Bellman Equation with Two Discount Factors

Question

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

Michael Greinecker · Accepted Answer

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.

Correct answer by Michael Greinecker on July 26, 2021

Bellman Equation with Two Discount Factors

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