I have some questions regarding optimal policy computation in Dynare:
Is it possible to compute optimal policy under commitment in a timeless perspective in Dynare? If yes, how? If I’m right the ramsey_policy command gives us optimal policy which is time inconsistent.
I tried to compute optimal policy using the command ramsey_policy. The model I use is linear - variables are expressed as percentage deviations from ss. The objective function I use is: y^2 + lambda*pi^2 with discount factor equal 1.
The procedure works well, but in the end I get the following results:
Approximated value of planner objective function
- with initial Lagrange multipliers set to 0: NaN
- with initial Lagrange multipliers set to steady state: NaN
What does it mean? Is there any problem?
Is there any way to compare the welfare losses achieved under optimal policy (ramsey_policy command with objective function y^2 + lambdapi^2 and discount factor equal 1) and under optimal simple rule (osr command with objective function var(y)+ lambdavar(pi))?
Dynare does not support the timeless perspective of Woodford. As far as I know, there is not operational definition of this concept that can be generally used.
Yes, there must be a problem. Note that using a linear model with a quadratic objective in the Ramsey command is generally not OK. It only works for efficient steady states. Otherwise, you will get spurious welfare rankings as discussed in e.g. Benigno/Woodford’d work.
No, I don’t think the two are equivalent. One is a sum of unconditional variances, the other of undiscounted conditional variance (I am not even sure the latter converges without discounting). But given the Dynare output, It should be easily possible to derive their values.
could you please clarify what kind of welfare analysis can I generally perform with a log-linearized model? Since the model is linear I cannot do second order approximation, but can I calculate optimal Ramsey policy then?
In general, this is only possible if you are working with a model that has a non-distorted steady state, i.e. where due to (close to) optimality the second order terms evaluate to 0. Please take a look at e.g. Woodford 2002 - Inflation Stabilization and Welfare.
I’m also interested in computing optimal policy using the command ramsey_policy in Dynare.
Let me give a concrete example. Suppose we use a well-known DSGE model developed by Adolfson et al. 2007: Bayesian estimation of an open economy DSGE model with incomplete pass-through.
The features of this model are the following:
model is log-linearized - variables are expressed as percentage deviations from SS, that is x_hat for a generic variable x
in SS x_hat =0 (by definition)
etc.
My objective function is linear-quadratic of the following form: L = lambda*y_hat^2 + pi_hat^2.
My question is: can I use the command ramsey_policy in the context of this model (i.e. Adolfson et al. 2007)?
My understanding is that this is not possible. The optimal policy depends on some cross-derivatives that would be absent due to the model already being linear. As far as I remember, their model has a distorted steady state so that the naive linear quadratic approach you are trying will result in wrong results.