Soc Choice Welfare (1995) 12:255-266

© Springer-Verlag 1995

A case for cardinal utility and non-arbitrary choice of commodity units*

Yew-Kwang Ng I, Jianguo Wang 2

1 Department of Economics, Monash University, Clayton 3168, Melbourne, Australia 2 School of Economics, University of New South Wales, Kensington 2033, Sydney, Australia

Received: 14 February 1994/Accepted: 7 November 1994

Abstract. With preference changes, cardinal utility is indispensable. For any necessity good, there exists an intermediate consumption level at which a change in preference intensity has no effect on utility, below/above which an increase in preference intensity decreases/increases utility. This is supported by an indicative empirical survey. At the intermediate consumption level, the utility from the relevant good is zero. For a Cobb-Douglas utility function, this intermediate consumption level equals one, making the choice of the unit of measurement non-arbitrary.

Introduction

It is rather remarkable that mainstream economics for half a century since Robbins has followed a way which is so different from what is goin9 on in the development of most sciences. Mostly science is followin9 reality instead of ignoring it. - van Praag (1991, p. 71). This paper argues for the compellingness of using cardinal utility (to represent either preference or welfare) for individual choices where changes in preference are involved. For the theory of social choice, one of us has argued elsewhere the necessity of using interpersonal comparable cardinal utilities. Contrary to popular belief, ordinal utility is insufficient even if a maximin social welfare function is used since the general possibility of interpersonal comparability of utility levels implies the comparability of utility differences (Ng, 1984a). Also contrary to popular belief 1, the cardinal utility functions derived by the von Neumann-Morgenstern

* We are grateful to a referee for some helpful comments 1 For example, Baumol (1977, p 431) writes: "what relationship, if any, does the N-M cardinal utility theory have to that of the neoclassical utility theorists? It is generally (though not universally) agreed that there is none- the two utility measures have nothing in common insofar as their cardinality is concerned"

256 Y.-K. Ng and J. Wang

expected utility hypothesis is identical to the subjective utility functions of neo- classical economists, explaining why individual attitudes towards risk is relevant for social choice (Ng 1984b) 2. (On the necessity of interpersonal cardinal utilities, see also Kemp and Ng 1977, 1987; Hammond 1991; Mueller 1989, Ch. 19.)

In Sect. 1, we argue that, for any necessity good, there exists an intermediate consumption level at which a change in preference intensity for this good has no effect on utility, below/above which an increase in preference intensity de- creases/increases utility. At this intermediate level of consumption, the utility from this good is zero. Moreover, if a Cobb-Douglas utility function is used, this intermediate consumption level should be defined as unity, making the choice of the unit of measurement non-arbitrary. Sect. 2 reports on an indicative empirical survey supporting the arguments of Sect. 1.

1. The Cobb-Douglas utility function and preference changes

With no change in preference, it is a classroom exercise that the utility-maximizing choice of a consumption bundle is invariant with respect to a monotonic trans- formation of the utility function. In this sense, ordinal utility is all that is required for the theory of consumer choice. Where there is a change in preference, the ordinal utility framework is unable to make inter-preference comparison of desir- ability. A cardinal welfare framework is required for this purpose 3. However, mainstream economists, especially those in the Chicago tradition, are not only hostile to cardinal utility but also adverse towards the concept of preference changes. In the words of two leading Chicago economists, "tastes neither change capriciously nor differ importantly between people. On this interpretation one does not argue over tastes for the same reason that one does not argue over the Rocky Mountains - both are there, will be there next year, too, and are the same to all men" (Stigler and Becker 1977, p 76). Instead, they explain apparent changes in tastes in terms of changes in variables (e.g., stock of human capital) and constraints in the household production function. While this approach is certainly important and insightful for many issues, the complete denial of changing tastes as arbitrary or methodologically suspect is excessive. For biological reasons, tastes changes

