Di Vta G., (2007) Capital accumulation, interest rate, and the income - pollution pattern - A simple...

8/10/2019 Di Vta G., (2007) Capital accumulation, interest rate, and the income - pollution pattern - A simple model.pdf

http://slidepdf.com/reader/full/di-vta-g-2007-capital-accumulation-interest-rate-and-the-income-pollution 1/11

Capital accumulation, interest rate, and the income– pollution

pattern. A simple model

Giuseppe Di Vita ⁎

Faculty of Law, University of Catania, via Gallo n° 24, 95124 Catania, Italy

Accepted 27 April 2007

Abstract

In this paper we use a modified Ramsey–Cass–Koopmans model to show that the inverse U-shaped income– pollution

relationship may be explained through the decreasing of the interest rate over time and with a growth in income. If the problem of

the benevolent planner is (whether and how) to implement a policy for environmental pollution abatement to maximize social

welfare, the rate of interest plays a fundamental role in determining the moment at which such a policy should be adopted. In

particular, growth and capital accumulation together reduce the marginal rate of return of savings (and capital), making it possible

to implement environmentally friendly devices at the moment when the country grows richer and its rate of interest becomes lower

than the social discount rate.

© 2007 Elsevier B.V. All rights reserved.

JEL classification: O41; Q20

Keywords: Environmental Kuznets Curve; Growth; Interest rate; Pollution emissions

1. Introduction

The income– pollution relationship is currently one of the most important topics of research in economic literature.

The so-called Environmental Kuznets Curve (hereafter EKC) is an issue in both theoretical and empirical studies

because of its immediate relevance for the policy-maker, who uses it to define the most appropriate measures to adopt

for environmental protection.Since the pioneering article of Grossman and Krueger (1991), which established an inverse U-shaped income–

pollution pattern, many other researchers have attempted to support this relationship with possible theoretical

arguments. Without meaning to be exhaustive, we can refer to the transit from an agricultural economy to an

industrialized one and later to the implementation of cleaner production processes; to government corruption, backstop

technologies, institution failures, satiation of consumers, increasing returns to scale in pollution abatement processes,

negative externalities on production or consumption, and so on (John and Pecchenino, 1994; Arrow et al., 1995; Selden

Available online at www.sciencedirect.com

Economic Modelling 25 (2008) 225–235www.elsevier.com/locate/econbase

⁎ Tel.: +39 95 230335; fax: +39 95 321654.

E-mail address: [email protected] .

0264-9993/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.econmod.2007.04.017

mailto:[email protected]

http://dx.doi.org/10.1016/j.econmod.2007.04.017

http://dx.doi.org/10.1016/j.econmod.2007.04.017

mailto:[email protected]



and Song, 1995; Jaeger, 1998; Stokey, 1998; Suri and Chapman, 1998; Rothman, 1998; Jones and Manuelli, 2000;

López and Mitra, 2000; Magnani, 2000; Panayotou, 2000; Andreoni and Levinson, 2001; Tahvonen and Salo, 2001; Di

Vita, 2004).1

In this paper we attempt to supply a new explanation for the inverse U-shaped pattern followed by some pollutants

in relation to a growth in income (Harbaugh et al., 2002), by means of capital accumulation and the dynamics of the

interest rate, which in the steady state should be equal to the marginal rate of return of capital.We assume the marginal rate of return of capital to be high at a lower level of income, such that individuals want to

invest in capital accumulation and to satisfy their present needs, without considering the negative externality

represented by pollution in their utility function. It is not until the country becomes wealthier and the marginal return of

capital decreases that agents will prove more willing to invest their savings in pollution abatement devices. This is also

because ‘impatience’ decreases as people grow richer (Chavas, 2004).

The theoretical justification for choosing to delay adoption of more environmentally friendly measures until the

interest rate drops may alone explain why pollution initially increases together with income and later falls as production

grows. By means of the simple mechanism of a decrease in the interest rate as the economy develops, well-known since

the paper of Ramsey (1928) and the theoretical framework elaborated by Cass (1965) and Koopmans (1965) (for the

sake of simplicity called the RCK model)we are able to explain the inverse U-shaped behavior of some pollutants as

income rises. The question that we bear in mind in writing this paper is: can the decrease in the interest rate as incomegrows explain the inverse U-shaped income– pollution pattern? We will show that pollution emissions first increase and

later decrease as income grows, depending on whether the interest rate is greater or lower than the discount rate.

