  b2 l þ l0 b2 ¼  l3 þ trðJÞl2  gðJÞl þ detðJÞ, where gðJÞ ¼ J 11 þ J 22 þ J 33 , J ik denoting the ði; kÞ minor of J. Therefore Eq. (10) holds if and only if ( gðJÞ40; detðJÞ ¼ gðJÞ  trðJÞ:

ð10Þ

(11)

Moreover it follows from (8) that detðJÞ, gðJÞ, trðJÞ are all linear functions of K n and E 1 . Precisely, utilizing Eq. (7), we obtain: 8   grð2a  1Þ > r½dbaðbþdÞ > K n þ að1bÞð2bdÞ ; > gðJÞ ¼ r E 1 þ < að1  bÞ grð2a  1Þ > > > : trðJÞ ¼ E 1 þ að1  bÞ K n þ r: So the straight lines ðL1 Þ  E 1 þ

grð2a  1Þ r½d  b  aðb þ dÞ Kn þ ¼0 að1  bÞ að1  bÞð2  b  dÞ

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and ðL2 Þ  E 1 þ

grð2a  1Þ Kn þ r ¼ 0 að1  bÞ

are parallel, and we can consider their intersections with the curve ðGÞ E 1 ¼ cðK n Þ, in order to detect intervals of K n where Hopf bifurcations can take place. Proposition 2. Let dob, 0oaa1. Then (i) when 0oao1, no Hopf bifurcation occurs; (ii) when a41, a Hopf bifurcation generically occurs, for some K n oK, if and only if E1o

grð2a  1Þ r½d  b  aðb þ dÞ . Kþ að1  bÞ að1  bÞð2  b  dÞ

(12)

In such a case a saddle with a two-dimensional stable manifold turns into a repeller and a ‘‘reachable’’ (i.e., endowed with a two-dimensional stable manifold) limit cycle may arise. Proof. Observe, first of all, that ð2a  1Þ=a_1 according that a_1: Hence (i) if 0oao1; the known term in L1 is negative and in fact the line does not intersect the curve G (in the positive quadrant). Thus gðJÞo0 along G; and no Hopf bifurcation can occur; (ii) let a41. Then L1 lies below L2 and both lines intersect G at exactly one point. Since gðJÞ40 implies trðJÞ40 also, no Hopf bifurcation is possible as K n 4K; where detðJÞo0, because of Eq. (11). e E g e Vice versa, if L1 intersects G at ðK; 1 Þ, with KoK, some Hopf bifurcation can occur e e e for K n 2 ðK; KÞ. Since detðJðKÞÞ40 ¼ detðJðKÞÞ, gðJðKÞÞ40 ¼ gðJðKÞÞ and finally e e trðJÞ40 in ½K; K, it follows that there exists at least one K n 2 ðK; KÞ for which conditions (11) are satisfied and thus a Hopf bifurcation generically occurs. This is easily seen to amount to condition (12), implying the proposition’s statement. & Fig. 5 shows a family of reachable limit cycles arisen via a Hopf bifurcation, using g as the bifurcation parameter. The parameter values are the same as in Figs. 2–4 and the bifurcation point is denoted by H, as in Fig. 2. It is interesting to note that the amplitude of oscillations increases as g increases. If the stock E becomes low, then an exogenous shock may generate an environmental collapse and, consequently, the unsustainability of the economic growth process. The same happens if the parameter a increases (i.e., if the economy experiments productivity shocks). 8. Conclusions The most interesting case we have examined is the a41 one (i.e., when the private and the environmental goods are substitutes). In such a case, the general prediction of the model is that the economy’s capital accumulation and consumption will be higher if g increases and E 1 decreases. Consequently, an exogenous reduction of E 1 or an exogenous

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0.8106 0.8104 0.8102 0.81 γ

0.8098 0.8096 0.8094 0.8092 205 1.995

200 E

1.99 1.985 195

x 104

1.98 1.975 190

1.97

K

Fig. 5. Family of reachable limit cycles arisen via a Hopf bifurcation.

increase of g fuels the economic growth. In other words, environmental degradation can be an ‘‘engine’’ of economic growth. Let us motivate the mechanism producing this result. Environmental degradation increases the ‘‘work motivation’’ of economic agents, leading them to concentrate their welfare sources more and more on expensive private goods, rather than on free access environmental goods. The consequent increase of private consumption produces further environmental degradation and so on. We have shown that, in such a context, economic growth may be undesirable; that is, capital accumulation levels and GNP may be inversely correlated with economic agents’ welfare. Another possible paradoxical result we have obtained is that, when a41, a productivity increase of physical capital and of labor (i.e., an increase of the parameter a) may reduce the economic agents’ welfare. The basic lesson emerging is that the aggregate level of consumption of the private goods may be a distorted index of individuals’ welfare. Economic growth policies, capable of achieving goals concerning capital accumulation and private consumption levels at high environmental costs, should be treated with great caution. Economic policies ought to be evaluated utilizing appropriate welfare indices, which take into consideration not only the level of aggregate consumption but also the environmental degradation.

References [1] A. Antoci, S. Bartolini, Negative externalities as the engine of growth in an evolutionary context, working paper no. 83.99, Fondazione ENI Enrico Mattei, Milan, 1999.

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