Hong Kong Economic Papers, No. 12, July, I9 78
THE SQUARE-ROOT
FORMULA
IN MONETARY
THEORY
Y. C. JAO*
It is well known that in the classical monetary theory, whether of the Fisherian
"equation of exchange” version or the Cambridge real-balance version, the demand
for money was seen strictly as a demand for transactions purposes, which in turn
was exogenously determined by institutional and technological factors.1 One of the
innovations which Keynes introduced into monetary theory was his explicit recognition
that the demand for money depends not only on income, but also on the yield on
some alternative financial asset, typically a government bond. Keynes [42] dichotomized the demand for money into two parts: the first part, the transactions and
precautionary demand depends, as in the classical model, on income, while the other
part, the speculative demand, was treated as a function of the market rate of interest.
Subsequently in the fifties, some economists began to argue that even the transactions
demand for money is sensitive to the rate of interest, and that there are important
economies of scale in the holding of cash for transactions purposes. This hypothesis,
crystallized in the famous square-root formula, has dominated monetary thinking since
the mid-fifties. Indeed, in some writings it has been referred to as the Square-Root
Law. In recent years, however, this formulation of the transactions demand has been
subjected to a critical re-examination from both the theoretical and empirical points
of view. The relevant literature has grown to such an extent that a brief survey is
called for.
The purpose of this paper is to re-appraise the inventory theoretic approach in
general and the square-root formula in particular, in the light of recent theoretical
discussions and empirical tests. Section I summarizes the formulation of the squareroot hypothesis and its extensions. Section II reviews recent critiques and tests of the
hypothesis. Section III then contains some concluding remarks.
I
Edgeworth [14, 15] is generally credited to be the first economist to anticipate
the square-root formula. His interest was, however, confined to the application of
* Reader in Economics, University of Hong Kong. The author wishes to thank an anonymous
referee for helpful comments, and Mr. S. K. Ma for drawing the figures
1 There have been some attempts by contemporary monetarists to establish the view that earlier
quantity theorists, notably Fisher, Marshall and Pigou, explicitly recognized the influence of
the interest rate on the demand for money. See especially Friedman [ 27] and Selden [ 63].
This view has however been disputed by Modigliani [50] and Patinkin [57]. In this paper, we
subscribe to the "mainstream” view that the interest rate did not enter into the classical
demand function for money.
38
probability theory to banking business only. In his two papers, Edgeworth argues,
by invoking the law of error, that a bank's cash reservesagainst customers’ withdrawals
vary, ceteris paribus, not in proportion to its deposit liabilities, but to the square-root
of such liabilities. This theory of banking, though backed by a number of economists,
notably Wicksell [75] and Fisher [22], did not appear to have attracted much
attention. A somewhat different variant of the square-root formula was independently
derived in the study of optimal inventory control, but this was a separate development
in the field of management science that bore little or no relationship to the
mainstream economic theory. Among the spate of expository writings following the
publication of Keynes’ General Theory, Hansen [32] hypothesizes that even
transactions balances will become interest-elastic at high enough interest rates. Apart
from these occasional insights, however, the interest-elasticity and economizing of the
transactions demand for money were not integrated into the corpus of monetary
theory until the fifties, as a result of the seminal work by Baumol [7] and Tobin [70].
Baumol's paper in particular has been widely regarded as representing the new
inventory approach because of the relative simplicity of his model and the explicit
derivation of the square-root formula. The contents of his paper are so well known
that only a very brief summary needs to be given. Assuming that transactions are
perfectly foreseen and expenditures occur in a steady stream, Baumol's problem is
how to minimize the cost of holding cash for transactions purposes. These costs are
the interest opportunity cost of investing cash balances, transactions costs (brokerage
fees) involved in switching from cash to interest-bearing securities and vice versa. In
the simplest case, where the transactor "lives off’ his capital by steadily withdrawing
cash from his investments to meet a predetermined value of transactions, the following
formulas can be derived by the usual optimizing procedure:
c*=
M*=
J
f-
i2bT
(1)
%
(2)
where C* is the optimal evenly-spaced withdrawal of cash, M* is the optimal average
cash balances, T is the money value of transactions, b is the fixed cost per transaction,
and i is the market rate of interest. In the more general case, where the transactor
retains part of his income in cash, R, and invests the rest in securities, I, the optimal
cash retained, R*, is given by
R* = C* + T&w + k~)
(3)
where kw is the variable cost of withdrawing funds and kd is the variable cost of
investment. By definition R = T - I.
Differentiating logarithmically equation (2) with respect to T and i, it can
readily be seen that the partial income elasticity of the transactions balances is 1/2,
while the partial interest elasticity is - 1/2. These elasticities have been widely
interpreted to mean that there are important economies of scale in the transactions
demand for money, and that even this demand is fairly sensitive to changes in the
interest rate. Furthermore, as long as the brokerage fee, b, (which can be regarded as a
short-hand symbol for all kinds of transaction and conversion costs between cash and
39
other earning assets) is positive, the demand for transactions balances will also be
positive, on the reasonable assumption that there is a finite limit to the rate of interest.
Baumol's model thus provides a rational motivation for money demand in an uncertain
world where information and other transaction costs are known to exist. Because of
the economies of scale in cash holdings, the model also suggeststhat monetary policy
might be more potent in unemployment situations than earlier thought, and that its
effect in turn depends on the distribution of income, since a higher concentration
implies larger economies of scale.
Tobin's 1956 paper is more concerned with the interest-elasticity of the demand
for cash at any given volume of transactions. His problem is to find the optimal
number of transactions that maximizes the transactor's net interest earnings, or given
the number of transactions, to find the optimal timing and amounts of such
transactions. Apart from stating that the optimal number of transactions and the
optimal share of bonds vary directly with the interest rate, while the share of cash
varies inversely with the interest rate, no square-root formula is explicitly presented.
However, Tobin himself states that his equations will produce "essentially the same
results” as Baumol's square-root formula. For this reason the square-root rule is often
referred to in the literature as the Baumol-Tobin model. There are however some
differences between the two authors. Firstly, Tobin permits the number of
transactions between cash and interest-bearing securities to take on only positive
integer values, while Baumol treats it as a continuous variable. Secondly, Tobin's paper
rigorously proves what Baumol only assumes,namely that cash withdrawals should be
equally spaced in time and equal in size. Thirdly, because Baumol's problem is
formulated in terms of the minimization of the cost function, he overlooks the
question whether interest earnings are high enough to justify any investment in bonds
at all. By contrast, Tobin expressly considers four different solutions for the optimal
number of transactions, depending on the relation between the interest rate and
transactions costs. Under one of the solutions, a corner solution for example, the
optimal number is zero, because transactions costs are assumed to be greater than
interest earnings. As we shall see, these differences have a bearing on later theoretical
developments.
The square-root formula has been developed implicitly in a closed economy
context, but there is no logical reason why it should not be extended to the open
economy case. A number of writers have in fact applied the rule to the field of
international finance. Thus Heller [34] argues that the customary criterion used by
national governments and international agencies to measure the adequacy of
international reserves - the ratio of official reserves to imports - is unsatisfactory.
