Bicriteria Decision Making and Financial Equilibrium: A Variational Inequality Perspective

In this paper we develop a framework for the study of financial equilibrium in the case of sectors in the economy, each of which is faced with two objectives/criteria in his portfolio selection decision making. In particular, we first present the bicriteria decision model of an individual financial sector, who seeks an optimal portfolio composition, given that the wishes to minimize his risk and to maximize his return. We utilize a value function approach to reformulate a sector's bicriteria optimization problem as a single optimization problem and argue that constant weight value functions may not adequately reveal a sector's preference over the return and the risk. Hence, we introduce state-dependent weights for the modeling of a sector's decision-making problem. We, subsequently, provide qualitative properties of the value function. We state the economic system conditions governing the instrument prices, define the financial equilibrium conditions, and show that they can be formulated as a variational inequality problem. Finally, qualitative properties of existence and uniqueness of the equilibrium are obtained. This work is the first to establish the connections among bicriteria problems (in the general setting of value functions), financial equilibrium problems, and variational inequality problems.