Optiquandary: A Practical Problem with Friedman’s “Optimum Quantity of Money”

Milton Friedman, optimum quantity of money, interest on excess reserves, interest on reserves, monetary base
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Milton Friedman, optimum quantity of money, monetary base, interest on excess reserves, interest on reserves

Fed economists defending interest on reserves have recently called on an unexpected quarter by reviving interest in Milton Friedman’s 1969 essay, “The Optimum Quantity of Money.”[1] As Ben Bernanke and Don Kohn put it, “Before the Fed paid interest on reserves, banks engaged in wasteful and inefficient efforts to avoid holding non-interest-bearing reserves instead of interest-bearing assets, such as loans.”[2] Laura Lipscomb, Antoine Martin and Heather Wiggens make explicit reference to Friedman’s paper in their recent post defending interest on reserves on the New York Fed’s Liberty Street blog.[3]

This discussion takes me back to my graduate student days at Chicago, when Friedman’s essay had just come out, and one witty student had nicknamed it the “Optiquan Model.”  “Optiquan” came out shortly after Friedman’s important 1967 AEA presidential address, “The Role of Monetary Policy,”[4] in which he refuted the notion, popularized by the Keynesians Paul Samuelson and Robert Solow, that positive inflation was beneficial to the extent that it reduced unemployment along a stationary Phillips Curve. Friedman convincingly argued instead that the Phillips Curve shifts up and down with expected inflation, so that the same “natural rate of unemployment” would arise at any sustained inflation rate.

But that paper left open the more academic question, what inflation rate really is theoretically optimal under a pure fiat money regime in which the central bank is not constrained by any parity to gold or silver? (Recall that 1968-71 was precisely when the US government severed the dollar’s external link to gold.)

In Optiquan, Friedman argued that, since fiat money is socially costless to produce, and since optimal inventory management induces money holders to incur real costs to keep down the foregone interest on their money balances, the first best optimum is to have  a real return on money high enough to drive the opportunity cost of holding money balances down to zero. One  way this could be achieved would be by paying interest on all forms of money — both currency and demand deposits — at the same rate that could be earned on short-term, riskless non-monetary assets. A second way would be to engineer a negative inflation rate just sufficient to drive the nominal interest rate on non-monetary assets to zero. In either case, agents would in theory hold such large money balances that at the margin they would be savings instruments and provide zero purely monetary services.

One big practical problem with the Optiquan Rule that bothered me at the time, and still does today, is that it would leave the price level indeterminate: Friedman’s Quantity Theory of Money predicts that the price level will gravitate to the level that equates the real value of the nominal money stock to the economy’s real demand for it. But this condition requires 1) that there be a predictable real demand for an appropriately defined monetary aggregate, and 2) that the central bank be able to control its quantity.

Friedman recognized that real money demand responds negatively to its opportunity cost which, assuming money pays zero interest (as was the case for currency and all checking accounts outside New England before the 1980 DIDMC Act), would be the nominal interest rate on safe non-monetary alternatives. He also recognized that nominal rates fluctuate owing to natural changes in real interest rates and also  to changes in expected inflation. However, inventory models of money demand such as the famous Allais-Baumol-Tobin model[5] predict that the money demand schedule is inelastic and therefore relatively steep at moderate nominal interest rates (when i is on the vertical axis and M/P is on the horizontal axis). Real interest rates are normally positive at all maturities, and must be so at most maturities to prevent the price of land from being infinite. While inflationary finance is always fiscally tempting, deflation is fiscally unattractive, and therefore not an important long-run concern under fiat money, even by accident under a zero-inflation target. Therefore, the demand for zero-interest money is reasonably predictable as long as nominal rates are positive and inflationary expectations don’t drive them excessively high. That is, it’s reasonably predictable as long as there’s an opportunity cost to holding money balances.

milton friedman, optimum quantity of money, interest on excess reserves, interest on reserves, monetary base
Figure 1

Figure 1 above illustrates the classical Quantity Theory of Money when money pays zero interest, with moderate uncertainty as to the relevant parameters. The blue line represents the inventory demand for real money balances M/P, with M/P on the horizontal axis and the nominal interest rate on the vertical axis.  Because the money demand function is only known approximately, it is shown as a thick line. In equilibrium, the nominal interest rate will average out to the equilibrium real interest rate r*, plus the inflation π that results from the central bank’s money growth policy. Since neither r* nor π is known with perfect certainty, their sum is depicted as a thick horizontal orange line. The intersection of the two curves determines real money balances and therefore the price level, but only imperfectly: Given a nominal money stock M0, the price level could be as high as P2 or as low as P1. So the price level is not determined precisely, but at least it’s reasonably bounded.