This is because official international reserves held by the monetary authorities of
various countries are for precautionary motives to finance temporary balance-ofpayments deficits. They are not held for transactions purpose because the monetary
authorities are not directly involved in day-today international payments. By contrast,
commercial banks which are directly involved in financing such transactions do have a
transactions demand for international means of payment. Thus in his view, the ratio of
commercial banks’ foreign exchange holdings to imports constitutes a more meaningful index of the adequacy of international transactions balances. Furthermore, citing
Baumol, he suggests that the commercial banks do not have to increase their foreign
balances in exact proportion to the volume of international trade to maintain a given
degree of adequacy. If the banks’ foreign exchange holdings relative to the square root
of imports are used as the criterion, then the coefficient of adequacy rose sharply from
12.6 in 1951 to 51.5 in 1966. The implication of this analysis is that the oft-repeated
complaint in the sixties about the shortage of international liquidity was exaggerated.*
In an essay published in the same year, Swoboda [68] makes a straightforward
application of Baumol's formula to a trader's transactions demand for a foreign
currency. Assuming that the importer has a commitment to meet a stream of foreign
currency payments over the relevant planning period, and that payments must be
made continuously and at a constant rate, it will be to his advantage to invest his funds
in domestic bonds, and convert them into foreign cash in lots of even size. The
following formula is then derived:
(4)
where F* is the optimal average holding of foreign currency, M is the value of imports
or commitments of foreign-currency payments, a is the fixed cost of converting
domestic bonds into foreign cash, and r is the interest return on domestic bonds. The
formula is the same as Baumol's with the terms appropriately redefined. Swoboda then
goes on to use the result to show the savings from single-currency (or vehicle currency)
denomination of international transactions. If there is no vehicle currency, the
importer is required to hold n different foreign cash balances, or a total of
ZF*r = &GZJZ’
i = 1,2,..*n.
(5)
If, on the other hand, there is a vehicle currency, then EMi =M, where the value
of all expenditures is expressed in terms of one single currency. The importer's foreign
currency cash balance is
F* =
a%
(6)
f 2r.
Assuming for simplicity that Mr = M2, ... , = Mi, ... , = M,, the saving in cash
balances, or the difference between (5) and (6), is positive. Thus
(ZF*l)-P
= (n-fi)Jw
= (n-fi)F:>O,forn>
1.
(7)
Equation (7) states that economies of scale can be realized by denominating
foreign currency payments in one single currency. These economies are directly related
to the number of currencies pooled and the square-root of the value of imports.
While the above two papers are concerned with scale economies in foreign
currency holdings that may be achieved by the private sector in international
transactions, Olivera [52] argues that even the precautionary reserves held by
monetary authorities are subject to the square-root rule. Such gold and foreign
exchange reserves are held to even out transitory external disequilibrium, but not to
finance transactions. He assumes that the monetary authorities choose a fixed
"coefficient of security”, i.e., the probability that the excess demand for foreign
2 This analysis is indirectly linked to the author's later monetarist stance that the rapid accumulation of international reserves was responsible for the acceleration of world-wide inflation in
the seventies. See Heller [ 35 ]
41
exchange reserves in any given time period will not exceed the amount of
precautionary reserves, and further, that the excess demand is normally distributed.
From this it is established that precautionary reserves must vary in proportion to the
standard deviation of excess demand. Further assuming that the exchange market
expands without any structural variation during the relevant time span, and making
use of the well-known proposition in statistical theory that the standard deviation of a
random sample from an infinite population is directly proportional to the square-root
of the size of the sample, Olivera concludes that "the (precautionary) demand for
reserves has an elasticity of 0.5 with respect to the volume of transactions”.3 This has
been subsequently [53] generalized by the same author into what he calls "the
square-root law of precautionary reserves” in a paper that synthesizes previous writings
on the probabilistic estimates of safety stocks of commodities and liquid reserves for
various purposes, including that of Edgeworth. One slight modification is that if
expected excess demand is not zero, then the elasticity of the total reserve with
respect to expected demand is between 0.5 and 1. The result follows from the
assumption of "constant structure” of excess demand, and that total reserve is defined
to be the sum of precautionary reserves and expected excess demand.
II
Since its explicit derivation in the fifties, the square-root rule has had a strong
influence on the micro-theory of the demand for money, and it was only during the
past decade or so that its underlying assumptions and empirical validity have been
increasingly questioned. This section is devoted to a review of theoretical critiques and
empirical findings.
We shall begin with a discussion of relatively minor criticisms or refinements
which do not affect the substance of the theorem, In a very short note Johnson [38]
observes that the Baumol-Tobin model assumes implicitly that any net interest
earnings or withdrawal costs are either added to or subtracted from the transactor's
stock of wealth, without affecting his current consumption. In other words, the model
assumes that the transactor maximizes his end-of-period wealth subject to a
consumption constraint. Johnson argues that a more acceptable approach is that the
individual maximizes his consumption expenditure, subject to an end-of-period wealth
constraint. On this alternative approach, a slightly different square-root formula is
derived. The income elasticity of the transactions demand for cash is still 1/2, but the
interest elasticity is changed from - 1/2to - [1/(2 + i)] . However, Johnson concedes
that the change isnegligible, if the interest rate on earning assetsper income-expenditure
period is trivial, as it may realistically be assumed to be, so that the modified interest
elasticity can be regarded as approximately equal to - 1/2.
This approach has been criticized by Miller [49] who argues that the
assumptions of both models are unduly restrictive. He maintains that it is more
consistent with standard consumer theory to assume that the individual maximizes
utility, which is a function of both consumption and wealth, subject only to the
constraint that his choice is attainable. In terms of graphical analysis, a consumption-
3 An alternative method of finding the optimal international reserves is provided by applying the
theory of Brownian motion. See Heller [ 33]1.
42
wealth frontier can be generated to the individual's opportunity set in the
consumption and end-of-period wealth space. The optimal consumption-wealth pair is,
as usual, determined by the tangency between the frontier and the individual's
indifference map. The optimal cash holding is then that level associated with the
utility maximization point. Algebraically, the individual seeks to maximize a utility
function of the form U(c, W) subject to W = f(c; M,, r, b) where c is consumption
expenditure over the planning period, W is the end-of-period stock of wealth, Mo is the
initial cash balances, r is the rate of interest per period, and b is the fixed transactions
cost in the bond market. More simply, the individual seeks to maximize the utility
function with respect to consumption, His money demand function is then
M*= z
J
(8)
where M* is the optimal average holding of transactions balances and c* is the optimal
consumption expenditure. As can be seen from this revised formula, optimal cash
holdings vary with the square-root of consumption expenditure rather than with the
volume of transactions. Moreover, the elasticity of transactions demand for cash with
respect to any other parameter (Mo, r, and b) is now a linear function of the elasticity
of consumption expenditure with respect to the same parameter. For instance, the
interest elasticity of the transactions demand for money will be equal to -1/2if and only
if the interest elasticity of consumption is equal to zero.
The specification of the cost function in Baumol's paper has also been critically
scrutinized lately. Morris [51] argues that Baumol's cost function for the general case
where a part of income is withheld as cash balances is incorrect, as the implied saving
in transactions costs involving the amount of cash so retained is not taken into
account. A term equal to kd(T - I) should therefore be deducted from the
opportunity costs of holding the retained income. When the cost function is thus
corrected, the resulting solution of the optimal cash withdrawn remains the same as
Baumol's, but the optimal cash initially withheld is higher. Morris’ note in turn has
given rise to two "counter-corrections” by Grace [31] and Ahmad [1] , the former
concluding that the optimal cash initially withheld should be even higher, while the
latter supporting Baumol's original formulation, The present writer's sympathy is with
Morris, though his result represents only a minor amendment.
