Consider the following variant of the Sargent and Wallace/Brock model of forward determination of the price level:
mt - pt = -α(rt + (Et(pt+1) - pt)) + yt (1.61)
yt = -βrt (1.62)
yt = d(pt - Et-1pt), (1.63)
where all notation is standard and all variables (except for rt) are in natural logs.
(a) Provide economic interpretations for these equations.
(b) Suppose the money stock has been growing for since time t = 0 at rate γ, and is expected to grow at that rate forever. Define m0 = m¯ . Solve for all endogenous variables at time t. (Hint: For this part and the next two, it isn't necessary to use recursive substitution to solve the expectational difference equation. Use intuition to get a much simpler solution.)
(c) Suppose the assumptions of the previous question hold, but unex- pectedly at time T the money stock jumps by an amount E. This jump is expected never to happen again. Solve for all endogenous variables.
(d) Now assume instead that at T , the subsequent growth rate of the money stock is expected to increase to γ + E and remain permanently at that rate. Solve for all endogenous variables.