Dear all,
I have a question regarding the timing of the cutoff criterion and the occurrence of the idiosyncratic shocks as I can’t wrap my head around the timing of the chain of events. From what I see the representative entrepreneur decides about capital stock that will be a state next period (K_t according to Dynare’s convention) as well as the cutoff criterion. However, many authors (not Bernanke et al. 1999 as they do not specify any timing) write it down as omegabar_t+1. Now here’s my difficulty in understanding the timing of events. The contract is concluded at the end of period t.

That means the idiosyncratic shocks have not occurred yet, right?

Do they hit the return to capital of the entrepreneur at the beginning of time t+1, along with whatever aggregate shocks are assumed?

If that’s the case, then in the appendix of BGG 1999 one sees that the entrepreneur maximizes its expected return subject to the participation constraint of the entrepreneur. But all that happens at the end of period t. That means, both the participation constraint of the bank as well as the expected return of the entrepreneur are……in expected terms. Right?

So at the end of period t the entrepreneur chooses the capital stock that will be rented out to the intermediate firm in period t+1. That means K_t is a state variable, as usual. But then it says that the entrepreneur chooses omegabar_t+1. But I’ve also read that it’s a control variable. How can it be a control variable when it’s chosen at the end of period t and not in t+1?

In the BGG case the participation constraint of the bank always holds. As a result, one would write it in dynare in period t only (i.e. no (+1)). And omegabar_t+1 adjusts so that the contract is always fulfilled with equality. But if that’s the case, what did the households chose at the end of period t? Only E_t Omegabar_t+1? That is to say, they are optimizing with respect to……hm, the expectation for the cutoff criterion? Why? Can’t they simply decide about this cutoff criterion when observing the idiosyncratic shocks at the beginning of t+1 so omegabar becomes purely static.

I know BGG are taking a short cut and do not use the normcdf function but just a parameter called “nu” for defining the external finance premium. Yet I’m rather inclined to follow those authors (I guess Christiano, Motto and Rostango) who specify the entire expressions for the Langrange multiplier associated with the participation constraint of the banker. This means, however, that a FirstOrder Condition with respect to omegabar will have to be specified in the dynare code. And hence, question 5.
Many thanks for taking the time to read my questions.
Best,
Peter