Multiplicative utility and the influence of environmental care on the short-term economic growth rate

Economic Modelling 16 Ž1999. 307]330

Multiplicative utility and the influence of environmental care on the short-term economic growth rate Nico Vellinga Wageningen Agricultural Uni¨ ersity, Weth. ¨ an Deutekomplein 106, 5706 TJ, Helmond, The Netherlands

Accepted 12 January 1999

Abstract This paper addresses the issue of determining under what circumstances economic growth rates are influenced by environmental care. The models used are extensions of the model by Lucas. The extensions consist of output leading to pollution and there is a stock of nature. There is also abatement to counter the effect of pollution. It turns out to be crucial not only to consider the specification of the utility function, but equally important is the specification of the temporary evolution of nature. A specification without a ‘carrying capacity’-term makes the growth rate dependent on environmental care in the long- and short-term. However, such a specification is unrealistic and may lead to an unstable model. If there is a ‘carrying-capacity’-term, the level of nature is constant in the long-term and there can be no influence of environmental care on the long-term growth rate. In the short-term the level of nature is changing and then, environmental care influences the growth rate. This influence can be positive or negative, depending Žamong other things. on the interaction between the level of nature and the marginal utility of consumption. It is then possible to have a ‘Win]Win’-situation because the environment improves and the growth rate of the economy increases at the same time. A specification with a stock of pollution instead of a stock of nature turns out not to be equivalent, as would be expected because these two entities can be transformed into one another. This is due to the fact that the interaction between the level of stock pollution and the marginal utility of consumption can only go one way and not two ways as with the stock of nature. Q 1999 Elsevier Science B.V. All rights reserved. Keywords: Economic growth rates; Environmental care; Pollution

0264-9993r99r$ - see front matter Q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 2 6 4 - 9 9 9 3 Ž 9 9 . 0 0 0 0 2 - 4

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1. Introduction The possible outcomes of an environmental growth model are largely determined by the model specification } but in what way does the model specification determine the model’s outcome? A related question is to determine under what circumstances the economic growth rate is influenced by care for the environment. Is it possible to have both an improving environment and a higher economic growth rate? In this paper we take a closer look at the Lucas Ž1988. growth model extended by nature and a flow of pollution. By nature is meant the aggregate amount of natural material, including the natural processes that take place within them. In this model the accumulation of knowledge is explicitly modelled and because knowledge is perhaps the most important factor contributing to economic growth this model specification has been chosen. We allow for different specifications of the utility function to analyse how the specification of the model determines the direction of the influence of environmental care on the economic growth rate. In this paper, natural resources are not direct inputs to production and they do not affect human capital accumulation. If this was the case, changes in the quality of the environment directly affect growth in the economy ŽMusu and Lines, 1993; Bovenberg and Smulder, 1995; Rosendahl, 1996; Smulders and Gradus, 1996.. Because natural resources are not direct inputs to production, the modelling of the utility function and the temporary evolution of nature are important. As shown by Michel and Rotillon Ž1995. the cross-effect of pollution on the marginal utility of consumption plays an important role. In their model, if an increase in environmental quality leads to a stronger desire for consumption, the growth rate of the economy can be higher if there is more environmental care. Michel and Rotillon consider a model in which there can be unlimited growth of pollution. They do that in order to be able to determine the impact of the specification of the multiplicative utility function1 on the way care for the environment influences the growth rate. The increasing level of pollution influences the marginal utility of consumption and ultimately it influences the growth rate. The reason why pollution can rise without limit is that there is no abatement in their model.2 Van der Ploeg et al. Ž1993. consider the possibility of unlimited growth of the quality of nature over time. They also consider a multiplicative utility function and note that it is possible to have a higher growth rate when there is more care for the environment. The approach of Michel and Rotillon and the approach of Van der Ploeg et al. are both half the story. In this paper a complete analysis will be carried out in which these two approaches both play a role. There is now abatement Žnot as a capital good but as 1

UŽ C1 , C2 ,..,Cn . s Ý nis1 f i Ž Ci . is a linear additive utility function. The utility function UŽ C1 ,C2 ,..,Cn . s n ai Ł is is a multiplicative utility function. 1Ci 2 They do consider the possibility of abatement. If abatement capital is ineffective, it does not pay to invest in abatement capital. If it is effective there can be unlimited growth of consumption and pollution, whatever the form of the utility function.

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a flow variable. and growth is accomplished through the engine of growth proposed by Lucas instead of the one of Rebelo which is used by Michel and Rotillon, and Van der Ploeg et al. Žalthough the latter also present models based upon Barro, 1990.. Instead of stock pollution, in this paper we look at the stock of natural capital. The reason for this will become clear later on. In this paper several functional forms are chosen for the equation describing the temporary evolution of nature. First, the one proposed by Van der Ploeg et al. and then one which is more realistic in that nature will be constant in the long-term, but in the short-term nature growth is either positive or negative. With this specification the level of nature is bound by some upper value, the so-called carrying capacity or saturation level of nature ŽClark, 1976. which results in a level of nature that does not go to infinity or equivalently, the level of pollution does not go to 0. In Section 2 the Lucas model extended by a flow pollution and a stock of nature is presented. The steady state is determined in Section 3. A long-term analysis is performed in Section 4.1. It is also shown that an ever changing level of nature does make the long-term growth rate dependent on care for the environment, but the steady state will be Žmost likely. unstable Žas shown in Appendix A.. A possible outcome with the engine of growth of Rebelo, instead of Lucas, and an ever changing level of nature is an unstable steady state, and only under special circumstances will it lead to a stable steady state as shown in Section 4.1. We therefore turn to the specification where nature will be constant in the long-term, but the level of nature changes in the short-term. As shown in Appendix C the corresponding steady state is locally stable. In the short-term analysis of Section 4.2 it is shown that care for the environment can influence the short-term growth rate. Under certain circumstances, care for the environment leads to higher levels of economic growth, contrary to what can generally be expected if part of the resources are used for cleaning the environment instead of investing in productive capital. Section 5 concludes this paper with a summary of the results obtained.