We are thus able to derive an inverse U-shaped EKC. Moreover, emissions increase in the rate of interest and

decrease in the pure time preference parameter index. Using the simple theoretical framework outlined above, we

demonstrate that even if some human resources are allotted to the pollution abatement sector and with increasing

returns to scale in the green sector, pollution emissions will not necessarily fall. It is only at some levels of income, and

for a lower marginal rate of return on capital (saving), that pollution emissions and income move in opposite directions.

This may also explain why pollution abatement policies cannot be adopted during the early stages of economic

development of a country.

The relevance of the interest rate in explaining the relationship between income and pollution is a recent topic and

has not been discussed in economic literature before now. The only paper concerning this question, by Gruver (1976),

was written before the EKC was discovered: in it the author introduces the portfolio choice between investments in productive capital and pollution control capital and underlines the fact that, at some optimal level of capital

accumulation, savings are allotted to the green industry to reduce emissions. The results of this paper are not general but

based on some configuration of parameters of capital and do not explicitly deal with the EKC (Pfaff et al., 2004). The

added value of the present paper is that here the interest rate is held to be endogenous, while in previous analyses it was

considered exogenous (see for example Di Vita, in press). This has allowed us to obtain more interesting results from

the analysis, because it is well-known that in real economy the interest rate declines with a growth in income. The next

step is to render the discount rate endogenous also. Other researchers have tackled the question using a very different

theoretical framework, less general than the model we have employed here.

The novelty of this simple but original explanation of the up-and-down behavior of some pollutants explains why

there are no econometric analyses considering the interest rate as an explanatory variable of pollution dynamics.

In the paper we use a very simple RCK growth model of a closed economy with an infinite temporal horizon,slightly modified in order to consider two sectors: the primary good is produced in the first; the second is the green

sector dealing with the pollution abatement policy. In the economy we have depicted in our model, if no labor time is

allotted to the emissions abatement sector only the industry in which inputs are transformed into a final output will be

considered. A very simple production function is used, exhibiting constant returns to scale in inputs, capital and labor.

The population is assumed to be constant, and normalized to one, for the sake of simplicity. In the hypothesis where

pollution reduction does take place, the labor will be divided between the two sectors. The emissions abatement

technology is described by a simple production function that exhibits increasing returns to scale (Andreoni and

Levinson, 2001). The utility function is additively separable into two areas: per-capita consumption and stock of

pollution (Hoel, 1978; Huhtala, 1999; Lusky, 1976; Plourde, 1972; Smith, 1972). We assume a functional form of the

utility function that satisfies the constant intertemporal elasticity of substitution (CIES) in both.

1 For a more detailed survey of the theoretical explanations of the EKC see Borghesi, 1999 and the introduction of Di Vita, in press.

226 G. Di Vita / Economic Modelling 25 (2008) 225 – 235



After this brief introduction, we set up the theoretical model in the section which follows. Section 3 sets out the main

results of the paper for a series of hypotheses. Comments and summary remarks conclude the work. Mathematical

details are produced in the Appendix.

2. Theoretical framework

To describe the model we start from the technology. The final output Y is obtained by combining two inputs,

physical capital K and labor effort v , allotted to the first sector of the economy. The production function is thus:

Y ¼ K a1 v a2 ; with a1 þ a2 ¼ 1; ð1Þ

which satisfies the so-called Inada conditions. Assuming the total labor force L available in our economy to be constant

and equal to one, (1−v ) is the number of workers involved in pollution abatement, if this activity takes place. The final

product Y may be utilized for consumption or investment purposes indifferently. This is why the amount of product that

is not consumed will be saved by the individuals in a closed economy. The law motion of physical capital is

d K ¼ Y C ; where K ð0Þ ¼ K 0 and K ðt Þ z 0: ð2Þ

(2) is a constraint which considers the change of K over time. For the sake of simplicity, depreciation in K is not

considered here. Aggregate consumption is denoted by C = xY , with 0b x≤1. Aggregate saving is S = sY = K ,̇ where

0≤ sb1, and s = (1− x).

