The Smooth Value of Lumpy Goods
Lumpiness and the Standard Picture
Economists often employ a convenient set of assumptions regarding the goods that individuals care about and the form of individuals’ preferences for these goods. For short, call this set of assumptions “the Standard Picture.” (1) Individuals’ preferences are “outcome-oriented,” in the sense that each individual cares about her own holdings of various goods—goods such as income, health, leisure, or environmental quality—rather than about the relation between her holdings and others’, or about the processes by which her holdings were produced. If there are M goods, an individual’s preference structure takes the form of a ranking of possible M-dimensional bundles of the goods. (2) An individual’s holding of each good is measured by a real number (for short, the good is numerically “quantifiable”). (3) Each good is infinitely divisible. That is, the range of possible holdings of the good corresponds to some interval of real numbers. (For example, an individual’s annual income might be any nonnegative real number. An individual’s daily leisure might be any real number between 0 and 24 hours, inclusive.) (4) An individual’s preferences with respect to bundles of goods can be represented by a real-valued utility function that is continuous, and that is monotonically increasing in each good. Roughly, a continuous function is one that doesn’t “jump.” A function is monotonically increasing in a given input if its value increases as the input value increases, holding constant the value of the other inputs.
One major theme in Lee Fennell’s rich and important new book, Slices and Lumps: Division and Aggregation in Law and Life, is that goods and preferences can depart from the Standard Picture. The departures that Fennell emphasizes are what she terms “lumpiness in supply” and “lumpiness in demand.” “Lumpiness in supply” arises when it is impossible to divide a good into indefinitely small pieces, or at least very expensive to do so. In other words, assumption three above in the Standard Picture fails. For example, pets come in discrete units; it’s not possible for me to have .57 of a pet dog.
“Lumpiness in demand” arises when an individual’s preferences are discontinuous. In other words, assumption four above in the Standard Picture fails. One type of discontinuity that Fennell describes arises with a step function. A step function involves one or more cutoffs with respect to the quantity of the good (on the x axis). When the quantity reaches the cutoff, utility (on the y axis) “jumps” up. A step function also involves a violation of monotonicity. Utility is flat, rather than increasing, until the cut-off level is reached.
Slices and Lumps explores the tricky problems in governmental policy design that arise by virtue of lumpiness in supply and demand. In this Essay, I wish to examine a different (although related) question—namely, whether lumpiness in demand and supply, and other departures from the Standard Picture, interfere with the valuation of governmental policy. Slices and Lumps doesn’t say much about this question. But it is, I believe, an important one.
Assume that departures from the Standard Picture are so severe that individuals’ preferences can’t be presented at all by utility functions (let alone utility functions that are continuous and monotonically increasing). This would create a major hindrance to policy assessment; the “decision costs” of policy assessment would rise dramatically.
Why? Assume that there are N individuals in the population, M goods, and we are considering a set of policies {P, P*, P**, …}. An “allocation” is an assignment of one or another M-dimensional bundle of the goods to each of the N individuals. Each policy, then, is a probability distribution over allocations.
(Note: If a policymaker is omniscient, she can conceptualize each policy as an allocation, namely the allocation that would result were that policy to be chosen. With a rule in hand to rank allocations, she can rank policies. If the policymaker lacks full information, a rule for ranking allocations doesn’t give her sufficient guidance for ranking policies. However, if the policymaker complies with basic axioms of rational choice, her information regarding the possible allocations resulting from each policy can be formalized as a probability distribution over allocations conditional on each policy. In short, then, each policy is a probability distribution over allocations.
If each individual’s preferences are representable by a utility function, then a given allocation corresponds to a vector (list) of utility numbers: one number for each of the N individuals, namely the utility of her bundle in that allocation. We can now assess policies by ranking probability distributions over utility vectors. This is a much more tractable task than ranking probability distributions over allocations. An allocation is an N x M matrix: some holding of each of M goods, for each of N individuals. Formulating, justifying, and applying a rule for ranking N-entry vectors (or probability distributions over vectors) is a lot easier than formulating, justifying, and applying a rule for ranking N x M-entry matrices (or probability distributions over those).
