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):

MV = PY
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 = k_d PY
where M = the quantity of money;
k or k_d = 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 k_d 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).

BIBLIOGRAPHY
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
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2184929

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.