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AN INVESTIGATION OF CASH MANAGEMENT
PRACTICES AND THEIR EFFECTS ON
THE DEMAND FOR MONEY
Michael Dotsey
I. INTRODUCTION

During the mid-1970s standard regressions explaining the demand for money underwent a- welldocumented shift. 1 This shift was largely attributed
to the adoption of more sophisticated methods of
cash management practices by firms. Specifically,
techniques were developed that allowed firms to perform a given level of transactions while holding lower
average money balances. Therefore, for given levels
of transactions income and interest rates the demand
for money was lower than that implied by any historical relationships.
This article investigates the effects that a number
of variables related to cash management have on the
demand for money. These variables are generally
suggested by analyzing specific methods of cash management and developing measures that incorporate
the intensity and sophistication of these methods. All
of the variables examined help explain the shift in
the demand for money. Most notably, the number of
electronic funds transfers made over the Federal
Reserve’s wire system restores stability to the estimated demand for money function.
Since the use of more sophisticated cash management techniques is believed to have its main effect on
demand deposit balances, this study concentrates on
demand deposits. The empirical work uses annual
data over the period 1920-1979. Annual data is used
to avoid controversy over the use of lagged dependent
variables. The starting date represents the earliest
date for which reliable data on all variables could be
obtained, while the terminal date was selected to
avoid the problems introduced by NOW accounts.
This relatively long sample highlights the significant
effects that more sophisticated cash management
methods have had on the demand for demand deposits.
For some examples see Lieberman [16], Kimball [12],
Porter, Simpson, and Mauskopf [19], and Simpson and
Porter [20].
1

The paper is organized as follows. Section II
provides an overview of some of the more popular
methods of economizing on cash balances. Section
III discusses a number of proxies that have been
employed in attempts to capture the effects of cash
management on the demand for demand deposits.
Section IV analyzes the relative ability of these
proxies to help explain cash management effects,
while Section V provides a summary and conclusion.
II.

POPULAR METHODS OF CASH MANAGEMENT

This section provides an overview of some popular
techniques used by firms in economizing on cash balances. These techniques are an outgrowth of changing technology, as well as a response to the higher
opportunity costs of holding transactions or cash
balances in the 1970s. The desire to economize on
the holdings of demand deposits has always been
present. However, changing economic conditions
alter the profitability of investing in new methods of
managing transactions balances. For example, lower
computer costs may make previously unprofitable
procedures profitable and advances in computer technology may make new methods in cash management
feasible. Firms that are attempting to minimize the
costs involved in carrying out transactions, costs that
involve the interest foregone on idle balances, will
respond to changes in their economic environment by
altering the levels and types of cash management
services. The degree to which cash management
technology is employed will be arrived at in a manner
analogous to the choice of any other investment
project.
The major changes that have spurred the growth
of more sophisticated and more widespread use of
cash management techniques in the 1970s have been
the improvement of computer technology, the lowering of computer costs, and the rise in market interest
rates. These changes have increased the benefit for
reducing transactions balances while lowering the

FEDERAL RESERVE BANK OF RICHMOND

3

costs of doing so. As a result, there has been a proliferation of ways in which firms manage their cash
balances. Examining these methods will indicate the
exact ways that a firm can reduce its demand for
money.
The various methods used in managing cash balances can be divided into three basic types. One type
speeds up the collection of receivables, another allows
firms to consolidate accounts, while a third helps
control disbursements. Most of the techniques are
linked with improved means of accounting, enabling
the firm to more efficiently monitor its cash position.
Also, many of the techniques are used together,
thereby providing a full range of complementary ways
for economizing on transactions balances.
Methods For Speeding Up Receivables
Lock Boxes Essentially a lock box is a centrally
located collection post office box selected to minimize
the mailing time taken to receive payments from customers. A firm will usually operate a number of
lock boxes in various areas of the country. Several
times a day, the firm’s local bank will open the lock
box, sort out the checks and deposit the money in the
firm’s account. The bank will then send the invoices
and a record of the deposit to the firm. Often photocopies of checks will be sent and, in many cases, the
information will be processed on magnetic tapes that
can directly interface with the firm’s accounting
system. For a given availability schedule, the firm
will have a good idea of the amount of money clearing
into each account on any given day. On average, lock
boxes can reduce mail float from one to four days.
Preauthorized Checks A preauthorized check is a
signatureless check used for accelerating the collection of fixed payments. The customer signs an
agreement with the corporation allowing the corporation or the corporation’s bank to write a check at
specified dates for specified amounts on his account.
The corporation, through the use of a computer file,
sends the bank the necessary information for performing this function. The bank then informs the
firm by means of a computer tape of the deposit and
the availability of the funds. This process lowers the
uncertainty in income flows as well as reducing mail
float.
Preauthorized Debit A preauthorized debit has
the same effect as a preauthorized check. In the case
of a preauthorized debit, the customer’s account is
automatically debited on a specified date and funds
are electronically wired from the customer’s bank to
the firm’s bank.
4

Consolidating Cash Balances
Concentration Accounts Concentration accounts
allow firms to pool the balances collected by or deposited in local banks. Local banks automatically
transfer funds, either by wire or by a depository
transfer check to a central concentration bank. This
process is advantageous for a number of reasons. It
allows the firm to consolidate its cash balances,
making it easier and less expensive to switch idle
funds into market instruments. It also reduces the
amount of total cash balances that need to be held
since it allows for some offsetting of local disturbances to transactions balances.
Depository Transfer Checks Depository transfer
checks, like preauthorized checks, are a signatureless
check. They are issued by the concentration bank
against one of the firm’s local collection banks based
on deposit information sent from the collection bank
to the concentration bank, usually over a data processing network. Specifically, the concentration bank
receives the deposit data and issues a check the same
day for collection. It then sends the information concerning the collected funds and their availability to
the firm.
Wire Transfers A wire transfer is a transfer of
funds most often sent over either the Federal Reserve’s wire system or a bank wire system. In this
case, the local bank transfers funds from the corporation’s account to the concentration bank. The
wire transfer’s advantage is that it allows for same
day use of funds, while its disadvantage is that it is
somewhat more costly than depository transfer
checks. Therefore, wire transfers are predominantly
used for larger transfers than are depository transfer
checks. For example, at an interest rate of 6 percent,
it would require a one-day transfer of $36,000 to
cover the typical $6.00 wire transfer cost. Naturally,
as interest rates rise, the minimum profitable level of
the transfer would fall.
There have been a number of innovations in the
use of wire transfers. Most notable is the ability of a
firm’s cash manager to initiate a wire transfer from a
computer terminal that either interfaces with the
bank’s computer controlled wire system or with the
data base of a third party that is used by the bank.
Often this service is linked with other cash management services, such as programs that forecast a company’s cash flows, and produces wire transfers that
are less costly and that provide hard copy verification
of funds transferred.