2 Samuelson (1947, p 228n) and Hahn (1982, p 195) both declare their failure to see why social choice (e.g. with respect to income distribution) should depend on individual risk aversion (with respect to income). This dependence is straightforward once the equivalence of the N-M cardinal utility function with subjective utility is recognized. The degree of risk aversion reveals the degree at which subjective marginal utility of income diminishes. Since social welfare is a function of individual subjective utilities, how rapidly marginal utilities diminish has obviously important effects on social choices that affect individual income levels 3 It is true that the approach of metapreferences (preferences over alternative situations with different preferences) may be used. But this leaves the reason for the metapreferences buried under technical terminology. We believe it is more straightforward and more revealing to use the cardinal welfare approach. Using the strictly ordinal preferences (including meta- preferences) approach, economists cannot even explain why children's relative preferences change away from food towards sex-related consumption and activities as they grow up. In contrast, the cardinal welfare approach provides a direct link to biology, providing the necessary explanation

Cardinal utility and commodity units 257

over the life time of an individual are largely independent of, though to some extent intertwined with, the consumption history. (For a survey of habits, addictions and related effects, see Becker 1992.) Also, for social and cultural reasons, different individuals have different tastes and different changes in tastes which cannot all be explained by the household production function approach. Moreover, taste changes need not be completely unanalysable. (See, e.g. Strotz 1955/56; Fisher and Shell 1968; von Weiz~icker 1971; Pollak 1976; Kapteyn et al. 1980; see also Schokkaert 1982 which combines the traditional and the household production function approaches.) Postulating changes in preferences does not seem more arbitrary than postulating changes in the household production function (Cowen 1989). We thus suggest that economists' reluctance to consider preference changes should also be reconsidered.

To commence our analysis, we first adopt a simple definition.

Definition 1. A preference pattern is Cobb-Douglas with respect to one (or more) good(s) X ( x l , x2 . . . . , x,,) if the utility function representing it can be expressed as

U(X, Y) = ~ lnX + F(Y) (la)

o r

U ( x l , x 2 , . . . , x , . , Y ) = ~ l l n x l + ~ 2 1 n x 2 + ... + ~ , , l n x , , + F ( Y ) , (lb)

where Y (which may be empty) is the set of all other goods other than X (x l , . . . , x,,), and the ~'s are preference parameters.

If a preference pattern is Cobb-Douglas with respect to all goods (or variables) under consideration, the utility function itself is said to be Cobb-Douglas, with U = Z~'= 1 ~, In x, or for the special case of two variables only, we have

U = ~ l n X + flln Y, (lc)

which is the familiar first-year case.

Proposition 1. I f a preference pattern is Cobb-Douglas with respect to any good: X before and after a change in preference intensity with respect to X, there exists an intermediate consumption level at which a change in preference intensity has no effect on utility, below~above which an increase in preference intensity decreases~increases utility. This intermediate level occurs at the level when X = 1, making the choice of the unit of measurement non-arbitrary.

Proof As U = ~ In X + F(Y) before and after a change in preference intensity with respect to X, the change in preference intensity can only be represented by a change in a. (An increase in a signifies a higher intensity.) Partially differentiating U with respect to ~, we have

~U/~a-- lnX > < 0 if X >_< 1. Q.E.D. (2)

Consider a shift in preference in favour of X in the 2-good case of (lc). This is reflected in (lc) by an increase in ~. A question is, what happens to fl? For the pure theory of consumer demand, it does not matter whether fl is held constant or decreased, say, to keep e + fl = 1, since what matters is only the relative value of

and ft. Thus, iffl is held constant, we may just increase e by a little more to achieve a desired increase in e/fl or ~/(ct + fl). However, if we want to know whether utility has increased or decreased after the change in preference, the ordinal utility framework cannot help. Nevertheless, the ordinalists may reply that, in the pres- ence of preference changes, utility comparison is either impossible or meaningless.

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titan

Fig. I.

The two sets of indifference curves intersect each other in general and no compari- son can be made. Such views are misleading and defeatist.

