HISTORY-ECONOMIC THOUGHT 125 (001) (Fall 2020)

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CHAPTER 3: CARDINAL UTILITY AND THE PROBLEM OF ITS MEASUREMENT

“The abstract idea of wealth or value in exchange …must be carefully distinguished from the accessory ideas of utility, scarcity, and suitability…which the word wealth still suggests in common speech. These ideas are variable and by nature indeterminate, and consequently ill suited for the foundation of a scientific theory.”

A. Augustin Cournot (1836) 1

B. A key innovation of the so-called “Marginal Revolution” was recognition that the prices for which goods exchange in the market result from the value subjectively placed on them by buyers rather than from their costs of production. As Schumpeter points out, the concept of use-value has been around since the time of Aristotle at least. By the time of the Scholastics, it had matured to the point where it lacked only the marginal framework. The concept of use-value then died out, only to be rediscovered in various pockets around Europe and Brittan.2 The problem addressed by the marginal revolutionaries who wrote between 1871 and 1906 was how to quantify so subjective a notion as use-value. The approaches taken by the various writers were driven by their varying concepts of what use-value really was. Following Aristotle, Karl Menger and Jules Dupuit retained the concept of it as a value judgment placed on an article due to its serviceability. Dupuit sought to operationally define the use-value of an object in terms of what the consumer was willing to pay for it. Jevons and his followers sought to attribute use-value to a quantifiable physiological response, such as the sensation of pleasure or the absence of pain. Beginning with Jevons, Bentham’s term “utility” came to be used for this response, though Bentham’s concept was quite different. During this period, the terms “use-value” and “utility” were used somewhat interchangeably, although their interpretations varied quite significantly. Alfred Marshall synthesized and extended the work of his predecessors. He retained Jevons’ view of utility as a sense of satisfaction, but he measured it by the consumer’s willingness to pay. Marshall, however, mistook the apparent Income Elasticity of Demand as an indication that the value one places on money varies with one’s income. As result, Marshall reasoned that money could be used as a measure of value only when the transactions involved did not substantially impact the consumer’s wealth. Marshall did not realize that a way around this problem was implicit in Menger’s work. With Pareto, the interpretation of utility regressed to that of simple ophelimity, a purely arbitrary phenomenon unworthy of quantification. From 1906 onward, the notion of use-value faded from

1 Cournot (1960) p.10 2 Shumpeter (1994) p. 1054

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view, as utility came to be seen only as an analytical means of describing a consumer’s tastes in terms ordinal preferences. This chapter will conclude by reassembling the work of the authors considered to show that abandonment of a cardinal measure of use-value was unnecessary. The mathematics needed to make such a measure workable was unavailable to the researchers at the time. A foretaste of the necessary mathematical approach, to be developed in the next chapter will be provided.

3.1) Utility as the Product of Human Physiology: The Work of Wm. Stanley Jevons and Francis Ysidro Edgeworth The project of William Stanley Jevons (1835-1882) was to found a theory of political economy on the workings of the human mind. Jevons’ viewpoint was a product of Mill’s hypothetico-deductive method, which held that knowledge of mental processes must necessarily be deduced from hypotheses derived from introspection. He broke from Mill’s position that the mind was completely separate from other natural phenomena. To Jevons, thought was a product of the brain, a physical organ subject to physical laws. Jevons was, by nature and by training, a natural scientist, having excelled in chemistry as a student and proven his analytical capabilities as a gold assayer in Australia. Jevons was keenly interested in the mechanics of human thought. He studied mathematics and logic under Augustus De Morgan, as well as the logic of Thomas Boole.3 Such logic, however, was not the logic used in rhetorical argument but the binary logic upon which computer circuits would later be based.4 As a means of demonstrating that reasoning could be carried out by a biological instrument, Jevons constructed his own Logical Machine, a mechanical computer that implemented Boole’s logic. This invention was an early foray into what would later come to be known as artificial intelligence. To model the brain’s ability to compare variable quantities such as pleasure and pain, Jevons followed current work in psychophysiology. In the opening passages of his Theory of Political Economy, Jevons declares that “value depends entirely upon utility”.5 His utility, however, is not that of Bentham, but a physiological sensation such as considered by Richard Jennings. Jennings’ Natural elements of Political Economy drew from contemporary work in psychophysiology from both Brittan and Germany.6 This work attempted to measure nervous system responses, whether it is the reaction time to the application of heat, as in the case of Carpenter in Britain, or the rate at which responses to repeated stimuli diminish, as in the case with Fechner in Germany. Jevons’ Degree of Utility, what Marshall would later call marginal utility, reflects Fechner’s diminishing response to stimulus. Jevons introduces his degree of utility with the following illustration: 3 Maas (2005) p.123-28 4 By the term binary what is meant is that all arguments are stated in binary form: they are either “true” or “false”; 1 or 0. 5 Jevons (1970) p.77 6 Maas (2005) p.10

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“Let us imagine the whole quantity of food which a person consumes on average during a twenty-four hour period to be divided into ten equal parts. If the food be reduced by the last part, he will suffer but little; if a second part be deficient, he will feel the want distinctly; the subtraction of the third tenth part will be decidedly injurious; with every subsequent subtraction of a tenth part his sufferings will be more and more serious until he will at length be on the verge of starvation. Now if we call each of the tenth parts an increment, the meaning of these facts is that each increment of food is less necessary, or possesses less utility than the previous one.”7

In Figure 3.1-1, reproduced from his Theory, Jevons represents the degree of utility obtained from each increment by the area of its corresponding rectangle. The total utility experienced is the sum of the areas of the rectangles. In Figure 3.1-2, he argues that if the number of increments becomes arbitrarily large and their sizes arbitrarily small, the total utility becomes the area under the curve corresponding to the degree of utility, between the first and the last increments of food consumed.

