Environmental & Resource Economics (2005) 30: 465–484

Ó Springer 2005

Population Density, Pollution and Growth CLAS ERIKSSON* and FICRE ZEHAIE Department of Economics, Swedish University of Agricultural Sciences, P.O. Box 7013, 750 07 Uppsala, Sweden; *Author for correspondence (e-mail: [email protected]) Accepted 3 November 2004 Abstract. We analyze a growth model where the damage of pollution depends on population density and the character of pollution. From the steady state rates of change, in the social optimum, of a neoclassical and a semi-endogenous growth model respectively, we conclude that the less responsive the damage of pollution is to population density, the more likely is a development path with positive growth in consumption per capita and declining perceived pollution per capita. Non-awareness of the character of pollution may thus give suboptimal solutions. In particular, the commonly held view that pollution is a pure public bad may lead to growth-rate targets that are lower than optimal. Finally, we find that the character of pollution does not influence the transitional dynamics qualitatively. Key words: economic growth, limits to growth, pollution, population density, semi-endogenous growth JEL classifications: O41, H41

1. Introduction The debate between growth optimists and various vintages of Malthusian pessimists has been going on for a long time. In hindsight, it seems that the fears of the Malthusians have been exaggerated. Malthus’ claim that rapid population growth would inevitably lead to declining standards of living was forcefully contradicted by the industrial revolution and later by the demographic transition. The predictions about the soon-to-come exhaustion of natural resources, made by for instance, Meadows et al. (1972), was followed by considerable new discoveries and improved extraction techniques. The debate has now shifted to the potential threats that pollution may pose to various ecosystems and thereby to growth (Meadows et al. 1992). The analysis of and debate about the environmental effects of economic growth will thus go on, and to clarify the discussion there is nothing more practical than a good theory. A theoretical literature on growth and pollution already exists1 , but we think that there is more to be learnt by looking closer at two aspects of the theory: (i) the character of pollution; (ii) the role of an increasing population in the growth process. In the existing literature it is typically assumed that

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pollution is a pure public bad, i.e. there is no rivalry in the suffering from it.2 When it comes to population growth, it is often ignored in this literature, although a larger population obviously tends to imply more pollution, due to greater volumes of material and energy throughput. When population growth is included (notable examples are Keeler et al. (1972) and Gradus and Smulders (1993)) it is typically combined with the assumption that pollution is a pure public bad. An increase in population by a factor t > 1 will then have a consequence that the pollution that reaches every individual is multiplied by a factor t (given that every individual generates the same amount of pollution, before and after the increase in population). This means that all pollution reaches all inhabitants of the economy. The purpose of this paper is to show that such an assumption will lead to unnecessary pessimistic conclusions about the limits to growth if the reach of pollution in fact is limited. We argue that the reach of pollution may be limited for at least two reasons. First, since pollutants may be less than purely public in character we need to consider that the quantity of pollution that affects an individual is less than the total pollution generated in the society. Second, the reach of pollution also depends on population density: the higher the population density the larger is the number of individuals who will be in the reach of pollution. Therefore the representative individual’s perceived pollution may be different from total pollution generated in the society. By making such a distinction between total pollution and the individual’s perceived pollution, we find that the elasticity of perceived pollution with respect to population density (EPD) plays a central role in the optimal long-run growth rates. This elasticity is determined by the interplay between population density and the character of the pollutant. We analyze the role of population growth in a one-sector growth model with pollution as an input in production and compare two slightly different versions of this model. In the first version, technology improves exogenously and there are constant returns to scale in production. In the second version, we allow for increasing returns to scale and drop the exogenous technological progress. This version is thus a semi-endogenous growth model (cf. Jones 1995; and Eicher and Turnovsky 1999). We show that the optimal choice of the society between consumption and pollution alleviation can be considerably affected by the characteristics of perceived pollution. The more responsive the perceived pollution is to population density (i.e. the higher the EPD), the more are the resources allocated to the environmental activities and the lower is the growth in consumption per capita. This implies that it is important to be aware of the nature of pollution as otherwise society may put too much (or too little) resources in the environmental sector than what is optimal. In the exogenous growth model, we derive the crucial condition for positive per capita growth and declining perceived pollution as a simple

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difference between two terms. This difference can be interpreted as a race between exogenous technological change, which is positively related to economic growth, and the drag on economic growth arising from the increasing pollution that the additional population brings with it. One central result in this paper is that (given that population grows) the lower the EPD is, the larger will the optimal growth rate of production and consumption be. The reason is that a higher gross pollution level can be tolerated when the relatively local character of pollution reduces the damage of pollution. The interesting feature of this result is that the debate between the neoMalthusians and the technological optimists is enriched by the introduction of the possible variation in the reach of pollution. In particular, it is interesting to note that the commonly held view that pollution has a pure public bad character (which corresponds to EPD ¼ 1) will put a rather large drag on economic growth and thus give strong support to Malthusian arguments. On the other hand, recognizing that perceived pollution sometimes is less responsive to population density would imply that pollution puts weaker limits to growth.3 In the semi-endogenous analysis, the crucial condition for positive growth is dependent on the output elasticities and the EPD. A key assumption in ordinary semi-endogenous growth theory is that there are increasing returns to scale (Turnovsky 2000, Chapter 14.5), which corresponds to the assumption that the sum of output elasticities of capital and labor is greater than one. In this paper, the condition is slightly more complicated due to the inclusion of emissions in the production function and the fact that we consider various magnitudes of the EPD. Let us first consider the extreme case when perceived pollution does not respond to changes in population density at all. Then, as in the previous literature, the crucial condition for sustainable growth is increasing returns to scale i.e. the sum of the output elasticities of capital, labor and emissions is greater than 1. In contrast to the previous literature, inclusion of pollution in the production function then allows for the sum of output elasticities of capital and labor to be less than one. On the other hand, if perceived pollution increases when population density increases, c:p:, then the condition for sustainable growth is more difficult to fulfil. Population growth now has two roles. First, it increases productivity and interacts with increasing returns to make sustained growth possible, as in the previous case. Second, it reinforces the damaging effect of pollution. In other words, sustainable growth may be jeopardized because the responsiveness of perceived pollution to increased population density counteracts increasing returns to scale. This becomes particularly evident when we consider the case when EPD
1. This will totally offset the productivity of pollution, or even worse. If the output elasticities of capital and labor sum to

