As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. The term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a utility function that represents a consumer's ordinal preferences over a choice set, but is not necessarily comparable across consumers or possessing a cardinal interpretation. This concept of utility is personal and based on choice rather than on pleasure received, and so requires fewer behavioral assumptions than the original concept.
Utility function
Consider a set of alternatives among which a person has a preference ordering. A utility function represents that ordering if it is possible to assign a real number to each alternative in such a manner that alternative a is assigned a number greater than alternative b if and only if the individual prefers alternative a to alternative b. In this situation someone who selects the most preferred alternative is necessarily also selecting the alternative that maximizes the associated utility function.
Suppose James has utility function such that is the number of apples and is the number of chocolates. Alternative A has apples and chocolates; alternative B has apples and chocolates. Putting the values into the utility function yields for alternative A and for B, so James prefers alternative B. In general economic terms, a utility function ranks preferences concerning a set of goods and services.
Gérard Debreu derived the conditions required for a preference ordering to be representable by a utility function. For a finite set of alternatives, these require only that the preference ordering is complete (so the individual is able to determine which of any two alternatives is preferred or that they are indifferent), and that the preference order is transitive.
If the set of alternatives is not finite (for example because even if the number of goods is finite, the quantity chosen can be any real number on an interval) there exists a continuous utility function representing a consumer's preferences if and only if the consumer's preferences are complete, transitive and continuous.
Applications
Utility can be represented through sets of indifference curve, which are level curves of the function itself, and which plot the combination of commodities that an individual would accept to maintain a given level of satisfaction. Combining indifference curves with budget constraints allows for derivation of individual demand curves.
A diagram of a general indifference curve is shown below (Figure 1). The vertical axes and the horizontal axes represent an individual's consumption of commodity Y and X respectively. All the combinations of commodity X and Y along the same indifference curve are regarded indifferently by individuals, which means all the combinations along an indifference curve result in the same value of utility.
Individual utility and social utility can be construed as the value of a utility function and a social welfare function respectively. When coupled with production or commodity constraints, by some assumptions these functions can be used to analyze Pareto efficiency, such as illustrated by Edgeworth boxes in contract curves. Such efficiency is a major concept in welfare economics.
Preference
While preferences are the conventional foundation of choice theory in microeconomics, it is often convenient to represent preferences with a utility function. Let X be the consumption set, the set of all mutually-exclusive baskets the consumer could conceivably consume. The consumer's utility function ranks each possible outcome in the consumption set. If the consumer strictly prefers x to y or is indifferent between them, then .
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 his utility function is u(nothing) = 0, u(1 apple) = 1, u(1 orange) = 2, u(1 apple and 1 orange) = 5, u(2 apples) = 2 and u(2 oranges) = 4. Then this consumer prefers 1 orange to 1 apple, but prefers one of each to 2 oranges.