The production and consumption of final output creates some pollution emissions E = Y γ, with 0bγ≤1 indicating

the rate at which emissions increase as income rises.

This functional form takes into account the empirical evidence for which, as income grows, the amount of total

output becoming emissions decreases, for example as a result of a structural change in production or output

dematerialization. E is the only control variable that governs the dynamics of pollution stock M over time, if no

pollution abatement policies are adopted. Otherwise, the behavior over time of M depends also upon the amount of

emissions that are abated E a by means of the labor effort devoted to this aim. The transition equation of the pollution

stock is thus:

d M ¼ r E E a p M ; ð3Þ

where 0bσ≤1 is a constant accounting for the emissions immediately absorbed by the ecosystem (with σ bγ). E a is

equal to Y γβ 1 (1−v )β 2, representing the pollution abatement technology, where β 1+β 2N1. Without loss of generality,

from here on we put β 1= 1 just to simplify the formal analysis. Finally, 0 bπ≤1 is the natural rate at which M decreases

over time. To distinguish better between σ and π, we can consider a pollutant, for instance CO2: the first parameter

regards the rate at which emissions are immediately absorbed by the atmosphere (for example, the concentration of

pollution is higher close to its source and lower at the upper level of the atmosphere); the second parameter considers

the rate at which the previously accumulated stock of pollutant dissipates over time.

It is worth spending a few words on E a and the assumption of increasing returns of scale. This is based on the

empirical evidence described by Andreoni and Levinson (2001) and on a simple consideration that the greater the

amount of pollution, the lower the marginal and average costs of abatement. Obviously if v =1, this implies that E a = 0.The instantaneous utility function of infinitely-lived households choosing between consumption and saving to

maximize their dynastic utility, subjected to an intertemporal budget constraint, can be indicated by

u ¼ uðC ; M Þ ¼ C 1h 1

1 h

M 1þx 1

1 þ x ; with h;g N 0: ð4Þ

(4) has continuous first and second partial derivatives, with uC N0, uCC b0, u M b0, u MM b0 and U C , M =0. Here θ

and ω are two parameters representing the elasticity of marginal utility with respect to consumption (θ) and pollution

stock (ω).

Not many words are necessary to clarify why utility depends on the per-capita consumption, whereas it could be

useful to explain why u depends on M . The reason for including the stock of pollution in the utility function is that the

accumulation of pollution has only negative effects on welfare (Keeler et al., 1971).

227G. Di Vita / Economic Modelling 25 (2008) 225 – 235



An exogenous social discount rate δN0 is applied to the flow of utility. The total welfare W associated with any

particular time path for C and M is derived by summing the discounted flow, as follows:

W ¼

Z l

0

uðC ; M Þedt dt ; ð5Þ

this reflects that current agents in this economy take into account the welfare and resources of their present or futuredescendants.

The function to be maximized is (5), subject to (1)–(4). The appropriate Hamiltonian for the problem is

H ¼ C 1h 1

1 h

M 1þx 1

1 þ x

edt þ k1ð K a1 v a2 C Þ k2fð K a1 v a2 Þg½r ð1 v Þb2 p M g ð6Þ

where λ i, i =1, 2, are the current values of shadow-prices of capital and pollution stocks, respectively. Note that it is

assumed that λ2b0 because pollution always reduces welfare, creating a negative externality on consumption.

