Given that I do not expect to see free banking in the immediate future, I would like to see one, or preferably more, central banks that now target inflation try targeting nominal GDP targeting instead. Targeting nominal GDP has some prospective advantages over inflation targeting. One is that nominal GDP targeting allows what seems to be a more appropriate behavior for prices over the business cycle, allowing "good" (productivity- rather than money supply-driven) deflation during the boom and "good" inflation during the bust.
Another is that inflation targeting as it has been both most widely proposed and as it has always been adopted has been a "bygones are bygones" version, with no later compensation for past misses of the target. During the Great Recession, many central banks undershot their targets, even allowing deflation to occur. They never corrected their mistakes. Nominal GDP targeting in the form that Scott Sumner and others have advocated it requires the central bank to undo its past mistakes. If it undershot last year's target, it has to increase the growth rate of the monetary base, other things being equal, to meet this year's target, which is last year's target plus several percentage points.
For all that, I am not an enthusiast of nominal GDP targeting. It is worth experimenting with, seeing as how that the widespread use of inflation targeting did not prevent the worst financial crisis since the Great Depression. (Indeed, inflation targeting aggravated the crisis with procyclical policies, according to accounts that I find convincing such as Robert Hetzel's book The Great Recession.) Anyone with a long memory, though, will have some skepticism towards nominal GDP targeting. In the late 1980s and the 1990s a wave of enthusiasm for inflation targeting swept through economists and policy makers. Inflation targeting was supposed to be superior to what had come before it. As with nominal GDP targeting now, its advocates had the luxury of contrasting an idealized hypothetical system with an actual system that, like all actual systems, had faults. (George Selgin's Less Than Zero was one of the few real criticisms of inflation targeting, comparing idealized inflation targeting to an idealized theoretical benchmark, the "productivity norm," in a consistent way.) Now that we have experience with inflation targeting, we can compare actual inflation targeting to other actual monetary arrangements, and obviously it has lost the luster it had when it was only hypothetical. In particular, advocates of inflation targeting did not anticipate that central banks would be so willing to allow such rapid, though brief, deflation.
With nominal GDP targeting it may well also happen that there will be flaws that only become apparent through experience. My reason for thinking that flaws are likely is that, like inflation targeting, nominal GDP targeting is an imposed monetary arrangement. It is not a fully competitive one that that people are at liberty to cease using at will, individually, the way they can cease buying Coca-Coca and start buying Pepsi or apple juice instead. Nominal GDP targeting when carried out by a central bank, which has monopoly powers, is a form of central economic planning subject to the same criticisms that apply to all forms of central planning. In particular, it does not allow for the occurrence of the type of discovery of knowledge that comes from being able to replace one arrangement with another through competition.