A cardinal utility approach for project evaluation

SOCM-Econ. Plan. SCI. Vol. 8. pp. 329-338

Pergamon Press 1914. Prmted in Great Britain.

A CARDINAL UTILITY APPROACH PROJECT EVALUATION*

FOR

J. K. STANLEY Commonwealth Bureau of Roads, Melbourne, Australia 3001 (Received 25 February 1974)

The paper describes a technique whereby individual preferences for monetary and non-monetary consequences of public projects might be combined into an aggregate impact index. The technique is built up from an attribute-based mode1 of consumer choice. It can be used as a complete evaluation tool, having certain similarities with matrix evaluation techniques, or as a method of imputing monetary values to particular non-monetary effects.

A CARDINAL UTILITY APPROACH FOR PROJECT EVALUATION It has recently been argued that project evaluation should encompass all project effects consistent with particular sets of moral notions.? Given such a set of

moral notions, project effects might generally be classified according to whether they are (i) market impacts (ii) physically quantifiable non-market impacts (iii) ‘intangible’ effects having no obvious physical or monetary dimension. Kaldor-Hicks hypothetical compensation approaches to cost-benefit analysis concentrate on market impacts, with some progress being made in certain fields towards inputing monetary values to particular physical non-market effects (e.g. valuation of private time savings). ‘Management science’ type approaches to C.B.A. might add ‘postulated prices’ for certain effects of type (ii) (e.g. lives saved or lost).$ In general, however, the Lichfield Planning Balance Sheet type of approach to evaluation is about as far as C.B.A. has gone with effects of types (ii) and (iii),0 subjective trade-offs being required to rank alternatives. * This paper was largely written while the author was on study leave at the University of Southampton. Thanks are due to the Bureau of Roads, Melbourne, Australia, for making this possible. Helpful comments have been received from Christopher Nash, David Pearce, Peter Simmons, Mike Common, Tony Flowerdew, John Hargreaves and David Hensher, although none of these persons is responsible for any remaining errors and shortcomings. t See Nash, Pearce and Stanley [19]. t On ‘postulated prices’ see Williams 1301. - _ # See Richfield [l?, IS]. 1)See, for example, Allen and Isserman f21 _ _ and Hensher

c1-a. TIThis work is. as yet, unpublished.

** For a much more detailed discussion of value judgments concerning individual preferences and evaluation see Nash, Pearce and Stanley [ 191.

With most large urban public projects today a substantial number of effects fall into the non-monetary category. This means that many subjective trade-offs are required if ranking of alternatives is to be achieved. Some technique of making these trade-offs explicit so that all project effects might be combined into an aggregate impact index will clearly be a great convenience for evaluation. Recent successes obtained using attitudinal data in travel mode choice studies suggest that attitudinal techniques might provide a useful method of aggregating individual preferences for market and non-market effects. In the mode choice studies, scaling techniques have been used to predict choice of mode, encouraging correlations existing between predicted and observed behaviour./l In addition, data recently collected in Australia for the Commonwealth Bureau of Roads has provided behavioural confirmation for price elasticities of demand for urban rail travel estimated from attitudinal data.7 This paper develops a framework for project evaluation using scaling techniques to provide a direct measure of cardinal utility. The framework is kept as simple as possible since the purpose is to derive an evaluation technique which might be made operational. The general presumption underlying the technique is that analysts should respect individual preferences concerning project effects.**

AN ATTRIBUTE-BASEDAPPROACH TO INDIVIDUAL UTILITY Following Lancaster [16], it is assumed that the ith individual derives utility not from commodities but from the characteristics or attributes of commodities. Thus a meal may be desired not for itself but (say) for

its nutritional and aesthetic attributes. Let x, and x2 be two of the commodities whose attributes yield utility for the individual, these commodities having m dif-

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J.K.

STANLEY

ferent attributes of which yj is typical.* The individual’s utility function can thus be written in general terms as: u’ = U’(y,(-x,, -uz), Y,(X,, -urX

3Y&l, %I. -3

(1)

where Z stands for all other arguments in the utility function. First-order conditions for utility maximization subject to a budget constraint may then be written as:

cIn au

ay;

(2)

ay; ax,

7--lp,=O

(

j=

j$ 5

2 - ip,=0

(3)

where pi is the price of the ith commodity and i. the marginal utility of income. In equilibrium the individual extends his purchases of x1 and x2 to the point at which the additional utility expected from the attributes of the commodities just equals the utility lost in paying the commodity price.‘!’ Let the initial quantify of x1 and x2 be designated as (0,O) and assume that the first-order constrained utility maximization conditions shown in equations (2) and (3) do plot hold. More particularly, assume that the individual’s purchases of x1 and x2 at the present time are less than optimal, i.e.

m aui ay; (0,O) ax, >

j5,ay;

lpl

(‘$1

buy an extra unit of both X, and .y3. However, if for some reason (6). (8). (9) and (IO) all hold starting from (4) and (5) but the individual can only buy one unit of s1 or one of .Y~.then it is assumed he will compare the net utilities he expects to gain from each and choose that alternative offering the larger net utility gain. Net utility is equal to the utility expected from the attributes possessed by each alternative less the utility lost by giving up its price. If then he should

then x, R’x, where R’ stands for “is ith individual”. Should tions in monetary terms (11) can be re-expressed

at least as preferred as, by the the individual do his calcularather than utility terms. then as

(4)

m dU’ 8yij(O,O) jF;

3

ax,

’ 3bp2.

