Econ30852 assignment - find analytically the minimum loss

Question

Assignment -

Consider the Poole (1970) model (W. Poole, Optimal choice of monetary policy instruments in a simple stochastic macro model, Quarterly Journal of Economics 84, 197-216):

y_t = y₀ - a_it + u_t (1)

m_t = m₀ + by_t - ci_t + v_t (2)

where u_t and v_t are independent random variables (i.e., Cov(u; v) = 0) with zero mean, and y₀ and m₀ are constant terms in the IS and LM curve respectively. Assume that the central bank wants to minimize the loss function

L_t = E(y_t)² (3)

Find analytically:

1. The minimum loss obtainable under an interest rate targeting regime.

2. The minimum loss obtainable under a monetary targeting regime.

3. Using your results in 1 and 2 above, discuss how the choice of the optimal policy instrument depends on the nature of the shock.

Particular attention must be given to the theoretical explanation and economic intuition of all your equations and results. Please provide the full derivation of your results, with explanations of all steps taken to arrive at each solution.