A DECISION ANALYSIS OF OBJECTIVES FOR A FOREST PEST PROBLEM
a a v i d E . B e l l December 1 9 7 5
R e s e a r c h R e p o r t s a r e p u b l i c a t i o n s r e p o r t i n g o n t h e work o f t h e a u t h o r . Any v i e w s o r c o n c l u s i o n s a r e t h o s e o f t h e a u t h o r , a n d d o n o t n e c e s s a r i l y r e f l e c t t h o s e o f IIASA.
A Decision Analysis of Objectives for a F o r e s t P e s t P r o b l e m David E. Bell
Abstract
The f o r e s t s of E a s t e r n Canada a r e subject to periodic outbreaks of a pest which devastates the t r e e s causing major disruption to the logging industry. This paper gives details of a study to find a c r i t e r i o n by which management policy alternatives could be evaluated i n con- junction with a simulation model of the forest. It d e s c r i b e s the manner in which the important decision f a c t o r s , o r attributes, were determined and how a value functioil and a utility function were a s s e s s e d over t h e s e attributes, taking into account t h e long t i m e horizon involved of
50-100 years.
The r e p o r t which follows d e s c r i b e s a n a t t e m p t t o d e t e r m i n e an.d quantify p r e f e ~ e ; 7 c e s f o r a f o r e s t region in New Brunswick, Canada. The f o r e s t i s subject zo outbreaks of a pest called the Spruce Budworm which d ~ e s g r e a t ciarnzge .to the t r e e s and thus to the logging industry, a m a j o r p a r t of the economy of the province. DDT has been sprayed extensiveiy f o r the l a s t twenty yeazs s o t h a t now if the spraying w e r e t o s t o p a widespread uutbreak would oc:cur. T ' l r Ecology P r o j e c t a t the International Institute f o r Applied Systems Analysis (IIASA) w e r e using a detailed s i m ~ l a t i o n m o d e l of the f o r e s t to examine possible s t r a t e g i e s f o r handling t h e pest, s e e
Holling
--
e t al. [5], the Methodology P r o j e c t contributing to t h e study by creating a ilymal-nic P r o g r a m m i n g Optimization Algorithm, Winkler [ l I.];a n d t h ~ study o ~ t l i ~ e d h e r e s t a r t e d when I attended a meeting of the Ecology and Metlmdology P r o j e c t s tcgether with some e x p e r t s f r o m the Canadian T o r e s t r y Commi.ssiorL. They w e r e t r y i n g t o e s t a b l i s h a n objective f,unction for the optimization n~odel. by fitting values c t o the l i n e a r formula
i
c ( E g g Density) ?- c 2 ( b t r e s s ) l
+
c ( P r o p o r t i o n of Old T r e e s )1 3
+
c ( P r o p o r t i o n of New T r e e s ) 4.
(I). I ).'; was disturSed by. this p r o c e s s f o r two r e a s o n s . F i r s t l y , they did not appear to have a v e r y a c c u r a t e way of a r r i v i n g a t t h e p a r a m e t e r s , and secoxldly the o n l y coricorn of t h e e x p e r t s s e e m e d to be the m o n e t a r y gains and l o s s e s to the logging i n d u s t r y -whereas I had always supposed t h a t our Zcology a.nd E n v i ~ o m x e a . t P r o j e c t would a l s o be concerned with the protecti.sn of wildlife zarl scenery. So 1 began t h i s study with two a i m s :
i ) to d e r i v e the p a r a m e t e r s c f o r the optimization model by i
diiferent nleaus a s a comparison,
i i t n discover the t r u e p r e f e r e n c e s of t h e m e m b e r s of the Ecology Projec-i- r sgarding t r a d e -of$s between profits, wildlife and t h e cnvirenment.
* ~ t ; . e s s i s z r~ieaouye s f the health of the t r e e s m e a s u r e d by the amount of defoliation in c a r r e n t and previous y e a r s caused by t h e budworm.
T h i s paper t e l l s of m y P r o g r e s s , s p r e a d over the next eighteen months, t o w a r d s achieving t h e s e a i m s . In performing the a n a l y s i s , inevitably many m i s t a k e s w e r e m a d e and i f I w e r e to r e p e a t t h i s on a s i m i l a r study, I would do a g r e a t many things differently, however I have chosen to d e s c r i b e h e r e what actually happened r a t h e r than to s e r v e up a neat exposition of decision analysis a t i t s best. It should be borne i n mind t h a t t h i s study was not planned i n detail ahead, r a t h e r it developed m o r e on a week by week b a s i s and was subject to constant i n t e r r u p t i o n s including two six -month s e p a r a t i o n s of analyst f r o m decision m a k e r .
The benefits of presenting it like t h i s , I hope, a r e t h a t on the one hand a number of t h e o r e t i c a l i s s u e s a r e r a i s e d to which s o m e attention should be paid and on t h e o t h e r it might encourage potential a n a l y s t s who ,may f e e l daunted by the imposing l i t e r a t u r e on decision a n a l y s i s to give i t a t r y t h e m s e l v e s .
The f i r s t section of t h i s r e p o r t deals with the initial investigztion I m a d e to check whether t h e coefficients of the l i n e a r objective function w e r e a c c u r a t e and r e c o u n t s t h e way i n which we attempted to r e s o l v e apparent d i s c r e p a n c i e s i n the p r e f e r e n c e s of the different Ecology group m e m b e r s by finding a l t e r n a t i v e s e t s of f o r e s t s t a t i s t i c s which better enabled the ecologists to a g r e e .
The second section d e s c r i b e s the way i n which I attempted to a s s e s s a value function f o r the p r e f e r e n c e s of one of t h e ecologists over a t t r i b u t e s which w e r e i m p o r t a n t to him. The difficulties a s s o c i a t e d with coila,psing i n d i c a t o r s over t i m e i s r a i s e d and discussed.
Section t h r e e r e p r e s e n t s stage two of the whole analysis. In t h i s a utility function i s a s s e s s e d for the s a m e "decision m a k e r ' ' wlGch i n -
c o r p o r a t e s m a n y of the complicating f a c t o r s which hindered the a s s e s s m e n t of the value function s u c h a s interdependencies of p r e f e r e n c e s for outcomes i n different periods.
Section four s u m m a r i z e s the p r e f e r e n c e a s s u m p t i o n s which w e r e used in the a s s e s s m e n t of the utility function. Section five p r e s e n t s a review of t h e whole p r o c e d u r e , d i s c u s s i n g s o m e of t h e i s s u e s r a i s e d and the pitfalls encountered
.
In o r d e r to k e e p t h i s paper to a reasonable length many of the
concepts used f r o m decision a n a l y s i s such a s value function, utility function
and various independence assumptions a r e d e s c r i b e d r a t h e r c u r s o r i l y , t h e r e a d e r who is not well acquainted with t h e s e definitions should
consult Raiffa [ l o ] o r Keeney and Raiffa [ 7 ] . 1. P r e l i m i n a r y Analysis
I began by asking five of the conference participants t o r a n k a l i s t of s t a t e s of the f o r e s t , exhibited in F i g u r e 1, by p r e f e r e n c e and a f t e r they had done t h i s , asked t h e m to give a value 0-100 to e a c h s t a t e indicating i t s "worth. I ' They w e r e to r a n k the l i s t by taking any pair of f o r e s t s t a t e s ( s u m m a r i z e d by the five data points) and decide which s t a t e they would p r e f e r the f o r e s t to be in, a s s u m i n g t h a t f r o m then on n a t u r e and m a n would be r e q u i r e d to deal n o r m a l l y with it. The value they gave to e a c h s t a t e could be derived by any reasoning they wished
save that the o r d e r i n g of p r e f e r e n c e s and of values should be the s a m e . I then used a s t a t i s t i c a l software package t o obtain r e g r e s s i o n coefficients, s e e f o r example [ 3 ] , f o r the linear f o r m u l a (0. 1) by using Egg Density, S t r e s s , P r o p o r t i o n of Old and Young T r e e s a s i n - dependent v a r i a b l e s and the value a s the dependent variable, deriving one f o r m u l a f o r e a c h of the five participants.