However, inventory models predict that if the opportunity cost of money actually falls to zero, real money demand will be literally unbounded. Of course if the resources of the economy are finite and money is not entirely costless — due to the risk of theft, loss, bank failures or unanticipated inflation — money demand will never actually reach infinity, but the point remains that the money demand schedule becomes virtually indistinguishable from the M/P axis over a wide range of values. Such a horizontal demand schedule for money  implies  an equally wide range of price levels capable of equating the supply and demand for money. In that case, the Quantity Theory would no longer predict either the price level or inflation.

Figure 2

Figure 2 above illustrates the inventory demand for money as a function of its net opportunity cost — net of any interest paid on money or real return from deflation — again as a thick blue line. As the opportunity cost approaches zero, this schedule coincides with the horizontal axis, for all practical purposes. The thick horizontal orange line depicts the approximately zero opportunity cost under the Friedman Optiquan deflation rule. Because the neutral real rate r* is uncertain and because the central bank cannot precisely control any deflation it engineers, this line again incorporates some uncertainty. Given a nominal money stock M0, the  price level is now indeterminate at any level between P2 and P1, and could be even lower than P1.

In order for the value of a fiat money to be determinate, therefore, some appropriate monetary aggregate must have a clear and positive opportunity cost relative to non-monetary assets. With all due respect to Milton Friedman, his “Optiquan” rule is therefore just an interesting academic exercise that is not implementable in practice. Its reasoning does argue against high-inflation policies, and does make a case for reducing the opportunity cost of at least the deposit component of M1 through interest on checking accounts. However, in order for the Quantity Theory to determine the price level, there must be a substantial aggregate controlled by the central bank that pays zero or at least greatly reduced interest. It certainly does not make a case for interest on excess reserves.

If the central bank believes it has better knowledge of the “natural rate of interest” (the real interest rate at which the non-monetary supply and demand for credit are equated [6]) than it does of the demand for real money balances, or if it has given up on trying to measure the narrow money supply (as when Sweep Accounts are not included in official M1), it may prefer to use a Taylor-type rule to manipulate the price level through short-term nominal interest rates. However, this only works to the extent that low or high real interest rates create an (albeit unobserved) excess supply or demand for money — but this is again indeterminate if no form of money has a clear and positive opportunity cost.

So long as currency pays zero interest, it has a clear opportunity cost (at least since early 2016 as nominal rates return to normal positive levels), and the price level must eventually equate the real value of its nominal quantity to its real demand. However, when banks are awash with zero-opportunity-cost excess reserves as at present, the Fed has no control over how much base drains from bank reserves into currency in circulation. Lately this currency drain has been steady, though lethargic, so that currency in circulation has almost doubled since 2007, while the nominal economy has only grown about 33%. This huge growth in currency should be a great cause for concern. However, more readily controllable bank excess reserves have a much more immediate impact on banks’ ability and willingness to lend, but only provided they likewise have a clear and positive opportunity cost.

While it is true that banks’ aggressive efforts to minimize zero-interest excess reserves are “wasteful” in the sense implied by Friedman’s “Optiquan” model, there is a flip side to this observation: If banks are being paid not to make loans they will be in no hurry to make (or cut back on) loans and thus to permit borrowers to engage in (or cut back on) the spending that stabilizes the price level on target as the Fed expands (or contracts) the base and therefore reserves. As I argued in my previous post on Alt-M, “The Rudderless Fed,” [7] the Fed has rendered itself essentially rudderless in its conduct of monetary policy as a result of its misguided policy of paying interest on excess reserves.


h/t George Selgin for the title of this post.

[1] Published in his collection, The Optimum Quantity of Money and Other Essays, Chicago 1969.

[2] Ben S. Bernanke and Donald Kohn, “The Fed’s Interest Payments to Banks,” Brookings blog, Feb. 16, 2016.

[3] Laura Lipscomb, Antoine Martin and Heather Wiggens, “Why Pay Interest on Required Reserve Balances?,” Liberty Street Economics blog, Federal Reserve Bank of New York, Sept. 25, 2017; see also Lipscomb et. al., “Why Pay Interest on Excess Reserve Balances?,”  Liberty Street Economics blog, Sept. 27, 2017.

[4] Published May, 1968 in the American Economic Review 58: 1-17.

[5] Willliam Baumol and James Tobin (1989). “The Optimal Cash Balance Proposition:  Maurice Allais’s Priority,” Journal of Economic Literature ,1160-2.

[6] See J. Huston McCulloch, “The Theory of Money and Credit,” class notes dated Sept. 2012.

[7] J. Huston McCulloch, “The Rudderless Fed,” Alt-M, Aug. 9, 2017.

