When I tell people that I have been doing research on why the government should drastically scale back the circulation of cash—paper currency—the most common initial reaction is bewilderment. Why should anyone care about such a mundane topic? But paper currency lies at the heart of some of today’s most intractable public-finance and monetary problems. Getting rid of most of it—that is, moving to a society where cash is used less frequently and mainly for small transactions—could be a big help.

There is little debate among...