Consider an economy that is characterised by the following New Keynesian model:
xt = Etxt+1 −1 σ[it − Etπt+1] + ut (6)

Economics 406 – Fall 2021 New Keynesian Model – sample questions

1. Consider an economy that is characterised by the following New Keynesian model:
   xt = Etxt+1 −1 σ[it − Etπt+1] + ut (1)
   where xt is the output-gap, it the nominal interest rate, πt inflation rate and ut a demand shock that follows a stationary AR(1) process:
   ut = γut−1 + νt,, 0 < γ < 1 (2)
   and νt ∼ N(0,σ2 u)
   πt = βEtπt+1 + κtxt + et (3) where et is a cost-push shock that follows a stationary AR(1) process:
   et = ρet−1 + ǫt, 0 < ρ < 1 (4)
   and ǫt ∼ N(0,σ2 ǫ) The objective of the central bank is to minimize the loss function:
   E0∞ X t=0βt(π2 t + λx2 t) (5)
   (i) Briefly comment where equations (1) and (3) come from [2 marks]
   (ii) Find the optimal policy under discretion and interpret your answer 5 marks Using the guess solution that πt = γet, show that xt = −h κ κ2+λ(1−βρ)iet and πt =h λ κ2+λ(1−βρ)iet [6 marks]
   (iv) Briefly comment why policy under discretion is inefficient compared to policy under commitment [2 marks]
2. Consider an economy that is characterised by the following New Keynesian model:
   xt = Etxt+1 −1 σ[it − Etπt+1] + ut (6)
   where xt is the output-gap, it the nominal interest rate, πt inflation rate and ut a demand shock that follows a stationary AR(1) process:
   ut = γut−1 + νt,, 0 < γ < 1 (7) and νt ∼ N(0,σ2 u) 1 πt = βEtπt+1 + κtxt + et (8) where et is a cost-push shock that follows a stationary AR(1) process: et = ρet−1 + ǫt, 0 < ρ < 1 (9) and ǫt ∼ N(0,σ2 ǫ) The objective of the central bank is to minimize the loss function: E0 ∞ X t=0 βt(π2 t + λx2 t) (10) (i) Derive the first-order conditions linking inflation and the output gap for the fully optimal commitment policy (ii) Explain why the FOC for time t differ from the FOC for time t + i for i > 0
   (iii) What is meant by a commitment policy that is optimal from a timeless perspective? [Explain in words]
   (iv) What is the FOC linking inflation and the output-gap that the central bank follows under the optimal policy from a timeless perspective?
3. Useful concept/theories to know:
   (i) How to derive the optimal policy under full commitment
   (ii) Optimal policy under discretion
   (iii) How and why optimal policy under commitment is different from discretion
   (iv) Stabilization bias and why it is different from the inflation bias
   (v) How can a central bank improve on the pure discretionary outcome?
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