Applications of ordinality and cardinality to economic utility theory


Published on

1 Like
  • Be the first to comment

No Downloads
Total views
On SlideShare
From Embeds
Number of Embeds
Embeds 0
No embeds

No notes for slide

Applications of ordinality and cardinality to economic utility theory

  1. 1. 93125705863566201855810072936065987648611791045334885034611365768675324944166803962657978771855608455296 Applications of54126654085306143444318586 Ordinality and Cardinality to76975145661406800700237877 Economic Utility65913440171274947042056223 Theory0538994561314071127000407854733269939081454664645880 12/16/201079727082668306343285878569 Benjamin Daniels830523580893306575740679545716377525420211495576158140025012622859413021647155097925923099079654737612551765675135751782966645477917450112996148903046399471
  2. 2. Introduction Economics is the science of choice and scarcity. Theres not enough stuff to go around, sohow do people choose what they buy? Perhaps even more importantly, how do governmentsdecide how to give out things like public schooling, tax rebates, and food stamps? Any decisionof this type requires comparing the costs and benefits of a particular transfer, but depending onthe underlying assumptions, it may not actually be possible to compare these things. Someconceptions of utility space can clearly order sets of individual preferences but cannot describethe distance between them, making the potential gains from trade difficult or impossible tomeasure. How can comparisons and decisions be made in this environment? What does thisfailing imply for theories of public choice? For theories of economic efficiency? Economic utility theory is the branch of economics concerned with answering questionsabout happiness. By examining these questions through the lens of utility theory, it is possible tosee that ordinality and cardinality play an important role in the theoretical constructions of utilityspace and interpersonal comparisons within it. In particular, economic utility theory seeks toanswer crucial questions about personal happiness, then quantify the results in order to provideuseful numerical outputs to policymakers and analysts. However, economists differ on thefundamental assumptions that can be made in the realm of utility theory. In order to build auseful model to aggregate happiness, it is important to first begin from the fundamentalconceptions of personal utility, then determine whether they can be successfully extended tointerpersonal and aggregate comparisons. In particular, an “indifference curves” approach is most useful when attempting to makesense of commodity and utility spaces. By applying basic assumptions about the order of thesespaces, it is possible to generate a set of equivalence classes that fully describe the space. Due to 1
  3. 3. the shape of these classes, it becomes straightforward to compare different goods and performLagrangian optimization tasks across them. However, the indifference curves approach is lesswell suited to social comparisons, unless it is possible to devise a method for makinginterpersonal comparisons. Even then, optimization is much less straightforward, because the setof inputs to a social utility function is much larger and their interactions more complex than thecomponents of individual utility. Policymakers can use various underlying mathematical assumptions about thecardinalization of utility space for individuals and for society as a whole in order to dramaticallysimplify the decision-making process. Indeed, without these simplifications, it is impossible toconstruct a decision rule that accurately reflects the desires of the society as a whole, as KennethArrow proved in 1950. However, politicians and policymakers must be aware of the assumptionsthey are making when utilizing different simplifying rules, particularly when the assumptionshave implications for overall efficiency and distributional equity. After establishing thetheoretical basis for the analysis and demonstrating the conflicts inherent in it, this essay willreview several such rules and analyze the strengths and weaknesses they exhibit with respect tothe overall theory of welfare maximization. From the popular construction of the theory to the practical implementation, one gap isoverwhelmingly obvious – a lack of concern for or understanding of problems of distributionalequity. While the theoretical formalization of the public choice problem readily admits thisfailing, the practical application generally assumes it away, and in doing so often implicitlyselects the most utilitarian limit case. If society wishes to design programs that do not conform tothis measure of societal well-being, a more thorough analysis of the topic will be needed ifdecisions are to be fully and completely analyzed. 2
