Consumption of Groundwater as A Private or A Public Good

423

CONSUMPTION OF GROUNDWATER AS A PRIVATE OR A PUBLIC GOOD

J . .PASQUUAL and I . ROCABERT Departament d'Economia A p l i c a d a , U n i v e r s i t a t Autdnoma de B a r c e l o n a , B e l l a t e r r a , Barcelona Spain ABSTRACT

Groundwater c o n s t i t u t e s a r e s o u r c e a s s e t t h a t g e n e r a t e s economic b e n e f i t s . These b e n e f i t s a r e a t t h e same time s u b j e c t t o e x t e r n a l i t i e s of one k i n d o r a n o t h e r . T h i s a r t i c l e examines t h e t r a d e - o f f i n v o l v e d i n u s i n g d e p l e t i v e o r n o n - d e p l e t i v e e x t r a c t i o n p o l i c i e s i n groundwater e x p l o i t a t i o n .

A p e r f e c t system of r e c h a r g e i n n o n - d e p l e t i v e u s e of groundwater means t h a t t h e t o t a l amount o f w a t e r a v a i l a b l e f o r use i s u n a f f e c t e d . R i v a l r y between consumers would n o t e x i s t . T h e r e f o r e , groundwater c o u l d be c o n s i d e r e d t o be a p u b l i c good, as d e s c r i b e d by Samuelson i n 1954. On t h e o t h e r hand, where e x t r a c t i o n i s d e p l e t i v e , i t must then be c o n s i d e r e d a p r i v a t e good. resources allocation t o The problem then l i e s i n t h e o p t i m i s a t i o n of t h e be e x p l o i t e d u s i n g e i t h e r one o f t h e s e two a l t e r n a t i v e s and t o d e t e r m i n e t h e optimum number of consumers f o r each o p t i o n . Once t h e problem has been a n a l y s e d from a P a r e t i a n p o i n t o f view, no i n t e r i o r s o l u t i o n e x i s t s . Maximum s o c i a l w e l f a r e must n e c e s s a r i l y be d e r i v e d from u s i n g e i t h e r one o r t h e o t h e r of t h e two s o l u t i o n s p o s s i b l e .

1 INTRODUCTION From

an

economic

point

of

view, groundwater c o n s t i t u t e s an asset whose

b e n e f i t s - t a n g i b l e and i n t a n g i b l e - are c l o s e l y consumer and

production a c t i v i t i e s .

related to,

and i n f l u e n c e d b y ,

Groundwater is a l s o t h e cause o f , and t h e

r e s u l t o f , e x t e r n a l economies and diseconomies. When e x p l o i t i n g an a q u i f e r , n o t o n l y

s h o u l d economic

agents determine the

s i z e of c a p i t a l investment r e q u i r e d as w e l l as t h e amount o f w a t e r t o

type and

be withdrawn. The i n f l u e n c e of e x t e r n a l i t i e s s h o u l d a l s o be determined i n terms of s o c i a l w e l f a r e i f t h e s e f i g u r e s a r e t o be o p t i m i s e d . The

results

will

property r i g h t s over possible t o

be

influenced

groundwater

by

existinglegislation- i n particular,

resources.

ik our

Moreover,

case,

it i s

d e c i d e whether groundwater i s a p r i v a t e o r a p u b l i c good u s i n g t h e

d e f i n i t i o n given by Samuelson ( 1 9 5 4 ) . Although t h e s i m p l e s t way o f d e f i n i n g

groundwater

is

as

a

pure p r i v a t e

good, t h e f a c t t h a t p r o p e r t y r i g h t s are n o t c l e a r l y d e f i n e d means t h a t problems may a r i s e similar t o t h o s e t h a t e x i s t i n t h e c a s e of p u b l i c goods. Where two o r more u s e r s

have t h e r i g h t t o e x p l o i t t h e same a q u i f e r , n e g a t i v e and r e c i p r o c a l

424 externalities appear ips0 facto. This situation is typical of "common ownership of natural resources", and, as Aguilera

policies to

correct

the

(1987), in

points out

situation, withdrawal

the absence o f

is uneconomical because the

amounts of groundwater withdrawn are excessive. Whether adequate legislation does o r

does not

exist, even

supposing an

ideal situation did exist, groundwater exploitation is always subject to the influence of externalities. The contamination of aquifers and between

health

the relationship

consumption are examples o f positive and negative

and water

externalities affecting groundwater exploitation. According to Bird (1987), differentiating between

a proper

analysis of

transferible

distinction should also be made

and

between

the problem

must involve

non-transferible externalities. A

depletable externalities

-acting as

good- and non-depletable externalities characteristic o f a pure

pure private public good

(Baumol and Oates, 1975).