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less than one, it implies that we cannot have a sustainable growth in per capita consumption in this case. The stability analysis of the two models offers few surprises. The exogenous growth model is unambiguously saddle-point stable. For the semiendogenous growth model, we find that if the output elasticity of capital is greater than one, the condition for a stable balanced growth path will be violated. This is consistent with Turnovsky (2000). The paper is organized as follows. Section 2 presents the components of the model. In Section 3 technological change is exogenous, whereas it is semiendogenous in Section 4.

2. Model To make the exposition simple, we present the economic problem and its solution as if we had a problem of a social planner at hand. However, we would like to think of the following expressions as resulting from decentralized optimization behavior, where externalities are properly internalized, due to for example taxes on pollution. As this has been done many times (e.g. Stokey 1998; Michel and Rotillon 1995), we leave such an analysis out of this paper. The representative dynasty is populated by LðtÞ ¼ ent individuals at time t. Each of them derives utility from consumption and disutility from pollution. Denoting aggregate consumption by C and ignoring distributional issues, consumption per individual is C=L, since consumption is a perfectly private good. Turning to disutility of pollution, we follow Copeland and Taylor (1994) and assume that the damage of pollution depends on pollution generated per individual and on population density. Therefore, we distinguish between aggregate gross pollution, P, and perceived pollution, p. The relation between these two variables is P ð1Þ p ¼ /ðdÞ ; L where d  L=T is the population density and T is the total amount of land in the economy. We assume that / ¼ CðL=TÞn , where C and n are constants. The parameter n can be interpreted as the elasticity of perceived pollution with respect to population density (EPD). Perceived pollution now is
n 1 P P  e: ð2Þ p¼C 1n T L L Without loss of generality we have here assumed that Tn ¼ C and 1  n ¼ e. It is plausible to assume that e  1 and possibly negative.4

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If population density is low enough, the community is so sparsely populated that individuals suffer only from their own pollution and perceived pollution is then just pollution per capita. Thus in Equation (2), for population densities below the threshold d
¼ C1=e , we assume that / ¼ 1. However, we will not be explicitly interested in such low levels of population density. If population density is high, all individuals suffer not only from their own pollution but also from the pollution generated by others. Thus, when population density is above the threshold population density d
¼ C1=e , /ðdÞ is an increasing function. Therefore, perceived pollution, p increases when d increases and all individuals are increasingly affected by the pollution generated by others. We can now study the properties of perceived pollution for different values of e. If e < 0 then n > 1, so / is a convex function of d. This implies that an increasing population density reinforces pollution damage at an increasing rate. If e ¼ 0 pollution perceived by every individual is equal to aggregate pollution: all pollution reaches everyone. This case coincides with the pure public bad5 formulation chosen by Keeler et al. (1972) and Gradus and Smulders (1993), as mentioned in the introduction. An example that is used often is the emission of carbon dioxide contributing to global warming.6 If e ¼ 1, perceived pollution is constant when population density increases. The underlying assumption must be that the pollutant stays very close to the one who creates it and never reaches anyone else.7 Although this is an unrealistic case, it provides a useful illustration of how the model works. An example of the case when 0 < e < 1 would be pesticides: sprayed on fruits-trees some of the pesticides get attached to the fruits and the person who sprays and are thus private in character; another part finds its way into watersheds and therefore becomes more public in character. Since this is an aggregate growth model, p and P are aggregates of many pollutants. Therefore, p is determined by the weight of each pollutant in aggregate pollution and their characteristics. We assume that the representative dynasty discounts future utilities at the rate q and seeks to maximize the utility function ( ) Z 1 1r 1þc ðC=LÞ p  dt eqt L U¼ 1r 1þc 0   Z 1 1r 1þc ðqnÞt nðr1Þt C eð1þcÞnt P e ¼ e e dt; ð3Þ 1r 1þc 0 where we have eliminated p from the lower line by use of (2). The constant r is the inverse of the intertemporal elasticity of substitution in consumption. There is a substantial amount of empirical evidence indicating that this elasticity is closer to 0 than to 1 (Hall 1988; Hahm 1998). Therefore we will assume that r > 1 throughout this paper. The parameter cð
0Þ captures the

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convexity of the disutility function of perceived pollution. The larger is c, the more rapidly will disutility increase when perceived pollution grows. The first constraint is the standard accumulation equation for capital, K. Assuming zero depreciation and denoting a time derivative with a dot it reads K_ ¼ YðK; L; EÞ  C;

ð4Þ

where Y is the production function and E is the outflow of emissions, generated in production. A final assumption is that pollution simply is equal to the flow of emissions P ¼ E:

ð5Þ

It would of course be more realistic to assume that a stock of pollution is accumulated by a flow of emissions. However, since there are examples indicating that this does not change the results significantly,8 at least in long run (e.g. Stokey 1998), we again favor simplicity of exposition by the assumption in Equation (5). Nevertheless, it should be remembered that the transitional dynamics probably would change significantly if there were stock pollution. We examine the optimal growth path of this pollution-extended standard growth model. The problem is to maximize utility (Equation (3)) subject to Equations (4), (5) and initial capital, K0 . Using k as the shadow price of capital, and substituting (5) into the utility function, we can form the presentvalue Hamiltonian H ¼ eðqrnÞt

C1r E1þc  e½qþnð1eð1þcÞÞt þ kðYðK; L; EÞ  CÞ 1r 1þc

ð6Þ

In the following two sections we will maximize this Hamiltonian for the exogenous and semi-endogenous models, respectively.

