Sorry for the basic question.
I am trying to model a representative firm that rents capital goods to firms producing intermediate variables. I need to introduce capital adjustment costs.

I know this is very basic, but I cannot understand how to get the F.O.C from the optimisation problem in the attached file (from Garcia and Restrepo (2007)). I know that Q is tobin’s Q, the lagrange multiplier that arises when maximising (26) s.t. (27). But why is P_t, the overall price level, in the f.o.c?

Any other reference you would recommend for the most basic and simple example of how to do capital adjustment costs?

The Case for a Countercyclical Rule-Based Fiscal Regime, Carlos J. Garcia and Jorge E. Restrepo, Central Bank of Chile.

What I don’t get is:

I know eq (28) is the maximisation of (26) s.t. (27) and Q is the lagrange multiplier. But I don’t know why P_t (aggregate price index) dividing P^I_t (investment goods price index) is there. Apart from that, my result is the same.

I know eq (29) is the choice of K, my results are somewhat similar but why the forward looking terms and -again- the price index?

My guess is that equation (26) is wrong. For example, the discounting of the future value is notably absent. Moreover, it is never clearly stated how Q is defined. It may well be that the Lagrange multiplier on the constraint has been defined as P_t*Q_t, which could explain the expression.

You might want to take a look at Basu/Bundick (2015): Uncertainty Shocks in a Model of Effective Demand. They have a similar setup (except for the relative price of investment)