The f o r m u l a s I derived f r o m t h e rankings of the two F o r e s t r y Commission m e m b e r s w e r e v e r y c l o s e to the p a r a m e t e r s c . actually
1
obtained a t the meeting (despite m y misgivings) but those of the t h r e e Ecology P r o j e c t m e m b e r s w e r e quite different f r o m the other two and f r o m e a c h other.
I d i s c u s s e d with the Ecologists the r e a s o n s f o r t h e i r differences.
The feeling e m e r g e d t h a t the s t a t e s i n F i g u r e 1 w e r e meaningless because the whole f o r e s t could not be composed uniformally. Indeed, i f it were, a l l the twenty-seven s t a t e s would be equally t e r r i b l e . So I a s k e d t h e m whether they could d e s c r i b e a new s t a t e vector which would be meaningful.
2 --
The f o r e s t c o v e r s about 15,000 s q u a r e m i l e s .
Figure 1. Forest S t a t e s .
L
1 2 3 4 5
Prop. o f Young Trees
- 1 0 . 1 5
. l o
020 - 1 0
Medium
Age Trees
. 3 . 3 5 . 4 05 . 3
O l d
' Trees . 6 .5 . 5 03 . 6
S t r e s s
0 0 0 20 1 0
Density Egg
0 . 3 8 . 6 0 . 5 1 . 0 0 . 1
1 . 1 Defining a Meaningful State Description
P r o f e s s o r Holling then devised a l i s t of seven typical endemic conditions of a s u b - f o r e s t ( F i g u r e 2) together with t h e i r a p p r o p r i a t e vector s t a t e c l a s s i f i c a t i o n a s i n F i g u r e 1. Then a new l i s t was drawn up ( F i g u r e 3 ) w h e r e t h e s t a t e s of the f o r e s t w e r e d e s c r i b e d by seven p a r a m e t e r s (summing to 1 ) giving the proportion o r mix of t h e t o t a l f o r e s t i n e a c h condition category.
All four m e m b e r s of t h e Ecology g r o u p w e r e then asked f o r t h e i r p r e f e r e n c e rankings of t h e s e twenty s t a t e s . 4 In addition I calculated t h e ranking implied by the objective function f r o m the stand m o d e l used i n t h e Dynamic P r o g r a m m i n g formulation which used t h e maximization of f o r e s t profits a s the objective. This i s labeled " F o r e s t I n d u s t r y t i in F i g u r e 4 which gives the c o r r e l a t i o n between t h e five rankings. The m a r k e d clifference between the ecologists and the " F o r e s t Industry"
p a r t l y r e f l e c t s t h e f a c t t h a t t h e e x p e r t s w e r e a s k e d t o think only i n t e r m s of the i m m e d i a t e f u t u r e w h e r e a s the m e m b e r s of the Ecology group w e r e thinking of the long t e r m implications of t h e various state..,
However, t h e r e w e r e s t i l l d i f f e r e n c e s i n p r e f e r e n c e s within the group. Those of Holiing and C l a r k w e r e e s s e n t i a l l y the s a m e , though they a r r i v e d a t t h e i r o r d e r i n g s in completely different ways. Holliag f i r s t c r e a t e d seven functions v1(p1), v2(p2),
. . . ,
v7(p7) which gave h i s subjective "value" t o having a proportion p. of t h e f o r e s t i n3 1
condition i. Hence he gave a value of
to f o r e s t s t a t e 2 i n F i g u r e 3, and then used t h e s e values t o obtain his ranking. C l a r k fixed h i s sights on having about 5 -10% of f o r e s t i n condition 4 (outbreak) and on keeping the predictability of t h e f o r e s t high (by having the proportions in conditions 3 and 7 low). He was aiming f o r a manageable f o r e s t .
:;<
The Canadian F o r e s t r y Commission e x p e r t s had r e t u r n e d to Cailada.
' ~ o t e that he h a s thus m a d e s o m e assumption of independence between the p a r a m e t e r s . F o r a discussion on t h i s topic s e e Section 3. 5 i n [7].
F i g u r e 2. C l a s s i f i c a t i o n a f P o s s i b l e S t a n d C o n d i t i o n s .
.
C o n d i t i o n o f Sub Region
P o s t Outbreak Endemic
Mid-Endemic P o t e n t i a l Outbreak T r i g g e r e d Outbreak Mid- O u t b r e a k
Disaster
Budworm E x t i n c t.
S t a t e N o .
1
23
4
5 6
7
0-9 Y r s .
.5 .4
- 1 5
. 1 5
. Z
. 31 5
S t r e s s
40 0
0
0
40 . 6
0 P r o n o r t i o n
10-30 Y r s .
. 3 .4
.35
.35
.4 .4
.35
-
Eggs
.03 .03
. 0 3
2
500 100
0 30-70+
Y r s .
.2 .2
. 5
. 5
.4 . 3
. 5
Figure
3. Types Of F o r e s tMixes.
F o r e s t Mix
No.
1 2 3
P r o p o r t i o n o f Land I n C o n d i t i o n C a t e g o r y
1 2 3
4 5
6 70 -0023 -0047
0 .0061 -0122
1.0 .975 .96
0 0 0 0
.0016 .0033
0 .0083 ,0165
0 -0017 .0033
F o r e s t Rashid C l a r k H o l l i n g J o n e s I n d u s t r y
F i g u r e 4 . C o r r e l a t i o n M a t r i x .
T h i s led to a g e n e r a l discussion of what was d e s i r a b l e . P r e - dictability s e e m e d t o be one preference. 4 Another w a s a d e s i r e to take the o b s e r v e d h i s t o r i c a l budworm outbreaks o v e r t i m e ( a cycle of the f o r e s t moving through conditions 1-6 sequentially) into t h e s a m e p a t t e r n over s p a c e t h a t is, have t h e , s a m e proportion of the f o r e s t i n each condition a t any given time: "Controlled Outbreaks. ' I
It was decided that t h e seven s t a t i s t i c s used w e r e not sufficient to d e s c r i b e the s t a t e of the f o r e s t and Holling s e t t o work t o c o m e up with a m o r e comprehensive l i s t of indicators. The a i m was to d e v i s e a s y s t e m whereby we could place a decision m a k e r i n a c h a i r w h e r e h e could wave a magic wand and place t h e f o r e s t i n condition A o r condition B, where A and B w e r e d e s c r i b e d by a s e t of s u m m a r y s t a t i s t i c s . Which s t a t i s t i c s would he like t o s e e to enable h i m t o m a k e a d e c i s i o n ?
If h e w e r e a logger he would want to know the amount of wood i n good condition f o r logging and the f o r e s t ' s potential f o r the next few y e a r s indicated by the l e v e l of budworm and s o on.
F o r any given decision m a k e r we would like t o build up a s e t of s t a t i s t i c s ( i n d i c a t o r s ) which t e l l s h i m a l l ( o r virtually a l l ) t h a t he wants to know i n o r d e r to choose between A and B f r o m his point of view.
To put t h i s into p r a c t i c e one m e m b e r of the group, Bill Clark, who is well acquainted with t h e problems of the a r e a w a s appointed a s a decision m a k e r . After Holling had drawn up a long l i s t of possible
i n d i c a t o r s we t h r e e had a meeting to d i s c u s s t h i s list with Clark. Which ones was h e i n t e r e s t e d i n ?