3. Exogenous Growth In the exogenous growth model we assume that the production function is Y ¼ AKa Eb L1ab ;

ð7Þ

where AðtÞ ¼ eat is a growing technology factor.9 The exponents of the production factors are positive and sum to one. Hence, the environment comes into the production function as a factor of production, as in e.g. Brock (1977).10 One way to think of this is to imagine an abatement sector that is separable from the output production sector. In this way, the choice of E determines how much capital and labor are to be allocated in the abatement sector. Then a decrease in E – i.e. stricter environmental regulation – will result in larger shares of K and L in the abatement sector and smaller shares

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of these resources being devoted to production of output, which therefore decreases.11 Substituting (7) into the Hamiltonian and following the usual Pontryagin procedure, we readily obtain the following necessary conditions for utility maximization: eðqrnÞt Cr ¼ k e½qþnð1eð1þcÞÞt Ec ¼ kAKa bEb1 L1ab k_ ¼ kAaKa1 Eb L1ab lim kK ¼ 0:

t!1

ð8Þ ð9Þ ð10Þ ð11Þ

Equations (4) (with (7) substituted into it), (5) and (8–10) can be used to determine the development of the five endogenous variables C, E, K, P and k along a balanced growth path (BGP). The following proportional rates of change are computed in Appendix A b¼K b ¼ ð1 þ cÞa þ n½bðr  1Þ þ ð1 þ cÞðbðe  1Þ þ 1  aÞ ; C bðr  1Þ þ ð1 þ cÞð1  aÞ

ð12Þ

aðr  1Þ þ n½ðr  1Þb þ eð1  aÞð1 þ cÞ : Pb ¼ bðr  1Þ þ ð1 þ cÞð1  aÞ

ð13Þ

b ¼ dC=dt. Since a A hat on a variable expresses proportional rate change, i.e. C C and b are fractions and r > 1, the denominator is positive. The first thing to note from these two expressions is that both aggregate consumption and pollution have two distinct driving forces: technological progress and population growth. It is probably more relevant to discuss the results by looking at the rates of change of the variables that enter the utility function, i.e. consumption per capita, C=L, and perceived pollution per capita, P=Le . Straightforward manipulations of Equations (12) and (13) yield bn ¼ C

ð1 þ cÞ½a  nbð1  eÞ ; bðr  1Þ þ ð1 þ cÞð1  aÞ

ð14Þ

and Pb  en ¼

ðr  1Þ½nbð1  eÞ  a : bðr  1Þ þ ð1 þ cÞð1  aÞ

ð15Þ

In both these expressions, the second parentheses of the numerators are crucial for the directions of change. They both constitute a race between technological progress and a drag on economic growth, which is due to the increasing pressure on the environment through pollution which results from a growing population. If

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a > nbð1  eÞ we have a steadily improving development with increasing consumption per capita and declining perceived pollution, P=Le . This is more likely if the population grows slowly and if emissions are not so important in production, i.e. b is small. As in previous works (e.g. Stokey 1998) a necessary condition for declining perceived pollution is that r > 1. Thus we need utility of consumption to approach an upper bound asymptotically, which can be interpreted as satiation in consumption. The higher is r, the faster does (relative) marginal utility decline when consumption increases. Thus high values of r tend to imply that a large share of the increase in productivity is used to improve the environment, while consumption is held back.12 Loosely speaking, consumption and clean environment are both normal goods. It is also interesting to note that e appears only in the term involving population growth, suggesting that the elasticity of perceived pollution with respect to population density (EPD) matters only when there is population growth. This is a consequence of our formulation of Equation (2): population growth can ‘deflate’ (or ‘inflate’ if e < 0, i.e. n > 1) the perception of a given aggregate level of pollution and e captures the extent to which this happens. Of course, e plays this role only if population does indeed grow. From the expressions above, it is clear that a higher EPD, i.e. lower e, makes the desirable good, C=L, increase more slowly (or even decline) and makes the non-desirable bad, P=Le , decrease more slowly or even increase. At a low e, Le grows slowly and so perceived pollution per capita increases fairly rapidly, because pollution from new members of society hits older members to quite a large extent. Thus, in this rather unfavorable situation, the economy holds back the increase in the per capita consumption. In a more advantageous situation, where e is large, it is possible and optimal to let this variable change at a higher rate.13 This can be illuminated by examining what happens when e is found at the two end points of the unit interval. In the extreme case when e ¼ 1, i.e. perceived pollution does not increase with higher population density, there is no drag on economic growth. New individuals, coming into the economy, will increase emission but this additional emission affects only one individual. Thus in per capita terms there is no increase in pollution and the pollution that enters the representative agent’s utility function is aggregate pollution per individual, P=L. This variable would remain constant, if it were not for the technological progress that makes it optimal to decrease P=L. The popular assumption that pollution is a pure public bad (Keeler et al. (1972) (Appendix) and Gradus and Smulders (1993)) means going to the other end of the unit interval, setting e ¼ 0 in our model. This implies a very