On the other hand, an aquifer may be

exploited where

fixed costs

are low

and variable costs high, or, where fixed costs are high and variable costs low. Where fixed costs

are high, groundwater may

be

considered a

public good

(Mueller 1979, and Baumol; Panzar and Willing 1982). It is, therefore, possible to convert a private good into a public good at any one time. Similarly, the withdrawal o f a particular quantity of water by one consumer may reduce

amount of water available f o r the use of other consumers

the total

(depletive extractions). Rivalry between consumers would then exist teristisic of

a private

the total amount

of

-a charac-

good. Alternatively, withdrawals may be made so that

water

available is not

affected

(e.g.

using perfect

recharge methods or non-depletive extraction policies)- a situation which would be characteristic of a

pure public

good. Again, the possibility

of deciding

whether to use groundwater as a private o r a public good arises. Groundwater is therefore a "transformable natural resource" (private f public). The situation, then, is a complex breaking

it

down

into

more

one which

simplified

can

best

be

understood by

forms, each o f which deals with one

relevant aspect of the whole. In this way, the situation can be

better defined

as a result of more detailed analysis. The following article is a study of groundwater as a transformable natural resource which may be used wholly or partially as a pure private good o r a pure public good without legislative restrictions of the optimum allocation of extractions within

available resources

the framework

of a

any kind. The aim is to find

for depletive o r non-depletive

general Paretian model. Similarly, an

effort will be made to optimise the number of consumers f o r each alternative.

425

2. THE MODEL Taking H as

identical individuals having

consumption of a transformable good X well as f o r the consumption of the same Individuals have

definite preferences f o r the

-groundwater- as a public good, Xn, as groundwater X as a private good,

also established preferences over

the

consumption of a

non-transformable pure private good, L. These preferences can be represented by the utility function, h h h h U (Xn, Xm, L ) ; h=l, which we

. .., H

suppose strictly concave and twice continually differentiable. Two

types of individuals are considered. On the one hand, one that consumes X only as a private good, m, whose utility function will be, Um(Xm, L), wM,

O<%H

and another that consumes X as

a collective good, n , whose utility function

will be, u"(Xn,Ln); nE(H-M), H>1. The problem consists of determining which

is the optimun size of each of

these groups -choosing M- and allocating the initial resources of the transformable good between consumption as a private and as a

collective good -choosing

Xm and Xn. Being w the initial resources of groundwater, the feasibility constraint will be, W-MX -X =O. m n This equation reflects that each of the M consumers consumes a quantity X

of X

as a private good, while X

represents that part consumed as a collective

good by the rest of consumers (H-M). With respect to the non-transformable private

good, there

and L resources, T. The individual consumptions L m variables. The associated feasibility constraint is,

exist initial

are additional decision

T-ML -(H-M)L =O The welfare in this

society will

be evaluated by using Bergson's social

welfare function,

L~ ) W = M U ~xm, ( L ~+ (H-M)u"(x~, ) Firstly, we

.

will consider the problem

of attaining

within a planned economy where decision-makers have at variables in

the problem

second-best optimum will be

a first best optimum

their disposal all the

as decision variables.Secondly Various problems of examined when

some of

the control variables are

predetermined.

3. THE FIRST-BEST PROBLEM 3.1 General Case The problem to be solved in its singlest version is as follows:

426

s . t . : W-MX -Xn (a0 rl m (H-M ) L

'

T-ML,-

n(3 r2 where X m , X n , Lm, L a r e t h e d e c i s i o n v a r i a b l e s a n d , H , w , T are p a r a m e t e r s .

a problem

This i s

of conditioned

n o n - l i n e a r programming i n which we know

t h a t i f t h e o b j e c t i v e f u n c t i o n as w e l l a s t h e c o n s t r a i n t s ( i n t h e

form 3 ) a r e

concave, t h e problem h a s a maximum ( i n t e r i o r ) . u t i l i t y f u n c t i o n s a r e s t r i c t l y concave, by h y p o t h e s i s i t is e a s y

Since the

t o prove t h a t t h e constraint

for

non-transformable

Conversely, the c o n s t r a i n t convex on ( X

W (.)

objective function the

m' Xn, M ) . i s no

Thus, t h e r e

is a l s o

corresponding

to

i n t e r i o r maximum,

concave. The f e a s i b i l i t y

good ( 3 ) , i s concave as w e l l .

private the

transformable choice is

hence t h e

good

( 2 ) is

limited t o the

s t u d y of two c a n d i d a t e p o i n t s , t h e two corners. These p o i n t s are Wo and W1,

