Bivariate FSD (BFSD)

Abstract

The classic expected utility paradigm of von-Neumann and Morgenstern assumes a univariate utility function where only one element, one’s own wealth, determines her welfare. Specifically, assuming a self-interest utility function U(w) where w stands for wealth, and in addition making a set of compelling axioms, the expected utility (EU) paradigm has been developed. Nowadays, many models in economics and finance rely on this EU paradigm. Yet, numerous experimental studies which analyze the choices made by subjects reveal some contradictions to the classic univariate EU paradigm (e.g., see Allais Paradox and Ellsberg Paradox). These “anomalies” in human behavior and the observed choices made by people indicate that at least in some situations some of the axioms underlying the EU model do not hold and other models, discussed in the previous two chapters, are suggested as substitutes to EU model, or as some modifications to EU model. Particularly, it is observed that the monotonicity axiom is violated in many cases: Experimental studies reveal that the subjects may be happier having a lottery prize of $1,000 rather than $2,000 provided that the peer group or close friends get even…. a lower prize, say, prize of $500 rather than $1,000. The explanation for such a phenomenon is that apart from one’s own wealth also emotions like envy and sense of fairness or relative wealth affects one’ welfare. In such cases the classic von-Neumann and Morgenstern univariate expected utility paradigm does not hold and some modifications in this paradigm are called for. A natural avenue to explain such a violation of the monotonicity axiom is to extend the EU model by assuming a multivariate preference U(x, y, z, … …), where x can be the individual’s wealth, as in the classic EU paradigm, y can be the wealth of one’s friend, z can be the typical climate in the city one decides to dwell etc., and the choice has to be done with respect to all these variables simultaneously. To simplify the analysis of choices in this EU extended framework and to present the essence of this multivariate utility model in the most transparent way, we focus in this chapter on the bivariate preference U(x, y), where x is always desirable, e.g., it stands for one’s own wealth and y can stand for various alternate variables, e.g., the wealth of your neighbor, outcome of the forgone prospect, quality of life, health and many more. With a utility which depends on two random variables the FSD discussed in previous chapters is replaced by bivariate FSD, denoted by BFSD.