Let X stand for sex and Y for food. In our pre-adolescent days, our ~ = 0 (approximately, with apology to Freud) and fl is fairly large. Now, in comparison, fl is a bit smaller but c~ is large. However, for the range of values of X and Y we actually have had, we have no hesitation in saying that we prefer the situation now than before. Moreover, this preference is not that we prefer the current situation only with our current preference pattern. Thus, we can expect that, when we become old and sick, our ~ might become zero again and our fl much more smaller. Then, even with our preference pattern then, we would still prefer our current middle-age situation to our pre-adolescent situation and also prefer the latter to our situation then (i.e. old and sick). How do we base our inter-preference compari- son of desirability? It is mainly based on our welfare which is defined as net happiness or enjoyment minus suffering with both defined to include all forms of positive and negative affective feelings respectively. The amount of welfare is the integration of affective feelings (which may differ in intensity) over time, with the zero line standing for neutral or no affecive feeling. As illustrated in Fig. 1, the welfare of an individual over a given period of time is the area above the neutrality line minus the area below 4.

However, the preference of an individual may not just depend on his/her welfare. It may also depend on the welfare of others and other (i.e. non-welfare) considerations. It can be argued (Ng 1990a) that non-welfare considerations are not ultimate but are just instrumental in the promotion of welfare. Nevertheless, this is a normative view. Moreover, factors like ignorance and irrational preference that cause a divergence between preference and welfare cannot be ruled out. Thus, we will not rule out non-welfare considerations. However, it is clear that personal welfare is an important and in most cases, the most important factor affecting the preference of an individual. Moreover, whatever other factors that are taken into account in the preference o f an individual, he can decide on the appropriate

4 With utility as a measure of subjective amount of happiness, it is clearly a cardinal amount, hence cardinally measurable in principle. While there are practical difficulties of actual measurement, they should not deny the cardinal measurability of utility in principle. In a review of Ng (t990b), Witzum (1992, p 173) doubts "that individuals have an a-priori knowledge of the strength of preferences (i.e. cardinal utilities) over the whole domain of the social choice function" as requied "if a social welfare function is based on individuals' cardinal utilities". In my view, this is a matter of practical difficulties due much to uncertainties and imperfect information. Moreover, this problem also applies to ordinal preferences. Thus, one may vote for party A and then regret having done so a few months after the election

Cardinal utility and commodity units 259

trade-offs between competing factors affecting preferences. Given the reasonable assumption of continuity in preferences, a cardinal utility representation of prefer- ence is still possible.

It is true that, where preference continuity is not satisfied (e.g. lexicographical preference), a real-valued utility function representing the preference of an indi- vidual over a multi-dimensional uncountable set may not be possible. However, uncountability is relevant only if the relevant variables are perfectly divisible. But with perfect divisibility, lexicographical preferences are not reasonable. The argu- ment of lexicographic preferences is mainly based on such a question as, "Is there some number of trinkets that will induce a starving coolie to part with one bowl of rice?" (Chipman 1960, p 221). The answer to such a question may well be negative, but this does not make his preference lexicographic. With divisibility, his preference can only be lexicographic if, given that he prefers more trinkets to less, there is no number of trinkets that will induce him to part with 0.0000000... 1 grain of rice. This is clearly unlikely. On the other hand, if we do not have perfect divisibility such that one grain of rice is the smallest unit, then preference representation is still possible.

While we do not rule out other factors affecting preferences, we will concentrate on changes in preference arising from changes in personal welfare. For such changes, the distinction between preference and welfare may be ignored.

With the above cardinal concept of welfare and preference, we may proceed to make inter-preference comparison of welfare or preference. For a shift of preference in favour of X, e increases. Whether fl decreases or stays unchanged depends on cases. For example, if our increased preference for sex is not accompanied by a decrease in the ability to enjoy food, fl remains unchanged. If there has been such a decrease,//decreases. Thus, in this cardinal concept of preference, we do not necessarily have ~ + fl -- 1. If ~ increases and fl decreases, we interpret this as an increase in the preference for X and a decrease in the preference for Y.

