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In early 1970 the Federal Reserve System modified some of the operating procedures it employs in
conducting monetary policy. Specifically, the Federal
Open Market Committee (FOMC),
which is the
System’s principal policymaking body, began to place
somewhat greater emphasis on what have since come
to be known collectively as “the monetary aggreand
gates” as operating variables in formulating
implementing monetary po1icy.l The monetary aggregates are various measures of the nation’s stock
of money.
During the 1950’s and 1960’s, in contrast, the FOMC had focused primarily on conditions
in the money markets, as indexed by member bank
reserve positions and certain key short-term interest
rates. This shift in procedural emphasis has generated a great deal of interest among and comment
from monetary economists, financial market participants, and other observers of System policy. Monetary economists found the change interesting because
it suggested that monetarist doctrine, which had
achieved considerable prominence in academic circles
in the E&O’s, had finally attained at least a degree of
acceptance in the halls of the nation’s principal moneMarket participants, on the other
tary authority.
hand, regarded the shift as important from the standpoint of evaluating past and present System policy
and making judgments about the likely future course
of policy.
The extent to which the Federal Reserve has in
fact altered its operating strategy since 1970 is the
subject of a spirited and sometimes heated debate
among economists.
Some monetarists claim that although the FOMC now gives lip service to the
monetary aggregates in its policy pronouncements, it
continues to focus mainly on financial market conditions in practice, thereby relinquishing potentially
useful control over the aggregates. Conversely, some
nonmonetarists believe the FOMC has paid too much
1 More specifically still. the FOMC began to express its operating
objectives more frequently in terms of the desired behavior of the
monetary aggregates in its instructions to the Manager of the
Open Market Account at the Federal Reserve Bank pf New
Acting as the FOMC’s agent, the Manager supervws the
System’s day-to-day purchases and sales of securities, or open
market operations. These operations constitute the FOMC’s princiPal tool for implementing monetary policy.
The FOMC normally
meets once each month. At the conclusion of each meeting it issues
a “Directive” to the Manager containing its operating instructions
covering the period until the following meeting.


attention to the aggregates to the detriment of the
credit markets and, consequently, the general economy, For their part, System officials have made it
plain in a number of public statements and articles
that as far as the System is concerned, the change
that occurred in 1970 represented a shift of emphasis
among alternative operating variables rather than
any official recognition of a change in economic doctrine. The monetary aggregates, while not emphasized, were by no means ignored prior to 1970.” Nor
have financial market conditions and interest rates
been ignored since 1970.
Whatever the merits of these arguments, it is clear
that the monetary aggregates presently play a more
important role than earlier, both in the formulation
and execution of monetary policy and in public discussions of policy. Perhaps the strongest indication
of the increasing prominence of the aggregates is
their central position in the Congressional resolution
concerning monetary policy passed on March 24,
1975. This resolution calls on the FOMC to maintain longer-run growth in the monetary aggregates at
rates consistent with the longer-run potential for
growth of the nation’s productive capacity.
resolution also requests the Federal Reserve to
inform the House and Senate Banking Committees
periodically of its targets for growth of the aggregates over the following twelve months.
The first
such hearings took place on May 1, 1975. At the
hearings Chairman Arthur F. Burns of the Federal
Reserve announced the System’s targets for certain
aggregates for the period March 1975-March 197L3
The hearings received considerable national attention.
The greater emphasis on the aggregates raises
some immediate questions. First, precisely what are
the monetary aggregates ? As the term implies, they
are essentially aggregations or summations of the
‘In 1966 the FOMC began supplementing its instructions in the
Directive regarding desired money market conditions with explicit
references to the desired behavior of certain monetary aggregates.
For an interesting discussion of the Committee’s attention to the
aggregates during the 1959% see Elmus R. Wicker, “Open Market
Money Supply Strategy.”
Qua+terly Jo~tnal of Ecfnunnics. 88
(February 1974), 170-g.
3 See the Statement by Arthur F. Burns, Chairman, Board of
Governors of the Federal Reserve System, before the Committee on
Banking, Housing and Urban Affairs, U. S. Senate. May 1, 1975,
reprinted in Federal Rese7ue BaUetin, May 1975, pp. 282-8.




public’s holdings of various financial assets that appear to function as “money” in household and business portfolios.
But this description raises more
basic questions.
What is money?
What are its
functional characteristics ? Exactly
which financial assets possess these characteristics ?
economists have not arrived at definitive answers to these questions.
As a result,
universally agreed definitions of money and the
money stock do not exist.
In the absence of such
definitions, the Federal Reserve has found it necessary to take an eclectic approach in the practical
of policy.
it has defined several monetary aggregates deemed relevant
to policy analysis. Each such aggregate is designated
by the letter M and a numerical subscript, higher
subscripts indicating more inclusive aggregates.
Table I defines the aggregates MO - Mr. Economists have traditionally focused on M1, Ma and, to a
lesser degree, Ms as the most useful definitions of
the money supply. Among these, M1 is the definition
most frequently referred to in public discussions of
money and monetary policy. The specification of the
higher numbered aggregates shown in the table is a
recent development reflecting the growing belief in
some quarters that advanced cash management techniques, the introduction of new financial instruments
such as large-denomination
negotiable certificates of
deposit, and other financial market innovations have
broadened the spectrum of assets that serve as
money.* For this reason, some students of monetary
policy believe that explicit consideration
of these
broader aggregates might increase the effectiveness
of monetary policy. 5 Others doubt this contention on
the grounds that the Federal Reserve would find it
difficult to control these aggregates and that their
behavior, in any event, bears a predictable relationship over time to the behavior of the narrower concepts such as Mr.
Whatever the outcome of this relatively technical
debate, it seems rather paradoxical that in a policy
environment where the money supply is such a central concept, there is no professional consensus as to
precisely what the money supply is. This article will
not attempt to answer this question.
Its purpose,
rather, is to indicate to nonprofessional
readersmany of whom probably take the existence of an



MO =







MB =



time deposits at commercial








ot commercial







banks other than large

of deposit

and credit




union shares








of deposit






















IJ. S. Government





* For more precise definitions the reader should consult the footnotes to the table titled “Measures of the Money Stock” in the
statistical section of any recent Federal Reserve Bulletin.

agreed money definition for granted-the
inherent in arriving at an unambiguous answer. The
article will also describe recent research aimed at
developing new money supply concepts superior to
those listed in Table I. It is hoped that this material
will assist the nonprofessional in critically evaluating
commentary in the financial press and elsewhere on
the use of a growing list of monetary aggregates in
the conduct of monetary policy.
The article contains four sections. The first section
reviews the earlier controversy
among economists
over the proper definition of money.
The second
section describes a general and highly flexible procedure for developing so-called weighted monetary
Such weighted aggregates are refinements of the conventionally-derived
aggregates listed
in Table I and, in the view of at least some economists, potentially
better measures of the money
supply. The third section reviews some preliminary
empirical efforts to estimate the weights that should
be attached to particular categories of financial as,sets
in developing operational weighted monetary aggregates.


~Mones supply statist& are published in the monthly Fe&d
Reserve Bulletin.
Series for MI. Mz, and M.T have been carried in
these tables for some time. MI and MS were added to the tables in
April 1975.

When it comes to definitions, money is a little bit
like ses appeal: everyone has a fairly clear intuitive
idea of what it is, but defining it in precise language
is difficult. Economists have been arguing about the
best way to define money for centuries.6

6For a concise official statement of this attitude. see the Statement
by Arthur F. Burns. Chairman, Board of Governors of the Federal
Reserve System. before the Committee on Banking. Cnrrency and
Housing, House of Representatives. July 24, 19’75. reprinted in
Reserve Bulletin, August 1975, pp. 491-7.

CAn excellent survey of the historical dialog is contained in Miltw
Friedman and Anna J. Schwartz, Monetary Stcrti~tic~ of the United
States: Estimates. Sources, Methods, New York: National B-u
Economic Research, 1970. pp. 89-198.






their inability to achieve a consensus, the question
cannot be abandoned either as a theoretical matter or
as a practical matter. Clearly, the concept of money
lies at the core of both monetary theory and monetary policy.
The effort to define money has been approached
from two directions during the postwar period. One
segment of the relevant literature has taken a theoretical approach and has sought to settle the issue
on the basis of general principles.
Analysts in this
camp have commonly begun by specifying their respective views regarding the purpose that money
serves from the standpoint of the economic unitshouseholds and business firms-that
hold money.
With these purposes delineated, the analyst has then
defined money to include the various categories of
deposits and other financial assets that appear to
serve the indicated functions.
The other approach
has been more heavily empirical.
Here, the choice
among alternative definitions has been made on the
basis of such criteria as the stability of the relationship between income and various candidate measures
of money as revealed by detailed statistical analysis.
As indicated
those who have taken a theoretical approach to defining money have often begun by asking what money is
used for, or, equivalently, why it is demanded. One
obvious response to this question is that money is
used to facilitate purchases: that is, money is a means
of payment.
Money should therefore be defined to
include those assets used directly in making purchases and to exclude other assets. On the basis of
this criterion, some economists have defined money
as the sum of currency in the hands of the public and
demand (checking) deposits at commercial banks, or
lM1. The appeal of this apparently straightforward
logic is so great that Ml has become the most widely
accepted definition of money in the eyes of the general public.’
L4 more thoughtful examination of these points,
however, suggests that neither the means of payment
criterion nor the M1 definition is necessarily preferable. From the standpoint of both economic analysis
and policy, money is interesting primarily because
changes in the stock of money held by the public are

likely to affect aggregate spending and hence broader
economic conditions respecting such things as the
level of output, employment, and prices. There is no
reason to believe that the stock of assets relevant to
spending decisions is limited to those assets that can
be used directly as payments media in the act of
exchange itself. For this reason, many economists
now regard the essential function of money as estending beyond its service as a means of payment to
include its use as a “temporary abode of purchasing
power,” that is, as a repository bridging the gap
between the receipt and disbursement of payments.s
This extension of the concept of money’s function
in the economy might seem at first glance to be a
minor refinement.
Actually, it constitutes a fundamental break with the narrower view of money as a
For although only a limited
means of payment.
number of assets can be used directly in effecting
payments, a wide variety of assets can be used as
temporary resen-oirs of purchasing power in anticipation of payments. It certainly seems reasonable to
suppose that a sizeable portion of household balances
in commercial bank time and savings deposits, in
mutual savings bank deposits, and in credit union and
savings and loan association shares are held in anticipation of specific payments.
On these grounds, the
view of money as a temporary store of purchasing
power suggests that M2 or MS, or at least some
portion of these aggregates, might properly be regarded as money.” Shifting the focus from households to business firms produces further possibilities.
It is well known that in the current business environment a major goal of corporate management is to
minimize noninterest-bearing
cash balances.
highly sophisticated
cash management
large corporations
are able to maintain a sizeable
fraction of what are effectively transactions balances
in various rnonez market instruments such as largedenomination certificates of deposit, short-term commercial paper, and short-term U. S. Government
It is on this basis that some analysts
would suggest that under present conditions at least a
portion of an aggregate as broadly inclusive as M;,
b Friedman and Schwartz, Monetary Statistics. pp. 106-7.

