TheInfoList

As a topic of
economics Economics () is a social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a bran ...

, utility is used to model worth or
value Value or values may refer to: * Value (ethics) it may be described as treating actions themselves as abstract objects, putting value to them ** Values (Western philosophy) expands the notion of value beyond that of ethics, but limited to Western s ...

. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of
utilitarianism Utilitarianism is a family of normative Normative generally means relating to an evaluative standard. Normativity is the phenomenon in human societies of designating some actions or outcomes as good or desirable or permissible and others as ba ...
by moral philosophers such as
Jeremy Bentham Jeremy Bentham (; 15 February 1748 Old_Style_and_New_Style_dates">O.S._4_February_1747.html" ;"title="Old_Style_and_New_Style_dates.html" ;"title="nowiki/>Old Style and New Style dates">O.S. 4 February 1747">Old_Style_and_New_Style_dates.htm ...

and
John Stuart Mill John Stuart Mill (20 May 1806 – 7 May 1873), also cited as J. S. Mill, was an English philosopher, Political economy, political economist, Member of Parliament (United Kingdom), Member of Parliament (MP) and civil servant. One of the most i ...
. The term has been adapted and reapplied within
neoclassical economics Neoclassical economics is an approach to economics in which the production, consumption and valuation (pricing) of goods and services are driven by the supply and demand In microeconomics, supply and demand is an economic model In econo ...
, which dominates modern economic theory, as a utility function that represents a single consumer's preference ordering over a choice set but is not comparable across consumers. This concept of utility is personal and based on choice rather than on pleasure received, and so is specified more rigorously than the original concept but makes it less useful (and controversial) for ethical decisions.

# Utility function

Consider a set of alternatives among which a person can make a preference ordering. The utility obtained from these alternatives is an unknown function of the utilities obtained from each alternative, not the sum of each alternative. A utility function is able to represent that ordering if it is possible to assign a
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
to each alternative in such a manner 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 someone who selects the most preferred alternative is necessarily also selecting the alternative that maximizes the associated utility function. Suppose James has utility function U =$\surd xy$ such that 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 $\surd\left(9 \times 16\right)$ = 12 for alternative A and $\surd\left(13 \times 13\right)$ = 13 for B, so James prefers alternative B. In general economic terms, a utility function measures preferences concerning a set of goods and services. Utility is often correlated with concepts such as happiness, satisfaction, and welfare which are difficult to measure. Thus, economists utilize consumption baskets of preferences in order to measure these abstract, nonquantifiable ideas.
Gérard Debreu Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist An economist is a practitioner in the social sciences, social science discipline of economics. The individual may also study, develop, and apply theories and concep ...
precisely defined the conditions required for a preference ordering to be representable by a utility function. 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 equal), and that the preference order is
transitive Transitivity or transitive may refer to: Grammar * Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects * Transitive verb, a verb which takes an object * Transitive case, a grammatical case to mark arg ...
. Very often the set of alternatives is not finite, because even if the number of goods is finite, the quantity chosen can be any real number on an interval. A commonly specified Choice Set in Consumer Choice is $R_+^n$, where $n$ is the number of goods. 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.

# Applications

Utility is usually applied by
economist An economist is a professional and practitioner in the social science Social science is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , pl ...

s to such constructs as the
indifference curve In economics Economics () is a social science Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among individuals within those societies. ...

, which plot the combination of commodities that an individual would accept to maintain a given level of satisfaction. Utility and indifference curves are used by economists to understand the causes of
demand curve In economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods ...

s as part of
supply and demand In microeconomics Microeconomics is a branch of that studies the behavior of individuals and in making decisions regarding the allocation of and the interactions among these individuals and firms. Microeconomics focuses on the study ...

