Your post “Does the zero lower bound exist thanks to the government’s paper currency monopoly” is very close to my answer in seminars of why private banks can’t undo what I propose central banks do.
For the record, I am not a fan of free banking. I am sympathetic with George Selgin’s claim (in a paper tweeted by David Beckworth) that the early days of central banking and 19th century US financial regulation may have been a step down in monetary stability and financial stability from free banking. But I believe that central banking with an electronic money system and monetary policy along the lines of what I discuss in my column “Optimal Monetary Policy: Could the Next Big Idea Come from the Blogosphere” is superior to free banking. I also give my view of the value of central banking in my post “Let’s Have an End to ‘End the Fed’” (Despite Ron Paul’s success in getting many people to chant “End the Fed” I think abolition of central banks and resumption of free banking is politically less likely–both in the US and in other nations–than the kind of electronic money system I recommend, so there is no strong argument for free banking as a politically easier solution to the zero lower bound problem.)
Many moons ago Matt Yglesias wrote that the “zero lower bound is a pure artifact of the existence of physical cash." In this post I’ll argue that the zero-lower bound, or ZLB, is an artifact of our modern central bank-managed monetary system, and not the existence of cash. In a free banking system in which private banks issue banknotes, competitive forces would force bankers to rapidly find ways to pierce below the ZLB, rendering the bound little more than a fleeting technicality.
What is the zero lower bound? When the economy’s expected rate of return drops significantly below 0%, interest rates charged by banks should follow into negative territory. But if banks set sub-zero interest rates on deposits, everyone will quickly convert them into central bank-issued paper currency. After all, why hold -2% yielding deposits when you can own 0% yielding cash? The inability to set negative interest rates is thezero-lower bound problem.
As I’ll illustrate, the threat of getting stuck at the zero-lower bound would impose such huge losses on private note-issuing banks that bank managers would quickly find creative ways to circumvent the problem. Central bankers, who aren’t beholden to the same financial motivations as private bankers, needn’t pursue these same zero-lower bound innovations with such zeal. This distinction has significant implications for the economy. Insofar as policies designed to remove the ZLB can prevent large macroeconomic distortions, central bankers are more likely to avoid such policies and destabilize the macroeconomy than private banker who, driven by bottom line concerns, will be quick to adopt ZLB-avoiding innovations.
Let’s set up our free banking system. Say that the Fed ceases issuing paper currency and only creates deposits. Into this void, private banks begin issuing their own paper dollar banknotes which can be exchanged for bank deposits at a rate of 1:1. This isn’t such a strange idea—for much of its history, Canada has enjoyed a privately-supplied paper currency. A few years later the economy nosedives and pessimism reigns. Private banks are desperate to decrease deposit rates into negative territory, say -4% or so. After all, banks earn income from the spread between the rate at which they borrow and the rate at which they invest. If, during bad times, a banker is investing at a -2% loss, he or she needs to be borrowing at -4% in order to earn spread income.
Unfortunately for our private banker, the intervening ZLB impedes rates from dropping into negative territory. Any attempt to cut to -4% and bank depositors will flock to convert negative yielding deposits into the bank’s 0% yielding banknotes. Very quickly the bank’s entire liability structure will be comprised of banknotes, a disastrous outcome since a bank that funds itself at 0% while investing at -2% will go broke very quick.
In a negative return world, profit-maximizing private banks would solve their ZLB problem using several strategies:
1. Remove Cash
If banks remove all of their already-issued cash from the economy in return for deposits, the deposits-to-cash escape route will be effectively erased, thereby clearing the way for banks to reduce deposit rates to -4%. One way to do this, courtesy of Bill Woolsey, would be for banks to issue cash with a call feature. Much like a convertible bond allows the bond issuer to force conversion upon investors, bank notes would carry a conversion clause permitting the issuing bank to call in all cash when it desires to reduce deposit rates below zero. 
