DE LA PARTICIPACION SOCIAL EN MATERIA FORESTAL CAPITULO I.

between the two prospects and to establish a u t i l i t y interval:

53

,U t i I es

Figure 2 . 1 : Total U t i l i t y for Good X.

Sinden (1974) argues that because the construction of the total u t i l i t y curve is dependent on the level of provision of the good Y, to obtain further total u t i l i t y curves for good X, i t is necessary to change the level of Y and reiterate the Ramsey

game. F i rs t , the combination of goods, X and Yb, which provides

a certain level of u t i l i t y , 3, is identified from the established

curve: in Figure 2.1, 3 utiles are said to be provided by the

combination Xf and Yb. This combination is entered into a Ramsey

matrix as Prospect I which is compared with a Prospect I I consisting of a new level of good Y, Yg, in position (iv) and the variable level of good X, Xh, in position ( i i i ) - the new decision matrix is displayed in Table 2.2.

Probabi1i t y of Prospect I Prospect 2 Occurrence

0.5 Xf Xh

0.5 Yb Yg

Table 2.2: Ramsey Model Decision Matrix f o r Subsequent U t i l i t y Curves. From the information on Xh provided by the respondent,

the ind if fe re n c e property between Prospects 1 and 2 is :

0.5 (U(Xf))+ 0.5 (U(Xb)) = 0.5 (U(Xh)) + 0.5 (U(Yg)) . . . (5) Sinden's assumption t h a t :

U(Xf) + U( Yb) = 3 . . . (6)

makes i t possible to deduce th a t

U( Xh) + U( Yg) = 3 . . . (7)

and from t h i s information i t is possible to est ablish a point on a new u t i l i t y curve f o r good X which uses Yg as a base: Figure 2.2

i l l u s t r a t e s t h i s new c u r v e , TU*.

U t i l e s

Figure 2 . 2 : The Sinden Derivation of a U t i l i t y Curve f o r X given Yg from the U t i l i t y Curve f o r X given Yb.

55

This horizontal s h i f t i n g process is used in Sinden's method to produce a fa m ily o f t o t a l u t i l i t y curves,from which can be derived

15 a set of ind if fe re n c e curves.

The method o f u t i l i z i n g the i nd if fe re nc e curves derived by the Sinden method has varied between the various case studies which have used the procedure. Sinden (1974) and Sinden and

Wyckoff (1976) used the marginal costs o f the two recreation a c t i v i t i e s which were involved - goods X and Y in the current notation - and

the budget each respondent all ocated to recreation over the study period of six weekends, to construct budget li n e s so th a t the compensated demand curve f o r recreation in the State Park could be constructed. Liesch and Sinden (1976) in t h e i r study of the value o f the "r u ra l way of l i f e " , a r e l a t i v e l y d i f f i c u l t concept to budget and cost, used a r e a d i l y valued good - overseas t r i p s - as the Y good to act as a numeraire from which income equivalents could be calculated.

Valuation using Sinden's adaption o f the Ramsey u t i l i t y model has the advantage of encouraging subjects to th in k c a r e f u l l y about the choices made, because of the de tailed judgements required. In a d d i t i o n , the d i f f i c u l t y some respondents have in giving d o l l a r answers to valuation questions is avoided. While no e x p l i c i t a n t i - str at egic-behaviour mechanism is employed, the d e ta il of judgements made and t h e i r non-monetary nature may have the e f f e c t o f lessening

15 The standard procedure o f holding u t i l i t y constant and observing the combinations o f X and Y which provide tha t level o f u t i l i t y applies. For instance, in Figure 2.2, (3 u t i l e s are provided by the combinations o f X and Y:

the problem of over-statement: this conclusion is drawn by Liesch and Sinden (1976) in a comparison of valuation techniques applied to a recreation valuation study - valuations were verified by a back-

projection t e st in which the relationship between valuation and actual use was tested for significance.

However an obvious limitation of the technique described,is the time and cost of undertaking such an extensive interviewing

programme - indeed the validity of conclusions concerning the presence of strategic behaviour drawn in all the u t i l i t y valuation studies

reviewed are somewhat dubious because only small numbers of respondents 16

had been interviewed.