: The most comprehensive effort to establish L& as the proper
definition of money on theoretical grounds is found in the work of
Pesek and Saving.
See Boris P. Pesek and Thomas R. Saving.
Money. Wealth, and Economic Theory. New York: The Macmillan
Company, 1967. PP. 39-254.
For a critique of this anabsis see
Milton Friedman and Anna J. Schwartz “The Definition of Money:
Net Wealth and Neutrality as Criteria,” Jozm-md of Money. Credit
and Banking, 1 (February 1969). l-14.

!‘Several recent innovations in the financial sector related to the
payments services provided by financial institutions to their eustomers further support this view.
For example, so-called NOW
(for negotiable order of withdrawal) accounts offered by thrift
institutions and banks in New Hampshire and Massachusetts permit
depositors to write ahat are essentially checks on interest-tearing
deposits. AIso. Federal regulatory authorities recently adopted new
regulations allowing hanks and thrift institutions to offer preauthorized bill-paying services to savings depositors.




might reasonably be considered money.
Nor does
the story necessarily end here. For example, bank
loan commitments to business firms are sources if
not abodes of purchasing power. None of the aggregates listed in Table I captures this additional source.
Although the view of money as a temporary abode
of purchasing power has considerably broader implications than the more restrictive means of payment
concept, both tend to focus attention on the relationship between money and current transactions.
somewhat different position regarding the basic function of money has been evident in part of the postwar
Taking their cue from Keynesian monetary theory, analysts in this group have emphasized
the role of money as a store of liquid wealth held to
meet unanticipated contingencies necessitating payments as well as expected transactions and to balance
illiquid assets such as long-term securities and nonfinancial assets in household and business portfolios.10
According to this view, “money” is synonymous with
“liquidity,” although the latter term has never been
specified rigorously.
Much of the analysis along
these lines was published in the late 1950’s and
1960’s. Writers in this vein argued that the transactions approach to defining money had tended to
restrict attention too narrowly to commercial bank
deposits, obscuring the significance of the postwar
shift of liquid balances from commercial banks to
other financial intermediaries
such as savings and
loan associations and credit unions. Unless money
were viewed more broadly as liquidity, and the liabilities of nonbank intermediaries considered part of
the money stock, monetary policy would be rendered
The more recent extension of the transactions appreach described above, which recognizes the possibility that transactions balances may well be held not
only in bank and nonbank deposits but also in a
variety of money market instruments, has blurred
some of the issues that were central to the earlier
debate and broadened the scope of the dialog. At
this point, many economists would probably acknowledge that ns a pztrely formal matter money might be
defined more broadly than M1, or perhaps more
broadly than l&f2 or M3. Beyond that, interest in
defining money on purely theoretical grounds appears
to have waned.

1” This strain of analysis began with the work of John G. Gurley
and Edward S. Shaw in the 1950’s. See John G. Gurley and Edward
S. Shaw, “Financial Intermediaries and the Saving-Investment Prccess,” Journal of Finance, 11 (March 1956). 257-76, and Gurley and
Shaw. Monay in a Theory of Finance. Washington. D. C.: The
Brookinns Institution. 1960. Similar views were put forward in the
Radcliffe Committee report on the British monetary system pub
lished in 1959.




Empirical Approaches
Since the theoretical
approach to defining money has failed to produce any
definitive agreement, it is not surprising that economists have attempted to settle the issue empirically.
Indeed,, Milton Friedman and Anna J. Schwartz, two
prominent participants in the discussion, have suggested that the question of the correct definition of
money cannot be separated from the question of the
practical uses to which such a definition would be
put by policymakers or others:
We conclude that the definition
of money is to be
for not on grounds
of principle
but on
of usefulness
in organizing
our knowledge
of economic relationships.
‘Money’ is that to which
we choose to assign
a number
by specified
it is not something
in existence
to be discovered like the American
it is a tentative scientific
to be invented,
like ‘length’
or ‘temperature’
or ‘force’ in physics.”

As suggested above, money is interesting to economists and policymakers primarily insofar as changes
in its stock affect basic economic variables such as
income, employment, and prices. From this standpoint, the best definition of money might be the definition producing the closest statistical correlation between money so defined and, say, national income. A
large number of statistical tests have in fact attempted
to determine which money definition yields the closest
Taken as a group, these studies have
shown a close relationship between income and several of the narrower money aggregates such as M1,
-J&, M3, and variants of these measures.
But they
have been contradictory
and inconclusive regarding
exactly which concept produces the best fit.‘” Ill
general, the results of these various tests have been
quite sensitive to the time period considered and the
exact form of the estimating equations used, especially their respective lag structures.
A related but nonetheless distinct empirical a.ppreach has focused on the degree of substitutability
among various categories of assets considered candi-

II Friedman and Schwartz. Monetary

Statistics, p. 1B’i.

12Representative examples are Milton Friedman and David Meiselman, “The ReIative Stability of Monetary Velocity and the Investment Multiplier in the United States? 1897-1958,” in Commission on
Money and Credit, Stabilization P&&s.
Englewood Cliffs, N. J.:
Prentice-Hall. 1963, pp. 165-268; George G. Kaufman. “More on an
Empirical Definition of Money,” American Ecmomi~ Review, 59
(March 1969). 78-87: Frederick C. Schadraek, “An Empirical Approach to the Definition of Money,” in Monetary Aggregates and
Mc-netanJ Policy. New York: Federal Reserve Bank of New York,
1974, pp. 28-34; and Jack L. Rutner. “A Time Series Analysis of
Income and Several Definitions of Money,” Monthly Review, Federal
Reserve Bank of Kansas City, November 1974, pp. 9-16.
Friedman-Meiselman and Schadrack studies concluded that Ms is
the preferable definition of money.
Kaufman’s results suggested
that a somewhat broader definition alonn the lines of Ms is slightly
better than either Ml or MS on the basis of certain evaluative
criteria. Rutner’s work suggested that the correIation between income and alternative money concepts is itself a function of the
time frame of the statistical analysis. broader a.cgrexates performing
relatively better over lonaer time horizons.



dates for inclusion in the definition of money. It is
generally agreed that demand deposits should be
included in any definition of money. A high degree
of substitutability between demand deposits and, say,
time deposits would suggest that time deposits can
satisfy at least partly the purposes for which demand
deposits are held and should therefore be considered
Statistically, the degree of substitutability
has commonly been measured by the sensitivity (in
technical language the “cross-elasticity”)
of the demand for agreed money assets, such as demand deposits, to variations in the interest rates paid on
candidate categories, such as commercial bank time
deposits and the liabilities of nonbank intermediaries.
these substitutability studies, like the
correlation studies discussed above,
have not produced conclusive results. Some studies
have found relatively low cross-elasticities and have
concluded that M1 is the appropriate
Others have found higher elasticities, suggesting that
Mz or Ma might be preferable.r3
To summarize, neither theoretical nor empirical
analysis has produced a concensus among economists
as to precisely what collection of financial assets constitutes “money.”
On reflection, this lack of agreement is not very surprising.
For one thing, a given
financial asset can serve its holder in more than one
fashion. For example, while a savings deposit provides its holder with a store of purchasing power, it
also produces income in the form of explicit interest
savings deposits as a class
might be partly money and partly something else.
There is no particular reason for insisting that the
definition of money either include or exclude the
entire stock of savings deposits outstanding.
basically, money is fundamentally a social phenomenon, and, like all social phenomena, is subject to
continuous change. What appears to be needed is not
some final, exclusive catalog of assets labeled money,
but a flexible framework aimed at helping analysts
and policymakers determine to what extent specific
asset classes are functioning as money at particular
points in time. The next section describes such a

13Two of the most widely discussed of these studies are Edgar L.
Feige, The Dentand for Liquid Assets: A Tempwal Cross Sectirm
Englewood Cliffs, N. J.: Prentice-Hall, 1964 and Tong
Hun Lee. “Substitutability of Non-Bank Intermediary Liabilities for
Money : The Empirical Evident,” Jouti
of Finance. 21 (Septemher 1966). 441-57.
Feige’s study indicated that demand deposits
and bank time deposits are weak substitutes, suggesting the
superiority of a narrow money definition.
Lee found significant
substitutability between thrift deposits and bank demand and time
deposits, indicating that a broader definition such as Ma might be
preferable. See also Franklin R. Edwards, “More on Substitutability
between Money and Near-Monies,” JOWVUIJ of Money. Credit and
Banking. 4 (August 1972). 651-71.
A fourth important study
dealing with substitutability, by Chetty. will be discussed later in a
somewhat different context.




