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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. It is devoid of its original interpretation as a measurement of the pleasure or satisfaction obtained by the consumer from that choice.

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 maximises 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.

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.

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 they were named 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}

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.

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.

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 is taken as ordinal, it is equivalent to the function , 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 is taken as cardinal, it is not equivalent to .

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** ranks each package in the consumption set. If the consumer strictly prefers *x* to *y* or is indifferent between them, then .

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 , and each package 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 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 **represents** a preference relation on X iff for every , implies . If u represents , then this implies is complete and transitive, and hence rational.

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]}

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:

- CES (
*constant elasticity of substitution*, or*isoelastic*) utility - Isoelastic utility
- Exponential utility
- Quasilinear utility
- Homothetic preferences
- Stone–Geary utility function
- Gorman polar form
- Hyperbolic absolute risk aversion

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]}

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. (Analyses 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 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:

More generally, for a lottery with many possible options:

where .

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 to mean 'A is weakly preferred to B' ('A is preferred at least as much as B'), the axioms are:

- completeness: For any two simple lotteries and , either or (or both, in which case they are viewed as equally desirable).
- transitivity: for any three lotteries , if and , then .
- convexity/continuity (Archimedean property): If , then there is a between 0 and 1 such that the lottery is equally desirable as .
- independence: for any three lotteries and any probability
*p*, if and only if . Intuitively, if the lottery formed by the probabilistic combination of and is no more preferable than the lottery formed by the same probabilistic combination of and then and only then .

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:

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 to a lottery if and only if, for the utility function characterizing that agent, the expected utility of is greater than the expected utility of :

- .

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.

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]}

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.

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]}

Let us assume that *x* is the amount of one's assets (for example, one's money minus one's debts). We shall use a common but a strong assumption

Let us assume that in a given decision problem, the possible outcomes are finite and known.

Linear transformations *x → ax + b* do not change the preference ordering of monetary expectations. Changing the money scale or changing the origin of the money axis neither changes that preference ordering. It is comfortable to make such a transformation that *x = 0* corresponds to the minimal possible outcome. Then, one can choose such an utility function

that

, and at the maximal value of *x*.

If the maximal possible value is *20*, then the corresponding linear utility function is shown on the Figure below.

All preferences of monetary expectations remain the same, if we use linear utility graphs like the one shown on the Figure.

That utility can be interpreted as a probability. The utility *u(x)* is the win probability in an utility-fair gamble, in which one can loose all one's money *x* and can achieve, as a result of win, the final assets equal to *20*.

For example, *u(12) = 0.6* . Thus, if one's initial assets are *x = 12* , then the following gamble would be monetary fair: with probability *p = 0.6* one wins *20 ‒ 12 = 8* units and with probability *1 ‒ p = 0.4* one loses *12* units of money.

However, the linear utility has some shortages.

First, if there is no upper bound of the possible wealth, then the linear utility meets with obstacles. If the point *A* on the Figure moves to the right, then the angle *α* decreases. In the limit, it diminishes to zero.

Second, the linear utility does not take into account the effect of risk aversion. Generally, the decision makers tend to avoid monetary fair bets. Losing *1* unit of money is considered as being more serious than winning the same amount of money. Also, when earning additional money, every next unit of money earned is considered as being less important than the previous unit earned.

Both of these obstacles can be overcame, if to assume that the utility function is concave - if to assume the diminishing marginal utility.

In 1738, Daniel Bernoulli [Bernoulli 1738, for the English translation see Bernoulli 1954] introduced the logarithmic utility function (also known as *log utility*):

This function is concave and it provides us with the risk aversion. More importantly, it also provides us with the *diminishing* risk aversion. According to Bernoulli, it is important, that than richer one is, the lesser is one's reluctance against tossing a coin in a fair gamble with the same fixed bet. An infinitely rich man should behave as if one's utility function is linear.

It is crucial to understand that the effect of risk aversion is the result of the concaveness of the utility function, while the effect of diminishing risk aversion is not.

Unfortunately, Bernoulli's utility function is without lower and upper bounds, which creates shortages. We can use the improved Bernoulli's function

Everything remains the same, except that the utility function has a lower bound now (see the Figure). Unfortunately, this improved utility still lacks the upper bound.

The risk aversion indicator or the Arrow-Pratt measure [see Pratt 1964] can be defined as follows:

(In some sources, the right side of this formula has been multiplied with a positive constant.)

If *λ > 0*, a local risk aversion appears. In the case of our logarithmic utility *u = ln(x + 1)*, the risk aversion is decreasing in *x* and it is diminishing to zero:

See the Figure.

Let us consider the following simple exponential utility:

where is a parameter.

This utility function is bounded:

and it can be interpreted as a probability. If *x* is the value of the assets of the decision maker, and *u(x)* above is one's utility function, then one regards one's assets *x* as equivalent to the gamble, in which with the probability *u(x)* one wins an infinite amount of money, while otherwise one loses everything. To prove this claim, suppose that *p* is the win probability in such an utility-fair gamble. Then

,

therefore

,

therefore

, which concludes the proof.