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large drag on growth of consumption per capita and on the decline in pollution. In other words, this assumption leads to a rather pessimistic outcome about the limits to growth. This growth-conservative position probably resulted because the neo-malthusians often set the agenda of the debate on limits to growth. However, this assumption can have negative welfare effects: growth targets may be set lower than they should, if in fact e > 0.14 To further explore the solution, we investigate the transitional dynamics of the solution to this problem. Define the variables z ¼ Y=K and x ¼ C=Y. In Appendix C we derive the proportional changes of these variables, b z ¼ D1 þ U1 z þ W1 zx

ð16Þ

xb ¼ D2 þ U2 z þ W2 zx:

ð17Þ

and The constant terms for Equation (16) are D1 ¼

1 fð1 þ cÞa þ bq þ n½ð1 þ cÞð1  a  bÞ 1þcþb  bð1  eð1 þ cÞg > 0;

U1 ¼ ð1  aÞ < 0 and W1 ¼

ð1 þ cÞð1  a  bÞ > 0: 1þcb

The corresponding constant terms for Equation (17) are D2 ¼

1 farð1 þ cÞ  q½ð1 þ c þ bðr  1ÞÞ ð1 þ c þ bÞr þrnð1 þ cÞða þ bð1  eÞÞg;

U2 ¼

að1  rÞ <0 r

W2 ¼

að1 þ cÞ > 0: 1þcb

and

Since in this section, we have constant returns to scale it follows that D1 ; W1 and W2 are positive while U1 and U2 are negative. To determine the sign of D2 we need further conditions to hold. A sufficient but far from necessary condition for D2 < 0 is that a > n. If that does not hold, we still have the q

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term, which should be substantial to guarantee convergence of the utility function. In steady state z_ ¼ x_ ¼ 0 and we can solve for the equilibrium points of z and x. For the output capital ratio, z, the equilibrium points are described by the equation ¼ zjz¼0 _

D1 : U 1 þ W1 x

ð18Þ

The corresponding equation for the consumption output ratio, x, is xjx¼0 ¼ _

U2 D2  : W2 W2 z

ð19Þ

The saddle path property of this system is formally demonstrated in Appendix D, by linearization of Equations (16) and (17) around the steady state. Thus we have confirmed what could be expected in view of earlier work (e.g. Van Der Ploeg and Withagen 1991): that this version of the neoclassical growth model qualitatively retains the transitional dynamics of the original model.

4. Semi-Endogenous Growth In the model of the previous section we showed that a necessary condition for positive long-run growth in per capita consumption was exogenous growth in the technology factor, A. We will now examine how this result changes if we do not allow for this exogenous technological change. To remain in the one-sector framework we simply allow for increasing returns to scale in production.15 Since there is positive population growth we have a semiendogenous growth model. This class of models have received considerable attention since they do not have two properties for which ‘pure’ endogenous growth models have been criticized: (i) they do not have the scale effect, which would imply that a larger economy grows faster; (ii) they do not have the knife-edge property, which would mean that sustained and non-accelerating growth is possible only if the output elasticities of reproducible factors exactly sum to one. Our model formulation makes it possible to analyze the conditions for sustained growth and stability while taking into account the concerns for the environment. A central feature of semi-endogenous growth models is that they (under certain assumptions about the returns to scale) imply that growing labor supply helps generating sustained growth. This is because the growing labor stock prevents the returns to reproducible inputs from falling below critical levels, which makes continued accumulation optimal.16 The balance between this positive effect of population growth and the negative environmental effect discussed above is central in this section.

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Formally, we now assume that the production function is Y ¼ Ka Eb Lg ;

ð20Þ

where the sum of exponents is no longer restricted to sum to one. The factor Ka is assumed to contain a positive externality from capital, as in Arrow (1962) and Romer (1986). For expositional reasons, the internalization of this externality is not modeled, but taken as given. Substituting Equation (20) into the Hamiltonian and solving the problem, gives some similar first order conditions as in the exogenous model. The two conditions that differ are e½qþnð1eð1þcÞÞt Ec ¼ kKa bEb1 Lg ; k_ ¼ kaKa1 Eb Lg :

ð21Þ ð22Þ

In this case, the optimal development of C, E, K, P and k along a balanced growth path can be determined by the equation K_ ¼ Ka Eb Lg  C;

ð23Þ

together with Equations (5), (8), (21) and (22). This is done in Appendix B where we in particular get b¼K b ¼ n½gðc þ 1Þ þ bðr þ eðc þ 1Þ  1Þ C bðr  1Þ þ ð1 þ cÞð1  aÞ

ð24Þ

n½ðr þ eð1 þ cÞ  1Þð1  aÞ þ gð1  rÞ : Pb ¼ bðr  1Þ þ ð1 þ cÞð1  aÞ

ð25Þ

and

Since a and b are fractions and r > 1, the denominator is positive. In per capita terms Equation (24) is rewritten as b  n ¼ nð1 þ cÞðg þ be þ a  1Þ ; C bðr  1Þ þ ð1 þ cÞð1  aÞ

ð26Þ

and from Equation (25) the perceived per capita pollution is nðr  1Þð1  a  eb  gÞ : Pb  en ¼ bðr  1Þ þ ð1 þ cÞð1  aÞ

ð27Þ

As in the previous section, the second terms of the numerators determine the directions of change. The critical condition for increasing C=L and declining P=Le is: a þ g þ eb > 1:

ð28Þ

Since this model can be regarded as an extension of the one-sector non-scale model in section 14.5 of Turnovsky (2000) it is reassuring that the conditions