As we have seen, the monetary aggregates presently monitored by policymakers (see Table I) are
simple summations of the total stocks of various financial assets.
The characteristic
feature of this
aggregation technique is that the stocks of all assets
included in a given aggregate carry equal and unchanging weights, namely unity. This is a convenient
procedure, of course, but it raises some rather pressing questions regarding the analytical usefulness of
these aggregates when they are expressed quantiSuppose, for example, that an analyst
wished to use M5 as a measure of the money supply.
This aggregate includes such diverse assets as currency, savings and loan shares, and large-denomination certificates of deposit. The weighting procedure
employed in deriving M5 would imply that each
dollar of each asset class serves as money to the same
This implicit assumption would probably
be invalid, whatever the analyst’s criterion for defining money might be. Therefore, any uncritical use
of 111s as a measure of the money supply would
almost certainly be analytically misleading.
This aggregation procedure is obviously a special
instance of a more general technique where the
weights attached to each asset category are permitted
to vary both among categories and over time. For
example, if the goal is an improved measure of the
money supply, an analyst might want to attach a
higher weight to demand deposits and a lower weight
to certificates of deposit in compiling M5. Edward J.
Kane has developed a general framework for the
weighted aggregation of monetary variables along
these lines, and it will be useful to recapitulate briefly
the main features of Kane’s technique here.l*
should be noted at the outset that Kane’s technique
requires that an analyst using it specify precisely
his criterion for determining the relative moneyness
of asset classes. Kane’s own criterion is the extent
to which assets are actually used, that is, liquidated,
to support expenditures.
It is this particular choice
among alternative criteria that gives Kane’s analysis
its substantive content and raises it above the level
of a purely mechanical exercise.
The following description of the framework employs elementary algebraic notation for generality and simplicity. No highpowered mathematics is involved.
Kane begins by defining the money balance held
by the jtb individual economic unit (perhaps a household or a business firm) as:

“Edward J. Kane, “Money as a Weighted Aggregate,”
fir7 Nationalokonomie, September 1964, pp. 222-7.






lllj =

2 wi5aii,

i = 1, * - -9 N;


where aij is the dollar amount of the i* asset (one of
N available assets) held by the jth unit (one of P
units in the economy) and wij is the weight. The wij
take on values between zero and unity. Any particular wij may be interpreted as the proportion of the ith
asset regarded by the jth unit as serving a money
function. We will adopt Kane’s money criterion and
regard the wij as signifying the proportion of the ith
asset actually used by the jth unit to support transactions during the time period in question.
number of alternative interpretations
of the weights
would be consistent with the framework.
The aggregate money stock, M, can be obtained
from (1) by summing over the P economic units in
the economy :






This expression






can be written





where Ai is the total dollar amount of the ith asset
outstanding in the economy. The weighted aggregate
is then:


M = TWiAi,

where :


wi = 1 wijaij .

(4) and (5) appear quite simple on
the surface, but they point out with great clarity the
fundamental problem facing analysts in monetary aggregation. That problem is to specify the determinants
of the individual unit weights (the wij) and, from
these, the determinants of the aggregate weights (the
Wi). In the absence of empirical evidence, one can
only speculate as to what these determinants might



be. Such things as interest rates and the price level
and expectations of future changes in interest rates
and the price level, however, are likely candidates.
Further, since both current and expected interest
rates and prices change over time, it seems reasonable
to suppose that the weights might change in some
systematic and therefore predictable manner over
Some simple examples might serve to illustrate the
potential analytical usefulness of the weighted aggregate concept. Suppose that some technological innovation or perhaps a regulatory change reduced the
cost and inconvenience to households of shifting funds
from savings accounts to demand deposits.
these circumstances, households would have an incentive to hold a greater portion of their transactions
balances in savings accounts. Abstracting from any
other factors affecting the total volume of savings
deposits held by households, the weight attached to
savings deposits in calculating the effective money
supply would rise.15
The preceding example suggests the kinds of factors that might alter the weights over the longer run.
A second example will indicate some of the factors
that might cause the weights to vary systematically
over the business cycle.
Suppose that during an
expansionary period a general increase in short-term
interest rates occurred.
would then
have a stronger incentive than during a period of
low rates to hold their transactions balances in the
form of money market instruments such as Treasury
bills or certificates of deposit. Under these circumstances, the monetary weights attached to these instruments would rise.
In view of these examples, it would appear that
weighted monetary aggregation of the sort suggested
by Kane’s framework might be useful in developing
improved measures of the money supply.
At the
same time, it is evident that efforts to apply the
technique in practice will confront difficult statistical
Nonetheless, the approach has been
sufficiently appealing to motivate several preliminary
empirical studies. The next section summarizes the
results of these studies.


To date, only a handful of studies have attempted
to measure statistically the weights that should be

15This example. it should be noted, is more than hupothetical, since
the Federal Reserve recently lifted its 39-year-old prohibition of the
use of the telephone for transferring funds between mvings and
demand deposits.



attached to individual asset categories in developing
monetary aggregates.
The results of these analyses
can only be considered preliminary.
The studies are
interesting, nonetheless, with respect not only to the
specific numerical estimates of various weights but
also to the methodologies employed.
This section summarizes four such studies published during the 1960’s. No attempt will be made
to evaluate the studies critically,
The purpose of
the summary is simply to convey the flavor of their
results. For brevity, the following notation is used:
CBDD = demand deposits at commercial


CBTD = time and savings deposits at commercial

weighted aggre,gates. To this end, Elliott employed a
cross-sectional analysis using per capita deposit and
income data by states to estimate the weight for a
composite group of assets consisting of CBTD, MSD,
and PSD. Three separate cross-sectional estimates
were derived for three separate years. The estimated
weights were .26 for 1929, .35 for 1937, and .65 for
Each of these estimates was significantly
different! statistically, from both zero and unity. A
separate time series analysis using aggregate national
data for the years lS97-1957 produced an estimated
weight of .37. This estimate was also significantly
different from zero or unity, and its magnitude was
consistent with those obtained from the cross-sectional tests.
Elliott’s regression

model was of the form:

MSD = mutual savings bank deposits

PSD = Postal Savings System deposits

Second, the studies employed different assumptions
and statistical procedures, and none adopted Kane’s
detailed framework and definitional criteria as a
starting point.
differences among the
estimated weights for particular assets across the
four studies reflect conceptual dissimilarities as well
as differences in the data and statistical models used.
Still, the underlying concepts are sufficiently alike to
permit comparison of the estimates,


where y is permanent income, and w, a, and b are
the parameters to be estimated? with w the desired
weight coefficient.‘” Several interpretations of w are
consistent with this model. One such interpretation
is that w measures the proportion of (CBTD +
3ISD + PSD j held to support current expenditures.
If this interpretation is adopted, Elliott’s three crosssectional results tentatively suggest that the moneyness, in this sense. of time and savings deposits was
increasing secularly over the time period considered.
Elliott’s results can be compared with the results
of two other studies employing roughly similar methodologies by (I> Richard H. Timberlake, Jr. and
James Fortson and (2) Gurcharan S. Laumas.‘!’
Both of these time series studies used the following
regression model :

The earliest of the four studies was a doctoral
dissertation completed by Roy Elliott at the University of Chicago in 1961. I7 The purpose of this study
was to investigate whether a money aggregate with
nonuniform weights displayed a more stable relationship with income than conventional, uniformly

‘“This debate grew out of Milton Friedman’s inclusion of consumer
time and savings deposits at commercial banks in his definition of
the money supply.
Virtually all empirical studies of the money
supply, including the four discussed here, include lk balances at
their full face value. That is, MI balances carry a weight of unity.
17Roy Elliott, “Savings Deposits as Money”
dissertation, University of Chicago, 1964).

+ w(CBTD

+ PsDjl = a + W-nW1,

SLS = savings and loan association shares
Several preliminary remarks are in order. First,
all bf these studies appear to have been stimulated by
the debate during the 1960’s over whether various
categories of consumer-type time deposits should or
should not be included along with Ml in the money
s~pp1y.l~ Thev, therefore focused on assets included
in the Mz and Ma aggregates of Table I. No effort
has yet been made to measure the weights that might
be assigned to other assets included in the more
broadly defined aggregates such as M4 and Ms.


(unpublished Ph.D.



AY = a + b(AM1)

+ c(AT),

where A indicates first differences in the variables,
Y is current aggregate income, M1 is as defined in
Table I? and T is time and savings deposits measured
in various ways as indicated below. Equation ( 7)
can be rearranged in the following manner :

AY = a + b(AM1 + c/b AT).

Is For statistical convenience. the equation Elliott actually estimated
was an approximation of (6) that was linear in the weight coefficient w. All variables in his tests were expressed in real per capita
“‘Richard H. Timberlake and James Fortson, “Time Deposits in
the Definition of Moues.” American Ec~~~omic Review, 57 (March
1967). 196-4: Gurcharan S. Laumas. “The Degree of Moneyness of
Savings Deposits.” Amc~ican Ecmmmic Review, 56 (June 1968),



