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In economics, utility is a measure of the happiness or satisfaction gained consuming good and services. Given this measure, one may speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior in terms of rational attempts to increase one's utility. In neoclassical economics, rationality is precisely defined in terms of utility-maximizing behavior, under economic constraints. The concept is applied by economists in such topics as the indifference curve, which plots the combination of commodities that an individual or a community requires to maintain a given level of satisfaction. The concept is also used in utility functions, social welfare functions, Pareto maximization, Edgeworth boxes and contract curves. It is a central concept of welfare economics. The doctrine of utilitarianism saw the maximisation of utility as a moral criterion for the organisation of society. According to utilitarians, such as Jeremy Bentham (1748-1832) and John Stuart Mill (1806-1876), society should aim to maximise the total utility of individuals, aiming for "the greatest happiness for the greatest number."
Cardinal and ordinal utilityThere are mainly two kinds of measurement of utility implemented by economists: cardinal utility and ordinal utility. Cardinal utility can be measured in terms of monetary units. Utility was originally viewed as a measurable quantity, so that it would be possible to measure the utility of each individual in the society with respect to each good available in the society, and to add these together to yield the total utility of all people with respect to all goods in the society. Society could then aim to maximise the total utility of all people in society, or equivalently the average utility per person. This conception of utility as a measurable quantity that could be aggregated across individuals is called cardinal utility. Cardinal utility quantitatively measures the preference of an individual towards a certain commodity. Numbers assigned to different goods or services can be compared. A utility of 100 units towards a cup of vodka is twice as desirable as a cup of coffee with a utility level of 50 units. The concept of cardinal utility suffers from the absence of an objective measure of utility when comparing the utility gained from consumption of a particular good by one individual as opposed to another individual. For this reason, neoclassical economics abandoned utility as a foundation for the analysis of economic behaviour, in favour of an analysis based upon preferences. This led to the development of tools such as indifference curves to explain economic behaviour. In this analysis, an individual is observed to prefer one choice to another. Preferences can be ordered from most satisfying to least satisfying. Only the ordering is important: the magnitude of the numerical values are not important except in as much as they establish the order. A utility of 100 towards an ice-cream cone is not twice as desirable as a utility of 50 towards a candy. All that can be said is that ice-cream cone is preferred over the candy. There is no attempt to explain why one choice is preferred to another; hence no need for a quantitative concept of utility. In principle, given a set of preferences which satisfy certain criteria of reasonableness, it is possible to find a utility function that will explain these preferences. Such a utility function takes on higher values for choices that the individual prefers. Note that a utility function to describe an individual's set of preferences clearly is not unique. If the value of the utility function were to be, for example, doubled, squared, or subjected to any other strictly monotonically increasing function, it would still describe the same preferences. With this approach to utility, known as ordinal utility it is not possible to compare utility between individuals, or find the total utility for society as the Utilitarians hoped to do. Utility functionsWhile preferences are the conventional foundation of microeconomics, it is convenient to represent preferences with a utility function and reason indirectly about preferences with utility functions. Let X be the consumption set, the set of all packages the consumer could conceivably consume. The consumer's utility function assigns a happiness score to each package in the consumption set. If u(x) > u(y), then the consumer strictly prefers x to y. For example, suppose a consumer's consumption set is X = {nothing, 1 apple, 1 orange, 1 apple and 1 orange, 2 apples, 2 oranges}, and its utility function is u(nothing) = 0, u(1 apple) = 1, u(1 orange) = 2, u(1 apple and 1 orange) = 4, u(2 apples) = 2 and u(2 oranges) = 3. Then this consumer prefers 1 orange to 1 apple, but prefers one of each to 2 oranges. In microeconomics 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 X = 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 rationalizes a preference relation <= on X if for every , u(x) <= u(y) if and only if x <= y. If u rationalizes <=, then this implies <= is complete and transitive, and hence rational. In order to simplify calculations, various assumptions have been made of utility functions.
Expected utilityThe [expected utility] model was first proposed by Daniel Bernoulli as a solution to the St. Petersburg paradox. Bernoulli argued that the paradox could be resolved if decisionmakers displayed risk aversion and argued for a logarithmic cardinal utility function. The first important use of the expected theory was that of John von Neumann and Oskar Morgenstern who used the assumption of expected utility maximization in their formulation of game theory. A von Neumann-Morgenstern utility function assigns a real number to every element of the outcome space in a way that captures the agent's preferences over both simple and compound lotteries (put in category-theoretic language, u induces a morphism between the category of preferences under uncertainty and the category of reals). The agent will prefer a lottery L1 to a lottery L2 if and only if the expected utility (iterated over compound lotteries if necessary) of L1 is greater than the expected utility of L2. Restricting to the discrete choice context, let be a simple lottery such that L(xi) = pi, where pi is the probability that xi is won. We may also consider compound lotteries, where the prizes are themselves simple lotteries. The expected utility theorem says that a von Neumann-Morgenstern utility function exists if and only if the agent's preference relation on the space of simple lotteries satisfies four axioms: completeness, transitivity, convexity/continuity (also called the Archimedean property), and independence. Completeness and transitivity are discussed supra. The Archimedean property says that for simple lotteries , then there exists a such that the agent is indifferent between L2 and the compound lottery mixing between L1 and L3 with probability p and 1 − p, respectively. Independence means that if the agent is indifferent between simple lotteries L1 and L2, the agent is also indifferent between L1 mixed with an arbitrary simple lottery L3 with probability p and L2 mixed with L3 with the same probability p. Independence is probably the most controversial of the axioms. A variety of generalized expected utility theories have arisen, most of which drop or relax the independence axiom. Discussion and criticismDifferent value systems have different perspectives on the use of utility in making moral judgments. For example, Marxists, Kantians, and certain libertarians (such as Nozick) all believe utility to be irrelevant as a moral standard or at least not as important as others such as natural rights. See also
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