YEW-KWANG NG

I N T E R P E R S O N A L L E V E L C O M P A R A B I L I T Y I M P L I E S

C O M P A R A B I L I T Y OF U T I L I T Y D I F F E R E N C E S *

It is generally believed that the possibility of making interpersonal com-

parisons of utility levels (as required, e.g., in applying the Rawlsian maximin criterion) does not require interpersonal comparison of utility differences (which involve cardinal utilities). Thus, if we are confined to level compar- ability, it is believed that no cardinal utility need be involved. Here, I wish

to show that, with some very general conditions, the general possibility of interpersonal comparison of utility levels implies interpersonal comparison of utility differences (or unit comparability, with suitable scaling). Thus, the

sufficiency of ordinalism is a mistaken belief. Interpersonal comparisons of utility levels involve saying whether one indi-

vidual is better or worse off than (or the same as) another individual in the

same or a different social state, e.g., UJ(x) > UK(x) or UJ(x) = UK(y). Inter- personal comparisons of utility differences involve saying whether the utility

difference between two social states for one individual is higher/lower than, or equal to, the utility difference between the same two or some other two

social states for another individual, e.g., Ua(x)- Ua(y)> Ur(x ) - U~:(y), or UJ(x)- UJ(y)= Un(z)- UK(w). 1 Obviously to be able to make such

comparisons of utility differences, we have to stick to a particular set of

cardinal measures of utility (one cardinal measure for each individual). The

utility index for each individual has to be unique up to a proportionate trans-

formation (i.e., ratio-scale measurability, more cardinal than the Neumann- Morgenstern index which is unique up to a positive afine transformation).

But level comparability has been taken as necessitating purely ordinal utility indices unique only up to a positive monotonic transformation, i.e., coordinal transformation.

The general belief that the Rawtsian maximin criterion involves only ordinal but not cardinal utilities is quite widespread. The following quotations from two of the most prominent writers in the area suffice to illustrate the point.

Theory and Decision 17 (1984) 141-147. 0040--5833/84/0172-0141500.70. �9 1984 by D. Reidel Publishing Company.

142 YEW-KWANG NG

Unlike the sum-of-utilities approach, however, this [i.e., Rawls's maximin criterion] does not require that the units in which different individuals' utilities are measured be com- parable, only that we be able to rank different individuals according to some scale of satisfaction . . . the maximin criterion requires only interpersonal ordinality, whereas the classical view requires interpersonally comparable units. (Arrow, 1973, pp. 253-254, italics added.)

Rawls's (1971) 'maximin' conception of justice . . . deliberately avoids comparisons of [utility] gains and losses . . . basing social preference on the levels of individual welfare without regard to cardinal measures that permit comparisons of gains and losses. . . (Sen, 1974, p. 393.) . . . we do not need at all a cardinal measure of individual welfare levels for the Rawls ordering (Sen, 1970, p. 393.)

I am not going to argue that the above quotat ions are wrong as I do need

two conditions to establish my result that the general possibility of inter-

personal comparison of utility levels implies interpersonal comparison o f

uti l i ty differences. However, since both conditions are very generally applic-

able, the above quotations, being writ ten by influential authors, may well be

very misleading to uncareful readers. (See also the subtle difference between

unit comparabil i ty and comparabil i ty of uti l i ty differences, a distinction

I owe to Arrow, discussed in Remark 1 below.)

EXISTENCE OF TWO POTENTIALLY "OVERLAPPING" INDIVIDUALS

(POI). (a) Among all individuals and all social states, there exist (at least) two

individuals such that one is judged bet ter off under some social state than

another under some (the same or different) social state and the reverse is true

for some other social state(s). (b) For at least one pair of individuals satisfying

Condit ion (a) neither individual is indifferent between all social states.

If Condit ion (a) in POI does not hold, all individuals can be ranked first,

second . . . . . such that, for all n, the nth individual dominates the (n + 1)th

in the sense that he is bet ter off in his worst social state than the other in his

best. This is obviously a very special case unlikely to apply for all practically

relevant cases. Condit ion (b) is almost certainly satisfied for all individuals,

not to mention just a pair. Thus, POI is a very general condition. Even if we

refrain from making interpersonal level comparison across different social

states, POI is still a very general condition. There may exist some individual

who is better off than another under any relevant social state. But this is

ex t remely unlikely to apply for all pairs of individuals in a large society.