The ratio c/b is then the collective weight for the
Since the
assets included in T in any given test.
tests, like Elliott’s test,
were based on the correlation of the monetary variable with an income variable, the interpretation suggested above for Elliott’s w might also be applied
to c/b.‘O
Although the Timberlake-Fortson
and Laumas
studies, respectively, were based on the same model,
they produced very different results. Using annual
data, Timberlake and Fortson estimated the weight
for various subperiods between 1897 and 1960 with
T defined, as in Elliott’s study, as (CBTD + MSD
+ PSD). Among the pre-World War II subperiods,
the estimated weight was positive only for the years
1933-1938. On these grounds the authors concluded
that time and savings deposits did not serve a money
function during most of the prewar period.
1933-1938 result was interpreted as implying that
the public associated a greater degree of risk with
demand than with time deposits during these years
in reaction to the rash of bank failures in the early
1930’s. Hence, money balances were held in the
form of time deposits during this period.
In the
postwar era, the weight was estimated at a relatively
low .15 for the 1953-1965 subperiod.
Laumas employed Timberlake and Fortson’s technique, but he restricted his study to the postwar era
(his tests covered the years 1947-1966), and he used
quarterly data and several specifications of T. His
results were as follows. Using the EIIiott-Timberlake-Fortson specification of T, (CBTD + MSD +
PSD), the estimated weight was .4S. It is worth
noting that this estimate falls about midway between
Elliott’s cross-sectional estimates for 1937 (.35) and
1954 (.65). Therefore Laumas’ results tend to substantiate Elliott’s.
With T more narrowly defined
as CBTD alone, the estimate increased to .58. Defining T more broadly as (CBTD + MSD + PDS
+ SLS) reduced the estimate to .32. These results
imply that the moneyness of CBTD exceeds that of
MSD and, in turn, the moneyness of MSD exceeds
that of SLS.
The final study, by V. Karuppan Chetty,
somewhat different approach.“l

measured the weights for individual time and savings
deposit categories (CBTD, MSD, and SLS, respectively) on the basis of prior estimates of their substitutability in demand for M1 balances.
Chetty’s conceptual procedure is illustrated in a
simplified manner by Figure 1. This diagram depicts the public’s allocation of its liquid balances between Ml-type assets, measured on the vertical axis,
and time deposits, measured on the horizontal axis.
The sloping line in the figure is what economists
refer to as an indifference curve. The curve specifies
various combinations of Ml balances and time deposit
balances that are equally satisfactory to the public.
It also indicates the rate at which the public is willing to substitute balances in one of the categories for
balances in the other.=
Let us suppose that the shape and position of the
indifference curve are known and that the public is
observed to be at point A on the curve.
At this
point it holds OM dollars of Mi balances and OT
dollars of time deposit balances. The curve indicates
that the public would be equally satisfied at point P,
where it would hold OP dollars of Mi balances and
no time deposit balances. This implies that the public
considers a combination of OM dollars of Mr balances and OT dollars of time deposit balances to be

curves are explained in most elementary economics
textbooks. See for example, Paul A. Samuelson, Economica, 8th ed.,
New York: McGraw-Hill Book Company, 1970, pp. 421-6.

8; ‘%

1 .FIGllREI,






took a


*I V. Karuppan Chetty.
“On Measuring the Nearness of NearMoneys,” Amen’ean Economic Review. 59 (June 1969). 270-81.



;. >

m While the concepts are similar, the statistical procedures and data
employed in the two sets of studies were vastly different.
present writer believes that Elliott’s procedures, as detailed on pages
35-40 of his study, were sounder and that his estimates are therefore
more reliable.









equivalent in moneyness
ances. That is:

to OP dollars of Ml bal-

Moneyness of ($OM + $OT)
Moneyness of $OP.


To complete the analysis, a measure of moneyness is needed. Let us arbitrarily assume that one
dollar of M1 balances contains exactly one unit of
Under this rubric the quantity
moneyness in any combination of Ml and time deposit balances is measured by the dollar value of the
M1 balance to which the combination is equivalent.
In the present example, equation (9) then indicates
that the combination of OM dollars of Ml balances
and OT dollars of time deposit balances contains OP
dollars of moneyness.
That is:


of ($OM + $OT)

= $OP.

Since an Ml dollar contains one unit of moneyness,
we know that:


of $OM = $OM.


$OM + Moneyness of $OT = $OP.
(12) can be rewritten
$OM + m($OT)


As previously stated, the results of these empirical
studies are tentative at best. As is common in statistical estimation of this sort, the numerical results
are quite sensitive to the methods and data used.2d
Nonetheless the similarities among some of the results shown in Table II are at least mildly encouraging. Moreover, the estimates fall generally within
a range that is intuitively plausible.
In short, the
results of these studies suggest that weighted monetary aggregation might be empirically feasible.
addition, they appear to justify further analysis aimed
at developing preliminary estimates of the weights of
some of the assets included in the broader aggregates
of Table I.
2’ As an example of this sensitivity, Franklin R. Edwards found
much less substitutability between MI balances and other assets
when he applied Chetty’s model to cross-sectional metropolitan area
data. See Franklin R. Edwards, “More cm Substitutability Between
Money and Near-Monies,” .7oumal of Money. Credit and Banking, 4
(August 1972 ) , 564-6.

It then follows that:

the three other studies discussed above.
weights were generally higher than those found in
the other studies. In particular, Chetty’s estimate of
the weight for CBTD was unity, implying that these
deposits should be included along with M1 balances
at their full dollar value in measuring the aggregate
money stock. Apart from this, it is worth noting that
Chetty’s estimates for the respective asset categories
were ordered identically to Laumas’ estimates.

as :

= $OP,

where m is the proportion of moneyness in a nominal
dollar of time deposits.
In other words, m is the
weight that would be attached to time deposit balances in monetary aggregation.
It is obvious from equation (13) that the dollar
values of the balances OM, OT, and OP are sufficient to determine m. Chetty used actual observations of OM and OT (along with interest rate data)
for the years 19451966 to determine, in effect, the
shape and position of the indifference
curve in
Figure 1. This procedure fixed the point P and
established the value of the hypothetical balance OP
from which the weight m was then derived.23
By deposit classes, the estimated weights were 1.00
for CBTD, 2% for MSD, and .62 for SLS. For the
reader’s convenience, Chetty’s results are shown in
Table II along with the postwar period results of










CBTD -i- MSD -i- PSD







time series,





series, 1947-l 966




time series,



1 .oo




1945- 1966



2~The foregoing description of Chetty’s technique is not precise,
but it is a close enough approximation for the purposes of the
present survey. In technical language, Chetty measured the weights
using a regression model derived by maximizing a CES utility
function having Ml. CBTD. MSD, and SLS as arguments.
the detailed derivation, see Chetty, “Near-Moneys,” pp. 272-8.



time and savings


Postal Savings

* All estimates





System deposits

loan association

assume a weight


at commercial


of unity for currency and demand





This article has attempted to provide an overview
of some of the major conceptual issues associated
with the process of monetary aggregation and, as a
consequence, with the use of monetary aggregates as
they are presently defined in the conduct of monetary
policy. The article reviewed the main features of the
debate among economists over the proper definition
of money. It then described a general framework for
weighted aggregation and suggested some of the
factors that might influence the weights and their
behavior over both the short and long runs.
third section reviewed preliminary efforts to estimate
the weights of a limited number of assets statistically.
A major aim of this discussion has been to suggest
that despite all of the current public comment about
the money supply, there is no firm agreement as to
precisely what it is or how it should be measured. As
we have seen, this state of affairs reflects the fact
that money is simply not as concrete and unambiguous a concept as is commonly believed.
what serves as money can change over time with
longer-run changes in financial technology, financial




regulations, and underlying social behavior, and POSsibly with variations in economic activity and financial conditions over the business cycle.
Do these observations imply that the use of the
various monetary aggregates shown in Table I is
analytically unsound ? Not necessarily.
They do
suggest, however, that the combined behavior of these
aggregates as a group may provide a more accurate
indication of the effect of monetary policy actions
than the behavior of any one of them.
This last comment is not intended to imply that
simply monitoring a larger constellation of aggregates’
is an ideal procedure.
Refinements are clearly possible. This is where research along the lines described
in Sections II and III is relevant.
True, the complexity of weighted aggregation and the measurement
difficulties associated with the technique will almost
certainly preclude employing any such aggregate as
an operational variable in the day-to-day conduct of
monetary policy.
Nonetheless, this research shows
promise of producing new insights that might suhstantively improve the ability of policymakers
interpret the behavior of the conventional aggregates.







Thomas M. Humphrey

Given the inherent complexity of the current inflation problem and the tendency of individuals to
differ in their interpretation of events, it is not surprising that a number of competing theories of inflation exist today. This article seeks to explain one of
these theories-namely,
the monetarist view-with
the aid of a simple dynamic macroeconomic model
developed by the British economist Professor David
Laid1er.l Laidler’s model is enlightening for reasons
quite apart from its monetarist orientation. Although
exceedingly simple, it nevertheless effectively conveys
all the essentials of dynamic process analysis-steadystate solutions, disequilibrium
dynamics, stability
conditions, etc. It is representative of a whole class
of models that deal not with levels but rather rates of
change of economic variables.
These models are
gradually supplanting the once-popular standard textbook or diagrammatical version of the Hicks-Hansen
IS-LM model, whose static equilibrium format is not
ideally suited to deal with the phenomenon of continuing inflation or with the dynamics of disequilibrium processes wherein economic variables evolve
and interact over time. Therefore, regardless of the
particular theory being expounded, Laidler’s model
can be viewed as an introduction to a distinctive form
of macroeconomic analysis that attempts to specify
the time paths of the inflation rate and related variables.

tarist view. The sole aim is to articulate the monetarist interpretation
within the framework
of a
mathematical model whose exposition constitutes a
useful exercise in its own right. It should be strongly
emphasized, however, that the model constitutes a
severe oversimplification
of a complex process and
thus would probably fit the statistical data poorly.
As used in this article, the model is intended solely
as an expository device and therefore purposely abstracts from many of the variables and behavior relationships that a well-specified empirical model would
model that purports to convey the essence of monetarism must embody certain key propositions or postulates that characterize the monetarist position. Not
all of these propositions, however, can be regarded as
exclusively monetarist. Some would be accepted to a
greater or lesser degree by nonmonetarists.
It is
therefore desirable to divide these propositions into
two groups, namely, those that are distinctively
monetarist and those that are not. A partial listing
of the uniquely monetarist propositions would include
the following.
hold that inflation is a purely monetary
that can only be produced by expanding
the money supply at a faster rate than the growth of
capacity output.
Thus at any given time the actual
rate of inflation is seen as reflecting
current and past
rates of monetary
of inflation-i.e.,
that attribute
rising prices to such alleged causes as
shifts in autonomous
government fiscal policies,
food and
fuel shortages,
the grounds that an increased
stock of money per unit of output is required
in all
cases and therefore
the true cause of
In short, the sole necessary
and sufficient
condition for the generation
of inflation is said to be

A word should be said at the outset about the
article’s position on rival theories of inflation.
Regarding the merits of alternative views, this article
takes a deliberately neutral stance. Neither monetarism nor any other theory is advocated as being the
most nearly correct.
No claims are made for the
superiority or indeed even the validity of the mone* Laidler presents his model in two papers: “The 1974 Report of the
President’s Council of Economic Advisers: The Control of Inflation
and the Future of the International Monetary System,” American
Economic Review. 64 (September 19741, PP. 535-43, and “The Influence of Money on Real Income and Inflation: A Simple Model
with some Empirical Tests for the United States 1953-72.” Manchaster School of Economic and Social Studies, 41 (December 1973).
pp. 367-95.
The version of the model contained in the present
paper differs from Laidler’s in at least five respects.
First, it is
simpler and employs a different notation,
Second, its numerous
close linkages with monetarism are identified. Third, it is employed
solely to explain the monetarist view of inflation.
Fourth. an
explicit derivation of the equations is provided. Finally, the model
and its components are expounded in considerably greater detail
than in Laidler’s rather terse treatment.

