Utility theory

In economics, utility is a measure of the relative satisfaction or desiredness from 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. A theoretical unit of measurement for utility is the 'util'.

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 $$u : X \rightarrow \textbf R$$ 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 $$\textbf R^L_+$$, and each package $$x \in \textbf R^L_+$$ 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 = $$\textbf R^2_+$$ 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 $$u : X \rightarrow \textbf{R}$$ rationalizes a preference relation $$\preceq$$ on X if for every $$x, y \in X$$, $$u(x)\leq u(y)$$ if and only if $$x\preceq y$$. If u rationalizes $$\preceq$$, then this implies $$\preceq$$ 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
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 $$u : X \rightarrow \textbf{R}$$ 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 $$L : X \rightarrow [0,1] $$ 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 $$L_1 \geq L_2 \geq L_3$$, then there exists a $$0 \leq p \leq 1$$ 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.

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.

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