We can now proceed to the discussion of more substantive objections and
modifications to the Baumol-Tobin model. To facilitate exposition, the critics can be
divided into two groups. The first group includes all those, while criticizing or
modifying individual aspects of the model, nevertheless retain the overall deterministic
framework in which the level of transactions is foreseen and cash disbursements occur
in a steady stream. The other group approaches the problem in a stochastic context,
with uncertainty in receipts and expenditures explicitly introduced. We shall discuss
them in turn.
Within the first group, one important issue concerns the existence of economies
of scale implied in the square-root formula. This has been sharply disputed by two well
known monetarists, Brunner and Meltzer [9], on both theoretical and empirical
grounds. They argue that, even accepting Baumol's assumptions, his model is not fully
developed. Specifically, since in a general formulation, cash holdings consist of R, the
initial amount withheld from investment, and C, the subsequent amount withdrawn
43
from investment, then the average holdings should be expressed as the weighted
average of these two components, the weights being the respective lengths of time for
which they are held, namely, (T - I)/T and I/T. The reformulated optimal weighted
average cash holdings are then
M* = dm
[l +(k,
+ k&/i]
+T/2
[(k, + kd)/i12
(9)
where bw and kw are the fixed and variable costs of withdrawing cash from bonds
respectively. From this, it can be shown that the elasticity of M with respect to T,
denoted emt, is
[l t (k, + kd)/i]
emt =2[1 +(k,
+kd)/i]
db,T/2i
db,T/2i
+ t(kw + kd)lil
2T
+ [(k, +kd)/il*T
.
(10)
The Baumol result that this elasticity equals 1/2obtains only when T approaches
zero, which is highly implausible.4 A more reasonable assumption is that bw is small
while T can increase indefinitely especially when the number of transactors is large. In
the limiting case, the elasticity in equation (10) approaches unity as T approaches
infinity. This, denoting absence of economies of scale, is of course the value implied in
the Quantity Theory of Money. Thus Brunner and Meltzer conclude that "the Baumol
and Tobin models are not alternatives to the familiar quantity theory. They both
imply it and as such perhaps furnish a firmer foundation for it”.
While Brunner and Meltzer are undoubtedly correct in asserting that the income
elasticity of the transactions demand for money will approach unity in a properly
specified model, they appear to have over-stated their case when they use this
particular result to vindicate the "familiar quantity theory” rather than simply
repudiate the square-root formula. For it can legitimately be asked whether unitary
income elasticity is the only hallmark of the Quantity Theory. Such a unitary
elasticity can indeed be unequivocably inferred from the Fisherian "equation of
exchange” or the Cambridge cash-balance equation, but there is an influential school
of thought within the quantity theorists themselves that money is a "luxury good”,
with an income elasticity much higher than one. Friedman [25] for instance shows
that the income elasticity of the demand for real balances was as high as 1.8 for the
United States in 1870-1954.5 More important, the Quantity Theory is distinguished
from the Keynesian theory principally by the fact that it regards the interest elasticity
of the demand for money as either equal to or not significantly different from zero.6
But this is where the attempt by Brunner and Meltzer to vindicate the Quantity
4 The result can be obtained from equation (10) by using L'Hospital's Rules.
5 This high value of income elasticity is of course an empirical finding that is not necessarily
implied by Friedman's monetary theory. Moreover, Friedman uses highly specialized definitions
of money (M2) and income (permanent income) to reach this result, which remains a matter of
unresolved controversy. Nevertheless, the point is that even among monetarists, there is no
unanimity that income elasticity must be unity.
6 Zero interest elasticity can readily be inferred from classical Quantity Theory of either the
Fisherian or Cambridge version. However, in the contemporary version of Friedman and others,
the interest rate does enter as an argument in the money demand function, though its elasticity
has been estimated to be very low. See Friedman [ 25 ; 26].
44
Theory gets them into trouble. As Ahmad [1] has shown, the interest elasticity that
can be derived from equation (9) has a limiting value of -2.0 when T tends to infinity
or bw tends to zero. The absolute value of this elasticity is much greater than that of
the Baumol-Tobin model, which is - 1/2. In the words of Ahmad, while Brunner and
Meltzer "have mended one flank of the quantity theory (even if only where income is
received in cash in a creditless economy) by eliminating the economies of scale, they
have, in the process, very much weakened its other flank by making the demand for
money even more sensitive to the rate of interest”.
Criticisms of the square-root formula from the Quantity Theory perspective are
therefore not entirely successful, and one must turn to other approaches that are not
concerned with the vindication of the Quantity Theory as such. One problem of the
Baumol model, to which we have briefly alluded earlier, is that the number of
transactions is not explicitly restricted to integer values, so that transaction
frequencies are assumed to react continuously to changes in such variables as interest
rates, transactions costs and so forth. By ignoring integer constraints, Baumol is able to
derive the various elasticities: the elasticities of money demand with respect to income
and transactions costs are both1/2, theelasticity with respect to the interest rate is - 1/2,
and the elasticity with respect to the receipt interval is zero. Barro [4] has shown that
if integer constraint is explicitly introduced, and assuming that the cross-sectional
distribution of family income can be approximated by the first-order gamma density,
then the elasticities predicted by the square-root formula would be close to those of
the aggregated, integer-constrained solution if and only if the average number of
transactions per payment period is above 1.6. Below that value, the square-root
formula would understate the responsiveness of aggregate money demand to income
and the receipt interval, and overstate the responsiveness to interest rates and
transactions costs. Put another way, the income elasticity would be higher (hence
economies of scale lower), the interest and transactions elasticities would be smaller,
and the receipt interval elasticity would be positive. Barro's conclusion is that,
although only a moderate value of the average number of transactions is required for
the square-root formula to perform well in the presence of integer constraints, it also
seems true from U.S. data that actual values of this number would be quite small.
However, Barro's results follow from a very specialized form of income distribution,
and it is not yet clear whether they would hold for more general forms.
An important refinement of the inventory theoretic approach is the incorporation of the value of time, apparently made independently at about the same time
by Barro and Santomero [6], Karni [40; 41], and Dutton and Gramm [12]. As the
paper by Barro and Santomero is primarily an empirical one, we shall defer discussion
of it until a later stage. In Karni's theoretical model, transactions costs consist of both
financial costs (broker's fees, transportation etc.) and forgone earnings due to the loss
of time in making conversions between cash and interest-bearing assets. The value of
time thus lost can be represented by real wages. Moreover, the value of time, proxied
by the real wage rate, is likely to change pari passu with real income. Once the
influence of the real wage rate is taken into account, the rigid income elasticity
implied by the Baumol-Tobin model is replaced by a wide range of possibilities, with
the value of one-half representing only a lower limit. Specifically, only when no time is
lost or the real wage rate is zero does the original Baumol-Tobin result hold. In all
other cases, the elasticity would be higher, or in other words, the predicted economies
45
of scale would be smaller. Karni also argues that the interest elasticity of minus
one-half neglects the possible effect of changes in the rate of interest on real income
and consumption. Again when this is taken into account, the absolute value of the
interest elasticity is normally smaller than one-half. Only when the proportion of
property income is zero does the original value of minus one-half hold, In short, the
validity of the square-root formula requires that the relevant variables - the volume of
transactions or income, the rate of interest, and the costs of cash withdrawals and
holdings are mutually independent. When the possible interdependence among the
variables is allowed for, the predicted elasticities differ considerably from that implied
in the formula. Dutton and Gramm also emphasize the fact that since the use of
money saves transactions time, the consumer's valuation of time, i.e. the real wage
rate, should be regarded as an additional determinant of the demand for money.
