Optimal sharing rule for a household with a portfolio management problem

We study an intra-household decision process in the Merton financial portfolio problem. This writes as an optimal consumption–investment problem in finite horizon for the case of two separate consumption streams and a shared final wealth, in a linear social welfare setting. We show that the aggregate problem for multiple agents can be linearly separated in multiple optimal single agent problems given an optimal sharing rule of the initial endowment. Consequently, an explicit closed form solution is obtained for each subproblem, and for the household as a whole. We show the impact of asymmetric risk aversion and market price of risk on the sharing rule in a specified setting with mean-reverting price of risk, with numerical illustration.