More from Alt-M on Interest on Reserves:


  1. " One big practical problem with the Optiquan Rule that bothered me at the
    time, and still does today, is that it would leave the price level
    indeterminate: Friedman’s Quantity Theory of Money predicts that the
    price level will gravitate to the level that equates the real value of
    the nominal money stock to the economy’s real demand for it. But this
    condition requires 1) that there be a predictable real demand for an
    appropriately defined monetary aggregate, and 2) that the central bank
    be able to control its quantity.."

    If money is backed and convertible into silver at $1=1 oz, then the above statement is completely backwards. The value of the dollar is fixed at 1 oz, and the quantity of money will automatically adjust to equal the quantity demanded.

    I expect most economists would object that the dollar is not backed or convertible. I disagree. There are many kinds of convertibility. A dollar might be returnable to the Fed in exchange for gold, for bonds, or for loan repayments. That's 3 kinds of convertibility right there. If the Fed suspends just one kind of convertibility (gold convertibility) for a finite period, then the dollar is still convertible, and still backed by the Fed's assets.

    I conclude that the US dollar is in fact backed and convertible. The price level is therefore determinate, and the quantity of money will automatically adjust to the public's demand for it. (Unless the Fed takes extraordinary measures to restrain the issuance of money)

    1. On a silver or gold standard, the value of the dollar would be determined by the international value of the monetary commodity in terms of other commodities, and we would no longer be dependent on the Quantity Theory (or the Taylor Rule) to determine its value, so that my objection would no longer be relevant. In that case, the Optiquan consideration would indeed argue for money paying as much interest as the market will bear.

      Interest on deposits is easy to arrange, and I have argued long ago that interest on fractional reserve currency could be paid with minimal transactions costs via periodic lotteries on the serial numbers. A few people might prefer the security of 100% specie reserves, in which case they would likely have to pay the bank a monthly storage fee, but most would prefer to receive positive interest with fractional reserves, even if that might mean an off chance of occasional brief "Scottish" style suspensions (as per Larry White).

  2. Baumol–Tobin model is wrong because the Fed holdings will have unpredictability, their model assumes stationarity. The Fed is cannot hold term debt,that is an insurance function, their model needs term lengths stable.

    The post is correct that the Fed must engage in asynchronous open market actions to maintain its holdings. The Fed can asynchronously buy and sell adjustable amounts of its own fiat to accomplish that, and get what Hu wants. The intersection of those two thick lines must be about the same area, its internals unknown and being discovered.

    1. Inventory models don't predict that a consumer with a $50 average money demand makes any effort to hold money balances at precisely $50. Rather, that's just the person's average money balances over time. However, it does purport to give the per capita money holdings at a given point in time when averaged over thousands or millions of consumers, when there's no excess supply or demand for money.

      The very simple Baumol-Tobin inventory model (which they concede was first published by Maurice Allais) has money just flowing out of the household, with occasional lumpy injections from sales non-monetary assets. The much nicer Miller-Orr model (QJE 1966) more realistically has money flowing in or out each period, randomly. This gives a constant interest elasticity of -1/3 rather than -1/2 as in Allais-Baumol-Tobin. This makes the money demand curve in Figure 1 even steeper, which is good for the Quantity Theory. However, it still gives the same asymptotic shape at low opportunity costs as in Figure 2, so that it does not solve the "Optiquandary" problem.

      The title "Optiquandary," BTW, was Alt-M editor George Selgin's excellent suggestion!

  3. "As Ben Bernanke and Don Kohn put it, “Before the Fed paid interest on reserves, banks engaged in wasteful and inefficient efforts to avoid holding non-interest-bearing reserves instead of interest-bearing assets, such as loans.”"

    This is the same old communist argument applied to money. Why waste time and money on a job search, we will assign jobs. Why waste time and money on advertising, we will assign production. Etc., etc., etc. Of course, everything will be done optimally!

    1. No. It is about increasing efficiency. Households and firms still buy and sell what they want, They still choose the banks they want. Transactions between customers of different banks are still cleared. But the banks have less incentive to undertake financial transactions to minimize their cash position. More fundamentally, there is no reason for households and firms to sacrifice all interest return to obtain the liquidity benefit of holding money. There is no reason for banks to sacrifice all interest return to obtain the liquidity benefit of holding clearing balances. The reason this has occurred in the past is to allow government to maximize its income from its monopoly on the issue of money.

      1. And, I suppose, in your view, there is simply no reason why agricultiral land owners should sacrafice income from their unused land? Thereby justifying subsidization from fellow tax paying citizens? Thereby curbing their need to limit, reduce or eliminate their land position? Retaining the benefit of future use?