Figure 3.1-18

7 Jevons (1970) p.106 8 Figure is adapted from Jevons (1970) p.107

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Figure 3.1-29 Jevons’ great accomplishment was of course recognizing the marginal relationship between the total and the degree of utility, and that the degree of utility diminishes with consumption. There is nothing in his argument that requires the degree to be defined as the derivative of the total. From his presentation, it would seem more natural to define the total as the integral of the degree. Additionally, he implies that, at least in some cases, his degree of utility can be measured by the sacrifice the consumer is willing to make. “I hesitate to say that men will ever have the means of measuring the feelings of the human heart…but it is the amount of these feelings that prompts us to buying and selling… and it is from the quantitative effects of these feelings that we must estimate their comparative amounts.”10 These “quantitative effects” are the ratios with which the consumer would be willing to exchange one good for another.

“Imagine that there is one trading body possessing only corn, and another possessing only beef. … Suppose for a moment, that the ratio of exchange is approximately that of ten pounds of corn for one pound of beef: then if, to the trading body that possesses corn, ten pounds of corn is less useful than one pound of beef, that body will desire to carry the exchange further. Should the other body possessing beef find one pound less useful than ten pounds of corn, this body will also be desirous to continue the exchange. Exchange will go on until each body has obtained all the benefit that is possible, and loss of utility would result if more were exchanged. Both parties, then, rest in satisfaction and equilibrium, and the degrees of utility have come to their level, as it were.” 11

9 Figure adapted from Jevons (1970) p.108 10 Jevons (1970) p.83 [emphasis is in original text] 11 Jevons (1970) p.139

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Were it not for his ability to equate the ratios of the degree of utility to the exchange prices, his work would never have gotten past the raised eyebrows of his contemporaries. Such ratios however, only provide partial information. We may know the magnitude of the consumer’s degree of utility for one good relative to another, but never know it for any good in absolute terms. In short, we know one relationship between two unknowns. This however is where Jevons’ physiological view of utility becomes problematic. It is because he insisted that utility actually exists in nature that its measurement in absolute terms becomes an issue. In the following section, we will see that Dupuit’s concept of value, which did not rely on a physiological source, was not subject to this problem. The physiological interpretation of utility becomes particularly problematic in the work of Francis Y. Edgeworth. In his Mathematical Psychics,12 Edgeworth references the same psycho-physiological work as does Jevons.13 Edgeworth is the first to emphasize the point that physiological responses are likely to vary unpredictably between individuals. According to Edgeworth, “If we were to follow Bentham’s precepts would this not mean that one’s compensation does (or should) be in proportion to his capacity to experience pleasure?” Edgeworth also speculates that some receive higher wages than do others due to their inherent greater capacity for enjoyment.14

3.2) Utility as Derived From Demand: The Marginal Revolution in France Leon Walras (1834-1910) and the tradition of French writers upon which drew, used the term “utility” to express the assessed value of a thing according to its serviceability, as understood since Aristotle. By leaving utility as an abstract notion, they were able to define it operationally and measure it indirectly in terms of demand. Walras himself made little use of utility per-se, building his theory on demand directly. In his Elements of Political Economy-Pure, Walras argued that there would normally be a unique set of prices, for which the demands for all goods in the marketplace would simultaneously equal their supply. Ingrao and Israel trace the origins of Walras’ approach back to the Physiocrats of the late Eighteenth Century.15 Due to their concern with practical matters, these writers were more empirically oriented than were their British counterparts. Early social choice theorists such as the Marquis de Condorcet (1743-1794), whose work will be discussed later in the chapter pioneered collection of demographic data regarding healthcare and the like. Walras’ precursors were often members of the French Engineering School.16 Such engineers were responsible for the maintenance of roads, bridges and other public works, and it was their responsibility to secure the necessary resources from the population for their construction. As result, they were concerned

12 Edgeworth (1967) 13 Edgeworth (1967) pp. 56-63, 14 Edgeworth (1967) p. 64 15 Ingrao and Israel pp.42-46 16 Blaug (1996) p.303

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that the benefits warranted the costs, particularly when such cost was paid in the form of the hated corvée, a form of conscripted labor17. From the time of the Physiocrats, these thinkers avoided speculating about human motivation. Turgot commented: “I do not wish to investigate how pleasure and pain…influence the determination of the will. I merely say that we find in experience only one principle productive of movement, and that is that the will of intelligent beings which is not primitively determined but determines itself.” 18 Rather than focusing on individual actors, these investigators drew analogies followed the flow of value through the economy, drawing analogies with the flow of blood through the body. According to Canard:

“The product of labor circulates in all channels of this system of ramifications like a liquid and everywhere attains its own equilibrium. Each vessel causing the product of labor to circulate and is accompanied by an analogous vessels causing money to circulate in the opposite direction; and the system of the circulation of money and labor as a whole resembles the circulation of blood.19”

