INTERPERSONALLY COMPARABLE UTILITY FUNCTIONS pdf

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1 1: "Harsanyi " by Matthew D. Adler Abstract. Over many years, interpersonal comparisons of utility have had a significant role to play in economics. Utility began as a basic concept on which Prances Hutcheson, Cesare Beccarla, Jeremy Bentham, John Stuart Mill, and Henry Sidgwick sought to build a general ethical theory that is simple yet profound. Advanced Search Abstract Benefitâ cost analysis BCA evaluates policy choices by summing unweighted monetary equivalents, and is insensitive to distributional considerations. This article provides an accessible overview of the topic of distributional weights, with a special focus on environmental policy. The intellectual foundation for weights is the concept of a social welfare function SWF. The article then considers two important objections to distributional weighting: BCA does not take into account whether those made better off by a policy have higher or lower incomes, or higher or lower levels of nonincome welfare-relevant attributes e. Distributional weights were adopted, for a time, at the World Bank Little and Mirrlees However, it appears that distributional weights have rarely if ever been used by BCA practitioners in the U. This article, which is part of a symposium on Distributional Considerations in Benefitâ Cost Analysis, provides an introduction to distributional weights. The SWF is a fundamental construct in many areas of welfare economics, including optimal tax theory, growth theory, and analysis of climate change. This is the view of BCA running through the literature on distributional weights, and is presented here. This account of BCA is quite different from the familiar view that sees BCA as a tool for implementing the criterion of Kaldorâ Hicks efficiency potential Pareto superiority. The Kaldorâ Hicks criterion has the advantage of avoiding interpersonal comparability, but has various flaws, described in a literature beginning with Scitovsky ; see also Gorman ; Chipman and Moore ; Sen ; Boadway and Bruce ; Boadway Rather, it aims to explain the key ideas. The article then discusses the functional form of weights matching these two SWFs. One concerns the possibility of interpersonal comparisons given heterogeneous preferences. The second is that distributional considerations are better handled via the combination of the tax system and unweighted BCA. The discussion aims to be accessible. The fundamentals of distributional weighting are illustrated with a simple, one-period model. Space constraints preclude a treatment of certain additional issues that arise in the intertemporal context, in particular the relation between distributional weights and discounting. Key mathematical formulas are provided in a Technical Appendix. A more rigorous, mathematical analysis of many of the topics discussed in the main text is provided in the online supplementary materials. The reader should consult these materials, along with cited references, as backup for the discussion. It now permeates many subdisciplines within economics although less so governmental practice. To illustrate, imagine that there are three people in the population and two outcomes are being compared. Jim has a particular bundle of attributes in outcome x and a different bundle in outcome y. The same is true of Sue. Laura has the same attributes in both outcomes, and thus is unaffected by the choice between them her well-being does not change. Thus outcome x is mapped onto the utility vector 10, 30, 40 and y is mapped onto the vector 11, 25, 40 â with the first entry the utility number for Jim, the second for Sue, and the third for Laura see table 1. Page 1

2 2: A Remark on Admissible Transformations for Interpersonally Comparable Utilities The ability to sum utility functions of different individuals depends on the utility functions being comparable to each other; informally, individuals' preferences must be measured with the same yardstick. Then the ability to create a social welfare function depends crucially on the ability to compare utility functions. This is called interpersonal utility comparison. The purpose of this paper is threefold: To present a formal framework for the analysis of paternalism, freedom and well-being. To use this framework in a discussion of endogenous preference adjustments such as the problem of cheap and expensive tastes. To explore under what circumstances it is defendable to use utility of money as an interpersonally comparable measure of well-being. It is argued that under some circumstances, the intrinsic value of freedom is the ethical position that justifies interpersonal comparisons. Key words and phrases. Interpersonal comparisons, well-being, freedom, paternalism. This is gratefully acknowledged. Introduction This note presents a formal framework for the analysis of paternalism, freedom and well-being, uses the framework in a discussion of the problems with cheap and expensive tastes and finally shows that the use of social welfare functions and inter- personally comparable utility functions can be motivated by certain paternalistic value judgements. The formal framework 2. A necessary condition for paternalism. Defining paternalism is not straight forward. In this paper, self perceived well-being will be denoted u and may be thought of as cardinal utility in the