2. The Lucas model with nature The model formulation is based upon Gradus and Smulders Ž1993., although they only consider pollution as a flow and they use a linear additive utility function. The Lucas growth model ŽLucas, 1988. is extended by a flow pollution, a stock of nature and there is also abatement: max c t ,¨ t ,u t

`

yr t

H0 L U Ž c , N . e t

t

t

s.t. Yt s c t L t q K˙t q A t Yt s K ta Ž u t Ht L t .

dt

K 0 is given

Ž1.

1y a

H˙t s j Ž 1 y u t . Ht

H0 is given

Ž2.

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310

N˙t s V Ž Nt , P Ž ¨ t .. P Ž¨t . s

N0 is given

Ž3.

¨ yg t

˙ t s p Lt L

L0

is given

Ž4.

A t s ¨ t Yt where 0 - a - 1, 0 - ¨ t - 1, 0 - u t - 1, j ) 0, r ) 0, p ) 0 and g ) 0. The Lucas growth model is chosen because it considers explicitly the creation of knowledge, perhaps the most important ingredient for economic growth. Instantaneous utility UŽ c t , Nt . is discounted at the constant rate of time preference r. Utility of a representative consumer depends positively on the amount of per capita consumption c t Žs CtrL t . and the level of nature Nt Žsee also Michel and Rotillon, 1995.. A consumer enjoys consumption, a rival good, but is also influenced by the level of nature, a public ‘good’, which influences all consumers equally. Total product Yt is used for consumption Ct Žs c t L t ., investment K˙t , and abatement A t . From the last equation it follows that the ratio of abatement over total product is nt Žs A trYt .. Consumers decide upon the proportion of non-leisure time Žstandardized to one. u t that they want to spent producing. The remaining proportion 1 y u t is spent studying and by studying consumers increase their level of human capital Ht . The value of j denotes the effectiveness of human capital accumulation. Nature growth is represented by a function V Ž.. which depends on nature and flow pollution P Ž... Flow pollution is a decreasing function of the ratio of abatement over total product. As shown in the last differential equation, L t grows exogenously at rate p.

3. Steady state solution of the model Determining the steady state reveals interesting intermediate results that can be used later on. Therefore, this analysis is considered in this section in more detail. We set up the Hamiltonian and apply the Maximum Principle to derive the first order conditions: 3 H s U Ž c t , Nt . q QtK K ta Ž u t Ht L t .

1y a

Ž 1 y nt . y c t L t

q QtH Ht j Ž 1 y u t . q QtN V Ž Nt , P Ž nt .. 3 The first order conditions are for an interior solution where the instruments are all larger than zero. In the sequel of this paper, whenever the first order conditions are derived an interior solution is assumed to exist or an interior solution exists because of the specification of the utility function. For instance, the level of consumption will be larger than zero with a log]utility function. With a zero level of consumption the marginal utility level is infinite and this guarantees that consumers will choose a positive consumption level instead of a zero consumption level.

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The first order conditions are: H ­H ­c t H ­H ­nt H ­H ­u t H ­H ­K t H ­H ­Ht

s Uc y QtK L t s 0

Ž5.

s yQtK K ta Ž u t Ht L t .

1y a

q QtN VP Pn t s 0

s QtK K ta Ž 1 y a .Ž u t Ht L t . s QtK a K tay1 Ž u t Ht L t .

1y a

ya

Ht L t Ž 1 y nt . y QtH Ht j s 0

Ž6.

˙ tK Ž 1 y nt . s Ž r y p . QtK y Q

s QtK K ta Ž 1 y a .Ž u t Ht L t .

ya

u t L t Ž 1 y nt . q QtH j Ž 1 y u t .

˙ tH s Ž r y p . QtH y Q ­H ­Nt

˙ tN s UN q QtN VN s Ž r y p . QtN y Q

We also have the four constraints ŽEqs. Ž1. ] Ž4... From the first three equations of the FOCs, it is possible to derive three relationships between the growth rates of the variables present in those equations: 4 gQ K q p s s g c q g

Uc N Nt Uc

gN

Ž7.

g Q K q a g K q Ž 1 y a .Ž g H q p . s g Q N q g V P q g P n

Ž8.

gQ K q a g K y a Ž g H q p . q p s gQ H

Ž9.

From the remaining equations we can derive the following growth rates:

gQ K s r y p y

a u1ya t

Ht L t

1y a

ž / Kt

Ž 1 y nt .

Ž 10.

gQ H s r y p y j g Q N s r y p y VN y

4

Ž 11. UN Nt c t L t VP Uc c t

K t Nt

Pn u ay1 t

Ht L t

ž / Kt

ay 1

Ž 12.

The expression Uc c c trUc is denoted by s. The growth rate of a variable in continuous time is defined as g X s X˙trX t s Ž dX trd t .rX t .

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312

g K s u1ya t

1y a

Ht L t

ž / Kt

Ž 1 y nt . y

ct Lt

Ž 13 .

Kt

g H s j Ž1 y u t . gN s

Ž 14 .

V

Ž 15.

Nt

And from these equations we can first derive the short-term growth rate of consumption. Substituting the expression for g Q K in Eq. Ž7. and solving for g c , we get: gc s

Ž Ht L trK t . r y p y a u1ya t

1y a

Ž 1 y nt . q Ž 1 q s . p

s

q

Uc N Nt Ž yUcc . c t

gN

Ž 16.

Here, g N is the growth rate of nature Žeither 0 or a function of nt given by V Ž Nt , P Ž nt ..rNt .. The ratios c t L trK t and Ht L trK t , and the variables u t and nt are constant in the long-term because of balanced growth. Therefore, the following equations hold in the long-term: gK s gH q p

Ž 17.

and g K s gC s g c q p

Ž 18.

