Utility

In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings.

In a normative context, utility refers to a goal or objective that we wish to maximize, i.e., an objective function. This kind of utility bears a closer resemblance to the original utilitarian concept, developed by moral philosophers such as Jeremy Bentham and John Stuart Mill.
In a descriptive context, the term refers to an apparent objective function; such a function is revealed by a person's behavior, and specifically by their preferences over lotteries, which can be any quantified choice.

The relationship between these two kinds of utility functions has been a source of controversy among both economists and ethicists, with most maintaining that the two are distinct but generally related.

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 must also choose one 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, and that the preference order is transitive.
Suppose the set of alternatives is not finite. In that case, a continuous utility function exists 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 individual demand curves derivation.
In an indifference curve, the vertical and 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 utility value.
Individual 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 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 = , and his utility function is u = 0, u = 1, u = 2, u = 5, u = 2 and u = 4. Then this consumer prefers 1 orange to 1 apple but prefers one of each to 2 oranges.
In micro-economic models, there is usually a finite set of L commodities, and a consumer may consume an arbitrary amount of each commodity. This gives a consumption set of, and each package is a vector containing the amounts of each commodity. For the example, there are two commodities: apples and oranges. If we say apples are the first commodity, and oranges the second, then the consumption set is and u = 0, u = 1, u = 2, u = 5, u = 2, u = 4 as before. For u to be a utility function on X, however, it must be defined for every package in X, so now the function must be defined for fractional apples and oranges too. One function that would fit these numbers is
Preferences have three main properties:

Completeness

Assume an individual has two choices, A and B. By ranking the two choices, one and only one of the following relationships is true: an individual strictly prefers A ; an individual strictly prefers B ; an individual is indifferent between A and B.
Either a ≥ b OR b ≥ a for all

Transitivity

Individuals' preferences are consistent over bundles. If an individual prefers bundle A to bundle B and bundle B to bundle C, then it can be assumed that the individual prefers bundle A to bundle C.
.

Non-satiation or monotonicity

If bundle A contains all the goods that a bundle B contains, but A also includes more of at least one good than B. The individual prefers A over B. If, for example, bundle A = , and bundle B = , then A is preferred over B.

Revealed preference

It was recognized that utility could not be measured or observed directly, so instead economists devised a way to infer relative utilities from observed choice. These 'revealed preferences', as termed by Paul Samuelson, were revealed e.g. in people's willingness to pay:

Utility is assumed to be correlative to Desire or Want. It has been argued already that desires cannot be measured directly, but only indirectly, by the outward phenomena which they cause: and that in those cases with which economics is mainly concerned the measure is found by the price which a person is willing to pay for the fulfillment or satisfaction of his desire.

Utility functions, expressing utility as a function of the amounts of the various goods consumed, are treated as either cardinal or ordinal, depending on whether they are or are not interpreted as providing more information than simply the rank ordering of preferences among bundles of goods, such as information concerning the strength of preferences.

Cardinal

Cardinal utility states that the utilities obtained from consumption can be measured and ranked objectively and are representable by numbers. There are fundamental assumptions of cardinal utility. Economic agents should be able to rank different bundles of goods based on their preferences or utilities and sort different transitions between two bundles of goods.
A cardinal utility function can be transformed to another utility function by a positive linear transformation ; however, both utility functions represent the same preferences.
When cardinal utility is assumed, the magnitude of utility differences is treated as an ethically or behaviorally significant quantity. For example, suppose a cup of orange juice 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. With cardinal utility, it can be concluded that the cup of orange juice is better than the cup of tea by the same amount by which the cup of tea is better than the cup of water. This means that if a person has a cup of tea, they would be willing to take any bet with a probability, p, greater than.5 of getting a cup of juice, with a risk of getting a cup of water equal to 1-p. One cannot conclude, however, that the cup of tea is two-thirds of the goodness of the cup of juice because this conclusion would depend not only on magnitudes of utility differences but also on the "zero" of utility. For example, if the "zero" of utility were located at -40, then a cup of orange juice would be 160 utils more than zero, a cup of tea 120 utils more than zero. Cardinal utility can be considered as the assumption that quantifiable characteristics, such as height, weight, temperature, etc can measure utility.
Neoclassical economics has largely retreated from using cardinal utility functions as the basis of economic behavior. A notable exception is in the context of analyzing choice with conditions of risk.
Sometimes cardinal utility is used to aggregate utilities across persons, to create a social welfare function.

Ordinal

Instead of giving actual numbers over different bundles, ordinal utilities are only the rankings of utilities received from different bundles of goods or services. For example, ordinal utility could tell that having two ice creams provide a greater utility to individuals in comparison to one ice cream but could not tell exactly how much extra utility received by the individual. Ordinal utility, it does not require individuals to specify how much extra utility they received from the preferred bundle of goods or services in comparison to other bundles. They are only needed to tell which bundles they prefer.
When ordinal utilities are used, differences in utils are treated as ethically or behaviorally meaningless: the utility index encodes a full behavioral ordering between members of a choice set, but tells nothing about the related strength of preferences. For the above example, it would only be possible to say that juice is preferred to tea to water. Thus, ordinal utility utilizes comparisons, such as "preferred to", "no more", "less than", etc.
If a function is ordinal and non-negative, it is equivalent to the function, because taking the square is an increasing monotone transformation. This means that the ordinal preference induced by these functions is the same. In contrast, if is cardinal, it is not equivalent to.