OR I G I NAL ART I C L E

The role of diminishing marginal utility in theordinal and cardinal utility theories

Chung-Cheng Lin1,2,3,4 | Shi-Shu Peng2

1Institute of Economics, Academia Sinica, Taipei,Taiwan2Department of Public Finance, National ChengchiUniversity, Taipei, Taiwan3Institute of Business and Management, NationalChiao Tung University, Taipei, Taiwan4Department of Economics, Tunghai University,Taichung, Taiwan

CorrespondenceChung-Cheng Lin, Institute of Economics,Academia Sinica, No.128, Section 2, AcademiaRoad, Nankang Dist, Taipei City 11529, Taiwan.Email: cclin@econ.sinica.edu.tw

We illustrate the role of the law of diminishing marginalutility in the two main modern utility theories, the ordinaland cardinal utility theories, using a generalised total util-ity function. In short, the ordinal utility theory, in whichutility is immeasurable, must abandon the law ofdiminishing marginal utility; the cardinal utility theory,although able to retain this law, suffers from keeping theunrealistic view of utility measurability, which Samuelsoncriticises as “infinitely improbable.” A new utility theorywith the advantages of the two mentioned theories (i.e. thenotions of both diminishing marginal utility and utilityimmeasurability) but without the disadvantages (i.e. thelaw of diminishing marginal utility is excluded and utilityis measurable) therefore still seems to be a Holy Graildeserving search and development by economists.

KEYWORDS

cardinal utility, diminishing marginal utility, ordinalutility

J E L C LA S S I F I C A T I ON

D01; D11

“… the problem of psychological measurement and its bearing on economic theory sofar has proved to be one of the most puzzling riddles.” (Bernardelli, 1938, p. 198)

“… the question whether recent attempts to construct … consumer theory will lead to areintroduction of marginal utility into analyses of consumer behavior… As of today,this question still remains open.” (Hudík, 2014, p. 690)

1 | INTRODUCTION

The notion of diminishing marginal utility is absent in many important microeconomics textbooks.For example, in Intermediate Microeconomics by Varian (1996), we can find only the terms

Received: 3 May 2018 Revised: 17 January 2019 Accepted: 20 February 2019

DOI: 10.1111/1467-8454.12151

Aust Econ Pap. 2019;58:233–246. wileyonlinelibrary.com/journal/aepa © 2019 John Wiley & Sons Australia, Ltd 233

diminishing marginal rate of substitution and diminishing technical rate of substitution, without theterm diminishing marginal utility.1 In Advanced Microeconomic Theory by Jehle and Reny (2011,pp. 4–5), the authors mention the term diminishing marginal utility in order to illustrate why this termwill not be used in the later analysis throughout the whole book. They said,

In additional, the “Principle of Diminishing Marginal Utility” was accepted as a psy-chological “law,” and early statements of the Law of Demand depended on it. These areawfully strong assumptions about the inner working of human beings.

Diminishing marginal utility, however, is a very common and normal concept which ordinarypeople experience in their daily lives as their marginal utility (or marginal satisfaction) becomeslower and lower, for example, when they eat more and more apples.2 It is also an important economicconcept adopted by classical economists such as Gossen, Jevons, Menger, Walras3 and Marshall.4

For instance, the famous Gossen law argues that marginal utility declines with the amount of goodsheld by an economic agent.5

Why is such a common and normal concept rarely mentioned in modern economics? And when itis mentioned, why is it criticised as an outdated, or even awfully strong assumption? How can werestore the law of diminishing marginal utility and at what cost?

As economists, we are trained to analyse the choice behaviour of an economic agent under theanalytical framework of maximising total utility subject to a budget constraint. But what on earth doutility or utility functions refer to? Utility is designed to depict the satisfaction or happiness which aconsumer obtains or feels, but how can we depict a feeling with utility numbers? What characteristicsand constraints will be caused when we use such utility numbers? This study is aimed to discuss thesomewhat surprising relationship between the main utility theories in terms of ordinal and cardinalutility and the law of diminishing marginal utility.

The first assumption of the ordinal utility theory is that an economic agent is capable of ranking dif-ferent bundles of goods in order of his preferences or utilities.6 The preference order of a utility functionshould be maintained under any positive monotonic transformation. Namely, the numbers in ordinaryutility matter only in terms of order, not magnitude.7 However, the signs of the second derivatives ofthe ordinary utility functions before and after a positive monotonic transformation might be different.The law of diminishing marginal utility therefore should be abandoned, since the signs of the secondderivatives cannot always remain the same after any positive monotonic transformation.8

Some economists such as Hicks (1946) support this abandonment by claiming that it is an advan-tage instead of a disadvantage of the ordinal utility theory, as one can develop an individual choicetheory with fewer conditions after the law of diminishing marginal utility is excluded. Some others,however, still do not agree that diminishing marginal utility should be abandoned in economics. Forexample, Bernardelli (1934, 1938) likens such abandonment to “amputation.” Rothbard (1956) evenstates that: “In doing so, they threw out the Utility baby together with the Cardinal bathwater.”9

The fact that the ordinary utility theory and the law of diminishing marginal utility cannot besimultaneously integrated in the same theoretical framework provides the cardinal utility theory theopportunity to be developed and to prevail.

