Consider a variation of the rational expectations monetarist model with time-varying velocity of money, vt (recall that the velocity of money is the ratio of nominal income to real-money balances).
mt - pt + vt = yt (4)
yt = -(wt - pt) (5)
wt = Et-1(pt) (6)
vt = vt-1 + t; 0 < < 1; t ε (0, 2) (7)
Recall: Et-1(xt) indicates the mathematical expectation of variable xt conditional on information available at time t-1.
i) Assuming that the monetary authority observes vt before it fixes the money supply at time-t (so that mt can be conditioned on vt), show that aggregate production/output can be solved as:
yt = mt - Et-1(mt) + t
ii) Suppose that the monetary authority announces that it will stabilize aggregate production/output at yt = 0 by using a money supply rule which implies the following:
mt = Et-1(mt) - t
Show that the equilibrium price-level is given by pt = Et-1(mt) + vt-1.
iii) Is the monetary authority's commitment to stabilize aggregate production/output enough to pin down the supply of money and the price level in this economy?
iv) Assuming that the monetary authorities have the following "loss function":
losst = -yt + (pt)^2/2
determine if discretionary monetary policy will result in an inflationary bias for this economy.