Because income in turn depends on the wage rate, it follows that the demand for
money is not determined by, but is simultaneously determined with the volume of
transactions or the level of income. Unlike Karni, however, Dutton and Gramm are less
concerned with the square-root formula than with the macroeconomic implications of
the income and substitution effects of a change in the wage rate on the demand for
money, In their analysis, for example, in a depression situation, consumers who are
over-supplied with leisure time due to involuntary unemployment will have an
incentive to reduce money balances, which have the generally accepted property of
reducing transactions time and providing leisure. And if wages fall in response to excess
labour supply, there will be a further reduction in the demand for money. The
combined effects of such reductions will hardly fail to have a stabilizing influence on
aggregate employment and output. Furthermore, their conclusion is much more
charitable than Karni's concerning the economies of scale implied in the square-root
formula. In their own words, "considering the net result of both the resource and
leisure time valuation effects, if a rise in income, wage rate constant, produced a
proportionate rise in the demand for money, then the total increase in the demand for
money, produced by an increase in the wage rate associated with a rise in income,
should be greater than proportionate to the increase in income due to the leisure
valuation effect. Thus, an income elasticity (which includes both the resource and
leisure valuation effects) of less than unity is not necessary for the confirmation of
William Baumol's and James Tobin's models which predict economies of scale in the
holding of transactions balances. If the leisure valuation effect is a significant
determinant of the demand for money, an income elasticity of one would indicate
economies of scale in transactions balances".7
The Baumol-Tobin analysis has also been criticized for treating bonds as the only
alternative to money in the wealth-holder's portfolio - a feature no doubt strongly
influenced by Keynes’ approach to monetary theory. Their model focuses attention on
the costs of exchange between money and bonds only, but ignores the costs of
exchange between money and commodities and hence neglects the holding of
commodity inventory. Moreover, by implicitly following the Keynesian assumption of
fixed prices, the model cannot deal with the effects of inflation on money and
7 In the quotation, "resource effect” is the income effect, while "leisure time valuation effect” is
the substitution effect of a change in wage rate on the demand for money. Both effects are
positive in the model.
46
commodity holdings. Feige and Parkin [18] break new ground by explicitly
incorporating costly transactions between money and commodities into the inventory
theoretic approach. However, they are more interested in the question of the payment
of interest on cash balances (demand deposits) as a means of inducing society to hold
the optimum quantity of money. Moreover, by abstracting from price changes they
fail to analyze the impact of inflation. It remains for Santomero [59] to work out a
more complete model that not only adds commodities as an alternative asset and
analyzes the effects of inflation, but also disaggregates money holdings into demand
deposits and currency in order to take into account their distinct characteristics. In
this model, a household has the option of holding working balances in four forms:
currency, demand deposits, savings deposits and commodities. An explicit rate of
return on demand deposits is allowed to capture the provision of bank services or the
remission of service charges in accordance with deposit balances.8 The rate of return
on commodities is treated as the difference between the expected inflation rate and
the rate of physical depreciation and storage costs. Given the respective yields on these
assets and the transactions costs between them, the household tries to maximize the
net return from its working balances over a given time period. The optimal average
money holding, M*, (the sum of currency and demand deposits) derived from the
solution of the model is:
M* =
jm
-G;F-G,*
(11)
where Y is consumption expenditure, a, is transactions costs of using saving deposits,
adc is transactions costs between demand deposits and currency, rs is the rate of return
on savings deposits, rd is the rate of return on demand deposits (rs > rd), Gd* is goods
purchased using demand deposits, and Gc is goods purchased using currency. These two
optimal quantities of goods purchased can in turn be expressed as:
G; =
G;
a&Y(l
-h)
J
2(‘d - ‘g>
J
WC -rg)
a,, Yh
=
(12)
(13)
where adg is transactions costs between demand deposits and goods, h is the fraction
of expenditures made in currency, rg is the rate of return on commodities, rc is the
rate of return on currency (assumed to be zero or negative in the model), and acg is
transactions costs between currency and commodities.
The major differences between this model and the Baumol-Tobin model are that,
by incorporating goods inventory as an alternative to money, an extra margin of
substitution between money and commodities is added and the impact of expected
inflation rate can be ascertained. Specifically, whereas in the square-root formula, a
rise in the bond rate i (see equation 2) will result in a fall in the demand for money, in
Santomero's model, the relevant variable is (rs - rd) so that a rise in rs will reduce the
8 A fuller explanation of allowing a nominal rate of interest on demand deposits, generally zero
in most countries, and a derivation of a time-series of such a rate, are given in a joint paper by
the same author and Barro. See Barro and Santomero [6]
demand for money if only rd is held constant. 9 A rise in the expected rate of inflation,
represented by rg, will, ceteris paribus, reduce money holding without the need of a
prior rise in the yield on financial assets.On the other hand, although money demand
is inversely related to the differential (rs - rd), an equal rise in both rs and rd, with rc
and rg constant, will have a positive effect on money demand. Another new dimension
is that the demand for money is inversely related to the commodity transactions costs.
The higher such costs, the less frequent will "shopping trips” be made, and hence
larger commodity inventory, or lower money balances, will be held. Only under the
restrictive condition that the costs of commodity transactions using either form of the
medium of exchange are zero, i.e., adg = acg = 0, will money demand approximate the
level suggested by the Baumol-Tobin analysis.
An alternative way of modifying the inventory approach to take account of the
inflationary factor is to treat the payments period as a function of the rate of
inflation. Barro [3] develops a model in which average cash holdings are reduced in
response to a higher inflation rate via the reduction of the time period between
transactions. The optimal payments period is derived as
T/n = dl(a/P)l
[Y/Wp
+ r*)l
(14)
where T is the given time horizon, n is the number of transactions, rp is the rate of
inflation, Y is nominal income, P is the price level, r* is the rate of discount, and a is
the costs of transactions. The corresponding optimal holding of real balances would
then be:
M*/P
= dz
(15)
Clearly a high and accelerating rate of inflation rp would drastically reduce both
the payments period T/n and real balances M*/P’. An additional mechanism by which
cash holdings could be reduced involves the substitution of some alternative assetsas
medium of exchange (foreign currencies, gold bars, physical goods etc.)” The
decision whether to employ money or its substitutes depends on the comparison of
the inflationary costs of using money with the benefits of money as a transactions
medium in terms of liquidity, convenience etc. However, once such substitutes are
used, average cash holdings can be reduced even if transactions periods (velocity)
remain constant.’ ’ The significance of this type of models is the demonstration that
the income elasticity of the demand for money cannot be derived in a straightforward
9 In this model, the yield on financial assetsis not represented by the usual bond rate, but by the
yield on savings deposits, rs.
10 This situation normally occurs only in extreme conditions of accelerating inflation. In China
during the late forties, for example, foreign currencies (mainly US and Hong Kong dollars)
and gold bars were regularly used as both means of payment and store of value in major cities
like Shanghai and Canton. Barro's own paper is devoted to a study of hyper-inflation in Austria,
Germany, Hungary and Poland during the early twenties.
