Why would optimal coefficient under Ramsey allocation be negative?

In Gali’s paper, “Monetary Policy and Unemployment,” the optimal coefficient on the output gap is negative. Gali did not explain it, but is there some theory that supports absorbing the output gap to stabilize inflation and output? I am doing some experiments, and I have a similar result. The optimal coefficient on the output gap is negative. Following an inflation shock, absorbing the output gap stabilizes inflation better. I have thought about it, but I can not figure out the theory. Or maybe we should put a lower bound on that coefficient to make it consistent with conventional theory (Here, I mean a positive optimal coefficient on the output gap)?

It will always depend on the model and the combinations of shocks. In most models, the planner faces a tradeoff for some shocks, e.g. between stabilizing inflation and the output gap. It’s not a priori clear what the sign for some coefficients in the rule should be. For that reason, I would accept negative values.

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Thanks. But are there any known plausible theoretical explanations that explain it?
Let’s say the optimal coefficient on the output gap is negative when all shocks are active in the model but positive when I drop productivity shock. Then productivity shock is causing or contributing to the optimal coefficient on the output gap being negative, right? And I still sort of need a mechanism to explain why absorbing the output gap following a productivity shock will maximize welfare. Maybe I can do research on what model features or parameters, or shocks cause this to happen :). Or have you come across any paper that talks about such things, if I may ask? Like how model features, assumptions, shocks, and parameters affect the optimal coefficient on the output gap.

Maybe not in your area, but I am curious how central banks feel about this result. Like if there is a negative supply shock, the central bank should raise the interest rate to maximize welfare (given that the optimal coefficient on the output gap is negative). In my mind, I think central bank practitioners would likely say no, and would likely reject such a solution…my thought, though. Or maybe not.

I am not sure the interpretation is that straightforward. The Taylor rule describes off-equilibrium behavior. In equilibrium, it’s not clear what the covariance between the output gap in the interest rate will be. See e.g.

But yes, it’s always a good idea to conduct a sensitivity analysis to find out what is going on in the model.

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But whether the covariance is positive or negative does not have theoretical underpinnings, right? Or maybe, we can do some reverse engineering:

  1. If the central bank’s goal is to stabilize the inflation rate and output, and finds the optimal coefficient on the output gap to be negative, then it means it will be more costly if the coefficient on the output gap was positive instead. Is it plausible that the optimal coefficient on the inflation gap undershoots the policy targets so that the Ramsey planner needs a negative optimal coefficient on the output gap to eliminate the undershooting?

Or should we treat the fact that the optimal sign on the output gap is unclear as a numerical plausibility without necessarily having theoretical underpinnings?