Gertler and Karadi (2010) type financial intermediation

Dear all,

I am currently extending the model of Bi, Leeper and Leith (2014) (which is still in its preliminary and incomplete version) to an open economy model. As a result, I am trying to understand whether net worth of the financial intermediary is a state or control variable in the period t (current period). That is, at the “end” of period t the bank decides how much shares and bonds to buy as a function of its net worth and deposits. However, the net worth itself is not decided by the bank in the current period, but rather it’s a function of the net worth at time t-1 plus all the revenues at time t (return on capital + return on bonds) minus all costs at time t (return to deposits). Then, based on this accumulated net worth, at the end of period t the bank decides about K_t+1 and B_t+1, which are predetemined variables. So I’d say N_t is a control even though no decision is made about it today, am I right?

Thanks a lot in advance,

The definition of a predetermined variable is not that a decision is made about it, but whether it can be changed in the current period. In your example, bonds and capital are states, because their quantity is determined in the previous period. But net worth should not be a state, because its value today depends on the revenues at time t.
Note, however, that this intuition can change depending on the timing you use. Take capital in the beginning of period notation:

k=(1-delta)*k(-1)+invest; y=k(-1)^alpha;
Here, capital k is a control and k(-1) is the state variable.

Lesson: just be consistent in your timing.

Thanks Johannes, this makes perfect sense.

I got a bit confused since net worth in time t is indeed a control since it depends on the return to both capital and bonds in period t, but we take it as given when we make the decision about capital and bond quantities to be used in the next period. I guess an easy way to think about it is that the bank decides about K and B at the end of period t and net worth has been accummulated at the beginning of period t so it’s given at the end of the period…hope my intuition is right.

Again thanks a lot for your help,