Anne-Robert Jacques Turgot (1721-1787) was among the first to challenge the early British notion that a good’s exchange value (market price) was determined by its cost of production.20 Turgot’s argument was conceptually the same as Jevons’ law of exchange, though less detailed. In Turgot’s model, each party to an exchange assesses in his own mind the relative value of the goods in question. When traders meet, each is willing to give up what he values less in exchange for what he values more. It is through the process of bargaining that the relative exchange-value is determined21. Turgot’s model was mathematically formalized somewhat by Achylle-Nicholas Isnard (1749- 1801), an engineer, and Nicholas-François Canard (1750-1833), a high school teacher of mathematics. The value that traders placed on the goods in question came to be measured implicitly by their willingness to pay for them. Canard postulated that for an exchangeable good, there would be a maximum price the purchaser would pay, and a minimum price for which the seller would part with it. Between these prices was a bargaining latitude within which the sale price will be negotiated.22 If, as Jevons were to later propose, the traders were to exchange the commodity in increments, Canard’s model would be reduced to Jevons’ law of exchange, but without reference to Jevons form of utility. The stance of Political Economy as an empirical science was considerably advanced by A. Augustin Cournot (1801-1877). Walras referred to Cournot as his mentor and referenced his

17 Jupp (1999) p.vi 18 Ingrao and Israel (1990) p.49 19 Quoted in Ingrao and Israel (1990) p.68 20 Jupp (1999) pp.i-x 21 Turgot (1999) pp. 14--15 22 Ingrao and Israel (1990) pp. 66-72

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work frequently. Jevons and Marshall were also significantly influenced by his work, with Jevons referring to himself as the first Englishman to have read Cournot.23 Cournot was not as concerned with expressing economic ideas in mathematical terms as he was with insuring such mathematics was used appropriately to express quantities that were empirically measurable. Cournot’s Law of Demand (or law of sales, since “…we do not see for what reason theory need theory need take account of any demand which does not result in a sale”24) was an empirically verifiable relationship between the prices and the quantities of goods sold. “Observation must be depended on for furnishing the means of drawing up between proper limits a table of the corresponding values of [sales quantities] and [prices]; after which by well known methods of interpolation or by graphic processes, and empiric formula or curve can be made to represent the function in question.25” Cournot’s dependence on mathematical functions, “ which may not be capable of algebraic expression”, was his means of articulating exactly what was, as opposed to what was not, knowable in the absence of data. Cournot’s demand curve (Figure 3.3-1) is an example of such reasoning. He argued that such a curve must be downward sloping from the simple observation that “the cheaper an article is, the greater ordinarily is the demand for it.” 26 He expressed more than slight irritation with those who rushed to apply a specific algebraic formula when such was not justified by the data27.

Figure 3.2-128 Cournot devoted the second chapter of his 1836 Researches into the Mathematical Principles of the Theory of Wealth to such discussion. He emphasizes the point that measurement is a relative process. Empirically, one cannot speak of some quantity A without there being at least one other quantity B against which it is measured. If there are n quantities, there are at most 23 Ingrao and Israel (1990) p.78 24 Cournot (1960) p.46 25 Cournot (1960) pp. 47-48 26 Cournot (1960) p.46 27 Cournot (1960) pp. 44-45 28 Adapted from Cournot (1960) Figure 1

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n −1 independent comparisons that can be made between them. As long as measurements can be made consistently, such a set of quantities can be measured relative to one another. One quantity is chosen as the standard and the values of the other quantities are given in terms of it.29 Cournot would have rejected Jevons’ attempt to attribute use-value to a physiological force as unnecessary, were it possible to define it operationally in terms of quantities already observed. Such definition is what Dupuit was to accomplish a decade later. Jules Dupuit (1804-1866) was a civil engineer charged with estimating the value provided by public works. Dupuit argued that the value one placed on the consumption of a good was measured by his willingness to pay for it. This value is generally greater than the price one is actually is required to pay for it. In his paper of 1849, he offered a spirited defense of the measurability of use-value, as he conceived it:

“Here is a person who needs a kilogram of meat, that is, who is willing to make a sacrifice to obtain it. The butcher says the kilogram of meat is worth one franc. Two things can happen: either he buys it or he does not. If he does, I shall ask him whether he would have bought it at twenty-one sous, then at twenty two, at twenty three, and so on. I do not think it would be an abuse of the word “obviously” if I say that, by so doing, I would gradually ascertain the maximum price at which he would be willing to buy his kilo of meat.”30

Figure 3.2-231 If the toll for a bridge were to be lowered from p to p ' , the public would certainly use it more. Dupuit argued that the public would be willing to pay the amount indicated by the shaded area under the demand curve in Figure 3.2-2, over and above what they were actually charged for its 29 Cournot’s chapter does not express this idea in quite this way. 30 Ingrao and Israel (1990) p.76 31 Adapted from Blaug (1996) p.305

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usage. This phenomenon, which is what Alfred Marshall would later call the consumer’s surplus32 can be measured, at least in principle, by having the consumer purchase his goods from a perfectly discriminating monopolist. As is the case with Marshall’s consumer’s surplus, Dupuit has defined this additional value to be the integral of demand curve from p ' to p , less the additional exchange value paid q(p '− p) . Dupuit’s “utility” is not that of either Jevons or Bentham It is a quantity that exists by virtue of its mathematical definition. We understand it in terms of the added opportunity cost a consumer would be willing to bear, but we have no knowledge of (or concern for) why the consumer would be willing to bear such added cost.

