I am trying to work out a formal solution for the central banks’ problem in the slightly modified IS-MR-PC model (popularized by Carlin & Soskice (2015) and Carlin & Soskice (2023) textbooks).

The objective function of the central bank is:

The baseline version of the model has the following equations:

x_t is output gap, r_t and r_e are actual and neutral interest rate, \pi_t is inflation.

One can formally find the CB’s FOC for this problem by plugging in the Phillips curve into the objective and differentiating w.r.t. x_t:

Then combine it with the IS curve to obtain the following interest rate rule:

Once you have the expression for the interest rate, all other variables can be calculated based on it, hence the system is solved.

I am trying to solve a similar model with a **forward-looking** IS curve:

However, I face difficulty in deriving the interest rate rule in a similar manner. It seems to me that the FOC sould be unaffected, but I do not understand 1) how to express r_t as a function of other variables 2) how to find a closed-form solution for the output gap x_t given that the equation includes a forward-looking component. Any help would be greatly appreciated!