1. Optimal monetary policy at the zero lower homid under discretion. Consider the standard New Keynesian model with a zero lower hound constraint E: = Err+1 - 34H}, Er'rH fr} ill W: = HEr''tH + HI: {3'} 1:1 : max [I], rt 'l' it"rr't + $1232] {3} Assume that r, follows a two state Markov prooess 155 P-( a I} where l] sf 5 at 1,11: 11:, a: [I in the low state and r1; : rH 2': [l in the high state. [a] Show that when we are in the high state such that rt = n; it must he the case that a} = 0,111: , and it = r3. [h] Show that when we are in low state such that r: = n, c: In that the zero lower bound must hind. (c) \"'hen the economy is in the low state, expectations can he calculated as Ert+1 =(1 Jl'L + 51TH: EI+1 ={1 151$]; + 51H. Dr, in other words, when in the low state, there is a l J prohahility of remaining in the low state next period and a d prohahility of returning to the high state. Using Equations (1] and [2] and the above expectations, solve for an; and 11' L as functions of parmeters. HINT: nm (a) it follows that an; = 3H = l]. [d] Using your answer to {c}, how does 15, the probability of exiting the low state, affect ination and the output gap? If {i becomes smaller, Le. the low state is expected to last longer, does the recession get hetter or worse