Everything about Utility totally explained
In
economics,
utility is a measure of the relative satisfaction from or desirability of consumption of
goods. Given this measure, one may speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior in terms of attempts to increase one's utility. For illustrative purposes, changes in utility are sometimes expressed in units called
utils.
The doctrine of
utilitarianism saw the maximization of utility as a moral criterion for the organization of society. According to utilitarians, such as
Jeremy Bentham (1748-1832) and
John Stuart Mill (1806-1876), society should aim to maximize the total utility of individuals, aiming for "the greatest happiness for the greatest number".
In neoclassical economics,
rationality is precisely defined in terms of imputed utility-maximizing behavior under economic constraints. As a hypothetical behavioral measure, utility doesn't require attribution of mental states suggested by "happiness", "satisfaction", etc.
Utility is applied by economists in such constructs as the
indifference curve, which plots the combination of commodities that an individual or a society requires to maintain a given level of satisfaction. Individual utility and social utility can be construed as the
dependent variable of a utility function (such as an indifference curve
map) and a
social welfare function respectively. When coupled with production or commodity constraints, these functions can represent
Pareto efficiency, such as illustrated by
Edgeworth boxes and
contract curves. Such efficiency is a central concept of
welfare economics.
Cardinal/ordinal utility
Economists distinguish between
cardinal utility and
ordinal utility. When cardinal utility is 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. An important example of a cardinal utility is the probability of achieving some target.
Utility functions of both sorts assign real numbers (utils) to members of a choice set. For example, suppose a cup of coffee 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 coffee of exactly the same amount is better than a cup of tea as the cup of tea is better than the cup of water.
It is tempting when dealing with cardinal utility to aggregate utilities across persons. The argument against this is that interpersonal comparisons of utility are suspect because there's no good way to interpret how different people value consumption bundles.
When ordinal utilities are used, differences in utils are treated as ethically or behaviorally meaningless: the utility values assigned encode a full behavioral ordering between members of a choice set, but nothing about
strength of preferences. In the above example, it would only be possible to say that coffee 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. As will be seen in subsequent sections, however, preference relations can often be rationalized as utility functions satisfying a variety of useful properties.
Ordinal utility functions are equivalent
up to monotone transformations, while cardinal utilities are equivalent up to positive linear transformations.
Utility functions
While
preferences are the conventional foundation of
microeconomics, it's convenient to represent preferences with a utility function and reason indirectly about preferences with utility functions. Let X be the
consumption set, the set of all mutually-exclusive packages the consumer could conceivably consume (such as an
indifference curve map without the indifference curves). The consumer's
utility function ranks each package in the consumption set. If u(x) ≥ u(y) (x
R y), then the consumer strictly prefers x to y or is indifferent between them.
For example, suppose a consumer's consumption set is X = assigns a real number to every element of the outcome space in a way that captures the agent's preferences over both simple and compound lotteries (put in category-theoretic language,
induces a morphism between the category of preferences under uncertainty and the category of reals). The agent will prefer a lottery
to a lottery
if and only if the expected utility (iterated over compound lotteries if necessary) of
is greater than the expected utility of
.
Restricting to the discrete choice context, let
be a simple lottery such that
, where
is the probability that
is won. We may also consider compound lotteries, where the prizes are themselves simple lotteries.
The expected utility theorem says that a von Neumann-Morgenstern utility function exists if and only if the agent's
preference relation on the space of simple lotteries satisfies four axioms: completeness, transitivity, convexity/continuity (also called the Archimedean property), and independence.
Completeness and transitivity are discussed supra. The Archimedean property says that for simple lotteries
, then there exists a
such that the agent is indifferent between
and the compound lottery mixing between
and
with probability
and
, respectively. Independence means that if the agent is indifferent between simple lotteries
and
, the agent is also indifferent between
mixed with an arbitrary simple lottery
with probability
and
mixed with
with the same probability
.
Independence is probably the most controversial of the axioms. A variety of
generalized expected utility theories have arisen, most of which drop or relax the independence axiom.
Utility of money
One of the most common uses of a utility function, especially in
economics, is the utility of money. The utility function for money is a nonlinear function that's
bounded and asymmetric about the origin. These properties can be derived from reasonable assumptions that are generally accepted by
economists and
decision theorists, especially proponents of
rational choice theory. The utility function is
concave in the positive region, reflecting the phenomenon of
diminishing marginal utility. The boundedness reflects the fact that beyond a certain point money ceases being useful at all, as the size of any economy at any point in time is itself bounded. The asymmetry about the origin reflects the fact that gaining and losing money can have radically different implications both for individuals and businesses. The nonlinearity of the utility function for money has profound implications in decision making processes: in situations where outcomes of choices influence utility through gains or losses of money, which are the norm in most business settings, the optimal choice for a given decision depends on the possible outcomes of all other decisions in the same time-period.
Discussion and criticism
Different value systems have different perspectives on the use of utility in making
moral judgments. For example,
Marxists,
Kantians, and certain
libertarians (such as
Nozick) all believe utility to be irrelevant as a moral standard or at least not as important as other factors such as natural
rights, law, conscience and/or religious doctrine. It is debatable whether any of these can be adequately represented in a system that uses a utility model.
Further Information
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