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

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.