1. Optimal monetary policy at the zero lower bound under discretion.  Consider the standard New Keynesian model with a zero lower bound constraint

xt    =   Etxt+1 − θ − 1 (it − Etπt+1 − rt)

πt    =   βEtπt+1 + κxt

it    =   max[0,rt+ ϕππt+ ϕ北xt]

Assume that rt  follows a two state Markov process

P =  ) ,

where 0 < δ < 1, rt = rL  < 0 in the low state and rt = rH  > 0 in the high state.

(a) Show that when we are in the high state such that rt  = rH  it must be the case that xt = 0, πt = 0, and it = rH .

(b) Show that when we are in low state such that rt  = rL  < 0 that the zero lower bound must bind.

(c) When the economy is in the low state, expectations can be calculated as Etπt+1 = (1 − δ)πL + δπH ,

Etxt+1 = (1 − δ)xL + δxH .

Or, in other words, when in the low state, there is a 1 − δ probability of remaining in the low state next period and a δ probability of returning to the high state. Using Equations (1) and (2) and the above expectations, solve for xL  and πL  as functions of parmeters. HINT: from  (a) it follows that πH = xH = 0.

(d) Using your answer to (c), how does δ, the probability of exiting the low state, affect inflation and the output gap? If δ becomes smaller, i.e. the low state is expected to last longer, does the recession get better or worse?

2. Technology shocks in the New Keynesian Model:  In this question, you will analyze technology shocks in the New Keynesian model following Ireland (2004; available on Canvas). You will need to download the .mod file Ireland NK.mod for this exercise. You will also need to download a copy of Ireland’s paper.

(a) How does Ireland model nominal rigidity?

(b) The taylor rule in Ireland’s models is given by

rt − ρrrt − 1 = ρππt + ρggt + ρ北xt + ϵr,t

Explain in words what the central bank is targeting with this rule.