Utility

It was recognized that utility could not be measured or observed directly, so instead economists devised a way to infer relative utilities from observed choice. These 'revealed preferences', as termed by Paul Samuelson, were revealed e.g. in people's willingness to pay:

Utility is assumed to be correlative to Desire or Want. 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

For financial applications, e.g. portfolio optimization, an investor chooses a financial portfolio which maximizes his/her own utility function, or, equivalently, minimizes his/her risk measure. For example, modern portfolio theory selects variance as a measure of risk; other popular theories are expected utility theory,[5] and prospect theory.[6] To determine a specific utility function for any given investor, one could design a questionnaire procedure with questions in the form: How much would you pay for x% chance of getting y? Revealed preference theory suggests a more direct method: observe a portfolio X* which an investor currently has, and then find a utility function/risk measure such that X* becomes an optimal portfolio.[7]

Cardinal utility states that the utilities obtained from consumption can be measured and ranked objectively and are representable by numbers.[8] There are fundamental assumptions of cardinal utility. Economic agents should be able to rank different bundles of goods based on their own preferences or utilities, and also sort different transitions of two bundles of goods.[9]

A cardinal utility function can be transformed to another utility function by a positive linear transformation (multiplying by a positive number, and adding some other number); however, both utility functions represent the same preferences.[10]

When cardinal utility is assumed, the magnitude of utility differences is treated as an ethically or behaviorally significant quantity. 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. With cardinal utility, it can 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. Formally, this means that if a person has a cup of tea, he or she would be willing to take any bet with a probability, p, greater than .5 of getting a cup of juice, with a risk of getting a cup of water equal to 1-p. One cannot conclude, however, that the cup of tea is two thirds of the goodness of the cup of juice, because this conclusion would depend not only on magnitudes of utility differences, but also on the "zero" of utility. For example, if the "zero" of utility was located at -40, then a cup of orange juice would be 160 utils more than zero, a cup of tea 120 utils more than zero. Cardinal utility can be considered as the assumption that utility can be measured by quantifiable characteristics, such as height, weight, temperature, etc.

Neoclassical economics has largely retreated from using cardinal utility functions as the basis of economic behavior. A notable exception is in the context of analyzing choice with conditions of risk (see below).

Sometimes cardinal utility is used to aggregate utilities across persons, to create a social welfare function.

Instead of giving actual numbers over different bundles, ordinal utilities are only the rankings of utilities received from different bundles of goods or services.[8] For example, ordinal utility could tell that having two ice creams provide a greater utility to individuals in comparison to one ice cream but could not tell exactly how much extra utility received by the individual. Ordinal utility, it does not require individuals to specify how much extra utility he or she received from the preferred bundle of goods or services in comparison to other bundles. They are only required to tell which bundles they prefer.

When ordinal utilities are used, differences in utils (values assumed by the utility function) are treated as ethically or behaviorally meaningless: the utility index encodes a full behavioral ordering between members of a choice set, but tells nothing about the related strength of preferences. For the above example, it would only be possible to say that juice is preferred to tea to water. Thus, ordinal utility utilizes comparisons, such as "preferred to", "no more", "less than", etc.

If a function ${\displaystyle u(x)}$ is ordinal, it is equivalent to the function ${\displaystyle u(x)^{3}}$, because taking the 3rd power is an increasing monotone (or monotonic) transformation. This means that the ordinal preference induced by these functions is the same (although they are two different functions). In contrast, if ${\displaystyle u(x)}$ is cardinal, it is not equivalent to ${\displaystyle u(x)^{3}}$.

For many decision models, utility functions are determined by the problem formulation. For some situations, the decision maker's preference must be elicited and represented by a utility (or objective) scalar-valued function. The methods existing for constructing such functions are collected in the proceedings of two dedicated conferences.[11][12] The mathematical foundations for the most common types of utility functions — quadratic and additive — were laid down by Gerard Debreu,[13][14] and the methods for their construction from both ordinal and cardinal data, in particular from interviewing a decision maker, were developed by Andranik Tangian.[15][16]

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:

• CES (constant elasticity of substitution).
• Isoelastic utility
• Exponential utility
• Quasilinear utility
• Homothetic preferences
• Stone–Geary utility function
• Gorman polar form
  • Greenwood–Hercowitz–Huffman preferences
  • King–Plosser–Rebelo preferences
• Hyperbolic absolute risk aversion

Most utility functions used for modeling or theory are well-behaved. They are usually monotonic and quasi-concave. However, it is possible for preferences not to be representable by a utility function. An example is lexicographic preferences which are not continuous and cannot be represented by a continuous utility function.[17]

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 less marginal utility than that brought 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. Furthermore, this is also used to analyze progressive taxes as the greater taxes can result in the loss of utility.