11 A related issue, which cannot be treated here, is the role of real balances as a factor. input.
See Fischer [ 21]. Given a positive role of real balances in the production function, there is a
finite limit to the reduction of real balances. This explains why, even under extreme conditions
of inflation, the function of money as a transactions medium does not entirely disappear.
48
manner independently of the rate of inflation.
Although the alternative definitions of money, M1, M2 , ... and so forth have
now been widely accepted in the statistical computation of money supply by
monetary authorities as well as quantitative studies by economists, their separate
treatment in the theory of demand for money has barely been developed. Santomero's
study cited earlier distinguishes between two components of M1, namely currency and
demand deposits, but not between M1 and M2. An important innovation in this
respect has been made by Claassen [11], who explicitly works out optimal balances
for both types of money. In his model, the economic unit can choose to hold its
average real income y/2 among three assets: m1, the real balances of money defined in
the narrow sense (demand deposits plus currency outside banks), w, liquid
interest-bearing claims, including time deposits, and g, physical consumer goods. Note
however that in his context, m2, real balances of money defined in the wider sense,is
equal to the sum of ml and w. Thus, it covers a much wider ground than M2 as
conventionally understood (M1 plus savings and time deposits at commercial
banks).12 Because of the economies of scale in inventory holding, m1 and g rise less
than proportionately with real income, while w rises more than proportionately with
it. This can be shown graphically in Fig. 1, where the hatched area denotes investment
in liquid claims w, which is equal to the difference between y/2 and (m1 + g).
The optimal balances for both types of money, expressed in nominal terms, are
given by:
MT = Pm1 = py+[(&)+
21
- (
2(sdP*))tl
M*2=Pm2=py/2-P(
ay ’
2(s - p*) )
(16)
(17)
where y is real income, P is general price level, a is transactions costs of commodities, b
is transactions costs of interest-bearing securities, i is interest on liquid assets, s is
storage costs of commodities, and p* is the expected rate of inflation. In comparison
with the Baumol-Tobin model, the optimal balance for M1 is smaller by the amount of
Pg because of the addition of physical commodities. The income elasticity of M1
remains equal to 1/2,but its interest elasticity is variable and no longer equal to -1/2, M2
has an income elasticity of greater than one and an interest elasticity equal to zero.13
Furthermore, the holding of both types of money is shown to vary inversely with the
expected rate of inflation, an effect ignored in the traditional inventory model. In
particular, the decrease in M1 in an inflationary situation results from two substitution
effects: one coming from w due to higher market rate of interest, and another coming
from g due to higher level of prices.
A forceful critique of the square-root formula, based mainly on the institutional
12 Thus the Federal Reserve System now has five definitions of money stock, but even the widest
one, M5, covers only M2 plus deposits at non-bank financial intermediaries and negotiable
Certificates of Deposits of large denominations. Claassen's definition of M2 in this paper
therefore approximates the "Radcliffe view”. See Meyer [461, Claassen [10], and Kwon [43].
13 This result follows from Claassen's rather special definition of M2 so that any income that is
not spent on consumer goods is necessarily invested in M2.9
49
characteristics of the modern banking and monetary system, has been made by
Sprenkle [64; 66; 67]. The formula is generally regarded as more relevant for large
economic units, but even this is disputed by Sprenkle. First, he points out that the
Baumol-Tobin model rests on the assumption that there is no pecuniary return on
money held in the form of demand deposits. This is by no means universally true; as in
many countries, banks still pay explicit interest on such deposits. Even in the United
States, where such a practice is legally prohibited, there are ample opportunities to pay
implicitly for such deposits by offering depositors a wide range of services, by providing
loans at favourable terms, and by remitting service charges, etc. This is a point
made also by a number of other economists. If a positive interest rate is allowed on
demand deposits, and if this rate is an increasing function of the total volume of
deposits (i.e., banks compete heavily for the largest deposit accounts) as well as the
yield on short-term securities, then the income elasticity and the absolute value of the
interest elasticity of optimal cash balances will both be substantially greater than that
implied in the square-root formula. Secondly, he shows that differences in decentralization of cash management and timing of receipts/expenditures can lead to enormous
disparity in optimal cash holdings. The Baumol-Tobin model implicitly assumes that
the economic unit has complete centralization of its cash management: all receipts
and payments flow in and out of the head office only without passing through any
other branch or office. Such an assumption is obviously unrealistic with respect to the
modern corporation. If a large economic unit has J branches or separate accounts, and
the jth branch or account receives xj per cent of total receipts T, then it can be shown
that optimal cash holdings under complete decentralization will be greater than that
under complete centralization by a factor equal to J,
the square root of the number
of branches or accounts. Clearly if J is large, the difference will be considerable: a firm
with 25 branches and complete decentralization will have to keep 5 times as much
cash holdings as that of another firm identical in all respects except having complete
centralization in cash management. Both are of course extreme cases,but as a general
statement it is still true that optimal cash balances vary directly with the degree of
decentralization and inversely with the degree of centralization. Thirdly, Sprenkle also
demonstrates that the minimal size of yearly transactions T for profitable switchings
between cash and securities to take place depends on the frequency of income receipts
during the year. The larger the frequency, the larger T will have to be. Thus taking the
parameters of a = $20, r = 0.05, where a is the fixed costs of switching between cash
and securities, and r is the annual interest rate on such securities, it can be shown that
for yearly receipts, T need only be equal to or greater than $3,200 for profitable
switchings to take place; but for weekly receipts, T will have to be equal to or greater
than $8.65 million. Even with a large economic unit, its cash balances for transactions
may well be held in relatively small accounts for which switchings hardly occur. The
implication of all these considerations is that by ignoring such institutional
characteristics, the square-root formula grossly underestimates the income elasticity of
the transactions demand for money and somewhat overestimates its interest elasticity.
The criticisms and modifications of the square-root rule so far reviewed still
retain the deterministic framework of cash flows assumed in the model, i.e., cash
expenditures and balances follow a smooth “saw-tooth” pattern as depicted in Fig. 2.
The other group of critiques denies this deterministic framework and replaces it with a
stochastic one. Cash flows in the approach are assumed to behave as if they were
generated by a trendless random walk.
CASH
t,
CASH
t
t2
51
The first model developed in this direction appears to be that of Orr and Mellon
[55] which specifically explores the effects of uncertainty in the cash flows of banks
on the expansion of bank credit. Taking a cue from the "banking game” proposed by
Edgeworth [15], they examine the problem of how far a profit-maximizing bank
should expand its credit, given the random nature of its cash flows and the reserve
requirement it must meet in order to avoid penalty loss. Their conclusion is that
optimal credit creation under uncertainty will generally be less than that postulated in
traditional static multiplier analysis where such an effect is not allowed for, or in other
words, that banks will tend to hold larger excess reserves. However, their model does
not explicitly criticize the square-root formula.
Much more explicit in its criticism and also much better known in the literature
is the stochastic model developed by Miller and Orr [47; 48]. In this model, the
random and driftless behaviour of the cash flows are assumed to conform to the
Bernoulli process, with the variance of daily changes in cash balances equal to m2 t,
where m is the change in cash balances (positive or negative), and t is the number of
operating cash transactions per day. The economic unit allows its cash balances to
fluctuate freely until they reach either the lower bound, zero, or an upper bound, h, at
which point a portfolio switch will be undertaken to restore the balances to a level of
z. This is a (h, z) cash management policy which requires that when the upper bound is
hit, there will be a lump-sum transfer from cash of (h - z) dollars, while when the
lower limit is reached, a transfer to cash of z dollars. This is shown in Fig. 3. The
economic unit's objective is then to minimize its expected costs (the sum of transfer
costs and opportunity costs) with respect to the control variables: the upper bound on
cash holdings h, and the intermediate return point, z. The solution of the model gives
the optimal average cash balances as:
where b is the fixed costs of transfer and i is the yield on securities.
