Hi Johannes and Michel
I think the reason why optimal policy in the two models is different has to do with the information sets that are used on impact. I wonder if this behavior is desirable.
But let me go step by step.
I looked at the equation systems that are solved, and if one ignores the information sets (which aren’t expicit in dynare language anyway), then the two models are isomorpic if one defines the following relationship between the model variables in the two versions (where lambda^8 is the planner’s 8th multiplier (MULT8) and v1 and v2 denotes the two versions).
Note: The first line explains the difference in the SS values that Johannes noted above.
And indeed if I use the solution of model version 2, lag the paths of r and MULT8 in line with the above formula the residuals are 0 for all periods but the first.
In order to understand why I checked the FOC of the planner wrt r. For the two versions, they are given by
According to the relationship in (1) the 8th multiplier’s in the two versions should be related as follows for the two models to be identical:
One can verify that this holds for all t>1. However, using the above definitions of lambda8 for t=1 and keeping in mind that t=0 is the SS we get:
there is a difference between the two lines: pi_1 vs pi_SS.
This inconsistency would disappear if the FOC wrt r for version 2 (see equ. 2) would instead read
where pi_t has been replaced by its expectation one period ahead.
I think it would be desiderable that the solution to model versions 1 and 2 were identical. My hunch is that the solution to version 1 is the correct solution. And ramsey could be djusted to deliver 5 for version 2, which would be identical.