The Review of Economic Studies, Ltd.

The Economic Welfare Function and Dr. Littles CriterionAuthor(s): Charles KennedySource: The Review of Economic Studies, Vol. 20, No. 2 (1952 - 1953), pp. 137-142Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2295847 .

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The Economic Welfare Function and Dr. Little's Criterion1

i. One of the less amiable characteristics of that interesting building site known as " Welfare Economics " is that the work of the builder on it tends to be more difficult and dangerous than the work of the demolition expert. This is perhaps particularly noticeable when the same contractor undertakes both demolition and building work. I had better confess at the outset that the present article is in the main a work of further demolition. Towards the end of the article, I shall attempt a very limited amount of building work myself ; but while I cannot expect that the result will be any more difficult to knock down than other constructions on the site, I comfort myself that it will be so small a structure that few self-respecting demolition experts may think it worth their while to do so. The task of demolition I am setting myself is possibly more important, for it concerns Dr. Little's basic criterion, which, in his own words, " underlies most of the analysis of his book."2 However, I should not wish to embark on this task without first paying a tribute to Dr. Little's very great skill in the trade-especially, one must add, in demolition work.

For the observer looking on from outside the palings, as well as for those gentlemen who from time to time survey the work in progress from the windows of the Cardinal Club nearby,3 it is intriguing to speculate on the various associations and relationships which appear to exist between the different enterprises at work on the site. On the evidence available, it would seem reasonable to suppose, for example, that Dr. Little's one-man firm is in close partnership with the powerful modern American concern, Functional Welfare Structures Inc., whose present managing director is Professor Samuelson.4 This is not to suggest that they are not to be seen having an occasional disagreement. Thus Dr. Little complains that the American corporation are over- ambitious and are intent on building a skyscraper where a smaller and more adaptable type of building would be of greater practical use5; while Professor Samuelson retorts that Dr. Little's designs seem to leave the building only three-quarters finished.6 Nevertheless, as far as basic principles of design are concerned, they appear to see very much eye to eye7 ; and it is well known that they co-operated in the task of dismantling the work of the other main firmn of contractors on the site, New Welfare

1 I have many acknowledgments to make. The ideas presented in this article were first formulated as a result of a discussion meeting of a number of Oxford economists, held under the auspices of Professor and Mrs. Hicks, at which Dr. Little read a paper on Professor Arrow's review article on Dr. Little's book (see later footnotes for references). At a later meeting, the ideas presented below were also discussed. I am, therefore, indebted to Dr. Little, to Professor and Mrs. Hicks, and to many other members of the group. I should mention especially Professor Hicks, to whom I am indebted for some suggestions which are-I think-partly embodied in paragraph 6 below, and Mr. C. R. Ross, who has given me invaluable advice at all stages in the preparation of the article. Finally, I am indebted to my colleagues, Mr. U. S. Haslam- Jones and Mr. A. D. Woozley, for help on mathematical and philosophical points respectively. No one but myself must be held responsible for any errors. 1

2 I. M. D. Little, A Critique of Welfare Economics, O.U.P., I950, Appendix, p. 273. See also especially Chapters VI and XII.

3 D. H. Robertson, " Utility and All That," Manchester School, May, I95I, p. I40. 4 A. Bergson, " A Reformulation of Certain Defects in Welfare Economics," Quarterly Journal of

Economics, February, I938; P. A. Samuelson, Foundations of Economic Analysis, H.U.P., I948, Chapter VIII; P. A. Samuelson, " Evaluation of Real National Income," O.E.P., January, I950.

5 Little, op. cit., p. ii6. 6 Samuelson, " Evaluation of Real National Income," p. 29n. 7 Samuelson, loc. cit. ; Little, op. cit., p. II2.

137

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I38 THE REVIEW OF ECONOMIC STUDIES

Productions, Limited, who-it will be remembered-had made the claim that all their work was done only on scientific principles.'

Now I believe that this appearance of partnership, however much it may have been fostered by Dr. Little and Professor Samuelson themselves, is largely superficial. Indeed, lying among the stores of Functional Welfare Structures, Inc., is a stick of dynamite all ready to blow up the foundations of Dr. Little's structure. If Professor Samuelson will allow me, I intend to make use of it in what follows.

