Under commitment the CB might follow this problem as Monetary Policy strategy: $$ \min_{\pi_t,x_t}=E_0\sum^\infty_{t=0}\beta^t \left(\frac{1}{2} ( \pi_t^2 + \alpha x_t^2 )\right) $$ $$ \text{s.t. }\text{ } \pi_t=\beta E_t[\pi_{t+1}]+kx_t+u_t $$ where $u_t$ follows AR(1): $u_t=\rho u_{t-1}+w_t$ ($w_t$ is white noise). $\pi$ is inflation and $x$ is output gap . I need help to solve this mathematically. I assume the Lagrangian looks something like: $$ L(x_t,y_t)= E_0\sum^\infty_{t=0}\beta^t \left(\frac{1}{2} ( x_t^2 + \alpha y_t^2 ) -\lambda_t(\beta E_t[x_{t+1}]+ky_t+u_t-x_t) \right) $$ Am I right? How can I find the derivatives wrt. $x_t,y_t,\lambda_t$?

I am confused by how to differentiate with expectations. also I don't know how to deal with $\partial E_t[\pi_{t+1}] / \partial \pi_t$

All the parameters are positive.


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