The proposition
of a
velocity or rate of turnover
2 Monetarists readily admit that nonmonetary influences*.g.,
wage pressure, monopoly (administered) pricing policies. OPEC
cartels. oil embargoes. crop failures, commodity shortages, and the
like--can directly affect particular prices.
But they argue that
without excessive monetary growth such nonmonetary-induced rises
in the prices of some commodities eventually would be offset by
declines in the prices of others, leaving the average price level




of money follows logically from the monetarist
that inflation
stems solely or largely from excessive
For if velocity were not a conmonetary
stant it would exhibit a non-zero rate of change that
would supplement
as a separate
and independent
of inflation.
It follows.
that monetarists
must assume that velocity
is at least a quasi-constant
if they are to assert that
stems solely or primarily
from changes
the stock of money per unit of output.
treat the quantity
and its rate of growth
as variables
are fixed outside the system.3
This view
with the nonmonetarist
of money
as an endogenous
within the system by the level of economic
and by the public’s preferences
for money and for
money substitutes.
The exogeneity
postulate implies that monetary
growth enters the system
as a datum to determine
the growth
rates of spending, prices, and nominal
The postulate
the monetarist
view of
as the independent
causal factor
the rate of inflation.
Implied by the exogeneity
this proposition
the notion
of passive
growth and asserts the monetarist
view of
the unidirectional
of influence
or flow of
running from money to spending to income
to prices.
growth is seen as entering
not as a dependent
or accommodative
variable responding
to prior income growth but
as the active independent
that precedes and causes inflation.
It is true that monetarists, in their asides and qualifications,
that income may influence
money indirectly
the policymakers’
to changes
in the economy.
But for the most part they have not incorporated
such policy
into their
formal models, and they continue
to treat monetary
policy as largely exogenous.

The preceding constitutes the group of uniquely
monetarist tenets. As for the remaining key propositions, i.e., those that monetarists share with at least
some nonmonetarists,
they can be listed briefly.
They include the following : (5) the non-neutrality of
money in the short run (i.e., the tendency for changes
in monetary growth to have substantial effects on real
output and employment in the short run) ; (6) the
long-run neutrality of money (i.e., the tendency for
changes in monetary growth to have no lasting
impact on real output and employment but only on
the rate of inflation) ; (7) the view of erratic and
voIatile monetary growth as the prime cause of business cycles ; (8) the inherent stability of the economy
(i.e., the view of the system as a self-regulating
mechanism, perturbations of which tend to generate
only damped cycles about full-employment
equilibrium) ; (9) the existence of long lags in the response
:‘The exoceneity condition applies only to the nominal and not to
the real (price-deflated) stock of money. Unlike the nominal stock,
the real stock is treated as an endogenous variable determined by
the public’s demand for real balances. The public, via the impact
of its spending on the price level. can make the rea1 value (purchasing power) of any given nominal stock of money conform to
whatever magnitude it desires.




of inflation to changes in the rate of monetary
growth ; and finally ( 10) the importance of inflationary expectations in determining market wage- and
price-setting behavior.
As shown below, Laidler’s
model is capable of accommodating all these propositions.
The Model and Its Components
The model
itself is composed of three equations, the first being
the wzonetary growth equation. A dynamic version
of the static Cambridge cash-balance formula, this
equation relates the rate of growth of real (pricedeflated) cash balances to the growth rate of real
output. The second relation in the model is a priceadjustment
equation that explains the determination
of the current rate of inflation. The third component
is an e.zpecta.tions-formation
equation that embodies a
particular hypothesis about how people formulate
their expectations
of the future rate of inflation.
Using these three equations one can solve for the
three endogenous variables of the model, namely,
(1) the current rate of inflation, (2) the expected
rate of inflation, and (3) an excess demand variable
represented by the gap between actual and capacity
real income. In addition to these endogenous variables there is one exogenous variable, the growth rate
of the nominal money stock, and one exogenous constant, the growth rate of full-capacity real income.
This treatment of the monetary variable reflects the
monetarist view of the money stock and its growth
rate as largely exogenous magnitudes determined by
an autonomous central bank via its control over a
base of so-called high-powered money, consisting of
currency and bank reserves. It also effectively rules
out any reverse-causation
feedbacks running from
income to money.
The assumption of a fixed capacity growth rate also squares with monetarist doctrine, which holds that the long-run path of potential
output is independently determined by fundamental.
real economic conditions including technological progress and labor force growth.
Three other features of the model should be mentioned at the outset.
First, all relations are linear
and are expressed in logarithmic form. There is a
specific reason for this formulation.
Modern monetarist analysis is usually stated in terms of percentage
rates of change of the relevant variables. And since
the percentage change of any variable over a given
interval of time can be represented mathematically
by the first time difference of its logarithm, it follows
that a log-linear formulation facilitates the analysis.
A second feature of the model is the introduction
of time delays in the form of lagged relationships
among the variables.
These lags reflect the moneNOVEMBER/DECEMBER


tarist view of the many delays or frictions inherent
in the inflationary process.
Their inclusion also
permits the analyst to describe the time paths taken
by output and prices following a monetary disturbance.
The third feature of the model is its extreme simplicity, as manifested by the minimal number of variables it contains. In particular, the model possesses
neither an interest-rate
variable nor a variable to
represent a discrepancy between actual and desired
real cash balances. As a result, the model ignores
two potentially important elements in the inflationary
process, namely, (1) changes in the rate of interest
and (2) the transitory rise in real cash balances (or
the temporary fall in the velocity of money) that
occurs at the beginning of inflationary periods immediately following a rise in the growth rate of money.
These elements could of course be explained in a
more complex model, but such a model would lose
in simplicity, manageability, and ease of comprehension what it gains in completeness.
Moreover, Laidler’s model, despite its simplicity, is capable of explaining a large part of the inflationary process,
namely, how variations in the growth rate of the
money stock are divided between changes in real
output and prices both in the short and the long run.
As for notation, the model employs the following
symbols. Let m, be the money stock, y actual real
income, yc standard or normal capacity real income,
s the excess demand variable represented by the
difference between actual and capacity income, i.e.,
s = y - yc, and p the price level-with
all variables
expressed as logarithms.
Actual real income, y, can
exceed capacity, yc, because the latter is defined not
as the absolute physical limit or maximum ceiling
level of output but rather as the output associated
with the economy’s normal or standard level of operation. This concept of capacity or potential output
corresponds roughly to the monetarist notion of the
nntz& rude of cfnenzploywzent, i.e., the unemployment
rate that, given the inevitable frictions, rigidities, and
market imperfections existing in the economy, is just
consistent with equilibrium between demand and
supply in the labor market.
The superscript e denotes the expected value of a variable, and the subscripts -1 and -2 denote time lags of one and two
periods, each defined as being a year in length.4 The
symbols A and P appearing before a variable denote
first and second time differences, respectively, so that
the model is effectively expressed in terms of proportional rates of change and rates of acceleration or
’ The time yeriud in Laidler‘s mud4 is: not specified but is defined
here as one year to conform to the monetarist interpretation that
thia article is developing.


deceleration of those rates of change. Finally, a bar
over a variable indicates that it is exogenous, i.e.,
determined outside the system.
The Monetary Growth Equation
tion of the model is the monetary

am -

The first equagrowth equation:

Ap = by = Ax + G.

This equation states that the rate of growth of the
real money stock-i.e., the percentage rate of nominal
money growth, Am, less the percentage rate of price
inflation, Ap-determines
the percentage change in
real expenditure and hence real income, Ay, that
occurs during the given period.
More precisely, a
rate of growth of the real money stock, Am - Ap, in
excess of the growth rate of capacity output, ?&
causes the growth rate of actual output, by, to
deviate from the capacity growth rate, where the
deviation is represented
by the variable Ax, i.e.,
Ax = Ay - z.
Equation ( 1) implie-3 a constant unitary income
elasticity of demand for real (price-deflated)
balances. This condition follows from the notionassociated with the old Cambridge cash-balance
version of monetarism--that
people desire to maintain a stable (constant)
between their real cash balances and real income.
If the ratio of real balances to real income is to
remain fixed, then both elements of the ratio must
grow at the same percentage rate, as in equation
( 1 j. The monetary growth equation also expresses
the strong monetarist view of a stable equiproportional relationship
between changes in nominal
money and nominal income and likewise between
changes in nominal money per unit of output and
the price level.
The equation predicts that a
given percentage change in nominal money will be
matched by an identical percentage change of nominal
income. The same holds for percentage changes of
nominal money per unit of output and of prices.
Kate, however, that the equation, by itself, is incapable of expressin g a stable predictable short-run relationship between nominal monetary growth and the
inflation rate.
This is because, in the short run,
monetary growth may stimulate output as well as
prices. And one cannot determine from equation ( 1)
alone the proportions in which the stimulus will be
divided between price changes and output changes.
One has to supplement equation (1) with the priceadjustment equation to explain this division.
Equation (1) may also be interpreted as embodying a crude monetarist view of the direct expenditure
whereby monetary impulses are trausmitted directly to income via a prior effect on the


demand (spending) for goods. The direct mechanism should be contrasted with the indirect interestrate mechanism-often
stressed by nonmonetaristsin which monetary changes influence income indirectly via a prior effect on the rate of interest.” As
shown in Appendix A, the money growth equation is
derived from the celebrated Cambridge cash-balance
equation and assumes that the velocity of money (or
the Cambridge K) is constant and that the money
market clears with sufficient rapidity to maintain
equality between money demand and supply.
assumption is what insures that
given rates of monetary growth, real and nominal,
will be matched in equation (1) by corresponding
identical rates of income growth, reaI and nominal.
The Price-Adjustment
The second
equation of the model explains how the current rate
of inflation is determined, i.e., the rate at which
businessmen mark up their product prices.
equation is written in the foIlowing
way :

