Suppose a central bank cares about inflation and output

Question

Topic: Monetary Theory

1. Suppose a central bank cares about inflation and output such that it minimizes a loss function:

L= (1/2) E_{t i=0}∑^∞βⁱ [π_t+i² + λx_t+i²]

subject to a Phillips curve relationship: π_t+i = βE_tπ_t+i+1 + κx_t+i + e_t+i

where x_t is the output gap around the natural rate, π_t is inflation. σ, λ, κ, and β are coefficients, e_t is a cost shock.

a) Solve the problem for the first order condition under discretion and determine the value of x_t.

b) Suppose now that the central bank is uncertain about the true value of κ and it now takes on a time varying form: κ_t = κ + v_t. What kind of uncertainty does the central bank have?

c) What are the first order conditions in this case? (Note: v_t and x_t are uncorrelated).

d) Show how this kind of uncertainty will affect the impact of e_t on x_t.

e) Why do some people refer to this result as 'gradualism'?

2. Suppose a central bank faces a utility function of the form U = (λ/2)(y_t - y_n) - ½ π_t² where y_t is output around, y_n, the natural rate, π_t is inflation and λ is a coefficient.

Assume also that the central bank faces a supply function y_t = y_n + a(π_t - π_t^e) + e_t, where π^e is expected inflation and e is a supply shock; and π_t = Δm_t + v_t therefore inflation is driven by the money growth rate Δm_t and a velocity shock, v_t.

a) Calculate the optimal inflation rate, the expected level of inflation π^e, output and the expected output, and then interpret the results.

b) Show that the central bank suffers from time inconsistency and explain what this means.

c) Explain how in a multi-period game with reputation, a central bank may have the incentive to resist the temptation to cheat.

Please answer question as clearly as you can showing all the math work and steps if needed.