INTERPERSONAL LEVEL COMPARABILITY 143

SEMI-CONNECTEDNESS AND CONTINUITY. (a) Semi-connectedness of

social states: For each individual i (or at least for each of some pair of indi- viduals satisfying POI), if xPiy, there exists another social state z such that yliz and there exists a continuum of social states connecting x and z. (b) Con-

tinuity of individual preferences.

Semi-connectedness, though not perfectly general, is a considerably less

restrictive condition than the usual one of connectedness. Semi-connectedness does not require all aspects (or dimensions) of (the space of) social states to

be divisible. Essentially, if there exists one (or several) divisible aspect (e.g., personal income) that is sufficiently important to .the individual, Semi-

connectedness will be satisfied. As illustrated in Figure 1, if other aspects

are indivisible, the domain of social states consists of vertical lines (hyper- spaces) disconnected from each other. But if the divisible aspect is sufficiently important, then for any xPiy, if we reduce that aspect (his income) from x by a sufficient amount, we will reach a social state z such that zIiy, assuming the usual continuity of preference which is Part (b) of the above condition

requiring that individual preferences do not jump if the social state is changed by an infinitesimal amount (in the divisable aspect). There may exist some individual of high principle who prefers, say, social states without the use of

Personal income

X

Z

Fig. 1.

--ui

Other aspect(s)

144 YEW-KWANG NG

torture irrespective of what happens to his income and the income levels of all other individuals, such that Semi-connectedness does not apply to him.

But we require Semi-connectedness and Continuity to be satisfied only for a

certain pair of individuals satisfying POI. The condition may thus be expected to apply for practically all relevant cases.

PROPOSITION: Given the Existence of two Potentially "Overlapping" Indi-

viduals (POI) and Semi-connectedness and Continuity, the general possibility

of interpersonal comparison of utility levels implies interpersonal comparison

of utility differences.

Proof. From POI, there exist two individuals J and K and social states x, x' (may be the same or distinct) and y, y' such that :2

(1) VJ(x) > V~'(x'),

(2) uK(y) > uJ(y'),

and such that xPay ', and yPKx' (due to Condition (b) in POI).

From Semi-connectedness, there exists z such that zlJy ' and there exists a continuum of social states connecting x and z. (For illustration, see Figure 2 which is not necessary for the proof.) Thus, as we move along (or within) this

, $

i - u f

J's indifference map

Y

\u~

Fig. 2.

K's indifference map

I N T E R P E R S O N A L L E V E L C O M P A R A B I L I T Y 145

continuum from x towards z, U J is changed continuously from UJ(x) and

(at least eventually) reduced towards UJ(z) = UJ(y'). Similarly, there exists

w such that x ' IKw and there exists a continuum of social states connecting

y and w. As we move within this continuum f r o m y towards w, U K is changed

continuously from UK(y) and eventually reduced towards UK(w) = UK(X').

Thus, there exist social states s and t (distinct or otherwise) such that

(3) u J ( s ) =

or so it is judged, and there exist social states q and r such that

(4) UJ(s) > UJ(q) = ut((r) < uK(t).

This holds since the s and t in (3) can be chosen such that UJ(s) > UJ(z) and UK(t) > UK(w).

From (3) and (4),

(5) UJ(s) -- UJ(q) = uZ((t) -- ug(r),

which is interpersonal comparison of utility differences. Q.E.D.

Remarks

(1) After establishing the above proposition, I concluded in the first draft o f

this paper that, with our two reasonable conditions, the general possibility

of interpersonal comparison of utility levels implies interpersonal compara-

bility of utility differences or units. However, following a helpful comment

from Arrow, we may distinguish a subtle difference between comparability

of utility differences and comparability of utility units. For a particular set

of scales for individual utility functions, we may have, say,

(5') ug(s) - ug(q) = 20 utils = Un(t) -- UK(r) = 10 utils.

