Continuous Utility Functions in Mathematical Economic Models

In Economics, preference relations (or preorders) are often described by means of utility functions. We are interested in continuous utility representations of non-total (partial) preorders. Non-total preorders are usually considered in the context of decision-making under uncertainty and risk. In this paper, we present two mathematical models described by non-total preorders. The first model is the Lorenz ordering, a preorder defined by the Lorenz curves on the set of all distributions of wellbeing of a given population. The Lorenz curve is an effective way to show the inequality of the income within and between populations. An example of utility function representing the Lorenz ordering can be given by the Gini coefficients. The second mathematical model is the Finite dimensional State Preference Model where a not-total preorder is defined by linear operators on financial markets. Both models can be extended in submetrizable k_omega-spaces (when the size of the population varies or when the number of goods in the portfolios varies) and the preorders can be described by the application of generalized representation theorems for submetrizable k_omega-spaces.