Question 1 (100 points) Cagan model with output growth

Refer to lecture notes "Seigniorage and inflation: The Cagan model". It is     straightforward to extend the model in the lecture notes to allow for output growth.  All you need to do is to introduce real GDP, Y, into the real money demand function and the seigniorage function:

(i)    Mt  / Pt  = AYt exp(−aπt +1 *),   a > 0       Demand for real balances (ii)    πt +1 * −πt * = b(πt  − πt *),   0 < b < 1     Adaptive expectations     (iii)  St  =   ln(Mt  / Mt −1 )   Seigniorage

In (i) money demand is assumed to be proportional to output (income). In (iii)             seigniorage is measured relative to output -- as a share of lagged GDP. For this           exercise, assume output grows at an exogenously given rate n, i.e. ln(Yt  / Yt −1 ) = n , and ab < 1.

a)   Suppose money grows at a constant rate, i.e. ln(Mt  / Mt −1 ) = σ > n > 0. Rewrite the money demand equation (i) in growth rate form. What is the steady state     inflation rate? Modify Figure 1 to trace out the dynamics of (πt , πt *) .

b)  Derive the steady state seigniorage as a function of the money growth rate σ .       Find the seigniorage-maximizing money growth rate σmax   and the maximal           seigniorage S . Does the Cagan rule still hold? What about the inflation tax Laffer curve? Is it still true that a given amount of seigniorage can be collected at either a high or low rate of inflation?

c)   Assume 1/ a > n. Modify Figure 3 to study the dynamics of money growth and   inflation, given a fixed amount of seigniorage to be collected. Hint: Modify (3. 1)

(i) and (3.2) (i). The locus of (σt , πt *) that implies ∆πt +1 * = 0 is no longer the 45- degree line passing through the origin.

d)  Apply (c) to study the impact of a fall in the exogenous output growth rate n,    assuming the economy is initially in the low inflation equilibrium. Demonstrate that such negative real shocks can even lead to ever-increasing hyperinflation.