Consider an economy where aggregate supply is given by the Lucas supply curve: y= (-), where is actual inflation rate, and Te is private sector's expectation on inflation. Let yn denotes the socially optimal level of output, and inflation above some level, *, is costly. The central bank's objective function is characterized by U=A (y-yn) - (*), A> O (1) Finally, the policy maker controls money growth rate, and thus . (a) (3 points) Show the First-Best monetary policy (Pareto optimal allocation). (b) (3 points) What is equilibrium policy with commitment? Does it achieve the First-Best allocation? Explain. (c) (3 points) Show that the policy maker always has incentive to deviate from the allocation in (b), if there exists commitment issue. (d) (4 points) Solve the equilibrium policy under discretion and show that policy under dis- cretion is inferior to that under commitment. Now consider the environment with real shocks. The Lucus supply curve now changes to: - y=yna (TT) +, a > 0, where is an output shock (E (E) = 0, var (e) = 2). We assume that the output shock is observed by central bank but not by the public. The monetary-policy instrument is denoted by z and thus the inflation rate is affect by policy instrument as follows: T=2+V, where v is a shock not observed by the central bank and E (v) = 0, var (v) = 2. i.e., z controls imperfectly. The central bank's objective function is expeted version of (1), i.e. U = X(-n) - (*), - where is the target rate of inflation (note that NO expectation on terms of U, but central bank maximize E[U]), and A> 0. Now consider the case where the central bank chooses z using discretion rather than a rule. (e) (4 points) Show that the optimal setting of zis: * = a +". (f) (4 points) What is the equilibrium inflation? Show the inflation bias exists. Discuss briefly. (g) (4 points) What is y and E[U] in equilibrium. Does the extra inflation generate additional output or improve expected utility? If not, any explanation for the extra inflation?