How does the relationship between deflation and the optimal quantity of hand-to-hand currency apply to a regime of free banking?
Consider a free banking system with a limited number of large, nationally-branched banks. Each bank issues both deposits of various sorts and hand-to-hand currency. Assume that that the deposits pay interest and that the nominal interest rate on the hand-to-hand currency is zero.
A reduction in the expected inflation rate should reduce the interest rates on nonmonetary assets and the nominal interest rates banks pay on deposits. Both reduce the opportunity cost of holding zero nominal interest currency. Under normal conditions, the more important effect would be a shift out of deposits into currency. The currency deposit ratio would increase.
Unlike a regime with a government monopoly on currency issue, the banks simply change the composition of the liabilities they use to fund their asset portfolios. There is no need to shuffle assets between the Fed, the banks, and the nonbanking public. In a free banking system, much of what today counts as "vault cash" is unissued banknotes. These aren't bank assets or liabilities, and they have no yield that would impact bank profitability. Clearinghouse balances instead form the key part of reserves, and like bank deposits, nominal interest can be paid on these reserves.
There are no complications involving the time path of government seniorage, because a free banking system generates little or no seniorage revenue for the government. Instead, a decrease in the expected inflation rate and so the nominal interest rate on earning assets reduces the profits banks earn by borrowing through the issue of currency for which they pay a zero nominal interest rate.
It is natural to assume that banks can profit by borrowing at a zero nominal interest rate and holding earning assets. However, redeemability limits each bank's issue of currency to the quantity its customers choose to hold. A reduction in nominal interest rates on both deposits and earning assets reduces the quantity of currency supplied by banks--that is, the amount the banks would like to supply. The lower nominal interest rates on deposits raises the quantity of currency demanded by their customers. The actual quantity of currency issued by banks matches the demand by their customers. By lowering nominal interest rates, a decrease in the expected inflation rate should bring the quantity supplied and quantity demanded closer to equilibrium.
That is what Friedman's argument about the "optimum quantity of money" amounts to in a free banking system. The quantity supplied and demanded of zero-interest currency may be brought into equilibrium by the correct expected inflation rate--likely a deflation rate. The quantity supplied and demand of deposits, presumably the bulk of the quantity of money in any free banking system, could clear through changes in nominal interest rates paid on the deposits.
If the cost of intermediation through the issue of currency is the same as the cost of intermediation for checkable deposits, then the optimal rate of deflation would be the negative of the competitive interest rate on checkable deposits. If currency is more expensive to print and maintain in circulation, then slightly less deflation would be optimal. As long as the rate of deflation generated by the productivity norm is less than (closer to zero) the rate of deflation that brings the supply and demand for currency into balance, then the productivity norm improves the performance of the zero nominal interest rate currency market relative to price level stability or mild inflation.
If there were a sharp drop in the real interest rate on earning assets, or perhaps a change in risk premia or the term structure of interest rates so that the real cost of intermediation rose, then the expected deflation rate generated by the productivity norm would make it more likely that the issue of currency would become unprofitable. If that were to happen, then banks would stop issuing zero-interest currency and customers would have to make do with deposits.
What if nominal interest were paid on currency? Consider the following scenario--when a depositor withdraws currency from his or her deposit, the principal amount is debited from the account, but the bank continues to credit interest to the account until the currency clears.
Consider an analogy with a paper check. Suppose a depositor writes a check to pay a bill, puts it in the envelope, but leaves it on his desk. The funds in the deposit continue to earn interest. After a few days, the check is mailed. The funds in the deposit continue to earn interest. The check is received and processed. The funds continue to earn interest. Eventually, the check is presented for payment. The bank finally debits the account on which the check was drawn, and no longer pays interest on the funds.
By treating currency in this fashion, continuing to pay interest until withdrawn currency clears, depositors can withdraw currency without opportunity cost. When currency is spent, the expected pattern would be that those receiving the currency, typically retailers, deposit it at their own banks. Currency, like paper checks or electronic payments, would be cleared. The person who withdrew the currency no longer earns interest and the retailer begins earning interest.
It would be possible, of course, for someone to spend currency and the person receiving the currency to hold it, and perhaps spend it again. Currency can circulate. The person who withdrew the currency and introduced it into circulation, earns interest until someone deposits the currency and it clears. Holding currency received in payment does involve an opportunity cost. The solution for an individual bearing an opportunity cost from holding currency is to deposit it.
Given this scenario, what is the advantage of lowering the expected inflation rate and so the nominal interest rates paid on earning assets, deposits, and currency? Paying interest on currency until it clears creates private incentives to deposit currency received in payment rather than hold and then spend the currency. These additional clearings involve real costs. By lowering the nominal interest rates, people will be more inclined to hold and spend whatever currency received in payment, which saves on clearing costs.
In a free banking system, individual banks cannot create an excess supply of currency because of the reflux. Those receiving currency in payment deposit the currency in their own banks. The currency returns to the issuing bank through clearing and results in a reserve drain on the issuing bank. A possible loss in the effectiveness of the reflux in controlling the over issue of currency must be weighed against any savings in clearing costs resulting from the expected deflation implied by a productivity norm.