Consider the following monetary regime:
The monetary authority targets a growth path for nominal GDP.
The monetary base solely takes the form of mutual fund claims against the monetary authority's asset portfolio.
The monetary authority charges a management fee, so that the yield on those mutual funds is less than the yield on the monetary authority's asset portfolio.
Banks offer a variety of checkable deposits and interbank claims are settled using mutual fund balances at the monetary authority.
From the point of any particular bank, its deposits are subject to being redeemed with mutual fund balances at the monetary authority.
The sole hand-to-hand currency is issued by banks. Depositors withdraw currency from ATM machines or teller windows at their banks, hold it or spend it as they choose. Retailers deposit the currency along with any paper checks they receive at their own banks. The currency is cleared like the checks.
From the point of view of any bank issuing hand-to-hand currency, the currency is subject to being redeemed with mutual fund balances at the monetary authority.
Under "normal" circumstances all nominal interest rates are positive other than interest rates on hand-to-hand currency which remains zero. Banks keep balances at the monetary authority to settle interbank claims.
They earn a return on those balances equal to the yield the monetary authority earns on its asset portfolio less the management fee. Banks have no reason to hold balances other than for settlement purposes because by holding the assets directly, they earn a higher yield and the risk is the same. Any losses on the monetary authority's portfolio are passed on to the banks because the balances at the monetary authority are mutual funds.
Banks issuing currency are able to fund part of their asset portfolios at a zero nominal interest rate. This is usually attractive to a bank, but the banking system is competitive, and the result is that the difference between the interest rates paid on deposits and the interest rates charged on bank loans shrinks to compete away all of the profits from issuing currency. This happens because banks issue their own currency to their depositors (and borrowers, if they want to use currency.)
The monetary authority uses open market operations to adjust the quantity of base money however much is needed to keep nominal GDP growing on its target growth path.
From time to time, interest rates throughout the economy fall substantially. This is due to a temporary increase in the supply of saving and/or decrease in the demand for investment. Another way to frame the phenomenon is that there is a temporary increase in the supply of credit and/or a decrease in the demand for credit.
The yields on earning assets held by banks fall. In a conventional banking system, this would motivate banks to reduce the quantity of credit supplied and hold more reserves. However, the yield on the monetary authority's earning assets fall as well, and so does the yield that banks earn on the balances they keep with the monetary authority.
The banks are less willing to supply deposits and hand-to-hand currency. For the deposits, the result is a decrease in the interest rates banks pay on deposits. However, that doesn't happen with currency. The interest rate on currency remains at zero, and so with the lower opportunity cost of holding currency, the demand to hold that currency rises. The banks, benefiting less from issuing currency, lower the interest rates on deposits more than in proportion to the decrease in the yields on their earning assets.
Once the interest rates on deposits are so low that funding earning assets is cheaper by issuing deposits than issuing currency, then banks will cease issuing currency. However, for those using currency and deposits, as the interest rates on deposits fall, using (and holding) currency becomes more attractive than holding deposits.
The result is a shortage of hand-to-hand currency.
The lower the equilibrium interest rate on deposits, the worse the shortage. Once all banks are at a point where none of them are issuing currency, it is solely the increase in the demand to hold currency at lower deposit interest rates that exacerbates the shortage.
Assuming banks have call provisions on their currency, the lower the yields on earning assets and the longer this period of low interest rates is expected to last, the more likely banks are too exercise their call option. As banks call the currency, the quantity outstanding shrinks, further exacerbating the shortage of currency.
Nominal GDP, however, continues on target, with all purchases of goods and services being made by check, either paper or electronic. There is no shortage of clearing balances, the monetary authority issues them as needed. There is no shortage of credit, the banks can fund loans by borrowing using deposits with still lower interest rates. There is no shortage of money in the form of deposits. The yields on them fall to a point where the demand to hold them matches the quantity banks are willing to issue.
The "problem" of low interest rates solely creates a problem of a shortage of hand-to-hand currency. Banks stop issuing currency because it is not profitable to issue it.
Suppose that instead of a general increase in saving supply and/or decrease in investment demand, or alternatively, increase in credit supply and decrease in credit demand, there was a change in the willingness to bear risk. For example, investors sell off corporate bonds and buy short term government bonds--T-bills.
To the degree that the monetary authority holds T-bills as an earning asset, the decrease in the yields on T-bills results in a decrease in the yields banks earn on their balances. There is no tendency to build up larger reserve balances. If anything, there would be a motivation to hold smaller balances as well as to reduce T-bills held directly and instead purchase other sorts of securities and expand lending. On the other hand, the lower yields on T-bills held either indirectly as mutual fund balances with the monetary authority or directly, would result in a decease in willingness of banks to supply currency and deposits, and so, a lower interest rate on deposits.
More importantly, the lower yield on T-bills would tend to raise the demand for currency and deposits. The decrease in the supply and increase in the demand for deposits results in lower yields on them, but the yield on currency remains zero. As long as issuing currency is profitable based upon the yields on earning assets, the banks will issue added currency to meet the demand.
Because bank deposits and currency are the liabilities of commercial banks, it is possible that the desire for safety would drive the yields on T-bills below zero. While this would raise the demand for currency, it would be mistake to carry over conditions that would apply to a system of government currency rather than private currency. In particular, when interest rates on T-bills are zero, privately-issued hand-to-hand currency is not a perfect substitute for T-bills. Assuming the government, with its power to tax, is more creditworthy than any particular bank, then privately-issued currency is inferior to T-bills with a zero yield. And so, if market clearing requires negative yields on T-bills, then the yields on T-bills will fall below zero.
Now, the lower T-bill yields fall, the greater will be the increase in the demand for the deposits and currency issued by sound banks. If banks are typically poorly capitalized or hold risky earning assets, then T-bill yields could become quite negative. On the other hand, a more conservative banking system would limit the decrease in T-bill yields.
If the increase in the demand for bank deposits and currency is large enough that the interest rates banks can pay on deposits fall low enough, then it will stop being profitable for banks to expand their issues of currency to meet the demand. They will stop, and a shortage of currency will develop.
If all banks are identical, then all of them would cease issuing currency at the same time. However, consider a situation where depositors perceive that different banks have different risks. Those banks perceived as least risky will see an increased demand for their deposits and currency. As low risk yields are lower, it would be the bank perceived as least risky that would find that it can borrow by issuing deposits at interest rates so low that issuing currency is no longer profitable.
While there is, of course, a shortage of the least-risky bank's currency, there would remain plenty of currency issued by those banks perceived as relatively risky. From their perspective, issuing currency at a zero-nominal interest rate rather than the higher deposit interest rates they must pay would be profitable. While those seeking to substitute bank deposits for T-bills whose yields have been driven to less than zero would hardly find currency issued by relatively risky banks attractive as a safe investment, those using currency for small face-to-face transactions would still have plenty of currency. Given the small amount of currency held for that purpose, risk is not a major concern.
Of course, if the demand for bank deposits, including those issued by the relatively risky bank is large enough, then no bank will find it profitable to issue currency. The result is a shortage of currency.
A shortage of currency is an undesirable situation. However, reductions in nominal expenditure on output are worse. The advantage of currency privatization is that during occasional periods of low interest rates, there is no tendency for spending on output to fall. Instead, there are shortages of hand-to-hand currency.