ECONOMIC REVIEW, SEPTEMBER/OCTOBER 1984

Methods For Controlling Disbursements
Controlled Disbursement A firm may also exert
more control over its cash balances by being able to
better predict disbursements on a day-to-day basis.
The firm can achieve this by using a bank that receives only one shipment of checks from the Federal
Reserve each morning. The bank informs the firm
of the value of checks drawn on its account and the
firm then knows, usually before noon, how much of
its balances are unnecessary.
Zero Balance Accounts This procedure is a special
case of controlled disbursement that allows the firm
to maintain zero transactions balances at a number
of banks from which it writes checks. When the
value of checks presented against the firm’s account
is tabulated, the appropriate amount of funds are
wired from a central account. This allows the firm
to greatly economize on the level of balances held at
each disbursing bank, and provides centralized data
on transactions.
Summary of Cash Management Services
It is clear from the description of the methods used
in managing cash balances that many of these procedures will be simultaneously employed.
For
instance, a firm is likely to have zero balance arrangements with local banks that also provide lock box
services. Also, the firm will use both depository
transfer checks and wire transfers to facilitate the
quick movements of funds. Crucial to the desire to
economize on transactions balances is the ability to
invest these funds in short-term market instruments
at relatively low costs. Otherwise there would be no
reason to incur the costs involved in reducing the
average level of balances.
Ill. METHODS OF CAPTURING
CASH MANAGEMENT EFFECTS IN
MONEY DEMAND EQUATIONS

The preceding discussion described how various
cash management techniques are able to reduce the
demand for money. Therefore, failure to incorporate
cash management effects in a demand deposit regression will result in a misspecified equation. Since the
degree to which cash management procedures are
used is a choice Variable of the firm and is related to
the cost and benefits of investing in these procedures,
this misspecification will have serious consequences
for any estimated equation.

For example, in inventory models of money demand, either stochastic or nonstochastic, some of the
investment in cash management services can be
viewed as lowering transactions costs. For instance,
a firm having a cash management system that allows
it to perform investments in repurchase agreements
from a computer terminal has invested in a procedure
that greatly reduces transactions costs. In stochastic
inventory models, many of the cash management
services can be viewed as ways for reducing the
variance of cash flows (see Porter and Mauskopf
[18]). Therefore, some key elements of the demand
for money, namely transactions costs and the variance
of cash flows, are not exogenous variables from the
standpoint of individual deposit holders, but are variables that can be influenced by the level of cash management sophistication.
This line of reasoning implies that firms are simultaneously choosing the level of investment in cash
management services and their average deposit balances. Since a number of the parameters that influence the demand for demand deposits are functions of
the level of cash management, the level of cash management should appear in the demand for money
equation. 2
Failure to include a measure of the effects of cash
management in the demand for demand deposits will
therefore result in a seriously misspecified regression.
As a result, coefficient estimates will be biased and
predictions from the regression will be inaccurate in
periods when cash management practices are changing. Further the regression will appear to be unstable
(leading one to believe that the demand for money is
unstable), when in fact the instability is totally due to
an omission on the part of the econometrician.
In this section a number of variables for capturing
cash management technology are examined. These
candidates are generally related to the actual methods
used in cash management and to the underlying costs
and benefits associated with investing in techniques
that help economize on transactions balances.
A Time Trend (T) The first and simplest way to
represent cash management innovations in a money.
demand equation is by use of a time trend. This was
initially employed by Lieberman [16]. The motivation behind this variable is that the adoption of new
technology will be fairly uniform and proceed at an
exponential rate. This procedure explicitly, treats the
2

The result of the optimization process by which firms
choose the level of cash management services and average
demand deposit balance is a two-equation system that is
recursive. For more detail see Dotsey [9].

FEDERAL RESERVE BANK OF RICHMOND

5

process of changes in cash management practices as
exogenous. It therefore omits from consideration any
economic forces, such as changes in costs or returns,
that would be expected to alter the rate at which cash
management techniques are implemented. However,
it serves as a useful benchmark for comparing the
effects that more sophisticated methods of incorporating the consequences of cash management have
had on the demand for money. One practical problem
in using a time trend is choosing the starting date
for the trend.
A Ratchet Variable (RATCHET) In general, a
firm would adopt new methods of cash management
if the expected benefits outweigh the costs. That is,
investing in a new cash management system would
involve the same considerations as investing in any
other project. The motivation behind the use of a
ratchet variable constructed from interest rates is to
capture some of the economic conditions that would
lead to firms’ implementing more sophisticated cash
management techniques.
Since much of the costs of employing innovations
in cash management are start-up costs (e.g., putting
in the necessary computer hardware and software),
it follows that once a new cash management system is
in place it will remain in operation until it is replaced
by more advanced technology. For the investment to
be profitable, the interest rate savings incurred from
lower average money balances must be substantial
and expected to last for some time. One would
therefore expect that major innovations would occur
when long-term interest rates are high relative to
their past history, and that these innovations would
continue to affect the demand for money once they
are initially adopted. Long-term rather than shortterm interest rates are the relevant variable, because
they indicate that a movement in interest rates is expected to persist. One is also interested in the movement of long-term rates with respect to its past, since
upward movements will spur new investment in cash
management due to the increased return obtained
from economizing on transactions balances.
The preceding discussion suggests that a nondecreasing variable based on long-term interest rates,
which increases (or ratchets up) when rates are relatively high, would be helpful in explaining changes in
cash management practices and hence changes in the
demand for money. The specific formulation investigated in this study is the one derived by Simpson and
Porter [20]. Specifically,
6

where r t is the long-term bond rate (Moody Aaa)
and the + sign indicates that only positive values of
the expression in parentheses are used. The variable
is somewhat sensitive to the value of n chosen, so
variables using n = 3, 4, 5, 6 were constructed. All
gave similar results and only the values for n=4 are
reported. A graph of the RATCHET is depicted in
Figure 1.
One can see that the formulation given by equation 1 captures the ideas behind the ratchet variable.
For example let n=4. Then the current value of
RATCHET is equal to last period’s value plus an
additional term. The additional term reflects the
value of today’s long-term interest rate relative to its
average over the latest four periods. If today’s rate
is higher than this average, then RATCHET increases indicating an increase in investment in cash
management services. If today’s rate is lower than
the average, then RATCHET remains the same as it
was last period. This implies no new investment in
cash management technology, and that today’s level
of technology is the same as last period’s level.
Although the ratchet variable possesses some useful
features, it does have certain limitations. It only
considers the potential benefits of new technology but

Figure 1

THE RATCHET VARIABLE
1920 - 1979

ECONOMIC REVIEW, SEPTEMBER/OCTOBER 1984

not its cost. Furthermore, the benefits are only
potential, since one doesn’t know how much economization occurs as a result of new technology. Also,
the variable does not consider depreciation.
The Price of Office Computing and Accounting
Equipment (P) The discussion in Section II makes
it clear that much of the use of cash management
techniques involve computers and accounting equipment. Therefore the costs of this equipment will be
closely related to the costs of cash management. In
constructing a variable that captures these costs, it is
important that the variable take into account adjustment in quality. For example, a new computer model
may cost slightly more than the one it replaces, but
it may be able to perform many more operations in
much less time. In terms of what the computer
actually does, the newer model is much less expensive
than the older model even though its price may be
somewhat higher. A true index of the computer’s
cost will take account of the change in quality. Such
an index is referred to as a hedonic price index.
As the cost of technology falls, more firms will
adopt the technology thus reducing the demand for
demand deposits. Therefore the price of office computing and accounting equipment could help to explain shifts in the demand for money induced by cash
management. However, the price variable does have
certain limitations. It does not account for technology already in place, nor does it reflect depreciation. Further it does not consider changes in the
benefits that occur from the implementation of new
cash management services. Therefore, it would be
natural to use this variable in conjunction with a
ratchet variable.
For the years 1956-1979 data on the hedonic price
of office computing and accounting equipment was
obtained from McKee [17]. Although his procedures
are somewhat rough, they are the best available. For
the time period 1920-1955, it is assumed that the real
cost of technology remained constant at its 1956 level.
A graph of this variable is given in Figure 2.
The Number of Electronic Fund Transfers (EFT)
The motivation for this variable is largely attributed
to Kimball [12]. The use of many of the cash
management techniques discussed in Section II involves the rapid movement of money so that it may
be invested in short-term market instruments. In
many cases idle transactions balances may only be