As in the Baumol-Tobin model, the demand for transactions balances is an
increasing function of the costs of transferring funds to and from the earning
portfolio, and a decreasing function of the interest rate or opportunity costs of
cash.14 The novel feature is the fact that optimal cash balances vary positively, not
with the total volume of transactions, but with its variance, m2 t. Miller and Orr justify
this by saying "that there is a relation between total sales and the variance of changes
in the cash balance is clear enough since total sales are approximately the positive
changes in the cash balance summed over a time interval”. But they also point out that
"the relation is a loose one, and no precise value can be established for the elasticity of
the demand for cash with respect to sales that is implied by our model”. Specifically,
if the increase in sales or income is due to the increase in the size of each transaction
but the frequency of transactions remains constant, then the income elasticity will be
in the order of 2/3.On the other hand, if transactions frequency increases with sales or
14 M* is defmed by Miller and Orr as the "discretionary” transactions balances (net of the minimum
balances agreed with the bank) to capture an important institutional characteristic of the
American banking system. It is of course applicable also to other banking regimes where
there are various requirements for compensating balances.
52
income, while the size of each transaction remains constant, then the predicted value
of the income elasticity will be only 1/3.In the words of the authors, "the existence of
such a wide range for the sales elasticity in our (h, z) model is in sharp contrast to the
prediction of the Baumol model where the elasticity of average cash holdings with
respect to sales (assuming constant prices) is always and precisely 1/2". The
21 and
attractiveness of their model therefore lies in the wider range of elasticity val
es,
as such offers a more plausible and testable explanation for the observed systematic
interindustry differences in cash holdings.15 At the same time, the predicted interest
elasticity is - 1/3,implying that the demand for transactions balances is much less
sensitive to changes in interest rates than assumed by the square-root formula if
uncertainty is incorporated into the model.
In their 1966 paper, Miller and Orr still retain the two-asset framework, the
only alternative asset to cash being a separately managed portfolio of short-term liquid
assets. In their 1968 paper, a third asset, long-term securities with less liquidity but
higher return, is introduced. However, they demonstrate that the money demand
predictions of their earlier paper are not significantly affected by this modification.16
They do admit however that negotiated compensating balances may distort estimates
of the relevant income and interest elasticities, and the practice of maintaining
numerous cash accounts may similarly obscure the presence of significant scale
economies. The first point has been taken up by Frost [29], who develops a model
that incorporates both the Miller-Orr stochastic structure as well as an explicit demand
for banking services in the form of minimum compensating balances. In contrast to
Miller and Orr, who regard minimum compensating balances in lieu of service charges
as completely exogenous to the model, Frost shows that such compensating balances
and transactions balances should not be treated independently if the bank takes into
account the firm's average deposit level or the range over which the firm allows its
deposits to vary in determining the supply of bank services. Only when the bank
supplies services in proportion to the minimum cash balances maintained by the firm
do the original Miller-Orr results remain unscathed. If however the supply of bank
services is a function of both the minimum balances and the average balances, then
complications arise and the original results may not hold.
Tsiang [72] argues that both the inventory theoretic approach to the
transactions demand for money of Baumol and Tobin, and the portfolio theoretic
approach to the asset demand for money of Tobin [71] and Hicks [36] neglect the
precautionary demand for money. Moreover, in a society with highly developed
money markets, there are always some very liquid assets that "dominate” cash in the
sense that they are virtually riskless and yet have a positive yield.” Thus, the asset or
15 This variable range in elasticity values should be stressed.Thus, the interpretation by Barro and
Fischer [ 21] that "the differences between the Miller and Orr elasticities and the Baumol
elasticitiesinvolve a substitution of l/3 for 1/2" is not strictly correct.
16 The 1968 paper by Miller and Orr also contains an elaborate defence of the symmetric Bernoulli
process that underlies their 1966 paper. A formal proof of the (h, z) decision rule is given by
Eppen and Fama [17] who, however, treat transactions costs as proportional rather than as a
fixed lump sum.
17 This point is also made by Hicks [ 37 ] and Modigliani [ 50].
53
speculative demand is likely to be less important than precautionary demand in
explaining cash holdings. Precautionary cash balances are defined as those held in
order to meet contingencies which are not expected to happen in the normal course of
events, but which, if they should occur, would call for cash payment. In Tsiang's
model, the expenditure rate is assumed to be stochastic, while the income receipt
pattern is still lumpy and deterministic. There are two assets other than cash:
short-term bills and long-term bonds. Any delay in the receipt of funds from forced
liquidation of such assets to meet contingencies will involve penalty costs. The
expected yield of holding precautionary balances, in addition to transactions balances,
is the avoidance or reduction in such costs. Tsiang's conclusions are that the optimal
number of cash replenishments during the income receipt period is fairly insensitive to
moderate changes in planned expenditure. Given the short-term interest rate and the
number of cash replenishments, the optimum precautionary cash balances would then
vary roughly with the square-root of the expected rate of expenditure.‘* However, if
the sum of the initial holdings of short-term assets and precautionary cash balances is
not a sufficiently large multiple of the square-root of the minimum required cash
replenishments, then the economies of scale in holding precautionary balances would
be less. The effect of a change in interest rate on the demand for precautionary cash
balances is even less straightforward. For an increase in interest rate would, on the one
hand, increase the opportunity cost of holding precautionary balances, but on the
other hand also raise the marginal yield on such balances by increasing the penalty for
running out of cash. In normal situations, however, a rise in interest rate would
probably reduce the demand for precautionary reserves. The major difference between
Tsiang's model and the Baumol-Tobin model is therefore that in the former, where the
rate of expenditure flow is uncertain, no precise predicted values of income and
interest elasticities can be inferred. Note that although Tsiang distinguishes conceptually between transactions and precautionary demand for cash, he recognizes that
they are not independent of each other: as a matter of fact, his model provides for a
joint determination of the optimal quantities of the two.
Incidentally, this joint determination of transactions and precautionary cash
balances also raises a more fundamental issue in monetary theory. The square-root
formula rests on the implicit assumption that transactions balances, or at least the sum
of transactions and precautionary balances, can be neatly separated from the asset or
speculative balances. Many economists, of both monetarist and Keynesian persuasion,
have questioned whether this dichotomy is valid. Thus Friedman, in his famous 1956
restatement of the Quantity Theory, explicitly rejects this dichotomy. More recently,
Hicks [37], Modigliani [SO], Borch [8] , Feldstein [20], Tsiang [73] and Thorn [69]
have expressed dissatisfaction with Tobin's [71] risk-aversion or mean-variance
analysis of the demand for money. The gist of the critics’ argument is that the
mean-variance indifference curve does not necessarily have the convexity property
ascribed to it, and that transactions costs and unforeseen requirements for cash
expenditures provide a surer foundation on which to rationalize the holding of cash
balances than risk-aversion. This controversy is however beyond the scope of this
18 A similar conclusion is reached in Dvoretzky [ 13] where the increased volume of transactions
is generated by an increase, not in the average value of each contract, but in their number. The
precautionary reserves then vary positively with the square-root of the number of transactions.