2. Much of the argument of this article rests on a self-evident proposition of a mathematical nature. It will be best to state this proposition straight away and in the most general form possible: its relevance to welfare economics will be explained later. Let E be an ordinal function of a number of variables U, V, etc., which are themselves ordinal functions of some or all of a number of other variables x, y, etc., so that we may write:

E=E (U, V, . . .). Let us define a situation as being a given configuration of the variables x, y, etc., and denote two different situations by the subscripts I and 2. Then, if the field of each of the variables U, V, etc., is completely ordered, so that for any two situations we can always say, for example, that U2 is greater than, less than or equal to U1, the E field may or may not be completely ordered according to the nature of the function. We shall be more interested, however, in the case where one of the variables-say U- is only partially ordered. If, for any two situations, U2 and U1 are not ordered, what can be said about the ordering of E2 and E1 ? Clearly, there are two possibilities

either (i) E2 and E1 are not ordered, or (ii) E2 and E1 are ordered;

but, for (ii) to be true, it is obvious that the variables V, etc. (i.e. excluding U) must, themselves be sufficient to determine the order of E2 and E1. This is the i equired proposition, which we may state as follows:

For any two situations, if the values of one variable are not ordered, then the values of the function cannot be ordered unless the other variables are sufficient to determine the order of them.2 3. How can this be applied to the problems of welfare economics ? First of all,

we must be careful to specify which particular problem we are considering. To begin with, we shall confine ourselves to the problem of the valuation of the social income. In the literature,3 this problem has usually been discussed with reference to national income comparisons for different years. However, in order to avoid the special difficul- ties which arise as a result of the lapse of time, we shall be concerned simply with the problem of comparing economic welfare in different situations. Any results we obtain will be applicable also, with a number of added qualifications, to the comparison of situations at different times. Moreover, our results will serve as a basis for discussion of the other closely related problem of welfare economics, the problem of finding

1 N. Kaldor, " Welfare Propositions of Economics and Interpersonal Comparisons of Utility," Economic Journal, September, 1939; J. R. Hicks, " The Foundations of Welfare Economics," Economic Journal, December, 1939. See also Hotelling, " The General Welfare in Relation to Problems of Taxation and Railway and Utility Rates," Econometrica, July, I938 I; Hicks, " The Rehabilitation of Consumer's Surplus," REVIEW OF ECONOMIC STUDIES, I940-I ; de Scitovsky, "A Note on Welfare Propositions in Economics," REVIEW OF ECONOMIC STUDIES, 194I-2; Baumol, "Community Indifference," REVIEW OF ECONOMIC STUDIES, I946-7; Kaldor, "A Comment." ibid.

2 Notice that it is not necessary that the other variables should be sufficient to order the values of the function for all situations, but only for situations where the one variable is not ordered.

3 Hicks, " The Valuation of Social Income," Economica, May, 1940; Kuznets, " On the Valuation of Social Income," Economica, February and May, 1948 * Hicks, " A Comment on Professor Kuznets' Reflec- tions," Economica, August, I948; Little, op. cit., Chapter XII.

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ECONOMIC WELFARE FUNCTION AND DR. LITTLE'S CRITERION I39

criteria on which to decide whether a given economic change would be desirable. But since this latter problem is in some ways a more complicated one, we shall postpone any discussion of it until later in the article.

The applicability of paragraph 2 above to the problem under consideration is immediately apparent as soon as we identify E with economic welfare and U with the total utility function. Then our problem becomes one of ordering the values of E for different situations ; while nearly all the difficulties of the problem arise from the fact that, so long as interpersonal comparisons of utility are not permitted, the U- function is incompletely ordered. If we make the simplest hypothesis about the nature of the Economic Welfare Function and suppose that it depends only on total utility, so that we may write:

E=E (U), our problem is quite straightforward. If the values of U are ordered for any two situations, the values of E will also be ordered1; but if the values of U are not ordered, it will be impossible to order the values of E. Thus, if we suppose that the total utility function depends on the different utilities for the individuals in the community and that interpersonal utility comparisons are inadmissible, it will clearly be impossible to compare the economic welfare of two situations where some persons have greater utility in the one situation, and some have less. No consideration of hypothetical or actual compensation either in the one situation or the other-neither the Kaldor-Hicks nor the Scitovsky test2-can possibly help us to get over this basic difficulty.

At this point, Dr. Little proposes a solution to the difficulty by introducing the " plain man " into the problem and asking him to form a judgment about the " dis- tribution of welfare " in the two situations. He then concludes that economic welfare will be greater in situation 2 than in situation i if two conditions hold

(i) the Scitovsky test is met, (ii) the " distribution of welfare " is no worse in 2 than in i.