The fundamental assumption of the cardinal utility theory is that an economic agent is assumed tobe capable of ranking not only different bundles of goods in order of his preferences or utilities, butalso different transitions between any two bundles in order of his preferences or utilities. The prefer-ence order of a cardinal utility function is preserved under any positive linear transformation, and thesigns of the second derivatives of the cardinal utility functions before and after such transformationare the same, offering a space for the law of diminishing marginal utility to be preserved. To allow

234 LIN AND PENG

only positive linear transformation implies that an economic agent is capable of distinguishingbetween the relative sizes of any two different utility values, which in turn implies that utility is mea-surable.10 Samuelson (1938, p. 70), through a mathematical proof, points out that the probability ofadding the assumptions required by the cardinal utility theory in real life is almost zero, and thus crit-icises the cardinal utility function as being “infinitely improbable.”

The modern ordinal and cardinal utility theories both seem to have serious flaws. This leads to adilemma whenever we write down a model maximising total utility. For example, Bernardelli (1952,pp. 255–256) has emphasised:

But this identification leads to a very awkward dilemma…that either we are talkingnonsense when we speak of decreasing marginal utility…or that we (illicitly) assumeutility to possess a cardinal measure. Prima facie this disjunction is not very plausible.

To develop a new utility theory that has the advantages but no disadvantages of the two men-tioned theories seems to be a mission impossible so far. Some economists such as Bernardelli (1938,1952) made great efforts to solve this riddle but received severe criticism.11 Bernardelli (1938,p. 198) stated, “the problem of psychological measurement and its bearing on economic theory so farhas proved to be one of the most puzzling riddles.” This riddle seems difficult to be solved, as notedby Lancaster (1953, p. 259), who emphasised that the classical theory “… does suffer from those dis-abilities, and there is nothing that Dr. Bernardelli, or anyone else, can do about it.”

The modern utility-maximising microeconomic and macroeconomic theories with a micro-foundation are, then, built on an imperfect and controversial basis. Some economists do not exactlyknow whether the economic theory they apply belongs to ordinal or cardinal utility theory, while somedo not know the two types of utility theories suffer from these great flaws. Therefore, as stated byBernardelli (1952, p. 267), they do not know that “… In fact, all writers … have been forcing the prob-lems of economics into the straightjacket of preference scales and of marginal rates of substitution…”.

The ordinal total utility theory launched and advocated by Pareto, Allen, Hicks, Samuelson andother great economists who participated in forming and debating its fundamental theories thereforehas never successfully materialised the dream or ideals of the ordinal utility revolution. This isexactly why the ordinal total utility theory is often absent in the fields of macroeconomics, growth,international finance, expected utility theory and other various models. As a result, in many fields ofapplied economics, the ordinal utility theory exists in name only.12

In what follows, we will use a generalised utility function to elaborate the role of the law ofdiminishing marginal utility in the modern ordinal and cardinal utility theories. Through these illus-trations, one will be able to clearly understand that the utility ordinality (which is unique up to anypositive monotonic transformation of utility functions) and the law of diminishing marginal utilityinherently cannot be simultaneously integrated in a theoretical framework, while the utility cardinal-ity (which is unique up to any positive linear transformation of utility functions) preserves this law,but is infinitely improbable.

Although this issue has been raised, discussed and even debated sporadically in the literature, it isdifficult to find an introductory article giving an overview for readers who are not familiar withit. The purpose of this paper is to offer as complete knowledge as possible related to this issue for thereaders who are interested in the very foundation of most economic theories and for the teachers whoare cautious in passing down economics knowledge to future economists.

The remainder of this paper is structured as follows. In Section 2, we show that the ordinal utilitytheory must abandon the notion of diminishing marginal utility, using a generalised utility function.In Section 3, we show that the cardinal utility theory, although keeping this law, fails to retain the

LIN AND PENG 235

notion of utility ordinality, that is, the critical and precious property of utility immeasurability.Section 4 concludes.

2 | THE ORDINAL UTILITY THEORY

In this section, we will illustrate why the law of diminishing marginal utility and the notion of ordinaltotal utility are mutually exclusive, using a generalised utility function.

2.1 | Sign ambiguity of second derivatives of total utility

Let us assume a utility function U(x, m) and its positive monotonic transformed utility function V(x, m) with the transformation function F(�):

V x,mð Þ=F U x,mð Þð Þ;F0 >0,F00⋛0, ð1Þwhere x and m are the two choices facing a consumer, representing, say, the quantity of commodity Xconsumed and the cash or income held by this consumer. The property of positive monotonic trans-formation is reflected by the first derivative being positive, that is, F

0(�) > 0, which means that all

rankings of bundles (x, m) in U(�) are preserved in V(�). The sign of the second derivative will notaffect the positive monotonic transformation at all, so it can be positive, negative, or zero, namely,F

0 0(�) ⋛ 0. The two utility functions U(x, m) and V(x, m) then represent the same utility, and there

should be an infinite number of utility functions that depict the same preference.Taking the first and second derivatives for the two utility functions, one will obtain the following

equations:

Vx =F0Ux, Vm =F0Um, ð2ÞVxx =F0Uxx +F00UxUx,Vmm =F0Umm +F00UmUm,

Vxm =F0Uxm +F00UxUm, Vmx =F0Umx +F00UmUx,ð3Þ

where Ux and Um denote the first derivatives of U(�) with respect to x and m; Uij with i, j = x, mdenote the second derivatives of U(�) with respect to x and m; and the same meaning is applied to Vx,Vm, and Vij with i, j = x, m.

