<p>Here is the general intuition:</p>
<p><strong>1. Finite Horizon Case</strong></p>
<p>It is necessary to include the time subscript here because <span class="math-container">$V_t(k_t)$</span> and <span class="math-container">$V_{t+1}(k_{t+1})$</span> have different number of time periods ahead of them. Suppose everything ends at time <span class="math-container">$T$</span>, then the former has <span class="math-container">$T-t$</span> periods ahead, and the latter has <span class="math-container">$T-t-1$</span> periods ahead of it. As is the case for slide 15.</p>
<p><strong>2. Infinite Horizon Case</strong></p>
<p>Here you can use the recursive form. There is no end in sight i.e for both <span class="math-container">$V_t(k_t)$</span> and <span class="math-container">$V_{t+1}(k_{t+1})$</span>, the time ahead of them is infinity. So the value function are same because they face the same future (The argument inside the value function is NOT i.e <span class="math-container">$k$</span> <span class="math-container">$\neq$</span> <span class="math-container">$k'$</span>). We say the value function is time invariant. This is not true for finite horizon case.</p>
<p><strong>3. Summary</strong></p>
<p>Drop the time subscript if the value function agents face are the same, which happens in stationary environments. So for example if you were to solve an OLG model in a recursive form, you would not drop the time subscript, because value functions are different for each individual since they are at different ages in different periods.</p>
<p><strong>4. Further Information</strong></p>
<p>Formally, this is discussed in Stokey and Lucas textbook in either Chapter 3 or 4.</p>

CC BY-SA 4.0
2022-07-02T11:25:31.483Z
2022-07-02T11:25:31.483Z
Rumi
<p>when I learned Bellman equations for macro, my lecturers were pretty loose about notation and switched between recursive and time-series notation. However, Prof Ben Moll says in his slides on Slide 15 below, that the time-subscript notation should be avoided, but I'm not sure why.</p>
<p><a href="https://benjaminmoll.com/wp-content/uploads/2021/04/Lecture2_EC442_Moll.pdf" rel="nofollow noreferrer">https://benjaminmoll.com/wp-content/uploads/2021/04/Lecture2_EC442_Moll.pdf</a></p>
<p>Can anyone explain why, please?</p>
<p>Thanks in advance!</p>

CC BY-SA 4.0
2022-07-02T10:57:52.097Z
It’s literally explained on the next slide
1muflon1
<macroeconomics><mathematical-economics>
Why should time-subscript notation be avoided in Bellman equations?
NuancedPaul
# Why should time-subscript notation be avoided in Bellman equations?

(Asked by: username_0 on 2022-07-02)

when I learned Bellman equations for macro, my lecturers were pretty loose about notation and switched between recursive and time-series notation. However, Prof Ben Moll says in his slides on Slide 15 below, that the time-subscript notation should be avoided, but I'm not sure why.

<https://benjaminmoll.com/wp-content/uploads/2021/04/Lecture2_EC442_Moll.pdf>

Can anyone explain why, please?

Thanks in advance!

**Tags:** macroeconomics, mathematical-economics

> **Comments:**
> **username_1**: It’s literally explained on the next slide  

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## 1 answers

### Answer by username_2 on 2022-07-02. Score: 6 | Accepted answer

Here is the general intuition:

**1. Finite Horizon Case**

It is necessary to include the time subscript here because $V\_t(k\_t)$ and $V\_{t+1}(k\_{t+1})$ have different number of time periods ahead of them. Suppose everything ends at time $T$, then the former has $T-t$ periods ahead, and the latter has $T-t-1$ periods ahead of it. As is the case for slide 15.

**2. Infinite Horizon Case**

Here you can use the recursive form. There is no end in sight i.e for both $V\_t(k\_t)$ and $V\_{t+1}(k\_{t+1})$, the time ahead of them is infinity. So the value function are same because they face the same future (The argument inside the value function is NOT i.e $k$ $\neq$ $k'$). We say the value function is time invariant. This is not true for finite horizon case.

**3. Summary**

Drop the time subscript if the value function agents face are the same, which happens in stationary environments. So for example if you were to solve an OLG model in a recursive form, you would not drop the time subscript, because value functions are different for each individual since they are at different ages in different periods.

**4. Further Information**

Formally, this is discussed in Stokey and Lucas textbook in either Chapter 3 or 4.


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