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This is a preprint of an article published in The Journal of Money, Credit, and Banking, v. 20, iss. 3,
pp. 463-74, copyright 1988 by the Ohio State University Press. All rights reserved. Reprinted with permission.
Working Paper 86-1
THE DEMAND FOR CURRENCY IN THE
UNITED STATES
Michael Dotsey*
Federal Reserve Bank of Richmond
March 1986
*I would like to thank Marvin Goodfriend, Robert King, and Tony Kuprianov
for their valuable comments. Excellent research assistance with respect to
implementing a number of the econometric tests was provided by Thomas Hahn
and Rob Willemese. The views expressed in this paper are solely those of
the author and do not necessarily reflect the views of the Federal Reserve
Bank of Richmond or the Federal Reserve System.-
THE DEMAND FOR CURRENCY IN THE UNITED STATES
Michael Dotsey
ABSTRACT:
The idea of totally deregulating the financial system and implementing
monetary policy through currency control has received renewed
attention.
An important aspect concerning the desirability of using
currency as the instrument of policy is the behavior of the demand for
currency.
If currency demand is not well behaved, then a policy of
controlling the nominal supply of currency could produce drastic
swings in the price level and interest rates.
The adverse consequence
of such effects could outweigh the benefits of deregulation. It is
therefore crucial that the behavior of currency demand be well
understood before unequivocally advocating total financial
deregulation and currency control. This paper takes a step in that
direction by analyzing the demand for currency over the period
1921-1980.
1
I.
Introduction
The idea of totally deregulating the financial system and implementing monetary
policy entirely through the control of currency has recently received attention. Fama
(1980, 1983) has resurrected the idea suggested by Patinkin (1961) that a well
determined nominal policy can be achieved through the control of currency, and that
other forms of financial regulation with their associated welfare losses need not be
tools of monetary control.
However, an important aspect concerning the desirability
of using currency as the sole instrument of policy is the behavior of the demand for
currency.
If currency demand is not well behaved, then a policy of controlling the
nominal supply of currency could produce drastic swings in the price level and
interest rates.
The adverse effects of the additional noise in these variables could
very well outweigh the benefits of deregulation.
It is therefore crucial that the behavior of currency be well understood before
one can unequivocally advocate total financial deregulation. This paper examines the
behavior of real currency balances over the period 1921-1980 and finds that currency
behaves in a systematic and explainable manner over this period.
This is somewhat
encouraging with regard to the policy advocated by Fama, since this fairly long sample
is characterized by a variety of monetary regimes.
These include a gold standard with
sterilization in the 1920's followed by a gold standard without sterilization in the
1930's.
The 1940's and early 1950's may best be described as a policy of a pure
interest rate peg, while monetary policy over the remainder of the sample followed an
adjustable peg.
However, in view of the Lucas critique, one still needs to be
cautious since direct control of currency was not attempted during the sample and
explicit interest rates on demand deposits were regulated over most of the sample.
The point of departure for analyzing the demand for currency is an inventory
model of the transactions demand for money.
The model developed in this paper~is
based on a number of extensions to the original Baumol-Tobin model. Specifically, the
2
work of Feige and Parkin (1971) and Barro and Santomero (1972) are jointly
incorporated to yield a framework that includes two media of exchange as well as
goods.
Therefore, the model is analytically similar to Santomero (1979) and Dotsey
(1983), and is rich enough to explain movements in the relative importance of currency
and demand deposits as transaction media.
That is, both the proportion of
expenditures made with currency and the average amount of currency held are
investigated.
Indeed the two are closely related.
The behavior of currency is empirically examined by employing Laurent's (1970)
methodology for constructing estimates of the value of currency transfers.
These
estimates, coupled with data on debits to demand deposits, are used to construct a
time series on the percentage of expenditures made with currency, which for simplicity
will also be referred to as the expenditure ratio.
The behavior of the expenditure
ratio and the demand for currency are then jointly investigated.
The behavior of both
variables show remarkable stability and the empirical work can be viewed as generally
supportive of the inventory theory of money demand.
Further, the use of annual data
in the empirical work removes any need for considering lagged dependent variables, and
thus avoids a growing dispute in monetary economics over the interpretation of these
variables.
A very important implication of the inventory model is that the proportion of
transactions income spent using currency is the relevant transactions variable for
determining the demand for currency, and that GNP, which is typically used in most
studies of currency demand, is likely to be a poor proxy for this variable.
Therefore, a comparison between regressions using these two transactions variables is
analyzed.
The poor performance of GNP casts doubt on the empirical procedures used in
a number of other studies on currency.