At X = 1, a change in e does not affect utility. This result is rather annoying to economists who have been used to the idea that units of measurement are arbitrary. For weight measurement, one may use micro-gram or a million tons as a unit. So, for the same amount of a good X, it may either be smaller or larger than one, depending on the unit of measurement. So, how could whether X is larger or smaller than one matter? An ordinalist may say again that this just shows how meaningless is utility comparison in the presence of preference changes. However, as argued below, (2) is sensible and implies that the unit of measurement is not arbitrary, at least for the purpose here.

Suppose X is food. If one's consumption of food is very inadequate (X < 1), one is suffering from starvation. If one's preference for food (~) increases, one suffers more from starvation if food remains inadequate. While one now gets more utility from the consumption of the available food, one also suffers much more from not being able to consume an adequate amount of food. Thus, utility actually de- creases. On the other hand, if food is abundant (X > 1), one is well fed. Then, if one's preference for food increases, the utility from eating increases. This is illustrated in Fig. 2.

From (la), ~?u/OX = ~/X. The marginal utility (MU) curve is thus a rectangular hyperbola as illustrated in Fig. 2. The total utility need not equal the area under the MU curve. If we ignore utility from Y, total utility equals zero at X = 1. Thus, total utility from X equals the area under the MU curve minus the area under the MU curve to the left of the vertical line at X = 1. For example, with X = 2, total utility equals the area ABCD; with X = ½, total utility equals the negative of the area

260 Y.-K. Ng and J. Wang

aX

...... . l i t

0

0 , i D X 1/a 1 2

Fig. 2.

ADEF. Thus, the area under the MU curve to the left of the vertical line at X = 1 may be called the disutility of inadequate consumption of X.

Now consider an increase in ~ to ~'. This increases the utility of consuming whatever the available amount of X. But it also increases the disutility of inad- equate consumption of X. If X < 1, e.g. if X = ½, the increase in utility (the shaded area) is less than the increase in the disutility of inadequate consumption of X by the area DHIE. Thus, the increase in ~ to ct' actually decreases total utility by the area DHIE. On the other hand, i fX > 1, e.g. ifX = 2, the increase in the area under the MU curve exceeds the increase in the disutility of inadequate consumption, thus total utility increases with the increase in ~ to ~' by the area CGHD.

We discovered the above contrasting responses of total utility to a change in preference intensity by chance in our study of the effects of preference changes. Once put in the above interpretation, we find the contrasting responses accord perfectly with common sense. Obviously, for a preference which one can amply satisfy, one gains in having an increase in preference intensity. In contrast, for a preference one has inadequate means to satisfy, an increase in preference intensity actually increases the frustration. Consider a real-life example. One of us was encouraging a single early middle-aged man to get more interested in sex and marriage. His response was, "Suppose you have an aching finger. After you have made it no longer a bother to you, why should you recreate the problem?" Obviously, this person's adaptation to a prolonged period of non-satisfaction is to reduce the value of ~t (preference intensity) to near zero in order to minimize the negative utility. More generally, the Buddhist philosophy of eschewing wants may also be regarded as a sensible way to reduce disutility in countries where the masses have little chance of satisfying most wants adequately. Thus, utility can be in- creased without increasing consumption. The issue of preference changes is an important topic with far-reaching policy implications. (see Wang and Ng 1993.)

