Growth and seignorage, and an alternative explanation of the inflationgrowth relationship. (Friedman, 1971.) Suppose that money demand is given by ln (M/P) = a − bi + ln Y, and that Y is growing at rate g_Y. What rate of inflation leads to the highest path of seignorage?

Sol:

We can focus on a situation in which g_M, p, i, and rare constant and in which p^e = p. Although not technically correct – since Y and thus M/P are growing – such a situation will be referred to as a steady state in what follows. Under these assumptions, it is therefore reasonable to assume that output, and the real interest rate are unaffected by the rate of money growth and that actual and expected inflation are equal. Taking the exponential function of both sides of the money demand function, which is given by 

The nominal interest rate is given by i=r+p^e. In steady state, pe and rare constant and thus so is the nominal interest rate. Thus in steady state, the quantity of real balances must grow at the same rate as Y(t). In other words,