  4. 4. A full analysis of every proposed decision rule is crucial because Arrow’s ImpossibilityTheorem, stated as follows, shows that any social decision rule will be imperfect: Any decisionrule that respects transitivity, independence of irrelevant alternatives, and unanimity is adictatorship. Transitivity is defined the usual way. Unanimity is defined to mean that if everyindividual prefers A to B, then option A will be preferred by the decision rule to B.Independence of irrelevant alternatives (IIA) is defined to mean that the societal relativeordering of A and B is affected only by the relative orderings of A and B by individuals. Proof. Let option B be chosen arbitrarily. By unanimity, in any preference profile inwhich every voter puts B at the very top or very bottom of his rankings, so too must society.Suppose to the contrary that for such a profile and for distinct A, B, and C, the social preferenceranked A > B and B > C. By IIA, this would hold even if every individual moved C above A,because that could be arranged without changing the relative order of A and B or C and B(because B is at the very top or very bottom of every profile). By transitivity the social rankingwould then place A > C, but by unanimity it should also place C > A, leading to a contradiction. In order to avoid the contradiction, there must be some voter n who is pivotal in the sensethat by changing his vote he can move B from the bottom of the social profile to the top (whetherthis happens immediately or after some 1-n voters have changed their votes is irrelevant). In thiscase, voter n is a dictator over any arbitrary pair of options (A, C) where A, C ≠ B, since theordering of A and C relative to B, and therefore to each other, is determined by the location of A,B, and C in voter n’s individual preferences. But because A, B, and C are chosen arbitrarily, wecan see that there must be such a dictator for any pair of options. In fact, this dictator will bevoter n for every pair, since the relative societal rankings of any other option B are dependent onthe rankings of A and C, which voter n also controls. So the system is a dictatorship.▪ 3
  5. 5. Ordering Preferences and Making Decisions To avoid dictatorship, we must accept flawed systems. But to fully analyze these systems,we must be able to discuss their impact on individuals. The basic building blocks of individualutility are commodities. When economists wonder how happy a person is, it is generallyaccepted to define the level of happiness as some function Uj of the various commoditiespossessed by a person j. For simplicity, we examine only that subclass of commodities known as“goods” – commodities for which the utility Figure 1. Commodity space for guns and butter,function has a positive-valued partial and an indifference curve IB.derivative. The simplest possible case, andperhaps the most illustrative, is the two-good economy. In this case, the quantity ofeach good can be mapped clearly onto acommodity space as shown in Figure 1. It is natural to make several assumptions that lead to the well-ordering of the commodityspace. First, order is imposed; that is, for any two points A and B, exactly one of the followingmust be true: A is preferred to B, B is preferred to A, or the individual is indifferent between thetwo points. Second, preferences are transitive; if A > B and B > C, then A > C. Third, becausewe have restricted our examination to “goods”, preferences are positive; moving in a rightwardor upward direction (ie, obtaining more of one good without sacrificing any of another) is alwayspreferable to the reference point. Lastly, preferences are convex, illustrated by the “indifferencecurve” moving away from Point B in the diagram; in order to convince an individual to sacrificemarginal units of one good, it requires an increasing amount of the other good as compensation. 4
  6. 6. Together, these rules yield a set of equivalence classes in commodity space known asindifference curves and denoted by Iij with index i for the individual and j for the utility level.These curves are dense in the space and, as one moves upward and rightward, represent ever-increasing levels of personal utility; j increases with utility. The conception is readily generalizedto many-dimensional commodity spaces and may even be extended to include commodities forwhich Uj has a negative partial derivative with some adjustment to the underlying assumption ofpositivity. In addition, because indifference curves themselves represent a new well-orderedspace, utility space, it becomes trivial for a mathematically-inclined individual to maximize hisor her utility: a simple Lagrangian process incorporating the prices of various goods and a budgetconstraint allows the individual to select an allocation that achieves the highest level of utility. Although individual utility space is well-ordered as a result of these assumptions, it is notyet a cardinalized space. That is, while we can easily say that Iim > Iin whenever m > n, we havenot established a method for determining the value of the ratio Iim/Iin. This is made more difficultby the fact that indifference curves are dense, and therefore has cardinality equal to that of thecontinuum, c; the indices are not natural numbers. To establish a base point for these ratios, it iscommon to select an arbitrary reference curve Ii1, then cardinalize the space such that Iim/Iin =m/n. By this method the ratio of utilities of any two commodity vectors can be determined,although there is still no conception for determining ratios between individuals, such as I1m/I2m. Given these relatively simple assumptions and the system they imply, the next step is toattempt to extend the approach to a social level. Since not all the decisions of society are made inopen commodity markets, the characteristics and limitations of these methods when applied tosocial decisions, public goods, and the like will be crucial for policymakers to understand theimpacts and restrictions of various types of analysis. 5