Ap = ax-l

+ Ape-1

where Ap is the current rate of inflation, x-~ is
excess demand lagged one period, and Ape-I is the
rate of inflation expected to prevail in the present
period as of the preceding period. The price-adjustment equation expresses a short-run relationship between the rate of inflation, Ap, and excess demand, x,
the latter measured by the gap between actual and
potential (i.e.* normal capacity) output.
The existence of a gap implies that businessmen are straining
productive capacity in an effort to meet demand.
Spare plant and equipment are being drawn into use
and increasing resort is being had to overtime and
marginal labor. In brief, resources become increasingly scarce relative to demand as production approaches and then surpasses standard capacity output. The size of the gap measures the pressure of
resource scarcity on prices. The larger the gap! the
greater the pressure.
As the gap expands, wages
are bid up, labor-hour productivity falls, unit costs
rise, bottlenecks develop, and the backlog of unfilled
orders mounts.
All these forces combine to cause
prices to rise at an increasingly rapid rate.
inflation accelerates as the gap expands.
From the preceding discussion, it is evident that
the price-adjustment
equation is similar to so-called
equations that state a trade-off relajModern monetarists acknowledge that interest rate effects are
alwavs present.
They view the direct mechanism merely as an
empirical proxy for the indirect mechanism in which many specific
interest rate effects cannot be captured statistically either because
they are implicit and hence unobservable or because they are too
weak and too brief to be measured.




tionship between the rate of wage increase and the
unemployment rate. In the price equation, however,
excess demand replaces the unemployment
rate as
the indicator of the level of economic activity, and
the rate of price inflation replaces the rate of wage
inflation as the dependent variable. It is! of course,
assumed that rates of wage increase in excess of
productivity growth eventually tend to be incorporated in rates of price inflation as businessmen raise
their prices to cover increases in unit labor costs.
equation the
According to the price-formation
rate at which businessmen mark up their prices depends upon two influences, namely, tile level of excess
demand, s, and the expected rate of inflation, Ape.
The equation states that if aggregate supply and
demand are equal so that there exists no excess
demand (x = zero), then actual price inflation will
just equa1 expected inflation.
If, however, product
demand exceeds supply at the economy’s natural or
normal capacity level of operation,
eventually will react to the excess demand by raising
prices at a faster rate than the expected rate of
This price response, however, is not inFor a while, quantities rather than
prices tend to absorb the impact of excess demand as
businessmen temporarily expand output and perhaps
allow their inventories to be depleted.
These quantity changes signal the desirability of raising the rate
at which prices are marked up. Later, therefore.
businessmen respond to the excess demand by raising
prices. The one-period lag-again
defined as a year
-on the excess demand variable is meant to account
for the time it takes for a shift in demand to affect
The coefficient a, attached to the excess
demand variable, measures the magnitude of the impact that any given volume of excess demand has on
the rate of inflation. The higher the numerical value
This coefficient. of
of a, the greater the impact.
course, must be a positive number, i.e., a > 0.
The third
equation of the model is the expectations-formation
equation. It is written as follows:

Ape = bAp +

or, alternatively,

Ap” -


as :
Ape-, = I~(Ap-Ap”-I).

This implies that the change in the expected rate of
inflation, Ape-ApeBXY is proportional to the amount
by whi& this period’s actual inflation? Ap, deviated
from expected inflation as forecast one year ago.
Al)“-,, with the factor of proportionality,
b, having a
valrle between zero and unity.





Embodied in the equation is a particular theorythe so-called adaptive-expectations
or error-learning
how inflationary
formed. According to the error-learning hypothesis,
people formulate expectations about the inflation rate,
observe the discrepancy between the actual and anticipated rates, and then revise the anticipated rate
by some fraction of the error between the actual and
It can also be shown that the
anticipated rates.
hypothesis is equivalent to the
theory that people formulate price-expectations
looking at a geometrically-weighted
average of current and past rates of inflation with the weights
diminishing exponentially
as time recedes.
weighting scheme implies that people assign higher
weights to more recent phenomena when forming
How realistic is the error-learning
hypothesis ?
Some economists claim that it is not an accurate
description of how anticipations are formed. These
analysts argue that expectations are as likely to be
generated from direct forecasts of the future as from
mere projections of the past. Moreover, they assert
that people probably base anticipations at least as
much on current information about a variety of
developments as on old data pertaining solely to
past price changes. There is undoubtedly much truth
in these observations.
Nevertheless, the error-learning formulation will be retained in this article subject
to the caveat that purely extrapolative price forecasts may be modified by additional information.
The Complete


Taken together, the ~zoney
and expectations-formation
form a simple three-equation system that
a monetarist view of the inflationary prorecapitulate, the complete system is written


cess. To
as follows

hm -

Ap = Ax + z


AP = ax-l



= By

+ Ape-l

bAp + (l-b)


0 < b < 1

The variables in this system of equations interact
to determine the rates of expected and actual inflation
and the short-run growth rate of real income. The
logic of the system implies that variations in the
money growth rate initially affect excess demand,
thereby inducin, m real income to deviate from its fullemployment path. Lagged excess demand interacts
with lagged price-expectations
in equation (2) to
determine the current rate of inflation. The current
rate of inflation enters equation (3) to influence the
expected rate, which in turn feeds back into equation

(2) to become a determinant of next period’s inflation rate. Finally,. in equation (1) the current rate
of inflation interacts with the given rate of monetary
growth to determine the growth rate of real income.
In this manner the system and its constituent elements determine the division of monetary growth,
Am, between price and output growth, Ap and Ay .
Less formally,
causal chain.
1. Inflation

is determined

2. Inflationary
ous inflationary
excess demand.
3. Excess

the model implies



is created



the following



are generated
by previand hence by previous
by excessive


4. Therefore,
excessive monetary
the root cause of inflation.


The Long Run and the Short It is useful at this
point to distinguish between the long-run and the
short-run properties of the system of equations. This
dichotomy, of course, corresponds to the two main
stages or phases of the inflationary process, i.e., the
temporary or transition phase in which changes in
monetary growth affect real output and employment
and the final or permanent stage in which the sole
impact is on the rate of inflation. It also corresponds
to the monetarist distinction between the long-run
neutrality and the short-run non-neutrality of money.
In the context of the model, the long run refers to
the equilibrium or steady-state solution of the system
after it has completely adjusted to a monetary disturbance.
By contrast, the short run refers to the
period between successive long-run equilibria. Regarding the
long run, the relevant question is whether a monetary
shock has any lasting impact on real variables, i.e.,
is there a permanent trade-off between inflation and
As for the short run, one should focus on
the type of monetary shocks that initially disturb the
system and upon the subsequent reaction of the
system to those shocks.
Does a monetary disturbance affect output as well as prices in the short run?
What types of time paths do the variables describe
in disequilibrium ? How do the variables interact to
produce these paths ? Finally and most important,
do these paths tend to converge on the long-run
equilibrium, i.e., is the system stable?
Solution of the System
According to monetarist doctrine, long-run monetary
equilibrium is characterized by the following conditions :




1. Equality
actual and expected
rates of
price change,
the long-run
to correctly
and fully
adjust to it;

The short-run analysis involves at least two steps.
First, because interest centers on the time-paths of
( 1) inflation and (2) the excess demand gap between
actual and capacity income, one must derive expressions for the dynamic behavior of these two variables.
This derivation is accomplished in Appendix B.
Second, the resulting expressions must be analyzed
to determine whether the system is dynamically
stable, i.e., whether the variables will eventually converge on their long-run equilibrium values.

2. The absence
of any trade-off
and output,
the tendency
of monetary
shocks to have no lasting impact on real variables
but only on prices;
3. A constant
rate of inflation equal to the difference
between the growth rate of the money stock and the
growth rate of capacity output;
4. Attainment
of full-capacity
real income reflecting
the long-run
of actual output to adhere to
its full-employment

Does the model yield these conditions?
Only a
look at its steady-state properties will tell, i.e., the
model must be analyzed at its long-run equilibrium
The concept of equilibrium, of course,
implies equality between aggregate
demand and
supply, i.e., a state of zero excess demand.
the excess demand variable, x, equal to zero in the
equation yields Ap = AP”-~. Thus,
actual and expected inflation are equal, as required.
Moreover, the zero numerical value of the excess
demand variable (an index of real economic activity)
in the price equation signifies the absence of long-run
as required.
growth has a neutral long-run impact on real variables, at least in the model.
The next step is to set the first difference of excess
demand, Ax, at zero in the money growth equation.
Doing so enables one to solve for the steady-state
rate of inflation, which is Ap = bm - G.
In brief,
the model does yield the monetarist conclusion that
the equilibrium rate of inflation is the difference between the respective growth rates of the money stock
and full-capacity income. The final step is to recognize that when excess demand goes to zero, actual
output growth converges on its full-capacity path,
consistent with the fourth condition of monetary
Therefore, the model contains all the
equilibrium conditions required by monetarist doctrine.
of the System in the
Short Run
So much for equilibrium
which is a relatively simple and straightforward exercise. The next stage is disequilibrium
analysis. Unfortunately,
the analytics of the shortrun disequilibrium behavior of the system are somewhat more complex and involved. For one thing, the
excess demand variable does not drop out of the
short-run analysis as it does in the long-run equilibrium case ; nor is the current rate of inflation
stationary and identical to the expected rate.



As shown
in Appendix B, the expressions for the respective
short-run time paths of the inflation rate and excess
demand are :

Ap = ax-l



+ AP-~




- A2yC + (2-a)x-1
[ l-a( I-b)]x-2.