It may be asked, why is the amount of a good at which a change in peference intensity has a neutral effect on utility happens to equal to one? The answer to this question is to turn the perspective the other way round. Since an increase in preference intensity increases/decreases utility if the amount of the good consumed is large/small and preference is continuous, there exists an intermediate amount at which a change in preference intensity has no effect on utility. If we are using a Cobb-Douglas utility function, a change in the preference parameter ct has no effect on utility when X = 1. Thus, we have to define the intermediate amount of

Cardinal utility-and commodity units 261

X (or any other good) as equal to one to make the Cobb-Douglas utility function consistent with the fact of contrasting responses of utility to a change in preference intensity depending on the amount of X. To accommodate this, the choice of unit is not arbitrary but has to depend on the subjective utility responses of the individual in question. This creates some practical difficulties but economists are not born to lead an easy life. However, this non-arbitrary choice of unit is needed to facilitate the analysis of preference changes and may not be required elsewhere.

While Proposition 1 is proved only with respect to Cobb-Douglas patterns of preference, it has more general applicability as is clear from the discussion above. However, it must be admitted that it may not be applicable to non-necessity goods. For example, suppose X is attending theatre performances which is regarded by a person A as a non-necessity good. H e has a mild preference for attending performances but does not feel frustrated or deprived if he attends none. Then it is possible that his utility from X is zero, not negative, when X = 0 and that an increase in his preference intensity for X (e.g. by being able to appreciate theatres more) may increase utility at all positive levels of X. However, for such non- necessity goods, the preference pattern is clearly not Cobb-Douglas. For a good with the Cobb-Douglas pattern of preference, a zero amount of consumption will decrease utility to minus infinity. For our purpose here, we need only to define a good X as a necessity if a low enough (positive) level of consumption involves a negative utility from X. Commonsense necessities such as food, clothing, and shelter clearly qualify. If we take into account factors such as frustration, envy, etc., many more goods qualify as necessities. With negative utilities prevailing at low levels, an increase in preference intensities thus decrease rather than increase utility as discussed With reference to Fig. 1 above.

A question arises as follows. If we define "the intermediate amount of X at which a change in preference intensity for X has no effect on utility" to be equal to one, does the total utility derived from X equal zero as required by ~ lnX? Our answer is affirmative. At this level of X, an increase in preference intensity for X increases the utility derived from consuming X by as much as it increases the disutility from inadequate consumption. If the Cobb-Douglas form remains ap- plicable both before and after the change, the change in preference is proportionate throughout. Thus, the utility derived from consuming X equals the disutility from inadequate consumption, making the total utility derived from X zero. However, for the more general non-Cobb-Douglas case, this question deserves further studies.

Conjecture 1. At the intermediate consumption level of X where a change in prefer- ence intensity for X has no effect on utility, the utility from X is zero.

If we are confined to Cobb-Douglas functionsl Conjecture 1 becomes Propo- sition 2.

Though the Cobb-Douglas form has been widely used and recognized as the benchmark case, it is a special form. For one thing, it requires additive separability. In the more general case of non-separability, the intermediate level of consumption of a good (at which a change in preference intensity does not affect utility) may depend on the amounts of other goods consumed. This will create further difficul- ties and complications. We must leave the analysis of such complications to more qualified researchers. Here, it may be noted that, the broader is the commodity grouping, the more likely that the Cobb-Douglas pattern is a close approximation and that the essence of our proposition need not be applicable only to strictly Cobb-Douglas cases.

262 Y.-K. Ng and J. Wang

2. An empirical survey

An empirical survey was undertaken to check the practical relevance of the argument in Sect. 1. Due to the small sample size, non-perfect representation, and the possibility of framing effects, the survey should be taken as indicative than conclusive. The survey was conducted in two parts. Part A was done in Melbourne with a sample of 33 respondents selected more-or-less randomly in shopping centres,and the streets. Part B was done in Beijing with a class of 40 graduate students in economics. The crucial Question 1 asked the respondents to consider the following five alternative situations regarding the amount of food for the respondent's own consumption,

(A) only possible to keep you from dying, but insufficient to avoid hunger and malnutrition. (B) just sufficient to avoid hunger but not adequate to avoid malnutrition. (C) just sufficient to avoid both hunger and malnutrition, but not adequate to provide variety and nice taste. (D) sufficient to provide variety and nice taste but insufficient to provide exoticism (like eating rare and expensive items, dining in good restaurants frequently). (E) sufficient to provide variety, nice taste, and exoticism.