The first equation reveals that signUx = signVx and signUm = signVm, which means that the signsof the marginal utility of x and m will remain the same after any positive monotonic transformation.The notions of positive and negative marginal utility can therefore be used in the ordinal utilitytheory.

More importantly, the second equation reveals that it is possible that signUxx 6¼ signVxx and signUmm

6¼ signVmm. Recall that F0(�) > 0 and F

0 0(�) ⋛ 0. If the original utility function U(�) has the property of

diminishing marginal utility (Uxx < 0), the transformed utility function V(�) can have the property of con-stant, increasing and diminishing marginal utility, depending on the sign and size of F

0 0(�)UxUx. In other

words, the sign of Uxx can be different from that of Vxx, and this holds as well for Umm and Vmm.

2.2 | The decision model under a generalised utility function

We further analyse the above consumer's choice with a generalised utility function in the framework ofordinal utility. Assume that a consumer with wealth or income M faces commodity X with price p, anddecides to buy a certain number of units of x and keeps a certain amount of his wealth or income m,under the consumer model of maximising ordinal total utility. His maximisation problem is as follows:

236 LIN AND PENG

maxx,m

U x,mð Þ; s:t:px+m=M: ð4Þ

Plugging the budget constraint m = M − px into the utility function yields:

maxx

U x,M−pxð Þ: ð5ÞLet A ≡ U(x, M − px), and we can represent the first-order condition as:

Ax =Ux x,M−pxð Þ−pUm x,M−pxð Þ=0orUx x,M−pxð ÞUm x,M−pxð Þ = p: ð6Þ

That there exist interior solutions (note that Uxm = Umx) requires the second-order conditionto be:

Axx =Uxx−2pUxm + p2Umm <0: ð7ÞOne can easily solve for the effects of a change in income and price on the purchased quantity of

commodity, respectively, as follows:

xM =Uxm−pUmm

−Axx, ð8Þ

xp =Um + x Uxm−pUmmð Þ

Axx: ð9Þ

Now let us consider the maximisation problem after the positive monotonic transformation isapplied to the original utility function:

maxx,m

V x,mð Þ=F U x,mð Þð Þ;F0 >0, s:t: px+m=M: ð10Þ

Again, we plug the budget constraint m = M − px into the utility function and obtain:

maxx,m

V x,M−pxð Þ=F U x,M−pxð Þð Þ;F0 >0: ð11Þ

Let B ≡ V(x, M − px), and we can represent the first-order condition as:

Bx =Vx x,M−pxð Þ−pVm x,M−pxð Þ=0orVx x,M−pxð ÞVm x,M−pxð Þ = p: ð12Þ

The existence of interior solution (note that Vxm = Vmx) requires the second-order condition to be:

Bxx =Vxx−2pVxm + p2Vmm <0: ð13ÞAgain, one can easily solve for the effects of a change in income and price on the purchased

quantity of commodity, respectively, as follows:

xM =Vxm−pVmm

−Bxx, ð14Þ

xp =Vm + x Vxm−pVmmð Þ

Bxx: ð15Þ

Now we will show that the results before and after the positive monotonic transformation are thesame. First, the first-order conditions beforehand (Ax = Ux − pUm = 0) and afterwards (Bx = Vx −pVm = 0) are the same, since:

p=Vx

Vm=

F0Ux

F0Um=

Ux

Um: ð16Þ

The two second-order conditions hold at the same time, since:

LIN AND PENG 237

Bxx =F0 Uxx−2pUxm + p2Umm� �

=F0Axx, ð17Þin which the relations of Vij = F

0Uij + F

0 0UiUj with i, j = x, m and the first-order condition of Ax =

Ux − pUm = 0 are applied.The (overall) results of comparative statics should be the same, as the utility functions before and

after the transformation should represent the same preference. We can actually prove that these twoeffects are the same in Appendix A, that is,

Vxm−pVmm

−Bxx=Uxm−pUmm

−Axx, ð18Þ

Vm + x Vxm−pVmmð ÞBxx

=Um + x Uxm−pUmmð Þ

Axx, ð19Þ

where Bxx = F0(Uxx − 2pUxm + p2Umm) = F

0Axx.

