Consider an economy with the following structure: Aggregate demand: yt =Mt Pt +t (A quantity theory type

Consider an economy with the following structure:

Aggregate demand:

yt =Mt −Pt +μt (A quantity theory type equation)

Aggregate supply:

yt = yf t +γ (Pt −Pe t )+ηt where the symbols have the usual meanings and are in logs. μ and η are random errors.

Expectations are formed in one of the following alternative ways:

(a) Rational expectations:
Pe t = Et−1Pt

(b) Adaptive expectations:
Pe t −Pe t−1 = (1−λ)(Pt−1 −Pe t−1)
(i) Given rational expectations, solve for Pt and yt (in terms of the money supply and random errors, etc.).
(ii) Given adaptive expectations, solve for Pt and yt (in terms of the money supply, etc.).
(iii) For the two expectations hypotheses, derive the time patterns of the response of the price level to a unit change in the money supply.
(iv) For the two expectations hypotheses, derive the time pattern of the response of real output to a unit change in the money supply.
(v) Discuss the differences implied by the two hypotheses for the impact of monetary policy on real output and prices.