When to use a lagrange multiplier?

Question

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.

Answer 1

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.