  7. 7. Aggregating Preferences and Making Social Decisions The social analysis begins from much the same place as the individual analysis, forobvious reasons: society as an entity is made up only of the individuals in it. As far ascommodities are concerned, a society can utilize exactly the same process as an individual formapping preferences and optimizing outputs. However, this is not the whole story for acollective decision to be made: The distribution of wealth is important for determining values and shaping production, and it can even be maintained that a country with one and the same amount of general wealth may be rich or poor according to the manner in which that wealth is distributed. (Schumpeter)So although society, like the individuals who constitute it, benefits from a greater number ofgoods, it is also important to take into account the impact of those goods on various individualsby way of a distributional analysis. Given this information, the utility space for a society gains a number of dimensions equalto the number of people in the society. This construction represents the fact that, in addition tomaking tradeoffs among different types of goods, society must also make tradeoffs about theallocation of those goods. However, it is not sufficient to quantify these tradeoffs amongindividuals in commodity terms, or even in dollar terms; from individual indifference-curveanalysis it is clear that different individuals have different valuations for various goods. Instead,the tradeoffs among people must be represented in utility terms. The crucial cardinal expressionneeded for this construction – the slope Iim/Ijm – is presently undefined, because there is noaccepted conception of individual utility that allows interpersonal comparison. There existvarious methods for addressing this problem in practice, but in order to fully appreciate them,their implications must first be examined through theories of distributional equity. 6
  8. 8. One key question facing decision-makers is the shape of the social indifference curvewith respect to distribution of goods among individuals. Theoretically, this parameter is the mostimmediate method for assessing the importance of distributional equity to the overall societalutility function. The analysis dates back to the utilitarian conception advanced by JeremyBentham, who asserted that distributional concerns were entirely useless; in this conception, agood is valued at exactly the same societal worth, no matter who possesses it. The diametricallyopposed theory was later advanced by John Rawls, who claimed that society should only bejudged by its worst-off citizens; additional goods were worthless if they were distributed toanyone else. Graphically and algebraically, these arguments represent the upper and lowerbounds on the space of indifference curves most simply expressed by the formulaRawls demonstrates the limit as α approaches -∞ and Bentham the case as α approaches 1.Figure 2 illustrates the space of indifference curves generated by this conception over a simple 2-person society. A Rawlsian society more concerned with equality will distribute goods moreevenly whereas, in the limiting Benthamite case, the additional interpersonal dimensions arerendered irrelevant to the maximization by the flat shape of the indifference curves. Figure 2. Rawlsian and Benthamite societal indifference curves. R represents the social choice made with Rawlsian assumptions and B the Benthamite choice. 7
  9. 9. In Practice: CBA, QALYs, and the U-Index When designing actual social policies, legislators and executives must handle these twinhurdles of utility measurement and distributional analysis. They must decide how to measureindividual utility, how to measure it against the utility of others, and what weight to put on thecompeting concepts of maximization and equity. Three practical and theoretical mechanismshighlight the various choices that could be made by a policymaker: cost-benefit analysis, theQALY system, and the U-Index. Each of these emphasizes a different aspect of the socialdecision and, together, they demonstrate the fundamental fact that tradeoffs must be made whichcannot be quantitatively compared. The time-honored practice of cost-benefit analysis (CBA) calls on the most utilitarianassumptions to make public choices. It weights total costs directly against total benefits, entirelywithout regard for distributional impacts. Most public accounting, whether directly or indirectly,utilizes CBA in order to justify budgets and often includes explicit costs and benefits only; forexample, Congressional Budget Office scores are often manipulated by legislators by shiftingexpenditures to states or individuals or by making advantageous assumptions about the value ofvarious services. However, CBA is not without merits; it does, in the broadest sense, capture theeconomic ideal of maximization by providing a clear quantitative guideline for the value of apublic project or a benefit schedule and allowing for sound derivative-based optimization. Whencosts and benefits are clear and distribution is of minor concern, CBA can be an appropriatechoice for decision-makers. Another method of decision-making is demonstrated by the QALY system, a method ofapportioning health benefits utilized by the British National Health Service. To construct theQALY system, researchers used an extensive series of interviews to determine the relative values 8
