One of the characteristics of the New Monetary Economics, and especially Greenfield and Yeager's BFH payments system, is that all money is "inside" money. While it is an asset to those holding it, it is a liability to the issuer. (At first, Greenfield and Yeager emphasized mutual fund type means of payment, but if those are money market type funds, then the financial assets directly held by the funds and indirectly held by the depositors would be liabilities to those issuing them.)
Friedman's argument that the optimal quantity of money requires a deflation rate equal to the risk-less interest rate is based on the assumption that the cost of issuing money is the printing cost. This is approximately zero, and so the opportunity cost of holding money should be approximately zero as well.
However, banks creating deposits can hardly treat the cost of issuing that money as being the cost of printing the checks.
Investment projects take time and involve risk. The risk of the investment projects doesn't disappear. It can only be shared. For example, when a bank or other financial institution issues inside money, the depositors are able to spend the proceeds whenever they want. Someone else, for example, the stockholders of the bank, is bearing the risk that the projects funded by the depositors have yet to mature or will fail to pay off as hoped.
Now, the deflation rate that Friedman proposed is the risk-less interest rate, and so, it is necessary to abstract away from the possibility that investment projects may not pay off as expected. All that is left is that the depositor may choose to use the funds before the project has matured. This is a liquidity risk.
With a pure inside money system, like the BHF payments system, all money is inside money. All money is used to fund some kind of real investment. Abstracting away from the risk of loss on the investment, there still remains the risk that depositors want to spend their money before the investment projects mature. This risk can be diversified and shifted, but those bearing the risk, such as bank stockholders, must be compensated. This suggests that a competitive pure inside money system, like BFH, would pay lower yields on money because those using the money can spend it at any time, and the projects being funded only mature over time.
Friedman's optimal quantity of money applies to a pure outside money. It isn't used to fund anything. It is an asset to those holding it, and a liability to no one. He assumed that the nominal interest rate paid on the money is zero because he focused on hand-to-hand currency, and the real rate earned by those holding it is equal to the rate of deflation. Friedman's argument is that the rate of deflation should equal the risk-less rate of interest.
However, with a deposit-only monetary system, it would seem that interest could be credited to those holding outside money deposits at a rate sufficient to pay the riskless interest rate. For example, if velocity is constant, so the demand to hold money is growing with real output, and further, the risk less real interest rate is equal to the growth rate of real output, then by crediting a nominal interest rate equal to the real interest rate to deposit accounts, the nominal quantity of money would grow with the real demand to hold money, leaving the price level stable. The nominal and real interest rate paid on money would equal the growth rate of the economy, which is the risk less real interest rate.
Only by identifying money with zero-interest hand-to-hand currency is there an implication that the real interest rate on money must be generated by deflation. In the example above, a constant quantity of money, with constant velocity, so the real demand for money is growing with real output, would generate a deflation rate equal to the growth rate of real output, which is the risk less real interest rate.
Friedman, then, is arguing that the government can go into competition with the private sector in providing deposits. The private sector would pay a lower real interest rate on deposits than the government could provide, because those bearing the risk that depositors might want to spend their money before investment projects mature require compensation. The government, creating money fiat money out of thin air, has no need to worry about whether those holding money want to spend it before investments projects mature.
While the argument seems plausible enough, I think it is an illusion. If velocity is assumed constant and real output always grows at a constant rate, then this is the same thing as assuming that there is never a problem of people wanting to spend more now. On the other hand, if velocity should rise, then the demand for money relative to real income falls. If money continues to be created to credit to deposits at a rate equal to the real rate of interest, then some other means must be used to contract the quantity of money. Taxes might be increased or else interest bearing debt can be issued (and taxes collected later to pay that interest.) In other words, it is the taxpayer that is bearing the risk that people might want to spend before the "investment" projects mature. But, of course, the only investment project in this scenario is lending to the government.
Similarly, suppose some natural disaster reduces real output and part of the existing capital stock. Real output declines, and so, given velocity, the real demand to hold money falls. But because the capital stock is lower, the real interest rate rises. The amount of money credited to deposit accounts would need to rise at a faster rate at the same time the quantity of money needs to fall. Higher taxes now or in the future, due to an increase in government debt, would be necessary.
If the focus is on creating deflation so that the real return on hand-to-hand currency remains at the appropriate rate, there is little difference, really. If velocity rises, then the deflation rate would slow or perhaps even turn to inflation. Taxes or sales of interest bearing debt would be needed to keep deflation at the real interest rate. If there was a natural disaster reducing output and the capital stock, higher taxes or sales of interest bearing debt would be necessary to accelerate deflation while reducing the quantity of money to match the lower real demand.
Another way to look at the situation is that if we lived in a world where output always grows at a constant rate, velocity never changes, and the real interest rate is fixed as well, the risk of funding real investment projects with deposits would be minimal. What would be the problem? It could be nothing other than competition. Any one bank would suffer risk that people would move their money to other banks. But that would also mean that those other banks would have the ability to fund the projects. Clearly, this process would be imperfect, and so centralizing this function would obviously be beneficial. Well, no.
In my view, having money fund real investment projects is desirable. No doubt it is a classical bias that saving and investment are somehow "good" and result in higher standards of living in the future. Imagining that we instead just give everyone some fiat currency at the beginning of time, so that a significant (perhaps huge?) fraction of wealth is some kind of quasi-ponzi scheme seems less desirable. In my view, it is wealth to each individual, who can spend fiat money balances when desired. But it is not wealth to the community, because no provision is being made to produce the consumer goods. As for giving the windfall to politicians, and having them produce lots of government goods at the beginning of time--well that seems even worse to me. Real investment projects or whatever "public" goods politicians select with what appears to be free money?
Of course, we aren't at the beginning of time. Implementing Friedman's scheme by just freezing the quantity of base money now, generating expectations of deflation for the future, and then allowing a massive deflation to bring real balances up the the desired level, would be a disaster. Expanding the quantity of money today and promising future deflation seems a bit problematic, but it would certainly be a less disruptive approach to expanding the quantity of real balances today than a one time drop in the price level. And what will the government do with all the new money? As at the beginning of time, they could give it away or else spend it on government projects. But under current conditions, there is a large national debt that could be purchased.
But, of course, there is no real need to generate deflation, (leaving aside trying to generate the optimum currency deposit ratio.) As explained above, just pay interest on reserve balances equal to the risk free equilibrium real interest rate and buy up as much of the national debt as necessary to meet the added demand for real balances. Sounds familiar.
If velocity is constant and real output always grows at a constant rate, what could go wrong? In effect, much of the national debt would be funded at a term to maturity of near zero. Everyone would have all the "liquidity" they desire provided by the taxpayer. And given the assumptions, the taxpayer never has to pay up. It is really little different from arguments that as long as the interest rate is no greater than the growth rate of the economy, the national debt imposes no burden since interest costs can be funded by new deficits.
But what if things do go wrong? What if the government decides it needs to create money to spend, rather than just credit reserve balance accounts? Suppose instead of government fiat money paying the riskless real interest rate as promised, the government generates inflation so that those holding all of that money suffer losses? And worse, what if people begin fear that the government will decide to create money to spend? Remember, much, if not all, of the national debt would have a zero term to maturity.
In the end, I reject Friedman's optimum quantity of money. I favor privatization of money. The interest rate paid on money should be lower to account for the cost of providing liquidity in a world where investment projects take time (and are not sure things either.) This is a real risk, and while Friedman's approach has a large element of truth, there is a good bit of ignoring a taxpayer subsidy by making special assumptions where it never needs to be be used.