We have shown that any positive monotonic transformation of the total utility function does notaffect a consumer's optimisation conditions, nor does it change the overall effects of the comparativestatics. This verifies the characteristics of the indifference curve analysis: the utility functions beforeand after any positive monotonic transformation represent the same preference, and thus the samebehaviour, as stated by Varian (1996, p. 67):

… a utility function is a way of assigning numbers to the different indifference curvesin a way that higher indifference curves get assigned larger numbers. … As long asindifference curves containing more-preferred bundles get a larger label than indiffer-ence curves containing less-preferred bundles, the labeling will represent the samepreferences.

In his famous textbook The Structure of Economics: a Mathematical Analysis, Silberberg (1978,p. 227) said:

The demand curves are independent of any monotonic transformation of the utilityfunction; i.e., they are independent of any relabeling of the indifference map. … Theabove derivation also makes clear why the concept of diminishing marginal utility isirrelevant in modern economics.

We need to emphasise that although the overall results of the comparative statics are the same,the signs and levels of the two terms within the overall results are not always the same. Let us illus-trate this by taking the income effect as an example. The two terms within the overall income effectbefore and after the positive monotonic transformation are not always the same, shown as follows,respectively:

Vxm

−Bxx=F0Uxm +F00UxUm

−F0Axx6¼ Uxm

−Axx, ð20Þ

pVmm

Bxx=pF0Umm + pF00UmUm

F0Axx6¼ pUmm

Axx, ð21Þ

unless F0 0= 0. This implies that we actually cannot assign realistic economic meanings to these two

terms.Without the law of diminishing marginal utility, the ordinal utility theory itself seems enough for

developing a consumer theory. Hence some economists think that the ordinal utility theory is surely

238 LIN AND PENG

an adequate theory, and the law of diminishing marginal utility is a redundant or even meaninglessconcept. Besides, this law is a kind of psychological law, and the psychological phenomenon thislaw governs is actually unobservable since these phenomena are subjective instead of objective.Abandoning such a psychological law with the features of unobservability and subjectivity thereforeshould be viewed as a symbol of theoretical progressiveness in economics.

In his book Value and Capital: An Inquiry into Some Fundamental Principles of EconomicTheory, Hicks (1946, p. 18) supports such a viewpoint by applying the principle of Occam's razor,and said:

The quantitative concept of utility is not necessary in order to explain market phenom-ena. Therefore, on the principle of Occam's razor, it is better to do without it. For it isnot, in practice, a matter of indifference if a theory contains unnecessary entities. Suchentities are irrelevant to the problem in hand, and their presence is likely to obscure thevision.

Still some economists do not agree, and think that the law of diminishing marginal utility is a rea-sonable psychological phenomenon, and is also the theoretical foundation of classical economics.The notion that the ordinal utility theory must abandon the law of diminishing marginal utility iscriticised as a kind of “amputation” or “throwing out the Utility baby together with the Cardinalbathwater.”13

In fact, some economists do not agree on abandoning the law of diminishing marginal utility. Tokeep this reasonable psychological phenomenon, they turn their attention to the cardinal utilitytheory.14

3 | THE CARDINAL UTILITY THEORY

As mentioned earlier, the fundamental assumption of the cardinal utility theory is that an economicagent is assumed to be capable of ranking not only different bundles of goods in order of his prefer-ences or utilities, but also different transitions between any two bundles in order of his preferences orutilities. The preference order of a cardinal utility function is maintained under any positive lineartransformation, and the signs of the second derivatives of the cardinal utility functions before andafter such transformation are the same. The concept of diminishing marginal utility is able to beretained if the signs are both negative.

In what follows, we will illustrate the role of diminishing marginal utility in the cardinal utilitytheory, as well as the derived characteristics and controversies, using a generalised utility function.

3.1 | Sign unambiguity of second derivatives of total utility

Let us again assume a utility function U(x, m), but now make a positive linear transformation toobtain V(x, m) so that:

V x,mð Þ= αU x,mð Þ+ β;α>0,β⋛0, ð22Þwhere U(x, m) and V(x, m) represent the same cardinal utility preference; α and β are constants. Theabove equation suggests the following relations:

Vx = αUx, Vm = αUm, ð23Þ

LIN AND PENG 239

Vxx = αUxx, Vmm = αUmm, Vxm = αUxm, Vmx = αUmx: ð24ÞThe first equation reveals that signUx = signVx and signUm = signVm, which implies that the

signs of the marginal utility of x and m will remain the same after any positive linear transformation.The notions of positive and negative marginal utility can thus be used in the cardinal utility theory.

More importantly, the second equation reveals that signUxx = signVxx and signUmm = signVmm.If the original utility function U(�) has the property of diminishing marginal utility (Uxx < 0), thetransformed utility function V(�) must have the same property of diminishing marginal utility. In otherwords, the sign of Uxx will always the same as that of Vxx, and this holds as well for Umm and Vmm inthe cardinal utility theory.