Benthamite sense. Imposed well-being will be denoted w. In this case the individual is the sole judge of her own well-being. Note that by focusing on w, we focus on paternalistic perceptions of well-being rather than paternalistic actions. Each individual is assumed to have preferences over the elements of A. This allows for several elements to be chosen from the choice set. Wether this is reasonable or wether C should be assumed to be a singleton depends on how the elements of A are defined, a question we will return to later on. Positive and negative freedom. The following simple but useful distinction between negative and positive freedom will be made: Positive freedom is related to the size of the choice set, and increases as the choice set is expanded. The liberal view that each individual is entitled to the most extensive basic liberty compatible with a similar liberty for others Rawls, is essentially a restriction on the permissible elements of individual choice sets. In this note we assume that all individuals enjoy these basic liberties and focus on the relation between positive freedom and well-being. Intrinsical and instrumental freedom. When judging well-being, freedom can be valued intrinsically or instrumentally. The formal definition of intrinsical and instrumental freedom is as follows: Any well-being function that puts an intrinsical value on freedom of choice must be at least partially paternalistic. To see this assume the following: Formally, let A denote the number of elements in A and suppose that the self-perceived well-being u can be written 2. Cheap and expensive tastes In this section we formalize the idea of a true self, and introduce the concept of endogenous preference adjustments. This simplifies the discussion of cheap and expensive tastes. Sen, discusses how the battered slave, the tamed housewife, the bro- ken unemployed and the hopeless destitute have cheap tastes, in the sense that they form plans of life that are too modest. The implication is that it is not advisable to judge the well-being of these people only by their self-perceived well-being u. According to Roemer, the concern about cheap tastes was introduced by Elster However, variations on the problem are likely to be found much earlier. The problem of expensive tastes PET is formulated by Arrow The core of both cheap and expensive tastes is the notion that people adjust their expectations to their actual situation. Such endogenous preference adjustments can 1 Choice-aversion may arise if the individual finds more alternatives bewildering, making the choice less easy. The opposite occurs if the individual preferes choosing from a larger set even if the chosen element would be the same. For the same reason, the revealed preference approach does not always reveal the correct preferences. Through the process of cognitive dissonance, people learn to like what they are accustomed to or what is available to them. In the extreme case, this would mean that given the options available, all individual achieves the same level of utility when choosing the most preferred of available alternatives. Clearly, this would not be true if the only alternative for some individuals would be starvation, but for other situations it seems plausible. This is to some extent paternalistic: For pure utilitarians, endogenous preference adjustments are simply not problems. The concept Page 2

3 of true selves and endogenous preference adjustments can be formalized as follows: Sup- pose, initially, that we know the well-being of the true self, call it v A. Some would probably argue that the ideal type of paternalism is to recover the preferences of the true self and thus the well-being imposed on this individual would ideally be equal to the well-being of the true self: However, through endogenous preference adjustments it may be that she would be equally happy in both cases. The big problem for policy making founded in the idea of ideal paternalism is that g is unknown and likely to vary on an individual basis. Note that this observation does not necessarily supports the view that the idea of a true self is dangerous, as argued by among others Isaiah Berlin 5. Melberg argues convincingly that the possibility of politicians and others misusing the concept of true selves is not an argument against the existence of the concept. Now suppose that the following is true the reader is encouraged to think of the alternatives a, b and c as life as a tamed housewife, living an ordinary life, living a life in luxury respectively: This condition says that the individual will achieve the same level of utility no matter how big or restricted is her choice set. This is a very strong assumption. Perhaps, it is more likely that preferences adjusts only partially: Definition 4 Partial adjustment of preferences, PAP. Let ua represent self per- ceived well-being when the individual is faced with choice set A. The problems of cheap and expensive tastes illustrated from a in the expanded choice set compared to the utility from a in the original choice set. When the problem of cheap tastes occurs we want to impose a well-being that is lower than the individual perceives it to be: Thus, if all choice sets are arranged in order of increasing well-being, it is possible to illustrate the situation graphically: It is important to note that the relations between w, u and A need not be