(5)

If the individual considers moving towards equilibrium by purchasing one extra unit of x, or x2, he should purchase x1 if the inequalities shown in (6) and (7) hold m du’ ayj(l,O) jg,

ayj

In avi j&, 3

ax,

a Ipl

(6)

then x1 R’x? where

8yj(O, 1) < 8X2

’ Ip2

On the other hand, if (6) holds together lowing inequalities

(7) with the fol-

(8) * Not all commodities (inc. services, etc.) will be perceived to possess all attributes, of course. t It is assumed that the second-order conditions for a maximum are fulfilled. $ Theoretical origins for the concept of ‘generalized cost’ in transport user benefit assessment can be found in this expression, tLeve1 is not strictly appropriate here since not all attributes will be physically measurable. The word is used intuitively.

is the marginal rate of substitution between the ,jth attribute and income for the ith individual. If the ith attribute were travel time this expression would be the individual’s value of travel time savings.1 Changes in the ith individual’s utility level are thus assumed to be a function of changes in the attribute ‘levels’ he experiences. these changes relating back to changes in the commodities he consumes (as in equations (2) and (3) for example).4 It should be noted that attributes can range widely so as to include such things as the distribution of income in society and personal health. Since inequalities (4t( 11) show that commodity price is relevant in the assessment of utility changes, for convenience price will also be regarded as an attribute. For simplicity reference to commodities can now be dropped. since in project evaluation one

A cardinal

utility approach

works directly in terms of project attributes (project effects). Also, since it is essential for this study not to be restricted to small changes in continuous variables, utility changes may be linearly approximated as:

(13) It is assumed that the second-order cross-partial derivatives of the individual’s utility function all equal zero.* so changes in the ith individual’s utility level can be approximated by the sum of independent changes in utility derived from each attribute he experiences. The additive independent formulation for changes in the ith individual’s utility level is quite restrictive. However, Shepard [25] has noted how frequently decision-making models which assume independent structures are validated in behavioural prediction. Operationally perhaps the most compelling argument for the independent model is provided by Edwards and Tversky [S] who argue that no alternative tractable formulations that describe non-independent structures have been successfully developed. Fortunately the additive independent model does have empirical consequences which can be tested, so the formulation is subject to some verification.?

for project

evaluation

to be associated

Ay;

PROJECT

EVALUATION

A2U’ = 0 * i.e. ~ AY; AY; t See pp. 336-331.

with this plan can all be set to zero. , = Ay',, =...=

Ayi,, zz 0

(14)

where Ay’;j = change in attributej perceived by the ith individual to be associated with project 1 (.j = 1,2,. . .,m). The perceived changes in attributes associated with the remaining projects are denoted by A& (p = 2.3. .,q;.j = 1,2. .,m). These changes may be either positive or negative as compared to the do-nothing alternative. In using a cardinal utility approach to project evaluation attribute changes would be presented to the individual for assessment. Hence there is no need to talk further of perceioed attribute changes since in practice all respondents will perceive the same changes. The expected changes in attribute observations associated with each project can be set out in tabular form. For simplicity the problem of time is ignored, all projects being assumed to have only immediate consequences. Table 1 presents the attribute changes, these being the various project consequences or impacts. Table

Projects

Various attributes which are arguments in the ith individual’s utility function will be affected by the projects which are to be evaluated. This is saying no more than that all projects have consequences of one sort or another. Let Y be the set of all possible combinations of attributes relevant to the ith individual’s utility function and assume there are m such attributes. Y is thus equal to Y, x Y, x . x Y,. Let yij be the particular observation of attribute Yj which the ith individual perceives to be associated with project p. The set of attributes the individual perceives to be associated with p will then be (ybi, y~,,...,y&,). Assume there are q alternative projects to be evaluated. Clearly some attributes will not be affected by any of the projects to be evaluated. in which case they will be irrelevant to the evaluation. Project evaluation has two major functions: (i) to rank alternatives; and (ii) to suggest whether there might be a case for undertaking no project at all. This being so, one of the alternatives to be evaluated will be the base or ‘do-nothing’ option. Evaluation can then concentrate on differences in attributes and corresponding cardinal utilities compared to this donothing option. If project 1 is regarded as the do-nothing option the changes in attributes as perceived by the ith individual

331

1. Project impacts on attributes

Y,

1 2

Ayz,

4

Aill

0

Aypj = change in attribute than the ‘do-nothing’ plan.