The question then asked: "Suppose your preference for food increases while your preferences for other goods and things remain unchanged, and your con- sumption of food and of other goods also remain unchanged, other things being equal, what will happen to your subjective welfare (happiness) if you are in: (please tick one box for each situation)." A respondent has to tick, for each situation, whether welfare increases substantially, slightly, remains unchanged, decreases slightly, or substantially. Apart from these five choices, a respondent may also tick, for each situation, either of the following: "Impossible to answer due to insufficient information", "Impossible to answer as the question is meaningless". It is notable that, for respondents in Melbourne, only one ticked "insufficient information" for all situations and only one other ticked "insufficient information" for situation A, and for respondents in Beijing, only one ticked "insufficient information" for situation E. Presumably, the Melbourne respondent had never experienced situ- ation A (hunger) and the Beijing student had never experienced situation E (exotic). Also remarkably, only one respondent in Melbourne and no respondent in Beijing ticked "meaningless". If utility is not cardinal, one would expect more people ticking this box.

To provide a summary of the answers to Question 1, we use a weight of 2 for the answer of "Welfare increases substantially", a weight of 1 for "Welfare increases slightly" a weight of 0 for "Welfare remains unchanged", a weight of - 1 for "Welfare decreases slightly", and a weight of - 2 for "Welfare decreases substan- tially". Admittedly, this is somewhat arbitrary and used only to give a quick summary. The average scores for each situation for both groups of respondents are reported in Table 1.

The results perfectly match the argument of Sect. 2. Cross culturally and economically, people expect their welfare to decrease/increase with an increase in preference for food if their food consumption level is low/high. Moreover, an increase in the food consumption level has a monotonic (and positive) effect on the degree of welfare improvement as preference for food increases. Even more remark- ably, the level of food consumption at which a change in preference for food has no effect on welfare is almost identical for the two very different groups of respondents.

Cardinal utility and commodity units 263

Table 1. Effects of an increase in preference for food on welfare

Situation Melbourne Beijing

A: Awful (Hunger) - 1.25 - 1.62 B: Bad (Malnutrition) - 1.00 - 1.05 C: Common - 0.08 - 0.18 D: Delicious + 0.33 + 1.03 E: Exotic + 0.42 + 1.28

Table 2. Percentages of respondents' food consumption situation

Situation Melbourne Beijing

A: Awful 0% 7.5% B: Bad 0% 36.25% C: Common 10.7% 51.25% D: Delicious 53.6% 5% E: Exotic 35.7% 0%

From Table 1, it may be inferred that such a level of food consumption is just a little more than "just sufficient to avoid hunger and malnutrition, but not adequate to provide variety and nice taste", i.e. just sufficient to provide a little variety and nice taste.

The only significant difference in answers by the two groups is that much higher proportions of the Beijing students than the Melbourne respondets reported substantial welfare increase (as the preference for food increases) when food consumption was at the Delicious and Exotic levels, accounting for the much higher positive average scores of 1.03 and 1.28 (versus Melbourne's 0.33 and 0.42 respectively). Though we have no hard evidence for this, we suspect that this may be due to the concern by some Melbourne respondents on overweight and other health problems of over-eating, making some of them ticking substantial decreases in welfare when food consumption is at the Exotic level.

The answers on "the amount you now normally spend on food corresponds roughly to situation . . . " are summarized in Table 2.

According to the answers, the Melbourne/Beijing respondents spent an average of 26.3%/66.6% of their total expenditure on food. The "minimum percentage of your current total expenditure such that, other things being equal and after an initial period of adjustment, your overall subjective welfare (happiness, taking everything into account) will be on average roughly zero" averages to 28.7%/ 55.7% for the Melbourne/Beijing respondents respectively. Ten out of the 33 Melbourne respondents did not answer this question, four of which indicated they did not understand the question. Only two out of the 40 Beijing students did not answer this question. Overall, this suggests that the concept of zero welfare appears meaningful to most people, especially highly educated people.

