Optimal saving in the presence of two small risks

We examine optimal saving in the presence of two small risks: income risk and a background risk. First, we compute the necessary and sufficient condition for a positive precautionary saving, showing that it depends on two terms capturing respectively the direct effect of income risk and the interaction between the two risks. Secondly, we examine the necessary and sufficient condition for a positive extra-saving due to the contemporaneous presence of the two risks. We show that this condition also depends on a term capturing the direct effect of background risk and that it can hold independently of the previous one.