linear, nor must w always be increasing in u. As illustrated in figure 1, both problems can occur even without CAP. Are cheap and expensive tastes two sides of the same coin? Roemer argues that expensive tastes are voluntarily cultivated whilst the tamed housewife merely reacts to a situation she has not chosen. Under the assumption that it is possible to control ones preferences, this means that the two problems are not two sides of the same coin. If utility is ordinal, interpersonal comparisons of well-being must be made using preferences over choice sets. The basis for these preferences is strictly normative. A well-being function that values freedom intrinsically is an example of such preferences. Sometimes the problems with unknown or non-comparable utility functions are dealt with by assuming that all individuals receive indirect utility from money, and that this utility function is interpersonally comparable. If the choice set that an individual gets from a certain amount of money is a subset of the choice set available to the same individual with a larger amount of money, then the indirect utility for money the represents the kind of preferences over choice sets described by ITPF This approach also means that preference adjustments like the THP and the PET are ignored when judging well-being. Does the concave utility function measure well-being? So far very little has been said about the definition of the elements in A. How- ever, this is a crucial point indeed when it comes to preferences over choice sets. Robbins argued that interpersonal comparisons are strictly normative, and the general response among economists was reverting to ordinal utility functions, trying to avoid making in- terpersonal comparisons. However, the normative assumptions that would justify using a concave utility function of money as an interpersonally comparable measure of well-being may not be unreasonably strong. Identifying these assumptions re- quires a moments reflection on among other things the definition of the elements in the choice set. Is well-being increasing in money? The first task is to motivate the sign of the first derivative, that is: Other things being equal, the well-being is increasing in individual income. Let Ayi be the choice set available to individual i with income y. Note that A0i is non-empty if there are alternatives with no monetary cost such as nude sunbathing. The following condition states that everything available to i with a given income is still available should her income increase: Definition 5 Monetary monotonicity of choice sets, MMC. Then w is interpersonally comparable. Assuming monetary monotonicity of choice sets may be more restrictive than initially seems, due to a number of well-known paradoxes in rational choice theory that depend intimately on how the elements are described. A well-known example is when the following choices are observed: Attempting to rescue rational choice theory from this and related paradoxes, Dowding, Revealed preference and external reference, forthcoming in Rationality and Society argues that the paradox is a result of improper descriptions of the elements in the choice set: To my understanding, the solution proposed by Dowding renders rational choice unfalsifiable. If elements Page 3

4 may be defined by among other things the size of the set that contains them, sets are no longer contractible: This is avoided if the elements of A can be defined independently of the choice set that contains them. In this case, the definition of every element in A changes when income changes: Definition 6 Income-dependent definition of elements, IDE. In this case it is slightly harder to motivate why well-being should be increasing in income. These all imply the stability axiom. See Suzumura for futher details. Definition 7 Monetary monotonicity of well-being, MMW. Another solution is to define the elements in the choice set in a way that is not vulnerable to the problem caused by IDE. Page 4

5 3: Social choice theory - Wikipedia -Utility is cardinal, that is, scale-measurable by observation or judgment.-preferences are exogenously given and stable.-diminishing Marginal utility.-all individuals have Interpersonally Comparable Utility functions With these assumptions, it is possible to construct a social welfare function simply by summing all the individual utility functions. Social welfare function Save In welfare economics, a social welfare function is a function that ranks social states alternative complete descriptions of the society as less desirable, more desirable, or indifferent for every possible pair of social states. Inputs of the function include any variables considered to affect the economic welfare of a society. One use of a social welfare function is to represent prospective patterns of collective choice as to alternative social states. The social welfare function provides the government with a simple guideline for achieving the optimal distribution of income. One point of a social welfare function is to determine how close the analogy is to an ordinal utility function for an individual with at least minimal restrictions suggested by welfare economics, including constraints on the number of factors of production. There are two major distinct but related types of social welfare functions: A Bergsonâ Samuelson social welfare function considers welfare for a given set of individual