Utility

For other uses, see Utility (disambiguation).

In economics, utility is a measure of the relative satisfaction from or desirability of consumption of goods. 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. For illustrative purposes, changes in utility are sometimes expressed in units called utils.

The doctrine of utilitarianism saw the maximization of utility as a moral criterion for the organization of society. According to utilitarians, such as Jeremy Bentham (1748-1832) and John Stuart Mill (1806-1876), society should aim to maximize the total utility of individuals, aiming for "the greatest happiness for the greatest number".

In neoclassical economics, rationality is precisely defined in terms of imputed utility-maximizing behavior under economic constraints. As a hypothetical behavioral measure, utility does not require attribution of mental states suggested by "happiness", "satisfaction", etc.

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. When coupled with production or commodity constraints, these functions can represent Pareto efficiency, such as illustrated by Edgeworth boxes and contract curves. Such efficiency is a central concept of welfare economics.

Cardinal/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. On the other hand, ordinal utility captures only ranking and not strength of preferences. 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 than 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. The argument against this is that interpersonal comparisons of utility are suspect because there is no good way to interpret how different people value consumption bundles.

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. In the above example, it would only be possible to say that coffee is preferred to tea to water, but no more.

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.

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. Let X be the consumption set, the set of all mutually-exclusive packages the consumer could conceivably consume (such as an indifference curve map without the indifference curves). The consumer\'s utility function $u :\; X\; \backslash rightarrow\; \backslash textbf\; R$ ranks each package in the consumption set. If u(x) ≥ u(y) (x R y), then the consumer strictly prefers x to y or is indifferent between them.

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 microeconomic 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 $\backslash textbf\; R^L\_+$, and each package $x\; \backslash in\; \backslash textbf\; R^L\_+$ 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 = $\backslash textbf\; R^2\_+$ 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 $u :\; X\; \backslash rightarrow\; \backslash textbf\{R\}$ rationalizes a preference relation $\backslash preceq$ on X if for every $x,\; y\; \backslash in\; X$, $u(x)\backslash leq\; u(y)$ if and only if $x\backslash preceq\; y$. If u rationalizes $\backslash preceq$, then this implies $\backslash preceq$ is complete and transitive, and hence rational.

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

• CES (constant elasticity of substitution, or isoelastic) utility is one with constant relative risk aversion
• Exponential utility exhibits constant absolute 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. There are some important exceptions, however.

Lexicographic preferences cannot even be represented by a utility function.

Expected utility

Main article: Expected utility hypothesis

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 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.

A von Neumann-Morgenstern utility function $u :\; X\; \backslash rightarrow\; \backslash textbf\{R\}$ 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 $L\_1$ to a lottery $L\_2$ if and only if the expected utility (iterated over compound lotteries if necessary) of $L\_1$ is greater than the expected utility of $L\_2$.

Restricting to the discrete choice context, let $L :\; X\; \backslash rightarrow\; [0,1]$ be a simple lottery such that $L(x\_i)\; =\; p\_i$, where $p\_i$ is the probability that $x\_i$ 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 $L\_1\; \backslash geq\; L\_2\; \backslash geq\; L\_3$, then there exists a $0\; \backslash leq\; p\; \backslash leq\; 1$ such that the agent is indifferent between $L\_2$ and the compound lottery mixing between $L\_1$ and $L\_3$ with probability $p$ and $1-p$, respectively. Independence means that if the agent is indifferent between simple lotteries $L\_1$ and $L\_2$, the agent is also indifferent between $L\_1$ mixed with an arbitrary simple lottery $L\_3$ with probability $p$ and $L\_2$ mixed with $L\_3$ 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.

Utility of money

One of the most common uses of a utility function, especially in economics, is the utility of money. The utility function for money is a nonlinear function that is bounded and asymmetric about the origin. These properties can be derived from reasonable assumptions that are generally accepted by economists and decision theorists, especially proponents of rational choice theory. 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 nonlinearity 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. J.O. Berger, Statistical Decision Theory and Bayesian Analysis. Springer-Verlag 2nd ed. (1985) ch. 2. (ISBN 3540960988)

Discussion and criticism

Different 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 other factors such as natural rights, law, conscience and/or religious doctrine. It is debatable whether any of these can be adequately represented in a system that uses a utility model.

See also

• Allais paradox
• behavioral economics
• Choice Modelling
• consumer surplus
• convex preferences
• cumulative prospect theory
• decision theory
• efficient market theory
• expectation utilities
• Ellsberg paradox
• game theory
• list of economics topics
• marginal utility
• microeconomics
• prospect theory
• risk aversion
• risk premium
• Transferable utility
• Utility Maximization Problem
• utility (patent)
• utility model

References and additional reading

• Neumann, John von and Morgenstern, Oskar Theory of Games and Economic Behavior. Princeton, NJ. Princeton University Press. 1944 sec.ed. 1947
• Nash Jr., John F. The Bargaining Problem. Econometrica 18:155 1950
• Anand, Paul. Foundations of Rational Choice Under Risk Oxford, Oxford University Press. 1993 reprinted 1995, 2002
• Kreps, David M. Notes on the Theory of Choice. Boulder, CO. Westview Press. 1988
• Fishburn, Peter C. Utility Theory for Decision Making. Huntington, NY. Robert E. Krieger Publishing Co. 1970. ISBN 978-0471260608
• Plous, S. The Psychology of Judgement and Decision Making New York: McGraw-Hill, 1993
• Virine, L. and Trumper M., Project Decisions: The Art and Science. Management Concepts. Vienna, VA, 2007. ISBN 978-1567262179

External links

• Anatomy of Cobb-Douglas Type Utility Functions in 3D
• Anatomy of CES Type Utility Functions in 3D

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