Lagrangian: when to discount budget constraint?

Question

I am getting a bit confused about setting up the Lagrangian in intertemporal constrained optimization problems. The confusion is as to when is the one-period budget constraint multiplied by the discount factor Beta and/or when is it included in the summation? Here is what I mean. In this first example:

the constraint is not multiplied by Beta, nor part of the summation. Now, in example 2:

the constraint is multiplied by the discount factor Beta and part of the intertemporal summation. Why this difference? Aren't the two cases pretty much equal, both maximizing subject to one-period constraints? The only difference I see is when the interest rate on the assets is realized, but that should not matter I believe(?) I also found a third case (will upload an image later if I find it) in which the one-period constraint is summed up intertemporally, but not discounted by Beta. So to conclude: what is the rule here? What am I missing?

EDIT:

By looking at the FOCs that follow example 1, it becomes clear that the author means the Lagrange Multiplier to be part of the summation, but not discounted by Beta. This is clear from the following:

So this really looks to me like a different formulation to the second example. The "funny" thing, however, is that in both cases you get the same result (i.e. the same Euler Equation).

Answer 1

The constraint is part of the objective function in each period. So, naturally it is subject to discounting, and so it must "be multiplied by beta".

So the fist setup above is wrong (or just a typo: one should check whether in the specific publication the solution given follows the above image or in reality it implies that the constraint is multiplied by beta).

The "third case" the OP mentions in his post is a different matter. There we "compact" the constraint into one, so here they are outside the sum and the intertemporal discounting, with the understanding that the single multiplier is a present-value multiplier in this case. This is ok, it is done for example on certain macroeconomic models related to optimal taxation.