3.2 | The decision model under a generalised utility function

We further analyse the consumer's choice with a generalised utility function in the framework of car-dinal utility. This analysis before the positive linear transformation has been shown in Equations (4)–(9), so here we only proceed to analysing the one after the transformation. The consumer'smaximisation problem after the positive linear transformation becomes:

maxx,m

V x,mð Þ= αU x,mð Þ+ β, s:t: px+m=M: ð25Þ

Plugging the budget constraint m = M − px into the utility function yields:

maxx

V x,M−pxð Þ= αU x,M−pxð Þ+ β: ð26Þ

Let B ≡ V(x, M − px), and we can represent the first-order condition as:

Bx =Vx x,M−pxð Þ−pVm x,M−pxð Þ=0orVx x,M−pxð ÞVm x,M−pxð Þ = p: ð27Þ

The existence of interior solutions requires the second-order condition to be:

Bxx =Vxx−2pVxm + p2Vmm <0: ð28ÞOne can easily solve for the effects of a change in income and price on the purchased quantity of

commodity, respectively, as follows:

xM =Vxm−pVmm

−Bxx, ð29Þ

xp =Vm + x Vxm−pVmmð Þ

Bxx: ð30Þ

Similarly, we will now show that the results before and after the positive linear transformation arethe same. First, the first-order conditions beforehand (Ax = Ux − pUm = 0) and afterwards (Bx =Vx − pVm = 0) are the same, since:

p=Vx

Vm=

αUx

αUm=

Ux

Um: ð31Þ

The two second-order conditions hold at the same time since:

Bxx = α Uxx−2pUxm + p2Umm� �

= αAxx, ð32Þin which the relations of Vij = αUij with i, j = x, m are applied.

The (overall) results of comparative statics are the same, since:

240 LIN AND PENG

Vxm−pVmm

−Bxx=α Uxm−pUmmð Þ

−αAxx=Uxm−pUmm

−Axx, ð33Þ

Vm + x Vxm−pVmmð ÞBxx

=αUm + xα Uxm−pUmmð Þ

αAxx=Um + x Uxm−pUmmð Þ

Axx, ð34Þ

where Bxx = α(Uxx − 2pUxm + p2Umm) = αAxx.We have shown that the positive linear transformation to the total utility function does not affect

a consumer's optimisation conditions, nor does it change the overall effects of the comparativestatics.

We need to emphasise that not only are the overall results of the comparative statics the same,but also the signs and levels of the two terms within the overall results are the same. Let us still illus-trate by taking the income effect as an example. The two terms within the overall income effectbefore and after the positive linear transformation are the same, shown as follows, respectively:

Vxm

−Bxx=

αUxm

−αAxx=

Uxm

−Axx, ð35Þ

pVmm

Bxx=αpUmm

αAxx=pUmm

Axx: ð36Þ

This implies that we can assign realistic economic meanings to these two terms, which willenhance the explanatory power and theoretical richness.

3.3 | Infinitely improbable measurable cardinal utility

The properties of the cardinal utility theory that signUxx = signVxx and signUmm = signVmm implythat the signs of the rate of change of marginal utility of the commodity and money remain the sameregardless of the positive linear transformation. Hence the cardinal utility theory is able to maintainthe law of diminishing marginal utility that is abandoned in the ordinal utility theory.

However, the concept of utility measurability is embedded in the cardinal utility theory. Lange(1934) illustrates this notion, using a figure of indifference curves, as shown in Figure 1. He repre-sents three indifference curves by I, II and III and assumes that an economic agent is able to tellwhether the difference in utility between I and II is greater or smaller than, or equal to the differencein utility between II and III. Then he adjusts indifference curve III until the mentioned two utility dif-ferences are exactly the same, which he calls “equivalent.”

Apparently, in this case, we can say that the utility difference between consuming I and III is twotimes that between consuming I and II or II and III. Through such a thought experiment, Lange pro-poses a way to construct a determinable utility function. In particular, in Figure 1 we can see that hefocuses on two commodities and constructs such a utility function.

In Figure 1, based on the assumption that one is capable of ranking different bundles of goods inorder of his preferences or utilities, the index values assigned to the indifference curves increase withmore preferred bundles of goods. This is the only condition this assumption requires for the utilityindex. Then, picking any two indifference curves (say I and II), we are able to find a third curve withwhich the increase in utility from II to III is exactly the same as that from I to II. This is possibleaccording to the fundamental assumption of cardinal utility, and we call such a curve an equivalentindifference curve.

Now we represent the utility index by the height of vertical segments. We use an arbitrary verticalsegment P1Q1 with the length of B to represent the index of the indifference curve I, and assumeanother arbitrary vertical segment with the length of A to represent the index difference between curves

LIN AND PENG 241

I and II. Curve II can then be represented by a vertical segment P2Q2 with the length of B + A. Noticethat the difference between the values of the two indices of curve II and III is no longer arbitrary. Basedon the assumption that “people are capable of ranking different transitions between any two bundles inorder of his preferences or utilities,” the length of this difference should be equal to P2Q2 − P1Q1, thatis, A. The index of curve III should thus be represented by a segment of length B + 2A, that is, P3Q3 inFigure 1. Suppose now we consider another equivalent indifference curve IV, then the length of the ver-tical segment representing its index should be B + 3A (i.e. P4Q4 in Figure 1).