(Note: A rule for ranking utility vectors is a social welfare function (“SWF”), for example a utilitarian SWF or, alternatively, a prioritarian SWF, giving greater weight to those at lower levels of utilities. Each SWF is associated with various possible “uncertainty modules,” namely rules for ranking probability distributions over utility vectors. In order to function as inputs to an SWF, each individual’s utility numbers must be adjusted via scaling factors, so as to enable interpersonal comparisons. This aspect of utility functions is beyond what can be discussed here.
The dominant economic methodology used within government to evaluate policy choice is not the SWF framework, but cost-benefit analysis (“CBA”). CBA assesses policies by summing individual monetary equivalents. The existence of individual utility functions also greatly facilitates CBA, by facilitating the calculation of monetary equivalents.)
Fortunately, lumpiness in demand and supply do not interfere with the construction of utility functions. In particular, as I’ll discuss in this Essay, von-Neumann/Morgenstern (“vNM”) utility theory offers a framework for constructing utility functions that is remarkably robust to a range of departures from the Standard Picture—both supply or demand lumpiness, and other violations of assumptions one to four. vNM theory yields a utility scale which is not merely real-valued, but smooth and cardinal (both of these to be explained below). In short, the irregularities that Fennell describes are consistent with the existence of a smooth, cardinal scale of individual value. These irregularities pose major challenges to policy design, but do not cut so deep as to jeopardize the very measurement of value.
This is good news for Fennell, I think. Her goals are constructive (to offer sophisticated advice in devising good policy, advice that takes account of lumpiness) rather than skeptical. The news I’m delivering is positive, not skeptical, as regards value measurement, and so bolsters Fennell’s constructive project.
In what follows, I first describe a kind of departure from the Standard Picture so severe as to block the existence of a utility function. I then consider two distinct ways to ensure utility measurement: the Debreu framework and the vNM framework. Both are widely employed in economic theory. The Debreu framework is not well suited to Fennell’s project. The vNM framework is, since its axioms are fully consistent with the departures from the Standard Picture that Fennell identifies and so richly explores.
I’ll simplify matters by generally assuming that individuals’ preferences are outcome-oriented. Fennell herself assumes as much, or at least I read her as doing so. Her challenge to the Standard Picture is targeted elsewhere. In fact, vNM theory is not committed to outcome-oriented preferences. If the theorist wants to drop this premise, too, he can. (See below.) But since dropping the premise isn’t Fennell’s aim, vNM theory’s flexibility on this count is not a feature of the theory that I’ll discuss at length.
Monica the Lexical Materialist: A Problem for Utility Measurement
Consider an outcome-oriented preference held by some individual—a preference that takes the form of a ranking of possible bundles of M goods. A utility function u(∙) assigns numbers to bundles—u(a) is the number assigned to bundle a—and it does so in a manner that represents the preference structure. This means the following. Whenever a is preferred to a*, u(a) > u(a*). And whenever the individual is indifferent between a and a*, u(a) = u(a*).
To avoid some tangential distractions, let’s assume that each of the M goods is quantifiable (if not necessarily infinitely divisible). That is, for each good m (one of the M goods), there is a scale sm for measuring holdings of good m that fully captures everything about the good everyone cares about. In order for an individual to ascertain where a bundle including good m fits in her preference ranking of bundles, it suffices for her to know that she has a certain number of units of that good on scale sm (rather than needing a “thicker,” nonnumerical description of the holding). An individual’s holdings of money are quantified in dollars (or euros or yuan or …), of leisure in hours, and so forth.
Clearly, a necessary condition for a preference to be represented by a utility function is that the preference be complete and transitive. However (and this is not at all intuitive), it turns out that completeness and transitivity are not sufficient for the existence of a utility function. Consider the following case.
Monica the Lexical Materialist: Monica cares about two things, money and leisure. Money is infinitely divisible. Leisure may or may not be.
Monica has a complete, transitive preference ranking of money/leisure bundles which gives lexical priority to money. She always prefers the bundle with more money; if money amounts are equal, she prefers the bundle with more leisure.
Monica’s preferences (it can be shown) cannot be represented by any utility function. See Appendix.
