Correlated risks, bivariate utility and optimal choices

Abstract

In this paper, we consider a décision-maker facing a financial risk flanked by a background risk, possibly non-financial, 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 order dimension (environmental quality or health improvement). In the first case, we show that the optimal amount of savings decreases as the pair of risks increases in the bivariate increasing concave dominance rules of higher degrees which express the common preferences of all the decision-makers whose twoargument utility function possesses direct and cross derivatives fulfilling some specific requirements. Roughly speaking, the optimal amount of savings decreases as the two risks become "less positively correlated" or marginally improve in univariate stochastic dominance. In the second case, a similar conclusion on optimal investment is reached under alternative conditions on the derivatives of the utility function.

Pigouvian tax, abatement policies and uncertainty on the environment

“…41 Similarly, the role of U cqq in decisions under risk has recently been emphasized by many papers (e.g. Courbage and Rey 2007;Menegatti 2009a,b;Denuit et al 2011;Chiu and Eeckhoudt 2010) which consider a two-argument utility. In this field, our conclusions confirm the role of third-order cross derivatives of the utility function in the determination of optimal choices under risk.…”

“…In a bivariate utility two-period model, in particular, the signs of U qqq and U cqq are shown to determine precautionary saving in different contexts together with U ccc (see Courbage and Rey 2007;Menegatti 2009a,b). Similarly the same derivatives are relevant for the determination of saving in the presence of correlated risks (see Denuit et al 2011). Finally, similar conditions affect the optimal choice of labor supply in a one period bivariate utility model of choice between consumption and leisure (see Chiu and Eeckhoudt 2010).…”

“…Courbage and Rey (2007) and Menegatti (2009a,b) analyse the effects of background risk on precautionary saving respectively under some specific assumptions on bivariate risk distribution and in the case of small risks. Finally Denuit et al (2011) generalize the analysis of optimal saving and investment in the presence of two correlated risks, financial and background. This paper is organized as follow.…”

Precautionary saving under many risks

“…Menegatti (2009b) analyzes the same problem in the presence of small risks, introducing the concept of 'two-source precautionary saving'. Finally, Denuit et al (2011) study this issue when financial and background risks are positively correlated.…”

“…The comparison between Problems (1) and (5) is analyzed by Courbage and Rey (2007), Menegatti (2009aMenegatti ( , 2009b and Denuit et al (2011), given different assumptions on the distribution of the two risks or in the case of small risks. These papers show that, in this framework, precautionary saving not only depends on prudence, but also on cross-prudence (which captures the effects on utility of the interaction between the two risks) and on the size and sign of the correlation between them.…”