The first-order conditions are straightforward and easy to calculate:

A H

AC ¼ edt

C h

k1 ¼ 0; or edt

C h

¼ k1; ð7Þ

A H

Av ¼ a2k1

Y

v k2Y g

a2grð1 v Þ a2gð1 v Þb2 ð1 v Þ þ ð1 v Þb2b2v

ð1 v Þv

" # ¼ 0;

or k2 ¼ a2k1Y 1g

½a2gr a2gð1 v Þb2 þ ð1 v Þb21b2v

; ð8Þ

d k1 ¼

A H

A K ¼ a1k1

Y

K þ

k2ga1Y g½r ð1 v Þb2

K ; ð9Þ

d k2 ¼

A H

A M ¼ M xedt pk2; ð10Þ

The growth rates of the dynamic multipliers are

g k1 ¼

d k1

k1

¼ a1

Y

K þ

k2

k1

ga1Y g½r ð1 v Þb2

K

" #; ð11Þ

g k2 ¼

d k2

k2 ¼

M xedt

k2 p

: ð12

Þ

Differentiating Eq. (7) logarithmically, the result will be:

g k2 ¼ h g c; ð13Þ

The transversality conditions are:

limt Yl

edt H ðt Þ ¼ 0; ð14Þ

limt Yl

½edt k1ðt Þ K ðt Þ ¼ 0; ð15Þ

limt Yl½edt k2ðt Þ M ðt Þ ¼ 0: ð16Þ




First of all, we want to prove that our model exhibits a locally stable saddle point. To this aim we have to find the

optimal values of physical capital, pollution stock and consumption. We know the equations that describe K ˙ and M ˙,

thus we want to derive the motion equation of consumption. To this aim we can calculate the partial derivative to

time of (7) and again substitute it for λ1 in (9); after eliminating λ2 by means of (8) and with a little algebra, we

obtain:

d C ¼

C

uWðC ÞC =u VðC Þ d r

a2grð1 Þ ð1 Þb2 ðb2 a2gÞ

ð1 Þb2ðb2 a2gÞ

" #( ); ð17Þ

where r =α1Y / K is the rate of interest (that in the steady state should be equal to the marginal rate of return of

capital, Cass, 1965; Romer, 1996). From our specification of the utility function (CIES) we know that u″(C )C / u′

(C ) =−θ and letting α2γσ (1−v )− (1−v )β 2 (α2γ−β 2v ) / (1− v )β 2 v (β 2−α2γ) =φ (where φN0), to simplify the

analysis, we may rewrite (17) as:

d

C ¼

C

h fr u dg: ð18Þ

Given that v is constant in the steady state, we can see that (18), together with (2) and (3), constitutes a system of

three unknowns in three equations that we can solve mathematically. By means of (8) and (10) it is also possible to

unite the two differential equations of physical capital and pollution stock in a single one, by integrating (3) with

respect to time and using the two equations mentioned above.

The result is:

d K ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffið K a1a2 Þ1þð2g1Þx p2a2C h

ðg1Þx

q C : ð19Þ

Letting ˙C = K ˙ =0 we may thus calculate the optimal values of consumption and capital respectively. By means of

(18) and (19), we find ∂ ˙C / ∂ K b0 and ∂ K ˙ / ∂C b0, from which we now possess all the information necessary to sketch

our phase diagram in ( K , C ) space, as in the figure below (Fig. 1).

This shows that we have a locally stable saddle point equilibrium. Moreover, we can prove this result by calculating

the characteristic roots of the system of equations represented by (18) and (19), evaluated in the steady state, to find that

we have two real roots, with opposite signs (see Appendix for analytical details).

Fig. 1. Phase diagram.




3. Decreasing marginal rate of return of capital and pollution dynamics

To study the relationship between the interest rate and the dynamics of pollution stock, we have to find the equation

that describes r . By means of (17) we obtain:

r ¼ d þ hd

cc

1u

; ð20Þ

where the term 1 / φN0 represents the effect on r of the internalization of the negative externality on the environment,

represented by pollution (we remember that φ combines the parameters of the model). In other words, the rate of

interest is lower when the negative externality of production is internalized. Without including the stock of pollution in

the utility function and the effort to reduce emissions in the model, the rate of interest would in fact be greater, because

equal to δ+θ (cd /c). This is because the marginal productivity of capital would be higher if the adverse effect of

production on the environment were accounted for.

