Cardinal utility

In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value u ( x i ) {displaystyle u(x_{i})} of one index u, occurring at any quantity x i {displaystyle x_{i}} of the goods bundle being evaluated, the corresponding value v ( x i ) {displaystyle v(x_{i})} of the other index v satisfies a relationship of the formCan we assign a set of numbers (measures) to the various entities and predict that the entity with the largest assigned number (measure) will be chosen? If so, we could christen this measure 'utility' and then assert that choices are made so as to maximize utility. It is an easy step to the statement that 'you are maximizing your utility', which says no more than that your choice is predictable according to the size of some assigned numbers. For analytical convenience it is customary to postulate that an individual seeks to maximize something subject to some constraints. The thing -or numerical measure of the 'thing'- which he seeks to maximize is called 'utility'. Whether or not utility is of some kind glow or warmth, or happiness, is here irrelevant; all that counts is that we can assign numbers to entities or conditions which a person can strive to realize. Then we say the individual seeks to maximize some function of those numbers. Unfortunately, the term 'utility' has by now acquired so many connotations, that it is difficult to realize that for present purposes utility has no more meaning than this.It might have happened to you that you were carrying a pile of papers, or clothes, and didn't notice that you dropped a few. The decrease in the total weight you were carrying was probably not large enough for you to notice. Two objects may be too close in terms of weight for us to notice the difference between them. This problem is common to perception in all our senses. If I ask whether two rods are of the same length or not, there are differences that will be too small for you to notice. The same would apply to your perception of sound (volume, pitch), light, temperature, and so forth...These terms, which seem to have been introduced by Hicks and Allen (1934), bear scant if any relation to the mathematicians' concept of ordinal and cardinal numbers; rather they are euphemisms for the concepts of order-homomorphism to the real numbers and group-homomorphism to the real numbers. In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value u ( x i ) {displaystyle u(x_{i})} of one index u, occurring at any quantity x i {displaystyle x_{i}} of the goods bundle being evaluated, the corresponding value v ( x i ) {displaystyle v(x_{i})} of the other index v satisfies a relationship of the form for fixed constants a and b. Thus the utility functions themselves are related by The two indices differ only with respect to scale and origin. Thus if one is concave, so is the other, in which case there is often said to be diminishing marginal utility.