3.3 Marshall’s Synthesis Alfred Marshall (1842 -1924) synthesized the work of his predecessors. Marshall retains Jevons’ notion of utility as “correlative to Desire or Want”,33 yet he concerns himself with it only as far as it produces an observable result34. Marshall does not assume, as might Jevons or Bentham, that one can predict a consumer’s behavior from an understanding of his thought processes. He in fact chastised Jevons for “…having led many of his readers into confusion between the provinces of Hedonics and Economics…”35 Marshall renames Jevons’ degree of utility as marginal utility since it measures the utility provided a consumer by the marginal purchase made when “he is on the margin of doubt whether it is worth his while to incur the outlay required to obtain it.”36 Marshall defines the price the consumer is just willing to pay for his marginal purchase as his marginal demand price. He then states his law of diminishing marginal utility: ”The larger amount of a thing that a person has, the less, other things being equal …will be the price he will pay for a little more of it: or in other words his marginal demand price for it diminishes.”37 If Marshall was willing to go so far as to actually define utility in terms of one’s willingness to pay, he could have replaced Jevons’ degrees of utility in Figure 3.1-1 with the corresponding marginal demand prices ri

38 that the consumer is observed to pay for each increment. The total utility of the food considered in Figure 3.1-1 would simply be the sum of the marginal demand prices for the respective increments:

U = ri i= II

X

∑ 3.3-1

32 In his development of consumer’s demand, Marshall references Dupuit’s work. See Marshall (1997) p.101 33 Marshall (1997) p.92 34 See Marshall (1992) p.16 35 Marshall (1992) footnote on p.101 36 Marshall (1997) p.93 37 Marshall (1997) p.95 38 r is used for marginal demand prices to distinguish them from market prices p

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If the increments of food F are taken arbitrarily small as shown in Figure 1.1-2, and we replace the incremental prices ri with a continuous marginal price function r(F) , we would have the integral:

U(F) = r(F)dF m

n

∫ 3.3-2 Had Marshall taken these steps, his utility would have simply been use-value, with no necessary connection to a sensation of satisfaction. Additionally, from Jevons’ law of exchange, we know that the rate at which the consumer is willing to exchange good xn for good xi (given by dxn dxi ) is the inverse of the ratio of their marginal utilities MUi MUn . At equilibrium, this equals the market price pi of xi in terms of xn .

dxn dxi

= MUi MUn

= pi 3.3-3

If xn is used as the numeraire, dxn dxi is also, by definition, the consumer’s marginal demand price ri . If there are n goods available, the consumers marginal demand prices would equal the market prices, or:

r1 p1

= r2 p2

= = rn−1 pn−1

= 1 3.3-4

Marshall wasn’t willing to go that far. His equimarginal rule states only that the consumers marginal utilities are proportional to market prices, and the constant of proportionality is the (unknown) marginal utility of the numeraire:

MU1 p1

= MU2 p2

= = MUn−1 pn−1

= MUn 3.3-5

Marshall leaves us open to consider that the marginal utility of whatever numeraire we might choose, varies with respect to some outside standard. Marshall recognizes two practical problems with using a numeraire good as the standard of measure. The first problem he was able to solve, and the second he was not. The first problem is that, depending on the numeraire chosen, the law of diminishing marginal utility may be violated. We can see why from a thought experiment provided by Irving Fisher. In this experiment, a consumer is asked to purchase increments of bread using milk as payment. 39 Depending on whether the consumer’s need for milk or bread diminishes faster, the consumer’s marginal utility for bread, as measured with respect to milk, may increase or increase.

39 Blaug (1996) pp.313-314

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The problem is solved by recognizing that money M is a commodity having properties distinct from other goods. Since its only value to the consumer is in its ability to be exchanged for other goods, such value is determined by whatever good the consumer may wish to purchase. If the law can be reworded to say that the marginal utility of any good diminishes with respect to consumption in general, then the use of money as numeraire solves the problem. Here is the second problem: Can the diminishing marginal utility of money have any meaning? If so, with respect to what might the marginal utility of money diminish and how can we detect it? 40 In Marshall’s view, it appears to vary with respect to other goods when the consumer’s wealth changes. While this income effect, familiar from basic microeconomics is very real, Marshall’s interpretation of it is due to an illusion. Marshall observes that poorer individuals are much less willing to spend money on luxuries than richer ones. “We have seen how a clerk with £100 a year will walk to business in a heavier rain than a clerk with £300 would.”41 He recognizes that an extra schilling can meet a greater need for the man who is poorer. “A rich man in doubt whether to spend a shilling on a single cigar, is weighing against one another smaller pleasures than a poor man, who is doubting whether to spend a schilling on a supply of tobacco that will last him a month. …If a poorer man spends [£1], he will suffer more from the want of it afterwards than the richer would.42…A stronger incentive will be required to induce a person to pay a given price for anything if he is poor than if he is rich”.43 From this assessment, Marshall concludes that: “… every diminution of his resources increases the marginal utility of money to him, and diminishes the price he would be willing to pay for any benefit.”44 The problem with this analysis is one that will figure prominently when we consider welfare analysis in Chapter 5. What Marshall, and all of his contemporaries except Menger, failed to consider is that humans generally prioritize their consumption. Without a minute’s supply of air, most of us would perish. It is the most important consumption good to any of us, and it is the good we consume first. Water would rank a close second in importance. Since these are at least for the moment nearly free goods, we are free to save our resources for goods that are of lesser importance. Recall that the marginal utility of any good to a consumer is a function of all the goods she already has. Since the rich man has previously satisfied more of his basic needs than has the poor man, he appears to value a schilling less. In reality, he solves a different problem than does the poor man.45 To get a better understanding of this issue, we look back to the work of Karl Menger.

