Policy rules, rational expectations, and regime changes. (See Lucas, 1976, and Sargent, 1983.) Suppose that aggregate supply is given by the Lucas supply curve

and suppose that monetary policy is determined by m_t = m_t−1+a + ε_t, where ε is a white-noise disturbance. Assume that private agents do not know the current values of m_t or ε_t; thus πe t is the expectation of  Finally, assume that aggregate demand is given by y_t = m_t − p_t.

(a) Find y_t in terms of m_{t −1}, m_t, and any other variables or parameters that are relevant.

(b) Are m_{t −1} and m_t all one needs to know about monetary policy to find y_t? Explain intuitively.

(c) Suppose that monetary policy is initially determined as above, with a > 0, and that the monetary authority then announces that it is switching to a new regime where a is 0. Suppose that private agents believe that the probability that the announcement is true is ρ. What is y_t in terms of m_{t −1}, m_t, ρ, y, b, and the initial value of a?

(d) Using these results, describe how an examination of the money-output relationship might be used to measure the credibility of announcements of regime changes.