Utility

The word utility describes the fact of being used for something, of being usable, quite simply useful.

• One speaks about public utility , when the realization of a Ouvrage (road, bridge, factory) or other is declared thus, following an investigation, with the favorable results, near the Citoyen S concerned.

Economy

At the beginning, in economy, the concept of utility was primarily related to the taking risk. The “theory to the measure of the risk” of Daniel Bernoulli (1700 - 1782), and in this one, the Paradoxe of Saint-Pétersbourg were at the economic base of the theories and financial of the Aversion to the risk, the Allowance for risk and the Utilité

The concept of utility became more largely a measurement of the wellbeing or Satisfaction obtained by consumption, or at least obtaining, of a good or a service. It is related to the concept of Besoin

The concept is used in the functions of utility, functions of social, optimum utility within the meaning of Wilfredo Pareto, boxes of Edgeworth. It is a central concept of the economics of welfare. Pareto did not like the term besides, that he considered charged with too many considerations morals. He proposed to use that ophelimity , étymologiquement are equivalent.

Utilitarianism and economy

The Utilitarisme is a moral philosophy which maintains the complex reports/ratios of cousinage with the economy. Jeremy Bentham (1748-1832) poses that the men are beings which seek the pleasure and which the promotion of greatest happiness should be the moral criterion of the good. John Stuart Mill, in her book Utilitarianism (1861), insists on the fact that utilitarianism is an ethical hedonism in the sense that an individual action is moral if it takes as criterion the greatest happiness of the greatest number and not the individual interest.

The closest links between utilitarian philosophy and the economy must be required in the economics of welfare. On the other hand, the roots of the revolution marginalist, initiated by Gossen, Jevons, Menger and Walras, can be connected in no obvious way to utilitarian philosophy.

Cardinal utility and ordinal

Within the neo-classic school, a central problem of the Théorie of the consumer is the construction of a function of request which can be it parallel of the function of offer resulting from the Théorie of the producer. This difficulty was solved in two times, initially by supposing a cardinal utility , measurable and comparable between the goods, then a ordinal utility , slightly less less constraining.

Cardinal utility

See also: cardinal Theory of the utility

The precursors of the revolution marginalist (Walras, Jevons, Menger) conceived the utility like the feeling of pleasure associated with consumption with a good. They defended the idea of a cardinal measurement of the utility by supposing that the consuming was able to give an evaluation of the utility which any combination of goods brought to him. This faculty was the exact mirror of the capacity supposed of the producing to predict the production for any combination of inputs given, and simplified the analysis considerably. For teaching reasons, it was also used, with some reserves, by Alfred Marshall.

For example, if the consumption of a $q\_A$ quantity of a good has gives a satisfaction of 100 and one $q\_B$ quantity of a good B gives a satisfaction of 10, $q\_A$ is equivalent to 10 times $q\_B$.

Ordinal utility

The example above illustrates the conceptual problem of the cardinal utility: there does not exist objective scale of the measurement of the utility. This is why Wilfredo Pareto, successor of Marshall proposed a formulation in terms of ordinal utility .

Within the framework of the ordinal utility, it is requested from the consumer to be able to reasonably classify the goods or baskets of goods according to the utility brought. It is enough for him thus to know if it prefers $q\_A$ with $q\_B$, $q\_B$ with $q\_A$ or if he is indifferent between the two. In mathematical terms, it is thus enough to be able to describe a complete préordre on the space of the baskets of goods: the relation preferably must thus be complete (one can compare any couple of baskets), reflexive (a basket is preferred with itself) and transitive (if the basket has is preferred with the basket B and the basket B with the basket C , then has is preferred with C ).

One finds like principal tenors of this ordinal design Wilfredo Pareto, Eugen Slutsky, taken again by Paul Samuelson and John Hicks.

Limits of the concept

That it is of the ordinal utility or the cardinal utility, the concept of utility would be altogether only one artifice to give a simple formulation to complex behaviors. The experiments of experimental economy indeed show that even on a restricted whole of goods, the agents are often unable to compare all the baskets two to two (non-complétude), and seldom propose a classification which respects transitivity.

However, the power of this tool as description of the behaviors is such as it remains very largely used.

Function of utility

The cardinal utility directly provides an evaluation of the utility of a basket of goods, which makes it possible to treat them like a mathematical size. For the passage to the ordinal utility, it is necessary to build an object which makes it possible to bring back each basket to a number reflecting the relation preferably subjacent. It is the function of utility .

Construction

One thus builds thus a mathematical Fonction U energy of the space of the goods in $R^+$ such as $U\; (A)>U\; (B)$ implies that the basket has is preferred with the basket B . One can thus build curved of indifference gathering the baskets which leave indifferent the consumer when it compares them two to two. Because of complétude and transitivity , these curves can then be classified according to a total order.

The function of utility, by associating an index with each basket, is not single. If U is a function of utility for an individual, then $G\; \backslash \; circ\; U$ is also if G is a strictly increasing function of $R^+$ in $R^+$ . So the ordinal utility is not comparable between the individuals, which makes impossible to deduce a social Utilité directly from it as wished it the utilitarian .

Properties

One in general supposes a certain number of properties to the function of utility in order to restrict itself with a probable class, and especially mathematically gérable, functions. Most of the time, one restricts infinitely a priori with derivable functions (functions of class $C^\; \{\backslash \; infty\}$). One can note that this restriction supposes the infinite divisibility of the well S consumed, which is less absurd than it appears to with it if it is considered that the consumption of these goods can be split in several units of time.