One further di f f i cul t y, involving the theoretical structure of the u t i l i t y valuation method, limits i t s use to cases in which the u t i l i t y of good X can be simply added to the u t i l i t y of good Y to form the joint u t i l i t y function U(X,Y), that is:

U(X) + U(Y) = U(X, Y) . . . (8)

To explain the reasons for this limitation, i t is useful to return to the description of the Sinden approach, and particularly the situation depicted in Table 2.1 which is expressed in Equation (1). When an individual is considering the expected u t i l i t i e s to be gained from Prospects 1 and 2, he realizes that the outcome will involve only one of Xa, Yb, Xc or Zd: for instance, i f Prospect 1 is chosen, there is a f i f t y - f i f t y chance that either Xa or Yb will occur. To represent Equation (1) more rigorously, i t is therefore necessary to specify that the u t i l i t y gained from Xa, U(Xa),will not be associated with any provision of other goods: this can be

16 Sinden (1974) and both the studies reported in Liesch and Sinden(1976) use only five subjects - the interview time for each respondent was between two and three hours. However Sinden

(1980) reports that an undergraduate student at the University of New England was able to reduce interviewing time to 20 minutes using a simplified procedure.

57

achieved using the n o t a t i o n U(Xa,Yo) t o rep re se nt the u t i l i t y o f a

u n i t s o f X consumed w i t h o u t any u n i t s o f Y. Hence Equation (1) becomes: 0 . 5 ( U ( X a , Y o ) ) + 0.5 (U(Yb,Xo)) = 0.5 (U(Xc.Zo)) + 0.5 (U(Zd,Xo))

The thr ee p o i n t s on the u t i l i t y curve can then be c a l c u l a t e d using the a r b i t a r y s e t t i n g o f U(Yb,Xo) - U(Zd,Xo) to some s p e c i f i e d l e v e l o f u t i l i t y , a , and U(Xa,Yo) as a base. However, i t i s c l e a r t h a t the curve so c o nst ru ct ed represents l e v e l s o f X, a, c, and e, which are enjoyed w ith o u t the p r o v i s i o n o f any good Y. Sinden (1974)

i n t e r p r e t s t h i s curve as being dependent on the consumption o f good Y a t l eve l b ,but f o l l o w i n g Kennedy's (1980) e x p l a n at i o n o f the problem, Sinden (1980) i s prepared to r e - s p e c i f y his i n i t i a l u t i l i t y curve

i n the Oregon Park case study as being based on zero consumption o f the a l t e r n a t i v e good - c l e a r l y t h i s problem i s overcome,but i t i s not u n t i l the second stage o f Si nden's method o f appl yi ng the Ramsey model t h a t the t e c h n i q u e ' s l i m i t a t i o n s are e v i d e n t . Consider once more the prospects d e t a i l e d in Table 2 . 2 , which i l l u s t r a t e s the f i r s t stage o f the u t i l i t y v a l u a t i o n method's approach to g ene ra ti ng f u r t h e r i n d i f f e r e n c e cur ves, but r a t h e r than i n c l u d i n g Yb in p o s i t i o n ( i i ) , the c o r r e c t Yo outcome i s added. I f the u t i l i t y curve f o r X given one u n i t o f good Y, Y^, i s to be d e r i v e d , then by the Sinden method, Y-j must be i ncl uded as outcome ( i v ) : Table 2.3 d e p i c t s t h i s new s cenar io.

. . . (9) P r o b a b i l i t y o f Prospect 1 Prospect 2 Occurrence 0.5 0.5 Xf Yo Xh Y,

Table 2 . 3 : Ramsey Model Decision M a t r i x f o r D e r i v i n g the U t i l i t y Curve f o r X given Y-j.

Equation 10 expresses rigorously the condition of indifference between the two prospects:

0.5 (U(Xf,Yo)) + 0.5 (U(Xo,Yo)) = 0.5 (U(Xh,Yo)) + 0.5 (U(Xo, Y-j))

and hence: ’ * *

U( Xf ,Yo) + U( Xo ,Yo) = U(Xh,Yo) + U(Xo ,Y-j) . . . (1 1)

Assuming that U(Xo,Yo) is equal to zero, i t is possible to rewrite (11) as:

3 = 5 + U( Xo, Y-,)

because the values for U(Xf,Yo), 3, and U(Xh,Yo), 6, can be observed from the original u t i l i t y function, TU, is il lu s t ra te d in Figure 2.2. Therefore:

U(Xo,Y-,) = 6 - 6 . . . (12)