Attributes Yz

Kl

0

0

AY,, Yq2

j if project

AYh .”

p is executed

.V,,

rather

The criterion of relevance for inclusion in Table 1 is that the attribute which is expected to change ‘level uis-a-uis the attribute ‘level’ for the do-nothing project must be an argument in the utility function of at least one individual. In this study it will now be assumed that there are only three person types affected by the project being evaluated. The reason for this simplification is that a cardinal utility method of evaluation in practice could only be attempted on a stratified random sample of persons because of the costly nature of data collection at the individual level. The change in cardinal utility that a given attribute change leads to may of course, differ as between person types. Let ai be the number of persons of type i (i = 1,2,3). The change in cardinal utility experienced by the ith person type in consequence of the pth project being implemented rather than the do-nothing project can be expressed as (15)

allj + k.

This calculation needs to be carried out for each person type and for the q projects being compared.

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J.K.

It should be noted that if one wishes to exclude individual preferences for certain project effects (e.g. for reasons of poor information),* this can be done by defining the relevant attribute set for evaluation purposes as excluding such effects. The assumption of additive independent utilities for assessing utility changes means that the assessment of remaining attributes will not be affected by such exclusions. Total change in cardinal utility if plan p is introduced rather than the do-nothing plan (AWp) is calculated as: AWp =

i aiAUb. i= 1

(16)

No adjustments to cardinal utilities have been made at this stage for any ‘equity’ reasons. The project evaluation may be set out in matrix form as:

STANLEY

Project ranking then requires summation across the rows of [a,AUk] in (17) if no explicit weighting for equity is required. Inspection of the entries in [AW,] is needed if social weighting of individual utility changes is adopted. So long as a project registers a positive score in the final analysis it is ranked ahead of the do-nothing option (i.e. AW, = 0). Clearly if one project dominates all others. in the sense of having larger ‘good’ attribute effects and smaller ‘bad’ effects than all others there is no need to conduct an evaluation. Generally this will not hold, however, so some means of trading-off effects is required. This study suggests a cardinal utility approach to the trade-off. The evaluation framework has been outlined, it now remains to consider how a cardinal utility measure might be introduced. Since the emphasis is on utility differences for evaluating, an interval scale cardinal utility measure can be used.

where

(171 AL? MEASUREMENT

and where aiAUb = total change in cardinal utility to persons of type i if project p is implemented rather than the do-nothing project. Summing across the rows of [uiAUb] will provide the total utility effect (=AWp in (16)) for each project compared to the do-nothing project based on individual valuations. An E vector can be added should one wish to explicitly weight utility gains in accordance with particular value judgments concerning the social worth of utility gains to various person types.? Thus Ei is the ‘equity’ weight to be attached to a utility gain by the ith person type.$ Allowing for such equity weights gives:

a,AUi ’

a,AU: ’ 1

r”‘l

OF UTILITY

THE II\;DIVIDI’AL

In the measurement of cardinal utility based on an ordering of utility differences, admissable transformations are limited to the set of positive linear transformations. Since the concern is with utility differences. the question of arbitrary origin can be ignored, the main problem being one of scale unit. With reference to equation (13) the scales for the nr attributes must be so aligned that they are consistent for the individual. It will be convenient to work with a somewhat revised form of equation (I 3): n, AU’ = 1 [;As; (19) ;= I

[Aid

(18) where the [E] vector is the vector of equity weights. * See Nash, Pearce and Stanley [ 191. t For a fuller discussion of equity in evaluation see Nash, Pearce and Stanley [19]. $ If no explicit equity weights are introduced then implicitly Ei = 1 for all i and one has what might be termed a naive utilitarian decision rule as in (17). This is just as much a value judgment as varying the Ei by person type.

and where: 1; = a measure of the importance of attribute j to the ith individual in a given choice situation; Sj = a scaled measure of the satisfaction the ith individual derives from the jth attribute, where scaling procedures apply the constraint that 0 d .Si < K for all j. K being an arbitrary constant. 1;i is a positive constant in a given choice situation which may differ for each j, and for continuous attributes .Sj is a real-valued function defined over the values taken by the attribute. The need to split the A;/Ay; from equation (13) into two parts arises because of the experimental

A cardinal

utility approach

scaling methods to be used to measure utility. Ideally one would like to map attributes directly into utility, but problems of feasibility exist in niulti-alternativemulti-attribute choice situations. However, several writers have considered the possibilities of a twostage process of utility measurement.* The AS: in equations (19) and (20) can be viewed as representations of AU: in equation (13) except for scale unit. A scale unit can be set for each ASj and if one uses a standard rating scale approach to measure satisfaction derived from various attributes each ASj may be measured on the same scale range (i.e. 0 < Sj < K with K an arbitary constant). Thus an individual asked to select between various commodities on the basis of several attributes possessed by each may be asked to rate each commodity out of ten points for each attribute according to the degree of satisfaction expected. However, adding up, for each commodity, its satisfaction score out of ten across all attributes is unlikely to provide the correct overall preference ranking of commodities based on expected utility. It will in general only provide the correct ranking if the individual weights all attributes equally in his choice process. More formally, the individual’s satisfaction scale ranges are unlikely to be consistent with his utility scale ranges. He may be completely satisfied with a commodity in two different attributes (i.e. Sj = sg = IO)yet expect to derive different utility from each attribute. Satisfaction scale ranges are equal for all attributes by construction, but the corresponding utility scale ranges may be generally expected to differ. To make the satisfaction scales consistent with the AUj scales, some measure of the relative importance of each attribute in a given choice situation is needed. If one wishes to proceed in this way from attributes via satisfaction to utility, it is possible to re-write equation (13) as follows:

