d'arge r., kogiku k., (1973) economic growth and the environment.pdf

8/10/2019 D'Arge R., Kogiku K., (1973) Economic growth and the environment.pdf

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.. .(1)

conomic rowth

and

the

nvironment 1,2,3

R. C. D ARGE and K. C. KOGIKU

University

of

California, Riverside

The premise that man resides withina bounded and closednatural environment, an environ

ment with relatively fixed dimensions, has been discussed in many forms for at least three

centuries. The idea of a spaceship

earth

has recently caught the imagination and

interest of both economists and environmentalists in particular, and the public in general

[3], [5], [6] and [17].

basic tenet of this viewpoint is that whatever has been produced,

consumed, not recycledand discarded within the sealed spaceship earth is still here and

willcontinue to be. Of course, energydispersionmay be offsetby nuclear fusion or breeder

reactors so that what was previously discarded may be completely reused without seriously

impinging on the earth s natural energy reservoirs. Economically and technically-feasible

extraterrestrial transportation systems, destruction of wastes by symmetrical anti-wastes,

or increased efficiencies in solar energy conversion also may become realities. However,

such energy-augmenting technological changes, at least at this time, appear to be in the very

distant future. What is of pressing current importance, if we accept the premise of a

nearly sealed capsule earth with relatively fixed dimensions in terms

of

mass-energy and

waste assimilative capability, is how rapidly economic growth should proceed.

Rather than attempting to dissect the spaceship

earth

premises regarding limited

energy reserves and assimilative capability, we take these as given in this paper. We then

construct several simple paradigms of an economy confronted with a bounded, closed

natural environment and then we analyze how such an economy should optimally produce

and consume over time. The next section of the paper deals with an extremely simple

model of waste generation, based on the conservation of matter-energy principle, and with

the consumption behaviour of the economy s inhabitants assumed to be predetermined.

In subsequent sections, the model is generalized to an optimal control problem where

consumption and waste generation are allowed to be regulated, and an attempt is made to

integrate the non-mutually exclusiveprocesses of resource extraction and waste generation.

The materials balance view of a closed resource system indicates that tonnages of raw

materials extraction utilized by an economy are approximately equal to tonnages of waste

products generated by the economy in the long-run [3]. A basic identity derivable from

the principle of conservation of matter-energy, given the assumption of materials balance

and no recycling is:

R ~ W f W

F=

W

1 First version received December 1971; final version received Apri 972 1s.).

2

A portion

of

the research reported here was financially supported by Resources for the Future,

Inc., with no responsibility for results or interpretation. A previous draft of this paper was presented at

the 1970Econometric Society Meetings in Detroit, Michigan.

3

The authors wish to acknowledge the very helpful comments and suggestions of K. Oddson, K.

Hamada, G. Heal, and two referees.

61


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62

REVIEW OF ECONOMIC STUDIES

where R, F, and W denote material extraction, production consumed by consumers, and

total waste flows, respectively. Wi and W

J

are the amounts of waste flow originating in

the production and consumption sectors.

Given these fundamental identities and assumptions weare able to write an additional

identity provided the flow F of material is assumed homogeneous. Basically,we assume

that there is only one type of consumer good. Then:

F

=

y

.. .

2

where y is per capita income in units of material flowper capita and N is total population.

Finally, if production and consumption are joint products of waste emissions, total waste

flows can be assumed to be proportional to final product [3], [5]:

W

=

gF. ... 3

Then, by assuming that each variable previously defined refers to time

r ,

waste flows are

linearly related to total income measured in material units:

Wet = g .y t .N t . . ..

4

Thus, we obtain a relation between waste flows and output per capita. Actually, the

assumption

of

proportionality is even more restrictive than one might at first presume,

since it completely specifies an implicit technology relating output to raw materials. By

definition,

Wet

=

R t ,

so

Wet

=

g

.y t .N t yields yet = R t . Thus, in this most

g N t

simple case, production results only from the magnitude of raw materials and there is no

substitution between labour and raw materials in the production process. In terms

of

crude empirical approximation the number

g

yet for theconsumption sectori.e.,

Wit /

N t

is estimated to be greater than one ton

per

capita per year for the United States at current

production levels.

Our next task is to

U model

the environment in some semi-realisticand yet manage

able way. Environmental pollution, at least in its quantitative dimensions, is usually

expressed in terms of concentrations, i.e., parts per million ppm) of dissolved solids or

DDT, ppm suitably indexed) of carbon monoxide or oxides of nitrogen concentrations in

the air, or tons per cubic acre of solid wastes. Thus, a natural single unit of waste concen

tration appears to be average density. Of course, concern with densities only may be

misleading, particularly with such pollutants as carbon monoxide or methane gas generated

from solid wastes. or now, however, wewill assume that density is a reasonable abstract

measure of waste concentration.