W (M=O,X = O , L = O , ( H - M ) = H , X = w , L =T/H) 0 m m n n W 1( M = H , X = w / H , L m =T/H,(H-M)=O,Xn = O , L n = O )

W

0

measures t h e s o c i a l w e l f a r e when

mable good expression W

t r a n s f o r m a b l e good

resource6 w

all the

of the transfor-

by a l l t h e i n d i v i d u a l s as a p u r e c o l l e c t i v e good. The

a r e consumed measures

(4) (5)

the

social

welfare

when

all

the

resources

a r e consumed by a l l t h e i n d i v i d u a l s as a p u r e p r i v a t e good.

A s a l l t h e i n d i v i d u a l s a r e i d e n t i c a l , we can do w i t h o u t t h e non-transformable

good, L ,

t h e consumption o f

t o e v a l u a t e t h e c a n d i d a t e p o i n t s . The d e c i s i o n

depends e x c l u s i v e l y upon t h e s i g n of D: det m D= U ( w / H ) - U n ( w ) Consequently, a preferred i n

of t h e

(6)

sufficiency condition

t o ensure

that it

i s not s o c i a l l y

any c a s e t o consume as a p r i v a t e good a good t h a t can be consumed

a s a c o l l e c t i v e one, i s ,

U ~ ( X ~ ~ U " )( , X f o r e v e r y

x,\
g i v e n t h a t w/H
~ < Q u ~ ( x ~ ) \ < u ~f o( rx ~e v)e,r y x&c xn

(7)

The r e v e r s e is n o t t r u e . I f t h e p r e f e r e n c e s a r e such t h a t : U ' " ( X ~ ) ~ U " ( X ~ )f o r e v e r y

t h e n n o t h i n g can be preferences. I n

x,axn

s a i d about

order t o

(8) the sign

of D

w i t h o u t knowing

exactly those

o b s e r v e t h e r e s u l t of d i f f e r e n t p r e f e r e n c e s o v e r t h e

consumption of X e i t h e r as a p r i v a t e good o r as a c o l l e c t i v e good, a s i m u l a t i o n was e f f e c t e d , as shown below.

427 b U n ( . ) = ( l / b ) X n + L n ; O
n s . t . : W-MX -Xn ~0

This

problem,

X > O , X > O , MaO. n' f o r t h e same r e a s o n s

as t h o s e e x p l a i n e d e a r l i e r f o r t h e

g e n e r a l c a s e , does n o t have an i n t e r i o r maximum. I n Table I , t h e v a l u e s t a k e n

for

the

social

welfare

function

f o r each

c o r n e r s o l u t i o n a r e shown f o r v a r i o u s v a l u e s of a and d i v e r s e r e l a t i o n s between

a and b. We w i l l suppose t h a t

the

of

value

the

parameters

the initial

of

r e s o u r c e s of t h e t r a n s f o r m a b l e good w and t h e p o p u l a t i o n H i s w=200 H=100.

A s follows

from t h e

Table I , whether o r n o t t h e o p t i o n of

o b s e r v a t i o n of

consuming i t

consuming X as a pure c o l l e c t i v e good w i l l dominate t h e o p t i o n o f

as a

pure p r i v a t e good w i l l depend upon t h e r e l a t i v e s t r e n g t h of t h e p r e f e r e n -

c e s between b o t h t y p e s of consumption. The goodness of each a l t e r n a t i v e cannot

"a p r i o r i "

be e s t a b l i s h e d

but i t

s h o u l d be computed f o r each c a s e , s i n c e i t depends on i n d i v i d u a l p r e f e r e n c e s .

4.

SECOND BEST PROBLEMS

A s we have seen i n t h e above s e c t i o n , t h e f i r s t - b e s t problem h a s a maximum, which is unique b u t i t i s a c o r n e r s o l u t i o n . There

It i s

for this

are closer

are no

interior solutions.

reason t h a t i t i s worth l o o k i n g a t some second b e s t c a s e s t h a t

t o more

r e a l i s t i c situations.

