Friday, September 12, 2014

The Various Versions of the Quantity Theory

This issue is vexing me at the moment, as I am writing an article in the course of which I am reviewing the different versions of the quantity theory as an explanation of inflation.

The following post is a work in progress, to help me summarise the history of the quantity theory.

In essence, the quantity theory comes in various versions, as follows:
I. Equation of Exchange Versions
(1) Irving Fisher’s equation of exchange in his book The Purchasing Power of Money: Its Determination and Relation to Credit, Interest and Crises (1911) (Fisher 1911: 24–28, and particularly 27):
MV = PT,
where M = the money supply;
V = the velocity of circulation (or the number of times money changes hands);
P = the average price level;
T = the volume of transactions of goods and services.
Fisher also gave this form of the equation:
MV = ΣpQ.
where ΣpQ is the sum of the price multiplied by quantity bought of every good in the economy (Fisher 1911: 26).
But Fisher also thought that ΣpQ could be written as PT:
“We may, if we wish, further simplify the right side by writing it in the form PT where P is a weighted average of all the p’s [prices], and T is the sum of all the Q’s. P then represents in one magnitude the level of prices, and T represents in one magnitude the volume of trade.” (Fisher 1911: 25).
(2) Milton Friedman’s version of the equation of exchange in his paper “The Quantity Theory of Money: A Restatement” (1956):
where M = the quantity of money;
P = the price level;
Y = aggregate income or value of aggregate output;
V = velocity.
Under equilibrium conditions where Q = Y, it can be written as:
MV = PQ.
II. Cambridge Cash Balance Equation Versions
(3) Alfred Marshall’s reformulation of the quantity theory as the cash balance approach in the 1870s. It is unclear to me whether Marshall already had an equation form of the quantity theory in the 1870s.

I have read that this was Marshall’s version of the Cambridge Cash Balance Equation:
M = KY
where M = aggregate money supply;
Y = aggregate real income;
K = the proportion or fraction of real income which people hold in the form of money/cash balances.
The value of money is then explained by the following equation:
where P is the purchasing power of money.
But where this was given in Marshall’s works is not yet clear to me.

It seems that the final form of Marshall’s version of the Cambridge Cash Balance Equation was given in his book Money, Credit, and Commerce (1923).

(4) Arthur C. Pigou’s version of the Cambridge Cash Balance Equation in his paper “The Value of Money” (1917: 52):
where P = the purchasing power or value of money;
k = proportion of R (real income) held in the form of money/cash balances;
R = aggregate real income;
M = aggregate money stock or money supply.
(5) J. M. Keynes’ version of the Cambridge Cash Balance Equation in A Tract on Monetary Reform (1923: 77).

The basic form that Keynes gives is this:
n = pk
where n = currency notes or other forms of cash in circulation with the public;
k = consumption units of cash on hand;
p = the index number of the cost of living.
There is also another version that Keynes gives in which he included bank deposits in the total quantity of money, as follows:
n = p(k + rk′),
where n = quantity of money, or currency notes or other forms of cash in public circulation;
p = the index number of the cost of living;
k = consumption units of cash on hand;
k′ = money people want to be available in banks in the form of their demand deposits or checking accounts;
r = cash reserves of the banks.
In this version, Keynes thinks that as long as k, k′ and r remain unchanged, if n rises, then p will rise too (Keynes 1923: 77).

At the time he wrote A Tract on Monetary Reform Keynes had no doubts about the truth of the quantity theory (Keynes 1923: 74).

(6) Marshall’s final formulation of the Cambridge Cash Balance Equation in Money, Credit, and Commerce (1923).

(7) Dennis H. Robertson’s version of the Cambridge Cash Balance Equation in Appendix A of the 1928 edition of his book Money (rev. edn. 1928: 150; later edition 1964: 150):
where M = the quantity of money;
k = proportion of T against which people hold cash or money balances;
P = the price level;
T = the total amount of goods and services purchased.
From this, it is easy to derive the standard form:
M = PkT
Now the “original” version of the Cambridge Cash Balance Equation (or so I have been told) is usually written as:
M = kPT
and this seems to be Dennis H. Robertson’s version.

