What Is a Utility Function?

The utility function is usually a function used to represent the quantitative relationship between the utility obtained by the consumer and the combination of goods consumed, in order to measure the degree of satisfaction that the consumer obtains from the consumption of a given combination of goods.

effectiveness function; utility function; utility function used;

Explanation of "Utility Function" in Reference

1. A function that represents the quantitative relationship between the utility that consumers obtain in consumption and the combination of goods consumed. It is used to measure the degree to which consumers are satisfied from consuming a given combination of goods. use

Since von Neumann and Morganstein proposed the expected utility theory (Von Neumann, Morganstein, 1944), because of its simple and standardized expression of axioms, the calculation of utility function capacity and formally better reflects the type of people's risk behavior (such as The attitude and degree of risk can be expressed by the convexity and concavity of the utility function and the Arrow-Pratt measure), and the theoretical and applied research of uncertain economics has been carried out under this basic framework. Modern game theory is also developed on this basis However, the assumption of independence axioms, which plays a key role in the theory of expected utility, does not always hold, and the famous Allais paradox states that in some cases people systematically violate this axiom. At present, it is sought to exclude this axiom. The utility function is

nowadays

The existence of the utility function expresses two characteristics of the utility function with a mathematical formula: the utility increases with the increase in the number of individual commodities, and the marginal utility of a single commodity decreases. The sum of the utility generated by X and the product combination Y is greater than the utility produced by the product combination X + Y. The existence theorem of the utility function in Western economics: It is assumed that consumer preferences are complete, recurrent, transitive, continuous, and strongly monotonous Sex, then, there is a continuous utility function that can represent that preference.

Under the above assumptions, Western economics first constructs a unit consumption bundle e consisting of 1 unit of all commodities (e is a vector in an n-dimensional real space Rn where each component is 1), and then all consumption The bundle is compared with this unit consumption bundle, and it is "proved" that all of these consumption bundles are indistinguishable from a certain multiple of the unit consumption bundle, so that this multiple can be used to represent utility, that is, the utility function exists.

However, the proof of the existence of the utility function in Western economics is a self-circulating argument. This is because those assumptions of the existence theorem of utility function are not based on facts, but based on the need for mathematical proof. To satisfy these assumptions, the existence of a utility function must be required in advance. In fact, without the pre-existence of the utility function, it would be impossible for consumers to judge the preferences of completeness, transitivity, and continuity for various infinite combinations of millions of commodities. This is the root cause of the lack of transitivity in the choices of those who have not previously set utility functions in psychological experiments.

Thus what Western economics proves is a theorem: Assuming that consumer preference is represented by a continuous utility function that can be mathematically proven to exist, then it can be proved that there is such a continuous utility function that can represent the preference.

Further, the utility function "proved" by the above existence theorem is continuous, and thus is a base utility, rather than a discontinuous ordinal utility. In other words, the existence of ordinal utility has not been proved. The biggest problem with cardinal utility is how to determine the "utility unit". Western economics has never answered the question of how much a "utility unit" is. In fact, judging from the process of "proving" the existence of the utility function in Western economics, Western economics actually implicitly treats the consumption utility brought by a unit consumption bundle, that is, one unit for all commodities, as a unit of utility . However, the rich will not eat the "pearl emerald white jade soup" of the poor. The soup has a positive effect on the poor and a negative effect on the rich. As a result, the poor and the rich have different consumption sets and different unit consumption bundles. So which consumption bundle should be counted? Especially for those whose wealth changes every day, such as those who are still white-collar workers today and become unemployed tomorrow.

In fact, if one consumption unit of all the commodities in one unit consumption bundle cannot be determined, the existence of the "proof" of the utility function also lacks a realistic basis. In addition, we notice that the utility of two unit consumption bundles, 2e, is exactly twice the utility of one unit consumption bundle, and the utility of ne is exactly n times that of e. That is to say, if a unit consumption bundle is regarded as a comprehensive commodity, the marginal utility of the comprehensive commodity is constant, which contradicts the diminishing marginal utility of western economics. (Decreasing margins is not a rule that applies to all situations.)

Furthermore, Western economics only "proved" the existence of the utility function, and did not find a specific utility function. But Western economics has thus gained the power to set utility functions at will. For example, we can see the following utility function form from Western economics textbooks

u ( x , y ) = x y

Among them, x and y are the consumption of two commodities, respectively, and U (x, y) is the utility brought to the consumer by such a consumption bundle, a> 0, b> 0. The above form of mathematically unproven utility function has the problem of considering a hungry and thirsty person. Let x and y represent the consumption of water and bread, respectively. The above utility function means that giving this consumer a bread crumb and an infinite amount of water, or giving this consumer a drop of water and an infinite amount of bread, can make The consumer gets infinite utility. However, in real life, the infinite utility of the above two consumption belts is not as good as the limited utility of two glasses of water and two breads, which is more suitable for his needs. This example shows that Western economics not only abuses the existence of the so-called utility function, it cannot even give a concrete form of utility function that does not conflict with people's actual feelings. (The utility function above is just an example.)