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. Using a mathematic equation, ${\displaystyle MRS=-\operatorname {d} \!x_{2}/\operatorname {d} \!x_{1}}$keeping U (x1,x2) constant. Thus, MRS is how much an individual is willing to pay for consuming a greater amount of x1.

MRS is related to marginal utility. The relationship between marginal utility and MRS is: ${\displaystyle MRS={\frac {MU_{1}}{MU_{2}}}}$[18]

Von Neumann and Morgenstern addressed situations in which the outcomes of choices are not known with certainty, but have probabilities associated with them.

A notation for a lottery is as follows: if options A and B have probability p and 1 − p in the lottery, we write it as a linear combination:

${\displaystyle L=pA+(1-p)B}$

More generally, for a lottery with many possible options:

${\displaystyle L=\sum _{i}p_{i}A_{i},}$

where ${\displaystyle \sum _{i}p_{i}=1}$.

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.

This is termed the expected utility theorem. The required assumptions are four axioms about the properties of the agent's preference relation over 'simple lotteries', which are lotteries with just two options. Writing ${\displaystyle B\preceq A}$ to mean 'A is weakly preferred to B' ('A is preferred at least as much as B'), the axioms are:

1. completeness: For any two simple lotteries ${\displaystyle L}$ and ${\displaystyle M}$, either ${\displaystyle L\preceq M}$ or ${\displaystyle M\preceq L}$ (or both, in which case they are viewed as equally desirable).
2. transitivity: for any three lotteries ${\displaystyle L,M,N}$, if ${\displaystyle L\preceq M}$ and ${\displaystyle M\preceq N}$, then ${\displaystyle L\preceq N}$.
3. convexity/continuity (Archimedean property): If ${\displaystyle L\preceq M\preceq N}$, then there is a ${\displaystyle p}$ between 0 and 1 such that the lottery ${\displaystyle pL+(1-p)N}$ is equally desirable as ${\displaystyle M}$.
4. independence: for any three lotteries ${\displaystyle L,M,N}$ and any probability p, ${\displaystyle L\preceq M}$ if and only if ${\displaystyle pL+(1-p)N\preceq pM+(1-p)N}$. Intuitively, if the lottery formed by the probabilistic combination of ${\displaystyle L}$ and ${\displaystyle N}$ is no more preferable than the lottery formed by the same probabilistic combination of ${\displaystyle M}$ and ${\displaystyle N,}$ then and only then ${\displaystyle L\preceq M}$.

Axioms 3 and 4 enable us to decide about the relative utilities of two assets or lotteries.

In more formal language: A von Neumann–Morgenstern utility function is a function from choices to the real numbers:

${\displaystyle u\colon X\to \mathbb {R} }$

which assigns a real number to every outcome in a way that represents the agent's preferences over simple lotteries. Using the four assumptions mentioned above, the agent will prefer a lottery ${\displaystyle L_{2}}$ to a lottery ${\displaystyle L_{1}}$ if and only if, for the utility function characterizing that agent, the expected utility of ${\displaystyle L_{2}}$ is greater than the expected utility of ${\displaystyle L_{1}}$:

${\displaystyle L_{1}\preceq L_{2}{\text{ iff }}u(L_{1})\leq u(L_{2})}$.

Of all the axioms, independence is the most often discarded. A variety of generalized expected utility theories have arisen, most of which omit or relax the independence axiom.

Castagnoli and LiCalzi (1996) and Bordley and LiCalzi (2000) provided another interpretation for Von Neumann and Morgenstern's theory. Specifically for any utility function, there exists a hypothetical reference lottery with the expected utility of an arbitrary lottery being its probability of performing no worse than the reference lottery. Suppose success is defined as getting an outcome no worse than the outcome of the reference lottery. Then this mathematical equivalence means that maximizing expected utility is equivalent to maximizing the probability of success. In many contexts, this makes the concept of utility easier to justify and to apply. For example, a firm's utility might be the probability of meeting uncertain future customer expectations.[20][21][22][23]

One use of the indirect utility concept is the notion of the utility of money. The (indirect) utility function for money is a nonlinear function that is bounded and asymmetric about the origin. The utility function is concave in the positive region, representing the phenomenon of diminishing marginal utility. 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.[24]

Rational consumers wish to maximise their utility. However, as they have budget constraints, a change of price would affect the quantity of demand. There are two factors could explain this situation:

• Purchasing Power. Individuals obtain greater purchasing power when the price of a good decreases. The reduction of the price allows individuals to increase their savings so they could afford to buy other products.
• Substitution Effect. If the price of good A decreases, then the good becomes relatively cheaper with respect to its substitutes. Thus, individuals would consume more of good A as the utility would increase by doing so.