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333

evaluation

be attached to the jth attribute must equal AU,/AS$ the rate at which satisfaction changes with the jth attrlbute generate utility changes. Fishburn ([7], p. 437) calls these weights “scale transformation parameters”, since their function is to take the ASj scales and stretch or compress them until they are consistent with the AU; scales. The value of one such weight can be set equal to unity and the remaining weights assessed relative to this. If an individual rates attribute x twice as important as attribute y in choosing between several objects that possess both attributes, it is assumed that this is because on average over the relevant attribute ranges? AU; yZ2A$. AS;

(22) Y If the individual were given a set number of points to allocate between attributes x and y according to how important each attribute is in his choice between projects 1,2,. . .,q, it is assumed that the following relationship holds

Substituting the 1; values in equation (21) gives equations (19) and (20). Since measurement unique up to a positive linear transformation is required for utility differences, one can apply the constraint that .. m

1

z; = 1.

This constraint stresses the relative nature of the utility scores and should thus help the individual to think in trade-off terms about his allocation. It is thus hypothesized that with respect to each attribute, changes in the individual’s cardinal utility A(,i = f “1 “: (21) are a function of: jz AS’: Ayj (a) changes in the degree of satisfaction he derives where 0 < Sj < K, K being an arbitrary constant, for from each attribute; and, all j. It can thus be said that the importance weight to (b) the relative importance he attaches to each attribute in the given choice situation. * See Fishburn [7]. One expected exception to this general approach t If A2Uj/AS;2 # 0, then it is assumed the individual now needs to be noted, the case where only two alterwill do his calculations of relative importance based on natives are being eva1uated.f It is hypothesized that in the average value of Avj/ASj over the relevant attribute this situation the individual provides relative weights ranges. This is why 1; in (19) was assumed to be a of various attributes depending on the expected carconstant in a given choice situation. dinal utility dif3^erencebetween the alternatives on each $ For example, a project and its ‘do-nothing’ alternative. Each alternative may be characterized by any number of attribute. The weight of attribute Yrelative to attribute X, in the choice between alternatives A and B, would attributes. then be equal to9 $ Note that condition (24) is insufficient to fix the Ayj

1

absolute Imposing

level of importance a constraint such as

scores,

which

is arbitrary.

f 1; = K, where K is some constant, does serve to fix scores. It Assuming a particular cardinalization of the utility function.

Ii/Ii YX

=

U’.rA U’

XA

-

uy, u:;

Thus if the expected utility difference between A and B on attribute Y is ten utils,(l whilst the difference on X is expected to be two utils, then it is argued that the ith indivi.dual would rate attribute Yas five times as im-

334

J. K. STANLEY

portarzt as attribute X in choosing between alternatives A and B. If people do in fact allocate importance scores in accordance with (24) they are giving cardinal utility indicators directly since (24) may be written I;.lIj, = lAu;lA%

ACJpAS;[AS;/Ay,] Ay, = AU~AS~cAS~,Ayv,,Al>s

(25)

To multiply I, I, by ASJAy,, Ayr etc. would then clearly be double counting: All that is necessary is that the cardinal utility indicators be signed and summed. This can be achieved by using the S functions to indicate the preference relation between the alternatives on each attribute, The less desirable level of the .jth attribute can be set at Sj = 0, the more desirable level at SJ = IO.* these scores being attached to the relevant alternatives. Multiplying the Sj values for each alternative by the relevant 1; weights and summing will then indicate the more preferred alternative overall. In other words, with the additive model:

where if J’j,

then S,i, = 10. S,: = 0

if BP;A then S Aj = 0. S,: = 10 and where P’ is a preference relation for the ith individual meaning ‘is preferred to’, 9;’ being the equivalent relation defined over the jth attrtbute. It is thus hypothesized that an individual in a binary choice situation assesses relative attribute importance by considering utility d@rences between the alternatives on each attribute. In the multi-alternative case it is necessary, if importance scores and satisfaction scores are to be used to assess choice problems, that importance weights be based on relative rates of change of utility with respect to satisfaction changes on the different attributes. Since essentially different judgments with respect to S are required with those two approaches it is possible to test which of the two procedures is being used in binary choice problems (since both approaches could conceptually be used in such cases). Only the second approach can be used in the multi-alternative problem unless the individual * Scale ranges of O-10 are assumed. t Importance weights would then generally be expected to differ as between each pairwise comparison. In the multialternative situation, an average value AU’/ASJ over the relevant attribute range is assumed to be the 1: value reported. 1 See Hansen [9]. #Remember that for convenience we are considering prlcc as an ‘attribute’ in assessing importance-satisfaction scores although it is formally distinct from an alternative’s attributes as seen in (27). Utility scores are clearly needed for attributes and price. IIThis section is an adaptation of Torgerson’s basic model. See Torgerson [28] pp. 62-3.

reduces that problem to a number of binary choices. relying on transitivity to reduce decision time.? Psychological studies indicate that time required for decision-making does not increase in any regular fashion as the number of alternatives to be considered increases. Rather, decision-time tends to be much reduced after some point, suggesting pairwise comparisons is not the method of solving such complex problems. $ All of this discussion on individual utility, importance-satisfaction and choice can now be summarized. For clarification, inequality (11) is repeated in a rearranged form (using delta notation): If

-I(p,

- p2) 3 0 then xi R’x,.

