Prof. John Munro

Department of Economics

University of Toronto

MODERN QUANTITY THEORIES OF MONEY: FROM FISHER TO FRIEDMAN

Most economic historians who give some weight to monetary forces in European economic history usually employ some variant of the so-called Quantity Theory of Money. Even in the current economic history literature, the version most commonly used is the Fisher Identity, devised by the Yale economist Irving Fisher (1867-1947) in his book The Purchasing Power of Money (revised edn. 1911). For that reason we cannot avoid it, even though most economists today are reluctant to use it without significant modification.

M = stock of money in coin, notes, bank deposits (‘high-powered’)

V = the velocity of circulation; the rate at which a unit of money circulates in effecting transactions in course of one year; the average number of times it ‘turns over’

P = some measure of the price level; e.g. Consumer Price Index

T = the total volume of monetary transactions that take place in the economy during the course of that same year.

a) This is more of an identity () or tautology than it is a causal equation. it simply states that total spending, in terms of the money stock multiplied by the rate of its turnover or circulation, necessarily equals total spending in terms of the total volume of monetary transactions multiplied by the current price index. The two values on each side of the sign are necessarily identical.

b) Problems with the Fisher Identity:

i) M and P, it has been argued, are extremely difficult to estimate or calculate. For the medieval, early modern, modern, and present day eras this is a form of nitpicking that in no way invalidates the model. Good proxies can be provided for most of these eras, certainly good enough to indicate general movements of both prices and monetary stocks. The other two objections are far more important.

ii) T really is quite impossible to calculate for any period or even to comprehend. That is, even if we could attach a numerical value to T, it would be rather meaningless: T = the total volume of all transactions in the economy, both intermediate and final, from raw materials to fully manufactured products along with all services. How can we resolve the problem of multiple counting? How can we add up all the transactions involving so many different commodities and services: with what common denominator? Adding together apples and oranges (as pieces of fruit) is a very simple task by comparison.

iii) V, as a measure of the velocity of circulation or turnover of money, is not in fact an independent variable, but rather a residual one, which has to be calculated algebraically by first knowing the other three. Thus we can calculate V only by this formula: V = (P.T)/M

2. The Cambridge Cash Balances Equation: M = k.P.T

This is a lesser-known rival to the Fisher Identity that emerged during the 1920s at Cambridge, with a formula that resolved at least the problems concerning Velocity:

a) Its originators at Cambridge (especially A.C. Pigou) asked two principal questions :

(1) how much ‘high-powered’ money (usually called M1), do people currently wish to hold in the form of cash balances (money held in coin, notes, bank deposits), rather than being spent or invested?

(2) What, therefore, is the ratio of those cash balances to the total money value of all transactions in the economy?

b) That ratio is indicated by the letter k; and this form of the Quantity equation now becomes: M = k(P.T). The letter k thus indicates the proportion of the total value of all monetary transactions that the public chooses to hold in cash balances; and thus it tells us the necessary amount of M that is required for that level of P.T (total spending). Note that P times T again equals the total monetary value of all transactions; and thus suffers from the same problems of estimating the value of T, as indicated above for the Fisher Identity.

c) Liquidity Preference: a concept further developed by Keynes, who asked a fundamental question. Why do people wish to hold cash balances, instead of immediately spending or investing that money? He suggested three motivations.

(1) transactions motive: people hold a stock of ready cash in order to meet their day to day needs in buying goods and paying for services, etc. This is deemed to be the major need for holding ready cash.

(2) precautionary motive: to have ready cash on hand in order to meet some unforseen emergency, as a contingency fund for future needs.

(3) speculative motive: to have ready cash to take immediate advantage of some special investment opportunity — a cash fund to speculate with.

d) Cash Balances and Opportunity Cost:

What is the cost of holding these cash balances? The true cost is the opportunity cost: i.e. the interest or other investment income foregone by not investing those balances. Consequently, we should find that cash balances are to some extent interest-sensitive, and vary with interest rates. That is, the proportion of national income held in cash balances (k ) should fall as real interest rates rise, because rising interest rates will increase the opportunity cost of holding those balances; and conversely that proportion k held in cash balances should rise with falling real interest rates.

e) Note that mathematically, the Fisher and Cambridge Cash Balances equations are related. k is the reciprocal of V; V is the reciprocal of k

Why is k a more useful variable than V? Because k is much more ‘predictable;’ and conceptually k is an ‘active’ variable — i.e. we should be able to predict roughly what proportion of total national expenditures people wish to hold in cash balances. But V. on the contrary, is a passive (i.e. resulting from) or ‘residual’ variable, calculated as noted only by first knowing M, P, and T. Thus one might say that k (cash balances) is a predictive measure of velocity, while V measures only resulting velocity.