The value of nt is constant in the long-term, so we must have that g P n is 0. Therefore, the expression g V P will be equal to Ž VP N NtrVP . g N . We end up with 11 equations ŽEqs. Ž7. ] Ž18.. and 11 unknown entities Ž u t , Ht L trK t , c t L trK t , nt , g K , g H , g N , g c , g Q K , g Q H , g Q N . Reducing these 11 equations to four equations with four unknowns Ž u t , Ht L trK t , c t L trK t and nt ., we get: 5 ut s

1 j

VN q

ut s 1 y ct Lt Kt Ht L t Kt

s s

UN Nt VP Uc c t Nt

rypyj js

q

Pn

j Ucc c t

a

ž

a

Ž1r1ya .

/

jqp

Kt

1 Uc N Nt

Ž 1 y a .Ž j q p . jqp

c t L t a Ž 1 y nt .

y

VP N Nt VP

gN

gN

Ž 19 . Ž 20 .

q j ut 1

ž / ut

Ž 1 y nt . y1r1ya Ž

.

5 With the specifications of the functions in the sequel the remaining ratios turn out to be parameter values of the functional forms chosen.

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These equations together determine the steady state of the model. The long-term growth rate Žof capital. can be determined by substituting Eq. Ž14. and Eq. Ž20. in Eq. Ž17.. We arrive at: gK s p q

rypyj s

y

Uc N Nt Ucc c t

gN

Ž 21.

Because of the definition of balanced growth, we have a constant growth rate g N , either 0 or, as before, a function of nt . 4. Environmental care and the growth rate Two types of analysis are considered } a long-term analysis and a short-term analysis. We start with the long-term analysis and use several specifications for nature growth. The first is one where nature is allowed to grow forever, but this is likely to lead to an unstable steady state. An alternative is one where there is a natural upper limit to nature growth. Also, a number of different specifications are chosen for the utility function. A linear additive and a multiplicative utility function are analysed further. Finally, a short-term analysis is carried out to show that it is possible to have a Žshort-term. growth rate which depends on the strength of environmental care. This influence can be both negative and positive. 4.1. Long-term analysis The long-term growth rate of the economy has already been derived Žsee Eq. Ž21... With a linear additive utility function ŽUc N s 0. the long-term growth rate of the economy is equal to p q Žr y p y j .rs. The marginal utility of consumption does not depend on the level of nature, or equivalently the appreciation of more consumption is not influenced by the level of nature. For instance, the following linear additive utility function has a cross-effect which is 0: U Ž c t , Nt . s log Ž c t . q u Ž Nt . where s s Ucc c trUc s y1. Growth in the economy equals j y r and is clearly independent of any parameter with respect to nature in the utility function, so g K does not depend on these parameters. Therefore, in the long-term, these parameters cannot have an influence on growth in the economy. From the equation for g K we conclude that for the growth rate to depend on these parameters it must be true that Uc N / 0 and g N s V Ž.. Nt / 0. This last expression means that nature indeed changes Ž g N / 0.. Because of Uc N / 0, we know that the marginal utility of consumption then changes also. These are necessary conditions for the growth rate to depend on the strength of environmental care. What is the reason for the growth rate being influenced only by environmental care if the level of nature changes and the marginal utility of consumption depends

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314

on the level of nature? For this we have to look more closely at the first order conditions derived earlier. The equation Hu t s 0 is the one we have to look at. To be precise, Eq. Ž6.. This equation states that optimally we equate the costs and benefits of transferring one unit of u t from human capital accumulation to production: QtK K ta Ž 1 y a .Ž u t Ht L t .

ya

Ht L t Ž 1 y nt . s QtH Ht j

The value on the right hand side is the cost of having one unit less of u t Žor Ht j . valued at its shadow price QtH. The benefits of this extra unit of u t in production result in more consumption and this is shown by the value on the left. This value is measured in terms of the shadow price of capital Ž QtK . which is equal to the marginal utility of consumption divided by the labour force ŽUcrL t , see Eq. Ž5... If Uc depends on the level of nature, the value of the optimal level of u t also depends on the level of nature. The growth rate of the economy depends on u t , and depends therefore also on the level of nature. For Uc to change because of Nt it must be due to Nt changing and therefore g N s N˙trNt s V Ž..rNt is non-zero. To have a growth rate which is influenced by the strength of environmental care we could have taken the following specification of V Ž..: N˙t s V Ž Nt , P Ž nt .. s c

Nt P Ž nt .

Ž 22 .

with c ) 0. The level of flow pollution determines the speed at which the level of nature improves over time. Van der Ploeg et al. Ž1993. use a similar equation in their Rebelo-based model. Abatement lowers the level of flow pollution and this in turn has a positive influence on the rate at which nature improves. In this equation there is no ‘carrying capacity’-term ŽClark, 1976.. Unfortunately, this formulation Žprobably. does not lead to a locally stable steady state. It is very difficult, if not impossible, to find a set of parameter values as shown in Appendix A, which result in a steady state which is locally stable. 4.2. Model by Van der Ploeg et al. For the Rebelo model used by Van der Ploeg et al. Ž1993., the long-term level of nature will rise to infinity if the steady state value of abatement over total product Ž nt . is between 0 and 1 } but there is not a steady state with a constant positive ratio of consumption over capital. The only possible steady states are with a ratio of consumption over capital which is 0 and the steady state value of n is between 0 and 1 Žincluding the two boundaries.. To show that this is the case let us look at the model with a linear technology: max c t ,¨ t

`

rt

H0 e y

FCtw c NtW N dt

N. Vellinga r Economic Modelling 16 (1999) 307]330

s.t K˙t s a K t Ž 1 y nt . y Ct N˙t s Nt cntg

315

K 0 is given N0 is given

For this model we can set up the Hamiltonian and derive the first order conditions and from these derive a system of differential equation for the ratio CtrK t Ždenoted by X 1 t . and nt Žboth of which will be constant in the long-term; for details, see Appendix B.: X˙1 t s X 1 t

˙nt s nt

ž

r y wN cntg wc y 1

1 1yg

X1 t 1 y

ž

y a Ž 1 y nt . wN cg wc a

ntgy1

wc wc y 1

q X1 t

/

/

Ž 23 . Ž 24.

