The utility function associated with each consumption bundle x is a number u(x) such that the basket is preferred to the cart z if and only if u(y)>u(z).
It is very important to note that the number u(x) has psychological significance for members of the current utilitarian. They agree in fact that if u(x) is twice as high as u(y), then it means that x twice provides satisfaction than y (this is called “Theory of cardinal utility” ). Such a view was materialized by the Austrian economist Carl Menger in what is called the Mengertable. This approach has drawn widespread criticism, including Vilfredo Pareto, in favor of “ordinal utility theory.” Indeed, Pareto against the idea of cardinal utility the concept of subjective utility for each consumer, called “ophelimity.”
In the neoclassical theory of the consumer, the utility function is only used to rank the baskets of goods based on consumer preferences. The number u(x) has no special meaning. Note also that there is an infinite number of different utilities functions representing the same preference relation. If u is a utility function representing the preferences of an agent i and g a function strictly increasing, then the composite g(u) is also a utility function representing the preferences of the agent i.
There are always no utility function representing preferences. For example, the lexicographical preference relation on R2 admits no utility function.
We are indebted to Gerard Debreu the proof of the theorem in 1954 characterisant the preferencial relationshipsfollowing the basic assumptions that can be represented by a utility function. Essentially, it suffices that the preferences are continuous (that is to say that the preferences are consistent with the convergence of sequences of points, that is to say that the preferences do not change when changing the margin a basket of goods).