However, the standard form of the Cambridge Cash Balance Equation as used today is usually given as follows:
M = kPY or
M = kd PY
where M = the quantity of money;
k or kd = the amount of money held as cash or money balances;
P = the general price level;
Y = real value of the volume of all transactions entering into the value of national income (that is, goods and services).
The variable k was held to be equivalent, but superior, to Irving Fisher’s “velocity of circulation” concept V (which is why it is held that 1/k = V), because, unlike V, k is supposed to be empirically measurable.

M and P are causally related, if kd and Y are constant (Thirlwall 1999).

It is interesting that, while Keynes had formulated his own version of the Cambridge Cash Balance Equation – no. (5) above – he had by 1933, as stated in a letter to Dennis Robertson, come to the view that no version of the Cambridge Cash Balance Equation had any serious use in economic analysis:
“In my present state of mind, however, I doubt that either version of the Cambridge equation is of any serious utility, and I can’t remember that I have ever come across a case of anyone ever using either of them for practical purposes of interpretation. Thus, whether my version is slightly better than yours, or whether I ought to yield to your criticisms, I am not prepared to put up a serious case in defence of either. All this section is really a survival of the time when I was trying to make some practical use of the Cambridge equation, an attempt I have long since given up.” (Keynes, Letter to Dennis Robertson, 3 May, 1933 in Keynes 1971: 18).
Finally, one should note that mathematical statements of the quantity theory were apparently already being given in the 19th century, as Irving Fisher noted:
“An algebraic statement of the equation of exchange was made by Simon Newcomb in his able but little appreciated Principles of Political Economy, New York (Harper), 1885, p. 346. It is also expressed by Edgeworth, ‘Report on Monetary Standard.’ Report of the British Association for the Advancement of Science, 1887, p. 293, and by President Hadley, Economics, New York (Putnam), 1896, p. 197. See also Irving Fisher, ‘The Role of Capital in Economic Theory,’ Economic Journal, December, 1899, pp. 515-521, and E. W. Kemmerer, Money and Credit Instruments in their Relation to General Prices, New York (Holt), 1907, p. 13. While thus only recently given mathematical expression, the quantity theory has long been understood as a relationship among the several factors: amount of money, rapidity of circulation, and amount of trade.” (Fisher 1911: 25, n. 2).
Dimand, Robert W. 2002. “Patinkin on Irving Fisher’s Monetary Economics,” The European Journal of the History of Economic Thought 9:2: 308–326.

Fisher, Irving. 1911. The Purchasing Power of Money: Its Determination and Relation to Credit, Interest and Crises. The Macmillan Company, New York.

Fisher, Irving. 1920. The Purchasing Power of Money: Its Determination and Relation to Credit, Interest and Crises (rev. edn.). The Macmillan Company, New York

Friedman, Milton. 1956. “The Quantity Theory of Money: A Restatement,” in Milton Friedman (ed.), Studies in the Quantity Theory of Money. The University of Chicago Press, Chicago. 3–21.

Friedman, Milton. 1968. “Money: the Quantity Theory,” in D. Sills (ed.), International Encyclopedia of the Social Sciences (vol. 10). Macmillan Free Press, New York. 432–447.

Humphrey, Thomas M. 2004. “Alfred Marshall and the Quantity Theory of Money,” FRB Richmond Working Paper No. 04–10,
December 1, 2004

Keynes, John Maynard. 1923. A Tract on Monetary Reform. Macmillan, London.

Keynes, John Maynard. 1971. The Collected Writings of John Maynard Keynes. Volume XXIX. The General Theory and After. A Supplement (ed. by D. Moggridge). Macmillan, London.

Laidler, David E. W. 1999. Fabricating the Keynesian Revolution: Studies of the Inter-War Literature on Money, the Cycle, and Unemployment. Cambridge University Press, Cambridge.

Marshall, Alfred. 1923. Money, Credit, and Commerce. Macmillan, London.

Marshall, Alfred. 1926. Official Papers (ed. by J. M. Keynes). Macmillan, London.

Newcomb, Simon. 1885. Principles of Political Economy. Harper, New York.

Pigou, A. C. 1917. “The Value of Money,” The Quarterly Journal of Economics 32.1: 38–65.

Robertson, Dennis Holme. 1928. Money (rev. edn.). Nisbet, London.