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for sustained growth in per capita consumption coincide when the productivity of pollution is assumed to vanish (b ¼ 0). The requirement is then that a þ g > 1, i.e. there must be increasing returns to scale to labor and capital together. A nice property of this condition is that it does not require a to exactly equal one, i.e. the knife-edge property is avoided. In fact, sustained growth is consistent with a wide range of a, at given g. Moreover, when pollution has a role in production (b > 0) positive growth is possible at even lower values of a, given that e > 0. The less responsive perceived pollution is to population density, i.e. the larger e is, the more likely it is that the condition in (28) will hold. If e ¼ 0 then the condition will hold only if a þ g > 1, i.e. there must be increasing returns to capital and labor alone. The assumption, common in the previous literature, that pollution is a purely public bad would lead to this strong requirement. On the other hand, if e ¼ 1, returns to scale just slightly above 1 in all the three production factors together is enough to get the desired development. Again the higher the EPD the more unlikely is sustainable increasing consumption per capita. It is also interesting to compare the role of population growth in the models, by looking at Equations (14) and (26). First positive population growth is necessary for positive per capita consumption growth only in the semi-endogenous growth case. This finding is consistent with non-environmental growth models. More interestingly when n increases, the growth rate of per capita consumption decreases in (14) but increases in (26) (given that (28) holds). Of course, the drag increases in both cases, but in the latter case this is dominated by the fact that the higher growth rate of population allows for sufficiently high returns to capital at higher growth rates of capital and output. As in the previous section we investigate transitional dynamics and use the same definitions of z ¼ Y=K and x ¼ C=Y. Following the same steps as in the previous section, the proportional changes can be described by Equations (16) and (17) while the equilibrium point can be described by (18) and (19). The only differences are that the constants change to D1 ¼ n=ð1 þ c  bÞ W1 ¼ fð1 þ cÞð1  a  bÞg=ð1 þ c  bÞ, fb þ ð1 þ cÞðg  ebÞg, U1 ¼ a  1, D2 ¼ nfð1 þ cÞð1 þ ð1  gÞðc  bÞÞ  b½bð1  eÞg=ð1 þ c þ bÞ  qð1 þ cÞ=r, U2 ¼ fð1 þ cÞð1  arÞ þ brg=ðð1 þ cÞrÞ and W2 ¼ fað1 þ cÞ  bð1  bÞg= ð1 þ c  bÞ. Since we assume increasing returns to scale in this section the sum of a; g and b is higher than 1. This opens up the possibilities that one parameter alone or a subsume of the parameters may be higher than one.17 Thus it is relevant to investigate the stability conditions for some critical range of a. In Appendix D, we investigate the behavior of the dynamic system as a increases from values below 1 to values above 1. Defining the critical value a ¼ 1  b=ð1 þ cÞ, there are three basic cases:

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 Case 1: For a < a , we have U1 < 0 and W1 > 0. We investigate the system of differential equations and show that in this case we have saddle path equilibrium.  Case 2: For a < a  1, we have U1  0 and W1 < 0. Also in this case we find a saddle path equilibrium.  Case 3: For a > 1, we have U1 > 0 and W1 < 0. We then find that the equilibrium is an unstable improper node. Thus a condition for stability is that a  1, which is exactly the same as in the benchmark one-sector semi-endogenous growth model in Turnovsky (2000) (Section 14.5). Thus, introducing pollution in the production and utility functions (and allowing for different values of EPD) does not change the central condition for stability. To summarize, we need that a þ g þ eb > 1 for balanced growth whereas stability requires that a  1. Obviously, there is a wide range of parameters at which both these requirements are fulfilled. Moreover, the presence of e and the b contributes to make it likely that the first inequality will hold. Thus, after the introduction of the environment in the production function in a semi-endogenous model, we are still able to retain stability as well as a balanced growth. It is interesting to compare these results with those in Groth and Schou (2002) who investigate a case when the environmental variable in the production function is assumed to be a non-renewable resource, the use of which does not affect utility. Groth and Schou need a minimum level of n or a minimum level a for a positive growth in consumption but we do not. The reason seems to be that the extraction of the limited resources must decline at a certain rate to avoid the resource from being exhausted in finite time. To prevent the returns to capital from falling to much, as a consequence, a certain growth rate of labor is needed. In our model there is no constraint on how much pollution one can ‘use’ in production. Therefore, if n is very low, the decline in E can be very low as well. Thereby, the returns to capital are kept at a level that is sufficiently high to give incentives for a positive rate of capital accumulation.

5. Conclusion In this paper, we have built a model to show that the traditional assumption about pollution as a pure public bad may lead to unnecessarily pessimistic conclusions about the limits to growth, if in fact pollution has a limited reach. This may be due to a low population density and that pollution is to some extent private in character. We derive this result in the neoclassical and semiendogenous versions of the one-sector growth model.