The framework of the paper is as follows.
model.
Section II develops the theoretical
Section III discusses the empirical implementation of the model, while in
3
Section IV the empirical results are discussed.
Section V critically examines some
past empirical work on currency, and Section VI contains a brief summary.
IIa.
The Model
The framework used to determine the optimal method of purchasing goods is an
extension of the basic inventory models of money demand developed by Baumol
and Tobin (1956).
(1952)
In order to investigate the way that individuals choose to make
expenditures, at least two media of exchange are needed.
Therefore, both currency and
demand deposits are included.-/ In this regard, the model is similar to Barro and
Santomero (1972).
In order to motivate the use of one media of exchange over another,
goods must also be explicitly introduced into the model.
Different characteristics of
goods with respect to transactions costs are employed to help generate a preference
for using one media of exchange over another.
Also, the model includes a short-term
store of value, called a savings account, that allows an individual to economize on
transactions balances.
The individual's problem is to maximize the profits (w) of managing his savings
balance, his transactions balances, and his goods inventories given the transactions
costs associated with going to the bank, cashing a check, and purchasing goods, as
well as the different rates of return on each asset.
The choice variables for the
individual are the number of trips he makes to the bank (n ), the number of times he
cashes a check in between trips to the bank (nd), the number of trips to the store
made in between check cashings (n), and the medium of exchange used in purchasing a
particular good.
notation.)
(To facilitate the exposition, Table 1 provides a summary of
Because the average holdings of each asset are determined by the choice
variables, the results of the profit maximization also yield expressions for the
various asset demands.
Since, the paper is concerned with currency, its average
balance is given by equation (1),
4
2n (n +1) [e
s d
where
YTe
n(nd +1
n
cp(i)di]
1) is the amount of currency withdrawn at the start of each
goods purchasing cycle and
2n YTe
f p(i)di are the average goods
2s (nd + 1)n c
inventories held when the goods are purchased with currency. Here p(i) is the
percentage of income, YT, spent on a particular good and fcp(i)di integrates this
percentage over all goods purchased with currency. Therefore, YT cp(i)di represents
the value of goods purchased with currency.
From (1), it is clear that the
maximization of transactions profits in terms of nS, nd, n, and e will yield a demand
for average currency holdings in terms of the exogenous variables of the model.
Formally, an individual maximizes
(2)
7 = r T§ + rdTD + r TC + r TG - a n
-
-adn nd
(l +A(E) + B(h))ns(nd + 1)n
where rs, rd. rcs and rg are the nominal yields on average savings balances, S.
average demand deposit balances, D, average currency holdings, C, and average goods
inventories, G.
The transactions cost of going to the savings deposit is as while the
transactions cost of cashing a check is ad.
The term (T + A(h) + B(h)) represents the transactions cost associated with the
purchase of goods.
transactions costs.
demand deposits.
Each good is associated with two different proportional
One is associated with the use of currency and the other with
A(h) and B() represent the total proportional transactions costs
incurred using currency and demand deposits, respectively.
Further, each shopping
which includes the value of an individual's time as well
trip incurs a fixed cost,
T,
as transportation costs.
One can therefore think of the model in terms of going to a
fixed location (e.g., a shopping mall), and making purchases at a number of different
5
TABLE 1
TABLE OF NOTATION
T
= an exogenously determined payments interval
X
= the individual's total expenditure on goods over a period of length T
Y
= X/T, the rate at which an individual spends
z(i) = the amount spent on good i on each shopping trip
p(i) = the proportion of total expenditure made on good i
e
= the proportion of total expenditure made with currency, also referred
to as the expenditure ratio
S
= the average savings account balance
D
= the average demand deposit balance
C
= the average amount of currency held
G
= the average goods inventory held
ns
= the number of trips made to the saving account
nd
= the number of check cashings made in between trips to the bank
n
= the number of trips to the store made in between check cashings
as
= the cost of going to the bank
ad
= the cost of cashing a check
a(i) = the proportional transactions cost associated with purchasing
good i with cash
a(i) = o(i)z(i), the transactions cost associated with good i on each
shopping trip
O(i) = the proportional transactions cost associated with purchasing
good i with a check
b(i) = S(i)z(i)
T
= the lump sum transactions costs associated with a trip to the store
rs
= the interest paid on savings
rd
= the interest paid on demand deposits
r
= the interest paid on currency
rg
= the interest earned on goods including depreciation
6
shops, where for simplicity each shop sells a particular type of good.