preferences or welfare rankings. An Arrow social welfare function considers welfare across different possible sets of individual preferences or welfare rankings and seemingly reasonable axioms that constrain the function. The object was "to state in precise form the value judgments required for the derivation of the conditions of maximum economic welfare" set out by earlier writers, including Marshall and Pigou, Pareto and Barone, and Lerner. The function was real-valued and differentiable. It was specified to describe the society as a whole. Arguments of the function included the quantities of different commodities produced and consumed and of resources used in producing different commodities, including labor. Necessary general conditions are that at the maximum value of the function: Bergson showed how welfare economics could describe a standard of economic efficiency despite dispensing with interpersonally-comparable cardinal utility, the hypothesizaton of which may merely conceal value judgments, and purely subjective ones at that. Earlier neoclassical welfare theory, heir to the classical utilitarianism of Bentham, had not infrequently treated the Law of Diminishing Marginal Utility as implying interpersonally comparable utility, a necessary condition to achieve the goal of maximizing total utility of the society. Irrespective of such comparability, income or wealth is measurable, and it was commonly inferred that redistributing income from a rich person to a poor person tends to increase total utility however measured in the society. VI argued that how or how much utilities, as mental events, would have changed relative to each other is not measurable by any empirical test. Nor are they inferable from the shapes of standard indifference curves. Hence, the advantage of being able to dispense with interpersonal comparability of utility without abstaining from welfare theory. A practical qualification to this was any reduction in output from the transfer. Auxiliary specifications enable comparison of different social states by each member of society in preference satisfaction. These help define Pareto efficiency, which holds if all alternatives have been exhausted to put at least one person in a more preferred position with no one put in a less preferred position. Bergson described an "economic welfare increase" later called a Pareto improvement as at least one individual moving to a more preferred position with everyone else indifferent. The social welfare function could then be specified in a substantively individualistic sense to derive Pareto efficiency optimality. Paul Samuelson, p. As Bergson noted, a welfare improvement from the social welfare function could come from the "position of some individuals" improving at the expense of others. That social welfare function could then be described as characterizing an equity dimension. Samuelson, p. He also presented a lucid verbal and mathematical exposition of the social welfare function, pp. As Samuelson, p. Samuelson further sharpened that distinction by specifying the Welfare function and the Possibility function, pp. Each has as arguments the set of utility functions for everyone in the society. Each can and commonly does incorporate Pareto efficiency. The Possibility function also depends on technology and resource restraints. It is written in implicit form, reflecting the feasible locus of utility combinations imposed by the restraints and allowed by Pareto efficiency. The Welfare function ranks different hypothetical sets of utility for everyone in the society from ethically Page 5

6 lowest on up with ties permitted, that is, it makes interpersonal comparisons of utility. Welfare maximization then consists of maximizing the Welfare function subject to the Possibility function as a constraint. For a two-person society, there is a graphical depiction of such welfare maximization at the first figure of Bergsonâ Samuelson social welfare functions. Relative to consumer theory for an individual as to two commodities consumed, there are the following parallels: The respective hypothetical utilities of the two persons in two-dimensional utility space is analogous to respective quantities of commodities for the two-dimensional commodity space of the indifference-curve surface The Welfare function is analogous to the indifference-curve map The Possibility function is analogous to the budget constraint Two-person welfare maximization at the tangency of the highest Welfare function curve on the Possibility function is analogous to tangency of the highest indifference curve on the budget constraint. Arrow social welfare function constitution Kenneth Arrow generalizes the analysis. Arrow finds that nothing of behavioral significance is lost by dropping the requirement of social orderings that are real-valued and thus cardinal in favor of orderings, which are merely complete and transitive, such as a standard indifference curve map. The earlier analysis mapped any set of individual orderings to one social ordering, whatever it was. This social ordering selected the top-ranked feasible alternative from the economic environment as to resource constraints. Arrow proposed to examine mapping different sets of individual orderings to possibly different social orderings. Here the social ordering would depend on the set of individual orderings, rather than being imposed invariant