Utility

A utility function is a mathematical function that is used in economics and, particularly, into the microeconomics to model the preferences of economic agents ( households, individuals ). It assigns any combination of goods is a number such that commodity combinations that are considered by the economic subject for better than others, get a higher number. This number then called the benefit of the particular type of product combination.

In modern microeconomic theory utility functions include only statements about the ranking of commodity combinations: Returns a property combination - a so-called bundle of goods - a higher value than others, so may it be only concluded that those from the corresponding budget summary " better" than this is; what are the numbers or how great the distance between them, it has no meaning. We call such utility functions as an ordinal utility functions because they only specify an ordering of the bundle of goods. The concept of ordinal utility function is based on a different theoretical foundation than the so-called cardinal utility functions, where also the difference between the benefit value of two goods is interpretable.

The concept of the utility function is used both directly in microeconomics and in the context of macro-economic issues.

• 2.1 Utility Concept and transformations of the utility function
• 2.2 existence of a utility function
• 2.3 Functional properties

• 4.1 Marginal Utility
• 4.2 Marginal Rate of Substitution

• 5.1 Quasi - linear utility function
• 5.2 Cobb -Douglas and CES utility function
• 5.3 Intertemporal utility function
• 5.4 Von Neumann -Morgenstern expected utility function 5.4.1 Risk aversion
• 5.4.2 HARA utility functions

Definition

In the following, each assumes only ordinal measurability of utility and the utility function is introduced, as it is constructed in the Financial theory.

Illustrative definition in the two-goods case

Is limited initially to simplify the scope of the bundle of goods on two goods, one can, for example, a bundle of goods A present, which is composed of two types of goods: Kiwi ( Good 1 ) and cherries ( Good 2 ). In A bundle of goods is now a certain amount Kiwi - designated - and a certain amount of cherries - designated included; to write for this bundle of goods shortly. Analogously, one imagines a second bundle of goods from B kiwi and cherries, which is represented accordingly by. With concrete values ​​can be thought of, for example, that, that is bundle of goods A are two kiwi and cherries contain six while. Assuming, as usual, that preferences are monotonic ( casually: "more is better " ), the budget B over A should prefer. There are endless Nutzenfuntionen which may reflect the preferences, since they only need to ensure that the function value at the point is greater than the. At the point For example, you could use a function with and. Negative values ​​are possible: Be a different utility function and respectively, then this utility function is consistent with the preferences of the household.

Analog to commodity combinations that likes the same household, will also get the same utility values ​​. For example, if the bundle of goods is well regarded as the same as the bundle of goods, then it must also apply to any utility function that.

Formal definition

In microeconomic theory, it is assumed that economic agents have preferences over them potentially available selection alternatives. Mathematically, such preferences ( which can be very general) represented as binary relations. For example, as agreed as preference - indifference relation. Be now and vectors of goods from a set of alternatives, then is expressed by that the bundle of goods is at least as good as or better than rated. In order to keep this information in the corresponding utility function, the function value of the utility function for the same or higher must be greater than that of there. This leads to the following exact definition:

Definition: A function is a utility function that reflects the preference - indifference relation, if each bundle of goods.

Utility function make it so certain indifference preference relations are equivalent functionally to represent (see also the section existence of a utility function in this article). Their advantage lies in the comparatively much simpler mathematical handling.

Just as in the analysis of preference relations also the indifference and strict preference can be derived from the preference - indifference relation. The definition of strict preference is: For two alternatives, and if and only if (1), but ( 2) not at the same time. If it is now in a utility function, the following applies with its definition above because of ( 1), and because of that ( 2) that not what precisely implies that if the strict preference actually. Analog is also for the indifference relation that it is thereby brought according to the above definition of just in the utility function expresses that for two equivalence considered bundle of goods. What is the distance between the function values ​​is or how high are the function values ​​themselves, is without significance.

Classification and properties

Benefit concept and transformations of the utility function

The for the above definition underlying interpretation is quite broadly defined in such a way that the concrete benefits values ​​are not interpretable in itself - it's the comparison of goods bundles only about how behave the two corresponding utility values ​​to each other, that is, whether a larger, equal or smaller than the other. This is based on the approach to interpret the measurability of utility as only ordinal. The benefit concept of modern household theory is based on this assumption, as in the preference relations is no more information contain (pairwise comparison of alternatives ). It is thus intuitively understood that utility functions in the sense defined above also can be transformed any positive strictly monotone, ie that contains the same information as if only strictly increasing in.

