Write out the steps in the derivation of the optimal rate rule stated in the following slides and include comments to explain what you are doing in your derivation.

Monetary Policy The desire of a central bank to minimize the loss function L = (yt - YP) +B(It - TT ) using the IS curve yt - y" = -Sy (rt-1 - r*) is frustrated by the Phillips curve Tit = Tit-1+1 (yt -JP Central banks minimize the loss function subject to the constraint of the Phillips curve.Monetary Policy We can implement this constrained minimization by writing the loss function as 2 L = (yt - P)"+B Tt- 1+7 (yt - yp ) TTT and finding the minimum OL ayt = 2 (yt - yP ) + 2By(It-1 + 7(yt - yp) - IT) = 0 from which ( yt - YP ) = - By(Tt - IT) This is the optimal path for the central bank.Monetary Policy: The Optimal Rate The central bank at time t wants to have (yt+ 1 - yp ) = -By(It+1 - TT) The central bank predicts inflation using the Phillips curve Titt1 = nitty ()+ +1 - yP) and can control the output gap in the next period via the IS curve yt+1 - y = -Sy (re - r*).Monetary Policy: The Optimal Rate Substituting the Phillips curve Witt1 = nitty (yz+1 - JP) into the optimal path (yt+1 - yP ) = -By(Tt+1 - 7) , solving for the output gap, and clearing the output gap with the IS curve, we get H It = (It - 7 T ) . Sy ( 7 + 1 yB This is the optimal rate rule