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Utility 

Utility
This article is about the economic concept. For other uses, see Utility (disambiguation).
In economics,utilityis a representation of preferences over some set of goods and services. Preferences have a (continuous) utility representation so long as they are transitive, complete, and continuous.
Utility is usually applied by economists in such constructs as the indifference curve, which plot the combination of commodities that an individual or a society would accept to maintain a given level of satisfaction. 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, under some assumptions, these functions can be used to analyze Pareto efficiency, such as illustrated by Edgeworth boxes in contract curves. Such efficiency is a central concept in welfare economics.
In finance, utility is applied to generate an individual's price for an asset called the indifference price. Utility functions are also related to risk measures, with the most common example being the entropic risk measure.
Quantifying utility
It was recognized that utility could not be measured or observed directly, so instead economists devised a way to infer underlying relative utilities from observed choice. These 'revealed preferences', as they were named by Paul Samuelson, were revealed e.g. in people's willingness to pay:
Utility is taken to be correlative to Desire or Want. It has been already argued that desires cannot be measured directly, but only indirectly, by the outward phenomena to which they give rise: and that in those cases with which economics is chiefly concerned the measure is found in the price which a person is willing to pay for the fulfilment or satisfaction of his desire. (Marshall 1920:78)
Cardinal and ordinal utility
For more details on this topic, see cardinal utility.
Economists distinguish between cardinal utility and ordinal utility. When cardinal utility is http://www.iwebtool.com/pagerank_checker?domain=http://www.ilapak.pl 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.
Utility functions of both sorts assign a ranking to members of a choice set. 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. When speaking of cardinal utility, it could be concluded that the cup of orange juice is better than the cup of tea by exactly the same amount by which the cup of tea is better than the cup of water. One is not entitled to conclude, however, that the cup of tea is two thirds as good as the cup of juice, because this conclusion would depend not only on magnitudes of utility differences, but also on the "zero" of utility.
It is tempting when dealing with cardinal utility to aggregate utilities across persons. The argument against this is that interpersonal comparisons of utility are meaningless because there is no simple way to interpret how different people value consumption bundles.[potrzebne źródło]
When ordinal utilities are used, differences in utils are treated as ethically or behaviorally meaningless: the utility index encode a full behavioral ordering between members of a choice set, but tells nothing about the relatedstrength of preferences. In the above example, it would only be possible to say that juice 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. However, preference relations can often be represented by utility functions satisfying several properties.
Ordinal utility functions are unique up to positive monotone transformations, while cardinal utilities are unique up to positive linear transformations.
Although preferences are the conventional foundation of microeconomics, it is often convenient to represent preferences with a utility function and analyze human behavior indirectly with utility functions. LetXbe theconsumption set, the set of all mutuallyexclusive baskets the consumer could conceivably consume. The consumer'sutility function[img]//upload.wikimedia.org/math/7/b/3/7b3f906d2f01fb048956761f8d08dab4.png"/> ranks each package in the consumption set. If the consumer strictly prefers xtoyor is indifferent between them, then , and each package andu(0,Â 0)Â =Â 0,u(1,Â 0)Â =Â 1,u(0,Â 1)Â =Â 2,u(1,Â 1)Â =Â 4,u(2,Â 0)Â =Â 2,u(0,Â 2)Â =Â 3 as before. Note that foruto be a utility function onÂ X, it must be defined for every package inÂ X.
A utility function on X iff for every implies , then this implies
More generally, for a lottery with many possible options:
to mean 'A is weakly preferred to B' ('A is preferred at least as much as B'), the axioms are:
 completeness: For any two simple lotteries , either (or both).
 transitivity: for any three lotteries and .
 convexity/continuity (Archimedean property): If between 0 and 1 such that the lottery .
 independence: for any three lotteries if and only if
which assigns a real number to every outcome in a way that captures the agent's preferences over simple lotteries. Under the four assumptions mentioned above, the agent will prefer a lottery if and only if the expected utility of :
is a morphism between the category of preferences with uncertainty and the category of reals as an additive group.
Of all the axioms, independence is the most often discarded. A variety of generalized expected utility theories have arisen, most of which drop or relax the independence axiom.
 CES (constant elasticity of substitution, orisoelastic) utility is one with constant relative risk aversion
 Exponential utility exhibits constant absolute risk aversion
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