The steady state values for X 1 t and nt are denoted by X 1 and n and they are: X1 s

1 1 y wc

Ž r y a Ž 1 y n . wc y wN cn g .

and ns

ž

cg wN a wc

1r1yg

/

For an economically viable solution these steady state values have to be positive and this leads to a number of restrictions on the allowed set of parameter values. The Jacobian matrix of the system of Eqs. Ž23. and Ž24. is also derived in Appendix B: Js

X1

0

0

X1

As can be readily seen, the eigenvalues of this Jacobian matrix are both equal to X 1 Ž) 0.. They are both positive and the steady state with a positive X 1 t is locally unstable. To show what is going on let us draw a phase-diagram for one particular case. The following parameter values are used: r s 0.12, a s 1.1, g s 0.5, c s 0.6, wC s 0.05 and wN s 0.15. The equation for ˙ nt s 0 is nt s n, so this is a horizontal line in the n]X 1-plane. The line X˙1 t s 0 is given by X 1 t s Žyr q wN cntg .rŽ wc y 1. q a Ž1 y nt . wcrŽ wc y 1.. These curves are displayed in Fig. 1: From the arrows we see that the steady state given by X 1 and n is indeed unstable. The best that can happen is to start somewhere in areas I and III and end up on the X 1 t s 0-axis and have a value of nt between 0 and 1. The purpose of the economy is merely to supply resources for nature to grow. Because X 1 t tends to 0 in the long-term, capital will grow faster than consumption. Capital is being

316

N. Vellinga r Economic Modelling 16 (1999) 307]330

Fig. 1. Phase diagram of the model by Van der Ploeg et al.

accumulated, not to provide future consumption, but to let nature grow. We can conclude that consumption and nature are very good substitutes Žsee also the discussion at the end of Appendix A.. It is possible for a certain starting point to end up with a n of 1. So this time there is no investment in capital, all resources are used to have a growing level of nature. With a X 1 t of 0 in the steady state, the level of capital will become 0, due to consumption. For the X 1 t s CtrK t to go to 0, the level of consumption decreases faster than the level of capital. Again, we have that nature is a good substitute for consumption. It is also possible to have a steady state with nt s X 1 t s 0 in the long-term, but this is not an economically viable solution. If nt is 0 in the long-term, the level of nature will be constant in the long-term. With a X 1 t of 0 we have either a lower growth rate of consumption compared to capital or capital will become constant and consumption will be 0; but both these cases are not economically viable either because the stock of capital will become larger and larger compared to consumption, or because there is a positive amount of capital, but no consumption. Both cases are not desirable. All other remaining cases are either not realistic Žcapital growth is higher or lower than consumption growth. or not attainable Žthe nt will be larger than 1 in finite time.. The overall conclusion is that it is possible for the economy to end up in an economic viable solution only for certain initial starting points. The purpose of the economy is then to supply the resources for nature to grow. Nature is apparently a good substitute for consumption. For all other starting points there is no economic viable solution. The Rebelo model together with Eq. Ž22. leads to an unstable steady state in

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317

some cases. It seems that the Lucas model together with Eq. Ž22. has the same problem. It is only much more difficult to show, as has been done for the Rebelo model, that this actually happens. 4.3. The model used in the sequel For economic growth to be influenced by care for the environment, the conditions mentioned earlier are that the marginal utility of consumption depends on the changing level of nature. The following utility function which is multiplicative and concave in Ž c t , Nt . makes sure that the second condition is satisfied: U Ž c t , Nt . s Fc tw c Nt w N

Ž 25 .

Two cases can be distinguished. The first with F ) 0, 0 - Ž wc , wN . - 1 and wc q wN - 1, leads to a positive cross-effect of nature on the marginal utility of per capita consumption Uc N ) 0. If F - 0 and wc and wN are both smaller than 0, we have the second case with a negative cross-effect ŽUc N - 0.. The s Žs Ucc c trUc . for this utility function is wc y 1. A changing level of nature occurs Žin the short-term. when the following function V Ž.. is used: V Ž Nt , P Ž nt .. s Nt Ž m y P Ž nt . Nt .

Ž 26 .

where m ) 0. Nature evolves according to a logistic growth curve ŽClark, 1976.. Low initial levels of nature result in a positive increase in nature, while high initial levels of nature result in a decrease in nature. In the long-term nature will be stationary at some particular value which is also influenced by the level of flow pollution, to be precise mrP Ž nt .. This is termed the carrying capacity or saturation level of nature ŽClark, 1976.. The flow of pollution is taken to be nyg and t therefore depends negatively on the level of nt . A higher value of nt means that more resources are used for cleaning up Žabatement . and the flow of pollution decreases. For the flow of pollution function to be convex, g has to be positive. The specifications given by Eqs. Ž25. and Ž26. for the functions UŽ.. and V Ž.. will be used in the sequel. Eq. Ž26. can be transformed in terms of stock and flow pollution, if one defines Žthe quality of. nature as the reciprocal of the stock of pollution S t , so Nt s 1rSt . We get: S˙t s P Ž nt . y mSt s nyg y mSt t

Ž g ) 0; m ) 0 .

Now the change in the stock of pollution is equal to the flow of pollution minus natural cleaning, which is assumed to be proportional to the stock of pollution ŽVan der Ploeg and Withagen, 1991.. With this specification, the value of the stock of pollution will be constant in the long-term. This can be seen if one looks at the differential equation for S t and notices that if nt is constant, the value of S t will approach a constant value in the long-term. Therefore, the level of nature will also be finite and constant. The reason for this result is that there is abatement in this model.

