Utility Function
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Utility Function

Within economics, the concept of utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or satisfaction within the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. The term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a utility function that represents a consumer's preference ordering over a choice set. Utility has thus become a more abstract concept that is not necessarily solely based on the satisfaction or pleasure received.

## Utility function

Consider a set of alternatives facing an individual, and over which the individual has a preference ordering. A utility function is able to represent those preferences if it is possible to assign a real number to each alternative, in such a way that alternative a is assigned a number greater than alternative b if, and only if, the individual prefers alternative a to alternative b. In this situation an individual that selects the most preferred alternative available is necessarily also selecting the alternative that maximizes the associated utility function. In general economic terms, a utility function measures preferences concerning a set of goods and services. Often, utility is correlated with words such as happiness, satisfaction, and welfare, and these are hard to measure mathematically. Thus, economists utilize consumption baskets of preferences in order to measure these abstract, non quantifiable ideas.

Gérard Debreu precisely defined the conditions required for a preference ordering to be representable by a utility function.[1] For a finite set of alternatives these require only that the preference ordering is complete (so the individual is able to determine which of any two alternatives is preferred, or that they are equally preferred), and that the preference order is transitive.

## Applications

Utility is usually applied by economists in such constructs as the indifference curve, which plot the combination of commodities that an individual or a society would accept to maintain a given level of satisfaction. Utility and indifference curves are used by economists to understand the underpinnings of demand curves, which are half of the supply and demand analysis that is used to analyze the workings of goods markets.

Individual utility and social utility can be construed as the value of a utility function and a social welfare function respectively. When coupled with production or commodity constraints, under some assumptions these functions can be used to analyze Pareto efficiency, such as illustrated by Edgeworth boxes in contract curves. Such efficiency is a central concept in welfare economics.

In finance, utility is applied to generate an individual's price for an asset called the indifference price. Utility functions are also related to risk measures, with the most common example being the entropic risk measure.

In the field of artificial intelligence, utility functions are used to convey the value of various outcomes to intelligent agents. This allows the agents to plan actions with the goal of maximizing the utility (or "value") of available choices.

## Revealed preference

It was recognized that utility could not be measured or observed directly, so instead economists devised a way to infer underlying relative utilities from observed choice. These 'revealed preferences', as termed by Paul Samuelson, were revealed e.g. in people's willingness to pay:

Utility is taken to be correlative to Desire or Want. It has been already argued that desires cannot be measured directly, but only indirectly, by the outward phenomena to which they give rise: and that in those cases with which economics is chiefly concerned the measure is found in the price which a person is willing to pay for the fulfillment or satisfaction of his desire.[2]:78

## Functions

There has been some controversy over the question whether the utility of a commodity can be measured or not. At one time, it was assumed that the consumer was able to say exactly how much utility he got from the commodity. The economists who made this assumption belonged to the 'cardinalist school' of economics. Today utility functions, expressing utility as a function of the amounts of the various goods consumed, are treated as either cardinal or ordinal, depending on whether they are or are not interpreted as providing more information than simply the rank ordering of preferences over bundles of goods, such as information on the strength of preferences.

### Cardinal

When cardinal utility is used, the magnitude of utility differences is treated as an ethically or behaviorally significant quantity. For example, suppose a cup of orange juice has utility of 120 utils, a cup of tea has a utility of 80 utils, and a cup of water has a utility of 40 utils. With cardinal utility, it can be concluded that the cup of orange juice is better than the cup of tea by exactly the same amount by which the cup of tea is better than the cup of water. Formally speaking, this means that if one has a cup of tea, she would be willing to take any bet with a probability, p, greater than .5 of getting a cup of juice, with a risk of getting a cup of water equal to 1-p. One cannot conclude, however, that the cup of tea is two thirds of the goodness of the cup of juice, because this conclusion would depend not only on magnitudes of utility differences, but also on the "zero" of utility. For example, if the "zero" of utility was located at -40, then a cup of orange juice would be 160 utils more than zero, a cup of tea 120 utils more than zero. Cardinal utility, to economics, can be seen as the assumption that utility can be measured through quantifiable characteristics, such as height, weight, temperature, etc.