This is Dr. Little's basic criterion, and it rests on the following argument.3 The meeting of the Scitovsky test implies that no lump-sum redistribution of income in situation i can lead to a situation in which everyone has greater utility than in situation 2. This in turn implies that, if lump-sum transfers are made in situation i such that a new situation (Ia) is reached in which the distribution of welfare is roughly the same as in 2, then everyone will have greater utility in 2 than in Ia-i.e. economic welfare is greater in 2 than in Ia. The second condition of Dr. Little's criterion then implies that the " distribution of welfare" is no worse in ia than in i. Thus, economic welfare is greater in 2 than in ia and no less in ia than in I. Hence it is greater in 2 than in I.

4. Some criticism of Dr. Little's criterion has already been made on the grounds that the " distribution of welfare " is not an altogether clear or unambiguous concept.4 Rather than follow up this line of approach and embark on a discussion of the logic of the phrase " a better distribution of welfare," we shall investigate instead the formal implications for the Economic Welfare Function of Dr. Little's suggestion. From this point of view, Dr. Little's proposal may be given one of two interpretations.

1 Assuming, of course, that E is a monotonically increasing function of U. 9 These tests are by now familiar enough not to require further explanation here. An account of them

is to be found in Little, op. cit., Chapter VI. 8 Little, op. cit., Appendix, pp. 273-4. 4 By Professor Arrow in " Little's Critique of Welfare Economics," American Economic Review,

December, I95I. With many of Professor Arrow's remarks I am in agreement; but it does not seem to me that he presses his criticism far enough. Indeed, at one point (p. 932) he appears to put forward a solution hardly different at all from that provided by Dr. Little in the Appendix to his booQl,

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I40 THE REVIEW OF ECONOMIC STUDIES

First, it may be interpreted as introducing a new variable-say V-into the Economic Welfare Function. We need not be concerned about the precise meaning of V in economic terms. Accordingly we may write:

E=E (U, V) where, once again, U is only partially ordered. Now our proposition in paragraph 2 tells us that, for any two situations for which U is not ordered, V must be sufficient to determine the ordering of the E-function, if the E-function is capable of being ordered at all. But this means that any " plain man " who is able to compare economic welfare in situations i and ia on the basis of the V-function must equally well be capable of comparing economic welfare between situations i and 2. The Scitovsky test seems, therefore, to be an unnecessary part of Dr. Little's criterion.

Secondly, Dr. Little's " plain man " may be interpreted as simply completing the ordering of the U-function, i.e. as making the interpersonal utility comparisons which the " economist " is inhibited from making.' Thus

E=E (U) where U is now completely ordered. But, in this case also, the " plain man " who is able to compare economic welfare between situations I and ia will be able to compare the economic welfare of situations i and 2 directly. Once again, the Scitovsky test is redundant.

The above two formal interpretations of Dr. Little's proposal are in a sense basic. While it would be easy to suggest more complicated interpretations by combining the two or by adding further variables to the Economic Welfare Function, we should still reach the same conclusion that the " plain man " would be able to compare situations i and 2 directly and would not need the assistance of the " intermediate " situation ia. Indeed, Dr. Little's error lies surely in his regarding situation ia as being essentially different from situation 2 in its relationship to situation I. In fact, the only distin- guishing feature of situation ia is that it is reached from situation i by lump-sum transfers. But this distinction would only be of significance if the " plain man " were to base his comparison of situations i and ia on a judgment about the means of moving from i to Ia. And this would surely be a dangerously misleading method of ordering economic welfare in different situations: there would be no certainty that a consistent welfare field would be obtained as a result.2

After an investigation of the purely formal properties of the Economic Welfare Function we are able to reach the conclusion that the Scitovsky test is an unnecessary part of Dr. Little's criterion. It must therefore be discarded-not merely on the principle of Occam's razor, but also because it is a positive nuisance, in that it inhibits us from making welfare comparisons in cases where the test is not met.3 This means,

ICf. Little, op. cit., p. IO9. 2 At this point a possible objection to the above analysis should be raised, which is best discussed with

reference to the familiar two-person utility map (see, for example, Little, op. cit., p. 274). Would it not be possible to define an economic welfare function such that E = E (U, V) where U determines the ordering of E for points lying in a S.W.-N.E. direction and V determines the ordering of E only for points lying along the utility-possibility curve ? In this way, the use of intermediate points would enable us to order pairs of points even in cases where neither the U-function nor the V-function were ordered, and a fairly complete welfare ordering could be built up. The weakness of such a scheme would be that the ordering of points would not necessarily be consistent. It could only 'be made so by the adoption of some far-reaching rule for the V-function-for example, that any moven~ent north-westward along the utility-possibility curve increased V and thus economic welfare. Even in this case, consistency is assured only because our economic welfare function reduces to one of a much simpler type. If all movements north-westward increase economic welfare as well as all movements north-eastward, this amounts to saying that all movements northward do so. In other words, economic welfare would depend only on the utility of one of the two persons.