We then r a n into a problem. When a decision m a k e r evaluates the s t a t e t h a t the f o r e s t i s i n now, he h a s t o look to the future. He h a s t o p r e d i c t how t h e f o r e s t will behave, keeping i n mind the p r e s e n t number of budworm, f o r example. Hence when he evaluates the f o r e s t condition h e a m a l g a m a t e s i n h i s mind how the f o r e s t w i l l develop i n t h e future. Now the way i n which t h e f o r e s t develops depends on the method of t r e a t m e n t , that i s , on the policies being used f o r logging, spraying and t h e like.
4~ r e c e i v e d a new perspective t o the p r o b l e m when I asked Holling why he ranked F o r e s t Mix Number 20 i n F i g u r e 3 l a s t . "Worst thing t h a t could possibly happen, ' I he said.
Now r e c a l l that we a r e looking f o r an objective function which we c a n optimize to find a best policy for treating the f o r e s t . But if the decision m a k e r had known of this "best policy" he might have evaluated the f o r e s t s differently, which changes the best policy. Right? As an example suppose that a simple device i s discovered which r e m o v e s a l l possibility of a budworm outbreak. The f o r e s t p r e f e r e n c e s of the decision m a k e r will be a l t e r e d . Although the r e s u l t of the optimization procedure m a y not be a s good a s t h i s "device" i t n e v e r t h e l e s s m a y change h i s p r e f e r e n c e s . What i s needed i s a s e t of s t a t i s t i c s such that p r e f e r e n c e s f o r t h e i r values a r e independent of the policy being used.
This was achieved by letting the decision m a k e r view a s t r e z m of s t a t i s t i c s about the conditions of the f o r e s t over a sufficiently long t i m e horizon. Hence the decision m a k e r need not p r e d i c t anything. He i s t o evaluate the s t r e a m of s t a t i s t i c s a s one single finished product and is not to w o r r y about how likely they a r e o r to wonder what policy
achieved them. Then it i s the job of the s i m u l a t o r to acljust i t s internal policies to maximize the value assigned by the decision m a k e r .
Note then that now the type of s t a t i s t i c s r e q u i r e d h a s changed. It i s not n e c e s s a r y to know t h e density of budworm a t any given t i m e ; that was only needed to get a n idea about t h e future s t a t e of the t r e e s . Since we can a l s o s e e the quantity of lumber obtained f o r the next 100 y e a r s and the amount spent on spraying, i t is i r r e l e v a n t t o know how much budworm is present. (Indeed, i t i s probably i r r e l e v a n t to know how much was spent on spraying- -a s i m p l e net profit o r l o s ~ m a y be
sufficient. )
1. 2 Finding t h e Attributes Relevant to our Decision Maker C l a r k went through Holling's l i s t of i n d i c a t o r s deleting, adding and modifying. Some w e r e d i s c a r d e d f o r being too m i n o r , that i s , not likely t o influence h i s decisions, o t h e r s because their implications w e r e too.difficult to understand ( p a r t i c u l a r l y standard deviations or? data over
space). The following l i s t e m e r g e d of s t a t i s t i c s for each year which C l a r k felt would affect h i s decisions.
F i n a n c i a l
X1 = P r o f i t of logging i n d u s t r y X 2 = C o s t of logging
X j = C o s t of s p r a y i n g
Logging P o t e n t i a l of F o r e s t
X4 = Amount of h a r v e s t a b l e wood
X5 = P e r c e n t of X4 a c t u a l l y h a r v e s t e d i n t h e given y e a r F o r e s t C o m p o s i t i o n
X6 = D i v e r s i t y , a m e a s u r e of t h e m i x t u r e of differing c l a s s e s , a g e type of t r e e s f o r r e c r e a t i o n a l p u r p o s e s . T h e h i g h e r t h e d i v e r s i t y t h e b e t t e r
X 7 = P e r c e n t a g e of old t r e e s
O b s e r v a b l e D a m a g e
X8 = P e r c e n t a g e of defoliated t r e e s X = P e r c e n t a g e of d e a d t r e e s
9
10 = P e r c e n t a g e of logged a r e a s (no t r e e s , s t u m p s , e t c . ) S o c i a l
X l l = Unemployment ( m e a s u r e d by t a k i n g a c e r t a i n logging l e v e l a s f u l l mill c a p a c i t y )
I n s e c t i c i d e
X12 = A v e r a g e d o s a g e p e r s p r a y e d p l o t .
I n a d d i t i o n t o t h e list a b o v e , a v a r i a n c e f o r t h e s e s t a t i s t i c s t a k e n o v e r t h e 265 s t a t e s w a s a l s o included i n s o m e c a s e s .
Ignoring the v a r i a n c e s for a moment t h i s s t i l l l e a v e s 12 x T s t a t i s t i c s f o r a h i s t o r y cf T p e r i o d s . Indeed, eight of t h e s e s t a t i s t i c s w e r e originally intended f o r each s i t e which would have given
(4 t 265
x
8 ) T s t a t i s t i c s .Two f i f t y - y e a r h i s t o r i e s w e r e g e n e r a t e d by t h e simulation model with a n i n i t i a l s e t of i n t e r n a l policies and t h e s e s t a t i s t i c s generated.
C l a r k studied t h e s e l i s t i n g s and, following h i s e a r l i e r p r o c e d u r e f o r o r d e r i n g the l i s t i n g 03 F i g u r e 3 , e s s e n t i a l l y Picked a few key s t a t i s t i c s which he d e s i r e d to maintain a t a c e r t a i n l e v e l and then checked t o s e e that t h e o t h e r s w e r e not s e r i o u s l y out of line.
The i d e a a t t h i s s t a g e was to give h i m a sequence of twelve o r s o such f i f t y - y e a r listillge sf s t a t i s t i c s and a s k h i m t o o r d e r them. Then he wolrld be given the complete simulation outputs and asked to r a n k t h o s e ; then the two l i s t s wou.ld be compz.red. In this way the l i s t of
s t a t i s t i c s would be modified and he would l e a r n b e t t e r what w e r e t h e i r i m p l i c a t i o n s , s o t h a t eventually he would be a b l e to a r r i v e a t the s a m e o r d e r i n g s f o r t h e complete l i s t i n g s and t h e reduced s e t of' s t a t i s t i c l i s t i n g s
.
Owing t o t h e m e c h a n i c a l difficulty of keeping IIASA's computer i n o p e r a t i o n and l a c k of t i m e t h i s was not done. F o r the s a k e of outlining t h e full p r o c e d u r e , let us assurrie t h a t t h i s was done.
We then s e t about the r e m a i n i n g l i s t of s t a t i s t i c s ( X I to X12) t o r e d u c e it to a s i z e of a t m o s t five o r six p e r y e a r .
I s u c c e s s f u l l y ai-gued t h a t s i n c e t h e potentia!ly h a r v e s t a b l e wood, potentially h a r v e s t a b l e wood h a r v e s t e d , c o s t of spraying and i n s e c t i c i d e
w4,
Xg' X,3 P X 1 2 ) w e r e given o v e r a l l p e r i o d s , if t h e s e four a t t r i b u t e s w e r e going s e r i o a s l y wrong i t would show up eventually s o m e w h e r e e l s e . The c o s t of logging could be deduced a p p r o x i m a t e l y fro-m t h e profitf i g u r e and the unemployment l e v e l (which i s proporti.onal t o wood h a r - v e s t e d ) .
T h i s left Pro.fit, Diver sit)-, Old T r e e s , Defoliation, Dead T r e e s , Logging Effects and Unemployment. It s e e m s c l e a r t h a t a l l but the f i r s t and l a s t a r e r e i a t c d to r e c r e a t i o n a l , visual and e n v i r o n m e n t a l con-
s i d e r a t i o n s . Could not t h e s e five s t a t i s t i c s be a m a l g a m a t e d into a single statistit:: oi r e c r e a t i o n ? Then we would have:
DEFOLIATION GGING OLD TRv DIVERSITY
DAMAGE STAND COMPOSITION
\/'
\VlSUA L RAT IN G
RECREATIONAL POTENTIAL
REC REATIONAL VALUE
F i g u r e 5.