Figure 2

REAL PRICE OF
OFFICE COMPUTING EQUIPMENT
1920 - 1979

invested overnight: To implement this type of activity often requires the use of immediately available
funds. Therefore, much of the transfer of money is
done over either the Federal Reserve’s wire system
or over Bank wire.
For instance, a firm may use a number of lock
boxes, have a zero balance account with a disbursement bank, and a consolidation account with another
bank. On any given day, funds would be wired from
the lock box collecting banks to the bank maintaining
the consolidation account and from the consolidation
account to the zero balance account. Funds may also
be wired to another bank for the purpose of executing a repurchase agreement if it can not be done
with the consolidating bank. In general there is good
reason to believe that the number of electronic funds
transfers is largely determined by the degree of cash
management practices. Because of this relationship,
the number of electronic fund transfers is a logical
variable for helping explain the shift in the demand
for money (for more detail see Dotsey [9]).
The value for the number of electronic fund transfers used is restricted to funds transfers made over
the Federal Reserve’s wire transfer system and is
depicted in Figure 3. Since there are other wire
transfer systems this value is not totally accurate.
However, it is believed that the time series properties
of the measure is not much different than what would
be observed if data on total wire transfers could be
obtained.

FEDERAL RESERVE BANK OF RICHMOND

7

Figure 3

N U M B E R O F
ELECTRONIC FUNDS TRANSFERS
1920 - 1979

eludes installment retail credit, noninstallment retail
credit, credit outstanding on bank credit cards, credit
owed to gasoline companies, and check credit. The
letter e refers to the disturbance term.
Consumption expenditures are used to represent
transactions income, while RD captures the desirability of holding a demand deposit.3 RS and RCP are
used to capture the return earned on alternative assets
held by different classes of economic agents. The
real wage rate is a proxy for the value of time and is
therefore related to transactions costs, while PCR
attempts to net out the percent of transactions income
spent via credit.4
3

The use of Klein’s rate involves some empirical issues
that make interpreting its effect difficult (see Carlson and
F r e w [ 6 ] ) . However, to the extent that one believes
that corporations earn a competitive rate on their deposits, omission of RD leads to specification bias. (For
more detail see Dotsey [9], especially footnotes 6 and 7.)
4

Since the emphasis of this article is to illustrate the
effects of cash management practices on the demand for
demand deposits, equation 2 is not discussed in detail.
For a full discussion see Dotsey [S].

IV. EMPIRICAL RESULTS

In order to appreciate the severity of the effect of
cash management on the demand for demand deposits, a regression explaining demand deposit behavior is run over the period 1920-1965, a period in
which cash management innovations are believed to
be unimportant. (An examination of Figures l-3
indicates that the various proxies are fairly constant
over this time span.) This regression is then rerun
over the extended sample period (1920-1979), and
the results are compared. This is depicted in Table I.
The regression equation examined is based on an
inventory model of the demand for money used in
Dotsey [8], [9]. Specifically,

Table I

REGRESSION ANAYLSIS FOR DEMAND DEPOSITS

where the letters LN refer to the natural log of a
particular variable (i.e., LNX equals the log of X).
The letter D represents the level of real demand deposits, C represents the level of real consumption
expenditures, RD is the own rate of return on demand deposits calculated using Klein’s [14] methodology, RS is a weighted average of the interest rate
on passbook savings accounts and money market
mutual fund shares, RCP is the commercial paper
rate, W is the real wage rate, and PCR is the ratio of
credit to consumption where the level of credit in8

ECONOMIC REVIEW, SEPTEMBER/OCTOBER 1984

The results of the regression run over the period
1920-1965, yield coefficients that are consistent with
an inventory model of money demand. The error
term does not exhibit any serial correlation and one
can not reject the stability of the regression. The
tests for stability used were a standard F-test, a test
using the cusum of squares statistic developed by
Brown, Durbin and Evans [5], and a test for stability using the varying parameters model of Cooley
and Prescott [7]. 5 The regression coefficients also
converge fairly quickly to their full sample values,
when the sample period is continually extended
from 1928 to 1965. This combination of evidence
strongly implies that the specification in equation 2
is a well-behaved representation of the demand for
demand deposits over the period 1920-1965.
When the sample period is extended through 1979
this is no longer the case, and equation 2 is no longer
an accurate model of the demand for demand deposits. Most importantly, the error structure of the
regression changes. This is evident from the low
value of the Durbin-Watson statistic, implying serial
correlation in the errors. This means that the standard errors of the regression coefficients are biased
making it impossible to state whether the coefficients
in column 2 of Table I are significant. After correcting for serial correlation the wage variable
becomes insignificant. Also, the presence of serial
correlation is often indicative of a missing variable
or variables. A good candidate, or candidates, for
this missing variable would be variables that take
into consideration the effects of cash management.
The reduction of the coefficients on consumption
and the real wage is consistent with the omission of
variables that represent a general decrease in transactions costs, or a lowering of the variance of cash
flows associated with a given level of business. Consider the graph in Figure 4. The locus of points
labelled D, represents a relationship between real
consumption C, and real demand deposit balances
with all other variables (i.e., interest rates, PCR,
and W) held fixed. The locus D’ represents the
same relationship depicted for a more sophisticated
use of cash management. As shown, less real balances are held for any level of consumption, interest
rates, real wages, and the intensity of credit purchases. Now as consumption rises (as it did over the

period 1965-1979), instead of moving from point A
to point B along D, there is a movement from point
A to point C. Thus, excluding the incorporation of
cash management implies that demand deposits will
appear to be less sensitive to changes in consumption.
A similar argument would apply to the real wage
rate. With respect to interest rates, however, the
effect of omitting cash management could be ambiguous. This is because interest rates both rose and fell
over the period 1966-1979. For example, consider
the commercial paper rate. Failure to explicitly
include the effects of cash management would imply a
greater sensitivity of demand deposits to increases in
the commercial paper rates when this rate was rising,
while just the opposite would occur when the rate
was falling.
Adding Proxies For Cash Management If a movement toward more sophisticated cash management
techniques is the sole or primary reason for a shift in,
the demand for demand deposits, then incorporating
variables that accurately account for this movement
should have a pronounced effect on estimated demand
for demand deposit equations. Specifically, the errors
should be white noise and the coefficients should
return to the values found in the regression run over
the 1920-1965 period. Further, stability of the regression over the extended 1920-1979 sample should
Figure 4

AN EXAMPLE OF HOW CASH
MANAGEMENT AFFECTS THE
MEASURED RESPONSE OF DEMAND
DEPOSITS TO CONSUMPTION

5

Descriptions of these test statistics are quite technical
and are therefore omitted. The interested reader can
find a more detailed discussion in Dotsey [8] or can read
the referenced articles. An excellent summary can also
be found in Boughton [4].
FEDERAL RESERVE BANK OF RICHMOND

Demand Deposits

9

of first order serial correlation of the residuals 6
Since RATCHET proxies for the potential benefits

not be rejected and there should be a significant
increase in the predictive power of the equation. The
various proxies described in Section III will be
examined with respect to all of the properties just
listed.
First, examine the regressions in Table II. All of
the proxies return the coefficients approximately to
the values estimated in column 1, Table I. However,
only in the case of EFT, can one reject the presence

6

The discussion of EFT in Section III indicates that its
effects should be examined within the context of a simultaneous system. Since the analysis indicates that this
system is recursive, simultaneity bias will occur only if
the errors in the two equations are correlated. A twostage least squares estimation technique gave similar
results to those obtained using OLS, implying that simultaneity bias is not a problem. For a more detailed discussion see Dotsey [9].