Therefore, if we start from any arbitrary indifference curve, consider all possible equivalent cur-ves, and represent their indices by vertical segments as mentioned above, the vertices of these seg-ments will then form a utility surface. On this surface, only two parameters are arbitrary: the fixedmeasurement unit A and the origin point of the fixed measurement B. The function of this surfaceG = G(x, y) is thus the utility function.

Based on the above analysis, Lange claims that the cardinal utility function is one under the con-cept of measurability. In fact, Allen (1935) argues that the positive linear transformation is essentiallya concept of length, such as transforming metres to centimetres. Hence, to allow only the positive lin-ear transformation implies that an economic agent is able to distinguish between the relative sizes ofany two different utility values, which in turn implies that the utility is measurable.

However, nothing is given without a disadvantage in it. It is an obvious restraint that we can onlyapply the positive linear transformation to a utility function. As pointed out by Allen (1935), theparameter of α in such a transformation is equivalent to the “unit of measurement,” while the otherparameter of β is equivalent to the “zero mark” in the length measurement concept. The positive lin-ear transformation is thus a concept like the transformation between length measures such as metresand centimetres. Samuelson (1938), through a mathematical proof, demonstrates that the probabilityof adding the assumptions required by the cardinal utility theory in real life is almost zero, and thuscriticises the cardinal utility function as “infinitely improbable.”

IIIIII

IV

FIGURE 1 Lange's determinable utility function [Colour figure can be viewed at wileyonlinelibrary.com]

242 LIN AND PENG

The ordinal and cardinal utility theories both seem to have serious flaws, leaving the modern util-ity theory in the dilemma, as noted by Bernardelli (1934, p. 71), that “at this point economics reachescross roads” and also by Bernardelli (1952, p. 255) that “But this … leads to a very awkwarddilemma.” On the one hand, we may constrainedly take the view of the ordinal utility theory with theadvantage that preference requires ranking different bundles of goods only, and without the disadvan-tage of utility measurability, but we have to give up related classical theories such as the law ofdiminishing marginal utility. On the other hand, we may reluctantly take the view of the cardinal util-ity theory with the advantage of keeping related classical theories including the diminishing marginalutility, but we have no choice but to embrace the strong concept of utility measurability. Both, how-ever, have great flaws.

4 | CONCLUSION

The two main modern utility theories both have serious shortcomings. To develop a new best-of-both-worlds utility theory with the advantages but without the disadvantages of the two theories isworth striving for but seems to be a mission impossible so far. The enormous modern economic theo-retical system is therefore built on controversial grounds, since the utility theory is the very founda-tion of the choice theory, and in turn of a great many economic theories. The purpose of this paper issimple: we hope to help the readers who are not familiar with this issue to deeply understand theplight facing the modern utility theory and microeconomic theory.

The ordinal total utility theory launched and advocated by Pareto, Allen, Hicks, Samuelson andother great economists who participated in building and debating the fundamental economic theorieswas never able to successfully materialise the dream or ideals of the ordinal utility revolution.

The concept of ordinal utility is generally accepted, but the concept of diminishing marginal util-ity is also generally experienced. That these two concepts cannot be simultaneously integrated in atheoretical framework implies that there might be some problems with the foothold of the ordinal util-ity revolution. It is possible that there is no such ordinal utility theory in which the two concepts canbe simultaneously integrated, but it is also possible that there is indeed such a Holy Grail awaitingeconomists.

Bernardelli (1938, p. 198), who participated in the debate about the ordinal and cardinal utilitytheories, stated that “the problem of psychological measurement and its bearing on economic theoryso far has proved to be one of the most puzzling riddles.” This riddle is still unsolved today, as Hudík(2014, p. 690) noted: “the question whether recent attempts to construct … consumer theory will leadto a reintroduction of marginal utility into analyses of consumer behavior… As of today, this ques-tion still remains open.” Nowadays it seems that people have taken for granted the analysis of realityusing controversial theories.

This critical unsolved riddle, however, is awaiting someone's endeavour so that the law ofdiminishing marginal utility can be properly included in the ordinal utility theory. Given that the util-ity theory is the foundation of the choice theory in economics, solving this riddle might fundamen-tally re-develop the choice theory and the enormous and various economic theories behind it.

ACKNOWLEDGEMENT

We thank two research assistants, Hsiao-Pei Lin and Yun-Jung Chiang, for their efficient assistance.