P = P r o f i t
U = Unemployment
R = Recreational Value of F o r e s t a s attributes f o r each time period.
The g e n e r a l plan used by Clark for producinga r e c r e a t i o n a l index i s shown i n F i g u r e 5.
The r e c r e a t i o n a l potential is a value assigned by the Canadian F o r e s t r y Commission to each region of the f o r e s t , indicating i t s accessibility to t o u r i s t s and quality of surroundings ( s t r e a m s , lakes, gorges). Each region h a s a value 0, 30, 70, o r 100.
F o r all the a t t r i b u t e s i n F i g u r e 5 Clark divided the possible range into t h r e e classifications, f o r example, for defoliation a stand with 0- 15% defoliation was good, 15-45% medium, 45- 100% bad. Then where two a t t r i b u t e s w e r e combined in F i g u r e 5 he used the r u l e displayed in F i g u r e 6.
F i g u r e 6.
GOOD MEDIUM BAD
Hence a stand would be given a visual rating equal to the w o r s t rating of i t s components. The final composition of r e c r e a t i o n a l potential and visual rating was achieved by F i g u r e 7.
GOOD GOOD MEDIUM BAD
F i g u r e 7.
I
GOOD MEDIUM BAD
MEDIUM MEDIUM MEDIUM BAD
BAD BAD BAD BAD
0 BAD BAD BAD
3 0 MEDIUM MEDIUM BAD
70 GOOD MEDIUM BAD
100 GOOD MEDIUM BAD
B e c a u s e s o m e of t h e r e g i o n s of t h e f o r e s t a r e not s u i t a b l e f o r r e -
c r e a t i o n even under t h e best of conditions, t h e following a r e t h e n u m b e r of r e g i o n s p o s s i b l e in e a c h r e c r e a t i o n category.
0 s GOOD 38
0 s MEDIUM s 262
3 5 BAD s 2 6 5
.
Since t h e t o t a l n u m b e r of regions i s fixed (265) i t i s only n e c e s s a r y to specify two of t h e above c l a s s i f i c a t i o n s ; hence t h e final l i s t of s t a t i s t i c s to be tabulated f o r e a c h period i s :
P = P r o f i t
U = Unemployment
G = Number of Good R e c r e a t i o n a l Regions B = Number of Bad R e c r e a t i o n a l Regions.
2. A s s e s s i n p a Value Function
The a i m now i s t o d e r i v e a f o r m u l a which t a k e s t h e s t a t i s t i c s ( P t , U t , Gt
,
Bt) t = 0, 1, 2. . . ,
and p r o d u c e s a value V s u c h t h a t if f o r e s t h i s t o r y a is p r e f e r r e d t o f o r e s t h i s t o r yP
t h e nOver r e c e n t y e a r s a g r e a t d e a l of r e s e a r c h h a s gone into devising good techniques f o r the a s s e s s m e n t of value functions [7, l o ] . T h e s e
techniques w e r e not t r i e d on t h i s problem. At t h e t i m e of t h e study t h e methodology g r o u p a t IIASA w a s experimenting with l i n e a r p r o g r a m m i n g (L. P. ) s o f t w a r e and w a s e a g e r for e x a m p l e s with which to work. I combined our two a i m s and used t h e following l i n e a r p r o g r a m m i n g a p p r o a c h t o find value functions.
C o n s i d e r a value function V having two v a r i a b l e s x , y. Suppose t h e d e c i s i o n m a k e r h a s s a i d t h a t i n the following p a i r s t h e f i r s t one i n e a c h i s p r e f e r r e d by h i m t o the second:
Thus
V(3, -7)
-
V( 1, 1) > 0 andV(0, 2) - V(-1, 2) > 0 ,
Suppose we approximate V with a quadratic polynomial
2 2 .
V(a, y) = ax t by
+
cxy+
dx+
ey,
then we have that
a r e n e c e s s a r y r e q u i r e m e n t s for V to be a valid function. Examples of polynomial e x p r e s s i o n s whose coefficients satisfy (2. 2) a r e :
By obtaining m o r e p a i r s of p r e f e r e n c e o r d e r i n g s , the s e t of possible coefficient values ( a , b, c , d, e ) m a y be reduced, for example, i f we now find that in addition
then only the f i r s t of t h e t h r e e examples above i s s t i l l valid.
If t h e r e a r e many alternative value functions for a given data s e t a n L. P. algorithm will a r b i t r a r i l y choose one of t h e m unless i t i s given s o m e selection c r i t e r i o n . Supplying an objective function for the L. P.
problem gives the advantage that with t h e s a m e data s e t t h e L. P. will
always choose the s a m e value function; hence a s the data s e t a l t e r s slightly (because of new o r d e r i n g s ) i t i s e a s i e r to s e e i t s effect on the resulting value function.
Note that if ( a , b, c , d, e ) i s a solution of ( 2 . 2) then so i s any positive multiple of it; hence the a r b i t r a r y constraint
was added to bound the problem. 5
The objective c r i t e r i o n used was t o maximize the minimum gap between p r e f e r e n c e rankings. In t h e exzmple used above the gaps between the left hand side of (2. 1) and the r i g h t hand side ( z e r o ) using V1 a r e 35, 26, 2; f o r V2 a r e 24, 51, 1; and f o r V3 a r e 30, 40, 1.
Hence the minimum gap i n each i s 2, 1, 1, and s o the maximum m i n i m u m gap i s 2 and V 1 would be the p r e f e r r e d polynomial f r o m that list.
In g e n e r a l , f o r a, l i s t of p r e f e r e n c e s
(> r e a d s "is p r e f e r r e d t o 1 ' ) the full l i n e a r p r o g r a m would be
s a; = Max s
Note that a valid function e x i s t s if and only if s* > 0 . If s': 0 the decision m a k e r would be questioned inor e closely on doubtfl.11 o r d e r i n g s , o r if he i s r e s o l u t e , a higher o r d e r approxima,tion should be taken.
Returning to our study, with four a t t r i b u t e s (P, U, G, B) per t i m e period two qualitative assumptions w e r e made by C l a r k (with my prompting) that w e r e felt to be reasonable ( i n the f i r s t c a s e ) o r
n e c e s s a r y ( i n t h e second).
5 ( a 1 m e a n s t a if a > 0, - a if a < 0 .
a ) P r e f e r e n c e s for profit and unemployment were
'
tindependent"f r o m those of recreation. That i s , the relative orderings of (P, U ) pairs were independent of the level of the recreation so long a s i t was the same in each case.
6
The r e v e r s e was also felt to be t r u e , that preferences for recreational alternatives were independent of profit/unemployment levels so long a s these remained constant.b) C l a r k ' s preferences for profit and unemployment levels in a year depended on what those levels were l a s t year and would be next year. F o r example, a drop in profits to gain fuller employ- ment i s not too serious i f compensatingly l a r g e r profits a r e made in the surrounding years. Also, a n unemployment level of 10%
i s worse i f i t follows a year of full employment than i f it follows a year of 10% unemployment; that i s , he p r e f e r s a steady level to one which oscillates.
Clark felt that if we replaced P a s a statistic by t
we might better justify a separable value function such a s
where Vt i s a value function based on the figures for year t alone.
These assumptions enabled us to work with a value function
allowing us t o calculate a value function f o r recreation independently of that for profit and unemployment.
Figure 8 shows the rankings given by Clark for the two value functions X, Y for any time period. Note that for (Q, U ) i t i s an
'preferential Independence, s e e Chapter 6 of [7].
o r d e r e d l i s t a n d t h e r a n k i n g s f o r r e c r e a t i o n i n c l u d e s o m e e q u a l i t i e s . T h e l a s t t h r e e i n t h e r e c r e a t i o n list w e r e a d d e d w h e n I d i s c o v e r e d t h a t t h e f i r s t p o l y n o m i a l e x p r e s s i o n w a s n o t s u i t a b l y m o n o t o n i c f o r e x t r e m e v a l u e s .
