# Bellman Equation with Two Discount Factors

For the following social planner's problem

$$\max \mathbb{E_{0}}\sum_{s=0}^{\infty}\beta_{1}^{s}(\alpha U(C_{s}^{1}))+\beta_{2}^{s}((1-\alpha)U(C_{s}^{2}))$$ $$s.t.\ \text{some constraints including C, K, N, etc.}$$

There are two different households with different discount factors, $$\alpha$$ and $$1-\alpha$$ are assigned weights for HH types 1 and 2, respectively. Is the following Bellman equation a correct formulation to solve the planner's problem? $$V(K) \equiv \max\ \alpha U(C^{1})+(1-\alpha) U\left(C^{2}\right)+\beta_{1} \alpha \mathbb{E} V\left(K^{\prime}\right)+\beta_{2}(1-\alpha) \mathbb{E} V\left(K^{\prime}\right)$$

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 $$\beta_1>\beta_2$$; 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.
• Thank you for the explanation. It is clear now. However, what if these weights were implied by a stationary distribution of two different $\beta$s. I mean $\beta$ is exogenously changing over time for each agent. In this case, there is a probability that each agent's discount factor changes in each period in the future. Will the value function for continuation still be independent of the lower $\beta$? Feb 23, 2021 at 20:10