In the case

which can be achieved by changing the money unit, our utility function has a particularly simple form (see the Figure)

in which case the disutility

is bisected after every additional unit of money obtained.

Such an utility is also called as an *utility with constant risk aversion*. The indicator of risk aversion (the Arrow-Pratt measure) *λ* is constant:

This result shows that while the concavity of the utility function produces the effect of risk aversion, it is not enough to secure that this risk aversion is *decreasing* in *x* and that it is diminishing to zero in the infinity of *x*. However, this is a serious obstacle when trying to model mutually motivated contracts, for example, mutually motivated insurance contracts.

Already Daniel Bernoulli [Bernoulli 1738], who introduced the notion of utility function, regarded it as important to be able to explain the insurance contracts. Why a rich man is willing to provide the insurance and a poor man is not? The answer consists not in the existence of the risk aversion, but in the effect of *decreasing* risk aversion. Incidentally, Bernoulli's logarithmic utility *u = ln(x)* is not only concave, providing the risk aversion, but also with decreasing risk aversion, as we have seen above. This risk aversion is diminishing in the infinity. However, Bernoulli's utility function is unbounded.

Since then, it has often been asked about whether there was any simple utility function that was bounded and which had a decreasing risk aversion, moreover, a risk aversion diminishing to zero in the infinity (see, for example, [Pratt 1964]). It has been asked about "everyman's utility function", but usually only the logarithmic or exponential utilities have been mentioned.

In the book *Utility Function u = x/(x + 1)* [Eintalu 2019] a very simple utility function has been investigated:

,

where is a parameter.

This utility function is bounded:

and it can be interpreted as a probability.

By changing the money unit, the condition *b = 1* can be achieved and this utility function obtains a particularly simple form (see the Figure above):

The Arrow-Pratt measure (or the indicator of the risk aversion) is also very simple and it is diminishing in *x* (see the Figure below):

In [Eintalu 2019], the initial assumption is that the risk aversion indicator has to have the following, diminishing form:

,

where and .

In this way, the logarithmic utilities considered above can be obtained as the special cases. In this way, moreover, also a class of bounded utilities can be obtained, called in that study as "generalized Bernoulli utilities". The simplest utility function in that class is the function .

One group of authors [Ikefuji, *et al* 2013] have started from another set of axioms, deriving a set of functions they call as "Pareto utilities". The function can also be considered as a special case of Pareto utilities.

There have been presented various utility functions earlier (for example, in [Pratt 1964]), having as a special case the function .

The function has one amazing feature. Technically, at every point *x*, the value of that function *u(x)* can be calculated as the value of the linear utility in the case when there is still exactly one unit of money missing from the "ultimate happiness" corresponding to the value (see the Figure). Therefore, in [Eintalu 2019], the utility function has been characterized as "There is always one cent missing". However, as yet, there is no deeper explanation of this feature, which might as well turn out to be a coincidence.

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 been demonstrated empirically.

Another criticism comes from the assertion that neither cardinal nor ordinal utility is 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]}

**^**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.**^**Marshall, Alfred (1920).*Principles of Economics. An introductory volume*(8th ed.). London: Macmillan.**^**Von Neumann, J.; Morgenstern, O. (1953).*Theory of Games and Economic Behavior*(3rd ed.). Princeton University Press.**^**Kahneman, D.; Tversky, A. (1979). "Prospect Theory: An Analysis of Decision Under Risk" (PDF).*Econometrica*.**47**(2): 263–292. doi:10.2307/1914185.**^**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.**^**Ingersoll, Jonathan E., Jr. (1987).*Theory of Financial Decision Making*. Totowa: Rowman and Littlefield. p. 21. ISBN 0-8476-7359-6.**^**Castagnoli, E.; LiCalzi, M. (1996). "Expected Utility Without Utility" (PDF).*Theory and Decision*.**41**(3): 281–301. doi:10.1007/BF00136129.**^**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.**^**Bordley, R.; Kirkwood, C. (2004). "Multiattribute preference analysis with Performance Targets".*Operations Research*.**52**(6): 823–835. doi:10.1287/opre.1030.0093.**^**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.**^**Berger, J. O. (1985). "Utility and Loss".*Statistical Decision Theory and Bayesian Analysis*(2nd ed.). Berlin: Springer-Verlag. ISBN 3-540-96098-8.**^**Robinson, Joan (1962).*Economic Philosophy*. Harmondsworth, Middle-sex, UK: Penguin Books.**^**Pilkington, Philip (17 February 2014). "Joan Robinson's Critique of Marginal Utility Theory".*Fixing the Economists*. Archived from the original on 13 July 2015.**^**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.**^**"Revealed Preference Theory". Archived from the original on 16 July 2011. Retrieved 11 December 2009.**^**"Archived copy" (PDF). Archived from the original (PDF) on 15 October 2008. Retrieved 9 August 2008.CS1 maint: archived copy as title (link)**^**Klein, Daniel (May 2014). "Professor" (PDF).*Econ Journal Watch*.**11**(2): 97–105. Archived (PDF) from the original on 5 October 2014. Retrieved 15 November 2014.**^**Burke, Kenneth (1932).*Towards a Better Life*. Berkeley, Calif: University of California Press.**^**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.

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