T h e s e r a n k i n g s p r o d u c e t h e folLowing v a l u e f u n c t i o n s , a q u a d r a t i c a n d a c u b i c p o l y n o m i a l a p p r o x i m a t i o n being u s e d r e s p e c t i v e l y .
a n d
2 2
Y = (71.8 - 1.88G)G
-
B (5.88 t .00134B)+
GB(19.63-
0 . 5 9 7 G t 0.185B)F i g u r e 8
T h e n C l a r k g a v e t h e following o r d e r i n g s f o r s e t s of a l l f o u r a t t r i b u t e s ( F i g u r e 9 ) . T h e g r o u p s a r e 1j.sts w i t h e a c h m e m b e r of a g r o u p being p r e f e r r e d t o t h e o n e below it.
5, 4, I69 50
5, 7, 16, 3 0
5, 0, 16, 100
5, l o , 16,
o
0, l o , 16,
o
F i g u r e 9.
With t h e a i d of t h e f u n c t i o n s X, Y t h e s e l i s t s m a y be r e d u c e d t o lists of two a t t r i b u t e s ; f o r e x a m p l e t h e f i r s t list b e c o m e s :
T h e cubic a p p r o x i m a t i o n t e c h n i q u e w a s u s e d a g a i n t o find a c o m b i n e d value f u n c t i o n of
- 9 , 0 5 3 ~ L - 3 , 0 3 9 , 5 0 0 ~ - 1 9 5 , 1 9 7 ~
2. 1 The T i m e P r o b l e m
So f a r t h e a n a l y s i s h a s reduced the simulated h i s t o r y of t h e f o r e s t into a t i m e s t r e a m of values, one per year. F o r two simulated
h i s t o r i e s with output values
and
it i s reasonable to suppose that the decision m a k e r p r e f e r s the f i r s t
1 2
h i s t o r y to the second if V 2 Vk f o r a l l k and if t h i s ineqoality i s
7 k
s t r i c t f o r s o m e k.
But it i s not possible a t t h i s stage f o r t h e analyst to s a y whether C l a r k would p r e f e r a five y e a r h i s t o r y
to one of
(2, 3, -1, 4 , 8 )
because we have no r u l e s f o r i n t e r t e m p o r a l trade-offs. The onljr manageable model f o r s u c h t r a d e - o f f s i s a linear assumption t h a t
v
= a t V(Qt, U,, Gt, Bt)f o r s o m e coefficients a where presulmabljr a r a 2 0 f o r a l l t.
t ' t t t l
I Even t h i s dominance argument i s ooly valid because we a r e a s s u m i n g that t h e r e a r e no i n t e r p e r i o d dependencies of p r e f e r e n c e s . F o r example we could imagine that the 5 year s t r e a m
( 1 , 2 , 3 , 4 , 5) would be p r e f e r r e d to
( 1 0 , 9 , 8 , 7, 6 )
if the decision m a k e r a b h o r r e d a d r o p f r o m one period to the next.
Had t i m e permitted we could have found viable values for the 8 coefficients a by using the s a m e technique which led to the coefficients
t
i n the second value function
However, a t that stage we agreed that the simulation model should generate different histories using a variety of policies and calculate the value
for a range of constants a, 0 < a < 1 .
3. The A s s e s s m e n t of a Utilitv Function
Even i f we ignore the crude manner i n which the t i m e s t r e a m s of the attributes w e r e evaluated t h e r e remains another important element i n the effective evaluation of policies by use of an objective function.
The particular h i s t o r y generated by the simulator depends upon the initial condition of the f o r e s t , the many complex equations governing the growth of budworm, t r e e s , the effects of p r e d a t o r s and other f a c t o r s , but a l l of these a r e deterministic only if the weather pattern i s known.
Different weather patterns will produce different h i s t o r i e s and hence a policy cannot be judged purely on the r e s u l t s of one run, i t s effects must be considered under a l l types of weather futures. Fortunately, this problem m a y be overcome if a utility function i s used r a t h e r than a value function. A utility function not only h a s the properties of a value function, but in situations in which outcomes a r e uncertain, i t s expected value provides a valid quantity for making rankings.
That i s , if u(P, W ) r e p r e s e n t s the utility (or value) of the f o r e s t history which r e s u l t s f r o m using policy P when weather h i s t o r y W o c c u r s and f(W) i s the probability that weather pattern W does occur then
u p , W ) f ( W ) = V ( P )
where the sum i s taken over a l l possible weather patterns, i s a legitimate value function over policies Po
ill
Clark r e t u r n e d to Canada in July 1974.A s s e s s m e n t p r o c e d u r e s for utility functions a r e s i m i l a r to t h o s e for value functions except that i n addition, the decision m a k e r ' s
attitude towards r i s k taking m u s t be incorporated. As with value functions i t i s useful to recognize assumptions that will break down
the a s s e s s m e n t of one function with many a t t r i b u t e s into one of a s s e s s i n g s e v e r a l utility functions each having a t m o s t one or two attributes.
One such assumption i s utility independence. F o r a utility function u(x, y), where x and y might be vectors of a t t r i b u t e s , if t h e decision m a k e r ' s attitude towards r i s k taking in situations w h e r e only the outcome of x i s uncertain but y i s fixed and known, i s independent of what that fixed value of y i s , then a t t r i b u t e s X a r e said to be utility independent of Y. It i s i m p o r t a n t to r e a l i z e that X m a y be utility independent of Y even if X and Y involve f a c t o r s which in other r e s p e c t s a r e closely related. F o r m o r e information and examples s e e Chapter 5 of Keeney and Raiffa [7]. The functional statement of t h i s p r o p e r t y i s that f o r any
1 2
two values of Y, y and y say,
for some constants a and b, w h e r e b m u s t be positive.
In our problem which h a s four a t t r i b u t e s per y e a r , with a horizon of T periods ( T will be in the range 50-200) we r e q u i r e a utility function of 4T a t t r i b u t e s s o that some extensive assumptions will be required.
Meyer [9] f o r example h a s shown that f o r a utility function u ( x l , x 2 ,
. . . ,
xT)if each subset of a t t r i b u t e s [X 1,
. . . ,
Xt] is considered t o be utility independent of [Xt+ l ,. . . ,
XT] and vice v e r s a , then the utility function h a s either an additive f o r mf o r some positive constants a or a multiplicative f o r m t
for s o m e constants b and c where in each c a s e u ( x ) i s a utility
t t ' t t
function over X alone.
t
T h e s e f o r m s w e r e i n a p p r o p r i a t e for o u r c a s e p r i n c i p a l l y b e c a u s e C l a r k ' s a t t i t u d e t o w a r d s r i s k taking f o r l e v e l s of unemployment in one p e r i o d depended on t h e l e v e l s of unemployment in t h e y e a r b e f o r e , and t h e y e a r a f t e r , and hence M e y e r ' s a s s u m p t i o n s of utility independence did not apply.