Neoclassical economics has largely retreated from using cardinal utility functions as the basis of economic behavior. A notable exception is in the context of analyzing choice under conditions of risk (see below).

Sometimes cardinal utility is used to aggregate utilities across persons, to create a social welfare function.

### Ordinal

When ordinal utilities are used, differences in utils (values taken on by the utility function) are treated as ethically or behaviorally meaningless: the utility index encodes a full behavioral ordering between members of a choice set, but tells nothing about the related strength of preferences. In the above example, it would only be possible to say that juice is preferred to tea to water, but no more. Thus, ordinal utility utilizes comparisons, such as "preferred to", "no more", "less than", etc.

Ordinal utility functions are unique up to increasing monotone (or monotonic) transformations. For example, if a function ${\displaystyle u(x)}$ is taken as ordinal, it is equivalent to the function ${\displaystyle u(x)^{3}}$, because taking the 3rd power is an increasing monotone transformation (or monotonic transformation). This means that the ordinal preference induced by these functions is the same (although they are two different functions). In contrast, cardinal utilities are unique only up to increasing linear transformations, so if ${\displaystyle u(x)}$ is taken as cardinal, it is not equivalent to ${\displaystyle u(x)^{3}}$.

### Preferences

Although preferences are the conventional foundation of microeconomics, it is often convenient to represent preferences with a utility function and analyze human behavior indirectly with utility functions. Let X be the consumption set, the set of all mutually-exclusive baskets the consumer could conceivably consume. The consumer's utility function ${\displaystyle u\colon X\to \mathbb {R} }$ ranks each package in the consumption set. If the consumer strictly prefers x to y or is indifferent between them, then ${\displaystyle u(x)\geq u(y)}$.

For example, suppose a consumer's consumption set is X = {nothing, 1 apple,1 orange, 1 apple and 1 orange, 2 apples, 2 oranges}, and its utility function is u(nothing) = 0, u(1 apple) = 1, u(1 orange) = 2, u(1 apple and 1 orange) = 4, u(2 apples) = 2 and u(2 oranges) = 3. Then this consumer prefers 1 orange to 1 apple, but prefers one of each to 2 oranges.

In micro-economic models, there are usually a finite set of L commodities, and a consumer may consume an arbitrary amount of each commodity. This gives a consumption set of ${\displaystyle \mathbb {R} _{+}^{L}}$, and each package ${\displaystyle x\in \mathbb {R} _{+}^{L}}$ is a vector containing the amounts of each commodity. In the previous example, we might say there are two commodities: apples and oranges. If we say apples is the first commodity, and oranges the second, then the consumption set ${\displaystyle X=\mathbb {R} _{+}^{2}}$ and u(0, 0) = 0, u(1, 0) = 1, u(0, 1) = 2, u(1, 1) = 4, u(2, 0) = 2, u(0, 2) = 3 as before. Note that for u to be a utility function on X, it must be defined for every package in X.

A utility function ${\displaystyle u\colon X\to \mathbb {R} }$ represents a preference relation ${\displaystyle \preceq }$ on X iff for every ${\displaystyle x,y\in X}$, ${\displaystyle u(x)\leq u(y)}$ implies ${\displaystyle x\preceq y}$. If u represents ${\displaystyle \preceq }$, then this implies ${\displaystyle \preceq }$ is complete and transitive, and hence rational.