(27)

If one wanted to use the importance-satisfaction approach to measuring an individual’s expected utility changes if various public projects are implemented instead of a do-nothing alternative: (i) The alternatives would be known, these being the xp in (27) and any number of alternatives can be assessed (p = 1,2,..., 4). (ii) The values of all the terms inside brackets in (27) are assumed to be known and would be presented to the individual for assessment, i.e. the value of AJJ~/A.Y, is known for all j(i = 1,2...m) and all p(p = 1,2,..., 4) and so are the relevant prices at which the alternatives will be available to the individual ( = pp. p = 1.2,. .q).$ (iii) The individual is required to assess AS) as between the various alternatives for all attributes, including price. The A$ are all measured on scales of equal length. (iv) The individual is required to weight all attributes (inc. price) according to their relative importance to him in a given choice situation. It is hypothesized that a different method of assigning these weights will be used if two alternatives are to be assessed than if more than two exist. SCALING THE INDIVIDUAL’S SATISFACTION IN MULTI-ALTERNATIVE EVALUATIONS

In what Torgerson ([28], Chapter 4) calls the “subjective estimate scaling methods”, it is assumed that subjects can arrange members of any stimulus series in such a way that the ratios of distances separating stimuli on an underlying psychological continuum are equal to the ratios of differences between the numbers assigned to the stimuli on a rating scale. If the psychological continuum relates to satisfaction and the stimulus is (say) traffic noise, the following model is envisaged for the individual : 1) (i) satisfaction is a discriminable dimension of traffic noise and the individual is capable of making direct

A cardinal

utility approach

quantitative judgments of the amount of satisfaction he derives from any level of traffic noise; (ii) the judgment of the individual is assumed to be a direct report of the satisfaction value of the noise stimulus on a linear subjective continuum. The origin and units in which the judgments are expressed may be arbitrary but must remain constant for the individual. Heise [I I] reports a number of studies using semantic differential scaling* which meet the metric requirements implied by (i) and (ii) to a satisfactory degree. Recent work by Rule [24] confirms earlier suggestions that a scale effect may exist, such that the larger the scale number the larger the scale interval needed to denote a given distance on the psychological scale. There is some doubt about the ability of subjects to satisfy interval scale metric requirements, but Heise [1 1] argues that this doubt is not such as to prevent reasonable accuracy being obtained with scaling techniques. The successes achieved in behavioural prediction using attitudinal data in mode choice studies certainly support this position. Furthermore, the work of Cliff [4] indicates that the adverbial quantifiers ‘slightly’, ‘quite’ and ‘extremely’ should define rating positions which are about equi-distantly spaced, so if rating scales have these positions identified, interval scale metric requirements stand a better chance of being satisfied than might be the case if the only scale guidance given to the respondent were bi-polar adjectives and numbers defining scale positions.

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evaluation

335

designated most preferred and (0) least preferred on all attribute satisfaction rating scales. Other alternatives can then be rated on the attribute bearing in mind the interval scaling requirements. If only two alternatives are being evaluated, the Si scale for the more preferred alternative can still be set at (10) and that for the less preferred alternative at (O), as noted previous1y.g If an individual is indifferent between any alternatives with respect to a particular attribute in multi-alternative assessments he can place them in the same scale position. However, the scaling requirements this approach places on respondents would appear to be excessive. A convenient alternative is to allow each respondent to select his own satisfaction scale unit (adverbial quantifiers being used to assist him), the responses subsequently being transformed so that the most preferred alternative is rated (10) and the least preferred alternative (0), the remainder being adjusted so as to preserve the interval scale measure. This approach should yield more reliable responses. The transformation of responses then ensures that the satisfaction scale values reported for all individuals and all attributes will be consistent. If the subjective (satisfaction) continuum were the same for all subjects, their transformed scale values must be identical. However, equal S scale values do not necessarily imply equal utility values because the importance weights have yet to be added. IMPORTANCE

