I am trying to figure out the differences between dynare’s Ramsey policy function when using a linear model and a nonlinear model.
In the woodford methodology, a welfare loss function is minimized subject to the constraints of the linearized model (I kind of understand that there are caveats that the linearized model neglects certain terms that can lead to incorrect utility rankings). However, in the dynare example file (examples/Ramsey_Example.mod · master · Dynare / dynare · GitLab), a planner objective is maximized subject to the constraints of the nonlinear model, which can be solved to second or higher order.
I have two problems: I have a nonlinear model where a simple Taylor rule produces mean welfare (variable ‘welfare’ in OpPol_stickydiffpriceflexwage.mod) that is higher (-1040.0498) than the Ramsey policy targeting the interest rate (-1040.05399). They have the same parameterization and are solved to the first order. They also seem to have the same steady state. Dynare 4.6.4.
The second question is: I would ideally like to have an second-order expansion of the utility function with a linear model for ease of intuition. For example, if could perform a second-order taylor expansion and manipulate it into the form of -x1ygap^2-x2pi^2 (where x1 and x2 are derived weights), how would I operationally use this in dynare?
could I simply write:
steady_state_model, initval; etc.
If I have performed the linearization and the second-order expansion of the utility function correctly, would I get the same result as if I had instead used the non-linear model and replaced the planner objective with raw utility function (using stoch_simul order=1)?
Thanks very much for any aid to my understanding!