analysis, which is used to analyze the workings of
goods In economics Economics () is a social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant ...
markets. A diagram of a general indifference curve is shown below (Figure 1). The vertical axes and the horizontal axes represent an individual’s consumption of commodity Y and X respectively. All the combinations of commodity X and Y along the same indifference curve are regarded indifferently by individuals, which means all the combinations along an indifference curve result in the same value of utility.   Individual utility and social utility can be construed as the value of a utility function and a
social welfare functionIn welfare economics, a social welfare function is a function (mathematics), function that ranks social states (alternative complete descriptions of the society) as less desirable, more desirable, or indifference curve, indifferent for every possible ...
respectively. When coupled with production or commodity constraints, by some assumptions these functions can be used to analyze
Pareto efficiency Pareto efficiency or Pareto optimality is a situation where no individual or preference criterion can be better off without making at least one individual or preference criterion worse off or without any loss thereof. The concept is named after Vi ...
, such as illustrated by
Edgeworth boxIn economics, an Edgeworth box, sometimes referred to as an Edgeworth-Bowley box, is a graphical representation of a market with just two commodities, ''X'' and ''Y'', and two consumers. The dimensions of the box are the total quantities Ω''x'' and ...

es in
contract curves in an Edgeworth box. If the initial allocations of the two goods are at a point not on this locus, then the two people can trade to a point on the efficient locus within the lens formed by the indifference curves that they were originally on. The ...
s. Such efficiency is a major concept in
welfare economics Welfare economics is a branch of economics that uses microeconomics, microeconomic techniques to evaluate well-being (welfare) at the aggregate (economy-wide) level. Attempting to apply the principles of welfare economics gives rise to the fiel ...
. In
finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in the left corn ...

, utility is applied to generate an individual's price for an
asset In financial accounting Financial accounting is the field of accounting Accounting or Accountancy is the measurement, processing, and communication of financial and non financial information about economic entity, economic entities such a ...
known as the
indifference priceIn finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available w ...
. Utility functions are also related to
risk measure In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the downside risk, risks taken by financial institutions, ...
s, with the most common example being the
entropic risk measure In financial mathematicsMathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive an ...
. For
artificial intelligence Artificial intelligence (AI) is intelligence Intelligence has been defined in many ways: the capacity for abstraction Abstraction in its main sense is a conceptual process where general rules and concept Concepts are defined as abstra ...

, utility functions are used to convey the value of various outcomes to
intelligent agent In artificial intelligence Artificial intelligence (AI) is intelligence demonstrated by machines, unlike the natural intelligence human intelligence, displayed by humans and animal cognition, animals, which involves consciousness and emotio ...
s. 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, is used primarily when individuals make choices or decisions among different alternatives. Individual preferences are influenced by various factors such as geographical location, gender, cultures and education. The ranking of utility indicates individuals’ preferences. Although
preference In psychology Psychology is the science of mind and behavior. Psychology includes the study of consciousness, conscious and Unconscious mind, unconscious phenomena, as well as feeling and thought. It is an academic discipline of immense scop ...
s are the conventional foundation of
microeconomics Microeconomics is a branch of mainstream economics Mainstream economics is the body of knowledge, theories, and models of economics, as taught by universities worldwide, that are generally accepted by economists as a basis for discussion. Als ...
, it is often convenient to represent preferences with a utility
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 $u\colon X\to \R$ ranks each package in the consumption set. If the consumer strictly prefers ''x'' to ''y'' or is indifferent between them, then $u\left(x\right)\geq u\left(y\right)$. For example, suppose a consumer's consumption set is ''X'' = , 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 $\R^L_+$, and each package $x \in \R^L_+$ is a vector containing the amounts of each commodity. For 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 $X =\R^2_+$ 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 $u\left(x_, x_\right) = x_ + 2 x_ + 2x_ x_.$ Preferences have three main properties: * Completeness Assume an individual has 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 to bundle B , and prefers bundle B to bundle C, then it can be assumed that the individual prefers bundle A to bundle C. (If a ≥ b and b ≥ c, then a ≥ c for all (a,b,c)). * Non-Satiation (Monotone Preferences) All else being constant, 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 are on a higher indifference curve. In other words, all else being the same, more is better than less of the commodity. * When a commodity is good, more of it is preferred to less. * When a commodity is bad, less of it 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 relative utilities from observed choice. These 'revealed preferences', as termed by
Paul Samuelson Paul may refer to: *Paul (given name), a given name (includes a list of people with that name) *Paul (surname), a list of people People Christianity *Paul the Apostle (AD 5–67), also known as Saul of Tarsus or Saint Paul, early Christian mis ...

