In free banking systems, banks have frequently included an "option" clause with banknotes. Hand-to-hand currency was generally redeemable on demand with gold, but if a bank could not honor that obligation, it would have the option to suspend gold payments. Typically, banknotes had a zero nominal interest rate, but during a period of suspension, the bank would pay interest. From the bank's point of view, this would be a penalty for failing to maintain redeemability. From the depositor's point of view, this is a bonus providing compensation for the temporary absence of redeemability.
Ignoring, for now, the penalty/bonus interest on banknotes, consider a scenario where the natural interest rate is negative. The interest rate needed to keep saving equal to investment at a level of real income consistent with the productive capacity of the economy is less than zero. More importantly, it is less than the storage cost of gold.
Assuming that the banks are sound, banknotes and bank deposits are as good as gold from the point of view of depositors. However, with the low natural interest rate, at historic market interest rates, banks will find low credit demand and so market interest rates will fall. Once the market interest rate falls below the cost of holding vault cash, banks will begin accumulate gold reserves. As other banks contract lending, each bank will suffer a gold drain.
Suppose banks respond to a scramble for gold reserves by suspending gold redemption. Again, leaving aside the penalty/bonus interest on deposits, there would be no zero nominal bound. Nominal interest rates on deposits could fall below zero, which would allow banks to fund loans that have negative nominal yields. Similarly, banks paying negative deposit rates could bid up the prices of bonds so that their yields are negative.
Unfortunately, hand-to-hand currency, with a zero nominal yield, becomes unprofitable for banks to issue when nominal yields on earning assets fall to a sufficiently low level. And, of course, if banks pay (or really, charge,) negative interest rates on deposits, depositors would demand additional banknotes. Of course, in a free banking system, there is nothing obligating any bank to issue hand-to-hand currency. And so, the result would be a system where gold redemptions are suspended, there is a shortage of zero-interest, hand-to-hand currency, but deposits (with negative interest rates) provide a sufficient quantity of money to fund payments. The nominal market interest rate would be negative, matching the negative natural interest rate.
And what about gold? Given the increase in gold demand, and the suspension of payments, the result would be an increase the market price of gold. In other words, gold coins (if any) and gold bullion, would trade at a premium. How high would the market price go? How high would be the premium?
One process is the one already explored in previous posts. The price rises until the expected future rate of decrease is equal to the negative real interest rate. With the nominal interest rate equal to the natural interest rate, the market price of gold rises to a point where it is expected to decrease at the negative nominal interest rate.
Existing hand-to-hand currency, assuming it will be honored at par in the future, should perform similarly to gold. It's price would rise to a premium against deposits, with the market price rising to a point where the expected future rate of decease equal the real and nominal interest rate. (If banks have a call option on hand-to-hand currency, then the quantity of currency would fall to zero, and all money would take the form of deposits.)
Given that the market price of gold (and banknotes) during this period would be uncertain, and taking into account that there are a variety of interest rates and the negative natural interest rate most likely only applies to short and safe financial assets, then the increase in the market price of gold and banknotes would be somewhat dampened.
There is a second market process. Those who had already been holding gold before the suspension receive a capital gain, become wealthier, and are motivated to consume more. This is reduced saving, which tends to raise the natural interest rate--making it less negative.
There is no similar "Pigou" effect for the banknotes. While those holding them, (presumably, most people,) would earn a capital gain when they appreciated, this would be matched by the loss to the issuing banks. For example, suppose banks had been using their banknote issue to fund holdings of T-bills. When nominal interest rates on T-bills turn negative, the banks are funding earning assets with negative yields with zero nominal yield currency.
If the banks are expected to suspend payments when the natural interest rate is negative, then the increase in the market price of gold will motivate everyone--banks and members of the nonbanking public--to redeem bank liabilities for gold. This would, of course, hasten the suspension.
Without the penalty/bonus interest on banknotes, and presumably something similar for deposits, it isn't obvious how or why the banks would contract credit and money to return to redeemability. Avoiding the penalty interest is why banks would ordinarily maintain redeemability, and ending the penalty is why they would take the actions that allow them to resume redeemability. Naturally, the banks have no reason to want to pay any penalty, but the depositors are likely to insist on what to them is a bonus for putting up with periods of suspension.
The penalty/bonus on banknotes (and deposits) puts a floor under nominal market interest rates during the period of suspension. And so, even if the natural interest rate is negative, the nominal market interest rate will not only remain positive, but rise to a higher than normal rate.
Banks, then, will be motivated to contract lending and the quantity of money, for two reasons. Lower credit demand and returns on earning assets combined with higher costs for deposits. As before, with suspension, the premium on gold will provide a capital gain to all of those already holding gold (including those redeeming gold before the suspension occurs,) which raises their wealth, reduces saving, and raises the natural interest rate. This makes the natural interest rate less negative.
At the same time, the contraction of money and credit (or equivalently, the reduction in real expenditures due to a natural interest rate below the market rate) will result in decreased prices and wages. As explained in previous posts, the price level must fall below its long run equilibrium level. One the price level is low enough, the expected inflation generated by the anticipated return of the price level to "normal," results in a lower real market interest rate. One the real market interest rate is equal to the natural interest rate, (both negative, by assumption) then real expenditure will rise to a level consistent with the productive capacity of the economy.
There is no "Pigou effect" for the banknotes and deposits. While the lower price level does raise the real wealth of those holding them, it reduces the real wealth of the banks directly, and indirectly, of those who have borrowed from the banks. However, the decreased price level does provide a real capital gain to those holding gold (at any given nominal price of gold.) While it is the general deflation of prices (including wages) that brings the market price of gold back to par, and so, the assumption of a "given" price of gold is unrealistic, to the degree that the conditions leading to a negative natural interest rate are persistent, long run equilibrium will depend on a Pigou effect. The price level (including wages) must fall to a sufficiently low level, that the real wealth (in the form of real gold holdings) is high enough, so that saving is low enough, that the natural interest rate is positive and equal to the market rate.
On the other hand, if the problem is temporary, so that the negative natural interest rate is solely due to excessively high saving supply and diminished investment demand perhaps due to perverse expectations, then no such permanent decrease in the price level is necessary. Once the price level is low enough so that expected future increases reduce the market rate to that natural rate, then recovery will falsify the perverse expectations, resulting in reduced saving supply and increased investment demand, the natural interest rate will recover to normal levels as will the price level.
Admittedly, the above analysis is preliminary. However, it does suggestion that while suspending gold redeemability does provide at least a dampening of deflationary pressures, the penalty/bonus interest exacerbates them. (Perhaps option clauses could set the penalty/bonus relative to some market rate, reducing any perverse effect.)
Of course, I don't favor a return to a gold standard. While I favor the privatization of hand-to-hand currency for a variety of reasons, tying both currency and deposits to gold or any other precious metal is undesirable. For the time being, keeping them both redeemable in reserve balances at the Fed is the least bad option, with the Fed being required to keep GDP (money expenditures on output) on a slow, steady, growth path. Imposing index futures redeemability on the Fed, should be explored. If effective, shifting that obligation directly to the banks will likely be a better approach to full privatization rather than depending on gold.