When to use a lagrange multiplier? I have a question in regards to the use of Lagrange multipliers in macro models.

I have seen that when writing a Lagrange there are two ways to write the Lagrange multiplier: 1) Indexed or 2) Unindexed. For example, in order to solve the Ramsey model in discrete time, in the simplest possible version, one would write the Lagrange as follows:

$$
L=\sum_{t=0}^{\infty}\bigg\{\beta^t u(C_t)-\lambda_t[C_t+K_{t+1}-(1-\delta)K_t-w_t-r_tK_t]\bigg\}
$$

and here the Lagrange multiplier is indexed in time and written as $\lambda_t$.

But when solving a simple lifetime consumption problem for an individual, one would write the Lagrange as follows:

$$
L=\sum_{t=0}^{\infty}[u(C_t)+\lambda(A_0+Y_t-C_t)]
$$

and here the Lagrange multiplier is not indexed in time, so we write it just as $\lambda$. 

Hence, my question is when is it appropriate to use each type of multiplier? 

Thank you.
I wish I could leave this as a comment, but the first formulation is a solution to a maximisation problem of the form
$$
\max_{C_t} \sum_{t} \beta^{t} u(C_t) \\
\text{s.t. } \quad C_t +K_{t+1} \leq (1-\delta)K_{t} + w_{t} + r_tK_t \quad \forall t 
$$

The second formulation is a solution to a maximisation problem of the form
$$
\max_{C_t} \sum_{t} \beta^{t} u(C_t) \\
\text{s.t. } \quad \sum_{t} C_t \leq \sum_{t} (A_0 +Y_t) 
$$

In a way, the first is a sequential problem, while the second is a time zero problem. In some instances the approaches are equivalent, but I'm not an expert on this.