Conceivable - but not compatible with the above concept - however, are quite other types of utility functions. If you measure the benefits, for example, on a cardinal scale, it would be a transformation only permissible if it is positive affine if so. The more restrictive requirements of the cardinal scale, however, correspond to the extended possibilities of interpretation, for here it would be quite possible, from the fact that the utility value at the transition from commodity bundles to 10 increases, while increases in the transition from to 20 to conclude that the additional use of over twice as high as that of other.

If you measure the benefits on a ratio scale, it would be a transformation only permissible if it is positive linear, if so. Here could be the fact that the benefits of commodity bundles is twice as large as that of, conclude that the former bundle also twice as much benefit as the latter creates.

In the extreme case, no transformation allowed (absolute scale), in which case even the absolute benefit amount would be (for example) to interpret.

Existence of a utility function

In the above definitions already shows the close relationship between preference relation and utility function. Once you unlike the simplistic examples, the number of the considered bundle of goods can be greater, however, it may happen that the preferences of actors can no longer be represented by a utility function.

Representability by a utility function:

This is called a preference - indifference relation as rational if it is transitive 1) complete and 2). The first property ensures that the budget can make the effect a rating for each pair of bundle of goods, whether it one better than the other, or both the same place well. Transitivity assumes that there is no circular preferences If bundle of goods A is at least as good as B, and B is at least as good as C, then A is also at least as good as C, and that for any bundle of goods A, B and C. For the formal definition of the two properties are referenced to the article preference relation.

The reverse direction of the first property can be illustrated by means of two simple examples. If you look, for example, two bundle of goods and the budget and do not know ( or does not specify ), as to assess the bundle compared, then we can not know whether the following applies in a utility function, or. This illustrates that complete preferences for the construction of a utility function is essential. At the same time, one can imagine three commodity bundles for which it holds that ( ie " is better than, better and better "). This is non- transitive preferences, and in fact there is no corresponding utility function, since it is impossible that.

Functional properties

Based on the associated preference orderings can also be statements about the properties of a utility function meet.

Relationship between the properties of the preference relation and the properties of the utility function is constructed therefrom:

• Is strictly increasing if and only (and only if) the underlying preference - indifference relation satisfies the property of strict monotonicity.
• Quasikonkav is exactly then (and only then ), if the underlying preference - indifference relation is convex.
• Is even strictly quasikonkav exactly then (and only then ), if the underlying preference - indifference relation is strictly convex.

This is called a preference - indifference relation as a strictly monotonic if; as convex if and when strictly convex if. See more detail of item preference relation.

Indifference curve

Utility functions give as defined above to the level of utility that can generate certain bundle of goods. Considering the function from another page you can also specify a certain level of utility and ask for the goods bundles with which this can be achieved. This forms the basis for the concept of an indifference curve ( also benefit isoquant or iso- utility function ). Assuming a bundle of goods, then it is at a formal indifference curve to the set of all commodity bundles for which it holds that ( Indifferenzmenge to ).

In the two-goods case, indifference curves can quite easily visualize as shown opposite. The set of all possible bundle of goods is ( at any point in this area marks a specific combination of good 1 and good 2 ) between the horizontal and the vertical axis. On the indifference curve 2 are, for example, all points that the budget the same benefits as cause and B can be seen as, among other things, that the household is indifferent between C and B ( ie C and B is good place). Assuming as usual monotonicity of preferences in ( "more is better " ), then are indifference curves for a more higher level of utility, the further they are away from the origin - the bundle of goods on indifference curve 2 are therefore always better than those on curve 1

Mathematically, an indifference amount in the sense defined above a level set for the utility function. For example, a utility function, then include the bundle of goods, and points to the indifference curve for utility level 4, because.

From the property it also follows that indifference curves can not intersect. Had namely A and B are two really different indifference amounts and there would be a bundle of goods that is contained both in A and in B, then this would necessarily lead to a contradiction. By definition a Indifferenzmenge would in fact apply to all other goods bundle of A, that they ( 's because included in A) the same benefit as generate; for all other goods bundle of B in turn gölte the same ( as is contained in B), which leads to the fact that all commodity bundles in A and B have the same benefits as generate. But then the indifference amounts can not be real different - in contradiction to the assumption.