Table II

REGRESSION ANALYSIS FOR DEMAND DEPOSITS (1920-l 979)
WHEN VARIOUS PROXIES FOR CASH MANAGEMENT ARE INCLUDED
I

II

III

The numbers in parentheses are t-statistics.
* indicates significance at the 5 percent level.
**indicates significance at the 1 percent level.
RHO is the coefficient for first order autocorrelation.

10

ECONOMIC REVIEW, SEPTEMBER/OCTOBER 1984

IV

of cash management, while LNP attempts to capture
costs in adopting new technology, it would be natural
to use both variables simultaneously. This was
attempted, but only LNP retained its significance,
perhaps because these variables only reflect general
trends and are therefore only picking up an overall
tendency toward increasing cash management sophistication. Finally a regression including LNP,
RATCHET, and EFT was run with only EFT
retaining its significance.
Second, all of the proxies decrease the instability
of the regressions in the sense that the cusum of
square statistic is lowered. However, only by using
EFT could a lack of stability be rejected using the
Brown-Durbin-Evans test. Also, one could not reject
stability of the regression with EFT under the procedure developed by Cooley and Prescott. However,
when the sample was divided in 1949, stability was
rejected using a standard F-test. Given that EFT
only includes the number of wire transfers over the
Federal Reserve wire system, and is therefore an
imperfect measure of total wire transfers, the net
result of the stability tests is encouraging.
Third, an examination of one step ahead out of
sample forecast errors is depicted in Table III.
Again, all the proxies generally improve the forecasts,
with EFT performing the best. Using EFT resulted
in a reduction of the average absolute error of the
forecast by 52 percent and a reduction in the root
mean square error by 34 percent.
V. SUMMARY AND CONCLUSION

This article, in a somewhat different empirical
setting than that underlying most conventional studies
of money demand, presents confirmation of the recent
shift in the demand for money. The hypothesis that a
shift has taken place in the function is supported by
stability tests and the poor predictive performance of
the model in the mid- and late 1970s. The empirical
evidence combined with documentation on the increased use of sophisticated cash management practices by firms makes changes in cash management
techniques a probable explanation for the shift in the
historical demand deposit relationship.

Table III

THE LOGARITHM OF ONE PERIOD AHEAD
FORECAST ERRORS
None

T

RATCHET

LNP

EFT

1966

- .19

.04

- .15

.00

.05

1967

- .24

.04

- .13

.00

.11

1968

- .16

.16

.08

.18

.18

1969

- .15

.14

.16

.17

.23

1970

- .18

.15

.19

.17

.21

1971

- .14

.10

.06

.12

.17

1972

- .25

.05

- .12

- .03

.07

1973

- .17

.07

- .04

.10

.32

1974

- .26

- .02

- .08

.03

.20

1975

- .71

- .04

- .47

- .38

- .28

1976

- .94

- .62

- .72

- .53

- .26

1977

- .63

- .48

- .70

- .36

.02

1978

- .73

- .48

- .80

- .33

.05

1979

- .69

- .30

- .60

- .40

.41

Avg.
Error

- .38

- .12

- .24

- .07

.11

.29

.28

.35

.24

.19

RMSE

An attempt is made to capture this process by the
use of variables which are believed to be related to
innovations in the management of transactions balances. This process seems to be captured quite well
by the variable EFT. The other proxies perform
reasonably well in reducing the forecasting errors of
the demand deposit relationship, but were not in
general able to capture the entire movement in the
function. The results are by and large encouraging
enough to make future research into transactions
technology and its relation to money management a
potentially rewarding avenue in helping to explain the
current behavior of the demand for money.

FEDERAL RESERVE BANK OF RICHMOND

11

References
1. Barger, H. Outlay and Income in the United States
1921-1938. New York: National Bureau of Economic Research, 1942.
2. Baumol, W. J. “The Transactions Demand for
Cash-An Inventory-Theoretic Approach.” Quarterly Journal of Economics 66 (1952), 645-56.
3. Beehler, Paul J. Contemporary Cash Management.
New York: John Wiley and Sons, Inc., 1978.
4. Boughton, J. M. “Recent Instability of the Demand
for Money : An International Perspective.” Southern Economic Journal 48 (1981), 579-97.
5. Brown, R. L.; J. Durbin; and J. M. Evans. “Techniques for Testing the Constancy of Regression
Relationships Over Time.” Journal of the Royal
Statistical Society, Series B 37 (1975), 149-63.
6. Carlson, J. A., and J. R. Frew. “Money Demand
Responsiveness to the Rate of Return on Money:
A Methodological Critique.” Journal of Political
Economy 88 (1980), 598-607.
7. Cooley, T. F., and E. C. Prescott. “Systematic
(nonrandom) Variation Models Varying Parameter Regression : A Theory and Some Applications.”
Annals of Economic and Social Measurement 2
(1973), 463-73.
8. Dotsey, Michael. “The Effects of Cash Management
Practices on the Demand for Demand Deposits.”
Working Paper 83-2. Federal Reserve Bank of
Richmond, 1983.
9.

“The Effects of Cash Management
Practices on the Demand for Demand Deposits.”
1984. (Processed.)

12. Kimball, R. C. “Wire Transfer and the Demand
for Money.” New England Economic Review,
Federal Reserve Bank of Boston (March/April
1980), pp. 5-22.
13. Klein, B. “The Payment of Interest on Commercial
Bank Deposits and the Price of Money: A Study
of the Demand for Money.” Unpublished Ph.D.
dissertation. University of Chicago, 1970.
14.

. “Competitive Interest Payments on
Bank Deposits and the Long-Run Demand for
Money.” American Economic Review 64 (1971),
931-49.

15. Kuznets, S. National Income and its Composition.
New York: National Bureau of Economic Research, 1941.
16. Lieberman, D. “The Transactions Demand for
Review of
Money and Technological Change.”
Economics and Statistics 59 (1977), 307-17.
17. McKee, M. J. "Computer Prices in the National
Accounts: Are Our Economic Problems a Computer Error.” Council of Economic Advisors, 1981.
(Processed.)
18. Porter, R. D., and E. Mauskopf. “Cash Management and the Recent Shift in the Demand for
Demand Deposits.” Board of Governors of the
Federal Reserve System, 1978. (Processed.)
19.

; T. D. Simpson; and E. Mauskopf.
“Financial Innovation and the Monetary Aggregates.” Brookings Papers on Economic Activity
(1: 1979)) 213-29.

; S. Englander; and J. Partlan. “Money
Market Mutual Funds and Monetary Control.”
Quarterly Review, Federal Reserve Bank of New
York 6 (1982), 9-17.

20. Simpson, T. D., and R. D. Porter. “Some Issues
Involving the Definition of and Interpretation of
the Monetary Aggregates” in Controlling the Monetary Aggregates III. Federal Reserve Bank of
Boston Conference Series no. 23 (1980), pp. 161234.

11. Karni. E. “The Value of Time and the Demand for
Money.” Journal of Money, Credit and Banking 6
(1974), 45-64.

21. Tobin. J. “The Interest Elasticity of Transactions
Demand for Cash.” Review of Economics and Statistics 38 (1953), 241-47.

10.