A lexical element in someone’s preferences doesn’t automatically frustrate utility measurement. Contrast Monica with Henry, the lexical parent. Henry cares about two things: the number of his biological children, and his money. He gives lexical priority to the first. In ranking bundles of children and money, Henry always prefers the bundle with more children; if those numbers are equal, Henry prefers more money. Henry’s preferences can be represented by a utility function. See Appendix.
So it is not lexicality, per se, that blocks a utility function, but lexicality of a certain sort. Generalizing from Monica’s case, we can say this. Assume that at least one of the M goods is infinitely divisible. Moreover, there is at least one other good, such that the individual lexically prefers the first good to this second good. Then the individual’s preference ranking of the bundles, even if complete and transitive, can’t be represented by a utility function.
The challenge we now face is this. Having a complete and transitive preference ranking is not sufficient for utility measurement. We want to place additional conditions on preference that will guarantee a utility function but that do not take us all the way to the Standard Picture—which seems much too restrictive. How to do this?
Divisibility and Countability
One way to guarantee a utility function is to limit the number of items being ranked. The policy analyst is modelling policy choices as leading to allocations of bundles among a population of N individuals, and is doing so in light of the individuals’ preferences over the bundles. For these purposes, the analyst (implicitly or explicitly) constructs a set of possible bundles. A given bundle is “possible” if the analyst’s model allows for the contingency (doesn’t rule out from the get-go) that there is some policy which yields an allocation including this bundle. Each individual’s preference ranking is, then, conceptualized as a ranking of this set of possible bundles.
If the set of possible bundles is finite, then any complete and transitive ranking of the set can be represented by a utility function. This is straightforward. Take a least-preferred bundle (one which is not preferred to any other bundle) and assign the number 1 to that bundle and every bundle indifferent to it; assign the number 2 to a second-least preferred bundle (one which is not preferred to any other bundle except a least-preferred bundle) and every bundle indifferent to it; and so forth.
Moreover, if the set of possible bundles is infinite but countable, then—again—any complete and transitive preference ranking can be represented by a utility function. A countable set is such that there is a one-to-one correspondence between its elements and the set of natural numbers {1, 2, ….}.
Problems in representing a complete, transitive preference ranking with a utility function can arise only if the set of items being ranked (here, set of bundles) is uncountably infinite—as with the example of Monica the lexical materialist above. This suggests one strategy for ensuring a utility function: stipulate that the set of possible bundles is finite or countably infinite.
However, I don’t believe this to be an attractive strategy, for Fennell’s purposes. She wishes to allow for indivisible goods (lumpiness in supply), but certainly not to require that all goods be indivisible. Assume that at least one of the M goods is both quantifiable and infinitely divisible, that is, the range of possible levels of that good corresponds to some interval of real numbers. (For example: all possible quantities of dollars between 0 and $1 billion. All possible hours of leisure in a day between 0 and 24.) If so, the set of possible bundles will be uncountable.
The Debreu Framework for Utility Measurement
One standard route to guaranteeing the existence of a utility function where the set of items being ranked is uncountable derives from the work of Gerard Debreu. (On this topic, see generally, Kreps, Microeconomics Foundations (2012)). In a nutshell, Debreu shows that an ordering of a connected subset of Euclidean space can be represented by a continuous utility function if and only if the ordering is continuous. Applied to the problem at hand, the Debreu framework ensures that a continuous utility function representing the preference ranking of the set of all possible M-dimensional bundles will exist, even if that set is uncountable, as long as the following is true: (1) every good is infinitely divisible and the set of possible levels of the good corresponds to some interval of real numbers; and (2) the preference ranking itself is continuous, which means the following. A continuous preference ranking: Assume that bundle a is ranked above bundle b. Then, there is some region around a which is also preferred to b. (See Appendix for precise definition.) For example, if Radhika prefers earning $100,000 and having 3 hours a day of leisure to earning $100,000 and having 2 hours a day of leisure, then there is some ε > 0 and λ > 0 such that she prefers all bundles with income in the range ($100,000 + ε, $100,000 – ε) and leisure in the range (3 hours + λ, 3 hours – λ) to earning $100,000 and 2 hours leisure.