3 Cf. Little, op. cit., p. I07 and p. 275. It will be noticed that, if the Scitovsky test is discarded, the second condition of the criterion must be amended to read " the distribution of welfare is better in 2 than in I "; and this must be taken to mean no less than that 2 is a better situation than i.

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ECONOMIC WELFARE FUNCTION AND DR. LITTLE'S CRITERION I4I

however, that we are left to rely entirely on the judgment of the " plain man." The " economist " has no longer any part to play in this branch of welfare economics: he must resign himself to the trivial task of comparing pairs of situations in cases where everyone has greater utility, or everyone less utility, in one of the pair.

5. This completes our discussion of the problem of the valuation of the social income, which, as we have argued above, is the fundamental problem of welfare economics.L The problem of finding criteria on which to decide whether a given economic change would be desirable presents a number of further complications, which arise mainly as a result of a certain ambiguity attaching to the phrase " a given economic change." If we interpret " a given economic change" as meaning simply a change from one situation to another, then our problem is a relatively simple one and almost identical with the problem of the valuation of the social income. Our only added concern is to make sure that the method of moving from one situation to another is not in itself harmful.2 In Dr. Little's treatment of the subject, however, " a given economic change" is given a wider interpretation. He concludes that the same basic criterion is applicable to this problem also, but in this case the Scitovsky test has a quite different and more significant part to play. While it cannot help us to determine whether a move from situation I to situation 2 is desirable, it does tell us whether a move from i to 2 is more desirable than a move from I to Ia. In other words, Dr. Little is asking himself not one question but two-he is discussing the desirability of a move from either i or ia to 2.3 If the question is put in this form, no exception can be taken to Dr. Little's answer. Rather, we must ask ourselves whether, once the meaning of an " economic change" has been partly extended in this way, we should stop short where Dr. Little has done. No good purpose would be served by following up this line of thought here, since it has already been exhaustively examined by Professor Samuelson.4 In the end, it is difficult to escape the conclusion-nor would Professor Samuelson wish us to-that once the question " would such and such a change be desirable ? " is given its wider interpretation, it is no longer a very useful question to ask. May it not be precisely for this reason that discussions of political issues which take the form " should such and such a change be made ? " seem so unsatisfactory ? Very often, not one but many questions are being disputed, and it is no wonder that the disputants should be at cross-purposes.

6. Can a more useful type of question be asked ? I think it can. The task of the political economist and for that matter of the civil servant and the Royal Commission is not to give yes-or-no answers to questions fired at them by someone else ; but rather to make recommendations for policy and to formulate policy. If this is accepted, then a more useful type of question to ask in welfare economics would be " what ought to be done ? " or " what would you recommend ? " Even the " economist "-where the inverted commas signify an unwillingness to make interpersonal utility comparisons -is not precluded from making policy recommendations, so long as he prefaces them by some such proviso as " if you cannot think of anything better to do, .. . " or " if you do not think this or that would be a better thing to do, . . . ". Moreover, there is no reason why, in his search for suitable policies to recommend, he should not make use of the tests which the new welfare economics has formulated ; though, in this connexion, it is interesting to note that, since he will always be on the look-out

1 The great merit of the Bergson-Samuelson approach to welfare economics is that it immediately focuses our attention on this problem.

2 Some judgment about the means of moving from one situation to another is clearly relevant to this problem, just as it is irrelevant to the valuation of the social income problem.

3 Little, op. cit., p. I03. 4 Op. cit. See also Arrow, op. cit., pp. 930-I.

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142 THE REVIEW OF ECONOMIC STUDIES

for policies which make everybody better off, it is the Kaldor-Hicks test which will be of use to him and the Scitovsky test which will be irrelevant.'

However, when this has been said, it probably remains true that the " economist" will perform his task more usefully if he is prepared to drop his inverted commas if, in connexion with any policy he is considering, he is prepared to ask himself straight- forwardly whether the gain to the gainers outweighs the loss to the losers-and any other questions which he may think relevant.

Oxford. CHARLES KENNEDY.

1 If any suggested measure passes the Kaldor-Hicks test, this means that the measure plus compensa- tion will make everyone better off. The " economist " will, however, only be in a position to recommend the measure plus compensation if compensation can, in fact, be carried out,

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