utility - Cardinal and ordinal utility, Utility functions, Expected utility, Discussion and criticism

In computing, a term used for a computer program which carries out a specific function needed by a computer centre, but which is not sufficiently extensive to justify the development of a computer package. Normally a function such as sort and merge is provided as a utility.

Given this measure, one may speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior in terms of attempts to increase one's utility.

The doctrine of utilitarianism saw the maximization of utility as a moral criterion for the organization of society.

Utility is applied by economists in such constructs as the indifference curve, which plots the combination of commodities that an individual or a society requires to maintain a given level of satisfaction. Individual utility and social utility can be construed as the dependent variable of a utility function (such as an indifference curve map) and a social welfare function respectively.

Cardinal and ordinal utility

Economists distinguish between cardinal utility and ordinal utility. When cardinal utility is used, the magnitude of utility differences is treated as an ethically or behaviorally significant quantity. An important example of a cardinal utility is the probability of achieving some target.

Utility functions of both sorts assign real numbers (utils) to members of a choice set. For example, suppose a cup of coffee 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. When speaking of cardinal utility, it could be concluded that the cup of coffee is exactly the same amount better as a cup of tea as the cup of tea is better than the cup of water.

It is tempting when dealing with cardinal utility to aggregate utilities across persons.

When ordinal utilities are used, differences in utils are treated as ethically or behaviorally meaningless: the utility values assigned encode a full behavioral ordering between members of a choice set, but nothing about strength of preferences.

Neoclassical economics has largely retreated from using cardinal utility functions as the basic objects of economic analysis, in favor of considering agent preferences over choice sets. As will be seen in subsequent sections, however, preference relations can often be rationalized as utility functions satisfying a variety of useful properties.

Cardinal utility functions are unique. Ordinal utility functions are equivalent up to monotone transformations, while cardinal utilities are equivalent up to positive linear transformations.

Utility functions

While 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.

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. 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.

In order to simplify calculations, various assumptions have been made of utility functions.

CES (constant elasticity of substitution) utility is one with constant relative risk aversion quasilinear utility homothetic utility

Most utility functions used in modeling or theory are well-behaved. They usually exhibit monotonicity, convexity, and global non-satiation.

Expected utility

The 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.

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. A variety of generalized expected utility theories have arisen, most of which drop or relax the independence axiom.

Discussion and criticism

Different value systems have different perspectives on the use of utility in making moral judgments.