In
economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
and
finance, exponential utility is a specific form of the
utility function
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
, used in some contexts because of its convenience when
risk
In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environm ...
(sometimes referred to as uncertainty) is present, in which case
expected utility The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on the ...
is maximized. Formally, exponential utility is given by:
:
is a variable that the economic decision-maker prefers more of, such as consumption, and
is a constant that represents the degree of risk preference (
for
risk aversion
In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more c ...
,
for risk-neutrality, or
for
risk-seeking). In situations where only
risk aversion
In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more c ...
is allowed, the formula is often simplified to
.
Note that the additive term 1 in the above function is mathematically irrelevant and is (sometimes) included only for the aesthetic feature that it keeps the range of the function between zero and one over the domain of non-negative values for ''c''. The reason for its irrelevance is that maximizing the expected value of utility
gives the same result for the choice variable as does maximizing the expected value of
; since expected values of utility (as opposed to the utility function itself) are interpreted
ordinally instead of
cardinally, the range and sign of the expected utility values are of no significance.
The exponential utility function is a special case of the
hyperbolic absolute risk aversion In finance, economics, and decision theory, hyperbolic absolute risk aversion (HARA) (Chapter I of his Ph.D. dissertation; Chapter 5 in his ''Continuous-Time Finance'').Ljungqvist & Sargent, Recursive Macroeconomic Theory, MIT Press, Second Edition ...
utility functions.
Risk aversion characteristic
Exponential utility implies
constant absolute risk aversion (CARA), with coefficient of absolute risk aversion equal to a constant:
:
In the standard model of one risky asset and one risk-free asset,
for example, this feature implies that the optimal holding of the risky asset is independent of the level of initial wealth; thus on the margin any additional wealth would be allocated totally to additional holdings of the risk-free asset. This feature explains why the exponential utility function is considered unrealistic.
Mathematical tractability
Though
isoelastic utility
In economics, the isoelastic function for utility, also known as the isoelastic utility function, or power utility function, is used to express utility in terms of consumption or some other economic variable that a decision-maker is concerned with ...
, exhibiting
constant ''relative'' risk aversion (CRRA), is considered more plausible (as are other utility functions exhibiting decreasing absolute risk aversion), exponential utility is particularly convenient for many calculations.
Consumption example
For example, suppose that consumption ''c'' is a function of labor supply ''x'' and a random term
: ''c'' = ''c''(''x'') +
. Then under exponential utility,
expected utility The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on the ...
is given by:
:
where E is the
expectation operator. With
normally distributed noise, i.e.,
:
E(''u''(''c'')) can be calculated easily using the fact that
:
Thus
:
Multi-asset portfolio example
Consider the
portfolio allocation problem of maximizing expected exponential utility