Utility

: This article is about "utility" in economics and in game theory. For utility companies and similar concepts, see public utility. For utilities in computers, see computer software. See also Utility (patent)

Utility functions

While preferences are the conventional foundation of

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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.

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The consumer's utility function u : X ightarrow extbf R assigns a happiness score to each package in the consumption set. If u(x) > u(y), then the consumer strictly prefers x to y.

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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.

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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 extbf R^L_+, and each package x in extbf 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 = extbf 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.

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A utility function u : X ightarrow extbf{R} rationalizes a preference relation

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for every x, y in X, u(x)

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

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