### Revealed preferences in finance

In financial applications, e.g. portfolio optimization, an investor chooses financial portfolio which maximizes his/her own utility function, or, equivalently, minimizes his/her risk measure. For example, modern portfolio theory selects variance as a measure of risk; other popular theories are expected utility theory,[3] and prospect theory.[4] To determine specific utility function for any given investor, one could design a questionnaire procedure with questions in the form: How much would you pay for x% chance of getting y? Revealed preference theory suggests a more direct approach: observe a portfolio X* which an investor currently holds, and then find a utility function/risk measure such that X* becomes an optimal portfolio.[5]

### Examples

In order to simplify calculations, various alternative assumptions have been made concerning details of human preferences, and these imply various alternative utility functions such as:

Most utility functions used in modeling or theory are well-behaved. They are usually monotonic and quasi-concave. However, it is possible for preferences not to be representable by a utility function. An example is lexicographic preferences which are not continuous and cannot be represented by a continuous utility function.[6]

## Expected utility

The expected utility theory deals with the analysis of choices among risky projects with multiple (possibly multidimensional) outcomes.

The St. Petersburg paradox was first proposed by Nicholas Bernoulli in 1713 and solved by Daniel Bernoulli in 1738. D. Bernoulli argued that the paradox could be resolved if decision-makers displayed risk aversion and argued for a logarithmic cardinal utility function. (Analysis of international survey data in the 21st century have shown that insofar as utility represents happiness, as in utilitarianism, it is indeed proportional to log income.)

The first important use of the expected utility theory was that of John von Neumann and Oskar Morgenstern, who used the assumption of expected utility maximization in their formulation of game theory.

### von Neumann-Morgenstern

Von Neumann and Morgenstern addressed situations in which the outcomes of choices are not known with certainty, but have probabilities attached to them.

A notation for a lottery is as follows: if options A and B have probability p and 1 - p in the lottery, we write it as a linear combination:

${\displaystyle L=pA+(1-p)B}$

More generally, for a lottery with many possible options:

${\displaystyle L=\sum _{i}p_{i}A_{i},}$

where ${\displaystyle \sum _{i}p_{i}=1}$.

By making some reasonable assumptions about the way choices behave, von Neumann and Morgenstern showed that if an agent can choose between the lotteries, then this agent has a utility function such that the desirability of an arbitrary lottery can be calculated as a linear combination of the utilities of its parts, with the weights being their probabilities of occurring.

This is called the expected utility theorem. The required assumptions are four axioms about the properties of the agent's preference relation over 'simple lotteries', which are lotteries with just two options. Writing ${\displaystyle B\preceq A}$ to mean 'A is weakly preferred to B' ('A is preferred at least as much as B'), the axioms are:

1. completeness: For any two simple lotteries ${\displaystyle L}$ and ${\displaystyle M}$, either ${\displaystyle L\preceq M}$ or ${\displaystyle M\preceq L}$ (or both, in which case they are viewed as equally desirable).
2. transitivity: for any three lotteries ${\displaystyle L,M,N}$, if ${\displaystyle L\preceq M}$ and ${\displaystyle M\preceq N}$, then ${\displaystyle L\preceq N}$.
3. convexity/continuity (Archimedean property): If ${\displaystyle L\preceq M\preceq N}$, then there is a ${\displaystyle p}$ between 0 and 1 such that the lottery ${\displaystyle pL+(1-p)N}$ is equally desirable as ${\displaystyle M}$.
4. independence: for any three lotteries ${\displaystyle L,M,N}$ and any probability p, ${\displaystyle L\preceq M}$ if and only if ${\displaystyle pL+(1-p)N\preceq pM+(1-p)N}$. Intuitively, if the lottery formed by the probabilistic combination of ${\displaystyle L}$ and ${\displaystyle N}$ is no more preferable than the lottery formed by the same probabilistic combination of ${\displaystyle M}$ and ${\displaystyle N,}$ then and only then ${\displaystyle L\preceq M}$.

Axioms 3 and 4 enable us to decide about the relative utilities of two assets or lotteries.