SATISFACTION

SCALING

ACROSS INDIVIDUALS

Since the purpose of the exercise is to add up effects over individuals as well as for each individual, a constraint is required such that all individuals use the same scale unit. With a standard rating scale (e.g. 0 < ,C$< 10 for all j) there is no reason to expect all individuals to choose the same unit of measurement.? This applies whether or not adverbial quantifiers are used to define scale positions. Psychological research, however, does not seem to have determined how far differences in scale checking style are a function of personality differences between individuals,$ a significant consideration if one is seeking a measure of cardinal utility. Differences in scale-checking style can be overcome if, for each attribute, each individual is required to set Sj = 10 for that alternative which he prefers most and ,S; = 0 for the least preferred alternative, (10) being * Semantic differential-the term is used in a generic sense to refer to any collection of rating scales anchored by bi-polar adjectives. See, for example, Nunally ([20], pp. 53% 541) and Osgood rt al. [21]. t Peabody [22] for example, observed important differences between individuals in scale checking styles. 1 See Heise [I l] p. 409. $ See pp. 333-334. jl See. for example, Jessiman et al. [15] and the Weiner and Deak [29] exercise. Holmes’ [14] ordinal evaluation method has similar problems.

WEIGHTS

The importance weights which are required may be looked at in two different ways: (i) as indicating how important particular attributes are to the individual in general; or, (ii) as indicating how important particular attributes are to the individual in his choice between the particular alternatives being assessed. The former possibility is only mentioned because it is the approach implied in many matrix evaluation approaches which require relative objectives to be weighted before any quantitative assessment of how alternatives meet objectives is made.11 It is difficult to imagine, however, how one could say A is more important than B without data on the actual quantities of A and B to be compared. Given the individual choice model used in the cardinal utility technique, importance weights could be assessed prior to quantification if the value of 3 AU; AS; /- AS; remained constant for all j, k, a very limited possibility. In the two alternative comparison it would require constancy of uj,

- V$

UhA - U$ for all j, k, which is even more restrictive.

Hence the

336

J.K.

second approach to importance weights can be presumed to be required in evaluations, actual attribute ‘levels’ being presented to the individual so he can meaningfully assign importance weights.* Attribute importance weights may be scaled in two ways, the implications of each being markedly different for cardinal utility. These two different methods are: (i) constrained&each individual can be made to allocate importance weights to each attribute so that W1 1 ‘;==I ;- I (or any positive constant which is equal for all):? or, (ii) unconstrained-importance of each attribute can be rated on a 0~10 scale (say). with no requirement that WI ,T, Ii = ’ for all individuals. If one rates satisfaction so that Si = 10 for the most preferred alternative and Sj = 0 for the least preferred alternative on each attribute then uses the

constraint the etrect is to give each individual ten ‘votes’ (‘utility difference units’) to allocate across the attributes of the alternatives being compared. This approach to evaluation is thus only sensible when all respondents are given the same alternatives for comparison. Intensity of preference can be expressed in a ~ltrti~ sense by the individual placing a large number of his ten ‘votes’ between his most preferred alternative and the remainder. This can be done by indicating a high level of satisfaction with the more important attributes: and a lower level of satisfaction with the remainder. Project ranking depends on utility differcnces. so those whose relative preferences are most marked influence project ranking more than those with less marked preferences.8 The allocation of a constant number of ‘votes’ to each individual is in effect the equity value judgment underlying the approach. Should one prefer an evaluation methodology which allows for absolute intensity of preference, using the unconstrained method of deriving importance weights would vary the votes cast by different respondents. For * If one believes certain attributes should not be relevant to an evaluation. then implicitly I; = 0. This kind if decision is essentially a value judgment which should be made prior to quantification. It is a different matter, however. to assign importance weights to admissable effects prior to quantification. t As noted previously, this constraint may help the individual to think in trade-off terms about his allocation. See 1’. 333. $ Remembering the individual derives utility not from prc)Jects but from their attributes. 3 Some numerical examples are presented in Stanley [X].

1’See p. 335.

STANLIZ)

the unconstrained method to provide a measure of cardinal utility in the absolute sense would require that all individuals mean exactly the same thing in terms of intensity for any given score on a (say) &lo importance rating scale. An individual who feels everything intensely would thus be expected to use the scale extremes more frequently than less intense persons. Scale adverbs would have the task of providing this common scale unit across all individuals. As noted previously, 11 there is no evidence to suggest that a common scale unit can be obtained across all individuals. It was the lack of such evidence that led to the use of S scales where a common scale unit is ‘forced’ by transforming S responses. If intensity could be allowed for using scale adverbs the effect would be to give a larger number of ‘votes’ to the individuals who feel more intensely about the projects being evaluated. Little hope exists that the resulting answer would in practice measure what it is intended to measure. however. Nevertheless, experimentally there is much to be gained from obtaining importance scores using both the constrained and unconstrained methods. The unconstrained importance scores can be normalized as follows:

where

17 = unconstrained importance score for attribute j for the ith individual ~1 = normalized importance score for attribute ,j(individual i) when unconstrained importance weighting is used.

