<p>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 <span class="math-container">$\alpha=1/2$</span>. Assume that <span class="math-container">$\beta_1&gt;\beta_2$</span>; 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 <span class="math-container">$\beta_2$</span>, which it clearly is not in your suggested form.</p>

CC BY-SA 4.0
2021-02-23T20:10:46.573Z
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 $\beta$?
Giordano
CC BY-SA 4.0
2021-02-23T20:13:06.407Z
I guess not, but I'm still not sure whether the problem can be easily treated in a recursive way.
Michael Greinecker
CC BY-SA 4.0
2022-08-02T17:04:09.113Z
Dear Micheal can you please look at my question as well? This is related to Bellman equation for the economy with capital and land in the recursive world. https://economics.stackexchange.com/questions/52251/writing-down-bellman-equation
studentp
CC BY-SA 4.0
2021-02-23T18:15:41.237Z
2021-02-23T18:15:41.237Z
Michael Greinecker
<p>For the following social planner's problem</p>
<p><span class="math-container">$$
\max \mathbb{E_{0}}\sum_{s=0}^{\infty}\beta_{1}^{s}(\alpha U(C_{s}^{1}))+\beta_{2}^{s}((1-\alpha)U(C_{s}^{2}))
$$</span>
<span class="math-container">$$
s.t.\ \text{some constraints including C, K, N, etc.}
$$</span></p>
<p>There are two different households with different discount factors, <span class="math-container">$\alpha$</span> and <span class="math-container">$1-\alpha$</span> are assigned weights for HH types 1 and 2, respectively. Is the following Bellman equation a correct formulation to solve the planner's problem?
<span class="math-container">$$
V(K) \equiv \max\ \alpha U(C^{1})+(1-\alpha) U\left(C^{2}\right)+\beta_{1} \alpha \mathbb{E} V\left(K^{\prime}\right)+\beta_{2}(1-\alpha) \mathbb{E} V\left(K^{\prime}\right)
$$</span></p>

<macroeconomics><bellman-equations><recursive-macroeconomics>
Bellman Equation with Two Discount Factors
Giordano
# Bellman Equation with Two Discount Factors

(Asked by: username_0 on 2021-02-23)

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

**Tags:** macroeconomics, bellman-equations, recursive-macroeconomics

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## 1 answers

### Answer by username_1 on 2021-02-23. Score: 4 | 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.

> **Comments:**
> **username_0**: 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 $\beta$?  
> **username_1**: I guess not, but I'm still not sure whether the problem can be easily treated in a recursive way.  
> **username_2**: Dear Micheal can you please look at my question as well? This is related to Bellman equation for the economy with capital and land in the recursive world. https://economics.stackexchange.com/questions/52251/writing-down-bellman-equation  

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