In more formal language: A von Neumann-Morgenstern utility function is a function from choices to the real numbers:

${\displaystyle u\colon X\to \mathbb {R} }$

which assigns a real number to every outcome in a way that captures the agent's preferences over simple lotteries. Under the four assumptions mentioned above, the agent will prefer a lottery ${\displaystyle L_{2}}$ to a lottery ${\displaystyle L_{1}}$ if and only if, for the utility function characterizing that agent, the expected utility of ${\displaystyle L_{2}}$ is greater than the expected utility of ${\displaystyle L_{1}}$:

${\displaystyle L_{1}\preceq L_{2}{\text{ iff }}u(L_{1})\leq u(L_{2})}$.

Of all the axioms, independence is the most often discarded. A variety of generalized expected utility theories have arisen, most of which drop or relax the independence axiom.

### As probability of success

Castagnoli and LiCalzi (1996) and Bordley and LiCalzi (2000) provided another interpretation for Von Neumann and Morgenstern's theory. Specifically for any utility function, there exists a hypothetical reference lottery with the expected utility of an arbitrary lottery being its probability of performing no worse than the reference lottery. Suppose success is defined as getting an outcome no worse than the outcome of the reference lottery. Then this mathematical equivalence means that maximizing expected utility is equivalent to maximizing the probability of success. In many contexts, this makes the concept of utility easier to justify and to apply. For example, a firm's utility might be the probability of meeting uncertain future customer expectations.[7][8][9][10]

## Indirect utility

An indirect utility function gives the optimal attainable value of a given utility function, which depends on the prices of the goods and the income or wealth level that the individual possesses.

### Money

One use of the indirect utility concept is the notion of the utility of money. The (indirect) utility function for money is a nonlinear function that is bounded and asymmetric about the origin. The utility function is concave in the positive region, reflecting the phenomenon of diminishing marginal utility. The boundedness reflects the fact that beyond a certain point money ceases being useful at all, as the size of any economy at any point in time is itself bounded. The asymmetry about the origin reflects the fact that gaining and losing money can have radically different implications both for individuals and businesses. The non-linearity of the utility function for money has profound implications in decision making processes: in situations where outcomes of choices influence utility through gains or losses of money, which are the norm in most business settings, the optimal choice for a given decision depends on the possible outcomes of all other decisions in the same time-period.[11]

## Discussion and criticism

Cambridge economist Joan Robinson famously criticized utility for being a circular concept: "Utility is the quality in commodities that makes individuals want to buy them, and the fact that individuals want to buy commodities shows that they have utility."[12]:48 Robinson also pointed out that because the theory assumes that preferences are fixed this means that utility is not a testable assumption. This is so because if we take changes in peoples' behavior in relation to a change in prices or a change in the underlying budget constraint we can never be sure to what extent the change in behavior was due to the change in price or budget constraint and how much was due to a change in preferences.[13] This criticism is similar to that of the philosopher Hans Albert who argued that the ceteris paribus conditions on which the marginalist theory of demand rested rendered the theory itself an empty tautology and completely closed to experimental testing.[14] In essence, demand and supply curve (theoretical line of quantity of a product which would have been offered or requested for given price) is purely ontological and could never have been demonstrated empirically.

Another criticism comes from the assertion that neither cardinal nor ordinal utility are empirically observable in the real world. In the case of cardinal utility it is impossible to measure the level of satisfaction "quantitatively" when someone consumes or purchases an apple. In case of ordinal utility, it is impossible to determine what choices were made when someone purchases, for example, an orange. Any act would involve preference over a vast set of choices (such as apple, orange juice, other vegetable, vitamin C tablets, exercise, not purchasing, etc.).[15][16]

Other questions of what arguments ought to enter into a utility function are difficult to answer, yet seem necessary to understanding utility. Whether people gain utility from coherence of wants, beliefs or a sense of duty is key to understanding their behavior in the utility organon.[17] Likewise, choosing between alternatives is itself a process of determining what to consider as alternatives, a question of choice within uncertainty.[18]