In order to obtain a function that implicitly defines the optimal
proportion of goods purchased with currency, it is assumed that there exists an
infinite number of goods lying along the continuum from 0 to 1.3Y
Further, it
is assumed for simplicity that goods whose indices are elements of (0, h) are
purchased with currency and those whose indices are elements of (h, 1) are
purchased by check.-/The proportional cost of purchasing the i
good on any particular shopping
trip with currency can be expressed as a(i)z(i), where a(i) is the proportional
th
transactions cost associated with the i
good and z(i) is the amount spent on
the ith good on each shopping trip.
purchasing the i
Similarly, let the proportional cost of
good with a check be denoted as $(i)z(i). That is, different
goods may have different transactions costs for both currency and demand
deposits.
For instance, a small purchase such as a pack of chewing gum or a
newspaper would be much cheaper to perform with cash than with a check, while a
large purchase such as a refrigerator may be more efficiently done with a check.
Carrying large amounts of cash might be associated with a psychic discomfort of
possible loss, while sellers may not accept checks for small items.
Also, one
may wish to have a written record for large purchases, a service which
accompanies checks.
For notational ease let
a(i) = a(i)z(i) and b(i) = 0(i)z(i).
can be written as
T
Then the cost of any particular shopping trip
+ fh a(i)di + fjb(i)di where fh(fR) is the integral over the set of
all goods purchased with cash (demand deposits). Defining A(h) = f a(i)di and
and B(h) = fSb(i)di gives the proportional transactions costs incurred using
currency and checks, respectively.
Given the structure of the model, the various inventories of goods and
assets can be expressed in terms of the choice variables ns, nd, n and the
proportion of goods bought with currency, e.
(For a detailed description, see
7
the appendix.) The optimal values of n , ndo n, and R (and therefore C) are solved
for in two steps.
First, for a given h, the profits from managing the frequency of
transactions is maximized. This yields ns, nd, and n as functions of R and the
exogenous variables of the model.
It also gives expressions for the average holdings
of the various assets in terms of h.
In particular the average value of currency C(h)
is depicted by
3_
_
1/2y _
_
2(T + A(h) + B(h))
d
[ eY(rd-r) 1 LY((l-e)(rd-rg)
+
e(r -r))I|
One therefore observes that for any given h, average currency balances increase when
transactions income, the cost of cashing a check, and the opportunity costs of holding
goods (rd-r
and rC-r ) rise, and that average currency balances fall when the
opportunity cost of holding currency rd-r
and the transactions costs of making a
shopping trip increase.
A complete solution to the model is.obtained by substituting in the solutions for
n (h), nd(h), and n(h) into the expression for profits yielding an equation in which
profits are a function of h and the exogenous variables of the model.
This expression
is then maximized with respect to h, and the resulting first order necessary
conditions yield an implicit function of h.
Since e is directly related to
h by e = f 0 p(i)di, solving for the optimal value h yields the optimal proportion of
goods bought with currency.
The optimal E, h*, occurs at a value that exactly offsets changes in interest
earnings and transactions costs.
that a(h*) < b(h*).
Further, the first order necessary conditions imply
Since demand deposits pay a higher yield than currency, one would
only use currency if it had a lower transactions cost.
However, a check might still
be used for the ith good if a(i) < b(i) since there is an opportunity cost of holding
currency instead of demand deposits.
Also, the second order conditions for profit
maximization imply that the derivative of b(h*), b'(h*) must be less than a'(h*).
8
Therefore, if currency is to be used at all it must provide a relative advantage in
transacting for some goods, and the transactions costs of using currency must increase
faster with respect to the ordering of goods.
These conditions seem reasonable since
the cost of using a check does not seem to vary much over goods, while the
transactions costs of currency increases sharply with the amount spent on the good
(especially if one includes the cost of expected currency loss).
MIb.
Comparative Statics.
The effects of changes in the exogenous variables on E* (and hence e*) and the
demand for currency will now be examined.
Since the demand for currency will
experience both a direct effect as well as an indirect effect due to changes in
transactions patterns when exogenous variables change, the comparative static results
for the expenditure ratio will be examined first.
(For detailed results see the
Appendix).
The Effects on the Expenditure Ratio
Regarding the expenditure ratio, one first observes that changes in a., rs, and Y
do not affect the optimal method of purchase as long as they do not influence the
consumption bundle.
This is because a
5
and r
5
are only involved in determining the
optimal number of trips to the saving account and therefore are not related to
expenditure patterns, while transactions income serves solely as a variable that
scales the level of expenditure. However, to the extent that movements in these
variables change expenditure patterns (i.e., the distribution of p(i)), they would
affect the expenditure ratio.