Decrease of the marginal utility

It is rather intuitive to suppose that for the near total of the goods, an increase in the quantity of a good in a basket increases or leaves unchanged the withdrawn utility of this basket. This is why one imposes on the function utility to be increasing in each one of his arguments:

$\backslash \; forall\; I,\; \backslash \; frac\; \{\backslash \; partial\; U\}\; \{\backslash \; partial\; x\_i\}\; \backslash \; geq\; 0$

On the other hand, one can also think that this increase is not independent of the quantity of this good already available in the basket. Thus, if the first beer mouthful gets an unutterable pleasure, the second is already less good, and so on, until arriving at the moment when the desire is dried up. That means that the utility of each new beer mouthful is lower than that of the preceding one: the marginal utility is decreasing .

$\backslash \; forall\; I,\; \backslash \; frac\; \{\backslash \; partial^2\; U\}\; \{\backslash \; partial^2\; x\_i\}\; \backslash \; Leq\; 0$

In order to avoid the solutions in corners in the problems of optimization, one in general supposes that the utility of the last consumed unit never becomes null, property known as of not-saturation :

$\backslash \; forall\; I,\; \backslash \; frac\; \{\backslash \; partial\; U\}\; \{\backslash \; partial\; x\_i\}\; >\; 0$

The cogency of this assumption rests on the Rationalité of the agent: if the utility is well defined, the agent will never waste its time to consume something which is detrimental for him or which does not bring anything to him.

One can finally restrict oneself with the fields where the marginal utility is strictly decreasing:

There are then functions of concave utility S, and thus of the curved of indifference defining convex units.

Elasticity of substitution

In terms of Theory of the consumer, the elasticity of substitution plays a fundamental role in the analysis. Sometimes this is why one requests from the function utility to present a elasticity of constant substitution : for any couple of baskets of goods, a reduction of 1% of the quantity of good has can be compensated by the increase in $c\; \backslash \; \%$ of the quantity of good B , where $c$ is a constant independent of the couple of baskets. One then speaks about function THESE ( Constant Elasticity off Substitution ).

These functions have the property to translate a constant Aversion with the risk on behalf of the agent.

Function additivement separable

Even in the class of the concave functions of utility, one can have very complex functional forms. In order to obtain analytical solutions with the programs of optimization, one often uses functions of utility additivement separable :

$\backslash \; forall\; (I,\; J),\; U\; (x\_i,\; x\_j)\; =u\_i\; (x\_i)\; +u\_j\; (x\_j)$

Such a formulation supposes that the goods are not complementary between them, or that the complementary goods (a computer and its operating system) are gathered in a composite Bien.

Use

In practice, Alfred Marshall points out that the cardinal utility does not pose an insurmountable problem. He notices indeed that if the agents are sufficiently rational so that the concept of ordinal utility has a direction, one can also suppose that them Propension to pay (the maximum price which they are ready to pay for a given basket of goods) provides a good measure of the utility that they withdraw some, which also allows the comparison between the agents.

Utility into dubious

In Theory of probability was developed a concept similar to that of the function of utility. It is still a question of a function which associates a number with a couple (évemenent, probability of this event), in order to overcome the difficulties related to the practical application of the mathematical Espérance.

The paradox of the hope

Whereas it constitutes a good tool of forecast, the mathematical Espérance fails has well to describe the behavior of the agents vis-a-vis a lottery. Thus, an agent downpour with the risk prefers to gain 1000€ immediately rather than to play play where it has a chance on 100 to gain 100.000€, whereas the hope of both is identical. The economists systematize this argument in terms of aversion to the risk and decreasing marginal utility to explain this phenomenon.

Conversely, the player with the Lotto shows that he prefers to exploit with a chance 10 million to gain 5 million euros that to keep the euro which the ticket costs him. Emile Borel developed a sales leaflet explaining why this player could be right contrary to appearances: the loss of one euro will hardly change its life; the profit of five million has beautiful improbable being, it will transform this one in a qualitative way and not simply quantitative.

Utility of von Neumann Morgenstern

Proposed by John von Neumann and Oskar Morgenstern in Theory off Ranges and Economic Behavior (1944), this formulation of the utility was very generally adopted in the modeling of choice when the events are probabilisable S.

Instead of defining the function of utility well on a space of S, one defines it on a space of Loterie S, and one considers those which are linear S compared to the lotteries and confused with the functions of usual utilities on the subspace of the unquestionable lotteries: for a lottery $L=\; \backslash \; \{(p,\; Q),\; (1-p,\; Q\text{'})\; \backslash \}$,

$U\; (L)\; =U\; (\backslash \; \{(p,\; Q),\; (1-p,\; Q\text{'})\; \backslash \})\; =pu\; (Q)\; +\; (1-p)\; U\; (Q\text{'})$

Function of social utility

In a way similar to what is made for the consumer, it is trying to define a function of social utility which reflects the preferences of the company as a whole. Such a function was often postulated in the economics of welfare or the political economy, without justification of its construction.

In practice, the construction of such a function encounters multiple difficulties. On the one hand, it rests crucially on the assumption of cardinal utilities (see will infra ). In addition, the construction of a function preferably collective, from where to deduce a function from collective utility, runs up against the Théorème of impossibility of Arrow, showing impossibility of building a relation preferably collective having the properties wanted without supposing that a single individual imposes his preferences on all the others.

Indirect utility

See: Theory of the consumer

The function of indirect utility is a part of the dual problem of the consuming . For a level of initial resources and a vector of price given, the function of indirect utility gives the maximum value of the utility atteignable by this agent.

See too

Bonds

• Microéconomie
• Preferences
• cardinal Theory of the utility
• Theory of the consumer
• Theorem of impossibility of Arrow

Random links:

World cup of hockey 1996 | 1891 in football | Prelado | Gold Adidas shoe | Weapon sapphire | RS-465 | Shithead_(jeu_de_carte)

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org