The Debreu framework is not well suited to Fennell’s project because its assumptions are too restrictive. First, the assumption that every good is infinitely divisible preludes Fennell’s “lumpiness in supply”—her basic premise that some goods are indivisible.
Second, the assumption that preferences are continuous precludes “lumpiness in demand.” Discontinuous preferences are such that: in at least some cases, I prefer bundle a to bundle b but do not prefer everything close enough to a to bundle b. This is exactly what occurs with a step function. For example, imagine that Teddy mainly uses his leisure to watch the TV shows he loves. He prefers 2 hours of leisure to 1 ½ hours, since with 2 hours he can watch one more show (these beloved shows coming in ½ hour increments). Teddy is indifferent between 1 ½ hours of leisure and any amount greater than 1 ½ and less than 2. Teddy’s preferences violate the continuity condition: although he prefers 2 hours to 1 ½ hours, there is no λ > 0 such that he prefers any leisure in the range (2 − λ, 2 + λ) to 1 ½ hours.
Preferences that take the form of a step function exemplify discontinuity, but this can occur in many other ways. Tweak the above example slightly so that Teddy becomes bored with leisure and no TV, and he dislikes being bored. So he actually prefers having less leisure to more leisure within the range from 1 ½ to 2, exclusive. His preferences are no longer a step function but still fail Debreu’s continuity requirement.
The vNM Framework for Utility Measurement
vNM utility theory offers a different route to ensuring that preferences can be represented by utility functions, even if the set of items S at hand is uncountably infinite. The theory places no conditions at all on that set: it can be finite, countably infinite, or uncountably infinite, and it can contain items of any sort. vNM theory instead guarantees a utility representation via a kind of enlargement of the scope of preference. The individual with a preference over S is also posited to have a ranking of the set L-S of all lotteries over the elements of S. (Let s, s*, s**, etc. be items in S. Then a lottery l assigns probabilities, summing to 1, to the elements of S. For example, lottery l might assign probability .5 to s, .3 to s*, .2 to s**, and 0 to every other element. Lottery l+ might assign probability 0 to s, .75 to s*, .25 to s**, and 0 to every other element.)
Note that an item (an element of S) is essentially the same as a “degenerate” lottery which assigns that item probability 1. Let ls be the “degenerate” lottery which assigns 1 to item s (and 0 to all other items). Then, surely, for any two items s and s*, the individual prefers s to s* if she prefers ls to ls* (and similarly for indifference).
Thus, the individual is supposed to have a preference ranking of L-S as well as S, and the latter ranking can be seen as embedded within the former in the sense just described. Further, it is supposed that both rankings are complete and transitive. vNM theory now adds two crucial conditions regarding the ranking of the lottery set L-S. It must satisfy an Independence axiom and an Archimedean axiom. Independence: If the individual prefers one lottery to a second, then—for any mixture rate m—she must prefer an (m, 1 – m) mixture of the first lottery and any third lottery to an (m, 1 – m) mixture of the second lottery and the third lottery. (See Appendix for an explanation of a “mixture” of lotteries.) Archimedean: If the individual prefers one lottery to a second to a third, then (a) there is some rate of mixture m* such that she prefers an (m*, 1 – m*) mixture of the first and third over the second lottery, and (b) there is some rate of mixture m+ < m* such that she prefers the second lottery to a (m+, 1 − m+) mixture of the first and third.
vNM theory now demonstrates the following. If the ranking of the lottery set is complete and transitive, and satisfies the Independence and Archimedean axioms, then there exists a u(∙) which assigns utility numbers to the elements of S, and that expectationally represents the lottery ranking. If the individual prefers one lottery to a second, then its expected utility (as calculated using u(∙)) is greater; if the individual is indifferent between two lotteries, then the two have the same expected utility. (See Appendix for a precise statement.) Further, because the items in S are nothing other than “degenerate” lotteries, this utility function also represents the preference ranking of S itself. If the individual prefers s to s*, then u(s) > u(s*); if she is indifferent, then u(s) = u(s*).
