When can one drop time subscripts? Example from Angrist and Kugler (2003)

Question

Not the first time I am asking myself, but in this paper they actually start with a time dependent maximisation problem and then drop all time subscripts.

Background: They have profit maximisation problem that reads $$ \Pi = \phi^{t} \sum_{i=0}^\infty \left \{f[\theta_{g}(N_t,I_t)] - w_{Nt}N_t - w_{It}I_{t}-\lambda C_N N_{t-1}\right\} $$ where $\Pi$ is profits, $\phi^{t}$ is discount factor, $f[\theta_{g}(N_t,I_t)$ is output, $N_t$ and $I_t$ is different type of labour, $w_{Nt}$ and $w_{It}$ the respective wages, and $C_N$ a firing cost for the $\lambda$ workers that are replaced in each period.

Steps: They write "Adjustment costs are linear and there is no aggregate uncertainty, so time subscripts can be dropped and the objective simplified to be:" $$ \Pi = (1-\phi)^{-1} [f(\theta_{g}) - w_{N}N - w_{I}I- \phi \lambda C_N N ] $$

Question: I only knew this step with the Bellmann equation were you simplify the infinite horizon problem into basically a timeless problem of finding the optimal policy function, but here it seems to be literally just dropping the dimension like that. Does anybody know more generally when one can simplify a problem like this? This question interests me also more generally because I see many papers writing basically immediately version 2 of the problem and not considering optimizing the sum of the profits through time.

Reference: Angrist, J. D., & Kugler, A. D. (2003). Protective or counter‐productive? labour market institutions and the effect of immigration one unatives. The Economic Journal, 113(488), F302-F331.

Answer 1

Because "adjustment costs are linear and there is no aggregate uncertainty", the FOC for $N_t$ is $$f'\theta g_N(N_{t}, I_{t}) - w_N = \phi \lambda C_N$$. Notice that this is exactly the same form for each period. The same is for $I_t$. This means that a firm will choose the same labor inputs in all periods. In other words, the firm gets into the steady state immediately in this dynamic optimization problem. This is not the case when there is aggregate uncertainty so that $w$s can vary over time or the adjustment costs are nonlinear, e.g. $C_N(N_t-N_{t-1}).$

As a result, the profit at each period is constant. And thus the present value of the discounted profits is $$\Pi= \frac{\left[f\left(\theta_{g}\right)-w_{N} N-w_{I} I-\phi \lambda C_{N} N\right]}{1-\phi}$$.