Correlated Risks, Bivariate Utility and Optimal Choices

In this paper, we consider a decision-maker facing a financial risk flanked by a non-financial background risk such as health or environmental risk. A decision has to be made about the amount of an investment (in the financial dimension) resulting in a future benefit either in the same dimension (savings) or in the other dimension (environmental quality or health improvement). In this framework, we study the impact of the correlation between the two risks on optimal choices. In the saving problem, we find conditions ensuring that positive correlation between the two risks implies that the optimal amount of savings increases. These conditions involve specific requirements on the direct and cross derivatives of the two-argument utility function. Similarly, we find a different and specific set of conditions ensuring that the same conclusion on optimal investment for health (environmental) improvement is reached. The two sets of conditions determined support the conclusion that the signs of the derivatives of the two-argument utility function should alternate.