- Yes, it will produce a second order approximation, but the evaluation of the objective function is still missing. See https://github.com/DynareTeam/dynare/issues/564
- This would be equivalent to what is done at 1.
- No, you do not need to specify an instrument to compute an optimal allocation. You can see this e.g. in Gali’s textbook where optimal policy is directly computed without going through the nominal interest rate. Having an instrument only simplifies perturbation solutions by making it easier to find the steady state to the Ramsey problem via the provision of a conditional steady state file. Conditional on that steady state, finding the steady state of the Ramsey problem is just solving a linear equation system as opposed to the nonlinear system you would need to solve otherwise.
- No, this is the suggested way of handling this problem. See Questions about Ramsey optimal policy