12

ECONOMIC REVIEW, SEPTEMBER/OCTOBER 1984

ALGEBRAIC QUANTITY EQUATIONS
BEFORE FISHER AND PIGOU
Thomas M. Humphrey

Perhaps the most basic tool of monetary analysis
is the quantity equation of exchange MV = PQ,
where M is the money stock, V its average turnover
velocity, P the price level, and Q the quantity of
goods exchanged against money. This equation has
at least three alternative interpretations. Stated as
the identity MV = PQ (where velocity is defined as
V = PQ/M so as to render the equation a tautology), it reminds us that expenditures must equal
receipts, that the sum total of monetary payments
(MV) must just add up to the aggregate value of
goods sold (PQ). Written as M/P = Q/V or
M = (Q/V) P, where velocity is now defined independently of the other variables such that the equation is non-tautological, it states that the price
level P must adjust to equate the real or pricedeflated value of the given nominal money stock M
with the given real demand for it, this real demand
being the fraction l/V of real transactions Q that
the public wishes to hold in the form of real cash
balances. In other words, it states that the price
level P is determined by the nominal money supply
M and real money demand (Q/V), varying directly
with the former and inversely with the latter. Alternatively formulated as P = MV/Q, it says that
prices are determined by total expenditure (MV)
relative to output Q, that is by aggregate demand and
supply. Most often the equation is used to expound
the celebrated quantity theory of money, which says
that, given real money demand, changes in the money
supply cause equiproportional changes in prices.
The equation’s applications are of course wellknown. Not so well-known, however, is its origin and
early history. For the most part, textbooks typically
treat it as a product of 20th century monetary
thought, usually identifying it with Irving Fisher and
A. C. Pigou, its most influential 20th century formulators. Fisher, in his Purchasing Power of Money
(1911), wrote the equation in its transaction velocity
form :

where M is the stock of currency, V its velocity, M'
is the volume of checking deposits, V' their velocity,
EpQ is the sum of the quantities Q of goods and
services sold valued at their market prices p, P is the
weighted average of these prices or the general price
level, and T is the aggregate of real transactions or
the sum of all the Qs. Similarly, Pigou, in his 1917
article “The Value of Money,” wrote the equation in
its alternative Cambridge cash balance form :

where l/P, the inverse of the price level, is the value
(purchasing power) of the monetary unit ; R denotes
real resources; k, the reciprocal of velocity, is the
the proportion of those resources that people wish to
hold in the form of money; and M is the money
stock. 1 Neither Fisher nor Pigou, however, were the
first to write such equations. On the contrary, the
cash balance equation preceded Pigou by more than
thirty years, having been presented by Léon Walras
in 1886. Likewise, the transactions velocity equation
predated Fisher by more than 100 years, having been
fully enunciated in 1804.
In fact, quantity equations are even older than the
preceding discussion implies. For rudimentary prototypal versions began to appear as early as the late
17th and 18th centuries, followed by increasingly
sophisticated versions in the 19th and early 20th
centuries-versions that were often more elaborate
and complete than those associated with Fisher and
Pigou. These earlier contributions have been largely
overlooked. In an effort to correct this oversight and
to set the record straight, this article traces the preFisherian, pre-Pigovian development of the quantity
1

Pigou’s equation is virtually the same as the celebrated
cash balance expression presented by John Maynard
Keynes in his A Tract on Monetary Reform (1923).
Keynes’s equation is
n = pk
where n is the nominal money stock (Pigou’s M), p is
the price level (Pigou’s P), and k is that part of real
output the command over which people wish to hold in
the form of cash balances (Pigou’s kR).

FEDERAL RESERVE BANK OF RICHMOND

13

equation in the British, German, Italian, French, and
American monetary literature. It covers only writers
who presented the equation in explicit algebraic form.
Unmentioned are the host of analysts (including,
among others, Locke, Hume, Smith, Thornton, Ricardo, Mill, and Marshall) who employed the equation in merely arithmetic or verbal form. It shows
that earlier economists not only formulated the equation and specified its components; they also interpreted it as an equilibrium condition between the
price level (or value of money) and money supply
and demand. That is, they viewed it as an algebraic
model of equilibrium price level determination.
British Writers
The first rudiments of the quantity equation have
their origin in the 17th and 18th century British
monetary literature. To John Briscoe [3] in 1694
and Henry Lloyd [14] in 1771 go the credit for presenting the first such equation and also for being the
first to interpret it as a model of equilibrium price
level determination. 2
Their equation, however,
lacked a velocity term, being written in the form

where P denotes the price level, M the money stock,
and Q the quantity of goods exchanged for money.
They omitted the velocity term (or implicitly assigned
it a magnitude of unity) because they viewed prices
as being determined in a single transaction involving
the one-time exchange of the entire stock of money
for the entire stock of goods. They did not understand that price level determination is a continuous
process and that the stock of money turns over
several times per period in purchasing goods. Nor
did they realize that the volume of goods exchanged
against money is a flow and not a stock, and that
the stock of money must therefore be multiplied by
its average velocity of circulation to make it dimensionally comparable with the flow of goods. Despite
this shortcoming, they were able to draw correct
conclusions from their equation, namely that prices
vary in direct proportion with money and in inverse
proportion with output. Lloyd even gave an algebraic
proof of this latter conclusion, pointing out that if the
quantity of goods increases by a scale factor y while
the money stock is held constant, then prices will fall

by the inversely proportional scale factor 1/y according to the equation

Lloyd also viewed his equation as embodying an
aggregate demand/aggregate supply theory of price
determination, a theory in which M serves as the
demand variable and Q as the supply variable. That
is, he saw M as affecting P through demand just as
Q influences P through supply. Although his work
had no apparent impact on his fellow countrymen, it
did influence his Italian contemporary, the mathematician P. Frisi. The latter, in his review of Lloyd’s
equation, proposed multiplying the money/goods
(M/Q) ratio by the ratio of the number of buyers to
the number of sellers (a proxy for real demand and
supply) in a crude effort to account for all nonmonetary market forces affecting general prices.3 He failed
to see that Lloyd’s commodity (Q) variable already
comprehends these forces so that additional variables
are superfluous.
After Lloyd, the next British writer to present a
quantity equation was Samuel Turner, who, in his A
Letter Addressed to the Right Hon. Robert Peel with
Reference to the Expediency of the Resumption of
Cash Payments Fixed by Law (1819), wrote the
expression

in which a is the value of commodities exchanged
(that is, PQ) over a period of time such as a year,
b is the quantity of metallic money in circulation (or
M), and c is the circulating power of money or the
number of times it changes hands during the year (or
V). Turner’s formula does not divide the, nominal
transactions variable into its price and quantity components. 4 But it does incorporate a velocity term and
therefore constitutes an improvement over the primitive equations of Briscoe and Lloyd. Turner also
expanded his equation to include a term for paper
money, resulting in the augmented expression

where p is paper money and b is metallic coin. Here
is the first quantity equation to contain separate
variables denoting different components of the
money stock, each multiplied by the same velocity
coefficient c.
3

2

On Briscoe, see Schumpeter [25, pp. 314-5]. On Lloyd,
see Schumpeter [25, p. 315] and Theocharis [26, pp.
30-31].

14

On Frisi’s modification of Lloyd’s equation, see Theocharis [26, p. 31] and Marget [18, pp. 154, 270-1, 277].

4

On Turner’s formula, see Theocharis [26, pp. 120-1].