"If there is a 20% increase in the budget you can spend on food while the budget on other things remains unchanged, other things being equal, the level of your subjective welfare will probably increase by" 15.4%/34.4% in the short run, and 10.9%/18.4% in the long run for Melbourne/Beijing respondents. "If there is a 20% increase in the total budget of your expenditure (which you may use to buy any number of goods), other things being equal, the level of your subjective welfare

264 Y.-K. Ng and J. Wang

will probably increase by" 18.9%/36.3% in the short run and 18.0%/23.9% in the long run for Melbourne/Beijing respondents. In all these answers, the lower percentage figures for Melbourne respondents over Beijing respondents are clearly as expected, since the latter have lower incomes, supporting the idea of diminishing marginal utility of income. Also, the higher figures for the short run (versus the long run) are also consistent with expectation.

3. Concluding remarks

It is true that there are areas in economics where the concept of cardinal utility is not needed. Occam's razor may then suggest the abstraction of the cardinal aspect and just work with ordinal preferences. However, for areas where cardinal utility is helpful or even indispensable, to persist in refusing to accept cardinal utility is to commit the fallacy of misplaced abstraction. Take an analogy in mathematics. For topology, we may abstract away properties like length, size, shape and weight, and concentrate on properties that are invariant with respect to continuous deforma- tion. After learning topology, if we ignore the relevance of length, weight, etc. in the theory and practice of geometry, aviation, engineering, medicine, etc., we will either be getting nowhere or crash and poison ourselves to death.

It is also true that there are practical difficulties in measuring individual utilities cardinally and compare them interpersonally. However, the cardinal measurability of individual utility in principle is obvious. It does not just make sense for us to prefer $2 million and $1 million and prefer $1 million to $0; it also makes perfect sense for us to say that our preference of $1 million over $0 is stronger than our preference of $2 million over $1 million. In fact, with just ordinal utility, the concept of diminishing (or increasing) marginal utility of income is meaningless. That the concept is really meaningful means that utility is cardinal. The interpersonal positive (in contrast to normative) comparability of utility in principle is also compelling, at least if a materialist solution to the mind-body problem is enter- tained, as argued in Ng (1992). Moreover, while the practical difficulties of measurement and comparison are substantial, they are not insurmountable.

Choices involving risks provide partially cardinal (and subjecive) utility indices. These may be supplemented by the method of income evaluation (van Praag 1968; van Praag and Kapteyn 1973; van Praag and van der Sar 1988), happiness surveys (Veenhoven 1984 and referees therein) and indirect measurement (Ng 1975, Section 9) and other methods (see Diakoulaki and Koumoutsos 1991, Seidl 1988 and references therein). Moreover, if we use the just perceivable utility (Edgeworth 1881;Ng 1975) as the unit of measurement, the cardinal utility indices are interper- sonally comparable. (The reply to critics of this approach can be found in Ng 1981.) We hope that economists will spend more time tackling difficult but important problems of cardinal utility measurement and preference changes instead of ignor- ing them.

References

Baumol, WJ (1977) Economic theory and operation analysis (4th ed) Prentice-Hall, London Becker GS (1992) Habits, addictions, and traditions, Kyklos 45:327-345 Chipman JS (1960) The foundation of utility. Econometrica 28:193-224

Cardinal utility and commodity units 265

Cowen T (1989) Are all tastes constant and identical? A critique of Stigler and Becker. J Economic Behav Organiz 11:127-135

Diakoulaki D, Koumoutsos N (1991) Cardinal ranking of alternative actions: extension of PROMETHEE method. Eur J Oper Res 53:337-347

Edgeworth FY (1881) Mathematical psychics: an essay on the application of mathematics to the moral sciences. Kegan Paul, London