3. The Basic Suppositions Concerning the Older Quantity Theories of Money

a) The Demand for Money is chiefly a TRANSACTIONS DEMAND:

b) The Transactions Demand for Money will be proportional to the aggregate value of transactions (i.e. k as proportion of P.T); and this proportion will not vary in the short run;

c) The Supply of money is exogenously determined, determined independently of the economy (by some external authority or events).

d) Full Employment prevails: so that any increase in aggregate demand will not increase the volume of output or transactions (T);

e) Those with excess money will spend it on goods and services; those with insufficient supply of money will cut their expenditures on goods and services.

f) The Transactions Velocity of Money is, at least in the short run, very stable.

4. The Modern Form of the Quantity Theory: Friedman’s Income Version

a) While the Cambridge cash balances approach apparently resolved the problem of V, it did not resolve the quite intractable problem of T. Modern economists, however, have more or less resolved that problem by ignoring the total volume of transactions, and by looking instead at the Net National Income or the aggregate of net national expenditures.

b) To understand this, we can begin with the Gross National Product or its equivalent, the Gross National Income: as the total current money value of all final goods and services produced in the economy in a given year. From that dollar amount we deduct a sum for ‘depreciation’ (for depreciation of worn out, wasted capital stock) in order to arrive at Net National Product. Thus, just as Gross National Product (GNP) = Gross National Income (GNI), so Net National Product (NNP) = Net National Income (NNI), which is represented here by the capital letter Y. That letter Y will be familiar to anyone who has studied at least the rudiments of Keynesian economics:

Y = C + I + G + (X — M).

That is, Net National Income (Y) equals the sum of total national Consumption (C) plus total Investment (I) plus Government Expenditures (G) plus the net difference between total Export incomes (X) and total expenditures on Imports (M).

c) Since this value Y is usually expressed in terms of current dollars, we must now express that net national income in dollars of unchanging values, i.e. in what are called ‘constant dollars’ that reflect a constant or stable purchasing power, which has been adjusted for inflation (thus the term: ‘deflated net national income’). That value of a deflated NNI, or ‘real NNI,’ or ‘net national income in constant dollars,’ is expressed by lower-case y. Upper-case Y of course measures NNI in current dollars, which currently has meant a declining purchasing power, because of inflation.

d) This new value y or real NNI is obviously much more measurable than T. To calculate y: divide Y by P. That is, calculate the NNI by deducting depreciation from the GNP; and then divide that result (NNI) by some agreed upon price index (e.g. consumer price index): y = Y/P .

For example: the value of the Gross Domestic Product in the 2nd quarter of 1991 was $683.64 billion (just 1.8% higher than the corresponding figure for 1990. Divide that amount by the GDP Price Index (whose base is 1986 = 100), which is 121.8 — i.e. meaning that this price index is 21.8% higher than the weighted average of prices for all items in the price basket for 1986. The result (divided by 1.218) is $561.3 billion, which is the ‘real’ GDP for 1991 in constant 1986 dollars. (This amount is 1.75% lower than the corresponding figure for 1990, $571.33 billion). Unfortunately the data currently available are for GDP only, not for NNP; and these GDP data will have to serve as proxies for Y and y.

e) So, by using that ‘y’ value to express constant or deflated net national income (NNI), in place of unmeasurable T, in the two quantity theory equations, those Fisher and Cambridge equations now become:

i) Fisher: M.V = P.y

Thus V measures the income velocity of money: the rate at which a unit of money circulates in producing total net national income (or net national expenditures or net national product).

ii) Cambridge Cash Balances: M = k.P.y or, M = kPy

Thus k measures the proportion of aggregate national income that the population collectively holds in cash balances.

iii) While the Cambridge version is conceptually preferable, it is mathematically related to the much more widely used Fisher equation, or better the modern income version of that equation (k = 1/V). So you will presumably also prefer to use it: but at least please use it in this modernized form: M.V = P.y [MV = Py]