2. Cease conversion into cash
Note-issuing banks might simply close the cash conversion window while allowing existing cash to remain in circulation. This would cut off any rush to convert deposits into cash upon a reduction of deposit rates to -4%. The price of existing cash would jump to a high enough level such that it would be expected to decline at a rate of 4% a year. Conversion stoppages are not without precedent. In 18th century Scotland, banks often issued notes with an option clause that allowed them to cease redemption should a bank run begin.
3. Penalize cash
By penalizing cash, a bank imposes a large enough cost on cash holders so that negative yielding deposits are no longer inferior to cash. There are plenty of ways for a bank to do this. One way is to impose a negative interest rate on cash by requiring cash holders to pay to "update” their bank notes lest they expire. This update fee, which would amount to around 4% a year, would forestall depositors from making a dash for cash when the bank sets deposit rates at -4%. In times past, locally-issued “scrip” like Worgl have had negative interest rates attached to them.
Another creative way for a banker to penalize cash is to impose a capital loss on cash holders. Rather than offering permanent 1:1 cash-to-deposit exchanges, banks might commit themselves to buying back cash (ie. redeeming it) in the future at an ever worsening rate to deposits. As long as the loss imposed on cash amounts to around 4% a year, depositors will not convert their deposits to cash en masse when deposit rates are cut to 4%.
In sum, a number of innovative routes are available for note-issuing banks to let their borrowing costs drop into negative territory. By necessity, private note-issuing banks will adopt these strategies in order to protect their shareholders from the painful effects of mass conversion of cheap deposit funding into relatively costly 0% cash.
That’s all fine and dandy, but our note-issuing mechanism is run by a centralized monopoly, not competing private banks. Because the ZLB is no less binding for central banks than it is for free banks, over the last few years economists and pundits have come up with all sorts of draconian techniques for central banks to escape the ZLB. There have been calls to ban cash, penalize it, and destroy it. At first I was somewhat appalled by these ideas as they seemed to be gross infringements on people’s ability to use cash. Over time I’ve realized that these authoritarian solutions are, somewhat paradoxically, the very same innovations that competing bankers would devise in a free banking world in order to free themselves of the ZLB problem. In other words, we can back out what a monopolist currency issuer *should* be doing to combat the ZLB by imagining what a network of competing banks *would* do. 
For instance, in a negative rate world a central bank ban on paper currency would be the equivalent of competing note-issuing banks simultaneously calling in their entire issue of paper currency in order to protect their solvency. If free banks were to penalize cash by redeeming it at ever deteriorating rates, this would be exactly the same strategy that Miles Kimball advocates central banks adopt in order to escape the ZLB.
That central banks have been so slow to evolve strategies for escaping from the ZLB could be due to any number of factors. Central banks aren’t privately owned nor are they disciplined by competition, and central bankers don’t have a mandate to turn a profit. Free banks, burdened by all of these checks, would be forced to rapidly adopt ZLB-escaping strategies or perish.
Further hampering efforts to get central banks like the Fed to innovate solutions to the ZLB is that these efforts might conflict with other goals. Withdrawing cash, penalizing it, or limiting conversion will put an end to, or at least diminish, the circulation of US paper dollars overseas. It might even result in the circulation of some other nation’s 0% yielding currency in the US. But the universal circulation of greenbacks is one of the most potent symbols of US hegemony, real or perceived. In the interests of protecting this symbol, innovations for escaping the ZLB may get short shrift. In a free banking system, these sorts of non-pecuniary motives are unlikely to outweigh the profit and loss calculation that dictates the necessity of adopting such innovations.
So the zero lower bound problem isn’t a problem with cash per se, it’s just a function of monopolistic intransigence. If you really want to short circuit the ZLB, better to devolve the provision of notes to profit-seeking private banks. Until then, hopefully evangelists like Miles Kimball succeed in getting central banks to adopt free banking-style contingency plans in preparation for the next time we experience a crisis that necessitates sub-zero interest rates.
 The idea that harsh central bank policies like banning cash or penalizing currency might mimic free banking responses is a recurring theme on this blog. Here, I hypothesized that in a world characterized by free banking, legal tender laws might evolve naturally as the result of market choice. It’s a strange world.