Robertson, Dennis Holme. 1964. Money (rev. edn.). University of Chicago Press, Chicago, Ill.

Thirlwall, A. P. 1999. “Monetarism,” in P. Anthony O’Hara (ed.), Encyclopedia of Political Economy: L–Z. Routledge, London and New York. 750–753.


  1. Interesting post. I hadn't seen some of these. One quibble. You write:

    "k or kd = the amount of money held as cash or money balances or the demand to hold money per unit of money income"

    The latter part of that statement seems odd to me. Surely only when we join k up with P.Y we get "the demand to hold money per unit of money income". k alone does not seem to me to have this definition.

    Finally, I am with Keynes: the quantity theory is useless. Unless we assumed fixed V or k then it tells us nothing at all. The only point in laying it out is really to knock it back down.

    1. Thanks -- I have fixed the definition of k.

      So would it be better to say as a PK definition of inflation the following:

      "inflation -- or changes in the general price level -- are not explained by the monocausal and mathematical quantity theory.

      In an endogenous money world, although monetary factors play a role, changes in individual prices that cause inflation/deflation can also be driven by independent non-monetary causes, such as supply side factors, costs, wages, demand factors, without being driven by imagined exogenous money supply changes."?

  2. Yes. Majority are probably wage inflations though.

  3. "Inflation is always and everywhere a monetary phenomenon IN THE SENSE THAT IT IS AND CAN BE PRODUCED ONLY BY A MORE RAPID INCREASE IN THE QUANTITY OF MONEY THAN IN OUTPUT. … (emphasis mine)

    The full quote suggests that Friedman had a slightly more sophisticated view of inflation than post-Keynesians would have us believe. It stands to reason that he would admit that negative supply shocks, oil disturbances could cause inflation . But in the absence of government supply restrictions, such an increase in the price level would quickly peter out. That would be what someone like him would say.

    Phil, wage inflation IS demand push inflation. Wages are typically part in nominal income, which is a measure of MV.

    Lets perform a simple thought experiment as to why endogenous money theory is nonsense. Let us say that the uk or us central bank freezes production of M0 base money (currency plus reserves. In the US. case the United States Bureau of Engraving and Printing physically prints the notes, but at the behest of the US Federal Reserve.) The central bank sees an asset price boom in place, and is fearful of a bubble even though inflation is under control. It doesn't want to raise interest rates, but it doesn't want the boom to continue either, so it simply holds the line, neither selling securities nor buying them slowing the monetary base growth to absolute zero.

    Now what do you think would happen?

    I'll tell you, the effect would be dramatically contractionary. (there is no neutral monetary policy) Even though money created by private banks comprises the vast majority of money in the economy, (and so it LOOKS like endogenous money rules) part of the reason its accepted is because the people and banks believe that their checks can be exchanged for cash. Cash doesn't have to be landed out in the traditional money multiplier fashion (Indeed, I accept that Friedman was wrong about this and you guys are right.. but it still misses the point) for it to have value. People like to hold it for precautionary liquidity preference motives. If their ATMS stop spitting bills, "endogenous" and credit money will collapse, lending will slow. and a massive recession will ensue

    1. So what you are saying is that, if an endogenous money system did not exist, then... it would not exist?

      That is hardly a refutation.

    2. Seriously, just consider how your comment is incoherent.

      You are saying on the one hand: "endogenous money does not exist!", but at the same time, "well, if the endogenous money system that **actually functions now** did not function, then we would not have endogenous money!"

  4. Wow,
    You are an seriously the most LITERAL person Ive ever blogged at., LK. I'll spell out what's implied in my posts.

    Endogenous money theory, as espoused by the Post Keynesians considers the role of the central bank to be trivial and private credit to be where the action is. While they are right on the surface, they are wrong on the deeper significance. Cash does NOT have to be lent out à lá money multiplier for it to be important. Cash and reserves are important for precautionary and liquidity preference motives. (In a bank run and when people stand on line panicking in front of ATMs) Since the central bank controls both, it follows that the central banks's role is still important.

    Have I sufficiently clarified the market monetarist counter-response to the Post Keynesians?

    1. Is there any literature on that? Books, articles ect?

  5. And I never said "endogenous money does not exist" Just that Post Keynesians are wrong about its significance