, were revealed e.g. in people's willingness to pay:
Utility is assumed to be correlative to Desire or Want. It has been argued already that desires cannot be measured directly, but only indirectly, by the outward phenomena which they cause: and that in those cases with which economics is mainly concerned the measure is found by the price which a person is willing to pay for the fulfillment or satisfaction of his desire.

## Revealed preference in finance

For financial applications, e.g.
portfolio optimizationPortfolio optimization is the process of selecting the best portfolio (asset In financial accountancy, financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) ...
, an investor chooses a financial portfolio which maximizes his/her own utility function, or, equivalently, minimizes his/her
risk measure In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the downside risk, risks taken by financial institutions, ...
. For example,
modern portfolio theory Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of Diversification ...
selects variance as a measure of risk; other popular theories are
expected utility theory The expected utility hypothesis is a popular concept in economics Economics () is a social science Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relat ...
, and
prospect theory Prospect theory is a theory of behavioral economics Behavioral economics (also, behavioural economics) studies the effects of psychological, cognitive bias, cognitive, emotional, cultural and social factors on the decision making, decision ...

. To determine a 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 method: observe a portfolio ''X*'' which an investor currently has, and then find a utility function/risk measure such that ''X*'' becomes an optimal portfolio.

## Cardinal

Cardinal utility states that the utilities obtained from consumption can be measured and ranked objectively and are representable by numbers. There are fundamental assumptions of 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. A cardinal utility function can be transformed to another utility function by a positive linear transformation (multiplying by a positive number, and adding some other number); however, both utility functions represent the same preferences. When cardinal utility is assumed, 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, this means that if a person has a cup of tea, he or 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 can be considered as the assumption that utility can be measured by quantifiable characteristics, such as height, weight, temperature, etc.
Neoclassical economics Neoclassical economics is an approach to economics in which the production, consumption and valuation (pricing) of goods and services are driven by the supply and demand In microeconomics, supply and demand is an economic model In econo ...
has largely retreated from using cardinal utility functions as the basis of economic behavior. A notable exception is in the context of analyzing choice with conditions of risk (see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
). Sometimes cardinal utility is used to aggregate utilities across persons, to create a
social welfare functionIn welfare economics, a social welfare function is a function (mathematics), function that ranks social states (alternative complete descriptions of the society) as less desirable, more desirable, or indifference curve, indifferent for every possible ...
.

## 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. 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. 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 assumed 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''. For the above example, it would only be possible to say that juice is preferred to tea to water. Thus, ordinal utility utilizes comparisons, such as "preferred to", "no more", "less than", etc. If a function $u\left(x\right)$ is ordinal, it is equivalent to the function $u\left(x\right)^3$, because taking the 3rd power is an increasing monotone (or monotonic) transformation. This means that the ordinal preference induced by these functions is the same (although they are two different functions). In contrast, if $u\left(x\right)$ is cardinal, it is not equivalent to $u\left(x\right)^3$.

## Constructing utility functions

For many
decision model {{Multiple issues, {{refimprove, date=March 2013 {{no footnotes, date=March 2013 {{context, date=October 2010 A Decision making, decision method is a formal (axiomatic) system, starting with a decision model, that contains at least one action axi ...
s, utility functions are determined by the problem formulation. For some situations, the decision maker’s preference must be elicited and represented by a utility (or objective) scalar-valued function. The methods existing for constructing such functions are collected in the proceedings of two dedicated conferences. The mathematical foundations for the most common types of utility functions — quadratic and additive — were laid down by Gerard Debreu, and the methods for their construction from both ordinal and cardinal data, in particular from interviewing a decision maker, were developed by
Andranik Tangian Andranik Ozanian, commonly known as General Andranik or simply Andranik;. Also spelled Antranik or Antranig 25 February 186531 August 1927), was an Armenian military commander and statesman, the best known '' fedayi'' and a key figure of the A ...
.

## 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: *
CES CES may stand for: *Closed ecological system, isolated from the outside *Climate_change_policy_of_the_United_States#Clean_Energy_Standards, Clean Energy Standards *Constant elasticity of substitution, in economics, a feature of a particular class of ...
(''constant elasticity of substitution''). *
Isoelastic utility In economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behavio ...
*
Exponential utility In economics and finance, exponential utility is a specific form of the utility function, used in some contexts because of its convenience when risk (sometimes referred to as uncertainty) is present, in which case Expected utility hypothesis, expect ...
*
Quasilinear utility In economics Economics () is a social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant l ...

