Another Proof Of The Utility Function In Economics, Development Prospects And The Expected Utility Function Existence Theorem

In 1954, Debrue had showed the existence theorem of utility function on which utility theory base, playing important role in consumer theory, when he considered the problem of consumption choice under certainty.However, consumption choice in reality didn't hold under certainty at all. In the past decades, it had become the core of mathematical economics to construct individual behavior models. A number of literatures were related to the maximum of expected utility .In fact, the generation of expected utility theory and development in the consumer theory had been widely applied in the field of economics.The existence theorem of expected utility function was the most important conclusion in expected utility theory. In Karni and Schmeidler's paper (1991), they discussed "utility theory and uncertainty" and provided the detailed introduction on von Neumann and Savage's theory and its development, and emphasized the importance of relaxing independence axiom and some necessity principle.In this paper, we introduced development of utility theory especially for expected utility theory and main ideas in every period. Moreover, we summarized the applications of the theory in finance analysis, pricing and valuate, insurance and game theory, and so on; we still illustrated the prospect of utility theory in economics. Then we introduced some conceptions and properties in order to prove the expected utility theorem. In addition, we proved the properties, basing on Archimedes axiom and independent axiom. Finally, according to probability distribution is isomorphic to distribution function, we regarded a close convex set as the Euclid Space R(?), thereby proved the sufficient, necessity and unique of the theorem of e expected utility theory on the basis of the properties obtained above.