Let’s now apply vNM theory to the problem at hand: guaranteeing a utility function representing an individual’s preferences with respect to M-dimensional bundles of goods. The theory imposes no restriction whatsoever on the nature of the goods and the size of the set. It posits, instead, that the ranking is embedded within a complete, transitive ranking of all lotteries over the bundles in the set, and that this lottery preference satisfies the Independence and Archimedean conditions. If so, there exists a utility function u(∙) such that: If the individual prefers bundle a to bundle a*, u(a) > u(a*); if the individual is indifferent between the bundles, u(a) = u(a*).
It might be objected that individuals’ preferences are much too fragmentary to satisfy the requirements of vNM theory. For any set of bundles (even a finite set, let alone an infinite set), individuals don’t “have in their heads” a ranking of all possible lotteries over those bundles. But this objection assumes that a preference ranking is part of someone’s conscious thoughts. Instead, preferences are dispositional: to say that I prefer a to b is just to say that, were I consciously to attend to the comparison, I would favor a over b. So, in positing that an individual prefers lottery l to l*, we are positing that, were the individual asked to state which lottery he would rather face, he would pick l. Further, the vNM axioms of Independence and Archimedean, like the more basic axioms of completeness and transitivity, should not be thought of as features of individuals’ actual dispositions. As behavioral economists have exhaustively documented, individuals in actuality frequently violate all of these axioms. Rather, they are plausibly posited only as an aspect of rational preferences. If my dispositions to favor are rational, then it should be the case that they are complete, transitive, etc.
vNM theory is suited to Fennell’s purposes because it does not require preferences to conform to any of the assumptions of the Standard Picture. These premises simplify analysis and empirical testing, and for that reason are frequently adopted by economics, but they are not entailed by vNM theory itself. vNM theory doesn’t require preferences to be outcome-oriented. Rather than thinking of each bundle a as a bundle of goods, we can instead think of it, more abstractly, as a bundle of individual attributes. These attributes might include relational attributes (the attribute of being situated relative to others in some way, e.g., having lower income) or procedural attributes (the attribute of having one’s situation being produced via a certain type of process). vNM theory can then be applied to a set S of such attribute bundles and set L-S of all lotteries over S.
Assuming preferences that are outcome-oriented, vNM theory doesn’t assume that the M goods are each quantifiable. Turning to Fennell’s critical focus (assumptions three and four), vNM theory imposes no conditions on the divisibility of the various goods. All, some, or none of the goods may be infinitely divisible. As for assumption 4: vNM theory requires neither monotonicity nor continuity. Return to the example above of Teddy who loves his TV, and is bored by leisure without TV. Teddy prefers 2 hours leisure to 1 ½ hours leisure. In comparing two levels of leisure in between 1 ½ hours and no leisure, he prefers less leisure. Teddy’s preferences are perfectly consistent with vNM utility theory. They are represented by a vNM utility function that slopes downward from 1 ½ to 2 hours, and jumps up at 2 hours. The downward slope is a violation of monotonicity (a flat slope would be a less stark violation); the jump upward a violation of continuity.
A little more should be said about continuity. vNM theory allows for discontinuity in the space of bundles. To repeat: Teddy’s preferences are discontinuous because he prefers 2 hours to 1 ½ hours but does not prefer (2 –λ) hours to 1 ½ hours for any λ > 0. However, vNM theory does imply a kind of continuity requirement in the space of lotteries. If someone prefers one lottery to a second, then she prefers all lotteries which are sufficiently close in their probabilities to the first, to the second. For example, Teddy prefers 2 hours (a probability 1 of one hour leisure and 0 of any other duration) to 1 ½ hours. Consider now a lottery which gives him a (1 –π) probability of getting 2 hours, and π probability of some other duration dispreferred to 1 ½ hours, e.g., only 1 hour of leisure. Then if π is sufficiently small, Teddy will also prefer this lottery to getting 1 ½ hours.
While discontinuity in bundle space seems rationally permissible (there is nothing irrational in Teddy’s preferences), the kind of continuity in lottery space that vNM theory implies does seem to be a requirement of rationality. Would Teddy be rational in insisting that 2 hours of leisure is better than 1 ½ hours, but that a “risky” 2 (with a chance π of getting 1, however small π) is always worse? This seems irrational. Thus Fennell, I suggest, would be on good ground in accepting my invitation to split the discontinuity baby. Allow for lumpiness in demand (discontinuity in bundle space), but accept the kind of continuity in lottery space that vNM theory implies. This hybrid posture is not only well supported as a matter of preference rationality. It is also, obviously, congenial to Fennell’s project— since it permits her to allow for supply lumpiness without jeopardizing the existence of utilities.