40 If the author says something in the forest, and his girlfriend is not there to hear him, is he still wrong? 41 Marshall (1997) p.95 42 Marshall (1997) p.19 43 Marshall (1997) p.19 44 Marshall (1997) p.96 45 this of cours is apparent in modern indifference curve analysis. The two individuals have different budget constraints and hence different optimal bundles.

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3.4) Use-Value and Hierarchical Need: The Work of Karl Menger (1840-1921) For our purposes, the most significant contribution of Menger’s Principles of Economics is his handling of diminishing marginal value. He avoids the term “utility” completely, defining value as “a judgment economizing men make about the importance of the goods at their disposal for the maintenance of their lives and well-being”. 46 Such a definition does not presuppose a sensory mechanism for its determination. Menger’s version of the law of diminishing marginal utility anticipates mid 20th century psychologist Abraham Maslow47, Menger predicts that all humans consume goods according to hierarchy. People first seek goods that satisfy their survival needs, followed by others that improve the quality of life. Finally they seek goods that provide only a passing pleasure. The first quantities of the most urgent goods consumed such as food, satisfy survival needs, while further quantities satisfy less urgent needs. According to Menger:

“Men consume food for several reasons: Above all, they take food to maintain life; above this they take further quantities to preserve health, since a diet sufficient to maintain life is too sparing, as experience shows, to avoid organic disorders; finally, having consumed quantities sufficient to maintain life and preserve health, men further partake of food simply for the pleasure derived from their consumption.”48

Menger’s example with food closely parallels that given by Jevons (See Figure 3.1-1). Menger’s application of this argument to the types of goods individuals seek, however, provides an added dimension.

“We observe that men fear the lack of food, clothing, and shelter much more than the lack of a coach, a chessboard, etc. … The maintenance of life depends neither on having a comfortable bed nor having a chessboard, but the use of these goods contributes, and certainly in very different degrees, to the increase of our wellbeing. Hence there can be no doubt that, when men have a choice between doing without a comfortable bed or doing without a chessboard, they will forgo the latter much more readily than the former.”49

Menger illustrates this idea graphically with the table 3.4-1. Goods such as food (I) that meet the most basic of needs can provide a maximum value of 10. Housing (II), though not absolutely necessary for survival, is certainly vital to life’s quality and hence can provide a maximum satisfaction of, say, nine. A comfortable bed (III) might provide a maximum value of eight; a chessboard (VII) would provide four, while an evening at the opera (IX) might provide maximum value of only two. The key to understanding the tradeoffs that consumer’s make between goods 46 Menger (2003) p.448 47 Maslow (1943) 48 Menger (2003) p.449-50 49 Menger (2003) p.449

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can be found in the way the marginal values of all goods diminish with consumption. A destitute individual would first seek only food, obtaining a value of ten for his first increment. The second increment of food would provide a value of only nine, equaling the value of the first increment of bedding. Hence, after the first increment of food is consumed, the individual would be indifferent between an additional unit of food and an initial unit of bedding and would most likely consume both in equal quantities.

Figure 3.4-150 To see how this concept addresses Marshall’s diminishing marginal utility of money, consider his example of the clerk whose frequency of cab rides depends on his income. Assume that a cab ride (in the worst possible weather) provides the clerk a maximum value of seven, placing it under Column IV in the table. It does not sustain life as well as food, shelter, bedding or the like, but cab rides in inclement weather can provide some protection against pneumonia. To even consider a cab ride, the clerk must have sufficient funds to purchase all goods upon which he places a value of 8 or higher. Only at that point would he consider a cab ride, to which he would be indifferent to a second increment of bedding, a third increment of housing, or a fourth increment of food. If all goods were priced at $1 per unit, the clerk would need an income of $10 as shown by wealth line A in Figure 3.4-151 If the consumer’s wealth were increased to $28, he would be indifferent to taking a third cab ride as opposed to purchasing a chessboard. It is not that he would value a dollar any less if he has $28 as opposed to $10. It is rather the fact that his un-met needs would be fewer. For a man with $10, the opportunity cost of a cab ride (his first in this case) is a fourth increment of food. For a man with $28, the opportunity of a cab ride (his fourth) is a seventh increment of food, or his first chessboard.

50 Menger (2003) p.451 51 Note that in this illustration, the budget line has a different interpretation that in standard microeconomic analysis. The budget indicated by the line must be sufficient to purchase all goods within the set, i.e. to above and to the left of the budget line.

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In standard microeconomic analysis, this phenomenon would be accounted for by the shape of the use-value (or utility) function, not a change in the scale against which it would be measured. As will be shown in Chapter 5, deriving such a function will take a bit of work, and will require the techniques that will be developed in Chapter 4. The mathematics Marshall and many of his contemporaries used were rather crude by today’s standards. Marshall used an additive utility function. Such approach presumed that the utility gained from consuming any one good did not depend on the quantities of other goods consumed.