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An important result in this paper is that the less responsive perceived pollution is to increased population density the larger will the optimal growth rate of production and consumption be. When growth is exogenous, there is positive per capita consumption growth only if technological change wins a race against the drag on economic growth that arises from the increasing pollution that additional population brings with it. In that case, the more useful an additional unit of pollution is in production and the higher the population growth rate, the greater will the impact of the drag on economic growth be. If pollution is assumed to have pure public bads characteristic, which is the commonly held view, pollution will put a rather large drag on growth. In the semi-endogenous analysis, the crucial condition is dependent on the output elasticities and the elasticity of perceived pollution with respect to population density (EPD). A key assumption in ordinary semi-endogenous growth theory is that we have increasing returns to scale, which corresponds to the assumption that the sum of output elasticities of capital and labor is greater than one. In this paper, the condition is slightly more complicated due to the inclusion of emission in the production function and the fact that perceived pollution varies with population density. In the extreme case, when EPD ¼ 0 we find, as in the previous literature, that the crucial condition for sustainable growth is increasing returns to scale, i.e the sum of the output elasticities of capital, labor and emission is greater than 1. In contrast to the previous literature, inclusion of pollution in the production function thus allows the sum of output elasticity of capital and labor to be less than one, since there is a third production factor here. The observation that perceived pollution is influenced by population density implies that the condition for sustainable growth is more difficult to fulfill. This is because the effect of the EPD on growth works in connection to the output elasticity of emissions. The higher the EPD the less will be the combined effect of these two factors in the condition for sustainable growth. In other words, sustainable growth may be jeopardized because the influence of population density on perceived pollution counteracts increasing returns to scale. Concerning future research, it would be interesting to examine whether the result of this paper survives in more general models. We believe that they would if, a separate research sector were included and if pollution were a stock accumulated from emissions. It would also be interesting to endogenize population growth. Acknowledgements We are grateful for valuable comments from Rob Hart, Bengt Kristro¨m, Susanna Lundstro¨m, A˚sa Lo¨fgren, Sjak Smulders and seminar participants at the Departments of Economics at SLU and Go¨teborg University. The comments from two anonymous referees are gratefully acknowledged.

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Notes 1. See for example Keeler et al. (1972), Brock (1977), Tahvonen and Kuuluvainen (1993), Bovenberg and Smulders (1995), Michel and Rotillon (1995), Smulders and Gradus (1996), Stokey (1998) and Schou (2000). A survey is given in Xepapadeas (2003). 2. The notion of public good is in this case not appropriate as pollution is a bad. For this reason we refer to public bad when one individuals suffering from one unit of pollution will not impede other individuals from suffering. The reasoning applies to private good and private bad as well. In this case one individual’s suffering from one unit of pollution means that other individuals cannot suffer from it. 3. However, in this case growth may still be jeopardized because in the long run it can be sustained only if technological growth is greater than the drag on economic growth. 4. We may get similar results by using the simpler formula p ¼ P1e . The exponent 1  e then reduces the ‘exposure’ of a given emission, e.g. because of decay (or assimilation by nature) during the transport from the emitter to the receiver. However, since our aim is to examine the roles of population growth and population density (not least the tension between the good and bad effects of population growth in the semi-endogenous case) we choose to have L in (2). 5. Another way to put it is that there is no ‘rivalry’ in the exposure to pollution. Baumol and Oates (1988) p. 19 use the term ‘undepletable’, about this case of externality, as opposed to ‘depletable’ in case of a private bad. 6. It should be remembered that it is likely that there are winners and losers in terms of utility of global warming. However, it is beyond the scope of this paper to distinguish between winners and losers of global warming. 7. With alternative terminologies, the externality is now extremely depletable or the pollutant is a pure private bad. An example of such a case would be that the garbage thrown into your garden cannot at the same time be thrown into the garden of your neighbor. 8. This statement must be qualified if there is a threshold at which the capability of the environment to assimilate pollution is drastically reduced. However, in the optimistic scenarios that we mostly consider, perceived pollution and possibly aggregate pollution are declining with time, so the economy would get further and further away from the threshold. 9. The technology factor could be split and distributed to all three factors: if a ¼ aK a þ aE b þ aL ð1  a  bÞ, then production would be ðeaK KÞa ðeaE EÞb ðeaL LÞ1ab . Thus, technological progress can be regarded as capital-augmenting, emission-augmenting and labor-augmenting. It makes less of all three factors needed, to produce a given quantity of output. 10. Unlike ordinary production functions the environment is here not a zero priced factor of production. It has a price even if the price is determined by a social planner rather than the market itself. 11. Emissions is really an output, ‘produced’ jointly with Y. See Copeland and Taylor (2003) for an example of how you can get from two such joint production functions to an equation like (7). b ¼ a=ð1  aÞ and Pb ¼ 0, but when r > 1, then 12. When n ¼ 0 and r ¼ 1, then C b < a=ð1  aÞ and Pb < 0. Thus, the growth rate of consumption is held back from the C benchmark level, a=ð1  aÞ, for the benefit of pollution decline. 13. Another way to put this is that the increase in the parameter e makes the disutility of perceived pollution lower. To restore an optimal marginal rate of substitution between pollution and consumption the latter is increased at the cost of slightly more aggregate pollution. The total result is that both arguments of the utility function change to the better at all points of time.

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14. On the other hand, it should also be noted that the prospects of growth can be bleaker: allowing n to be larger than unity (i.e. e < 0) implies that the drag grows even further. 15. Alternatively, one could introduce a separate sector for development of new and better technology. However, the one-sector model displayed here seems sufficient to point out some central results. 16. In some papers (e.g. Jones 1995) the crucial role of population growth is that more people generate more ideas, thereby keeping the productivity in the innovation sector at a sufficiently high level. 17. Groth and Schou find that a necessary condition for balanced growth in a semi-endogenous model is that either the capital share of the economy, a, or the sum of the capital share of the economy and the labour share of the economy, a þ g, is above one (Groth and Schou 2002). 18. Recall that the constant terms differ between the two sections.