Not only t h a t but C l a r k wished t o m a k e a n a s s u m p t i o n of s t a t i o n a r i t y ( s e e Koo9mans [ 8 ] ) t h a t i s , h e wished t o t r e a t a l l y e a r s e q u a l l y , both with r e g a r d t o value o r d e r i n g s and i n r i s k taking. T h i s m e a n t t h a t t h e
c o e f f i c i e n t s a
t 7 bt' C t a n d the functions u would a l l be independent of t
t h e i r suffix t implying t h a t all. t i m e s t r e a m s which w e r e m e r e l y p e r - m u t a t i o n s of o n e a n o t h e r would be a s s i g n e d e q u a l utility, which w a s not t h e c a s e . F o r e x a m p l e , dealing only with l e v e l s of e m p l o y m e n t he
p r e f e r r e d the s t r e a m (100, 100, 90, 90, 100) t o (100, 90, 100, 90, 100) b e c a u s e of the r e d u c e d v a r i a n c e between y e a r s . 9
F i s h b u r n [4] u s e d a s s u m p t i o n s c a l l e d M a r k o v i a n dependence t o p r o d u c e a f o r m
w h e r e u (x
,
x ) i s a utility function o v e r the two a t t r i b u t e s x xt t t t l t' t t 1 '
Whilst t h i s d o e s a l l o w f o r s o m e i n t e r d e p e n d e n c y between a t t r i b u t e s i n neighbouring p e r i o d s C l a r k w a s quite f i r m in p r e f e r r i n g the l o t t e r y
t o t h a t of
9 ~ switched f r o m t a l k i n g in t e r m s of unemployment t o e m p l o y m e n t s o e t n a t t h e s y m b o l u woald n o t be c s e d s i m u l t a n e o u s l y f o r utility and a l e v e l of unemployment. E = 100 - U i s the new a t t r i b u t e .
t t
w h e r e t h e f i g u r e s a r e p e r c e n t a g e of employment in 5 s u c c e s s i v e y e a r s . F o r (3. 3) to be valid f o r C l a r k ' s p r e f e r e n c e s h e should have been
indifferent between the two l o t r e r i e s .
3. 1 Finding A p p r o p r i a t e Assumptions
To find a functional f o r m that would be a c c e p t a b l e to h i m I con- s i d e r ed a s s umptions involving conditional utility independence. T h i s condition s a y s , i n e s s e n c e , t h a t if the s e t of a t t r i b u t e s i s divided into t h r e e p a r t s X, Y and Z then X i s conditionally utility independent of Y i f whenever Z i s fixed a t s o m e level and we r e g a r d t h e problem a s now only having two a t t r i b u t e s X and Y then X i s utility indepen- dent of Y and t h a t t h i s i s t r u e for a l l fixed values of Z. F o r m o r e detailed expositions of t h i s concept s e e Chapter 6 , Keeney a n d Raiffa [7]
o r B e l l [2].
The idea was t o a s s u m e t h a t e a c h s u b s e t
[ x l , . . .
, X t - l } w a s conditionally utility independent of{xtt . . . ,
X T ] and vice v e r s a . T h i s i s quite s i m i l a r t o t h e a s s u m p t i o n s used by Meyer t o obtain (3. 1 ) and (3. 2) but d o e s not m a k e any a s s u m p t i o n of independence of p r e - f e r e n c e s f o r X o n e i t h e r X t - lt o r X
t t l '
T h e s e a s s u m p t i o n s led ( f o r T 2-4) t o t h e r e s u l t t h a t e i t h e r
w h e r e X i s a constant and
0 0 0 0 0
ut(xt,xtt
=
u ( x l , x 2 ,. . ,
X t - 1 , X t , X t f 1'Xtt2,. . ,
x T )w h e r e xo i s a n y fixed value of X , , so that f o r e x a m p l e
0 l 0 1'
Ut(:t7 X t t 1?= Ut+ 1 ixtt 1
'
X t t 2 ) ' and w h e r e u was s c a l e d s o t h a tu u
u ( x l , .
. .
, x T ) = 0. F o r a proof of t h i s r e s u l t s e e B e l l [2].
Note that ( 3 . 4 ) i s exactly (3. 3) but that (3. 5 ) not only allows interperiod dependencies but a l s o i s able to differentiate between the l o t t e r i e s L1 and L 2
.
Bill C l a r k r e t u r n e d to UASA for the s u m m e r of 1975 and I quizzed h i m on the a p p r o p r i a t e n e s s of the assumptions which led to (3.5). He a g r e e d t h a t they seemed appropriate and so we proceeded to a s s e s s h i s utility function over the attributes [Pt, E t , G t
,
B t), t = 1,. . . ,
T.Questioning soon established that h i s p r e f e r e n c e s f o r the r e c r e a t i o n t i m e s t r e a m s ( G I , B1, G2, B E , .
. .
,%,BT] w e r e mutually utility inde- pendent with those of profit and employment { P 1 , E I , P 2 , E 2 , .. . ,
P T , E T ) , enabling us to use the formula ( s e e Keeney [ 6 ] )where uR i s a utility function for r e c r e a t i o n and u a social utility S
function, k l and k2 being constants, k l a t e r being identified a s z e r o . 2
I should emphasize that C l a r k was not one to m a k e assunlptions out of expediency, whenever he agreed that a n assumption was valid, we had d i s c u s s e d the implications a t length and verified that his p r e f e r e n c e s reflected the r e q u i r e d pattern o r w e r e sufficiently close.
3 . 2 The Utility Function f o r R e c r e a t i o n
F o r the r e c r e a t i o n s t r e a m s , he felt that the assumptions of Meyer w e r e appropriate and i n addition that in any given t i m e period G and
T BT w e r e mutually utility independent. To d e t e r m i n e whether the
additive f o r m (3. 1 ) or multiplicative f o r m (3. 2) was the a p p r o p r i a t e one to u s e I a s k e d h i m if he had any p r e f e r e n c e between the following two l o t t e r i e s
where G1 and G2 a r e the number of good r e c r e a t i o n a l a r e a s i n two s u c c e s s i v e y e a r s and B1 and B2 a r e i n a l l outcomes a s s u m e d to
be fixed. l o If t h e additive f o r m (3. 1) was t o be appropriate he should have been indifferent between the two but i n f a c t he p r e f e r r e d t h e second l o t t e r y on the grounds that h e was v e r y a v e r s e t o having two v e r y bad y e a r s together. This meant that the f o r m of the r e c r e a t i o n a l utility function was
where the various constants a r e independent of the t i m e s u b s c r i p t because of the assumption of stationarity.
The m a r g i n a l utility functions u and u for the number of
G B
good and bad a r e a s w e r e a s s e s s e d i n the usual manner ( s e e f o r example Raiffa [ l o ] ) by asking questions of the f o r m "what value G = g
*
f o r c e r t a i n do you f e e l i s equally p r e f e r a b l e to a 50-50 gamble between G = 20 and G = 5?11Thus u ( g ) was a s s e s s e d a s i n F i g u r e 10 which was fitted quite G
closely by the exponential curve uG(g) = 1
-
exp(-0. 08g): The function u ( b ) was slightly m o r e complicated ( F i g u r e 11) being fitted i n two B pieces by.u (b) = -0. 3
+
0. 35(+1.463+
28. 222 exp(-0. 0164b)) BThe constants k l , k2, k3, k4 w e r e calculated by fixing k l f k2uG(40)
+
k3uB(o)+
k4uG(40h-lB(0) =and
and then using indifferent p a i r s given by C l a r k
l o R e c a l l that because G and Bt a r e mutually utility independent it i s not n e c e s s a r y t o speciry a t what level B1 and B2 a r e fiued.
Figure 10. Utility Function for the Number o f Good Recreational Areas.
Figure 1 1 . Utility Function for the Number o f Bad Recreational Areas.
to f o r m t h r e e m o r e equations in t h e k i l s Taking a l l the combinations of two p a i r s f r o m (3. 9 ) together with (3. 8) provided t h r e e solutions f o r the k ' s which a r e exhibited i n F i g u r e 12.
F i g u r e 12
The pair l t 3 s e e m e d to be the l e a s t r e l i a b l e of the t h r e e since it i n - volves two p a i r s that a r e quite s i m i l a r . Also since C l a r k always p r e - f e r s to i n c r e a s e the number of good a r e a s i f possible, the constraint
P a i r l t 2 2 t 3 l t 3
should be t r u e for a l l b, and f o r s i m i l a r r e a s o n s a l s o k2
-.
