Empirical analyses of Cagan’s money demand schedule for hyper-inflation have largely ignored the explosive nature of hyper-inflationary data. It is argued that this contributes to an (i) inability to model the data to the end of the hyper-inflation, and to (ii) discrepancies between “estimated” and “actual” inflation tax. Using data from the extreme Yugoslavian hyper-inflation it is shown that a linear analysis of levels of prices and money fails in addressing these issues even when the explosiveness is taken into account. The explanation is that log real money has random walk behaviour while the growth of log prices is explosive. A simple solution to these issues is found by replacing the conventional measure of inflation by the cost of holding money.