# Utility

Short description
Concept in economics and game theory

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 happiness 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 happiness or pleasure received.

## Utility function

Utility function indicates utility values associated with bundles of goods: U (X, Y). 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.

Suppose Jimmy has utility function U =$\displaystyle{ \surd xy }$ where x is the number of apples and y is the number of chocolates. Alternative A has x = 9 apples and y = 16 chocolates; alternative B has x = 13 apples and y = 13 chocolates. Putting the values x, y into the utility function yields $\displaystyle{ \surd(9 \times 16) }$ = 12 for alternative A and $\displaystyle{ \surd(13 \times 13) }$ = 13 for B, so Jimmy prefers alternative B.

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.

In some special applications, such as the conventional theory of Consumer Choice, the Choice Set is not usually finite. In fact, a commonly specified Choice Set in Consumer Choice is $\displaystyle{ R_+^n }$, where $\displaystyle{ n }$ is the number of perceived commodities in the market of consideration. In this case, there exists a continuous utility function to represent a consumer's preferences if and only if the consumer's preferences are complete, transitive and continuous.[2]

## Total Utility

Total utility refers to the quantifiable sum of satisfaction or happiness gained by consuming various units of particular goods or services, also used for economic analysis of consumer behaviors. In economics, the change of behavior and consumption are depended on the marginal trend, which means people are concentrated on the level of the utility.

• Rational Choice Theory

Rational Choice Theory is a theory that to maximize the utility in each consumption by a consumer and make a rational choice in their decision-making.[3]

The basic total utility formula:

          T(U)= U1 + M(U)2 + M(U)3 ...


where U is the utility and M(U) is the marginal utility.

• Example of Total Utility

John decides to purchase a cake and gain total utility from purchasing a cake is 20; purchase 2 cakes are 25 utils; purchase 3 cakes are 27 utils, and purchase 4 cakes are 24 utils. From each purchase, the total utility will raise until the maximum of 3 cakes. After that, the total utility would decrease in purchasing 4 cakes. We finally can calculate the marginal utility by using the data of total utility.

## 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.

A general diagram of indifference curve is shown as below (Figure 1). The vertical axes and the horizontal axes represent an individual’s consumption over commodity Y and X respectively. All the combinations of commodity X and Y along the same indifference curve make individuals indifferent, which means all the combinations long an indifference curve result in the same value of utility.

Figure 1

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.

## Preference

Preference, as human’s specific likes and dislikes, are primarily used when individuals make choices or decisions among different alternatives. The formation of individual’s preferences is influenced by varied factors such as geographical locations, gender, cultures and education. The ranking of utility indicates individuals’ 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 $\displaystyle{ u\colon X\to \R }$ 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 his utility function is u(nothing) = 0, u(1 apple) = 1, u(1 orange) = 2, u(1 apple and 1 orange) = 5, u(2 apples) = 2 and u(2 oranges) = 4. 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 $\displaystyle{ \R^L_+ }$, and each package $\displaystyle{ x \in \R^L_+ }$ is a vector containing the amounts of each commodity. In the example, there are two commodities: apples and oranges. If we say apples is the first commodity, and oranges the second, then the consumption set is $\displaystyle{ X =\R^2_+ }$ and u(0, 0) = 0, u(1, 0) = 1, u(0, 1) = 2, u(1, 1) = 5, u(2, 0) = 2, u(0, 2) = 4 as before. Note that for u to be a utility function on X, however, it must be defined for every package in X, so now the function needs to be defined for fractional apples and oranges too. One function that would fit these numbers is $\displaystyle{ u(x_{apples}, x_{oranges}) = x_{apples} + 2 x_{oranges} + 2x_{apples} x_{oranges}. }$

Preferences have three main properties:

• Completeness

Assume an individual is facing two choices, A and B. By ranking the two choices, one and only one of the following relationships is true: an individual strictly prefers A (A>B); an individual strictly prefers B (B>A); an individual is indifferent between A and B (A=B). Either a ≥ b OR b ≥ a (OR both) for all (a,b)

•  Transitivity

Individuals’ preferences are consistent over bundles. If an individual prefers bundle A over bundle B , and prefers bundle B over bundle C, then the individual weakly prefers bundle A over bundle B could be concluded. If a ≥ b and b ≥ c, then a ≥ b for all (a,b,c)

• Non-Satiation (Monotone Preferences)

All else hold constantly, individuals always prefer more of positive goods rather than negative goods, vice versa. In terms of the indifferent curves, individuals will always prefer bundles that lie on a higher indifference curve. In another word, all else being the same, more is better less of the commodity.

• When being good, a commodity more is preferred to less, and same level of consumption.
• When being bad, a commodity less is preferred more, like pollution.