60-.
57- . 4 5 1
-.
48-.
55-.
71although since k3 and k4 a r e positive t h i s i s not important. The s m a l l e s t value of u ( b ) is 0. 32 and hence the coefficients should be B chosen so that
which none of the solutions i n F i g u r e 12 satisfy. However extrapolating the f i r s t two s e t s of coefficients until (3. 10) was satisfied gave
coefficients of
k 3
.
15. 1 7 . 2 2
and the implied utility function using t h e s e coefficients made a l l of the equivalences i n (3. 9 ) hold a l m o s t exactly!
k4 1.42 1.37 1 . 2 3
To finalize the r e c r e a t i o n a l utility function now only r e q u i r e d t h e knowledge of a and Q i n (3. 7).
F o r this I asked h i m t o consider a t i m e s t r e a m in which a l l values a f t e r year 2 a r e a s s u m e d fixed and that the number of bad a r e a s i s fixed a t 100 f o r y e a r s 1 and 2. So, considering only v e c t o r s of the type (number of good a r e a s i n year 1, number of good a r e a s in year 2) he was t o give values g l , g2, g3, g49 g5 s u c h that
His a n s w e r s w e r e 9, 10, 12, 14 and 15 respectively. In attempting t o solve (3. 7) with t h i s information it became c l e a r that i n f a c t the
additive f o r m (3. 1 ) r a t h e r than the multiplicative f i t s (3. 12). Referring t h i s apparent inconsistency back to C l a r k we established t h a t h i s p r e - f e r e n c e between the l o t t e r i e s L and L4 was caused by the r a t h e r
3
e x t r e m e n a t u r e of the consequence i n L of two s u c c e s s i v e y e a r s with 3
z e r o good r e c r e a t i o n a l a r e a s . When I r e p l a c e d t h e z e r o s in L 3 and L4 with something positive he became indifferent. P e r h a p s t h i s should indicate a singularity i n the function u a t G = 0 but I chose t o ignore
G this.
Thus t h e r e c r e a t i o n a l utility function was established a s
3. 3
-
The Social Utility FunctionC l a r k having accepted the conditional utility independence assumptions n e c e s s a r y to validify t h e use of equation (3. 5 ) we chose
0 0
fixed l e v e l s of p = 0 million d o l l a r s p e r y e a r and e = 100 p e r c e n t
t t
employment. The m a i n t a s k was t h u s to a s s e s s , f o r e a c h t = 1 2,
. . . ,
T-
1,
the functiono r i n a shorthand notation w h e r e we o m i t explicit r e f e r e n c e of a t t r i b u t e s a t t h e i r fixed values, uS(pt, et, ptC1, et+l). Whilst previous assumptions about independence between a t t r i b u t e s had e i t h e r appeared f r o m
questioning o r had been prompted by m e , on t h i s occasion C l a r k volunteered the information t h a t when considering h i s p r e f e r e n c e s for employment i n a given y e a r , h e was only concerned with the l e v e l s of profit in t h e s a m e year and the levels of employment i n t h e previous and l a t e r y e a r , and that h i s p r e f e r e n c e s f o r profit i n a given y e a r depended only upon the level of employment in t h a t y e a r . This implied t h a t for t h e a t t r i b u t e s Pt,Et,
q+l,
Et+l we could a s s e r t t h a t7
was mutually conditionally utility independent with P and Et+l and s i m i l a r l y t h a tt+ 1
%+
was mutually conditionally utility independent (m. c. u. i. ) with P t and Et. This s e t of additional assumptions proved to be m o s t useful.Consider the assumptions leading to (3.4) and (3. 5) for T = 4. In full they a r e
and
(x1,x2]
m . c . u . i . X4Those that C l a r k had proposed w e r e
I
and
showing t h a t ( 3 . 4 ) o r (3. 5) was appropriate for the r e s t r i c t e d function us(pt, et' Pt+
'
et+ l ) .It i s e a s y to show ( s e t a l l a t t r i b u t e s a t t h e i r fixed level except f o r p2) that the assumption of stationarity f o r c e s uS(pt, et, pt+ l , e t+ ) and t h e full function uS(p, e ) e i t h e r both t o be additive o r both to be
-
multiplicative and i f multiplicative to have the s a m e p a r a m e t e r X. The non indifference between l o t t e r i e s L1 and L 2 showed the multiplicative f o r m to be the a p p r o p r i a t e one. Hence using a l l the d e c l a r e d independence
a s s u m p t i o n s , the s o c i a l utility function could be e x p r e s s e d a s
T T - 1
f o r s o m e constant A , w h e r e u
A and u% a r e each two a t t r i b u t e utility 0
A t ' t t, e t t l ) = u ( e o e ). Thus the functions for which u ( p e ) = u ( e
B t-1' t a s s e s s m e n t p r o b l e m r e s t e d o n f i n d i n g u u and A .
A' B 3. 3. 1 The I n t e r ~ e r i o d E m ~ l o v m e n t Function
We began with u R e c a l l that uB(et, et+ ) i s , i n effect,
0 0 0 B' 0 0
uS(2 , e l 7 .
. ,
e t-
l , et, et+ l , ett2,. . . ,
e T ) SO that when questioned about h i s p r e f e r e n c e s he w a s to c o m p a r e employment s t r e a m s of the f o r m (100, 100, .. . ,
100, et, ett l , 1 0 0 , .. . ,
100). I proceeded by fixing the l e v e l of E t a t s o m e valuee
and then a s s e s s i n g t h e one a t t r i b u t et
function u ( e 1 t t l
I
E t =e
t ). It a p p e a r e d t h a t f o r l o t t e r i e s involving l e v e l s of E t t l t h a t w e r e higher thanet
he was r i s k a v e r s e but was r i s k prone f o r l e v e l s of Et t l lower than
e
The r e a s o n was that t h e t 'previous y e a r ' s employment level r e p r e s e n t e d a goal o r a s p i r a t i o n level f o r the p r e s e n t y e a r , p a r t of h i s d e s i r e f o r stability i n employment levels. The only d e p a r t u r e f r o m t h i s was that if I fixed
et
a t anything higher than 80% he was n e v e r r i s k prone f o r values of e 2 80, becauset t l
a n y y e a r i n which employment was a t l e a s t 80% was "satisfactory".
Hence h i s "goal" was min{et, 801
.
A typical g r a p h of u l ( e t t l ( E t =et)
i s shown i n F i g u r e 13.
The two piece function w a s fitted again quite c l o s e l y by a n exponential c u r v e of the f o r m
- e x p t - 0 . 0 3 e }
t t l e 2 m i n { 8 0 , e t } t t l
and
- exp { + 0 . 0 3 e t t l } e 5 min{80,et}
t t l
In a s i m i l a r way u (e
I
E t t l = - e ) was a s s e s s e d , exhibiting much2 t t+ 1
the s a m e f e a t u r e s . Since E t - was fixed a t 100 t h e r e was a d e s i r e to achieve t h i s goal with e t but this was t e m p e r e d by the opposite d e s i r e not to exceed
e .
The r e s u l t s e e m e d t o be that C l a r k p r e -t+ 1
f e r r e d the p a t t e r n (100, 8 0 , 9 0 , 100) to (100,90, 80, 100); that although a d r o p f r o m 100 to 80 was s e r i o u s it was better to suffer that and follow i t with two y e a r s of improvement than be faced with two y e a r s of falling employment, even though t h i s was ultimately followed by a n i n c r e a s e f r o m 80 t o 100. The function u ( e
/
Ettl =-
e ) was of the2 t t + 1
f o r m
-
exp { - . 0 3 et) e 2 min( 80, ettl-
- 5)t (3. 16a)
and
-
- exp { + . 0 3 e t ) e 5 min(80, e t t l
-
5 )t (3. 16b)
That the exponential coefficients i n (3. 15) and (3. 16) a r e a l l shown equal to 0. 03 was because
they
w e r e a l l f a i r l y close and the implicationss e e m e d insensitive to t h i s p a r a m e t e r .