54
paper. For our purpose, we shall continue to assume that it is conceptually convenient
to distinguish between money held for transactions and other purposes, in order to
meet the square-root theorem on its own ground.
We now turn to a brief review of the results of empirical testings of the
square-root hypothesis. Since the Baumol-Tobin analysis is generally regarded as being
more relevant to large economic units, most direct tests relate to the observed cash
behaviour of business firms. Attention is focused on the estimated values of income
and interest elasticities, particularly the former.
One of the best-known tests is Meltzer's [45] study of cash balances of 126
business firms in 14 American industries for the period 1938-57. Meltzer uses sales as a
proxy for income and experiments with several alternative estimating equations. His
main finding is that the income elasticity of the demand for cash by firms is
approximately unity, thus implying neither economies nor diseconomies of scale. For
particular industries, he finds that in each of the nine years and in twelve of the
fourteen industries, the mean value of the income elasticity is greater than unity. The
26 casesin which this elasticity is smaller than unity are largely concentrated in only a
couple of industries. He thus concludes that "these data give little evidence of
economies of scale and little support to the Baumol-Tobin model as an explanation of
the demand for money by firms”. However, apart from asserting the importance of
interest rate as a determinant of the firms’ demand for money, no precise estimate of
the interest elasticity is given. Whalen [74] makes a cross-section study of 109
non-financial corporations in 8 American industries during the fiscal year 1958-59. In
the first test, the firms’ cash holdings are adjusted for investment balances, so that
they represent money held for transactions and precautionary purposes only. Using a
linear equation that treats the ratio of cash balances to total monetary assets as the
dependent variable, and the ratios of sales as well as the square-root of sales
respectively to total monetary assets as the independent variables, the author
concludes that no systematic economies of scale in cash management can be found,
though he is cautious enough to add a proviso that "required for this cross-section
approach is the assumption that firms in the same industry are homogeneous in all
respects except for the size of their business operations”. In the second test, he uses a
modified form of Meltzer's log-linear equation, but without adjusting for investment
balances. The results are similar to Meltzer's, but the omission of interest rates in the
money demand function, and the failure to adjust for investment balances are
acknowledged to be possible sources of bias. Frazer [23] also makes a cross-section
analysis of manufacturing corporations during 1956-61, but his paper is not strictly a
test of the square-root hypothesis, since he is concerned with precautionary cash
balances only. His main finding is that the larger-sized firms have smaller precautionary
cash balances, larger non-cash liquid assets, and less bank borrowings, all relative to
total assets, than smaller-sized firms, thus indicating some economies of scale with
respect to money held for precautionary purposes, though no elasticities are given.
Sprenkle [66] studies the actual cash holdings at the end of 1966 of 475 of the largest
500 American industrial firms as listed by the Fortune magazine. He finds that, for the
largest 50 firms, on the average only 1% of their cash balances is explained by the
square-root formula, and the greatest amount explained is 3.7%. For all the firms, the
median amount explained is only 2.5%, and for only 2 firms is more than 20%
explained. Thus, on the average, 97.5% of the actual cash holdings of these large firms
55
is unexplained by the model. Nor can allowance for decentralization of cash
management and the timing of receipts fully explain such observed balances. In
Sprenkle's view, the only convincing rationale for these excess balances is that they are
held to pay for bank services in the U.S. institutional setting. Aronson [2] finds that
cash balances held by American State and local governments in 1962 are grossly in
excess of that implied by the square-root formula, though he attributes this more to
mismanagement or decentralization of cash management than inadequacy of the
model.
Empirical tests of the hypothesis with respect to household demand for cash
have been relatively few. Barro and Santomero [6] use a modified form of the
square-root formula for their test, where the relevant interest variable is not the usual
yield on bonds, but the differential between the interest rate on deposits of savings
and loan associations and the interest rate on commercial bank demand deposits. Barro
and Santomero argue that remission of service charges and provision of various services
as a function of average balances should be regarded as a form of positive imputed
return on demand deposits. A novel feature of their study is the derivation of a timeseries of such yield for 1950-68 through a private survey of American commercial
banks. They find that the income elasticity is close to unity, though they do not
regard this as necessarily inconsistent with the square-root hypothesis, since the time
value of transactions costs is typically neglected. To quote the authors, "as transaction
volume increases, economies-of-scale are realized only to the extent that transactions
costs rise less than transactions volume. Since transaction costs depend largely on value
of time, and since value of time may increase even faster than transactions volume,
diseconomies-of-scale (money being a "luxury”)
is quite compatible with the
inventory approach”. As regards interest elasticity, if the differential between savings
deposit rate and demand deposit rate is taken as the relevant variable, the coefficient is
-0.549, strikingly close to the predicted value of the Baumol-Tobin model. On the
other hand, if savings deposit rate alone is used as the interest rate measure, the
coefficient becomes -1.092. The conclusion is therefore that a reconstructed inventory
model is more robust to empirical testing than the simple Baumol-Tobin version. The
Tobin formulation of the model also predicts that, if the average receipt interval is one
month, the maximum average holding of cash would be two weeks’ income. Barro and
Fischer [5] however report that "this figure substantially under-estimates household
money holdings in the United States, which are about 1 1/2months’ income in 1973”.
Aggregate money demand functions have also been used to test directly or
indirectly the hypothesis. Khan [39] employs the average real wage rate as a surrogate
for transactions costs in testing the model for the period 1900-65 in the United States.
He concludes that "although the coefficients of current income and the rate of interest
were found to differ significantly from the 0.5 value predicted by the theory, the
model was found to yield marginally better results than the Friedman-Laidler and
Keynesian models”. Liebermann [44] in his very recent study departs from the usual
procedure by treating debits to demand deposits rather than income as the measure of
transactions, and incorporating time as a crude estimate of the mean rate of
technological change, for the period 1947-73 in the United States. The main finding is
that there are very substantial economies of scale, as the debits elasticities tend to fall
in the 0.33 to 0.37 range. However, the interest elasticities tend to have a very small
absolute value, varying between -0.051 and -0.098. A potentially important metho-
56
dological issue has been raised in the author's observation that "previously published
transactions demand for money studies may have been misspecified”.
Most of the empirical studies reviewed above are apparently unfavourable to the
square-root hypothesis. Even when the results appear to be more favourable, they are
derived from modified or reconstructed models. At the same time, one must
acknowledge that the empirical evidence is by no means final and conclusive. One
important conceptual and methodological issue is that the Baumol-Tobin model
explicitly confines itself to the transactions demand for money. Observed cash
balances, whether held by firms, households or other economic agents may well include
balances held for other purposes - precautionary and speculative purposes for
example. Only if the observed cash balances are held entirely for transactions purposes
will the empirical findings be strictly relevant for evaluating the hypothesis. However,
in the studies cited above, only Whalen has adjusted the observed cash holdings for
investment balances (i.e., balances held for speculative purposes) in one of his tests.
The failure or inability to segregate transactions balances (or at least the sum of
transactions and precautionary balances) would seem to cast doubt on the validity of
much of the evidence. Nevertheless, critics of the Baumol-Tobin hypothesis have
defended their estimating procedure on the following grounds. Firstly, they argue that
in advanced contemporary economies where money markets are highly developed and
virtually riskless liquid assets (with respect to nominal capital value) with positive
yields are readily available, speculative balances can be regarded as trivial, if not zero.