Marginal benefit and marginal rate of substitution

Marginal utility

The first partial derivative of the utility function for a commodity is called the marginal utility of this good. Clearly indicates the marginal benefit, how much additional value would cause a marginal increase in the amount of good, the quantity of all other goods is left unchanged. A marginal utility of means that for this good saturation has occurred. Another unit of this good would (for a concave function curve ) cause no additional benefit.

It is important to remember that just as the utility function, the marginal utility function or the marginal utility of a good in itself has no significance. For example, looking at the two-goods case, a utility function, then is the marginal utility of good second A strictly monotone positive transformation of the utility function, however, leads to the fact that the marginal utility of good 2 now amounts to - so he will also tripled, making it clear that marginal utility values ​​can be transformed arbitrarily. It appears, however, that in contrast, the ratio of the marginal utility of different goods is very well interpreted, as the following section shows.

In some applications, it is assumed that the marginal utility of goods in the set is typically decreasing; Hermann Heinrich Gossen already presented in the context of its utility theory the assertion that the added value of additional units of a commodity will always be lower, the more units you own of the estate already (First Gossen'sches Act). However, it must be remembered that the assumption is not consistent with an ordinal utility theory as it was budgetary theoretical reasoning above applied. This is because the utility values ​​precisely have no meaning - the fact that this is a utility function with and equivalent to having or showing so but that corresponding allowable modifications of the utility function can significantly affect the marginal utility change, which implies that the to - or decreasing character when using an ordinal concept offers no interpretation possible.

Another common assumption is a strictly positive marginal utility, that is, each additional unit of a good generates added value. This assumption corresponds to the preference- theoretic foundation with the adoption of strict monotonicity of household preferences, which in any environment of a freight bundle a strictly präferiertes bundle of goods exists, in the remaining of all goods the same amount of at least one good but more is included.

Marginal rate of substitution

In the two-goods case is defined as the absolute value of the slope of an indifference curve as the marginal rate of substitution (MRS ). It is

(read: marginal rate of substitution of good 1 relative to good 2 ), that is just the ratio of the marginal utility.

This can be shown as follows: As in falls, and thus is also what the penultimate equation explains. Next is for a bundle of goods that the indifference curve lies in the plane so that they can record directly as a function for which. This can be seen as representing the bundle of goods and it is the definition of the indifference curve that (constant). The derivative of with respect to is now beyond that

( because it corresponds ), which together with exactly the listed equation of the GRS - which was to be shown.

The GRS indicates with which exchange ratio a household is willing to trade a marginal unit of good 2 for a of good 1. This marginal rate of substitution is invariant under strictly positive monotonic transformation. The concept can also be used for a wider range of goods, in which case according to any goods:

The GRS is usually assumed to be strictly monotonically decreasing, which is equivalent to the statement that indifference curves are convex and corresponds directly to the Konvexitätsannahme of preferences in the preference- theoretic foundation. Intuitively, this means in the two-goods case that you have to be compensated for giving up a marginal unit of good 2 with the more units of good 1, the less one has of good 2.

Examples of utility functions

Quasi- linear utility function

A utility function is a quasi -linear, when it has the form, again is a utility function. In the simplest case, and accordingly. The function is quasi- linear in, ie it is " partially linear". In the two-goods case, indifference curves differ from the quasi-linear utility functions graphically only by the height of the vertical intercept. Thus, all indifference curves have the same slope for a given amount of good 1. With quasi-linear preferences locally, there is no income effect, as long as the income is m large enough, ie, the change in demand due to a change in price of any good is there entirely due to the substitution effect.

Cobb -Douglas and CES utility function

As a Cobb -Douglas utility function is called usually a utility function of the form

With; and for all. Simplified but you meet the two-goods case, often the assumption that and that the exponent just add up to one thing constant returns to scale ensures:

With

The Cobb -Douglas utility function is a common subclass of the general CES utility function

With; and for all as well. It converges for straight against the Cobb - Douglas function.

Intertemporal utility function

An intertemporal utility function represents preferences over consumption alternatives that are available at different time points are available. With it can be explained, among other things, why and in what amount people save or borrow.

In accordance with empirically observable behavior is followed in intertemporal preferences often assume that individuals prefer a more timely consumption compared to a distant time consumption of the same amount; one speaks of a positive time preference. In utility functions, this positive time preference is frequently depicted by discount factors, we simplify often assumes a constant rate of time preference even when income changes.

