Precautionary paying for stochastic improvements under background risks

In a two-dimensional framework, we propose a general two-period decision model which extends the temporal precautionary saving and effort model. We relate the role of cross-prudence to the impact of background risks on paying for stochastic improvements of the future risk. We find that the effect of background risks introduced in the first period is consistent to signing cross derivatives of bivariate utility functions, which is independent of the type of stochastic improvements brought by additional paying; however, when the background risk occurs in the second period, that is not the case.