3.5) Vilfredo Pareto and the Demise of Cardinal Utility With publication of Vilfredo Pareto’s Manual of Political Economy, the concept of utility regressed to an even more narrowly hedonic interpretation than was proposed by Jevons. The rational judgment by which Aristotle, Marshall, and Menger believed use-value to be assessed was replaced with mere emotion. Pareto’s position seems to be influenced considerably by his somewhat cynical view of social processes in general. He regarded social “equilibria” to be more the product of arbitrary whim than of reason. Prior to his appointment as Lecturer at the University of Florence in 1886, the professional life of Vilferdo Pareto (1848-1923) had been occupied with engineering. Beginning in 1872 however he Wrote about sociology, economics, and philosophy as a sideline. Through Pareto’s early writings, Placido Bucolo traces Pareto’s growing disillusionment as he watched the idealism of the mid-19th Century devolve into plutocratic demagogy.52 Bucolo characterizes Pareto’s drama as “the drama of Europe, which – an impotent witness – watches its own destruction, ready to resign the mastery of the world and become the slave of its passions53.” Despite the public rhetoric of morality and rationality, Pareto observed that both public and private decisions were as often based on “non-logic” such as prejudice or taste, as they were on logical criteria. In his economic writings, Pareto emphasizes this point by replacing the term “utility” with ophelimity. Pareto defines ophelimity as satisfaction of emotional desire without regard to whether or not the consumer’s wellbeing is enhanced. To an addict, Pareto regards morphine as economically useful, “even though it is unhealthful, because it satisfies one of his wants”54. From this reasoning it is apparent why Pareto might regard ophelimity as not only immeasurable, but unimportant. The project of Pareto’s 1906 Manual was to build a theory of economic equilibrium that made (almost) no use of ophelimity. His system was based on indifference curves representing sets of bundles, to the consumption of which the consumer would be indifferent. Ophelimity serves only as an index identifying which of the curves are preferred over the others.

52 Bucolo (1980) p.3 53 Bucolo (1980) p.3 54 Pareto (1971) p.111

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For a consumer with the opportunity to choose between n goods xi i = 1,2,…n( ) , her set of indifference curves can be represented by a utility function U(x1, x2 ,…xn ) satisfying certain requirements. Each indifference curve55 may be represented by an equation of the form:

I =U(x1, x2 ,…xn ) 3.5-1 where I is a constant representing the ophelimity index of the respective curve. Since the indices represent only an ordering of the curves, there are many utility functions that could represent the same set of curves. Pareto demonstrates this as follows: For any utility function

U x1, x2 ,…xn( ) let F U x1, x2 ,…xn( ){ } be another function which is a positive transformation of U x1, x2 ,…xn( ) as shown in Figure 3.5-1. The equation defining the indifference curves represented by F U{ } is given by:

J = F U x1, x2 ,…xn( ){ } 3.5-2

Figure 3.5-1 The “shape” of the curves represented by U and F{U} can be deduced from the differentials of Equations 3.5-1 and 3.5-2 respectively. Pareto shows these to be equivalent56:

0 = d F U x1, x2 ,…xn( ){ }⎡⎣ ⎤⎦ = dF dU

∂U ∂x1

dx1 + dF dU

∂U ∂x2

dx2 ++ dF dU

∂U ∂xn

dxn

= ∂U ∂x1

dx1 + ∂U ∂x2

dx2 ++ ∂U ∂xn

dxn

= d U x1, x2 ,…xn( )⎡⎣ ⎤⎦

3.5-3

55 or hyper-surface if n>2 56 See Pareto (1971) pp. 392-393

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As figure 3.5-1 shows, transformation of U simply shifts the positions of the indifference curves along the U axis without changing their shape. Beginning with Pareto, the economics profession began to replace the notion of utility with the notion of ordinal preferences. One may be able to say that a consumer prefers a bundle lying on a higher indifference curve, but not by how much. While a utility function may be used to describe a consumer’s preferences, such a description is only an analytic convenience. In a reversal of his earlier efforts, which will be discussed in the next chapter, Pareto solved the problem of value by simply avoiding it. Though Marshall, Cournot, and Fisher are mentioned in Pareto’s Manual he, never discusses their work or makes significant use of their results. Pareto won the day because he was successful at devising a means of avoiding the increasingly thorny issues of utility’s measurement. This “advancement”, however comes at a considerable cost as will be discussed presently.

3.6) Ordinal Preferences and the Impossibility of Interpersonal Comparison Except for the following brief comments, this chapter will not explore the development of preference ordering theory due to limited space. While such an approach has been successful in analyzing the behavior of a single individual, attempts at aggregating or interpersonally comparing the ordinal preferences of multiple individuals has proven impossible. Kenneth Arrow demonstrated the problem with his classic Impossibility Theorem of 1950. Arrow’s theorem follows from Condorcet’s Paradox of Voting, which is illustrated as follows: Consider three individuals, Fred, Mary, and George, who are asked to rank, according to their preference, three bundles of goods, a, b, and c. Fred prefers a to b to c (written a  b  c ) Mary prefers b to c to a , and George prefers c to a to b . These orderings are summarized below:

�

Fred : a  b c Mary : b  c  a George : c  a b

The three are asked to vote their preference among each pair of alternatives. By a two-thirds majority, Fred and George prefer a to b . By the same majority, Fred and Mary select b over c . Finally, Mary and George select c over a . We find that the social “ordering” (shown below) is no ordering at all. It lacks transitivity and is unable to select any alternative as “best” or “worst”.