to them. Stunningly relative to a course of theory from Adam Smith and Jeremy Bentham on, Arrow proved the general impossibility theorem which says that it is impossible to have a social welfare function that satisfies a certain set of "apparently reasonable" conditions. Cardinal social welfare functions A cardinal social welfare function is a function that takes as input numeric representations of individual utilities also known as cardinal utility, and returns as output a numeric representation of the collective welfare. The underlying assumption is that individuals utilities can be put on a common scale and compared. Examples of such measures can be: For the purposes of this section, income is adopted as the measurement of utility. The form of the social welfare function is intended to express a statement of objectives of a society. The utilitarian or Benthamite social welfare function measures social welfare as the total or sum of individual incomes: Page 6

7 4: Social welfare function - Wikipedia freedom, choice and well-being: on the ethical interpretation of interpersonally comparable utility functions. andreas bergh work in progress. abstract. The object was "to state in precise form the value judgments required for the derivation of the conditions of maximum economic welfare" set out by earlier writers, including Marshall and Pigou, Pareto and Barone, and Lerner. The function was real-valued and differentiable. It was specified to describe the society as a whole. Arguments of the function included the quantities of different commodities produced and consumed and of resources used in producing different commodities, including labor. Necessary general conditions are that at the maximum value of the function: Bergson showed how welfare economics could describe a standard of economic efficiency despite dispensing with interpersonally-comparable cardinal utility, the hypothesizaton of which may merely conceal value judgments, and purely subjective ones at that. Earlier neoclassical welfare theory, heir to the classical utilitarianism of Bentham, had not infrequently treated the Law of Diminishing Marginal Utility as implying interpersonally comparable utility, a necessary condition to achieve the goal of maximizing total utility of the society. Irrespective of such comparability, income or wealth is measurable, and it was commonly inferred that redistributing income from a rich person to a poor person tends to increase total utility however measured in the society. VI argued that how or how much utilities, as mental events, would have changed relative to each other is not measurable by any empirical test. Nor are they inferable from the shapes of standard indifference curves. Hence, the advantage of being able to dispense with interpersonal comparability of utility without abstaining from welfare theory. A practical qualification to this was any reduction in output from the transfer. Auxiliary specifications enable comparison of different social states by each member of society in preference satisfaction. These help define Pareto efficiency, which holds if all alternatives have been exhausted to put at least one person in a more preferred position with no one put in a less preferred position. Bergson described an "economic welfare increase" later called a Pareto improvement as at least one individual moving to a more preferred position with everyone else indifferent. The social welfare function could then be specified in a substantively individualistic sense to derive Pareto efficiency optimality. Paul Samuelson, p. As Bergson noted, a welfare improvement from the social welfare function could come from the "position of some individuals" improving at the expense of others. That social welfare function could then be described as characterizing an equity dimension. Samuelson, p. He also presented a lucid verbal and mathematical exposition of the social welfare function, pp. As Samuelson, p. Samuelson further sharpened that distinction by specifying the Welfare function and the Possibility function, pp. Each has as arguments the set of utility functions for everyone in the society. Each can and commonly does incorporate Pareto efficiency. The Possibility function also depends on technology and resource restraints. It is written in implicit form, reflecting the feasible locus of utility combinations imposed by the restraints and allowed by Pareto efficiency. The Welfare function ranks different hypothetical sets of utility for everyone in the society from ethically lowest on up with ties permitted, that is, it makes interpersonal comparisons of utility. Welfare maximization then consists of maximizing the Welfare function subject to the Possibility function as a constraint. For a two-person society, there is a graphical depiction of such welfare maximization at the first figure of Bergsonâ Samuelson social welfare functions. Relative to consumer theory for an individual as to two commodities consumed, there are the following parallels: The respective hypothetical utilities of the two persons in two-dimensional utility space is analogous to respective quantities of commodities for the two-dimensional commodity space of the indifference-curve surface The Welfare function is analogous to the indifference-curve map The Possibility function is analogous to the budget constraint Two-person welfare maximization at the tangency of the highest Welfare function curve on the Possibility function is analogous to tangency of the highest indifference curve on the budget constraint. Arrow social welfare function constitution [ edit ] Kenneth Arrow generalizes the analysis. Arrow finds that nothing of behavioral significance is lost by dropping the requirement of social orderings that are real-valued and thus cardinal in favor of orderings, which are merely complete and transitive, such as a standard Page 7