References Arrow, K. J. (1962), ‘The Economic Implications of Learning by doing’, Review of Economic Studies 80, 155–173. Baumol, W. and W. E. Oates (1988), The Theory of Environmental Policy. Cambridge: Cambridge University Press. Bovenberg, A. and S. Smulders (1995), ‘Environmental quality and pollution-augmenting technological change in a two-sector endogenous growth model’, Journal of Public Economics 57, 369–391. Brock, W. A. (1977), ‘A polluted golden age’, in V. L. Smith, ed., Economics of Natural and Environmental Resources. New York: Gordon and Breach. Copeland, B. and S. Taylor (1994), ‘North-South Trade and the Global Environment’, Quarterly Journal of Economics 109, 755–787. Copeland, B. and S. Taylor (2003), Trade and the Environment. Theory and Evidence. Princeton: Princeton University Press. Eicher, T. S. and S. J. Turnovsky (1999), ‘Non-scale models of economic growth’, The Economic Journal 109, 394–415. Gradus, R. and S. Smulders (1993), ‘The trade-off between environmental care and long-term growth—pollution in three prototype growth models’, Journal of Economics 58, 25–51. Groth, C. and P. Schou (2002), ‘Can non-renewable resources alleviate the knife-edge character of endogenous growth?’, Oxford Economic Papers 54, 386–411. Hahm, J. H. (1998), ‘Consumption adjustment to real interest rates: Intertemporal substitution revisited’, Journal of Economic Dynamics and Control 22, 293–320. Hall, R. E. (1988), ‘Intertemporal substitution in consumption’, Journal of Political Economy 96, 339–357. Jones, C. I. (1995), ‘R&D-based models of economic growth’, Journal of Political Economy 103, 759–784. Keeler, E., M. Spencer and R. Zeckhauser (1972), ‘The optimal control of pollution’, Journal of Economic Theory 4, 19–34. Meadows, D. H., D. L. Meadows and J. Randers (1992), Beyond the Limits: Global Collapse or Sustainable Future. London: Earthscan. Meadows, D. H., D. L. Meadows, J. Randers and W. Behrens (1972), The Limits to Growth: A Report for the Club of Rome’s Project on the Predicament of Mankind. New York: Earth Island. Michel, P. and G. Rotillon (1995), ‘Disutility of pollution and endogenous growth’, Environmental and Resource Economics 6, 279–300.

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Romer, P. M. (1986), ‘Increasing returns and long-run growth’, Journal of Political Economy 94, 1002–1037. Schou, P. (2000), ‘Polluting non-renewable resources and growth’, Environmental and Resources Economics 16, 211–227. Smulders, S. and R. Gradus (1996), ‘Pollution abatement and long-term growth’, European Journal of Political Economy 12, 505–532. Stokey, N. L. (1998), ‘Are there limits to growth?’, International Economic Review 39, 1–31. Tahvonen, O. and U. Kuuluvainen (1993), ‘Economic growth, pollution, and renewable resources’, Journal of Environmental Economics and Management 24, 101–118. Turnovsky, S. J. (2000), Methods of Macroeconomic Dynamics . 2nd ed. Cambridge, MA: MIT Press. Van Der Ploeg, F. and C. Withagen (1991). ‘Pollution control and the Ramsey problem’, Environmental and Resource Economics 1, 215–236. Xepapadeas, A. (2003), Economic growth and the environment. Technical report, University of Crete.

Appendix A Equations (4), (5) and (8–10) determine the development of the five endogenous variables C, E, K, P and k . To compute the steady state growth rates we first differentiate (5), (8) and (9) logarithmicaly with respect to time

Pb ¼Eb b ¼b q þ rn  rC k b þ ðb  1  cÞEb q þ n½a þ b  eð1 þ cÞ ¼b k þ a þ aK

ð29Þ ð30Þ ð31Þ

Next, transform the LHS’s of (4) and (10) into growth rates:

b ¼AKa1 Eb L1ab  C K K a1 b 1ab b k ¼  AaK E L

ð32Þ ð33Þ

The proportional rates of change of these two equations are constant only if the combinations of variables on the RHSs are constant. We will use these conditions to infer some relations between growth rates on the balanced growth path. First, from (33), we have that AKa1 Eb L1ab is constant, so

b a þ bEb þ ð1  a  bÞn ¼ ð1  aÞK

ð34Þ

This also implies that C=K must be constant in (32) and thus

b b¼C K

ð35Þ

The five Equations (29)–(31) and (34)–(35) can be used to solve for the steady state growth rates of the five endogenous variables (rates of change). As a first step, use (30) and (35) in (31):

b ¼ a þ n½ðr  1Þ þ eð1 þ cÞ þ ð1  a  bÞ  ð1  b þ cÞEb ðr  aÞK ð36Þ

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b we Equations (29), (34) and (36) determine the growth rates in P, E and K. Solving (34) for E, get

1 b  a  ð1  a  bÞng Eb ¼ fð1  aÞK b

ð37Þ

Substituting this into (36), we obtain

b  aÞb þ ð1  b þ cÞð1  aÞ ¼að1 þ cÞ þ n½bðr  1Þ K½ðr þ ð1 þ cÞð1 þ be  a  bÞ

ð38Þ

b from which directly yields Equation (12) in the main text. Using that expression to eliminate K (37):

b Eb½ðr  aÞb þ ð1  b þ cÞð1  aÞ ¼abð1  rÞ þ nb½bðr  1Þ þ eð1 þ cÞð1  aÞ

ð39Þ

This is easily rewritten as (13).