An evolutionary psychology perspective is that utility may be better viewed as due to preferences that maximized evolutionary fitness in the ancestral environment but not necessarily in the current one.[19]

## References

1. ^ Debreu, Gérard (1954), "Representation of a preference ordering by a numerical function", in Thrall, Robert M.; Coombs, Clyde H.; Raiffa, Howard (eds.), Decision processes, New York: Wiley, pp. 159-167, OCLC 639321.CS1 maint: ref=harv (link)
2. ^ Marshall, Alfred (1920). Principles of Economics. An introductory volume (8th ed.). London: Macmillan.
3. ^ Von Neumann, J.; Morgenstern, O. (1953). Theory of Games and Economic Behavior (3rd ed.). Princeton University Press.
4. ^ Kahneman, D.; Tversky, A. (1979). "Prospect Theory: An Analysis of Decision Under Risk" (PDF). Econometrica. 47 (2): 263-292. doi:10.2307/1914185. JSTOR 1914185.
5. ^ Grechuk, B.; Zabarankin, M. (2016). "Inverse Portfolio Problem with Coherent Risk Measures". European Journal of Operational Research. 249 (2): 740-750. doi:10.1016/j.ejor.2015.09.050. hdl:2381/36136.
6. ^ Ingersoll, Jonathan E., Jr. (1987). Theory of Financial Decision Making. Totowa: Rowman and Littlefield. p. 21. ISBN 0-8476-7359-6.
7. ^ Castagnoli, E.; LiCalzi, M. (1996). "Expected Utility Without Utility" (PDF). Theory and Decision. 41 (3): 281-301. doi:10.1007/BF00136129. hdl:10278/4143.
8. ^ Bordley, R.; LiCalzi, M. (2000). "Decision Analysis Using Targets Instead of Utility Functions". Decisions in Economics and Finance. 23 (1): 53-74. doi:10.1007/s102030050005. hdl:10278/3610.
9. ^ Bordley, R.; Kirkwood, C. (2004). "Multiattribute preference analysis with Performance Targets". Operations Research. 52 (6): 823-835. doi:10.1287/opre.1030.0093.
10. ^ Bordley, R.; Pollock, S. (2009). "A Decision-Analytic Approach to Reliability-Based Design Optimization". Operations Research. 57 (5): 1262-1270. doi:10.1287/opre.1080.0661.
11. ^ Berger, J. O. (1985). "Utility and Loss". Statistical Decision Theory and Bayesian Analysis (2nd ed.). Berlin: Springer-Verlag. ISBN 3-540-96098-8.
12. ^ Robinson, Joan (1962). Economic Philosophy. Harmondsworth, Middle-sex, UK: Penguin Books.
13. ^ Pilkington, Philip (17 February 2014). "Joan Robinson's Critique of Marginal Utility Theory". Fixing the Economists. Archived from the original on 13 July 2015.
14. ^ Pilkington, Philip (27 February 2014). "utility Hans Albert Expands Robinson's Critique of Marginal Utility Theory to the Law of Demand". Fixing the Economists. Archived from the original on 19 July 2015.
15. ^ "Revealed Preference Theory". Archived from the original on 16 July 2011. Retrieved 2009.
16. ^ "Archived copy" (PDF). Archived from the original (PDF) on 15 October 2008. Retrieved 2008.CS1 maint: archived copy as title (link)
17. ^ Klein, Daniel (May 2014). "Professor" (PDF). Econ Journal Watch. 11 (2): 97-105. Archived (PDF) from the original on 5 October 2014. Retrieved 2014.
18. ^ Burke, Kenneth (1932). Towards a Better Life. Berkeley, Calif: University of California Press.
19. ^ Capra, C. Monica; Rubin, Paul H. (2011). "The Evolutionary Psychology of Economics". Applied Evolutionary Psychology. Oxford University Press. doi:10.1093/acprof:oso/9780199586073.003.0002. ISBN 9780191731358.