LIN AND PENG 243

ENDNOTES1In some other textbooks, for instance Katz and Rosen (1994), one will also find that the term “diminishing marginal utility” is excluded.2Another example is that Bernardelli (1952, p. 260) claimed that “the underlying principles of my nursery experience werethose of increasing total—and of decreasing marginal—utility.”3The three founders of marginal utility theory (Jevons, Menger and Walras) all assumed that the marginal utility of every com-modity is decreasing. See Moscati (2013a) for further discussion.4Marshall (1920, p. 93) pointed out that: “There is an endless variety of wants, but there is a limit to each separate want. Thisfamiliar and fundamental tendency of human nature may be stated in the law of satiable wants or of diminishing utility thus:The total utility of a thing to anyone (that is, the total pleasure of other benefit it yields him) increases with every increase inhis stock of it, but not as fast as his stock increases.”5The so-called first Gossen law states that marginal utility decreases with an increasing stock of goods in possession of a per-son, see Bernardelli (1938).6The so-called ordinal revolution is inaugurated by Pareto (1971) and virtually concluded by Hicks (1946)‘s Value and Capital;see Moscati (2013a) and its references.7The ideas of preference order and the notion that a utility function is unique up to any positive monotonic transformation wasintroduced in economics by Pareto (1971). A comprehensible review, see Moscati (2013b).8Allen (1934) and Hicks and Allen (1934) are the earliest papers to point out and discuss the fact that the signs of the secondderivatives of the ordinary utility functions cannot always remain the same after any positive monotonic transformation.9Being unable to assign economic meaning to the second and cross derivatives of a total utility function will cause problemsbeyond what one can imagine. For example, one who believes the ordinal total utility theory cannot claim that he is “gettingmore and more tired” when working longer and longer, even if he indeed feels so. He also cannot claim lemon to be a comple-ment for tea, even if he indeed feels more satisfied after adding some drops of lemon juice into his tea. In order to make thisclaim, he, instead, will need to figure out the impact of the price of lemon on his consumption of tea. Samuelson (1974,p. 1257) virtually comments on this: The same astonishment might be registered by most in-laws of economic Ph.D.'s.“What?” they will ask, “to discover that tea and lemon are complements must I accompany any change in their prices orquantities by a ‘compensating’ change in money income or in some other good or goods? Bless my soul, I'd never havesuspected that. Pray, why?” In addition, an inferior good cannot be defined as a good with its marginal utility falling withincome, but instead as a good with its quantity consumed falling with income. Silberberg (1978, p. 241), however, commentson this current definition: “Also, inferiority is of necessity a local concept. Goods cannot be inferior over the whole range ofconsumption, or else they would never be consumed in positive amounts in the first place!”10The meanings and implications of cardinal utility were thoroughly investigated in a debate that took place after 1934, mainlyin the Review of Economic Studies. See Lange (1934), Phelps Brown (1934), Bernardelli (1934), Allen (1935), Samuelson(1938) and Alt (1971). For a comprehensive review, see Moscati (2013a).11Bernardelli (1938)‘s first effort to solve this riddle was criticised by Samuelson (1939, p. 87), who said that “I am very muchafraid, therefore, that the answer is yes to Mr. Bernardelli's question, and that the text-books still need that rewriting which hehoped so much to avoid.” Bernardelli (1938)‘s second effort was commented on with disappointment by Lancaster (1953,p. 259), who stated that “It would be very neat for parts of economic theory if only the classical marginal utility approach didnot suffer from the disabilities emphasised by the modern value theorists. However, it does suffer from those disabilities, andthere is nothing that Dr. Bernardelli, or anyone else, can do about it.” Some other economists of the Austrian school on ordinalmarginal utility tried another approach to derive the marginal utility. Instead of postulating the existence of a total utility func-tion and then defining a marginal utility as a derivative or difference of total utility, they assume the existence of an ordered setof ends or wants, of which each end or want can be satisfied with a unit of goods, and then they derive marginal utility fromthe set of ordered ends. See Rothbard (1956), McCulloch (1977) and High and Bloch (1989) for details.12There are many examples of this type of studies in the field of applied economics, including Romer (2001) in macroeconom-ics, Koopmans (1965) in growth, Calvo (1981) in international finance, and Allingham and Sandmo (1972) in expected utilitytheory, among others.13The former is the criticism of Bernardelli (1934, pp.71-72): “… is similar to the behaviour of a man who cuts off one of hislegs, in order to see how he gets on as a cripple. And it is extraordinary how one can get on without the leg of the second postu-late, as the results of Pareto, and more recently of Allen and Hicks, prove. Yet this would seem insufficient reason for makinga virtue of such an amputation.” The latter criticism is made by Rothbard (1956, p. 10): “The ordinalist rebels, led by Hicksand Allen in the early 1930, felt it necessary to overthrow the very concept of marginal utility along with measurability. Indoing so, they threw out the Utility baby with the Cardinal bathwater.”

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14As mentioned earlier in endnote 11, some other economists such as the Austrian school on ordinal marginal utility proposedanother approach and believe that they can preserve both the utility ordinality and the law of diminishing marginal utility. SeeRothbard (1956), McCulloch (1977) and High and Bloch (1989) for details.