Let D t and v denote waste density at time

t

and total environmental waste holding

capacity, respectively. We, in effect, identify the closed resource system by a simple fixed

volumetric magnitude,

v.

Then using the definition that waste divided by volume equals

waste density and applying it to 4) yields:

D t

=

D O

+

ft

y t N t dt. ... 5

v 0

Given exogenously determined percentage rates of growth in population of

l n

and

income per capita of l P waste density can be easily related to initial population levels

and material flowper capita:

p+n t

D t =

D O + fl_- y O N O . ... 6

v f3+n

Increasing density of wastes within the closed resource systemis not necessarily bad per se.

However, if the natural environment is relatively small in comparison to current and

expected future sums of waste flows, people may suffer physiological, psychic, or other

damages. As an example, we could presume for each individual a monotonic, continuous


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64


(8)

to the form of the utility function in addition to separability which sidesteps the problem

of interpreting cross partial derivatives of the utility function:

uy>O

UD>O uT>O

UyyO UTT O

In essence, these conditions stipulate that per capita utility of material flowis increasing at

a decreasing rate, while

per capita

disutility generated by increasing waste densities is

increasing at an increasing rate. As the finiteplanning interval r is increased, it is assumed

that per capita utility at any time t is increased, but at a decreasing rate. This is perhaps

one of our most tenuous assumptions since it implies that, regardless of how close

t

is to

T,

individuals will be precommitted to applying a non-shifting evaluation to an increment

in r.

Consequently,

the optimal finite planning interval is deduced with no adjustment

occurring over time to account for potential future disobedience in the Strotz

[ ]

sense,

as

t approaches T(l).

From equation (5), we are able to write a relation between waste flow and waste

densities:

D= W

v

g

D

= -

yN

v

(9.1)

...(9.2)

Equation (9.2) is, in effect, a description of the changing state of the natural environ

ment as a result of the amount of material consumption at time t. Equations (7) and (9.2)

yield the pollution process model except for specification

of

initial and terminal con

ditions and inclusion of a social rate of time preference factor. The terminal time

r

is

conceptualized to be that period of survival which is optimum for the maximization to be

stated below. Given a presumed decimation density level (for example, 2,000

4,000 ppm

of carbon monoxide),

then

D r

=

To complete this exposition, the initial density is specifiedas

D O

=

l ,

and population

at

t as:

N t

=

N O e

nt

and the utility function is discounted with a discount factor a.

2

(10)

It

should

be

noted that the assumptions of a finite planning interval coupled witha finiteenvironmental

capacity for wastesare extremelyrigid. Ofcourse, infinite planning intervals in conjunction with an unlimited

natural environment assimilative capacity reduces to a trivial non-existent scarcity case unless scarcity is

introduced in some other way, i.e., available technology, knowledge, or capital. Alternatively, an infinite

planning interval in conjunction with a finite environmental waste assimilative capability appears to be

philosophically inconsistent, provided population is constant or increasing through time, material flow

per capita has a positive lower bound, rates of environmental waste assimilation are very low or zero, waste

generating technology is completely static in character, and wastes at some level of concentration become

lethal to the human species. These conditions quite obviously ensure that at some moment in the finite

future, decimation willoccur. The alternative case of a finiteplanning interval and infinitewaste assimilative

capability also appears to be philosophically inconsistent. Specifying an infinite planning interval for

mankind, of course, implies the imposition of a constraint on the decision process, namely that regardless

of how degraded man s habitat becomes, he must continue to exist. A more general and encompassing

decision process would stipulate that the survival period be optimally chosen with infinity as one possible

choice. We have chosen the course of specifying a finite planning interval and making it subject to choice

by current generations. However, such a decisionprecludes consideration of an infinite time horizon. The

marginal utility of an additional unit of time may approach or be at infinity for those concerned only with

preservation of the human species. In such a case, the length of

T

would dominate all other decisions in a

trivial manner since minimal consumption at each instant of time must prevail with finite resources,

specificallywaste assimilative capability.

2

Although weshall apply a discount factor which reflectsa positive or zero rate of time preference for

material-consumption flows,several potentially explosiveethical questions are brushed aside.

the human

race faces the possibilityof extinction bypollution (or other means)over a finiteinterval, are inter-generational

utility comparisons defensible? Within a different context, Dasgupta recently suggested that a small

positive discount of the

future

may be accepted as

ethical

[12]. However, if we assume that each


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D ARGE KOGIKU

GROWTH

AND THE

ENVIRONMENT 65

... (11.1)

Given the above definitions, an optimal control problem can be formulated as follows;

maximize

f

fT{

d 1_e-

at

}

J =

[u(y)-u(D)

u T)]e-atdt

=

[u(y)-u(D)]e-

at+

-

u t) dt

o 0 dt a

where

y

is the control variable and

is a state variable, whose equation of motion is

iJ

=

fl

yN = bye b

gN O)jv).

v

...(11.2)

The problem is to select the admissable control y so that a maximum integral of discounted

individual utility is achieved over the interval 0 t T where T is free to vary.

