1. Uptinial monetary policy at the zero lower bound under discretion. Consider the standard New Keynesian model with a zero lower bound constraint is = Et$t+1 51i5t E11T1+1 W] [1} 1H'1', = Ez''rH + HE: {E} if, = max [l]. T: + $1: 'iTt 'i' rmt] {3) Assume that n follows a two state Markov process 155 p(D I): where [I a: d a: l, n = 11, r: i} in the low state and r; 2 TH :3 i] in the high state. (a) Show that when we are in the high state such that r: = 1*}; it must be the case that r=, 111:6, and i1=rg. {b} Show that when we are in low state such that n = r; s: i} that the zero lower bound must bind. [c] 1|Iiiihern the economy is in the low state, expectations can be calculated as Err+1=l1 {il''c + Eng, E11+1=El :]L + 5:11:14. Dr, in other words, when in the low state, there is a 1 d probability of remaining in the low state next period and a d probability of returning to the high state. Using Equations [1) and {2) and the above expectations, solve for r; and an; as functions of parnieters. HINT: from. {a} it follows that 11'}; = 1.3:,- = i]. {d} Using your answer to (cl, how does 5-, the probability of exiting the low state, affect ination and the output gap? If 5 becomes smaller, i.e. the low state is expected to last longer, doe: the recession get better or worse

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