T o obtain the combined function u ( e
B t 7 e t + l ) I used t h e f a c t that
and
uB(et, e t t l ) = N e t )
+
Wet) u l ( e t t l l E t = e t )f o r s o m e functions f, g, h, k, and solving t h e s e gives
where a a Z , a 3 , a4 w e r e constants calculated in much the s a m e m a n n e r a s the constants k k Z , k g , k4 w e r e f o r the r e c r e a t i o n a l
4 0 function, and e
t
#
et was any constant, chosen t o be 50.F i g u r e 13. Utility Function for Employment, Conditional o n t h e Level of
Emnloyment in the Previous Year.
Figure 14. Profit/Employment Indifference C u r v e s for a Single Year.
3. 3. 2 The P r o f i t - Employment Tradeoffs
The next s t e p was to calculate u ( p e ). This could have been A t' t
done in the s a m e way a s f o r u but Clark found it e a s i e r to think in B
t e r m s of indifference c u r v e s between p t) e t p a i r s . Hence on g r a p h paper with a x e s of e f r o m 50 to 100 and of p t f r o m -10 to
+
30 wet l 1 2 2
located on it p a i r s ( p t , e t ) , ( P t ' e t ) between which C l a r k was in- different, again bearing i n mind that a l l other a t t r i b u t e s w e r e a t t h e i r fixed levels, and then fairing in sample indifference c u r v e s . The r e s u l t i s exhibited in F i g u r e 14.
Wh.at was delightful to m e a s the analyst was that if we d e s c r i b e t h e above indifference c u r v e s by the functional relationship
@ ( p t, et) = constant
f o r varying constants it was e m p i r i c a l l y observable that
for a l l values of e
.
T h i s meant t h a t quantification of C$ was easy. I used a polynomial c u r v e fitting p r o g r a m on one of the indifference c u r v e s , finding that a quadratic w a s sufficiently a c c u r a t e and by substituting i n the other c u r v e s confirmed d i r e c t l y the visual observation that p r o p e r t y( 3 . 20) held. 11
The indifference c u r v e s w e r e
$ ( p , e t t ) = et
+
1 . 9 pt-
0.04 p t 2 = constant.
The next a s s e s s m e n t t a s k was now to a s s e s s a utility function o v e r
,
the value function. Using t h e indifference c u r v e s , any p a i r ( p e )
>:c
*
t' tcould be r e p l a c e d by a n equivalent pair ( p 100) where @ ( p t , 100) =
t '
l l F o r aiq example of t h i s p r o p e r t y i n connection with t i m e s t r e a m s s e e Bell [1]
.
The one d i m e n s i o n a l utility function u ( p
A t' 100) had e a r l i e r been a s s e s s e d i n the u s u a l m a n n e r i n the r a n g e - 8 t o
+
26, the r e s u l tdepicted i n F i g u r e 15.
R e c a l l t h a t if C l a r k h a s been c o n s i s t e n t we should be able t o o b s e r v e t h a t u (0, e t ) = uB(et, 100). As a check I calculated the
A
i m p l i e d function u ( p 100) using uB(et, 100) and @ ( p t , e t ) . Actually, A t'
c o m p a r i s o n was o n l y p o s s i b l e between - 8 5 p t 5 0 but h e r e the
a g r e e m e n t w a s close. The f u l l implied function u A ( P t' 100) i s shown i n F i g u r e 16 f o r p c 0.
Note t h a t b e c a u s e @ ( - 9 . 15, 100) = @(0, 80) the implied function u A ( ~ t , 100) becomes r i s k prone f o r pt c - 9. 15. F r o m a consistency c h e c k point of view we w e r e p e r h a p s f o r t u n a t e t h a t t h e d i r e c t a s s e s s m e n t of uA(Pt, 100) did not involve a r a n g e t h a t low!
F o r l a t e r calculations the value of u A ( p t , e t ) w a s t a k e n to be
w h e r e
'#
@ ( p t , 100) = @ ( p t , e t ) and
f o r @ ( p t , et) < 100 w h e r e
~ ( 0 , e r ) = a p t , e t )
.
The functions u a n d u w e r e s c a l e d s o t h a t u (0, 100) = u (100, 100) = 0
A B A B
a n d u (0, 50) = uB(50, 100) = u (100, 50) = -1.
A B
3. 3. 3 Evaluatine t h e Constant X
T o complete t h e a s s e s s m e n t of u ( p , e ) it r e m a i n e d to calculate X , S -
-
t h e constant i n equation (3. 14). What t h i s constant c o n t r o l s i s t h e d e g r e e t o which the d e c i s i o n m a k e r p r e f e r s a m i x t u r e of good y e a r s and bad y e a r s t o a p p e a r i n bunches o r i n t e r s p e r s e d . So I began by asking C l a r k if he had t o a r r a n g e 50 good y e a r s a n d 50 bad y e a r s i n a
sequence of 100, how would he do it ? R e c a l l t h a t if we w e r e not using
F i g u r e 1 5 . l d a r g i n a l U t i l i t y F u n c t i o n f o r a S i n g l e Y e a r ' s P r o f i t s .
F i g u r e 1 6 . T h e M a r g i n a l U t i l i t y F u n c t i o n f o r P r o f i t I m p l i e d by F i g u r e s 1 3 a n d 1 4 .
functions with i n t e r p e r i o d dependencies such a question would not a r i s e s i n c e a l l permutations would be equally p r e f e r r e d since C l a r k i s
adopting a "no d i s c o u n t i n g t f policy. He c e r t a i n l y disliked both the options i n which good and bad a l t e r n a t e d and i n which a l l 50 good y e a r s c a m e together. As 1/A b e c o m e s l a r g e r t h e tendency i s for the
utility function to p r e f e r s m a l l e r blocks and a s it becomes s m a l l e r (and negative) to p r e f e r t h e l a r g e bunching.
I asked C l a r k to consider the following four s t r e a m s of seven year employment f i g u r e s
(ii) 100, 70, 70, 70, 70, 100, 100 (iii) 100, 70, 70, 70, 70, 70, 100
and t e l l m e what s t a t e m e n t s he could make regarding h i s p r e f e r e n c e s between them. He e s t a b l i s h e d t h a t ( i ) was the best, (iii) the w o r s t and f e l t t h a t ( i v ) w a s p r e f e r a b l e to (ii) "if anything. "
I d r e w the following graph ( F i g u r e 17) which shows the utility of ( i ) fixed a t 1, the utility of (iii) fixed a t z e r o and the corresponding utilities of (ii) and ( i v ) a s functions of 1/A using ( 3 . 14). Note that 1/A = 0 c o r r e s p o n d s t o t h e additive c a s e .
The n e a r indifference of (ii) and (iv) suggested t h a t l / h should be chosen to be about 1 but t h e r e w e r e other considerations. In o r d e r t o avoid discontinuities i n u S ( p , 2) it m u s t be t h e c a s e t h a t A t u B ( e t y e t t l ) and A
+
uA(pt, e t ) a r e e i t h e r always both negative o r always bothpositive. Lf both positive then
A 2 m a x [ -uB(50, 50), - u A ( - 2 0 , 5011 i s a constant and if both negative then
C l e a r l y it i s t h e f o r m e r c a s e which i s a p p r o p r i a t e h e r e and so A 2 1.487 o r 1/A 5 . 6735. F r o m F i g u r e 17 t h i s m e a n s t h a t to be consistent, C l a r k
F i g u r e 1 7 . T h e R e l a t i v e U t i l i t i e s o f F o u r T i m e S t r e a m s a s .i V a r i e s .