*
Homothetic preferences In consumer theory, a consumer's preferences are called homothetic if they can be represented by a utility function which is homogeneous function, homogeneous of degree 1. For example, in an economy with two goods x,y, homothetic preferences can be ...
*
Stone–Geary utility functionThe Stone–Geary utility function takes the form :U = \prod_ (q_i-\gamma_i)^ where U is utility, q_i is consumption of good i, and \beta and \gamma are parameters. For \gamma_i = 0, the Stone–Geary function reduces to the generalised Cobb–Dougl ...
* Gorman polar form ** Greenwood–Hercowitz–Huffman preferences ** King–Plosser–Rebelo preferences *
Hyperbolic absolute risk aversionIn finance, economics, and decision theory, hyperbolic absolute risk aversion (HARA) (Chapter I of his Ph.D. dissertation; Chapter 5 in his ''Continuous-Time Finance'').Ljungqvist & Sargent, Recursive Macroeconomic Theory, MIT Press, Second Edition ...
Most utility functions used for 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 preferencesLexicographic preferences or lexicographic orderings describe comparative preferences where an economic agent prefers any amount of one good (X) to any amount of another (Y). Specifically, if offered several bundles of goods, the agent will choose t ...
which are not continuous and cannot be represented by a continuous utility function.

# Marginal utility

Economists distinguish between total utility and marginal utility. Total utility is the utility of an alternative, an entire consumption bundle or situation in life. The rate of change of utility from changing the quantity of one good consumed is termed the marginal utility of that good. Marginal utility therefore measures the slope of the utility function with respect to the changes of one good. Marginal utility usually decreases with consumption of the good, the idea of "diminishing marginal utility". In calculus notation, the marginal utility of good X is $MU_x=\frac$. When a good's marginal utility is positive, additional consumption of it increases utility; if zero, the consumer is satiated and indifferent about consuming more; if negative, the consumer would pay to reduce his consumption.

## Law of diminishing marginal utility

Rational individuals only consume additional units of goods if it increases the marginal utility. However, the law of diminishing marginal utility means an additional unit consumed brings a less marginal utility than that brought by the previous unit consumed. For example, drinking one bottle of water makes a thirsty person satisfied; as the consumption of water increases, he may feel begin to feel bad which causes the marginal utility to decrease to zero or even become negative. Furthermore, this is also used to analyze progressive taxes as the greater taxes can result in the loss 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 a mathematic equation, $MRS=-\operatorname\!x_2/\operatorname\!x_1$keeping U (x1,x2) constant. Thus, MRS is how much an individual is willing to pay for consuming a greater amount of x1. MRS is related to marginal utility. The relationship between marginal utility and MRS is: $MRS=\frac$

# Expected utility

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 Daniel Bernoulli FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family Resour ...
in 1738. D. Bernoulli argued that the paradox could be resolved if decision-makers displayed
risk aversion In economics Economics () is a social science Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among individuals within those societies ...

and argued for a logarithmic cardinal utility function. (Analysis of international survey data during the 21st century has shown that insofar as utility represents happiness, as for
utilitarianism Utilitarianism is a family of normative Normative generally means relating to an evaluative standard. Normativity is the phenomenon in human societies of designating some actions or outcomes as good or desirable or permissible and others as ba ...
, it is indeed proportional to log of income.) The first important use of the expected utility theory was that of
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American Hungarian Americans (Hungarian language, Hungarian: ''amerikai magyarok'') are United States, Americans of Hungarian p ...

and
Oskar Morgenstern Oskar Morgenstern (January 24, 1902 – July 26, 1977) was a German-American German Americans (german: Deutschamerikaner, ) are Americans Americans are the Citizenship of the United States, citizens and United States nationality law, nation ...
, who used the assumption of expected utility maximization in their formulation of
game theory Game theory is the study of mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. ...
. In finding the probability-weighted average of the utility from each possible outcome: EU= r(z)×u(value(z)) r(y)×u(value(y))