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With nature instead of stock pollution it is possible to specify concave utility functions that have either a positive or a negative cross-effect of nature on the marginal utility of consumption. This is not possible for the model under consideration if we cast it in terms of stock pollution. A specification which allows for balance growth is UŽ c t , S t . s F c tw c S tw S , where S t is now the stock of pollution. It is possible to have a positive cross-effect, but a specification with a negative cross-effect is impossible. For the latter case we must have at the same time Uc ) 0, Ucc - 0, US - 0, USS - 0 and UcS - 0. This is impossible with the specification of the utility function just given. There is no combination of F, wc and wS that can satisfy these conditions simultaneously. In Appendix C it is shown that the steady state is a locally stable steady state with the specifications of UŽ.. and V Ž.. as in Eqs. Ž25. and Ž26.. 4.4. Short-term analysis In the short-term the following equation which gives us the general short-term growth rate of the economy is valid Žsee Eq. Ž16..: gc s

Ž Ht L trK t . r y p y a u1ya t s

1y a

Ž 1 y nt . q Ž 1 q s . p

q

Uc N Nt Ž yUcc . c t

gN

If the initial level of nature is lower than its optimal value, nature would increase monotonically over time and would end up at its optimal level. During the transition period the level of nature is changing and if care for the environment changes, this will have an effect on the growth rate of the economy. Whether this effect is positive or negative depends Žamong other things. on the sign of the cross-effect of nature on the marginal utility of nature ŽUc N ..6 The relationship between care for the environment and the resulting growth rate is determined by the sign of the term preceding the growth rate of nature, Uc N NtrŽŽyUcc . c t .. Because UŽ.. is a concave utility function, Ucc is negative. Therefore, the sign of the term is determined by Uc N or equivalently the sign of the cross-effect of nature on the marginal utility of consumption. If the cross-effect is negative, an increase in nature lowers the growth rate of the economy and the opposite conclusion applies if the cross-effect is positive. The overall effect is also influenced by the change in u t , Ht L trK t and nt . Neglecting the influence of u t , Ht L trK t and nt , if the cross-effect is negative an increasing level of nature exerts a negative effect on the growth rate of the economy. For the model under consideration we can set up its discrete time equivalent and trace out the time paths of the variables and determine the growth rates of capital and per capita consumption to see that this actually happens. For the numerical calculations, we set up a discrete time formulation of the 6

With a linear additive utility function there can also be growth effects in the short-term, but then only due to changes in u t , Ht L trK t and nt .

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319

continuous time model. The specifications of UŽ.. and V Ž.. Žas in Eqs. Ž25. and Ž26.. are used. The time horizon has to be finite and we assume that, at the last time period considered, the economy is in a steady state. This is only true approximately, but if a long enough time horizon is taken the error made will be negligible. By assuming that the steady state is reached in the last time period considered, utility in the last time period can be adapted such that it represents the infinite sum of utility levels from that time period onward. A number of extra constraints are added to the model to guarantee that the steady state is indeed reached at the last time period. The time paths of the variables of the discrete time model are determined using GAMS ŽBrooke et al., 1988.. Two separate cases are considered, the first is the case with a utility function with a negative cross-effect between the marginal utility of consumption and nature and the second case is one in which this cross-effect is positive. For both cases considered, the initial levels of capital and labour are arbitrarily chosen to be 3 and 1, respectively. The initial levels of the other two stock variables Žhuman capital and the level of nature . are arbitrarily chosen to be 5% below their starting levels if there was balanced growth from t s 0 onward. Inspection of the numerical results Žpresented later. show that with a time horizon of 50 years the error made Žassuming that the model is in a steady state in the 50th period. is negligible for both models. The discrete time formulation of the model is: `


Ý ts0

ž

1qp 1qr

t

/

Fc tw c Nt w N

s.t. K tq1 y K t s K ta Ž u t Ht L t .

1y a

Ž 1 y nt . y c t L t

K 0 is given

Htq1 y Ht s j Ht Ž 1 y u t .

H0 is given

Ntq1 y Nt s Nt Ž m y P Ž nt . Nt .

N0 is given

For this Žinfinite horizon. model, the steady state values follow from setting up the discrete time Hamiltonian and applying the Maximum Principle ŽFeichtinger and Hartl, 1986.. The steady state value for u t Žs u. is: ut s 1 q

1 j

y

1 j

ž

1rs

1qr Ž 1 q j .Ž 1 q p .

/

The long-term growth rate g is then: g s Ž1 q p .

Ž sy1rs .

ž

1qr 1qj

1rs

/

y1

The steady state values of the ratios Ht L trK t , c t L trK t and nt Ždenoted by X 1 , X 2 , and n, respectively. follow from solving the following three equations:

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320

1 q a u1ya Ž X 1 . 1 q u1ya Ž X 1 . 1 1qj

ž

1y a

1y a

Ž 1 y n . s Ž 1 q p .Ž 1 q j .

Ž 1 y n . y X 2 s Ž 1 q p . sy1rs

1qr Ž 1 q j .Ž 1 q p .

