Bellman Equation with Two Discount Factors
No. Consider the following problem: Each period, one total unit of consumption falls from the sky that can be distributed between the two agents in any way. Their per-period utility is simply how much they consume. We let $\alpha=1/2$. Assume that $\beta_1&gt;\beta_2$; the first agent is more patient. All optimal allocations have a very simple form: The consumption unit is arbitrarily distributed in the first period, in all subsequent periods, agent 1 gets everything. This implies that the value function for continuations is completely independent of $\beta_2$, which it clearly is not in your suggested form.