Optimal Monetary Policy without IS equation

Hi Forum,

when dealing with optimal monetary policy, central banker does not consider IS equation both under discretion and commitment? In most cases I see the following form,

\min \; \pi_t^2 + \nu x_t^2 s.t. \pi_t = \beta \mathbb{E}_t \pi_{t+1} + \kappa x_t

but no IS equation. Can someone please explain why? thanks a lot.

Do you have an example? It’s not true for e.g.

I am referring to Gali’s textbook (2nd edition) example of 5.2.1 and 5.2.2. The optimal conditions are derived without IS equations as one of the constraints.

As documented in the book further down below, the DIS equation is an implementability constraint. It describes how agents react to interest rates with their consumption. In the basic planner’s problem, the planner simply chooses consumption. Hence, the DIS equation is not needed.

Thank you very much Prof. Pfeifer