In much of the literature on vNM theory, the “magic” of an expectational representation is discussed. vNM theory imposes conditions on a lottery preference (Independence and Archimedean) such that there exists a number assigned to each lottery tracking its location in the preference ranking and these numbers can be calculated in a very convenient way, by calculating expected utilities. Here, I wish to stress a different “magical” aspect of vNM theory. The theory does not directly impose any conditions at all on the individual’s bundle preference. However, by embedding the bundle preference in a ranking of lotteries, and by imposing conditions on the lottery ranking, vNM theory ensures the existence of a utility function that represents the bundle ranking. That is to say, the elements of vNM theory, taken together, indirectly rule out a preference ranking of bundles that can’t be represented by a utility function—for example, the ranking of Monica the lexical materialist.
Conversely, by letting the lottery axioms “do the work” in securing a utility function, vNM theory doesn’t imply extra restrictions on bundle preferences—that, is restrictions above and beyond what is required for a utility representation. This is what makes vNM theory consistent with a wide range of non-standard preferences. Suppose that an economist posits axiom A as a requirement for the bundle ranking. However, axiom A is not a necessary condition for a utility function; there exists a ranking of bundles that violates axiom A but is still representable by a utility function. Then this ranking can be embedded in a lottery preference that satisfies vNM theory. vNM theory itself will not imply axiom A.
Finally, let me discuss the sense in which vNM theory yields a utility scale that is smooth and cardinal. While the mapping from bundles to utility numbers can be discontinuous, the utility scale itself is smooth. By this I mean that the scale itself has no gaps. Let U+ be the utility number assigned to the most-preferred bundle in the set of possible bundles (or, if there is no most-preferred bundle, the least upper bound of utilities). Let U– be the utility number assigned to the least-preferred bundle in the set of possible bundles (or, if there is no least-preferred bundle, the greatest lower bound of utilities). Then, for any number u* in between U+ and U–, there is some item with utility u*. Either a bundle has utility u* or, if not, there is a lottery with utility u*.
vNM theory leads to a cardinal scale of value in the following sense. Assume that u(∙) is a vNM utility function representing a preference ranking of bundles and lotteries. Then some other function v(∙) will also rank bundles and lotteries the same way as u(∙) if and only if v(∙) is a cardinal rescaling of u(∙), i.e., the v(∙) numbers are equal to the u(∙) numbers multiplied by a positive constant plus a constant. For example, imagine that the set of possible bundles consist of just three bundles {a, b, c}. Fang’s preference ranking of this set and all lotteries is represented (let’s assume) by the utility function “First” that assigns 10 to a, 20 to b, and 30 to c. Consider now a different utility function (“Second”) that assigns 100 to a, 200 to b, and 190 to c. This utility function doesn’t represent Fang’s bundle preferences. By contrast, the utility function (“Third”) that assigns 100 to a, 200 to b, and 350 to c does represent Fang’s bundle preferences, but it doesn’t represent her lottery preferences. (Note that Fang is indifferent between bundle b and a lottery with .5 chance of a and .5 chance of c. But Third doesn’t track this lottery preference.) Finally, the utility function (“Fourth”) that assigns 115 to a, 215 to b, and 315 to c does represent Fang’s bundle and lottery preferences. These numbers can be derived from the initial numbers by multiplying by a positive constant (10) and adding a constant (15). There is no such cardinal rescaling that transforms the First numbers into the Second or the Third, and thus neither is a vNM representation of Fang’s preferences.
Conclusion
Fennell’s aim in Slices and Lumps is not merely descriptive (to delineate various departures from what I have termed the Standard Picture in economics regarding goods and individuals’ preferences for goods), but constructive (to point us in the direction of designing good governmental policy notwithstanding these departures—notwithstanding lumpiness in supply and lumpiness in demand).
