Here is the question from an intermediate macroeconomics problem set. The topic is optimal rate rules and monetary policy. The question is attached and the Lecture slides

1. Write out the steps in the derivation of the optimal rate rule given on slides 2123 of lecture 12. Include a derivation of the the rst equation on slide 22 by minimizing the central bank loss function. In your derivation include comments that explain What you are doing. Monetary Policy Rules From Our Macro Model Working with a dynamic IS curve Yt = Y - Cyrt-1: The central bank: . knows Tit-1 and can work out Tit = Tit-1 + 7 (yt - yP). . determines the associated policy rate via Yt = Y - Cyrt-1. . knows the rate associated with equilibrium YP = Y - (yr* 2 The gap form of the IS curve is: yt - y' = -Sy (rt-1 - r* ) Lecture 12 - Monetary Policy Rules: R. J. Hawkins Econ 100B: Macroeconomics 21/ 30Monetary Policy Rules From Our Macro Model Working with a dynamic IS curve Yt = Y - Cyrt-1: ( yt - yP) = -yB(It - TT) At: Phillips curve -1 ( yt - yp ) = Tt Ity ( yt - y p ) -IT - ( *+ B (yt - y ) = mt- 1-1' Substituting yt - y? = -Sy (rt-1 - r*): It- 1 - r = 1 Tit- 1 - 7 T ) yB Lecture 12 - Monetary Policy Rules: R. J. Hawkins Econ 100B: Macroeconomics 22/ 30Monetary Policy Rules From Our Macro Model Solving for the Monetary Policy Rule: Writing this rule as: 1 HamWt) we see that when (y = "y = ,8 = 1 the rule becomes rt=r*+%(7rt7TT)