### 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.[4]:78

### Revealed preference 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,[5] and prospect theory.[6] 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.[7]

## 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

Cardinal utility states that the utilities obtained from consumption can be measured and ranked objectively and are representable by numbers.[8] There are fundamental assumptions under cardinal utility. Economic agents should be able to rank different bundles of goods based on their own preferences or utilities, and also sort different transitions of two bundles of goods. [9]

From mathematic perspective, cardinal utility function is unique up to a positive linear transformation. A utility function U(x) could be transformed to another function, by multiplying a positive number, and plus any number. Both utility functions represent the same preferences.[10]

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

Instead of giving actual numbers over different bundles, ordinal utilities are only the rankings of utilities received from different bundles of goods or services.[8] For example, ordinal utility could tell that having two ice creams provide a greater utility to individuals in comparison to one ice cream but could not tell exactly how much extra utility received by the individual. Under ordinal utility, it does not require individuals to specify how much extra utility he or she received from the preferred bundle of goods or services in comparison to other bundles. They are only required to tell which bundles they prefer.

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 $\displaystyle{ u(x)^3 }$.

### Constructing utility functions

In many decision models, utility functions are determined by the problem formulation. In some situations, the decision maker’s preference must be elicited and represented by a utility (or objective) scalar-valued function. The existing methods for constructing such functions are collected in the proceedings of two dedicated conferences.[11][12] The mathematical foundations for the most common types of utility functions — quadratic and additive — were laid down by Gerard Debreu,[13][14] and the methods for their construction from both ordinal and cardinal data, in particular from interviewing a decision maker, were developed by Andranik Tangian.[15][16]

### 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.[17]

## Marginal utility

Holding the quantity of one good constant, the change in utility for one unit change of another good is called marginal utility (MU). Marginal utility therefore basically measures the slope of the utility function with respect to the units changed in one good.[18] Marginal utility trends to decrease with the consumption, however, it would not gain zero, which need to depend on the good consumed. For example, marginal utility of good X is $\displaystyle{ MU_x=\frac{\bigtriangleup U}{\vartriangle X} }$  . This equation can be further transformed to  $\displaystyle{ \vartriangle U=MU_x\vartriangle x }$which represents the rate of change in utility. Thus, the marginal utility measures the rate of converting consumption units into utility units. If M(U) is greater than MC, their consumer will be consuming more of a good. In analyzing the marginal utility, when M(U)>0, that means this item brings additional happiness; when M(U)=0, it indicates that the item would not produce any extra happiness; when M(U)<0 means that consuming more would cause harmful.[19]

### Law of diminishing marginal utility

Rational individuals only consume additional unit of goods only if it increases the marginal utility. However, the law of diminishing marginal utility means an additional unit consumed brings a less marginal utility brought by the previous unit consumed. For example, drinking one bottle of water makes a thirsty person satisfied, as the consumption of water increasing, he may gradually feel bad which leads the marginal utility decrease to zero and even becomes negative. Furthermore, this is also usually utilized to analyze progressive taxes as the higher taxes can result in the losses of utility.

### Marginal rate of substitution (MRS)

Marginal rate of substitution is the slope of the indifference curve, which measures how much an individual is willing to switch from one good to another. Using mathematic equation, $\displaystyle{ MRS=-\operatorname{d}\!x_2/\operatorname{d}\!x_1 }$holding that U (x1,x2) constant. Thus, MRS is how much an individual is willing to give up for consuming a greater amount of x1.

MRS is related to Marginal utility. The relationship between marginal utility and MRS is: $\displaystyle{ MRS=\frac{MU_1}{MU_2} }$[18]

## Expected utility

Main page: Expected utility hypothesis

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.

In finding the probability-weighted average of the utility from each possible outcome:

 EU=[Pr(z)×u(value(z))]+[Pr(y)×u(value(y))]


### von Neumann–Morgenstern

Main page: Von Neumann–Morgenstern utility theorem

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:

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

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:

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

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.[20][21][22][23]

## 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.[24]

## Budget constraints

Individuals' consumptions are constrained by their budget allowance. The graph of budget line is a linear, downward-sloping line between X and Y axes. All the bundles of consumption under the budget line allow individuals to consume without using the whole budget as the total budget is greater than the total cost of bundles (Figure 2). If only considers prices and quantities of two goods in one bundle, a budget constraint could be formulated as $\displaystyle{ p_1X_1+p_2X_2 =Y }$, where p1 and p2 are prices of the two goods, X1 and X2 are quantities of the two goods.

Figure 2

Slope = -P(x)/P(y)

### Constrained utility optimisation

Rational consumers wish to maximise their utility. However, as they face budget constraints, a change in price would affect the quantity of demand. There are two factors could explain this situation:

• Purchasing Power. Individuals obtain higher purchasing power when the price of a good decreases. The reduction of the price allows individuals to increase their savings so they could afford to other products.
• Substitution Effect. If the price of good A decreases, then the good becomes relatively cheaper to its substitutes. Thus, individuals would consume more of good A as the utility would increase by doing so.

## 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."[25]: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.[26] 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.[27] 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.).[28][29]

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.[30] Likewise, choosing between alternatives is itself a process of determining what to consider as alternatives, a question of choice within uncertainty.[31]

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.[32]

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