Two monetarist features are immediately apparent
even from the most casual inspection of these equations.
First is the appearance of the second time
difference, As, of the money stock variable in the
excess demand equation.
This second difference, of
course, measures the rate of change (i.e., acceleration
or deceleration) of the money stock growth rate, Its
role in the equation as an active independent variable
and determinant of the excess-demand gap is consistent with several monetarist
squares with the monetarist view of variation in the
growth rate of money as the prime initiating cause of
business cycles. It corresponds with the monetarist
argument that sharp changes in money growth can
disturb real income in the short run. In general, it is
consistent with the monetarist focus on changes in.
the growth rate rather than the level of money as a.
key indicator of recent policy shifts and future price
The second conspicuous monetarist feature is the
appearance of lagged values of excess demand in the
price-change equation. The equation states that demand leads inflation by as much as two periods, each
defined as a year-another
of the
monetarist view of the tendency for shifts in demand
to influence quantities first, prices only later.
lead-lag relationship corresponds to the monetarist
notion of long and complex lags in the monetar,y
transmission mechanism.
The lag structure of the model carries some important policy implications.
Given the long lag in
the response of prices to changes in demand-not



mention additional delays in the influence of money
on demand-inflation
will be slow to respond to
policy. This is especially true if inflationary expectations have become firmly embedded
It is a generally-accepted
in behavior patterns.
principle that an inflation rate that comes to be anticipated will resist a period of deficient demand
much longer than a rate that is not anticipated.
reduce the actual rate of inflation one must reduce
the expected rate, since the latter is a determinant of
the former. This requires a recession during which
the actual rate falls below the expected rate. inducing a gradual downward revision of the latter.
According to the adaptive expectations hypothesis.
however, expectations are based on a weighted averAnd it
age of current and past rates of inflation.
may take a long time before the decelerating current
rate begins to outweigh the lagged influence on expectations of accelerating past rates.
During this
time there exists the danger that the authorities,
observing the failure of their actions to achieve quick
results, may be tempted to abandon monetary restraint as ineffective.
Monetarists, however, would
counsel perseverance, believing that contractionary
policy, if adhered to long enough, would eventually
bring down the rate of inflation. Monetarists would
argue, moreover, that there is no other option-coritinued monetary restraint is the only way to reduce
inflation permanently.
Stability Analysis of the System The last step
in the analysis of the model is to examine the d?;namic stability of the system. Here the term stability
means the tendency of the system when in disequilibrium to converge on its long-run steady-state equilibrium.
The concept of stability is central to the
rules versus discretion debate between monetarists
Some of the latter group claim
and nonmonetarists.
that the economic system may be inherently unstable
such that once disturbed it tends either to oscillate
ceaselessly about equilibrium in cycles of regular or
increasing amplitude, or alternatively, to move steadily away from equilibrium via a divergent monotonic
path. Other nonmonetarists believe that, while the
system is stable, the adjustment process takes too
long to be left to itself. These views lead to the
advocacy of discretionary stabilization policy to counter or smooth the cycle. By contrast, the monetarist
group views the economy as an inherently stable
self-regulating mechanism capable of restoring equilibrium without the intervention
of discretionary
policy. In fact, monetarists contend that tlue to the
esistence of long. varial)le. illltl
unpredictable lays in
the monetary transmission mechanisnl, tliscretionnr~.

stabilization policy has a capricious and often destabilizing impact on the economy, amplifying rather
This argument
than dampening cyclical swings.
forms the basis of the monetarist advocacy of a rigid
policy rule fixing the growth rate of the money stock.
What about the stability of the model?
output converge on its capacity growth path and will
the excess demand gap vanish as the monetarists
predict ? To answer these questions one must analyze
the excess clemand equation

s = FG - E
+ (2-a)s-1
- [ 1-a( l-b)]>;-?.

It is assumed that the initial monetary disturbance
has ended and that, consequently, money is now
growing smoothly at a constant rate. In other words.
the rate of clzangc of the money growth rate-pmis zero. IMoreover, it is also assumed that the growth
rate of capacity output is a constant, i.e., that the
rate of change of the capacity growth rate-Azyc-is
also zero. Setting these first two terms on the righthand side of the equation at zero leaves the secondorder difference equation :

s = (2--a)~-~



Specialists in dynamic models have worked out a
set of stability conditions for this type of equation.
These conditions are listed in Appendis C.
referring to the stability criteria, it can be shown that,
given plausible values of the coefficients a and b.
Depending upon the
the system will be stable.
specific magnitudes of the coefficients. the system
may approach lon g-run equilibrium either monotonically or cyclically, but it will always converge upon
it.” Hence the model conforms to the monetarist
specification of an inherently stable system.
Monetarist View of the Inflationary Process The
foregoing section completes the analysis of the
steady-state and disequilibrium dynamical properties
of the model. These properties were shown to be
consistent with the basic postulates of monetarist
It remains to compare Laidler’s formal
model with a leading monetarist’s verbal description
of the inflationary process to see if the two agree with
regard to treatment of timing. direction of causation,
and pattern of interaction of key variables.
GOnly the excess demand equation is examined here. Exactly the
same type of analysis can be performed on the difference equation
expressinK the behavior of the inflation rate following a step increase in the monetary growth rate. Such an analysis reveals that
the rate of inflation eventually stabilizes at a level equal to thr
difference between the new monetary rrowth rate and the growth
rate of capacity output.



Milton Friedman,
perhaps America’s
foremost monetarist,
the inflationary
process as a stylized sequence of events.
a hypothetical,
when monetary
has been proceeding for some time at a constant
rate so that the
public in general
has adjusted
to that rate.
in nominal terms will then be growing
at about the
same percentage
rate as M,, prices at about 3.0 to
4.0 percentage
points less.
Let the growth
rate of
M., accelerate.
For something
like six months,

main effect will be that actual balances will exceed
which may temporarily
rates but will have little other
effect. After about six to nine months, the rate of
of nominal
GNP will accelerate,
as holders
of the excess cash seek to dispose’ of it.
The increased
. . . will ‘excite
high nominal
demands treat the increase
as special to them and so
seek to expand
For a time they can do so,
because their suppliers
too, including
the increase
in demand
as special
and temporary
and do not alter their anticipations.
This, if you
will, is the temporary
output responds more quickly than prices.
In its
course, prices do respond,
rising more rapidly
before, and interest
rates stop falling
and start to
But it takes
output starts to quicken-or
two years after money
the main effect
to have shifted
from output to prices.
this period, anticipations
are changing,
in interest
rates, but even after prices have
to absorb
the bulk of the acceleration
money, anticipations
have not fully caught up. In
the next year or so they will, which will force a
decline in the rate of growth
of output back to or
below the ‘natural
level,’ producing
the stagflation

Friedman’s description clearly implies a chain of
causation running from money to spending to output
to prices to inflationary expectations, with deviations
between actual and expected rates of inflation feeding
back into the process to determine the division of the
increase in spending between price and output
Moreover, there are substantial time lags
operating in each link of the chain or stage of the
inflationary process. Together, these feedbacks and
time lags produce growth cycles, i.e., oscillations of
output growth about the equilibrium or full-capacity
growth rate.
How does the formal model compare with Friedman’s description ? Two differences are immediately
The first relates to the initial (moneyspending) link. Friedman asserts the existence
of a
six-to-nine month lag in the response of spending to
i “Rediscovery of Money-Discussion.”
65 (May 1975). 178.







monetary stimuli.
During this interval, the total
impact of the monetary shock is absorbed by a passive
rise in undesired cash balances; none of the shock
is transmitted to spending.
By contrast, the model
implies an instantaneous
response of
spending to changes in money growth.
The difference stems from the model’s simplifying assumption
that actual and desired real cash-balances are always
identical, implyin g the absence of an adjustment lag
As a second departure
for real balances.
Friedman’s version, the model-again
for purposes
of simplicity-contains
no interest rate variables and
therefore cannot describe the impact of inflation on
interest rates. In brief, Friedman’s description implies the existence of one additional time-lag and one
additional variable absent from the modeL8
As for (1) direction of causation and (2) pattern
of interaction of variables, however, the model is
quite similar to Friedman’s description.
runs from money to output to prices to expected
inflation and back again to real income. Specifically,
in the model the sequence is as follows.
(1) Accelerated



excess deto rise above its fullSee equation

mand, thus causing real output



(2) After a lag, excess demand begins to influence
the current rate of inflation, causing it to rise above
the expected
See equation
(3) The rise in the actual inflation
rate in turn influences the expected
rate, which win feed back into
next period’s actual rate.
See equations
(3) and (2).
(4) The rate of inflation
with the given
rate of money growth
to determine
the growth
of output.
See equation
since the
rate of inflation
itself is determined
by the level of
excess demand and by expected
these two
may be regarded
as determining
the division of monetary
output and price
level growth.
(5) Finally, current output growth
as determined
(1) feeds back into equation
(2) to become
s It should be noted that Friedman’s explanation of the expeetationsformation mechanism is consistent with the s+called rotio~~al e:rpact&ionshypothesis and thus may differ from the adaptbre
expectations model employed by Laidler. According to the rational
expectations hypothesis, the inflationary expectations that individuals formulate represent the most-accurate (unbiased) forecasts
given the available
market information on the stochastic process
By contrast, the adaptive expectations
generating the inflation.
hypothesis may imply nonrational forecasting behavior. That is, it
oan be shown that under certain conditions, the adaptive expectations mechanism will produce forecasts
that are sustematicz&
wrong. For example. suppose the monetary authority followsI;
policy rule of continually accelerating the rate of inflation.
this case the backward&ok&r
adaptive expectations model WiIl
yield a predicted rate of inflation that lags consistently behind the
actual rate, i.e., inflation will be systematically underestimated.
Adherence to the adaptive expectations model despite per&tint
forecasting errors implies nonrational behavior. Rational individuals
would revise their forecasting model to produce unbiased predictions.
Once rational individuals learn of the policy rule. they will adopt. it
Under other very restrictive
as their optimal forecasting model.
conditions,. however, the adaptive expectations model will yield
This would be the case if
rational (I.e.. unbiased) predictions.
the time path of inflation is generated by random shocks of a
The notion of the inflationpermanent and transitory nature.
generating process as a random-walk with noise superimposed
would eeemto correspond closely to the monetarist view of the
capricious and unpredictable impact of discretionary monetary
policy. If so then the adaptive expectations mechanism would be
consistent w&h rational behavior, at least within the context of
monetarist models.



a determinant of next period’s inflation rate, etc. As
in the preceding
section, this iterative
process is capable of producing
like those mentioned by Professor Friedman.