Let u s

v a r i a b l e s of t h e

t a k e some

g e n e r a l c a s e a s predetermined v a l u e s which means t h a t t h e y s h o u l d be c o n s i d e r e d

as a p a r a m e t e r . We w i l l s t u d y two c a s e s i n p a r t i c u l a r . F i r s t l y , when t h e duals, M,

consuming t h e

number of i n d i v i -

good as a p u r e p r i v a t e good i s i n s t i t u t i o n a l l y f i x e d ,

and s e c o n d l y , we w i l l suppose t h a t some o t h e r v a r i a b l e s a r e p r e d e t e r m i n e d . I n t h e f i r s t c a s e , t h e c o n s t r a i n t ( 2 ) , which was convex ( i n t h e form the general

case, w i l l

be transformed

t h e r e w i l l be an i n t e r i o r s o l u t i o n . I f p r i v a t e good

were p r e d e t e r m i n e d ,

a

)

the

the lack

quantity

Xm

as

consumed

a pure

of c o n c a v i t y of t h e c o n s t r a i n t ( 2 )

would p e r s i s t . Moreover, a c o r n e r s o l u t i o n (M=O) i s

not feasible

since i t not

p o s s i b l e t o f u l f i l t h e c o n s t r a i n t ( 2 ) f o r t h i s v a l u e . Furthermore, i f g i v e n X t h e necessary concavity of t h e o b j e c t i v e

function

fails,

the

study

l i m i t e d t o Case 1. Case 1. M g i v e n

I t would be t o s o l v e :

max .W=MUm( Xm,Ln) + ( H-M )Un(Xn,Ln )

in

i n t o a s t r a i g h t l i n e and c o n s e q u e n t l y ,

(9)

n' w i l l be

4 N -

m

TABLE 1

Measure of welfare f o r options W1 (water consumption as a private good) and Wo as a function of the preferences (a,b), being w=200 and H=100

(consumption as a public good)

: a I

1/100

10.069

1/50 1/20 1/15

5.069 2.070 1.571

1/10

1.072

1/7 1/5 l/4 1/3 5/12

772,s 574 475 7 378,l 320,Z 9 4 t

2.135

1.983

1.699

1.551

1.507

1.500

1.487

1.440 1.540 1.754 2.371 3.417

1.452 1.633 1.944 2.828 4.387

1.590

1.858

2.040

2.081

2.168

2.081

2.828

2.828 5.129 9.924

4.388

3.351 5.555 14.030 37.779

3.470 5.829

3.725 6.426

15.052 41.456

17.340

9.924 23.942

429

s.t.: w-MX -X =O r m n 1 T-ML -(H-M)L =O

r

where Xm, Xn, L

are control variables and M. H, w and T are parameters.

and L

(10)

(11)

2

The first-order conditions of this programme

are necessary and sufficient

for an overall maximum and they are: M(qm-rl)

=

M($m-;2)

= 0

0

(H-M)UXn-rl = 0

0

( H-M)(U"L-r2) =

That is, from (12) and 1=(13), and from (14) and (15):

which coincide with the optimality conditions f o r a pure private good (16) for M consumers and for a pure collective good ( 1 7 ) f o r vely. From (13) and (15) results,

(H-M) consumers, respecti-

Tm=qn.

as that from (16) and (17) it can

so

be written, That is, the utility at the optimum of increase in

the consumption of the

an individual due to

a marginal

good Xm has to be (H-M) times higher than

that corresponding to an individual n due to marginal consumption of X

.

5. CONCLUSIONS Groundwater may be considered to be an asset factors, each

influenced by

a multitude of

in itself of sufficient importance and complexity to justify a

separate analysis. This

article has

examined

the

social desirability of

depletive extraction policies (as a private good) as opposed to non-depletive extraction (as a public good) independent of other relevant aspects. I f it is considered that groundwater may be used wholly

private good

or as

or partially as a

public good, then independent of the social welfare func-

tion, a two-good solution can never be an optimum solution. That is

to

say that, depending upon

individual preferences

for one or

other system, it will either be socially beneficial to use the total amount of water available without depleting resources or to

carry out

it will

be socially beneficial

depletive extraction. Any in-between solution would not be as

socially beneficial. This conclusion is society organised

valid

for nations, as well

as any

in nations even though differing individual preferences may

exit. The analysis has been made from a Paretian point of view in nal vacuum.

an institutio-

430

The conclusions reached are independent of any property rights that may be established. The concession of

property rights must however be in consonance

system is chosen. If not, rivalries will develop between private

with whatever

and social interests

(Turvey 1963) which

could give

rise

to inefficient

allocation.

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Externalities, Public Outlays

and

"The The

theory

of

Environmental Policy:

Quality of Life". Prentice-Hall,

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1982. "Contestable Markets and the

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"Politicians, Legislation, and

the

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Survey Article",

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Samuelson, P.A.,

1954.

"The Pure Theory

o f Public Expenditure", Review o f

Economics and Statistics, 36: 387-389. Turvey, R., 1963.

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1983. "Local Provison of Public Services: The liebout Model after Five Years". Academic Press Inc. New York. 77877;).