1rs

/

s

ž

1qr 1qj

1rs

/

1 1 y m q Ž wN rwc . mg X 2ny1 u ay1 Ž X 1 .

ay 1

The time paths of the variables of the discrete time model can now be determined. We start with the negative cross-effect. 4.5. Negati¨ e cross-effect For the negative cross-effect ŽUc N - 0. the following parameter values are used: r s 0.040, p s 0.025, a s 0.200, j s 0.020, g s 0.500, m s 0.050, F s y1.000, wc s y0.400, wN Žlow. s y0.400 and wN Žhigh. s y0.450. If there is more environmental care Ž wN s wN Žhigh.., the resulting levels of physical and human capital are lower, as are the levels of consumption and investment. There are now less resources available for consumption and investment. In Fig. 2 we see that pollution is lower and the level of nature is higher. The value of nt is higher because more is spent on abatement, and more time is spent producing Žhigher value of u t . resulting in a lower growth rate of the economy. Because of the lower consumption desire, there is not much need for high levels of capital to provide the economy with the opportunity to consume at a high rate in the future. As a result of this, people do not want to invest more time in studying because this would lead to a high level of capital in the future. Finally, to ascertain the effect of care for the environment on the growth rate, Fig. 3 shows the growth rates of consumption and capital. Clearly the growth rates are lower if there is more care for the environment. If nature improves, this leads to lower levels of marginal utility of consumption. Consumption and nature are substitutes. 4.6. Positi¨ e cross-effect The parameter values for the case with a positive cross-effect ŽUc N ) 0. are: r s 0.040, p s 0.025, a s 0.200, j s 0.020, g s 0.500, m s 0.050, F s 1.000, wc s 0.400, wN Žlow. s 0.400 and wN Žhigh. s 0.450. If there is more environmental care, more resources are used for cleaning up and the levels of consumption and capital are lower; but due to the improving level of nature, the desire for consumption increases and the growth rate of consumption Žand capital. can increase. From Fig. 4 we conclude that there is more abatement Ž nt is higher., resulting in a lower flow of pollution and a higher stock of nature. The value of u t decreases, so less time is spent producing and more time is spent studying, resulting in a higher growth rate of the economy.

N. Vellinga r Economic Modelling 16 (1999) 307]330

321

Fig. 2. The time paths of flow pollution Pt , nature Nt , proportion of available time spent producing u t , and nt Žnu., are depicted for the case with a negative cross-effect ŽUc N - 0.. The solid line corresponds to a low Žnegative. value of wN . The dashed line is with a higher Žmore negative. value of wN corresponding to the case with more care for the environment.

Because of the higher consumption desire, it pays to devote a larger proportion of available time to studying because that leads to a higher stock of capital and higher consumption levels in the future. If the cross-effect of nature on the

Fig. 3. The growth rates of per capita consumption g Ž c . and capital g Ž K . over time are depicted for the case with a negative cross-effect ŽUc N - 0.. The solid line corresponds to a low Žnegative. value of wN . The dashed line is with a higher Žmore negative. value of wN corresponding to the case with more care for the environment.

322

N. Vellinga r Economic Modelling 16 (1999) 307]330

Fig. 4. The time paths of flow pollution Pt , nature Nt , proportion of available time spent producing u t , and nt Žnu., are depicted for the case with a positive cross-effect ŽUc N ) 0.. The solid line corresponds to a low Žpositive. value of wN . The dashed line is with a higher Žmore positive. value of wN corresponding to the case with more care for the environment.

marginal utility of consumption is positive, more care for the environment exerts a positive effect on the growth rate of the economy. We would expect that an improving environment could lead to higher economic growth, which is correct if one looks at Fig. 5:

Fig. 5. The growth rates of per capita consumption g Ž c . and capital g Ž K . over time are depicted for the case with a positive cross-effect ŽUc N ) 0.. The solid line corresponds to a low Žpositive. value of wN . The dashed line is with a higher Žmore positive. value of wN corresponding to the case with more care for the environment.

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323

The growth rates of consumption and capital increase for this parameter set. Consumption and capital start at a lower level but they increase at a faster rate. The growth rate increases despite more use of resources to improve the environment. If nature improves this leads to higher levels of marginal utility of consumption. Consumption and nature are complements. In a cleaner environment people enjoy consumption even more and it pays to invest in capital to have higher future levels of consumption. This situation is termed a ‘Win]Win’ situation ŽVan der Ploeg et al., 1993. because we have an improving environment, the growth rate of nature increases, and we have at the same time more economic growth.

5. Conclusions The model studied in this paper has the following features. It is based upon the Lucas model with three types of capital: physical, human and natural capital. There is abatement to counter the effect of flow pollution, and growth of the stock of nature is limited in the long-term by the carrying capacity of nature. This is different from Michel and Rotillon Ž1995. and Van der Ploeg et al. Ž1993.. They both consider the Rebelo model and in Michel and Rotillon’s model there is no abatement,7 whilst in the model of Van der Ploeg et al. there is unlimited growth of nature. The main conclusion based upon the analysis carried out in this paper is that: For realistic results we ha¨ e to include a ‘carrying capacity’-term in the equation describing the temporary e¨ olution of nature. Care for the en¨ ironment then only has a short-term influence on economic growth. Another main conclusion is that: The conditions under which en¨ ironmental care has an influence on the long-term economic growth rate are that the le¨ el of nature should not be constant and should influence the marginal utility of consumption. Only then is the growth rate influenced by the strength of environmental care. For this, both the functional form of the utility function, and the specification of the function describing the temporary evolution of nature are crucial. It turns out that: v

v

7

If we have a linear additi¨ e utility function environmental care cannot have any influence on Žlong-term. economic growth. If we include a so called ‘carrying capacity’-term in the equation describing the temporary evolution of nature, we end up with a constant level of nature in the long-term and again Žlong-term. economic growth is independent of environmental care.

See footnotes 1 and 2.

