Optimal discretionary monetary policy In class, we focused on the problem of optimal monetary policy when the central bank faces a trade-o' between stabilization of the ination rate and stabilization of the output gap. We found that, if the central bank cannot commit to a specic path of future policies, the equilibrium paths of ination and output gap follow a} = 0191111., :13; = mI"u, where u, = K:(yf y?) is a shock that makes the equilibrium with exible prices not efcient. When deriving the dierences between these paths and those obtained under commitment, we never discussed they ways these paths were implemented. We implicitly assumed that the monetary authority can choose its desired level of ination and the output gap at each point in time. Of course, in practice, a central bank cannot directly set either variable. One way to implement that policy is to adopt an interest rate rule that guarantees that those paths are attained. (a) Inserting the derived paths for if, and :5, into the households' IS equation 1 . 37t = E[? E1711\" T:)+]Et$t+1, derive the equilibrium nominal interest rate as a function of a": and at. Is this a desirable interest rule? [Hint: remember our discussion of the desire to have rules that lead to a unique equilibrium and do not allow for self-fullling equilibria.] (b) Consider an interest rate rule of the form ' E at = r, + thwart. Using the equilibrium path of ination (which depends on at), nd the coef- cient 11),, that makes this rule consistent with the interest rate rule derived in the previous