So rather than providing a point on the u t i l i t y curve for good X given one unit of Y, u(Xi ,Yq), the rigorously defined

u t i l i t y valuation method enables an estimation of the u t i l i t y curve for good Y, given no consumption of X, u(Xo,Y]). For the method to define u(Xi ,Yq) i t is necessary to substitute for Equation (11) the expression:

U(Xf,Yo) = U(Xh,Y-|) . . . (13)

Clearly then U(Xh,Y-|) is equal to 6 uti le s and U(Xh,Y^) is a point on the new u t i l i t y function u(Xi,Y-|). Because the u(Xi,Y-|) curve

is based on the variable consumption of X given the constant

consumption of Y, the function u(Xi,Y-|) should be v e r t i c a l l y above the function u(Xi,Yo) which is already derived, by a constant amount, the marginal u t i l i t y of one unit of Y: in the current example

this distance can be derived from Equation (12) to be 6-6 uti le s. Given the necessity of considering Equation (11) to be equivalent to Equation (13), what are the conditions under which equivalence can be assumed? Smith and Bennett (1981) set out two

59 c o n d i t i o n s w h i c h m u s t be s a t i s f i e d if t h e " s t r i c t a d d i t i v i t y " a s s u m p t i o n , n e c e s s a r y to m a k e E q u a t i o n s (11) a n d (13) e q u i v a l e n t , is to h o l d : (i) T h e r e is n o t h i r d g o o d a t w h o s e e x p e n s e th e c o n s u m p t i o n o f X a n d Y o c c u r s , a n d w h o s e m a r g i n a l u t i l i t y is n o t c o n s t a n t ; a n d , (ii) t h e u t i l i t y d e r i v e d f r o m c o n s u m p t i o n o f X d o e s n o t d e p e n d on t h e q u a n t i t y o f Y c o n s u m e d a n d 17 wee versa. T h e f i r s t c o n d i t i o n c a n b e e x p l a i n e d u s i n g P r o s p e c t 2 o f T a b l e 2.3. S u p p o s e t h e c o n s u m p t i o n o f X a n d Y is c a r r i e d o u t at t h e e x p e n s e o f a n o t h e r g o o d W , a n d t h a t t h e s u b j e c t ha s a t o t a l " b u d g e t " t o s p e n d on g o o d s X , Y a n d W o f n u n i t s . T h e e x p e c t e d u t i l i t y o f P r o s p e c t 2 c a n t h e n b e w r i t t e n r i g o r o u s l y as: 0 . 5 ( U (X h ,Y o ,W n - h )) + 0 . 5 ( U ( X o , Y ] , W n - l )) ... (14) N o w S m i t h a n d B e n n e t t ( 1 9 8 1 ) s u g g e s t t h a t if t h e m a r g i n a l u t i l i t y o f g o o d W i n c r e a s e s as t h e " a m o u n t o f W c o n s u m e d is r e d u c e d t h e n 't h e n e t u t i l i t y g a i n f r o m a combination o f (Xh) a n d (Y-j) wi ll b e s m a l l e r t h a n t h e a d d e d n e t u t i l i t i e s o f (Xh) a n d (Y-j) c o n s i d e r e d as m u t u a l l y e x c l u s i v e e v e n t s ' (p. 7, u s i n g c u r r e n t n o t a t i o n ) . H e n c e : U ( X h , Y ] , W n - h - l ) < U ( X h , Y o , W n - h ) + U ( X o , Y 1 , W n - l ) ... (15) a n d t h e " s t r i c k a d d i t i v i t y " a s s u m p t i o n w h i c h e n a b l e s t h e e q u i v a l e n c e b e t w e e n e q u a t i o n s (11) a n d (13) c a n n o t a p p l y . 1 7 A f t e r S m i t h a n d B e n n e t t ( 1 9 8 1 ) , p. 9.

Compliance with the second condition, that the u t i l i t i e s from X and Y are independent, according to Smith and Bennett (1981), is 'most likely i f (X) and (Y) satisfy different kinds of wants so that they are neither close substitutes nor close complements'

(p. 9). They i l l us t r a t e this by describing the case which would occur i f the goods were perfect substitutes: because the marginal u t i l i t y of a unit of X will be the same as that of an additional unit of Y, the u t i l i t y function u(Xi,Yo) would be equivalent to both

u(Xo, Yi) and u(Xi, Yi).

Clearly the study reported by Sinden (1974) and Sinden and Wyckoff (1976) meets neither of these conditions - the good Y in