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N. Vellinga r Economic Modelling 16 (1999) 307]330

In the short-term nature can change and then it is possible that growth is influenced by care for the environment. Only if we use a relatively unrealistic assumption can long-term growth be influenced by environmental care. If the level of nature improves continuously over time, and the flow of pollution determines the speed at which this happens, growth is influenced by concern for the environment, but then there is no real environmental problem. The environment will get better in the long-term, only the speed at which this happens is determined by consumers. It was shown numerically that: If the cross-effect of pollution on the marginal utility of consumption is negati¨ e, in the short-term (when the long-term steady state outcome has not yet been reached), economic growth can decrease if concern for the en¨ ironment increases. If the cross-effect of pollution on the marginal utility of consumption is positi¨ e, in the short-term, growth can increase if concern for the en¨ ironment increases. A more general conclusion is not possible, because this result also depends on the outcome for nt , u t and the ratio Ht L trK t . If nature improves over time and we have a multiplicative utility function, the value of the marginal utility of consumption will change. It either increases or decreases depending on the sign of the cross-effect of nature on the marginal utility of consumption. A positive crosseffect can be interpreted as follows: if there is more nature, the appreciation of more consumption increases, so only with a high level of nature will consumers care about additional consumption. If the level of nature is low, more consumption is less appreciated } or equivalently, the more Žstock. pollution there is, the less one cares about extra consumption. A negative cross-effect means that if the level of nature is high Ža low level of stock pollution., additional consumption is less appreciated } or, if the level of nature is low Ža high level of stock pollution., more consumption is highly appreciated. It is a matter of personal opinion on how one reacts to more consumption depending on the level of nature Žstock pollution.. I personally would not be very happy with a clean loaf of bread if the environment is heavily polluted Žlow level of nature . because I do not think it matters much, even though I have one extra clean loaf of bread. So my cross-effect is positive, but it is imaginable that other people would be very glad with this extra unit of bread, so their cross-effect is negative. If growth in the economy increases and there is at the same time an increase in the level of nature we have a ‘Win]Win’-situation, because we gain with respect to both economic growth and the environment. If one is setting up an environmental growth model, this paper can be of use if one tries to avoid setting up a model with a predictable outcome. Furthermore, a model with stock pollution and a model with a stock of natural capital, seem to be comparable. It turns out that it is not a trivial choice to decide which specification is chosen, because it matters with respect to the specification of the utility function. A negative and positive cross-effect is only possible with a stock of nature, not with a stock of pollution, so one should carefully choose between

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325

them and be aware of the consequences. In the long-term the stock of natural capital or stock of pollution is constant } at what level is decided by the preference of the consumers, and in both cases there is no influence of the preference for a clean environment on the long-term growth rate. The reason for this is that, with a constant level of stock pollution or stock of nature, the utility function becomes a utility function dependent only on consumption. The parameter with respect to stock pollution or natural capital does not play any role, and it does not matter whether the utility function is linear additive or multiplicative. The constant stock of pollution or constant stock of natural capital only introduces a constant factor in the utility function, so the long-term growth rate is independent of the preference for a clean environment. With the models based upon Rebelo this was quite different, there we indeed have an influence of care for the environment on the long-term growth rate. By setting aside resources for abatement, growth will be lower because there are less resources for investment in the only form of capital.

Acknowledgements The author would like to thank H.J.C. Huijberts from Eindhoven University of Technology for helpful comments on the stability analysis carried out in this paper.

Appendix A: Instability with ever growing level of nature In this appendix we determine whether or not the steady state for the model of Section 4.1, with the equation describing the temporary evolution of nature as in Eq. Ž22., is locally stable or not ŽGeorge, 1988.. The eigenvalues are such that, whenever a set of reasonable parameter values is given, all four have positive real parts, and none of them has a negative real part. Changing the values of the parameters does not lead to other signs of the eigenvalues, and the conclusion therefore is that the steady state of the model is likely to be locally unstable. The following variables and ratios approach constant values in the long-term: c t L trK t , Ht L trK t , u t and nt . On the basis of the first order conditions, we derive difference equations for c t , nt and u t . We already have the difference equations for K t and Ht , and we can derive difference equations for the ratios Ht L trK t and c t L trK t . Let us denote Ht L trK t } Ž HLrK . by x 1 t , c t L trK t } ŽcLrK . by x 2 t , nt y n by x 3 t , and u t y u by x 4 t . Then the difference equations for these variables x ti Ž i s 1,...4. are determined. This system of equations is linearized around the steady state x 1 t s x 2 t s x 3 t s x 4 t s 0, to arrive at ˙ x t s Jx t , where x t is equal to Ž x 1 t , x 2 t , x 3 t , x 4 t .T , ˙ x t is Ž ˙ x1 t , ˙ x2 t , ˙ x3t , ˙ x 4 t .T , and the matrix J, the Jacobian matrix, is ­Ž ˙ x1 t , ˙ x2 t , ˙ x3t , ˙ x 4 t .r­Ž x 1 t , x 2 t , x 3 t , x 4 t . Ž t s 0.. The Jacobian matrix is determined with Maple ŽHeck, 1993..

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326

With p s 0.015, r s 0.025, a s 0.1, j s 0.022, g s 0.5, c s 0.005, wc s 0.2 and wN s 0.4, we get the following Jacobian matrix: 0.339 1.979 Js y0.0853 0.0620

y0.0498 y0.333 y0.000221 0.000819

0.161 1.068 y0.000773 0.00287

y0.388 y2.634 y0.00172 0.0120

The eigenvalues of this matrix are: 0.00560, 0.00560, 0.00280 q j0.161 and 0.00280 y j0.161. Here j s 'y 1 . None of the eigenvalues has a negative real part } therefore, for this set of parameters, the steady state of the model is locally unstable. Consumption and nature are both goods which are appreciated. A high level of consumption can only be achieved if production is high, but this also leads to a high level of flow pollution. A high level of nature does not have this disadvantage. A high growth rate of nature can be realized with a high ratio of abatement over total product, or equivalently, with a high level of abatement and a low level of total product, leading to low levels of flow pollution } so consumers aim for a high value of nt . Consumption can be low because nature is a good substitute for consumption. Investment in capital, and the level of capital itself, can be low. The time spent producing will also be low to make sure that production will be low. Therefore, it is impossible to reach the steady state with a value of u t close to its long-term steady state value, because u t will be lower than this value. Also, nt will be higher than its long-term value. The conclusion therefore is that this economy will not start off at a point on the stable branch, and will not move towards its long-term steady state solution.

