When can one drop time subscripts? Example from Angrist and Kugler (2003)
Because &quot;adjustment costs are linear and there is no aggregate uncertainty&quot;, 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 $$\Pi= \frac{\left[f\left(\theta_{g}\right)-w_{N} N-w_{I} I-\phi \lambda C_{N} N\right]}{1-\phi}$$.