This constructive goal is, I suggest, considerably facilitated by the existence of individual utility functions, representing individual preferences. A utility function u(∙) represents the preferences of a given individual (“Sarah”), if the following is true: whenever Sarah prefers some bundle of goods a to some other bundle a*, u(a) > u(a*); and whenever Sarah is indifferent between two bundles a and a*, u(a) = u(a*). Indeed economists typically posit that individual preferences are represented by utility functions. But is this supposition robust to lumpiness in supply, lumpiness in demand, and other departures from the Standard Picture?
The answer is yes. The Debreu framework for utility measurement guarantees a utility function, but only if all goods are infinitely divisible and individuals’ preferences are continuous. This is not a good option for Fennell, since a central claim of hers is that goods can be indivisible (lumpiness in supply) and individuals’ preferences can be discontinuous (lumpiness in demand). A better option is the vNM framework. This approach is, to be sure, very well known by economists and decision theorists. The point stressed by this Essay, less often discussed in the literature, is that the vNM framework is very flexible regarding the nature of goods and individuals’ preferences for them. It doesn’t insist on any aspect of the Standard Picture, and in particular is fully consistent with lumpiness in supply and demand. vNM requires only that individuals’ preferences for lotteries over goods satisfy basic rationality axioms (Independence and Archimedean), and yields a smooth, cardinal, scale of individual value notwithstanding indivisible goods and discontinuous preferences.
Appendix
This Appendix provides mathematical backup for various parts of the Essay.
1. Monica the Lexical Materialist.
Let y denote money and l leisure. Assume that the set of (y, l) bundles includes all y amounts within some interval of real numbers. (We can assume this because money is assumed to be infinitely divisible.) Monica’s preferences are such that: if y* > y, she prefers (y*, l) to (y, l+) for any levels of l and l+; and she prefers (y, l) to (y, l+) if l > l+.
Assume that Monica’s preferences are represented by a utility function u(∙). Then u(y*, l) > u(y, l+) if y* > y and u(y, l) > u(y, l+) if l > l+. We’ll show that this assumption yields a contradiction.
Arbitrarily choose two levels of leisure l++, l+++, such that l++ > l+++. For any y, pick some rational number in between u(y, l++) and u(y, l+++). This is possible because the rationals are dense in the reals. Denote r(y) as the rational number assigned to money amount y. Because u(y*, l+++) > u(y, l++) if y* > y, it must be the case that r(y*) > r(y) if y* > y. So we have constructed a one-to-one mapping from an interval of real numbers onto the rational numbers. This is impossible, because an interval of real numbers is uncountable, while the rational numbers are countable.
The original insight that certain lexical preferences can’t be represented by utility functions is Debreu’s . For general discussion of when preferences can be represented by utility functions, see Kreps, Microeconomic Foundations I: Choice and Competitive Markets.
2. Henry the Lexical Parent
Let c represent the number of children, and y money. Children, of course, are not infinitely divisible; the number of possible children for Henry is either 0 or some positive integer. Pick any strictly increasing function f(∙) on the real line that is bounded below by 0 and above by 1. Let u(y, c) = c + f(y). Then u(∙) represents Henry’s preferences.
3. The Definition of a “Continuous” Preference Ranking
To say that a ranking of M-dimensional vectors of real numbers is “continuous” means: for any vector v, the set of vectors ranked better than v and ranked worse than v are each open sets. That is, if v* is ranked above v, then there is some region around v* such that every vector in that region is also ranked above v; and if v* is ranked worse than v, then there is some region around v* such that every vector in that region is also ranked worse than v. See Kreps (2013, p 14) for this and equivalent definitions of a continuous ranking.
4. vNM Theory
Lottery mixtures: For a given lottery l, let πl(a) denote the probability that the lottery assigns to bundle a. Then an (m, 1 – m) mixture of two lotteries, l and l*, 0 < m < 1, is a new lottery that assigns each a a probability mπl(a) + (1−m)πl*(a).
u(∙) expectationally represents the preference ranking. This means the following. l is preferred to l* if and only if Σπl(a)u(a) > πl*(a)(u)(a). And l is indifferent to l* if and only if Σπl(a)u(a) = πl*(a)(u)(a).