Society : a  b, b  c, c  a This problem is due strictly to the ordinal nature of preferences. Were the agents able to express the intensities of their preferences on a common cardinal scale, simple addition of the agents’ preference scores for each alternative would produce a consistent (transitive) ordering.

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Since publication of Arrow’s Theorem, social choice theorists have attempted without success to find a way around this problem. In his classic Collective Choice and Social Welfare Amartya Sen has essentially proven that the only means by which this dilemma can be overcome is by finding a surrogate of preference that can be cardinally measured on a common scale.57 Sen illustrates this problem with the following example: Agents’ preferences are scaled so that his most preferred bundle is assigned a utility level of “1” while his least preferred bundle is assigned a utility of “0”. Intermediate bundles are assigned utility levels in between, based on empirical testing. Sen shows that such utility values cannot be compared between agents. Generally, agents will not have the same best and worst alternatives. Additionally, we cannot assume that the intensity of preference for the best over the worst alternative is the same between agents. Should a new “best” or “worst” alternative appear for any agent, the intensity scale for all other alternatives would change. In this case adding the agents’ intensities of preference will not produce a consistent or meaningful result.

3.7) Conclusion and Critique: Utility vs. Use-value Without the ability to interpersonally compare preferences, economic theorists have no way of analyzing how changes in the economic environment might affect consumers in the aggregate. This limitation severely restricts the usefulness of economic theory as a tool for policy analysis. As we have seen, abandonment of a cardinally measureable form of utility (use-value) was premature. Pareto’s multiplicity of equivalent utility functions tells us nothing more than what is apparent from Marshall’s equimarginal rule, or Jevons’ law of exchange. This insight is nothing more than an acknowledgement that only the ratio of an individual’s marginal utilities can be observed. The following textbook58 example of Pareto’s insight serves as an illustration. Let MRSi− k be the rate by which the consumer would exchange goods xi for xk . From the definition of the marginal rate of substitution we know that:

MRSi− k  dxk dxi

≡

∂U ∂xi ∂U ∂xk

≡ MUx MUy

3.7-1

Defining another utility function V that is a positive transformation f U( ) does nothing but multiply the numerator and denominator of MRSi− k by a common factor:

∂V ∂xi ∂V ∂xk

=

∂ f U( )⎡⎣ ⎤⎦ ∂xi

∂ f U( )⎡⎣ ⎤⎦ ∂xk

=

df ∂U

∂U ∂xi

df ∂U

∂U ∂xk

=

∂U ∂xi ∂U ∂xk

= MRSi− k 3.7-2

57 Sen (1970) pp. 128-30 58 Suen and Silberberg (2001)

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Applying a positive transform to the utility function in Marshall’s equimarginal rule (Equation 3.3-5) has no impact, since it scales the marginal utilities for all goods equally, including the marginal utility for money. As we have seen from Menger’s argument, there is no reason why the marginal utility of money cannot be presumed to be unity. With this assumption, an empirically measurable use-value function can be defined. Assume that there are n +1goods from which a consumer may choose, the first n being consumption goods and the last being money M . Using the language of Pareto’s argument, assume the consumer’s preferences are described by a utility function U x1, x2 ,…xn ,M( ) . We define another function V = F U[ ] to be a positive transformation of U for which the marginal utility of money is one. The positive nature of the transform allows us to state its inverse: U = f V( ) . Recall that the consumer’s marginal price ri( ) for good xi is the ratio of the marginal utilities for money M and xi or:

ri 

dM dxi

= MUi MUM

3.7-3

Substituting in the differential expressions for U and V , and using the above condition that ∂V ∂M ≡ 1 , we have:

dM dxi

= MUi MUM

=

∂U ∂V

∂V ∂xi

∂U ∂V

∂V ∂M

= ∂V ∂xi

= ri (x1, x2 ,xn ,M ) (3.7-4)

Using Equation 3.7-4, we can write the total differential of U as:

dU =

∂U ∂V

r1dx1 + r2dx2 ++ rm−1dxn( ) = ∂U∂V dV (3.7-5) The quantity in parentheses in Equation 3.7-5 is the total differential of V . Since the differential is composed entirely of observable quantities, the function V must be observable as well. V does not pretend to measure the satisfaction the consumer obtains from consumption, only the value she is observed to place on goods by virtue of what she is willing to pay for them. The function V therefore describes the consumer’s use-value. The term dU dV states the manner in which the consumer’s sensation of satisfaction varies as a function of V . Without a means of measuring dU dV , we are best off discarding it altogether, which can be done by substituting V for U in economic analysis.

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The function V that solves Equation 3.7-5 is defined by the following rather curious looking expression:

V x − x 0( )  r (x)

x0

x

∫ • dx 3.7-6 Equation 3.7-6 is the multivariate equivalent of Equation 3.3-2. Interpretating this expression will take up much of the next chapter. The terms

x and x 0 are vector shorthand for the complete

bundles of goods (x1, x2 ,…xn ,M ) and (x1 0 , x2

0 ,…xn 0 ,M 0 ) respectively. Equation 3.7-6

measures in money the value the consumer places on some bundle of goods x relative to the

value she would place on a given reference bundle x 0 .

The set of terms ri (x1, x2 ,…xn ,M ) that appear in the differential of V (Equation 3.7-5) are components of the vector function

r (x) . While economists are used to thinking of vectors as arrays of independent objects, this mindset does not apply to vector functions. The components of r (x) are intimately interrelated, making it easier to think of them as parts of a single entity.