ECONOMIC REVIEW, SEPTEMBER/OCTOBER 1984

Twenty-one years after Turner, Sir John Lubbock
presented in his On Currency (1840) the first quantity equation to incorporate separate velocity coefficients for the different items comprising the media of
exchange. 5 Lubbock’s equation, which also includes a
term for transactions (such as gifts) that do not
involve market prices, is

the sum of transactions or transfers not involving
prices (such as gifts, tax payments, the repayment of
principal on debts, etc.), D is the amount of checking
deposits (Fisher’s M'), B is the total amount of bills
of exchange (Fisher’s M''), C the total amount of
cash, or money narrowly defined (Fisher’s M), and
1, m, and n are velocity coefficients corresponding to
the V', V'' and V terms of Fisher’s equation. Note
that Lubbock distinguishes between the money and
near-money (or money-substitute) components of the
media of exchange-the near-money component being
defined as deposits and bills of exchange. He further
decomposes the money or cash component C into its
bank note, coin, and cash-reserve constituents according to the equation

to monetary changes since such responses require
offsetting disproportionate movements in prices. In
particular, with E invariant to monetary shocks (a
situation Lubbock thought most likely) prices would
tend to move in greater proportion than money.
German Writers
In the 48-year interval separating the contributions
of Lloyd and Turner, German economists made significant advances in the formulation and analysis of
algebraic quantity equations. Claus Kröncke, in his
Das Steuerwesen nach seiner Natur und seinen Wirkungen untersucht (1804), was the first writer to
introduce a velocity term into the equation, doing so
fifteen years before Turner.7 Kröncke’s equation is

Here is the first appearance of the deposit expansion
multiplier and separate velocity coefficients in a quantity equation.
From his equation, Lubbock concluded as follows:
given output and velocities, prices move equiproportionally with changes in the means of payment-but
only if the volume of unilateral transfers E is zero or
also moves equiproportionally with the means of payment. In these cases the quantity theory holds. It may
not hold, however, if E responds disproportionately

where r is the money stock needed in a country, ø
is the nominal value of all goods sold during a certain
period of time, and m is the number of times on the
average that money turns over in purchasing goods
during the period. This is the same as the conventional quantity equation M = PQ/V, where r = M,
ø = PQ, and m = V. Although Kröncke did not
divide his nominal transactions variable into its
price and quantity components, he did state that if
output and velocity are given, prices must vary directly with the money stock. That is, he used his
equation to help illustrate the quantity theory of
money. He also recognized that monetary contraction could occur without depressing nominal activity
only if there were offsetting rises in velocity. Because
he wished to maintain the level of activity while
simultaneously minimizing the quantity of gold in
circulation (so that the excess could be exported for
consumption goods), he advocated policies to increase
velocity.
In 1811, Kröncke’s compatriot Joseph Lang made
two key contributions to quantity-equation analysis.8
He was the first to include separate terms for the four
crucial variables M, V, P, and Q, thereby improving
upon Kröncke’s three-variable formulation. He was
also the first mathematical economist to employ finite
difference notation in deriving the quantity theory
prediction that prices vary equiproportionally with
money. He writes the equation in his Grundlinien
der politischen Arithmetik (1811) as

5

On Lubbock, see Marget [17, pp. 11-12].

7

On Kröncke, see Theocharis [26, pp. 102-3].

6

See Marget [17, pp. 152-3].

8

On Lang, see Theocharis [26, pp. 109-10].

where f denotes bank notes in circulation, g denotes
coin in circulation, and D/k denotes the coin and
bullion reserves backing banks’ note and deposit liabilities, these reserves being expressed as the ratio of
deposits D to the deposit expansion multiplier k. 6
Substituting this last formula into the one immediately
preceding it yields the augmented quantity equation

FEDERAL RESERVE BANK OF RICHMOND

15

where according to his symbols, y is velocity, Z is
money, P is real output, and x is the price level. His
equation can be translated into the conventional formula MV = PQ. Having written the equation, he
then solves for the price level or

from which he concludes that prices P vary in direct
proportion to M and V and in inverse proportion
to Q.
Then, for the first time in the history of mathematical economics, he employs finite difference, or
delta (A), notation to demonstrate rigorously that,
with V and Q given, prices vary in exact proportion
to money.9 Starting with his equation

he supposes money to increase by a small amount
AM, where the delta symbol denotes an incremental
change in the attached variable. Assuming V and Q
fixed, he notes that only prices can respond. De-

quantity equation yields

Expanding this equation, subtracting the preceding
equation MV = PQ from the result, and then solving
for the increment in prices gives him the expression

which states that the incremental variation in prices
is exactly proportional to that of money, with the
ratio V/Q (the inverse of the demand for real balances) being the factor of proportion. Here is the
first rigorous algebraic statement of the quantity
theory of money.
Other 19th century German writers who employed
quantity equations include K. Rau and W. Roscher.10
Little need be said about them, however, as they
added virtually nothing to the earlier formulations of
Kröncke and Lang. Rau, in his Grundsätze der
Volkswirtschaftslehre (1841), stated the formula
MV = PQ, prompting Friedrich Lutz in 1936 to
suggest that it thereafter be called the “Rau-Fisher
9

equation.“ 11 Similarly, Roscher, in his Grundlagen
der Nationalökonomie (1854), presented an equation
similar to Kröncke’s, namely

where u is the monetary sum of transactions (or
PQ), m is the quantity of money (or M), and s is
the velocity of circulation (or V). Roscher’s threevariable formula, of course, was already obsolete at
the time he published it, having been superseded by
Lang’s four-variable formulation forty-three years
before. Nevertheless, the quantity equation’s appearance in the popular textbooks of Rau and Roscher
indicates that it had gained thorough acceptance in
Germany by the middle of the 19th century.
Italian Writers
At least three pre-twentieth century Italian writers
presented versions of the quantity equation. They
include P. Frisi in 1772, L. Cagnazzi in 1813, and
M. Pantaleoni in 1889. Of these, Frisi has already
been discussed above and for that reason will be
treated only briefly here. As previously mentioned,
his equation, as presented in his review of Henry
Lloyd’s An Essay on the Theory of Money (1771), is

where P is price, M is money, Q is quantity of goods,
C is number of buyers (a crude proxy for real demand), and V is number of sellers (a proxy for real
supply). In essence, Frisi’s equation constitutes a
naive attempt to decompose the price level into its
nominal (monetary) and real determinants. In this
connection, he argues that the ratio of money to
output M/Q captures the monetary factors affecting
prices while the ratio of buyers to sellers C/V captures the real factors. 12 What he overlooks is that
the real factors underlying prices are already accounted for by the output variable Q so that the other
variables C and V are unnecessary. This, plus the
omission of a velocity term, renders his equation
defective.
Also defective is Cagnazzi’s equation, but for a
different reason : it omits the price variable. No
price term appears in his formula, which he presents
in his Elementi di Economia Politica (1813), namely
11

See Theocharis [26, p. 109].

10

On Rau and Roscher, see Marget [17, pp. 10-11].

16

Marget [17, p. 10].

