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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!
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.