## von Neumann–Morgenstern

Von Neumann and Morgenstern addressed situations in which the outcomes of choices are not known with certainty, but have probabilities associated with them. A notation for a ''
lottery A lottery is a form of gambling Gambling (also known as betting) is the wagering something of Value (economics), value ("the stakes") on an Event (probability theory), event with an uncertain outcome with the intent of winning something e ...
'' is as follows: if options A and B have probability ''p'' and 1 − ''p'' in the lottery, we write it as a linear combination: :$L = p A + \left(1-p\right) B$ More generally, for a lottery with many possible options: :$L = \sum_i p_i A_i,$ where $\sum_i p_i =1$. 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 computed as a linear combination of the utilities of its parts, with the weights being their probabilities of occurring. This is termed the ''expected utility theorem''. The required assumptions are four axioms about the properties of the agent's
preference relationThe term preference relation is used to refer to orderings that describe human preferences for one thing over an other. * In mathematics, preferences may be modeled as a weak ordering or a semiorder, two different types of binary relation. One specif ...
over 'simple lotteries', which are lotteries with just two options. Writing $B\preceq A$ 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 $L$ and $M$, either $L\preceq M$ or $M\preceq L$ (or both, in which case they are viewed as equally desirable). # transitivity: for any three lotteries $L,M,N$, if $L\preceq M$ and $M\preceq N$, then $L\preceq N$. # convexity/continuity (Archimedean property): If $L \preceq M\preceq N$, then there is a $p$ between 0 and 1 such that the lottery $pL + \left(1-p\right)N$ is equally desirable as $M$. # independence: for any three lotteries $L,M,N$ and any probability ''p'', $L \preceq M$ if and only if $pL+\left(1-p\right)N \preceq pM+\left(1-p\right)N$. Intuitively, if the lottery formed by the probabilistic combination of $L$ and $N$ is no more preferable than the lottery formed by the same probabilistic combination of $M$ and $N,$ then and only then $L \preceq M$. 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: :$u\colon X\to \R$ which assigns a real number to every outcome in a way that represents the agent's preferences over simple lotteries. Using the four assumptions mentioned above, the agent will prefer a lottery $L_2$ to a lottery $L_1$ if and only if, for the utility function characterizing that agent, the expected utility of $L_2$ is greater than the expected utility of $L_1$: :$L_1\preceq L_2 \text u\left(L_1\right)\leq u\left(L_2\right)$. Of all the axioms, independence is the most often discarded. A variety of
generalized expected utilityGeneralized expected utility is a decision-making In psychology, decision-making (also spelled decision making and decisionmaking) is regarded as the Cognition, cognitive process resulting in the selection of a belief or a course of action among s ...
theories have arisen, most of which omit 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.

# Indirect utility

An indirect utility function gives the 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 Concave means curving in or hollowed inward, as opposed to convex. Concave may refer to: * Concave function In mathematics, a concave function is the additive inverse, negative of a convex function. A concave function is also synonymously called ...

in the positive region, representing the phenomenon of
diminishing marginal utility In economics Economics () is a social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant ...
. The boundedness represents the fact that beyond a certain amount money ceases being useful at all, as the size of any economy at that time is itself bounded. The asymmetry about the origin represents 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 by gains or losses of money, which are the norm for most business settings, the optimal choice for a given decision depends on the possible outcomes of all other decisions in the same time-period.

# 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 $p_1X_1+p_2X_2 =Y$, where p1 and p2 are prices of the two goods, X1 and X2 are quantities of the two goods. Slope = -P(x)/P(y)