This does not imply that 2 utils for J always equal 1 util for K throughout

all social states. Even if this happens to be true for a particular set of utility

scales, it cannot remain so under arbitrary but coordinal transformation of the utility scales. Thus, for some other social states, and/or for some other

set o f utility scales, we may have

(6) UJ(a) -- UJ(b) -- 5 utits = UK(c) -- UK(d) = 7 utils.

146 YEW-KWANG NG

and (5 ') and (6) need not be inconsistent. In this sense, a unit o f utility is

non-comparable. However, what is important (for social choice anyway)

is interpersonal comparability of utility differences between social states.

Once we have such comparability, we can always choose a set of utility

scales representing individual preferences such that the units are also com-

parable, a (2) It is not difficult to see that full interpersonal comparability of utility

differences or units (with suitable scaling) does not imply any level com-

parability. This is so since unit comparability is invariant with respect to

arbitrary addition to or subtraction of constants from individual utility

functions. (3) In response to the first draft of this paper, Sen makes the observation

that utilitarianism is informationally more demanding in a w6rld of more

than two persons than in a two-person world. For example, one might be

able to say that

U' (x) -- Ul (y) > U2(y) --,V~, (x) > V3(y) -- U3(x) > O,

without being able to say whether Ei Ui(x) >, <, = F-,i Ui(Y), though such

a comparison is possible for the two-person case. This is an interesting

observation that did not occur to me." However, it does not affect our con-

clusion. In the proof of our proposition, (5) is established as an equality,

not just an inequality, thanks to Semi-connectedness and Continuity. It

is true that, for a given pair of social states, the relevant comparison of

utility differences may be in the form of inequalities rather than equalities.

But if we can establish equalities of utility differences for other social states,

we could choose a set of utility scales such that we also have comparability

of utility units as remarked above. Then for any two utility differences

that are unequal, we do not just know which one is larger but also by

how many times one utility difference is larger than the other. This will then provide one with sufficient information to practice utilitarianism if one

chooses to. It is true that the actual comparison of utility differences may involve

many practical difficulties and will probably be very imprecise. (See, how- ever, Ng (1975) on ways of practical comparison.) But our proposition shows that, if this is so, the same must be true for the comparison of utility

levels.

I N T E R P E R S O N A L I, E V E L C O M P A R A B I L I T Y 147

N O T E S

* I am grateful to Kenneth Arrow and Amartya Sen for helpful comments. i For a more detailed discussion of interpersonal comparison, see Sen (1979). 2 If we refrain from making interpersonal level comparison across different social states, x andx ' are taken as the same social state, and so arey andy ' . 3 Basu (1982) makes the subtle distinction between comparability of utility differences in the traditional framework (his Axiom 2) and that in the modern social choice frame- work (Axiom 2*) and shows that the former implies cardinal utility while the latter implies cardinality with an additional assumption on the connectivity of u(x) which is satisfied given our assumption of Weak Connectedness and Continuity.

R E F E R E N C E S

Arrow, KennethJ . : 1973, 'Some Ordinalist-Utilitarian Notes on Rawls's Theory of Justice', Journal of Philosophy 7 1 , 2 4 5 - 2 6 3 .

Basu, Kaushik: 1982, 'Determinateness of the Utility Function: Revisiting a Controversy of the Thirties', Review of Economic Studies 49, April, 307-311.

Ng, Yew-Kwang: 1975, 'Bentham or Bergson? Finite Sensibility, Utility Functions, and Social Welfare Function', Review of Economic Studies 42, 545-570 .

Sen, Amartya K.: 1970, 'Interpersonal Aggregation and Partial Comparability', Econo- metrica 38, 393-409.

Sen, AmartyaK.: 1974, 'Informational Bases of Alternative Welfare Approaches', Journal of Public Economics 3 , 3 8 6 - 4 0 3 .

Sen, AmartyaK.: 1979, 'Interpersonal Comparisons of Welfare', in M. Boskin (ed.), Economics and Human Welfare: Essays in Honor of Itbor Scitovsky, Academic Press, New York. Reprinted in Sen, Choice, Welfare and Measurement, BlackweU, Oxford, 1982.

Department o f Economies,

Monash University,

Clayton, Victoria 3168,

Australia