  4. Rather than the demand for money being infinite, isn't it constrained by total wealth? The cost of accumulating money is sacrificed current consumption. In a simple one interest rate model, if money pays that interest rate, and it is perfectly liquid, at first pass it is better to hold than any other asset. The demand for money equals total wealth. To accomodate that demand, the central bank must hold all assets. Now, can it control the price level? I think it could create an excess demand for money by selling off assets. Money is less than total wealth, which is less than the amount desired. An excess supply of money could be created by purchasing assets at a price that results in a negative nominal interest rate. While no one else will lend at that interest rate, so what? No one else is lending anyway. The central bank holds all assets. There is an excess supply of money.

    Now, Friedman, I think, was imagining a pure fiat currency. There is no buying or selling assets matching the monetary base. It is like paper gold. Shifting to the optimum quantity of money requires a massive one time deflation. Or one time helicopter drops. Now, the real stock of this "paper gold" is equal to the total desired wealth… well, I am not sure that this leaves us in a good place. Isn't the capital stock zero?

    In the real world, there is not just one interest rate, so if the nominal interest rate on risk-less assets is zero, then everyone will hold money rather than risk less assets. It would seem that the central bank would need to buy all of those. It could create an excess demand for money by selling some. It could create an excess supply of money by buying riskless assets at a negative yield or buying some risky assets that have a positive yield. Really, then, an "optimal quantity of money" fully crowds out short term government debt rather than all assets. And while there won't be transactions with T-bills to minimize reserve or other cash holdings, there would still be such transactions with assets that have some risk and yield.

    In my opinion, Friedman's view of the optimum quantity of money required to special assumptions to make the cost of producing money zero. In particular, velocity is constant, the growth rate of the economy is constant and equal to the real interest rate. A frozen quantity of fiat money generates a deflation rate equal to real interest rate. The nominal interest rate is zero, and people hold the optimal quantity of money. Since keeping the quantity of money frozen is very low cost if not exactly costless, it all looks to add up. What a happy coincidence. Even so, the problem is that it seems that the demand for money would equal to total wealth. If all wealth is fiat money balances (like paper gold,) it would seem that the capital stock would be zero. If instead, it is claims to some kind of financial assets, then the central bank owns the entire capital stock. But as explained above, there is more than one interest rate.

    1. Yes, the demand for money wouldn't actually be infinite, just very large. Excess reserves are a good example of a money inventory. With zero opportunity cost as at present, bank demand for them is about 500 times as large as it was pre-2008. A factor of 500 is far short of infinity, but still very large.

      I don't see that the banks would have to buy up all the assets in the economy under the Friedman rule, since all the Fed has to do to induce a deflation is to slow the rate of growth of the nominal money stock below the rate of growth of money demand, which is the rate of growth of real income g times the income elasticity of demand for money e (e = 1/2 under the simple ABT inventory model). This would make a fixed nominal quantity of assets grow in real value without bound over time. A deflation of -r* would require money growth of (e)(g)-r* after velocity stabilizes, but as velocity falls in the transition, it would not have to be this low, due to what I call "velocity drag".

  5. Hi Edward,

    If you believe, and I do,

    "As my book says, if market forces are able to determine prices, including the price of credit, this optimises the use of the resource in question so that those with the best use for the resource are those able to afford it."

    …and believe, as I do, that there is no resson, it is futile, to go about trying to determine the OQM, then why the need to "manage" from there? It seems like you are saying the market is the best way to determine prices for and ultimately deliver food to those that want to purchase it, but then the government of Zimbabwe must step in to manage it.?

    1. Milton you may need to read my BOOK SUMMARY on my main blog


      but there is also a short essay on the South African Monetary Poliy problem here:


      This is what is now being circulated around the top people at the S AFrican TReasury.

      They are arranging to meet me.

      Zimbabwe may catch up after I return home later this month

      Incidentally we will be planning apilot webinar for a trial batch of students for this course. Those who do well may be invited to become seniors or examinors or tutors for the next course.


      South Africa may also run this course. It is to be discussed.

      1. Sorry I am typing from the back seat of a car. Typos I'm afraid.

        The fin24 essay explains the general idea for monetary policy.

        In terms of money creation the book is the authority.

        The idea is to auction deposits for lenders to bid for at interest. …
        There has to be the right mix of debt-based money nd debt-free money.

    2. Milton I am not quite understanding your question.

      There is nothing in my book which says anything much about what governments should do other than to provide the best possible financial framework for their economies.

      What I do say is that governments have social obligations and the means (tax revenues) with which to carry them out. A robust macr-economic design and management system will help their revenues and their ability to get relected.

      1. I disagree that individuals in government have social obligations. Second, their means are cages and deadly weapons, their end is tax dollars to serve themselves. State governments have no business in finance. Our views are quite different.

        1. My words referred to the responsibilities of governments, not how well they carry them out.

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