The economic meaning of such an interrelationship becomes clear when we consider substitute and complementary goods. If good xi is a substitute (or complement) for good xk , then xk must similarly be a substitute (or compliment) for good xi . As will be shown in Chapter 4, the mutual nature of substitute (complementary) relationships requires that the consumer’s marginal price functions satisfy: ∂ri ∂xk

≡ ∂rk ∂xi

3.7-7

As will be shown, if an analyst were to specify ri and rk so as not to satisfy Equation 3.7-7, he would obtain extremely peculiar results were he to evaluate Equation 3.7-6. Attempts to understand the meaning of such odd results sent the economics profession on a wild goose chase that lasted nearly 70 years! This conundrum is known in the literature as the problem of integrability, and will be discussed in Chapter 4. Even without going through the rigors of the next chapter, it can be seen that Equation 3.7-7 does in fact overcome the measurement problems cited earlier. First, the scale it provides (including “zero-point”) is common for all consumers. With ∂V ∂M ≡ 1 , V and r are all measured in units of M . The lower limit of integration establishes the value placed on

x 0 as the scale’s origin. Marshall’s income effect is accounted for by the bundle

x the consumer holds at the time he evaluates

r (x) . As will be shown in Chapter 5, the values of r (x) can be compared

between consumers, provided all consumers hold the same bundle x at the time the evaluation is

made.

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REFERENCES Antonelli, G. B. On the Mathematical Theory of Political Economy reprinted in Preferences, Utility, and Demand– A Minnesota Symposium Chipman, J. et al eds. Harcort, Brace, Jovanovich NY. (1971) Aristotle Politics (Benjamin Joett trans.) in The History of Economic Thought: a Reader (S. Medema, & W. Samuels ed.) Routledge NY (2003) Aristotle Nichomachean Ethics (W. D. Ross trans.) in The History of Economic Thought: a Reader (S. Medema, & W. Samuels ed.) Routledge NY (2003) Bentham, J. An Introduction to the Principles of Morals and Legislation in John Stuart Mill - Utilitarianism and On Liberty (M. Warnock ed) Blackwell Publishing (Oxford 2003) Blaug, M Economic Theory in Retrospect Cambridge University Press Cambridge UK (1996) Blaug, M. The Methodology of Economics 2nd ed. Cambridge University Press Cambridge UK (1992) Bucolo, P. Introduction to V. Pareto’s The Other Pareto (P. Bucolo and G. Bucolo Trans) St Martin’s (NY 1980) Cournot, A. Researches into the Mathematical Principles of the Theory of Wealth (N. T. Bacon trans.) Kelley (NY 1960) Dupuit, J. On the Measure of Utility Edgeworth, F. Y. Mathematical Psychics Kelley (NY 1967) Fisher, I Mathematical Investigations in the Theory of Value and Price Yale Univ. Press (1961) Ingrao, B and Israel, G The invisible Hand, Economic Equilibrium in the History of Science. (Ian McGilvray trans) The MIT Press Cambridge MA (1990) Jennings, R Natural elements of Political Economy Jevons, W. S. The Theory of Political Economy Penguin Books (Harmondsworth UK 1970) Jupp, K. Introduction to Turgot’s The Formation and Distribution of Wealth, Reflections on Capitalism Othila Press (London 1999) Little, I. Social Choice and Individual Values The Journal of Political Economy Vol. 60 No. 5 (Oct. 1952) Marshall, A. Principles of Economics Prometheus Books (NY 1997)

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Maslow, A. A theory of Human Motivation Psychological Review 1943 50 pp.370-96 Maas, H. William Stanley Jevons and the Making of Modern Economics Cambridge UVP (NY 2005) Menger, C. Principles of Economics reprinted in The History of Economic Thought: a Reader (S. Medema and W. Samuels Eds.) Routledge (NY 2003) Moscati, I. History of Consumer Demand Theory 1871-1971: A Neo-Kantian Rational Recostruction The European Journal of the History of Economic Thought 14:1 119-156 March 2007 Pareto, V. Manual Of Political Economy (A. Schwier trans. From French Edition of 1927) A. M. Kelly Publishers NY (1971) Robbins, L. An Essay of the Nature and Significance of Economic Science 2nd ed. McMillian London (1935) Schumpeter, J History of Economic Analysis Oxford Univ. Press. NY (1954) Sen, A. K. Collective Choice and Social Welfare Holden-Day (San Francisco 1970) Sen, A. K. Rational Fools Philosophy and Public Affairs Vol. 6, No. 4 (Summer 1977) pp.317- 344 Sen, A. K. Equality of What? Tanner Lectures on Human Values Vol. I (S. McMurrin ed.) Cambridge Univ. Press (1980) Sen, A. K. Well-Being, Agency, and Freedom: The Dewey Lectures 1984 The Journal of Philosophy 82 (Apr. 1985) pp. 169-221 Sen, A. K. On Ethics and Economics Blackwell Publishers (Cambridge MA 1987) Suen, W and Silberberg, E. The Structure of Economics, A Mathematical Analysis Irwin- McGraw-Hill (NY 2001) Turgot, A. The Formation and Distribution of Wealth, Reflections on Capitalism (K. Jupp Trans.) Othila Press (London 1999)