Appendix B To compute the growth rates along the BGP it is useful to take log of Equation (21) and differentiate with respect to time: b þ ðb  1ÞEb þ gn q þ n½1  eð1 þ cÞ þ cEb ¼ b k þ aK

ð40Þ

Also, (4) and (22) become b ¼Ka1 Eb Lg  C K K b k ¼  aKa1 Eb Lg

ð41Þ ð42Þ

b k is constant only if Ka1 Eb Lg is constant, which implies b bEb þ gn ¼ ð1  aÞK

ð43Þ

This also implies that C=K must be constant in (41) and thus b b¼C K

ð44Þ

The five Equations (29), (30), (40), (43) and (44) can be used to solve for the steady state b K, b E, b Pb and b growth rates of the five endogenous variables C, k. Using (30) and (44) in (40): b ¼ n½ðr  1Þ þ eð1 þ cÞ þ g  ð1  b þ cÞEb ðr  aÞK

ð45Þ

b we get Solving (43) for E, 1 b  gng Eb ¼ fð1  aÞK b

ð46Þ

Substituting this into (45): b  aÞb þ ð1  b þ cÞð1  aÞ ¼ n½bðr  1Þ þ beð1 þ cÞ þ gð1 þ cÞ K½ðr

ð47Þ

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b from (45), which gives Equation (24) in the main text. Finally we can use that to eliminate K which then can be solved for (25) in the main text

Appendix C Output–capital ratio is defined as z ¼ Y=K ¼ AKa1 Eb L1ab which implies: b þ bEb þ ð1  a  bÞn: b z ¼ a þ ða  1ÞK

ð48Þ

Further use (30) and (33) to get the following equality: b ¼ AaKa1 Eb L1ab q þ rn  rC

ð49Þ

Using the definition of z we can rewrite Equations (31), (32) and (49) as follows: b ¼z  C K K 1 b C ¼ faz þ rn  qg r 1 b b þ q  n½1  eð1 þ cÞ ½k þ b zþK Eb ¼ 1þc

ð50Þ ð51Þ ð52Þ

b Combining (48) and (50) we can eliminate K: 1 C ½a þ bEb þ ð1  a  bÞn  b z ¼ z  1a K which is equivalent to: C b z ¼ a þ bEb þ ð1  a  bÞn þ ð1  aÞ  ð1  aÞz K

ð53Þ

Now the consumption output ratio is x ¼ C=Y ¼ AKa EbCL1ab . This implies: b  a  aK b  bEb  ð1  a  bÞn xb ¼ C

ð54Þ

Equation (51) can be written as: b ¼ z  xz K

ð55Þ

b from (52): Using the fact that b k ¼ az and Equation (55) we can eliminate b k and K Eb ¼

1 ½zð1  aÞ  b z  xz þ q  n½1  eð1 þ cÞ 1þc

ð56Þ

b K b and Eb from (53) and (54). As a final step, we can use (51), (55) and (56) to eliminate C; Beginning with (53) we get: b z¼

1 ½ð1 þ cÞa þ bq þ n½ð1 þ cÞð1  a  bÞ  b½1  eð1 þ cÞ 1þcb  ð1  aÞð1 þ c  bÞz þ ðð1 þ cÞð1  aÞ  bÞxz:

ð57Þ

Turning to (54) also using (57) we get: xb ¼

1 ½nrð1 þ cÞða þ bð1  eÞÞ  q½1 þ c þ bðr  1Þþ ð1 þ c  bÞr þ zð1 þ c  bÞað1  rÞ  arð1 þ cÞ þ zxrað1 þ cÞ

ð58Þ

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CLAS ERIKSSON AND FICRE ZEHAIE

Appendix D The dynamics of the systems in sections 3 and 4 are18 : z_ ¼zðD1 þ U1 z þ W1 zxÞ x_ ¼xðD2 þ U2 z þ W2 zxÞ:

ð59Þ ð60Þ 



Linearizing the system around the steady state values z and x we need: @ z_  ð61Þ ¼z ðU1 þ W1 x Þ @z @ z_ ¼z2 W1 ð62Þ @x @ x_ ð63Þ ¼x ðU2 þ W2 x Þ @z @ x_ ð64Þ ¼x z W2 : @x Defining z_ ¼ z _ z and x_ ¼ x _ x , a first-order Taylor approximation of the system is:  


ðz  z Þ ðz _z Þ z ðU1 þ W1 x Þ ðz Þ2 W1 ; ¼ ðx  x Þ x ðU2 þ W2 x Þ x z W2 ðx _x Þ and characteristic equation is: l2  lK þ C ¼ 0; 



ð65Þ 

 2

where K ¼ z ðU1 þ x ðW1 þ W2 ÞÞ and C ¼ x ðz Þ ½U1 W2  U2 W1 . The eigenvalues are given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K K2  C: ð66Þ l1 ; l2 ¼  2 4 Considering first the system from Section 3, we can easily see that U1 =W1 > 1 > U2 =W2 . Therefore C < 0, which implies that the eigenvalues are of the opposite signs and we have a saddle point. Turning to the dynamic system of Section 4, we have three cases:  Case 1: a < a , which implies that U1 < 0 and W1 > 0. Both terms in C are negative and since the second term together with the minus sign is positive we cannot immediately determine the sign of C. However, if we compare the absolute values of the two terms we find that the first term is greater and we establish that C is negative. Then we calculate the eigenvalues and find two distinct eigenvalues of opposite signs which implies that we have a saddle path equilibrium.  Case 2: a < a < 1, so U1 < 0 and W1 < 0. In this case it is easy to see that C is negative and following the same arguments as in Case 1 we conclude that we have a saddle path equilibrium.  Case 3 : a > 1, U1 > 0 and W1 < 0. In this case the first term of C is positive and since the second term together with the minus sign is also positive, this implies that C is positive. Given this information we need to investigate the roots closely. In particular we need to establish the sign of K. We know that U1 and W2 are positive and W1 is negative. Then we need to determine the sign of the parenthesis. Since the absolute value of W2 is greater than W1 we conclude that K is positive. Thus we know that the eigenvalues have two positive roots which implies that we have an equilibrium characterized by an unstable node.