Like most monetary economists, I am aware that Milton Friedman argued that the "optimal quantity of money" generates a deflation rate equal to the equilibrium real interest rate. From the Fisher effect, the nominal interest rate on risk-less bonds would be zero. The opportunity cost from holding risk-less money rather than risk-less bonds would be zero. The amount of real balances that people would choose to hold would drive any implicit liquidity yield on money to zero. This is optimal because it is costless for the government to print money.
There is a certain plausibility to the argument. But I find it difficult to take seriously. Why? Because of the influence of the "New" Monetary Economics, and especially Greenfield and Yeager's Black-Fama-Hall Payments system.
While Greenfield and Yeager pitched their idea as a proposal for reform, they did mention that it would be helpful in understanding our existing monetary order by way of contrast. One of the characteristics of the New Monetary Economics is consideration of monetary regimes with no hand-to-hand currency. Consider that for just a moment. All money takes the form of checkable deposits. These could be debt instruments, with bank stockholders bearing substantial risk. Or perhaps they take the form of mutual fund shares so that all risk is borne by those holding the "deposits." That issue is not important here. What is important is that all money can pay interest.
In a monetary regime where all money bears interest, then Friedman's argument dissolves into nothing. If the real interest rate is 3%, then a stable price level and 3 percent interest rate on deposits results in a real interest rate on risk-less deposits equal to that on risk free bonds. The real demand for money drives any liquidity benefit from holding money to zero.
On the other hand, with a 10% inflation rate, a 13% nominal interest rate on money has that same effect. And with a 5% deflation rate, a minus 2% interest rate on deposits also has that same effect. Whatever the selected inflation or deflation rate, the quantity of money should adjust to the demand to hold it, with the nominal interest rate on money equal to the nominal interest rate on risk free bonds--equal to the chosen inflation rate plus the risk free equilibrium real interest rate.
Of course, in the real world we do have zero-nominal-interest-rate hand-to-hand currency. So, what does consideration of the world without "currency" tell us? That the so-called "optimum quantity of money" is really about the optimum performance of the price level, not for money as whole, but for only that portion of the quantity of money that takes the form of hand-to-hand currency. It is about generating a deflation rate to create the optimum currency-deposit ratio.
Considering the role of hand-to-hand currency in modern economies, is it really sensible to place much weight on this factor? With mild deflation, people will have higher currency deposit ratios, and so will have full wallets and purses when they need to make small purchases?
Interesting, but is this really an important issue in comparing monetary regimes? If 100 years from now, we really have a monetary order with little or no hand-to-hand currency, will the optimal performance of the price level change? Will the optimal quantity of money (or really, the optimal currency-deposit ratio) change?
I think the reason hand-to-hand currency receives such emphasis is that deposits are treated as claims to base money--chiefly hand-to-hand currency. Turn that around. Consider hand-to-hand currency to be a claim to interest bearing deposits. Even under current conditions, consider Federal Reserve notes to be claims to reserve balances rather than vice-versa.
While much of the new monetary economics considered monetary systems without hand-to-hand currency, Greenfield and Yeager (and others, including me) were soon adding in privately-issued banknotes as more or less an afterthought. Banks could issue hand-to-hand currency on the same terms of deposit accounts, and any excess issue would be returned through interbank clearings.
While in days of yore, when banknotes were common, writing down serial numbers and keeping track of who withdrew currency and when it was returned through the clearings would be difficult, with today's data processing and imaging technologies, already used for checks, allowing those who withdraw banknotes to continue to earn interest until they are spent and clear would be feasible.
Rather trying to make hand-to-hand currency attractive to hold through manipulating the price level, perhaps a better approach would be to find ways of paying interest on hand-to-hand currency. That way, the optimum currency-deposit would be consistent with the inflation rate and price level free to accomplish other purposes.