Optimal Monetary Policy in IS-MR-PC model

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:

L = \min x_t^2 + \beta (\pi_t - \pi^T)^2

The baseline version of the model has the following equations:

x_t = - b(r_{t-1}-r_e) \\ \pi_t = \pi_{t-1} + \alpha x_t + \varepsilon^\pi_t

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:

x_t = -\alpha \beta (\pi_t-\pi^T) \\ x_t = -\alpha \beta (\pi_{t-1} + \alpha x_t -\pi^T) \\ x_t = -\frac{\alpha \beta}{1+\alpha^2\beta} (\pi_{t-1} - \pi^T)

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

r_t = r_e + \frac{\alpha \beta}{b(1+\alpha^2\beta)} (\pi_t - \pi^T)

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:

x_t = E_t x_{t+1} - b(r_{t-1}-r_e) \\ \pi_t = \pi_{t-1} + \alpha x_t + \varepsilon^\pi_t

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!

Do you need an analytical solution? In the context of basic forward-looking rational expectations models, you can often use the method of undetermined coefficients to compute a solution.