Clearly

and a perfectly consistent respondent would provide values of Ii’ (j = I. 2, _, 01)such that rr; = I; for all j,

(29)

where 1,; = the constrained importance weight This internal consistency check on the importance data will be a useful indicator of the reliability of responses. Since no testing of reported I-S scores against actual behaviour will generally be possible by the nature of the exercise such internal consistency checks become important. SOME FURTHER

CONSISTENCY

CHECKS

It is possible to build a number of other limited tests for consistency into an evaluation analysis. Fishburn [7] for example, has shown how hypothetical paired comparisons can be used to check response consistency. Adams and Fagot [I] have shown how paired comparisons can be used to test whether an individual’s preference system satisfies r~r.s,sar~~ conditions

A cardinal

utility approach

for it to be represented by an additive independent utility function. Such tests are rather limited, however, in that they look at necessary not sufficient conditions.* In the last resort a certain amount of faith is needed if attitudinal techniques are to be used.

METHODS

OF APPLYING THE CARDINAL UTILITY APPROACH

Cost-benefit evaluation techniques seek to identify dollar effects at the individual level and add these up to an aggregate monetary impact measure, perhaps supplemented by a discussion or listing of non-monetary consequences. One would like to use the cardinal utility technique to extend assessment at the individual level to include all effects in the aggregate impact index. The difficulty here, however, is the constant votes per person feature of the C.U. approach. Since various persons are affected to differing extents by public projects, it seems unfair to give them all the same number of votes with which to indicate preference.t The CU. approach really requires all individuals to be assessing the same alternatives. Thus two methods of application emerge: (1) The CU. approach may be used as a method of evaluating project effects at the aggregate level. This application would be along similar lines to matrix evaluation techniques,$ aggregate project consequences being assessed by participants assessing the same consequences. Each respondent is thus put in the role of decision-maker and is required to make explicitly the kinds of subjective judgments a decisionmaker implicitly makes in trading off monetary and non-monetary consequences (say) in a Planning Balance Sheet. The approach is thus a kind of multidimensional voting procedure. Respondents would be randomly selected and need not necessarily be personally affected by the project. If respondents do not know whether or not they are to be personally affected, their responses may be of the impersonal or ‘ethical preference variety which the present author believes are relevant for project evaluation.$ Whilst being similar to matrix evaluation techniques in this application, the C.U. approach seems an * Fishburn’s [S] sufficient condition for an additive independent utility function is of no use experimentally. t This objection applies to most voting techniques. $ See, for example, Jessiman et al. [15), Weiner and Deak 1291. 5 See Harsanyi [lo], Rawls [23] and Nash, Pearce and Stanley [ 191. l/With the conventional indifference map this relative value varies depending on the particular location chosen in the individual’s preference system. 7 Thiswill require varying yj for the appropriate attributes such that f 1;; AS; = 0 j=* for the individual.

for project evaluation

337

advance in that it derives from a model of individual choice which brings out clearly the requirements for a matrix approach to be meaningful. In particular, it points to the need for importance weights (cf. objective weights in matrix methods) to be determined after the respondent has been presented with data on project effects. The failure to do this makes most matrix evaluation exercises to date rather suspect. Two difficulties with this application stand out. First, it is difficult to envisage how non-monetary effects (e.g. air pollution) over time could be presented to respondents. Many urban transport project evaluations are design year only exercises in which this problem would not arise. However, such exercises avoid the problem by avoiding the question of time, hardly a solution! Second, respondents will be required to assess aggregate magnitudes quite out of the range with which they will be familiar. One can charitably suggest that this is a reason why most matrix techniques use planners to provide relative objective weights. With ample evidence today that planners’ preferences often do not coincide with community preferences this alternative provides no solution for the analyst who believes the preferences of the latter group should generally be normative. (2) The individual choice part of the C.U. model can be used to impute monetary valuations to particular non-monetary effects, thus serving as a method of extending the domain of existing cost-benefit techniques. This is a more limited application, one which is not a full method of evaluation in any sense. Its advantages are that it works with effects at the individual level and problems of time do not arise. One merely looks for specific individual trade-off rates which can be used to impute dollar values to non-monetary attribute changes. These imputed values would be subsequently used as input in evaluation studies, just as values of private travel time savings are used in transport project evaluations. Note also that no need exists for a common scaling unit across individuals with this approach, monetary values being imputed for every individual separately. Economists define the value of a commodity to an individual in a relative sense in terms of the individual’s marginal rate of substitution between the commodity and some numeraire at a particular utility level.11 This approach can be applied to attributes. In , the general importance-satisfaction approach marginal rates of substitution may be identified moving along the individual’s existing indifference surface as follows AU’ =

2 j=l

Ii.

f$yj

’ Ay;

zz

0,

(30)

One can vary attribute levels (i.e. Ayj) for particular monetary and non-monetary attributes so that (30) holds,y thus identifying the rates at which the individual trades the non-monetary attributes for money.

338

J.K.

These marginal rates of substitution are the sought after imputed values.* It is possible to investigate individual preferences over a wider range of attribute levels with this C.U. approach than with the Hoinville [ 131approach, making the C.U. technique rather more genera1.t CONCLUSION

A technique has been offered for use in evaluating public projects having important non-monetary consequences, drawing on attitude measurement tools. Difficulties are seen in applying the technique to complete project evaluations, but existing techniques which seek a single aggregate impact index face all the same problems. Should the cardinal utility technique fail as a complete evaluation methodology, then matrix evaluation techniques must fail for the same reasons unless one is prepared to accept the paternalism involved in using planners’ preferences. At a lower level the cardinal utility technique can be used to extend existing cost-benefit methodologies by providing a method of imputing dollar values to particular non-market impacts. It seems likely that this second application is where the tool will find its main use. REFERENCES I. A. W. Adams

2. 3. 4. 5. 6.