12

ECONOMIC

REVIEW,

On Frisi’s equation, see the references cited in footnote 3.
SEPTEMBER/OCTOBER

1984

( 1 8 ) MC = D C
where M is the money stock, c its velocity of circulation, D the quantity of goods, and C their velocity
of circulation.13 Cagnazzi claims that his equation describes market equilibrium between the flow of money
and the flow of goods. Without a price term, however, his equation makes little sense since it equates
dimensionally dissimilar magnitudes. It equates one
flow having the dimensions dollars per unit of time
with another flow having the dimensions real quantity
per unit time. To render the latter flow dimensionally
comparable to the former, he should multiply goods
by their dollar prices.
Cagnazzi’s equation was the first to include a velocity coefficient on the goods variable. Conventional
quantity equations of course dispense with that coefficient (or implicitly assign it a magnitude of
unity). They do so on the grounds that since the PQ
side of the equation summarizes a continuing process,
i.e., an ongoing flow of physical goods and services
sold, each item transferred should be treated as if it
were sold but once before disappearing from economic
circulation. That is, each good should be treated as if
it had a turnover velocity of one. On this logic, items
transferred more than once are to be counted as additional goods each time they are sold. For example,
if a single item such as a house were sold four times
during the period for which PQ is measured, it would
be counted in the Q variable as four houses. In this
way, the goods variable itself registers commodity
turnover; no velocity coefficient is needed. Cagnazzi,
however, proposed that such transfers be registered
by a velocity coefficient. Here is the first appearance
in the equation of a term denoting the velocity of
circulation of goods, a concept later embodied in the
quantity equations of the Frenchmen Levasseur and
Walras and of the Americans Bowen and Kemmerer.
Maffeo Pantaleoni, in his Pure Economics (1898),
also endorsed the goods-velocity concept. The volume of business transactions, he said, resolves itself
into two elements: the quantity of goods offered for
sale and the number of times each good is bought
and sold for money. Having acknowledged the goods
turnover concept, however, he failed to assign it a
specific symbol in his quantity equation

where v is the value of the monetary unit (or inverse
of the price level l/P), m is the volume of business

transactions (or Q), q is the quantity of money (or
M), and r is its rapidity of circulation (or V). He
did, however, present his equation as a money demand/money supply theory of price level determination. He defined the numerator m of the right hand
side of his equation as real money demand and the
denominator qr as nominal money supply. Today
we would define m/r as real money demand and q as
nominal money supply. Their quotient-the ratio
of money demand to money supply-determines the
value of money and hence the price level. He also
stated the quantity theory of money according to
which, for given values of the transactions and velocity variables, the value of money varies equiproportionally with its quantity.
French Writers
Quantity equations made their debut in the English, German, and Italian literature no later than the
early 1800s. Not until the middle of the century,
however, were they first seen in French monetary
texts. E. Levasseur in his La Question de l’Or
(1858) was the first French writer to present a quantity equation. 14 Like Pantaleoni, he argued that the
value of money is determined by the ratio of real
money demand to nominal money supply, the former
defined by him as the quantity of goods for sale times
their rate of turnover and the latter defined as the
money stock times its circulation velocity. To illustrate this proposition he writes the equation

where P is the price level, T is the total sum of goods
and services for sale, C their circulation velocity,
(M-R) is the portion of the total quantity of precious metals M that circulates as money-the remainder R being reserved for nonmonetary uses-,
C' is the circulation velocity of metallic money, and
C r denotes credit instruments serving as nonmetallic
means of payment multiplied by their velocities.
Except for the inclusion of the velocity of circulation
of goods C, Levasseur’s equation is virtually the same
as Irving Fisher’s equation. This can be seen by
omitting the C variable and replacing Levasseur’s
terms T, (M-R), C’, and C r by their Fisherian
counterparts T, M, V, and M'V' to obtain Fisher’s
formula

13

On Cagnazzi, see Marget [17, p. 11] and Theocharis
[26, pp. 39-40].

14

On Levasseur, see Wu [31, pp. 191-3].

FEDERAL RESERVE BANK OF RICHMOND

17

which implies that the price level adjusts to equilibrate money demand and supply.
Sixteen years after Levasseur, Leon Walras, in the
first edition of his Eléments d’ économie politique
pure (1874), also presented a Fisherian equation.15
In addition, he formulated the quantity equation in
its alternative cash balance form, becoming the first
person to do so. Also, he augmented the latter equation with a base/multiplier component to account for
the relationship between high-powered (metallic)
money and the rest of the money stock. His contributions are outlined below.
Regarding the Fisherian equation, he derives it in
two steps. First, he assumes that the means of payment consists solely of metallic money so that the
equation is :

which, except for the v or goods-velocity term, is the
same as Fisher’s formula.
Walras’ next contribution is his cash balance equation. This states that the nominal stock of money M
must just equal the demand for it, this demand being
the aggregate nominal value of goods kPQ the command over which people desire to hold in the form of
cash. In his Théorie de la Monnaie (1886) he writes
the cash balance equation as

of the goods A, B, C, D . . . . the money value of
is the quantity of money needed to satisfy these reof the goods B, C, D.17 This expression is essentially
the same as Keynes’ famous cash balance equation
n = kp presented almost 37 years later in his Tract
on Monetary Reform (1923), where n is money, k is
the collection of goods the command over which
people desire to hold in money form, and p is the
price of those goods. Indeed, Walras elsewhere presents his equation in Keynesian form, writing it as

where H is the demand for real balances (Keynes’
n), and P a is the value of money (the inverse of
Keynes’ p). From this equation Walras reached
the strict quantity theory conclusion: given the demand for real balances H, the value of money P a
Finally, in the 2nd ed. of his Eléments and in his
1898 Etudes [30], Walras adds to his cash balance
e q u a t i o n the term F resulting in the augmented
expression

Expressed this way, Walras’ equation is :

As a second step, Walras adds to the left-hand side
of his equation the term F (or M’V’ in Fisher’s notation) to represent the value of exchanges effected by
means of fiduciary (nonmetallic) money. The result
is the augmented expression
15

What follows draws heavily from Marget’s [16] classic
study of Walras’ work on quantity equations.
16

Marget [16, p. 577].

18

where F is defined as the stock of fiduciary (nonmetallic) money in circulation. Denoting such fiduciary money F as a fixed multiple f of the stock of
metallic money Q (i.e., F = fQ) and substituting
this expression into the one preceding it yields

Here are four key ingredients of modern monetarist
analysis, namely the stock of high-powered or base
a money multiplier (1 + f), the demand
17

Marget [16, p. 580].

ECONOMlC REVIEW, SEPTEMBER/OCTOBER 1984

for real balances H, and the value of money P
or its inverse, the general price level.18 All this
in an equation presented in 1898, fully 13 and 19
years, respectively, before the appearance of Fisher’s
and Pigou’s equations.
Additional evidence that French monetary theorists
had fully developed algebraic quantity equations
before Fisher and Pigou comes from A. de Foville.
His book La Monnaie (1907) contains the expression
a

actions, and P the price level. From his equation he
concluded that prices vary equiproportionally with
changes in the money stock since the latter can have
no lasting effect on the steady-state levels of the real
variables R and K. Equilibrium values of these real
variables, he said, are immune to monetary change
such that the latter registers its full impact on prices
only. To explain how money affects prices, he constructs an aggregate demand function from the components of his quantity equation. Like modern monetarists who define real aggregate demand as money

where P denotes prices, M the money stock, V its
velocity, C the quantity of commodities exchanged
against money, and the upper-case and lower-case
letters refer to the magnitudes of these variables on
any two different dates. Written in ratio form,
Foville’s expression explains the relative change in
the price level between any two dates as the product
of the underlying relative changes in its money, velocity, and output determinants.
American Writers
After a late start, the quantity equation developed
rapidly in the United States, progressing from an
initial incomplete version in the mid-1850s to an
elaborate disaggregated version in the early 1900s.
The major steps in this progression can be outlined
briefly. In 1856 Francis Bowen presented in his
The Principles of Political Economy the equation

where g is the quantity of goods sold, s is the number
of times the goods are sold, m is the quantity of
money in circulation, and r is its rapidity of circulation. Bowen’s equation, expressing as it does an
equivalence between a flow of goods and a flow of
money, is the same as that presented earlier by Cagnazzi and suffers from the same defect, namely the

omission of a price-level variable necessary to render
the two sides dimensionally comparable.
Simon Newcomb corrected this defect in his “equation of societary circulation” which he presented in
his Principles of Political Economy (1885). Newcomb’s equation is

stock times velocity divided by prices (MV/P) he
writes the demand function as:

