I want to search the optimal monetary policy in an estimated model. I use Taylor Rule as: \frac{{{R}_{t}}}{R}={{\left( \frac{{{R}_{t-1}}}{R} \right)}^{{{\rho }_{R}}}}{{\left[ {{\left( \frac{{{\pi }_{t}}}{\pi } \right)}^{{{\rho }_{\pi }}}}{{\left( \frac{GD{{P}_{t}}}{GD{{P}_{t-1}}} \right)}^{{{\rho }_{y}}}} \right]}^{1-{{\rho }_{R}}}}e_{t}^{R}

However, after I set the objective, the result of optimal {\rho }_{y} is always close to 0 (which is the lower bond I set). I have no idea what the problem is. Would you please take a look? The code is sent through message.

Could also depend on the shock structure in the model. For example in Gali’s book, \rho_y \rightarrow 0 and \rho_{ \pi } \rightarrow \infty reduces household welfare losses following a technology shock. If there are no zero bounds, \rho_y can even be negative.

(1) I think it might be a problem because this optimal {\rho }_{y} is close to 0 only when the lower bond I set is 0. If the lower bond is -inf, then the optimal {\rho }_{y} will be negative. Is this normal? How should I interpret a negative {\rho }_{y} ?

(2) What is the right way to set the parameters in objective function, which may also influence the optimal results ? It seems that differnet empirical values are used in different papers.

Looking forward to your reply.
Thank you very much!

It can be negative…thus, accommodation of output variations instead of stabilizing it. Read this paper: Monetary Policy and Unemployment by Jordi Galí.

@kofiemma Is right. The output feedback can be negative. A priori it is almost impossible to judge what the optimal response of the central bank is for a given model. After all, frictions differ in their size, sign, and interrelation across models. Therefore, the optimal stabilization policy may look very different.

Gali, however, did not interpret in the paper or in his book why a strong positive response of interest rate to output gap is bad for consumer welfare.

Here is my thought…
A larger output gap is better for consumption because the consumer wants a booming economy, and so the central bank should accommodate a larger output gap and not stabilize it. At the same time, the consumer wants lower inflation (leading to larger consumption), so the CB should stabilize inflation.

If phi_pi is large enough to stabilize inflation, the consumer would then want some accommodation of a larger output gap…:). Something like that…? Only when the CB is unable to stabilize inflation (using phi_pi) should it use output gap stabilization…right?

Again, it depends on the model. A larger output gap is not necessarily good. Higher consumption is good, but the household has to work for that. It may be a bad time to work more if TFP is low. It may have to do with the shock combination, i.e. output signalling which type of shock is happening.

Many thanks!!!
In my model, for each shock, phi_y < 0 is optimal except for preference shock. So like for all the shocks, except preference shock, the benefit to the household (higher consumption) outweighs the cost (more work), and hence it is better for the central bank to accommodate a larger output gap instead of stabilizing it, right? Only for preference shock is the reverse.

Let me also mention that while the CB is accommodating the output gap, it is at the same time stabilizing the inflation rate in my model. I know there is a trade-off here. So I guess the accommodation of the output gap here does not pose a problem for stabilizing the inflation rate under the estimated optimal simple rule…right? Otherwise, the CB would not do it.

I am not following here. I don’t think there is a nice one-to-one mapping. Different shocks will result in a different mapping from variable movements to frictions.

Sorry for the confusion. I am just wondering if there is any reason(s) why phi_y < 0 can be optimal for the central bank. Because it improves the welfare of households in the model? But how?

There is only one friction, actually. Sticky price. So like phi_y < 0 being optimal is linked to prices being sticky? Lemme also say the estimated calvo parameter in the model is very low…about 0.02.

No, there is also monopolistic competition. Both together lead to deviations of output from efficient output and to price dispersion. Each shock in the model leads to different movements in these two gaps. The reaction to inflation and output simply reflect how to best address those.
Note that this is complicated by the fact that the Taylor rule describes off-equilibrium behavior as mentioned in