7. x.

and R. Fagot, A model of riskless choice, in Decision-Making (Ed. by W. Edwards and A. Tversky), Penguin, Harmondsworth (1967). W. B. Allen and B. Isserman, Behavioural modal split. Hiyh Speed Ground Transpn 6(2), I 8C I99 (1972). K. J. Arrow, Social Choice md Individual Values, 2nd Edn. Wiley, New York (1963). N. Cliff, Adverbs as multipliers. Ps~~ckol. Rev. 66( I ), 2744 (1959). W. Edwards and A. Tversky (Eds.). DmsiowMaking. Penguin, Harmondsworth (1967). M. Fishbein, Attitude and the prediction of behaviour. in Atfifudrs und B~/uciour, (Ed. by K. Thomas). Penguin. Harmondsworth (1971). P. C. Fishburn, Methods of estimating additive utilities. M,zgrl~t Sci. 13(7), 435-453 (1967). P. C. Fishburn, Preferences, summation and social welfare functions, Mngmr Sci. 16(3), I79- 1X6 (1969).

* In (12). they equal t Hoinville’s [I-i] ‘priority evaluator’ suitably modified would be a convenient way to collect the data required by the cardinal utility approach.

STANLEY 9. F. Hansen. C‘onsu~nc~ Cl~orcc Bci~tr~iour.. Collier-~ Macmillan. London (1972). IO. J. Harsanyi. Cardinal welfare. mdividualistic ethics. and interpersonal comparisons of utility. J. Polir. EUJ~I. 63, 309-321 (1955). I I. D. R. Heist. Some methodological tssucs in semantic differential research. Ps~~chol. B~rll. 72(h). 406 322 ( lY77). 12. D. A. Hensher. Consumer’s choice function. A stud! 01 traveller behaviour and values. Unpublished Ph.D. Thesis. School of Economics. University of New South Wales (1972). 13. G. Hoinvillc. Evaluating community prekrmxs. Emiron. Ph. 3(I). 33 50 (1971). 14. J. C. Holmes. An ordinal method of evaluation. I rh. Stud. 9(Z), 179-192 (1072). 15. W. Jessiman, D. Brand, A. Tumminia and C. R. Brussec, A rational decision-making technique for transportation planning, Highway Research Record No. 1X0. Washington D.C. (1967). 16. K. J. Lancaster. A new’ approach to consumer theory. J. /‘o/it. Ecorl. 74, 132 157 (1966). 17. N. Lichfield, Cost benefit analysis in plan c\aluation. T/i. F’lu!~/i. Rrr. 35, I60 I6Y ( 1964). IX. N. Lichfield. Cost-~benefit analysis in planning: a critique of the Roskill commission. Reg. Stud. S(3). I57-- I83 (1971). 19. C. A. Nash. D. W. Pearce and J. K. Stanley. An cvaluation of cost&benefit analysis criteria. Discussion Paper No. 73 I I. Dept. of Economics. University of Southampton ( 1973). Nc\\ ‘0. J. Nunally. Psychomctric~ T/wor~~. McGraw-Hill. York (1967). 21. C. S. Osgood. G. J. Suci and P. H. Tannenbaum. ‘rht, Mrasurrnw~t of M~mr~ir~g. liniv. of Illinois Press. Urbana (lY57). 22. D. Peabody, Two components in bi-polar scales: dircction and extremeness. Psycho/. Rrr,. 69(2), 65~-73 (1962). 23. J. Rawls. A Theory qf Jusricr. Clarendon. Oxford (I 972). 24. S. J. Rule. Comparisons of intervals between subjective numbers. Prrcpt. Psyc/wphys. 1 l( I B), 97-98 ( I Y72). 25. R. N. Shepard, On subjectively optimum decisions among multi-attribute alternatives. D~~~isio1f-~~trkiflU. (Ed. by W. Edwards and A. Tversky). Penguin. Harmondsworth (1967). 26. J. K. Stanley, A cardinal utility framework for project evaluation. Discussion Paper No. 7305, Dept. of Economics. University of Southampton (I 973). 27. L. L. Thurstone. Tl~c Mcusurrrnrnr of l’cdws. C’niv of Chicago Press, Chicago (1959). 2x. W. S. Torgerson. T/wor!~ trud Mct/wd\ of’Scrr/~mq. Wile), New York (1965). 29. P. Weiner and E. J. Deak, E/lrliro,l,,rc,rltu/ Furors II) Trunsportution Plmr~ir~g. Lexington, Heath (I 972). 30. A. Williams and E. J. Mishan. Cost-benefit analysis, J. Puhl. Ecm. I( 3,‘4), 39x 400 ( 1972).