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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? $$ V(K) \equiv \max\ \alpha U(C^{1})+(1-\alpha) U\left(C^{2}\right)+\beta_{1} \alpha \mathbb{E} V\left(K^{\prime}\right)+\beta_{2}(1-\alpha) \mathbb{E} V\left(K^{\prime}\right) $$

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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.

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  • $\begingroup$ 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 $\beta$? $\endgroup$
    – Giordano
    Feb 23 at 20:10
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    $\begingroup$ I guess not, but I'm still not sure whether the problem can be easily treated in a recursive way. $\endgroup$ Feb 23 at 20:13

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