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.