[cake_eating_problem] Some suggestions

Hi John @jstac

After reading the lecture cake_eating_problem.md, I have a few suggestions:

1. Two typos (marked in bold):

In particular, consumption of $c$ units $t$ periods hence has present value $\beta^t u(c)$

Suggestion: In particular, consumption of $c$ units in $t$ periods hence has present value $\beta^t u(c)$

The intuition here is essentially the same it was for the McCall model.

Suggestion: The intuition here is essentially the same as it was for the McCall model.

2. Possibly missing exercise:

In Section 51.5.1, the text states:

In the exercises, you are asked to verify that the optimal policy (51.7) does indeed satisfy this functional equation

However, the lecture currently has only one exercise (deriving the value function and optimal consumption policy). Should we add an exercise asking students to verify that the optimal policy satisfies the Euler equation?

3. Notation inconsistency:

Throughout the lecture, we use both { $c_t$ } (without time index) and { $c_t$ }_${t=0}^\infty$ (with time index) to denote the consumption sequence. Should we standardize this? If so, which notation would you prefer?

I'm happy to submit a PR once we've settled on the changes.

Best,
Longye