Basic question about investment adjustment costs

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.

  1. 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?

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

screen1.pdf (7.93 KB)

Please provide a full reference. Usually the Benchmark DSGE model is a good place to start:

economiaynegocios.uahurtado. … strepo.pdf

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?

Thanks a lot.

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)

Yes, it also looks wrong to me. I will check the reference you gave me. Thanks a lot.