Now, putting in evidence σ− (1−v )β 2 in (17) and then substituting in (3), we get the equation that links the

dynamics of the pollution stock with the interest rate:

d M ¼ ð K a1a2 Þg

r h g c d

r a2g

a2g r ð1 Þb21

h iþ ð1 Þb21

b2n o

p M : ð3′Þ

From the equation above we can see that M ˙ is a direct function of the interest rate. To express it simply, if the interest

rate is high the absolute change in pollution stock is also high and vice-versa. Now we can consider two different

hypotheses: in the first, no pollution abatement policy is adopted (i.e. v =1); in the second, some labor effort is devoted

to making the environment cleaner (i.e. v b1).

3.1. Case I: v=1

If we assume that no labor effort is allotted to the green sector (i.e. v =1), we have to rewrite the appropriate

Hamiltonian and derive the relative first order conditions. If we repeat the process outlined above to derive the equationdescribing the dynamics of pollution stock in terms of interest rate, pure time preference and growth rate of

consumption, we find:

d M ¼ ½r d h g C

Kedt u VðcÞ

k2ga1

p M : ð21Þ

Seeing that K ¼ Y 1a1 v

a2a1 from (1), and substituting in (21) we obtain:

d M ¼ ½r d h g C Y

1a1

a2a1

edt u VðcÞ

k2ga1

p M : ð22Þ

Calculating the first and second partial derivative of M ˙ with respect to the final output, we find:

Ad

M

AY ¼ Y

1þa1a1 ðr d h g C Þ

a2a1

edt u VðcÞ

k2ga21

; ð23Þ

and

A2d M

AY 2 ¼ Y

12a1a1

ða1 1Þ

a21

ðr d h g C Þ

a2a1

edt u VðcÞ

k2ga1

: ð24Þ

It is worth noting that ∂ M ˙ / ∂Y T0 and ∂2 M ˙ / ∂Y 2T0 depending on whether r Tδ+θ g C . Thus in the steady state, in

which r =δ+θ g C , the pollution stock will decline at the rate π. It is worth noting that this occurs for any positive value

of the term Ke−δt u′(c) / λ2γα1. In this way we are able to show that the pollution stock will decline without any




environmental policy or technological change process, merely as a result of growth. During the transitional dynamics,

M ˙ may be positive or negative depending on whether the first addend is greater or lower than the second in (22). We

thus obtain a fairly inverse U-shaped EKC in which the peak occurs for [r −δ−θ g C ] Ke−δt u′(c) / λ2γα1=π M .

Note that the declining behavior of the rate of interest has two effects. Firstly, income grows as a result of capital

accumulation, thus making it possible to implement more environmentally friendly devices, because the country

becomes richer. Secondly, the aggregate investment and production levels increase, and the level of pollution may riseso much that it could counterbalance the former effect.2

To understand better why we find that pollution emissions decline starting from a given income level, we can take

the partial derivative of pollution emissions M ˙ with respect to the interest rate, using Eq. (23) to obtain:

Ad

M

Ar ¼ Y

1a1

a2a1 edt u V

c

k2ga1

: ð25Þ

If we compare (23), that accounts for the effect of income growth on pollution (positive for the environment), and

(25), that considers the effect of an increase in production and in the level of emissions (negative for the environment),

knowing that Y 1a1 c=ðk2ga1ÞbY

1þa1a1 c=ðk2ga

2

1

Þ, we are able to say that, when r b (δ+θ g C ), the effect of an income

increase on the pollution emissions level is greater than the effect of the reduction in the interest rate. In other words,

when capital accumulation is low (the country is poor) the two effects work in the same direction (for r Nδ+θ g C ).

When the interest rate becomes low and capital accumulation is high, the two effects work in opposite directions and

the first is greater in magnitude than the second, such as to explain the declining branch of the EKC.

3.2. Case II: v b1

Now we can consider the case in which some resources are allotted to improving the quality of the environment (i.e.

v b1). In this case the dynamics of M are described by (3′), by means of which we calculate the first and second partial

derivative of M ˙ with respect to Y , to obtain

A

d

M AY

¼ Y g r h g c dr a2

a2g r ð1 Þb21

h iþ ð1 Þb21

b2n o

; ð26Þ

and

A2 d M

AY 2 ¼ Y g2

g 1ð Þ r h g c d

r a2

a2g r ð1 Þb21

h iþ ð1 Þb21

b2n o

: ð27Þ

It is worth noting that ∂ M ˙ / ∂Y T0 depending on whether r Tδ+θ g c. At the very first stage of the economy it will

probably be true that ∂ M ˙ / ∂Y N0 basically for three different reasons. i) The rate of interest is high as a consequence of

low accumulation of capital; ii) few resources are devoted to pollution abatement (i.e. u is very close to 1); iii) g c is far

from its steady state value.