8 indifference curve map. The earlier analysis mapped any set of individual orderings to one social ordering, whatever it was. This social ordering selected the top-ranked feasible alternative from the economic environment as to resource constraints. Arrow proposed to examine mapping different sets of individual orderings to possibly different social orderings. Here the social ordering would depend on the set of individual orderings, rather than being imposed invariant to them. Stunningly relative to a course of theory from Adam Smith and Jeremy Bentham on, Arrow proved the general impossibility theorem which says that it is impossible to have a social welfare function that satisfies a certain set of "apparently reasonable" conditions. Cardinal social welfare functions[ edit ] A cardinal social welfare function is a function that takes as input numeric representations of individual utilities also known as cardinal utility, and returns as output a numeric representation of the collective welfare. The underlying assumption is that individuals utilities can be put on a common scale and compared. Examples of such measures can be: For the purposes of this section, income is adopted as the measurement of utility. The form of the social welfare function is intended to express a statement of objectives of a society. The utilitarian or Benthamite social welfare function measures social welfare as the total or sum of individual incomes: Page 8

9 5: Social welfare function Revolvy The social welfare function methodology is, in principle, compatible with any theory of well-being. By this I mean that if an interpersonally comparable utility measure corresponding to the theory can be constructed, outcomes can be conceptualized as vectors of utilities of that sort. For example, the social welfare function approach can be coupled with a happiness view of well-being if we have available utility numbers that reflect levels of happiness. Individual preference can be modeled in terms of an economic utility function. Then the ability to create a social welfare function depends crucially on the ability to compare utility functions. This is called interpersonal utility comparison. Following Jeremy Bentham, utilitarians have argued that preferences and utility functions of individuals are interpersonally comparable and may therefore be added together to arrive at a measure of aggregate utility. Utilitarian ethics call for maximizing this aggregate. Lionel Robbins questioned whether mental states, and the utilities they reflect, can be measured and, a fortiori, interpersonal comparisons of utility as well as the social choice theory on which it is based. Consider for instance the law of diminishing marginal utility, according to which utility of an added quantity of a good decreases with the amount of the good that is already in possession of the individual. It has been used to defend transfers of wealth from the "rich" to the "poor" on the premise that the former do not derive as much utility as the latter from an extra unit of income. Robbins, pp. Apologists of the interpersonal comparison of utility have argued that Robbins claimed too much. John Harsanyi agrees that full comparability of mental states such as utility is never possible but believes, however, that human beings are able to make some interpersonal comparisons of utility because they share some common backgrounds, cultural experiences, etc. In the example from Amartya Sen, p. Harsanyi and Sen thus argue that at least partial comparability of utility is possible, and social choice theory proceeds under that assumption. Sen proposes, however, that comparability of interpersonal utility need not be partial. A starving peasant may have a particularly sunny disposition and thereby derive high utility from a small income. This fact should not nullify, however, his claim to compensation or equality in the realm of social choice. Social decisions should accordingly be based on immalleable factors. Sen proposes interpersonal utility comparisons based on a wide range of data. We can proceed to make social choices based on real variables, and thereby address actual position, and access to advantage. The initial results emphasized the impossibility of satisfactorily providing a social choice function free of dictatorship and inefficiency in the most general settings. Later results have found natural restrictions that can accommodate many desirable properties. 6: A Remark on Admissible Transformations for Interpersonally Comparable Utilities. - CORE However, the normative assumptions that would justify using a concave utility function of money as an interpersonally comparable measure of well-being may not be unreasonably strong. Identifying these assumptions re- quires a moments reflection on (among other things) the definition of the elements in the choice set. Page 9

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