new keynesian model with technology shocks

Question

New Keynesian model with technology shocks: Consider a New Keynesian economy with equilibrium conditions given by

where and are in ation and aggregate output in period t, x_t = y_t- y_tⁿ is the output gap, y_tⁿ is the natural rate of output, Ρ is the exogenous discount rate, and σ > 0, k > 0 and 0< β < 1. Monetary policy is described by a simple rule of the form

with Φ_Π > 1. Labour productivity is given by

where is employment and is an exogenous technology parameter that evolves according to

where is a mean-zero process and persistence parameter p_aε [0 1). The underlying RBC model is assumed to imply a natural output level proportional to technology

where Ψ_y>1
(a) Discuss the real output and in ation expressions verbally.

(b) Determine the equilibrium response of output, employment and in ation to a technology shock.

(c) Describe how these responses depend on the value of Φ_Π and k. Provide economic intuition. What happens when Φ_Π -> ∞ ? What happens when the elasticity of labour supply (HINT: one of the ingredients of parameter k) increases ?

(d) Analyse the joint response of employment and output to a technology shock and discuss briefly the implications of the role of technology as the source of business cycles.