This implies, as suggested by the general theory of growth, that if the rate of interest is greater than the discount rate,

amplified by the elasticity of marginal utility of consumption, pollution will increase over time. The opposite occurs

when the country becomes wealthy and may devote some resources to making the environment cleaner. Note that

∂2 M ˙ / ∂Y 2T0 depending on whether r Tδ+θ g c. In this way a quite inverse U-shaped income– pollution pattern is

generated as income grows and the rate of interest decreases with time, as we report in Fig. 2, below.

The upper part of Fig. 2 shows the relationship between the discount rate (that is constant) and the interest rate; it is

easy to understand that the peak of the EKC occurs at the level of per-capita income at which r =δ. The shape of the

EKC, first concave increasing and subsequently convex decreasing, is consistent with the findings of some other

scholars (John and Pecchenino, 1994; Jaeger, 1998; Stokey, 1998),3 as well as with some econometric analyses (Brock

2

I am grateful to one of the referees for this suggestion.3 The analytical condition to get a different shape of the EKC is to assume that there are not increasing returns to scale in pollution (i.e. β 1+β 2b1).




and Taylor, 2005; Galeotti et al., 2006; Panayotou, 2000). The exact shape of the relationship between the per-

capita income and harmful emissions in the real world, however, probably depends on the kind of pollutant accounted

for.

Eq. (3′) is also interesting because it allows us to clarify the relationship between pollution dynamics and the interest

rate. Calculating the appropriate partial derivative we get:

Ad

M

Ar ¼ Y g

h g c þ d

r 2a2ga2g r ð1 Þb21

h i þ ð1 Þb21b2n o N 0; ð28Þ

such that we may affirm unequivocally that M ˙ and r are positively related. In other words, when the country has a high

rate of interest as a result of limited availability of capital, the pollution stock increases. It is only when the marginal return

of capital decreases such that r bδ+θ g c, that the country will allot more labor effort to the green industry. Finally, we

can also demonstrate that there is an inverse relationship between M ˙ and the pure time preference, deriving [3′] with

respect to δ:

Ad

M

Ad ¼

Y g

r a2ga2g r ð1 Þb21

h iþ ð1 Þb21

b2n o

b 0: ð29Þ

From (29) it is evident that the dynamics of pollution are decreasing in the discount rate, as highlighted in a recent

article (Di Vita, in press).

Fig. 2. Income– pollution pattern.




4. Final remarks

Without an environmental policy, pollution dynamics show a fairly inverse U-shaped pattern. The up-and-down

behaviour e of emissions depends on the level of the rate of interest. When the latter declines to a value at which M ˙ is

zero, a further increase in income will imply a drop in pollution accumulation.

Using both versions of the model we may prove the existence of a direct relationship between the interest rate and pollution emissions dynamics. This is why it is only when people become rich that they are prepared to allot resources

to the green sector of the economy.

The abatement effort is not necessary to reduce emissions and to explain the falling branch of the income– pollution

pattern. The same occurs for the assumption of increasing returns to scale in the pollution abatement function. It is only

when the interest rate becomes lower than the discount rate that the country is willing to implement some environmentally

friendly devices. In cases where an environmental policy is launched, the question is whether or not this policy anticipates

the time at which the peak in the income– pollution pattern is reached; it is basically an empirical matter.

In this paper we assume that the discount rate is exogenously given, but we know that “impatience” in Fisher (1930)

means a decrease in income growth (Chavas, 2004; Das, 2003), such that our results could be biased. An interesting

topic for further research is to make the pure time preference index endogenous.

By means of the mechanism outlined in this paper we are able to explain why we find a positive relationship between income and pollution emissions in developing countries and a negative one in wealthy nations.