ORCID

Chung-Cheng Lin https://orcid.org/0000-0002-3025-9246

REFERENCES

Allen, R. G. D. (1934). A comparison between different definitions of complementary and competitive goods. Econometrica, 2(2), 168–175.Allen, R. G. D. (1935). A note on the determinateness of the utility function. Review of Economic Studies, 2, 155–158.Allingham, M. G., & Sandmo, A. (1972). Income tax evasion: A theoretical analysis. Journal of Public Economics, 1, 323–338.Alt, F. (1971). On the measurability of utility. In J. S. Chipman, L. Hurwicz, M. K. Richter, & H. F. Sonnenschein (Eds.), Preferences,

utility and demand (pp. 424–431). New York, NY: Jovanovich.Bernardelli, H. (1934). Notes on the determinateness of the utility function: II. Review of Economic Studies, 2, 69–75.Bernardelli, H. (1938). The end of the marginal utility theory?Economica, 5(18), 192–212.Bernardelli, H. (1952). A rehabilitation of the classical theory of marginal utility. Economica, 19(75), 254–268.Calvo, G. A. (1981). Devaluation: Levels versus rates. Journal of International Economics, 11, 165–172.Hicks, J. R. (1946). Value and capital: An inquiry into some fundamental principles of economic theory (2nd ed.). Oxford: Clarendon Press.Hicks, J. R., & Allen, R. G. D. (1934). A reconsideration of the theory of value. Economica, 1, 52–76, 196–219.High, J., & Bloch, H. (1989). On the history of ordinal utility theory: 1900–1932. History of Political Economy, 21(2), 351–365.Hudík, M. (2014). Reference-dependence and marginal utility: Alt, Samuelson, and Bernardelli. History of Political Economy, 46(4), 677–693.Jehle, G. A., & Reny, P. J. (2011). Advanced microeconomic theory (3rd ed.). India: Pearson Education.Katz, M., & Rosen, R. (1994). Microeconomics (2nd ed.). Burr Ridge, IL: Richard D. Irwin Inc.Koopmans, T. C. (1965). On the concept of optimal economic growth. In The Economic Approach to Developing Planning. Amster-

dam: Elsevier.Lancaster, K. (1953). A refutation of Mr. Bernardelli. Economica, 19, 259–262.Lange, O. (1934). The determinateness of the utility function. Review of Economic Studies, 1, 218–225.Marshall, A. (1920). Principles of economics: An introductory volume. London, England: Macmillan for the Royal Economic Society.McCulloch, J. H. (1977). The Austrian theory of the marginal use and of ordinal marginal utility. Journal of Economics, 37(3/4), 249–280.Moscati, I. (2013a). How cardinal utility entered economic analysis: 1909–1944. The European Journal of the History of Economic

Thought, 20(6), 906–939.Moscati, I. (2013b). Were Jevons, Menger and Walras really Cardinalists? On the notion of measurement in utility theory, psychology,

mathematics, and other disciplines, 1870–1910. History of Political Economy, 45, 373–414.Pareto, V. (1971). Manual of political economy. New York, NY: Kelley.Phelps Brown, H. E. (1934). Notes on the determinateness of the utility function: I. The Review of Economic Studies, 2(1), 66–69.Romer, D. (2001). Advanced macroeconomics (2nd ed.). New York, NY: McGraw-Hill.Rothbard, M. (1956). Toward a reconstruction of utility and welfare economics. In M. Sennholz (Ed.). Princeton, NJ: D. Van Nostrand.

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How to cite this article: Lin C-C, Peng S-S. The role of diminishing marginal utility in theordinal and cardinal utility theories. Aust Econ Pap. 2019;58:233–246. https://doi.org/10.1111/1467-8454.12151

LIN AND PENG 245

APPENDIX A: THE MATHEMATICAL PROOF

We first prove that the results of the following two equations are the same:

xM =Uxm−pUmm

−Axx, ðA8Þ

xM =Vxm−pVmm

−Bxx: ðA18Þ

We will first discuss the denominators and then numerators. First, for the denominator in (18),namely, Bxx = Vxx − pVxm − pVmx + p2Vmm, by plugging Vij = F

0Uij + F

0 0UiUj with i, j = x, m into

this equation above, we obtain:

Bxx =F0 Uxx−pUxm−pUmx + p2Umm� �

+F00 UxUx−pUxUm−pUmUx + p2UmUm� �

:

By plugging the first-order condition Ux = pUm into this equation, we obtain:

Bxx =F0 Uxx−pUxm−pUmx + p2Umm� �

+F00 UxUx−UxUx−UxUx +UxUxð Þ:We finally obtain Bxx = F

0(Uxx − pUxm − pUmx + p2Umm) = F

0Axx.

For the numerator in (18), by letting K = Vxm − pVmm, and plugging Vij = F0Uij + F

0 0UiUj with i,

j = x, m into K, we get:

K =Vxm−pVmm = F0Uxm +F00UxUmð Þ−p F0Umm +F00UmUmð Þ,which can be rewritten as the equation below:

K =F0 Uxm−pUmmð Þ+ Ux−pUmð ÞF00Um:

By plugging the first-order condition Ux = pUm into this equation, we obtain:

K =F0 Uxm−pUmmð Þ:We therefore find that Equations (8) and (18) are the same, since:

Vxm−pVmm

−Bxx=F0 Uxm−pUmmð Þ

−F0Axx=Uxm−pUmm

−Axx:

By the same logic, one can easily show that the following two equations are the same:

xp =−Um−x Uxm + pUmmð Þ

−Axx, ðA9Þ

xp =−Vm−x Vxm + pVmmð Þ

−Bxx: ðA19Þ

246 LIN AND PENG