Utility
In economics, utility is a measure of a certain person's satisfaction from a certain state of the world.
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.
In this situation, someone who selects the most preferred alternative must also choose one that maximizes the associated utility function.
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.
The vertical and horizontal axes represent an individual's consumption of commodity Y and X 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.
These 'revealed preferences', as termed by Paul Samuelson, were revealed e.g. in people's willingness to pay:
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.
[4]: 78 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.
A notable exception is in the context of analyzing choice with conditions of risk (see below).
In order to simplify calculations, various alternative assumptions have been made concerning details of human preferences, and these imply various alternative utility functions such as: Most utility functions used for modeling or theory are well-behaved.
[10] Rational individuals only consume additional units of goods if it increases the marginal utility.
However, the law of diminishing marginal utility means an additional unit consumed brings a lower marginal utility than that carried by the previous unit consumed.
For example, drinking one bottle of water makes a thirsty person satisfied; as the consumption of water increases, he may feel begin to feel bad which causes the marginal utility to decrease to zero or even become negative.
Marginal rate of substitution is the slope of the indifference curve, which measures how much an individual is willing to switch from one good to another.
(Analysis of international survey data during the 21st century has shown that insofar as utility represents happiness, as for utilitarianism, it is indeed proportional to log of income.)
By making some reasonable assumptions about the way choices behave, von Neumann and Morgenstern showed that if an agent can choose between the lotteries, then this agent has a utility function such that the desirability of an arbitrary lottery can be computed as a linear combination of the utilities of its parts, with the weights being their probabilities of occurring.
In more formal language: A von Neumann–Morgenstern utility function is a function from choices to the real numbers: which assigns a real number to every outcome in a way that represents the agent's preferences over simple lotteries.
A variety of generalized expected utility theories have arisen, most of which omit or relax the independence axiom.
An indirect utility function gives the optimal attainable value of a given utility function, which depends on the prices of the goods and the income or wealth level that the individual possesses.
The boundedness represents the fact that beyond a certain amount money ceases being useful at all, as the size of any economy at that time is itself bounded.
The asymmetry about the origin represents the fact that gaining and losing money can have radically different implications both for individuals and businesses.
The non-linearity of the utility function for money has profound implications in decision-making processes: in situations where outcomes of choices influence utility by gains or losses of money, which are the norm for most business settings, the optimal choice for a given decision depends on the possible outcomes of all other decisions in the same time-period.
If only considers prices and quantities of two goods in one bundle, a budget constraint could be formulated as
However, as they have budget constraints, a change of price would affect the quantity of demand.
[12]: 48 Robinson also stated that because the theory assumes that preferences are fixed this means that utility is not a testable assumption.
[13][unreliable source] This criticism is similar to that of the philosopher Hans Albert who argued that the ceteris paribus (all else equal) conditions on which the marginalist theory of demand rested rendered the theory itself a meaningless tautology, incapable of being tested experimentally.
[14][unreliable source] In essence, a curve of demand and supply (a theoretical line of quantity of a product which would have been offered or requested for given price) is purely ontological and could never have been demonstrated empirically[dubious – discuss].
[17] There are many empirical works trying to estimate the form of utility functions of agents with respect to money.