To summarize,
both Friedman and Laidler agree
that, owing to the operation of lags in the monetary
transmission mechanism, the effect of money growth
on the rate of inflation is spread over substantial
periods of time. During the interim, quantities as
well as prices are affected, i.e., variations in money
growth can produce
business fluctuations.
changes in money growth have no lasting impact on
output. Ultimately, the entire effect is on the rate of
Policy Implications of the Model Since much of
the monetarist discussion of inflation tends to be
strongly policy-oriented, it is appropriate to close the
article with a brief mention of some of the policy
implications of Laidler’s model. From the point of
view of the policymaker, two features of the model
are of particular interest.
The first feature is the
time it takes for changes in the rate of money growth
to work through to the rate of inflation. The second
feature is the marked short-run impact of changes in
money growth on real output. These features combine to produce in the model dissimilar patterns of
response of output and prices to the monetary change.
These response patterns have important implications
for monetary stabilization policy.
First, owing to the slow response of inflation to a
monetary change, it necessarily takes a long time for
monetary policy to work.
monetary remedies for inflation do not exist. Moreover, since the first effect of a change in the growth
rate of money is on output and employment rather
than on prices, monetary restraint would almost
surely entail a recession or at least a marked retardation in the expansion of the economy. In sum, a
temporary but protracted period of high unemployment and sluggish growth would have to be tolerated
if monetary policy were to be successful in permanently lowering the rate of inflation.
Second, due to the difference in timing of the
responses of output and prices to a monetary change,
monetary policy may appear impotent or, even worse, counter-productive
and perverse.
Because inflationary movements tend to subside so
slowly, prices may continue to rise long after output
and employment have turned down. Thus inflation
can persist even in slack markets-a
condition variously known as inflationary recession, stagflation, or
During such periods, monetary restraint nlay be wrongly blamed for causing both the
slump and the accompanying inflation, and the tempFEDERAL RESERVE BANK

tation may be strong to abandon prematurely
policy of monetaq restraint as ineffective at best and
harmful at worst.
Third, the same asymmetrical pattern of response
-output first, prices only much later-may
create the
dangerous illusion that expansive policy in the upswing can achieve permanent gains in output and
employment at the cost of very little additional inflation. This view may have unfortunate consequences.
For monetarist reasoning teaches that stimulative
policy can peg output and employment above their
natural or equilibrium levels only by continuously
accelerating the rate of inflation. In any case, time
lags may well compound the problem of curbing inflation by leading to the undue prolongation of expansive policy, thus increasing the momentum behind
inflation when it finally occurs. In sum, given the
commitment to full employment, the tendency for
output to respond quickly and prices sluggishly to
both monetary ease and tightness is sufficient to bias
monetary policy toward inflation over the entire
policy cycle.
,4 fourth policy implication is that direct controls
cannot permanently reduce inflation within an environment of expansionary
and fiscal
policy. As previously mentioned, the elimination of


the eradication

of inflationary


pectations, since the latter are a determinant of the
former. According to the model, however, the only
way to dampen expectations is to create slack (excess
supply) in the economy, thus causing the actual rate
of inflation to fall below the expected rate, which in
turn leads to a downward revision of the latter.
Here direct controls are sometimes advocated as a
means of speedin g the fall of expectations and thus
reducing the duration and severity of the recession
necessary for the dampening of inflation.
The idea
is that controls would influence inflationary anticipations


of the



mechanism described in equation (3).
To be successful, however, the controls program must be supported by restrictive
policy that
eliminates excess demand. For, as shown in equations (2) and (3), unless excess demand is eliminated, actual inflation will lie above expected inflation
leading to an upward revision of the latter.
course controls might conceivably lower expectations
by reducing the current rate of inflation itself, but
only if people are convinced that the lowered rate
will likely continue after the controls are lifted. It is
useless to endeavor to dampen expectations via controls while simultaneously pursuing demand-expansion policies that lead inevitably
to their disappointnlent nnd subsequent resurgence.
In short, the


elimination of excess demand is the sine qzta non for
the success of a controls program.
And the excess
demand problem may be compounded by the inevitable shortages created by controls.
This article has expounded
the principal postulates of monetarist doctrine within the
context of Professor David Laidler’s three-equation
macroeconomic model. This model can account for
the phenomena of stagflation (i.e., the persistence of
inflation long after aggregate demand has slackened) f
for the entrenchment of inflationary expectations, for
the intractability or resistance of inflation to antiinflationary
monetary policy, and, finally, for the

output and employment effects of such a policy.
Since the model embodies virtually all of the monetarist predictions relating to the long-run neutrality
and short-run non-neutrality
of monetary disturbances, it can be interpreted as capturing the essence
of the monetarist view of the inflationary process.
Moreover, the very simplicity of the model renders
it a pedagogically useful introduction
to the economics of long-run steady-state equilibrium and of
short-run dynamic disequilibrium processes in which
economic variables interact and evolve over time. It
also provides a framework for stating clearly the
public-policy issues involved in the monetarist-nonmonetarist controversy.


of the Monetary

from the Cambridge

Let M be the money stock, P the price level, Y
the level of real national income, and K the desired
ratio of real cash balances to real income. This Cambridge K, the reciprocal of the income velocity
of money, is treated as a fixed constant.
elements comprise the Cambridge cash-balance equation, M/P = KY. This equation is interpreted as
the equilibrium solution of a three-equation demandSpecifically, the Cambridge formusupply system.
lation implies : (1) a relation expressing the demand
for real balances as a function of income, Md/P =
KY ; (2) an exogenously-determined
nominal money
M ; and (3) an equilibrium (marketsupply, Ms =
stating that nominal money
supply must equal nominal money demand, M, =
Md, resulting in the Cambridge cash-balance formula,
M/P = KY.



Cash-Balance Equation

To transform the Cambridge formula into the
money growth equation of the text, simply take the
logarithm of both sides of the formula.
Remembering (1) that the logarithm of a ratio is equivalent to
the logarithm of its numerator minus the logarithm
of its denominator, and (2) that the logarithm of the
product of two terms is equal to the sum of their
respective logarithms, one obtains log M - log P =
log K + log Y. Expressing the logarithms of the
variables as lower-case letters allows the preceding
relation to be expressed more simply as m - p =
k + y. Taking the first difference of this equation
yields Am - Ap = by, the money growth equation
of the text. The first difference of k is of course zero
and thus drops out of the money growth equation,
i.e., Ak = zero, since k is a constant by definition.

Derivation of the Expressions for the Disequilibrium
of the Inflation Rate



of the expression

First, lag equation

for hp.


The model in the text is



(4) into (2) to get



Ap = ax-1 + Ape-r


(3) one time period to get

Ape-r = bAp-r

Next, substitute

Time Paths

(Ap) and Excess Demand (x)

Ap’ = bAp +

Ap = ax-]

Then, rewrite



+ bAp-1 +

(2) as

Ape-I = Ap -





Then, take the first difference

Next, lag (6) one time period to obtain

Ape-z = Ap-1 -

Next, substitute



+ Ap-1.

Equation (9) is the expression for the disequilibrium
time path of the inflation rate that appears in the
text. Recognizing that Ap - Ap-r = A2p, one can
also express (9) as

A*p = ax-l



a( 1--h)x-a.


of the expression

First, start with equation


A2x = x -

Now, substitute





(8) and simplify to obtain


A*x = Ax - Ax-1 = (x-x-r)
- x-2x-r
+ x-2

Ap = ax-1 + bAp-1 +
- ax-a).

Ap = ax-r

Azm = A2x + A2y, f

Next, expand A2x to obtain

(7) into (5) to get

Finally, expand



of (1) to get

for x.

A*m =

+ x-2.

(12) and ( 10) into (11) to get
+ A2yC
+ [ax-l-

solve (13)


for x and simplify to obtain

- A2yC + (2-a)x-r

Equation (14) is the expression for the disequilibrium time path of the excess demand variable, as
stated in the text.

(1) again, i.e.,


Stability Conditions for Second-Order
Homogeneous Difference Equations

The general homogeneous second-order difference
equation x + alx-r + a2x-2 = 0 has two solutions
or roots (r) which can be found by solving the quadratic characteristic equutiovz r* + air + a2 = 0
corresponding to the difference equation. Depending
on the numerical values of the roots, the time path
of x will move toward, away from, or around equilibrium. It is not, however, necessary to solve for
the roots of the equation to determine if the system
is dynamically stable, i.e., tends to converge on equilibrium either via damped-oscillatory
or monotonic
paths. One needs only to refer to the stability conditions pertaining
to the difference equation.
stability, all of the following conditions must be met1 :

the stability conditions and the term [l-a (l-b)]
corresponds to coefficient a*. Substitution of these
terms for al and a2 in the stability conditions quickly
reveals that the first and second conditions are automatically satisfied as long as a > 0 and 0 < b < 1,
the range of values specified in the model of the text.
The third stability condition will be satisfied if a >

[J/P--b) I.
To determine whether the stable path is oscillatory
or monotonic, one must analyze the characteristic
roots of the system. The roots of the characteristic
equation r-a + air + a2 = 0 are



al +


> 0


> 0.

In the excess demand equation of the text, the
term -(2-a)
corresponds to the coefficient a1 of
1 Paul A. Samuelson. Foundations of Economic Analwk
Harvard University Press, 194’7. p, 436.



-C V a21 -



where al = -(2-a)
and a2 = [la(l-b)].
The system will exhibit oscillatory behavior if the
roots are cogqtplex, i.e., if 4a2 >a*r, or in terms of the
model, if [4 - 4a(l-b)]
> [-(2-a)]*.
latter inequality reduces to a2 < 4ab, hence oscillatory behavior is obviously possible for a > 0 and




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