For example, will usually be considered in Overlapping Generations models assume that individuals live exactly two periods: In the first period they have an income that they consume or to save. In the second period they then live on their ( yielding) savings and an additional smaller facilities (such as a government subsidy ). The individuals then maximize the benefits across the entire consumption during their lives, that is, they maximize an intertemporal utility function

Which in the example shown simplistic and in the absence of transfer systems and ( the consumption of a born in t individual is in period t, the consumption of a born in t individual is in period t 1 [ ie just his second period of life ], the interest rate is on the savings between period t and t 1).

The rate of time preference of an economic subject is the private rate of time preference, while the is referred to a society as a social rate of time preference. The concept of the indifference curve can be applied analogously.

Von Neumann -Morgenstern expected utility function

Decisions under uncertainty are microeconomic often modeled as a lottery. The benefits of choosing an alternative is not immediately known here. Instead of a utility function, therefore, an expected utility function for modeling the preferences of the agent is employed.

Here, the expectation value of a (typically one-dimensional ) utility function for each alternative as a benefit value is defined. Therefore, the utility function of each alternative and the probability distribution to determine the benefit of a lottery: expected utility is simply the expectation value of the benefits of the alternatives. Such a utility function is also known as von Neumann -Morgenstern ( expectation ) utility function.

Denotes the expected utility function of the random variable ( conditions that occur with different probabilities ) and is called the Bernoulli utility function as a function of. The von Neumann -Morgenstern utility function is nothing else than the weighted with the probabilities benefit from the different states that can result from the lottery.

However, the existence of an expected utility function requires stronger assumptions, in particular the controversial independence axiom, according to may have no influence on the result of the irrelevant alternatives. Regardless of the admissibility of an expected utility formulation economic actors can be classified as risk averse, risk neutral or risk averse.

Risk aversion

Utility functions in the expected utility theory differ according to the next expression in them the degree of risk aversion of individuals. We call an individual as risk averse, if he prefers a lottery with mean a secure income in the amount of a, for example, so if the individual prefers the safe amount of 50 euros compared to a lottery in which there with 50 percent chance of 100 euro, with 50 percent probability, but only 0 euros receives. One can show that, under normal assumptions an individual exactly then (and only then ) is risk averse if his von Neumann -Morgenstern expected utility function is strictly concave.

According to the Arrow -Pratt measure result from the utility functions of the following subclasses:

• CRRA: constant relative risk aversion
• IRRA: Increasing relative risk aversion
• DRRA: Decreasing relative risk aversion
• IARA: Increasing absolute risk aversion
• DARA: Decreasing absolute risk aversion
• CARA: constant absolute risk aversion

HARA utility functions

In the life cycle model of a financial planning Economics, the term HARA ( hyperbolic absolute risk aversion ) is combined class of utility functions used. By default, this the CRRA case is considered, since there can not occur bankruptcy in the model, it is empirically relatively adequate and is mathematically still relatively easy to handle. ( The seminal paper by Merton Although considered other cases, but the solutions were incorrect and included negative consumption. ) The classical Bernoulli -log utility function is a special case of CRRA. It can be proved that all CRRA utility functions belong to the class HARA.

The general form of the HARA - utility function

Wherein the amount of consumption is. The function must possibly be completed steadily with the rule of L'Hospital for the Bernoulli case () and the risk-neutral case ().

Indirect utility function

In the context of utility maximization problemes that arises in the design marshall shear demand functions, is often a special " utility function " is used, the indirect utility function called. It is usually denoted by and is designed so that it directly indicates the maximum level of utility as a function of the prices of goods and the household budget that would have resulted from the solution of the corresponding maximization problem under constraints.

For details, refer to the main article above.

Recoverability problem

As Recoverability problem is referred to the question, to be determined from a utility function, the order of preference that generates the utility function presented. This is the reverse of the problem, to find a preference ordering a utility function with specific characteristics.

Macroeconomic utility theory

In the macroeconomic context, see overall economic utility functions use to measure the profitability of certain political and economic developments for the overall economic development.

In macroeconomics, the concept is also used to model the behavior of economic policy actors. In this context, for example, utility functions for re election -oriented politicians are created in the context of public choice theory. Thus, politicians will choose the one political alternative, and she uses her re-election chances the most.