Appendix B: Calculations for the model of Van der Ploeg et al. For the Rebelo-based model of Van der Ploeg et al. Ž1993., in Section 4.1, we set up the current value Hamiltonian: H s FCtw c Nt w N q QtK w a K t Ž 1 y nt . y Ct x q QtN w Nt cntg x From this, we determine the first order conditions: H ­H ­Ct H ­H ­nt H ­H ­K t

s Fwc Ctw cy1 Nt w N y QtK s 0 s yQtK a K t q QtN Nt cgntgy1 s 0

˙ tK s QtK a Ž 1 y nt . s rQtK y Q

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H ­H ­Nt

327

˙ tN s FwN Ctw c Nt w Ny 1 q QtN cntg s rQtN y Q

From the differential equations of K t and Nt , and these first order conditions, we can derive the growth rates of Ct , K t , Nt , QtK , QtN and nt Žfor convenience, denote the ratio CtrK t by X 1 t .: gC s

1 wc y 1

Ž r y a Ž 1 y nt . y wN cntg .

Ž B.1.

g K s a Ž 1 y nt . y X 1 t g N s cntg

Ž B.2.

g Q K s r y a Ž 1 y nt . g Q N s r y cntg y 1

gn s

1yg

wN cg wc a

X1 t 1 y

ž

ntgy1 X 1 t

wN cg wc a

ntgy1

/

In case of balanced growth, we have a constant nt and X 1 t Žor equivalently a constant ratio CtrK t .. We can now derive equations for ˙ nt and X˙1 t : X˙1 t s X 1 t g X 1 s X1 t Ž gC y g K . s X1 t

ž

r y wN cntg wc y 1

y a Ž 1 y nt .

wc wc y 1

q X1 t

/

˙nt s nt g n s nt

1 1yg

X1 t 1 y

ž

wN cg wc a

ntgy1

/

The steady state values for X 1 t and nt , are denoted by X 1 and n. They follow from X˙1 t s nt s 0, and are equal to: X1 s

1 1 y wc

Ž r y a Ž 1 y n . wc y wN cn g .

and ns

ž

cg wN a wc

1r1yg

/

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We now have a system of equations depending only on X 1 t and nt , and this can be analysed in the usual manner ŽGeorge, 1988.. Let us denote X 1 t y X 1 by x 1 t , and nt y n by x 2 t . The two remaining equations X 1t and ˙ nt , are rewritten in terms of x 1 t and x 2 t :

˙x 1 t s

Ž x 1 t q X1 .

ž

r y wN c Ž x 2 t q n .

g

wc y 1

˙x 2 t s Ž x 2 t q n . Ž x 1 t q X 1 .

1 1yg

ž

y a Ž1 y x 2 t y n .

1y

wN cg wc a

wc wc y 1

Ž x 2 t q n . gy1

q x1 t q X 1

/

/

This system of equations is linearized around the steady state x 1 t s x 2 t s 0 to arrive at ˙ x t s Jx t , where x t is equal to Ž x 1 t , x 2 t .T , ˙ x t is Ž ˙ x 1t , ˙ x 2 t .T , and the matrix J x Žthe Jacobian is ­Ž ˙ x1 t , ˙ x 2 t .r­Ž 1 t , x 2 t . Ž t s 0., or: Js

X1

0

0

X1

Appendix C: Stability for Lucas model with nature On the basis of the first order conditions, derived in Section 3, we can derive additional difference equations for c t , nt and u t . We already have the difference equations for K t , Ht and Nt . We can then derive difference equations for Ht L trK t and c t L trK t . Let us denote Nt y N by x 1 t , Ht L trK t y Ž HLrK . by x 2 t , c t L trK t y ŽcLrK . by x 3 t , nt y n by x 4 t , and u t y u by x 5t . The difference equations for these variables x ti Ž i s 1,...,5. can be determined easily. This system of equations is linearized around the steady state x 1 t s x 2 t s x 3 t s x 4 t s x 5t s 0, to arrive at ˙x t s Jx t , where x t is equal to Ž x 1 t , x 2 t , x 3 t , x 4 t , x 5t .T, ˙x t is Ž ˙x 1 t , ˙x 2 t , ˙x 3 t , ˙x 4 t , ˙x 5t .T, and the matrix J, the Jacobian matrix, is ­Ž ˙ x1 t , ˙ x2 t , ˙ x3t , ˙ x4 t , ˙ x 5t .r­Ž x 1 t , x 2 t , x 3 t , x 4 t , x 5t . Ž t s 0.. Unfortunately, the elements of the Jacobian matrix for this system of differential equations are very messy. The determination of the eigenvalues has been ‘automated’ in that this process is translated into a file which is handled by the computer program Maple ŽHeck, 1993.. The following parameter values are used: p s 0.025, r s 0.04, a s 0.2, j s 0.02, g s 0.5, m s 0.05, wC s 0.4 and wN s 0.4. For the Jacobian we get: y0.0500 0.000 J s y0.252 y5.268 20.671

0.000 y0.180 y0.0598 0.277 y1.088

0.000 0.385 0.192 1.468 y6.343

0.00247 0.117 0.0511 0.780 y3.060

0.000 y0.126 y0.0394 0.183 y0.706

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The eigenvalues of this matrix are: y0.691, 0.703, y0.0443, 0.0560 and 0.0117. The corresponding eigenvectors are: 0.000700 0.000672 y0.0102 0.0106 0.00000000173 y0.184 0.181 0.0217 y0.0942 1.670 , , y0.0160 , y0.175 , y0.0567 0.0664 0.0000000141 . 0.182 0.205 y0.0235 0.453 y0.0000000426 y0.749 y0.875 y0.0936 0.0598 y2.531 Now a submatrix has to be formed of those eigenvectors which have an eigenvalue with a negative real part, in this case the first and third eigenvector.8 Of these two eigenvectors, a submatrix is formed consisting of the first two rows: y0.000700 y0.184

y0.0102 0.0217

This submatrix is invertible as required for local stability.

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8 This analysis is based upon a slightly adapted version of the results obtained by Huijberts and Withagen Ž1992..

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