The Hamiltonian function for the problem can be written:

d 1_e-

at

H =

[u(y)-u(D)]e-

at+

- u(t)+

pbenty

dt

a

d

e

at

= [u(y)-u(D) pbye n+a)t]e-

at

-

u t)

dt

a

...(12)

where p is the costate variable corresponding to

According to the theory of optimal control, optimality calls for the following con

ditions :

iJ = bye D O

= 2 D T)

= 5

. aH D) -at

p = - - = U e

aD

y*

maximizes

u(y)+

pbye n+a t

[u(y)-u(D)+ pbye ? n) ]e-

at

=

f

{a[

u(y(s-u(D(s]

d

1_e-

at

- npbye n+a)S}e-asds - -

u t)j

dt

a t

d e

at

[u Y T-u D T

pbye n+a)T]e-

aT

+

-

u t)/t = O

dt

a

If we define a new variable

z = -

bpe n+a t

... (13.1)

... (13.2)

... (13.3)

... (13.4)

...(13.5)

generation can be exactly separated from others, then current generations with higher rates of time preference

may actually eliminate the existence of some distant future generation. The faster we consume in a closed

resource system, the more rapidly extinction occurs. While the utility of distant generations may seem

valueless now, if we were that distant generation, we may very well value our continued existence at or

approaching infinity. Even very high rates of time preference over a finite interval would not make it

worthwhile to consume currently above some basic subsistence level, provided the last future generation s

utilities were considered. In any case, we shall assume that such inter-generational utility comparisons can

be contrived so that each generation would be equitably considered, including potential but excluded future

generations, and in the remainder of the discussion emphasize the results where all generations are treated as

equals, i.e.,

a = O

1 See, e.g., Michael Athans and Peter L. Falb, Optimal Control, New York, N.Y.: McGraw-Hill,

1966, Theorem 5-7, p. 293.

E---40/1


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66


... (12 )

... (13.1 )

... (13.2 )

(13.3 )

then (12) and (13.1)-(13.5) become

d

l e

at

H

= [u(y)-u(D)-zy]e-

at+

-

u(t)

dt

a

dD/dt

= benty

D O

=

f} D -r:

= i5

dzldt

= (n+a)z-bentuD

u(y)-zy u(w)-zw for all w such that 0 Vo w VI

where

V

o

is a lower bound and

VI

an upper bound for consumption;

u z

0

[u(y(t)- u D t -

z(t)y(t)]e-

= fe- {a[u(y(s-u(D(s]nz(s)y(s)}ds

d 1_e-

at

- -

u(t) t

(13.4 )

dt a

... (13.5 )

l e

at

[u(y(r-u(D(r)-z(t )y(r)]e-

at+

-

u(t)lt

=

dt a

For (13.3 ) the control should be:

y

= Vo

if z u,(vo)

= [Uy]-I(Z) ifu,(vt) z u,(vo) ...13.6

= if z u (Vt).

That is, Y = F(z), which is Lipschitz continuous for all z and non-increasing, should have

the graph shown in Figure 1. Using F z the system (13.1 )-(13.2 ) can be replaced with

dD/dt = bentp(z) D O

=

f}

.. (13.1 )

dzldt =

(n+a)z-bentuD

z O = Zo

(13.2 )

where

Zo

is free. A solution exists and is unique for all Several possible timepaths are

shown in Figure 2.

From

the preceding two equations:

: ~ = n : ~ ; : r U D ~ 0 ... (14.1)

depending on

y =

F(z)


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D ARGE KOGIKU

GROWTH AND THE ENVIRONMENT 67

z

z

b

n a

D

I V

o

)

I V

o

)

b

n

... (14.2)

V

b

n a

UD

D

0

D

0

FIG

2

dz 0

dD


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68 REVIEW OF ECONOMIC STUDIES

a substantial undiscounted shadow price should be applied to waste density. Following

the initial large-scale shift the shadow price decreases over time.

Re-writing (13.3 ) for an interior solution

Hence in this case

u, y = z.

(15)

(17)

Y

= Zfuyiy

...

16

and

y is increasing when z is decreasing. Equation (13.2 ) shows that z will be falling

when

n a z

0, uQQ

d'arge r., kogiku k., (1973) economic growth and the environment.pdf

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