Thus Modigliani [50] writes, "the proposition that people will flee from long-term
bonds when the price of bonds is deemed to be untenably high, seems valid enough,
but the obvious abode for funds accruing from moving out of long-term bonds should
be short-term ones, not cash”. Secondly, precautionary balances can also be treated as
negligible, particularly with respect to large economic units. Thus Sprenkle [66]
asserts that "the only uncertainty involved in a precautionary demand for money is
the uncertainty of payments and receipts between the present and next planned
purchase or sale of short-term assets. For a large economic unit this period can be as
little as one day. The amount of uncertainty over such a short period is apt to be small
indeed, and thus the precautionary demand for money should also be very low”. In
another paper [65], he argues that the precautionary demand might conceivably be
negative under certain circumstances. Consequently, observed cash balances are still
regarded as a good proxy for transactions balances.
The other reservation about empirical studies concerns the role of transactions
costs. It is being increasingly realized that these costs, which are nothing but the costs
of factor inputs, are not independent of the volume of transactions or income. As real
income rises, real wage rate representing these costs tends to rise puri passu. Hence
there will be an incentive to hold more real balances to save costs. As emphasized by
Barro and Santomero [6] and Dutton and Gramm [12] , when the value of time is
recognized, a measured income elasticity of money demand in excess of unity is not
necessarily inconsistent with the inventory theory. At the same time, as Enzler,
Johnson and Paulus [16] have pointed out, recent innovations in banking and finance,
such as overdraft credit lines, money-market mutual funds, third-party transfers from
savings accounts, bank-managed accounts etc. have the effect of reducing costs of
switching between cash and interest-bearing instruments. These developments work in
opposite directions on the demand for money, and the net effect is by no means clear.
57
Apart from the direct tests of the Baumol-Tobin hypothesis, there is of course a
vast empirical literature on the aggregate demand function for money. These studies
are however not concerned with the square-root formula as such, but with wider
macroeconomic issues such as the stability of the money demand function, the
efficacy of monetary policy, the substitutability of money and near-monies and so
forth. Again, no attempt has been made in such studies to isolate the transactions
balances as a component of total money holdings. Nevertheless, one useful by-product
of these investigations is the information on the range of income and interest
elasticities, which could serve as an approximate frame of reference for our more
specialized purpose. A convenient survey has recently been provided by Feige and
Pearce [19]. In summarizing the time-series studies, they report that "the mean
income elasticity is 0.57, consistent with the inventory theoretic prediction of
economies of scale in money holdings”. This statement seems to us to be unwarranted.
For a closer look at the data reveals that the standard deviation of these estimates is
quite high, with the coefficients ranging from a low value of 0.117 to a high value of
1.219. It is highly doubtful whether a crude arithmetic average can be taken to be a
valid confirmation of the square-root hypothesis. Moreover, when we turn to temporal
cross-section studies, we find that the mean becomes 0.887, ranging from a low value
of 0.62 to a high value of 1.12. Although some economies of scale are indicated, they
hardly provide a sufficient basis for vindicating the hypothesis. As to the interest
elasticities, results vary greatly, depending on which rate of return is chosen as the
relevant variable. The best performance is shown by the Treasury bill rate, with a mean
of -0.57, which is very close to the value predicted by the square-root formula.
Unfortunately, the standard deviation is again very high, with the estimates ranging
from a low value of -0.15 to a high value of -0.006. Furthermore, apart from the fact
that money demand in these studies relate to total demand rather than transactions
demand only, comparatively little work has been done to disaggregate money demand
by ownership category. In those few studies that do, it is found that elasticities may
differ significantly across sectors. In a recent important study by Goldfeld [30], for
example, while the income elasticity is estimated to be around 0.7 in the aggregate
money demand function, that of the household sector is found to be well above unity.
Goldfeld however finds himself unable to make sense of the money demand behaviour
of the business sector and the state and local government sector. As an aside, he
remarks that “this evidence suggests that a simple transactions model (especially if
couched in real terms) will have a hard time explaining money holding by business and
by state and local governments”.
Finally, all empirical tests of the square-root hypothesis, whether direct or
indirect, are based on domestic (predominantly American) money demand data. The
extension of the square-root formula to international transactions and/or precautionary reserves has so far not been formally tested, although Heller [34] has made
some ad hoc observations on the trend of official reserves and commercial banks’
foreign exchange holdings relative to imports during the period 1951-66. On purely a
priori grounds however, there are reasons to believe that the deterministic inventory
model is even less able to assessthe adequacy of international reserves, since they are
held not only to accommodate systematic and random fluctuations in current account
receipts and payments as well as capital flows, but also to cope with unpredictable and
often irrational speculative attacks, a need that is not obviated by the present
makeshift system of "managed floating”.
58
III
The inventory theoretic approach to the transactions demand for money marks a
new departure in the evolution of monetary theory by treating such demand, not as a
constant exogenously determined by institutional and technological factors (system
and customs of receipts and payments, degree of market specialization and industrial
concentration, efficiency in communications and transportation and so forth) but as a
result of voluntary choice by cost-minimizing or utility-maximizing wealth-holders. It
also represents a seminal attempt to demonstrate that positive transactions costs are a
necessary and sufficient condition for the holding of money balances.19 That it
constitutes an important innovation and contribution to the micro-foundations of
monetary theory can hardly be gainsaid.
This inventory approach, as pioneered by Baumol and Tobin, gives rise to the
famous square-root formula which purports to show that there are important
economies of scale and interest elasticity in the transactions demand for cash. The
simple though somewhat mechanical way in which the formula is presented
undoubtedly has considerable intuitive and heuristic appeal which explains its
popularity and influence for well over two decades. However, it is symptomatic of the
pace of developments in monetary economics that this simplistic model has in recent
years been increasingly challenged on both theoretical and empirical grounds. From
the theoretical point of view, it is clear from our survey that alternative inventory
models, based on more plausible assumptions, can yield quite different results. Perhaps
the greatest weakness of the square-root formula lies in the rigid predicted values of
income and interest elasticities, which to a large extent explain the poor performance
of the formula in empirical studies. Granted that these studies are by no means
conclusive due to the conceptual and methodological difficulties already described, it
nevertheless remains true that no study has come up with any firm support of the
model in its original form .20
In spite of these strictures, we canot concur with Sprenkle's [66] blanket
condemnation of all transactions models as being "useless”. A distinction needs to be
drawn between the inventory approach to monetary theory in general and the
particular forms which it might take, e.g. the square-root formula. In our judgment,
the inventory approach is still a fruitful and suggestive way of tackling the issue of
money demand with potentially rich rewards. The square-root formula, on the other
hand, is at best a tentative and unconfirmed hypothesis. It certainly does not deserve
the title of "law” given to it by some of its more enthusiastic supporters. We therefore
agree with Orr's recently expressed view and hope [54] that "the combination of a
more appropriately specified model and a more directly relevant data base will yield
sharply improved, albeit still improvable, predictions”.
19 For a formal discussion of this point, see Savings [ 60; 6 1; 62].
20 Some writers tend to regard a measured unitary income elasticity as the dividing line for
supporting or rejecting the square-root hypothesis. This seemsto be somewhat misleading.
For the hypothesis predicts substantial, not just some economies of scale, in the transactions
demand for money.
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