Fisher FM, Shell K (1968) Taste and quality change in the pure theory of the true cost-of-living index. In: Value, capital and growth: papers in honour of Sir John Hicks Wolfe JN (ed) Edinburgh University Press, Edinburgh

Hahn FH (1982) On some difficulties of the utilitarian economist. In: Utilitarianism and beyond. Sen A, Williams B (Eds) Cambridge University Press, Cambridge

Hammond PJ (1991) Interpersonal comparisons of utility: Why and how they are and should be made. In: Interpersonal comparisons of well-being Elster J, Roemer JE (eds) Cambrdige University Press, Cambridge

Kapteyn A, Wansbeek TI, Buyze J (1980) The dynamics of preference formation. J Economic Behav Organiz 1:123-157

Kemp MC, Ng Y-K (1977) More on social welfare functions: the incompatibility of individualism and ordinalism. Economica 44:89-90

Kemp MC, Ng Y-K (1987) Arrow's independence condition and the Bergson-Samuelson tradition. In: Arrow and the foundations of the theory of economic policy Feiwel G (ed) Macmillan, London, pp 223-241

Mueller DC (1989) Public choice II. Cambridge University Press, Cambridge Ng Y-K (1975) Bentham or Bergson? Finite sensibility, utility functions and social welfare

functions. Rev Economic Studies 42:545-569 Ng Y-K (1981) Bentham or Nash? On the acceptable form of social welfare functions.

Economic Record 57:238-250 Ng Y-K (1984a) Interpersonal level comparability implies comparability of utility differ-

ences. Theory Dec 17:141-147 Ng Y-K (1984b) Expected subjective utility: is the Neumann-Morgenstern utility the same

as the neoclassical's? Soc Choice Welfare 1:177-186 Ng Y-K (1990a) Welfarism and utilitarianism: a rehabilitation. Utilitas 2:171-193 Ng Y-K (1990b) Social welfare economic policy. Harvester Wheatsheaf London Ng Y-K (1992) Utilitarianism and interpersonal comparison, Soc Choice Welfare 9:

1-15 Pollak R (1976) Habit formation and long-run utility functions. J Economic Theory 13:

272-297 Pollak RA (1979) Bergson-Samuelson social welfare functions and the theory of social

choice. Quart J Economics 93:73-90 Samuelson PA (1947) Foundations of economic analysis. Harvard University Press,

Cambridge Schokkaert E (1982) A general framework for the analysis of preference variation. Eur

Economic Rev 19:159-179 Seidl CC (1988) Experimental utility functions of income and their associated tax schedules.

Institut ftir Finanzwissenschaft Discussion Paper No. 22 Sen A, Williams B (eds) (1982) Utilitarianism and beyond. Cambridge University Press,

Cambridge Stigler GJ, Becker GS (1977) De gustibus non est disputandum. Am Economic Rev 67:

76-90 Strotz R (1955-1956) Myopia and inconsistency in dynamic utility maximization. Rev

Economic Studies 23:165-180 van Praag BMS (1968) Individual welfare functions and consumer behaviour. North-

Holland, Amsterdam van Praag BMS (1991) Ordinal and cardinal utility: an integration of the two dimensions of

the welfare concept. J Econometrics 50:69-89 van Praag BMS, Kapteyn A (1973) Further evidence on the individual welfare function of

income: an empirical investigation in The Netherlands. Eur Economic Rev 4:33-62

266 Y.-K. Ng and J. Wang

van Praag BMS, van Der Sar NL (1988) Household cost functions and equivalence scales. J Human Resources, 23:193--210

Veenhoven R (1984) Conditions of happiness. Dordrecht, Holland von Weizs~icker CC (1971) Notes on endogenous changes in tastes. J Economic Theory 3:

345-372 Wang J, Ng Y-K (1993) The pursuit of relative utility in general competitive equilibrium.

Typescript Witzum A (1992) Review of Ng (1990b) Economic J 102:171-173