Obviously, more econometric analyses are necessary to support the results of this paper.

Acknowledgements

I am grateful to Reyer Gerlagh for encouraging me to write this paper, and to an anonymous referee, Roberto Cellini

and Efrem Castelnuovo for their useful suggestions and comments. An earlier version of this study was presented at

seminars held at the University of Catania, University of Padua, Third World Congress of Environmental and Resource

Economists, Kyoto (Japan) July 3–7, 2006, Eighth International Meeting of the Society for Social Choice and Welfare,

Instanbul (Turkey) July 13–17, 2006 and 62nd Congress of the International Institute of Public Finance, Paphos,

Cyprus, August 28–31, 2006. All errors are the Author's alone.

Appendix A

To check that our model exhibits a saddle point equilibrium, we have to calculate the characteristic roots of the

differential equation system represented by (18) and (19), to form the Jacobian matrix and evaluate it at the steady state

point ( K ⁎, C ⁎)

J ¼

P

Ad

K P

Ad

K

A K AC

P

Ad C P

Ad C

A K AC

264

375

ð K ⁎;C ⁎Þ

ðA1Þ

Calculating the four partial derivatives at K ⁎, C ⁎ we obtain:

Ad

K

A K ¼

ð K a1ð1þ2xgxÞa2ð1þ2xgxÞÞ 1

ðg1Þx

ðg 1Þx a1

1 þ 2xg x

K b 0; ðA2Þ

Ad

K

AC ¼ p

2ðg1Þx

ða2C hÞ 1

ðg1Þx

ðg 1Þx

h

C 1 N 0; ðA3Þ

Ad C

A K ¼ a1 K a12ða1 1Þa2 C

u

h b 0; ðA4Þ

Ad C

AC ¼ 1

h ðr u dÞS 0: ðA5Þ




Thus the qualitative Jacobian matrix takes the form

J ¼ þ

?

; ðA6Þ

The qualitative information we need about the characteristic roots ρ1

and ρ2

to confirm that we have an equilibrium

is conveyed by the result that

q1q2 ¼ J ð K ⁎;C ⁎Þ ¼ p2

ðg1Þxða2C hÞ

1ðg1Þx

ðg 1Þx

h

C

" # 1 K a12ða1 1Þa2

C

h

ðb2 1Þ

ð1 Þb2þ1r 1 þ b2

" #b 0:

ðA7Þ

This implies that the two roots have opposite signs, which establishes the steady state to be locally a saddle point.

References

Andreoni, James, Levinson, Arik, 2001. The simple analytics of the environmental Kuznets curve. Journal of Public Economics 80, 269–

286.Arrow, Kennet, Bolin, B., Costanza, R., Dasgupta, P., Folke, C., Holling, C.S., Jansson, B.O., Levin, S., Mäler, K.G., Perrings, C., Pimentel, D., 1995.

Economic growth, carrying capacity, and the environment. Science 268, 520–521.

Borghesi, Simone, 1999. The Environmental Kuznets Curve: A Survey of the Literature. Nota di Lavoro, vol. 85. Fondazione Eni Enrico Mattei,

Milan.

Brock, William A., Taylor, Scott M., 2005. Economic Growth and the Environment: A Review of Theory and Empirics. In: Durlauf, S., Aghion, P.

(Eds.), Elsevier.

Cass, David, 1965. Optimum growth in an aggregative model of capital accumulation. Review of Economics and Studies 32, 233 –240.

Chavas, Jean-Paul, 2004. On impatience, economic growth and the environmental Kuznets curve: a dynamic analysis of resource management.

Environmental and Resource Economics 28 (2), 123–152.

Das, Mausumi, 2003. Optimal growth with decreasing marginal impatience. Journal of Economic Dynamics and Control 27 (10), 1881 –1898.

Di Vita, Giuseppe, 2004. Another explanation of the pollution–income pattern. International Journal of Environment and Pollution 21 (6), 588–592.

Di Vita, Giuseppe, in press. Is the discount rate relevant in explaining the environmental Kuznets curve? Journal of Policy Modeling.