  10. 10. of various health states as judged by normal people; the results were averaged and normalizedsuch that one year of normal life has a value of one Quality-Adjusted Life Year (QALY). Thekey assumption underlying this valuation is that every life has the same value, which sidestepsthe problem of interpersonal comparison entirely. The QALY values are calculated bymultiplying expected life quality produced by expected longevity, so Value = Quality ∙ LongevityThe marginal value of a procedure is given by its expected impact on the value of a life, andthanks to the inherent Lagrangian process we can use it to assess the distributional impacts of theQALY system. The total derivative is given by: ∆Value = (Quality ∙ ∆Longevity) + (∆Quality ∙ Longevity)The inherent bias of this total derivative is to target longevity-increasing treatments to those witha good quality of life, and quality-enhancing treatments to those expected to live the longest.While this seems sensible from the abstract efficiency criterion, it is important to note that itwould also be possible to design a system that attempted to improve the quality of life for theworst-off class first – a system that would certainly be preferable for the very sick. The U-Index is a theoretical system proposed in order to avoid cardinalizing personalutility space altogether. In a similar manner as the QALY system, the U-Index obtains ordinaland relative measures of well-being from participants. It converts the responses into directlycomparable cardinal measurements by characterizing a respondent into binary groups of “happy”and “unhappy” based on responses, and then reports a “U-Index” for an activity based on theproportion of people who are unhappy during that activity. Like CBA and the QALY system, theU-Index provides a clear, cardinal, quantitative target for policymakers to optimize, but like theother systems it dodges the question of distributional impact by assuming utilitarian equality. 9
  11. 11. Conclusion The examples used to demonstrate practical applications of utility and social choicetheory make clear one crucial shortcoming of the theory: it is extremely difficult to explicitlyvalue interpersonal comparisons of utility. Generally, policymakers elect to assume them awayentirely, choosing to implement programs on a mostly utilitarian basis, and giving weight todistributional concerns only indirectly – by allowing measured benefits to increase with need.Indeed this is the largest gap in the underlying theory as well. Despite the obvious need for amethod of valuing interpersonal exchanges from the lens of society, theory and practice have notsolved the key question of cardinalizing interpersonal comparisons such that I1m/I2m has a well-defined value, nor have they attempted to devise simplifying assumptions that handle theproblem in a more nuanced manner than simply assuming the value to be equal to m/m = 1. However, the shortcoming may not be entirely the fault of the policymakers. Eventheories that are relatively liberal – the British healthcare system, for example – have defaulted tothe utilitarian conception implicitly, almost without realizing it. In addition, academic literaturebeyond Rawls and responses to him offers no guidance on how to better include equity concernsor how to “optimize” equity. It is a far easier task to maximize output than to ensure that it windsup in the right hands, and even today much work on inequality lacks the formal rigor that wasdeveloped for efficiency by the early Chicago school. Policymakers motivated to addressinequality will need a stronger set of theoretical and mathematical tools with which to developand assess policy options if they are to use anything other than the simplest criteria fordistributional equity. 10
  12. 12. ReferencesBarnett, William. “The modern theory of consumer behavior: Ordinal or cardinal?” Quarterly Journal of Austrian Economics 6, No. 1 (2003). 41-65.Baumol, William J. “The Cardinal Utility Which is Ordinal.” The Economic Journal 68, No. 272 (Dec 1958). 665-672.Fleming, Marcus. “A Cardinal Concept of Welfare.” Quarterly J. of Econ. 66, No. 3 (Aug 1952). 366-384.Geanakoplos, John. “Three Brief Proofs of Arrow’s Impossibility Theorem.” Cowles Foundation Discussion Papers 1123RRR, Cowles Foundation, Yale University. 2001.Harsanyi, John C. “Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility.” The Journal of Political Economy 63, No. 4 (Aug 1955). 309-321.Kahneman, Daniel, and Alan B. Krueger. “Developments in the Measurement of Subjective Well-Being.” J. of Econ. Perspectives 20 (2006). 3-24Kaldor, Nicholas. “Welfare Propositions of Economics and Interpersonal Comparisons of Utility.” The Economic Journal 49, No. 195 (1939). 549-552.Mandler, Michael. "Compromises Between Cardinality and Ordinality in Preference Theory and Social Choice."Cowles Foundation Discussion Papers 1322, Cowles Foundation for Research in Economics, Yale University. 2001.Mandler, Michael. “Cardinality versus Ordinality: A Suggested Compromise.” The American Economic Review 96, No. 4 (Sep 2006). 1114-1136.Mercuro, Nicholas, and Steven G. Medema. Economics and the Law, Second Edition: From Posner to Postmodernism and Beyond. 1997.Schumpeter, Joseph. “On the Concept of Social Value.” Quarterly J. of Econ. 23 (1908-9).Wold, H., G. L. S. Shackle and L. J. Savage. “Ordinal Preferences or Cardinal Utility?” Econometrica 20, No. 4 (Oct 1952). 661-664. 11