where D is the quantity of goods demanded, N is a
fixed constant, and V, R, and P are the volume-ofcurrency, rapidity-of-circulation, and price-level variables as defined above. This equation says that,
whereas demand varies directly with money and
inversely with prices, it is unaffected by equiproportional changes in both variables. Thus, according
to Newcomb, a monetary expansion initially puts
upward pressure on real demand. But the resulting
rise in demand subsequently bids up prices, which
eventually rise equiproportionally with money, thus

restoring real demand to its original level. In steadystate equilibrium, prices vary proportionally with
money, and the latter is neutral in its effect on real
variables-just as the quantity theory predicts.
While endorsing the quantity theory, however, he
was quick to point out that it holds only if prices are
flexible. He put his quantity equation VR = KP
to work in demonstrating that price inflexibility
would render monetary changes nonneutral in their
effect on real activity. For, with prices P slow to
adjust to monetary shocks, the real transactions K
term of the equation VR = KP would have to bear
some of the burden of adjustment to currency contraction. Also, he noted that, with velocity given,
autonomous rises in prices P engineered by monopolistic sellers would result in compensating falls in real
activity K if the money stock V were held constant.
Despite this, he warned that a policy of validating or
19

where V is the volume of the currency, R its average
rapidity of circulation, K the number of real trans18

Marget [16, p. 585].

More precisely, he writes

where F (“the flow of the currency”) is defined as
F = VR.
Substituting this latter expression into the former yields
equation 32 of the text.

FEDERAL RESERVE BANK OF RICHMOND

19

underwriting such price increases with money growth
in an effort to maintain full employment would only
serve to perpetuate inflation. The full employment
guarantee, he claimed, would encourage sellers to
raise prices repeatedly. Each time accommodating
money growth would follow. In this way prices and
money would chase each other upward ad infinitum
in a cumulative inflationary spiral. Newcomb’s work
strongly influenced Irving Fisher, who derived his
famous equation of exchange from Newcomb’s formulation and who dedicated his The Purchasing
Power, of Money (1911) to Newcomb.
Following Newcomb, the quantity equation appeared with increasing frequency in the U. S. monetary literature. Arthur Hadley helped to popularize
it by incorporating it into his well-known textbook
Economics (1896) in the form
(33) RM = PT
with M being money, R its rapidity of circulation, P
prices, and T the volume of real transactions. Similarly, Edwin W. Kemmerer employed it in his
Money and Credit Instruments in Their Relation to
General Prices (1907), stating it alternatively as a
price equation and a money supply/demand equation.
The price equation version he writes as

Five years before Kemmerer, John P. Norton, in
his Statistical Studies in the New York Money Market (1902), presented perhaps the most elaborate
version of the quantity equation to be found in the
literature. He includes separate terms for each type
of coin and currency in circulation. He distinguishes
between the velocity of demand deposits and the
velocity of coin and currency. He expresses the total
of demand deposits, in terms of its three underlying
components, namely bank reserves, the depositexpansion multiplier, and the proportion of maximum
allowable deposits that banks actually create. Lastly,
he shows the effect of loan extension and repayment
on the equation. He does all this in the following
way.
First, he starts with the equation’s money or MV
side, writing it as
(36) E = (MV+DU)T
where E is total monetary expenditure, M is money
narrowly defined (coin and notes), V is its velocity,
D is the volume of demand deposits, U is the turnover velocity of deposits, and T is the number of
units of time for which these variables are measured.
He then disaggregates the money variable M into its
constituent components, namely gold coin G, silver
coin S, silver certificates C, United States notes N,
National bank notes B, and all other forms of currency L. That is, he defines money M as
(37) M = G+S+C+N+B+L.

where P is the price level; M the money stock (coin,
currency, bank notes), R its average rate of turnover,
C the dollar amount of checks in circulation, R c the
average rate of check turnover, N and N c are the
number of commodities exchanged by means of
money and checks, respectively, and E and Ec are
the average number of exchanges of these goods (that
is, their velocities of circulation). This equation expresses the equilibrium price level as the quotient of
its monetary and real determinants, which he identifies with money supply and demand. He obtains his
alternative money supply/demand expression by rewriting the equation as

Third, having expressed money in terms of its coin
and currency components, he next expresses demand
deposits in terms of the reserves backing them. More
precisely, he defines such deposits D as the product
of the reserve base R, the deposit-expansion multiplier Z (which determines the maximum of deposits
per dollar of reserves), and the proportion K of
maximum allowable deposits that banks actually
create. In short, D = ZKR. He multiplies these
deposits by their turnover velocity U and aggregates
over the four classes of banks in existence in 1902 to
obtain the expression

( 3 5 ) M R + C Rc = P ( N E + Nc E c ) .
According to him, the left-hand side measures money
supply, the right-hand side measures money demand,
and the price level P adjusts to equilibrate the two.
Except for the inclusion of velocity in his concept of
the money supply, his analysis is the same as modern
monetarists’.
20

where the subscript i indexes the type of bank (country, reserve city, central reserve, and state). He then
substitutes this equation and the one immediately
preceding it into equation (36) to get his final expression for the MV side:

ECONOMIC REVIEW, SEPTEMBER/OCTOBER 1984

This equation, together with those of Newcomb, Hadley, and Kemmerer, prepared the way for the appearance of Fisher’s equation in 1911.
where E is total expenditure.20
He next attempts to show how bank loan extension
and repayment affects the equation. He argues that
loan repayment temporarily absorbs expenditures that
otherwise would be directed toward goods just as
loan extension expands them. To show the effect of
loan retirement, he adds to the goods or PQ side of
“spot” or current dollars M s as borrowers repay
loans. Similarly, to show how loan creation increases
expenditure he adds to the opposite side of the equaof loan assets or “claims to future dollars” Mf, each
such dollar valued at its discounted price (1-d),
where d is the discount rate on loans. For convenience, he then transposes this term to the goods side
of the equation such that the latter reads

where the first term denotes expenditure on goods
and the last two terms denote net debt repayment.
Finally, he equates both sides of the equation to
obtain the entire expression

20

Note that Norton defines the velocity V term of equation (39) as the average of the individual velocities of
each currency component weighted by each component’s
share in the entire stock. He writes

Here is another example of his elaborate derivation of the
equation’s components.

Conclusion
Irving Fisher and A. C. Pigou presented their
famous quantity equations in the second decade of
the 20th century. By that time, however, their contributions had already been largely or fully anticipated by at least 19 writers located in five countries
over a time span of at least 140 years. Except for
some primitive initial versions, these writers formulated equations that in all essential respects were
virtually the same as their Fisherian and Pigovian
counterparts, and in at least two cases were even
more detailed and sophisticated than the latter.
Not only did these earlier equations include the
same variables and possess the same properties as
their celebrated modern counterparts, they also embodied the same analysis. Their authors presented
them either as price equations expressing P as a
mathematical function of the variables M, V, and Q,
or as money-supply-and-demand equations expressing
an equilibrium condition between the money stock
and the underlying determinants of the demand to
hold it. In any case, earlier writers perceived their
quantity equations as functional relationships and not
as mere identities, just as Fisher and Pigou likewise
were to do. Recognition of this fact renders invalid
the typical textbook identification of Fisher and
Pigou as “the original sources of the equation of
exchange and the cash balance equation” [1, p. 98].
Far from being the source of such equations, those
writers were the recipients or inheritors of them. In
short, whereas quantity equations may have culminated in the writings of Fisher and Pigou, they did
not begin there. As documented in this article, their
source is to be found elsewhere.

FEDERAL RESERVE BANK OF RICHMOND

21

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