Fisher, Irving, 1930. The Theory of Interest. New York, Macmillan.

Galeotti, Marzio, Alessandro, Lanza, Pauli, Francesco, 2006. Reassessing the environmental Kuznets curve for CO2 emissions: a robustness exercise.Ecological Economics 57 (1), 152–163.

Grossman, GeneM., Krueger, AlanB., 1991.Environmentalimpacts of a NorthAmericanfreetradeagreement. NBERWorkingPaper No. 3914.November.

Gruver, G.W., 1976. Optimal investment in pollution control in a neoclassical growth context. Journal of Environmental Economics and Management

3 (2), 165–177.

Harbaugh, William T., Levinson, Arik, Wilson, David Molloy, 2002. Re-examining the empirical evidence for an environmental Kuznets curve. The

Review of Economics and Statistics 83 (3), 541–551.

Hoel, Michael, 1978. Resource extraction and recycling with environmental costs. Journal of Environmental Economics and Management 5,

220–235.

Huhtala, Anni, 1999. Optimizing production technology choices: conventional production vs. recycling. Resources and Energy Economics 21, 1–18.

Jaeger, William. 1998. “Growth and Environmental Resources: ATheoretical Basis for the U-shaped Environmental Path.” mimeo, Williams College.

John, A., Pecchenino, R., 1994. An overlapping generations model of growth and the environment. Economic Journal 104, 1393 –1410.

Jones, Larry E., Manuelli, Rodolfo E., 2000. Endogenous policy choice: the case of pollution and growth. Review of Economic Dynamics 4 (2),

369–405.

Keeler, Emmett, Spence, Michael, Zeckhauser, Richard, 1971. The optimal control of pollution. Journal of Economic Theory 4, 19–34.

Koopmans, Tjalling C., 1965. On the concept of the optimal economic growth. The Econometric Approach to Development Planning. North Holland,

Amsterdam.

López, Ramón, Mitra, Siddhartha, 2000. Corruption, pollution and the Kuznets environment curve. Journal of Environmental Economics and

Management 40 (2), 137–150.

Lusky, Rafael, 1976. A model of recycling and pollution control. Canadian Journal of Economics 9 (1), 91–101.

Magnani, Elisabetta, 2000. The environmental Kuznets curve, environmental protection policy and income distribution. Ecological Economics 32,

431–443.

Panayotou, Theodore, 2000. Economic growth and the environment. CID Working Papers with No. 56. Center for International Development at

Harvard University.

Pfaff, Alexander S.P., Chaudhuri, Shubham, Nye, Howard L.M., 2004. Household production and environmental Kuznets curves. Environmental and

Resource Economics 27 (2), 187–200.

Plourde, C., 1972. A model of waste accumulation and disposal. Canadian Journal of Economics 5, 119–125.

Ramsey, Frank, 1928. A mathematical theory of saving. Economic Journal 38, 543–

559.Romer, Paul M., 1996. Advanced Macroeconomics. McGraw-Hill, New York.




Rothman, D.S., 1998. The environmental Kuznets curves— real progress or passing the buck? A case for consumption based approaches. Ecological

Economics 25, 177–194.

Selden, Thomas M., Song, Daquing, 1995. Neoclassical growth, the J curve for abatement, and the inverted U curve for pollution. Journal of

Environmental Economics and Management 29, 162–168.

Smith, Vernon L., 1972. Dynamics of waste accumulation: disposal versus recycling. Quarterly Journal of Economics LXXXVI, 600–616.

Stokey, Nancy L., 1998. Are there limits to growth? International Economic Review 39 (1), 1–32.

Suri, Vivek, Chapman, Duane, 1998. Economic growth, trade and energy: implications for the environmental Kuznets curve. Ecological Economics25, 195–208.

Tahvonen, Olli, Salo, Seppo, 2001. Economic growth and transitions between renewable and nonrenewable energy resources. European Economic

Review 45 (8), 1379–1398.


Di Vta G., (2007) Capital accumulation, interest rate, and the income - pollution pattern - A simple...

Documents

Transcript of Di Vta G., (2007) Capital accumulation, interest rate, and the income - pollution pattern - A simple...