Microeconomics: Bernoulli and von Neumann-Morgenstern

These people did paramount work on utility functions and expected utility theory.

Rather than give their backgrounds, I thought that I would write their contribution instead.

First was Bernoulli, who formulated a utility function over an individual's wealth. Utility over wealth changes as the wealth level changes. Have a think at this: if you are broke, $100 is worth much more to you than if you were $1,000,000 rich, and were offered the same $100.
Bernoulli therefore put forth the notion that an individual's subjective valuation for a certain sum of money was not necessarily the same as the objective valuation.

What von Neumann and Morgenstern did was to extend Bernoulli's utility function to define expected utility over lotteries, but with several premises (see the previous post on the preference axioms). Their utility function is called the von Neumann - Morgenstern utility function, or more commonly, expected utility function.

In the first stages, one can only think of utility as being ordered (i.e. ordinal utility). For instance, if the utility of a certain commodity bundle, x (I shall use commodity bundles for the moment, as wealth must generally be mapped as a density function because of its continuous nature) is subjectively higher than the utility of another commodity bundle y, intuitively,

U(x) > U(y)

Fortunately, once we have the expected utility form (which satisfies the independence axiom, a.k.a. independence of irrelevant alternatives), we can then start to think of the notion of utility as being a cardinal form (i.e. one with real values). I will not go through the proofs here, but it is pretty easy to check them out on the web.

Hooray for the expected utility form!