## Constrained utility optimisation

Rational consumers wish to maximise their utility. However, as they have budget constraints, a change of price would affect the quantity of demand. There are two factors could explain this situation: * Purchasing Power. Individuals obtain greater 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 buy other products. * Substitution Effect. If the price of good A decreases, then the good becomes relatively cheaper with respect 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 Joan Violet Robinson (''née'' Maurice; 31 October 1903 – 5 August 1983) was a British economist An economist is a practitioner in the social sciences, social science discipline of economics. The individual may also study, develop, and ...
famously criticized utility for being a circular concept: "Utility is the quality in
commodities In economics, a commodity is an economic goods, good, usually a resource, that has full or substantial fungibility: that is, the Market (economics), market treats instances of the good as equivalent or nearly so with no regard to who Production ...
that makes individuals want to buy them, and the fact that individuals want to buy commodities shows that they have utility". Robinson also stated that because the theory assumes that preferences are fixed this means that utility is not a
testable Testability, a property applying to an empirical Empirical evidence for a proposition In logic and linguistics, a proposition is the meaning of a declarative sentence (linguistics), sentence. In philosophy, "Meaning (philosophy), meaning" is ...
assumption. This is so because if we observe changes of peoples' behavior in relation to a change in prices or a change in budget constraint we can never be sure to what extent the change in behavior was due to the change of price or budget constraint and how much was due to a change of preference. This criticism is similar to that of the philosopher
Hans Albert Hans Albert (born 8 February 1921) is a German philosopher. Born in Cologne, he lives in Heidelberg. His fields of research are Social Sciences and General Studies of Methods. He is a critical rationalist, paying special attention to rationality, ...
who argued that the ''
ceteris paribus ' or ' () is a Latin phrase meaning "other things equal"; English translations of the phrase include "all other things being equal" or "other things held constant" or "all else unchanged". A prediction or a statement about a ontic, causal, epist ...
'' (all else equal) conditions on which the
marginalist Marginalism is a theory of economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economic ...
theory of demand rested rendered the theory itself a meaningless tautology, incapable of being tested experimentally. In essence, a curve of demand and supply (a theoretical line of quantity of a product which would have been offered or requested for given price) is purely
ontological Ontology is the branch of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Ph ...
and could never have been demonstrated
empirically In philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, ...
. Another criticism derives from the assertion that neither
cardinal Cardinal or The Cardinal may refer to: Christianity * Cardinal (Catholic Church), a senior official of the Catholic Church * Cardinal (Church of England), two members of the College of Minor Canons of St. Paul's Cathedral Navigation * Cardina ...
nor
ordinal Ordinal may refer to: * Ordinal data, a statistical data type consisting of numerical scores that exist on an arbitrary numerical scale * Ordinal date, a simple form of expressing a date using only the year and the day number within that year * O ...
utility are observable empirically in the real world. For the case of cardinal utility it is impossible to measure the degree of satisfaction "quantitatively" when someone consumes or purchases an apple. For 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.). Other questions of what arguments ought to be included in a utility function are difficult to answer, yet seem necessary to understanding utility. Whether people gain utility from coherence of
wants The idea of want can be examined from many perspectives. In secular societies want might be considered similar to the emotion desire Desire is the emotion Emotions are biological states associated with all of the nerve systems brought on b ...
,
beliefs A belief is an Attitude (psychology), attitude that something is the case, or that some proposition about the world is truth, true. In epistemology, philosophers use the term "belief" to refer to attitudes about the world which can be either tru ...
or a sense of
duty A duty (from "due" meaning "that which is owing"; fro, deu, did, past participle of ''devoir''; la, debere, debitum, whence "debt Debt is an obligation that requires one party, the debtor A debtor or debitor is a legal entity (legal ...

is important to understanding their behavior in the utility
organon The ''Organon'' ( grc, Ὄργανον, meaning "instrument, tool, organ") is the standard collection of Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philos ...

. Likewise, choosing between alternatives is itself a process of determining what to consider as alternatives, a question of choice within uncertainty. An
evolutionary psychology Evolutionary psychology is a theoretical approach in the social Social organisms, including humans, live collectively in interacting populations. This interaction is considered social whether they are aware of it or not, and whether the exchan ...
theory is that utility may be better considered as due to preferences that maximized evolutionary fitness in the ancestral environment but not necessarily in the current one.

*
Law of demand In microeconomics Microeconomics is a branch of mainstream economics Mainstream economics is the body of knowledge, theories, and models of economics, as taught by universities worldwide, that are generally accepted by economists as a basis ...

*
Marginal utility In economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behavio ...
*
Utility maximization problem Utility maximization was first developed by utilitarian philosophers Jeremy Bentham Jeremy Bentham (; 15 February 1748 Old_Style_and_New_Style_dates">O.S._4_February_1747.html" ;"title="Old_Style_and_New_Style_dates.html" ;"title="nowiki/ ...
*
Decision-making softwareDecision-making software (DM software) is software Software is a collection of instructions that tell a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Mod ...