The effects of changes in the various transactions costs ad, b(h*), a(h*) are
straightforward. A decrease in the cost of cashing a check implies that more goods
will be bought with currency.
As check cashing becomes less expensive, a smaller
opportunity cost is incurred when goods are bought with cash since an individual can
economize on currency balances by using demand deposits as a store of value.
In the
limit as ad approaches zero, the model would be equivalent to having two means of
9
payment both stored as demand deposits.
In this case an individual's currency
balances would be zero, and he would purchase a good using cash or check depending on
whether a(i) < b(i).
An increase in b(h*) relative to a(h*) causes more goods to be
purchased with currency since the proportional transactions costs of using currency
becomes relatively cheaper.
The effect of a change in the fixed cost of going to the store cannot be
unambiguously signed, nor can the effects of changes in rd. rc and r .
However, there
is an exact relationship between the comparative static effects of these variables
that hinges on the sign of the following inequality:
(4)
(1/T)(Y/2)p(i*)(rd-r) > (b(R*) - a(h*))n (n
+ 1)n,
where ns (nd + 1)n is the number of trips made to the store over each payments periods.
The left-hand side of this expression represents the gain of using demand deposits
relative to currency, since the level of currency and deposit balances are affected by
the amount spent on the marginal good.
The right-hand side represents the additional
cost of using a check multiplied by the number of trips made to the store.
Increases in the fixed cost of going to the store, T. rg and rd will cause the
the expenditure ratio to rise, while an increase in rc will cause it to fall if the
left-hand side of (3) is greater than the right-hand side.
With respect to increases
in T and rg, both cause fewer trips to the store as well as a decline in the
additional amount of demand deposts and currency that would have to be held if an
additional good was purchased with either medium.
Therefore, the transactions benefit
of using currency declines, as does the opportunity cost of holding currency.
Whichever declines by more will determine the net effect on the expenditure ratio.
An increase in the own rate on currency causes the number of trips to the store
to increase making currency's use relatively more attractive. However, it also
increases the average holdings of currency reducing the attractiveness of using
10
currency.
If the gain due to transactions use outweighs the increased opportunity
cost due to higher balances, then more goods will be bought with cash.
An increase in the rate paid on demand deposits has two conflicting effects on
the expenditure ratio.
That is, an increase in rd should produce opposite results to
those encountered with r and r . It can be shown that the substitution effect
g
c
between demand deposits and currency dominates, if the left-hand side of (1) is
smaller than the right side implying an increase in the expenditure ratio.
Therefore, although an exact sign can not be given to the comparative static
results of a number of the exogenous variables, there does exist a well-defined
relationship between their various effects.
For instance, a rise in the fixed cost of
going to the store will cause the expenditure ratio to move in the same direction as
an increase in the yield on goods inventories or demand deposits, while an increase in
the yield on currency should cause the expenditure ratio to move in the opposite
direction. Therefore, although one can not predict the way in which the expenditure
ratio will shift when interest rates or
T
move, there is a well-defined relationship
between the effects generated by movements in these variables.
This is summarized in
table 2.
The Effect on Average Currency Balances
The effect on average currency balances of a change in an exogenous variable, x,
can be separated into two components. One is a direct effect for any given
expenditure pattern, while the other involves the effect of a change in expenditure
patterns. Therefore, it is important to establish the shape of C(h) with respect to
h.
Initially, as one begins to use currency, currency balances rise.
In the
neighborhood of h*, C(h) will be increasing as the number of goods purchased with
currency increase if (l/T)(Y/2)½ p(1 *)(rd-rc) < (b(h*) -a(R*))(ns(nd(h*)+l)n(h*),
which turns out to be the case empirically (for more detail see Appendix A).
Satisfying this inequality implies that the amount of currency held in between check
cashings rises by more than the average inventories of goods, implying that average
11
currency balances rise.
dC(h*)
dx
Notationally,
aC(K*)
ax
=
+ aC(h)
h*
+
A*
ax
The inequality that determines the sign of Bc(h*)/Dh* is the same inequality that
determines the sign of ah*/Dx.
Therefore, it will turn out that the expenditure ratio
effects embodied in the second term on the right-hand side of (4) will reinforce the
direct effects of the exogenous variables if column 2 of table 2 is relevant.
Since
this turns out to be the case empirically, the comparative static effects of changes
in exogenous variables can be summarized by
(6)
C(as, dd3 eY, e, rs, rd. rc, b(h*)/a(h*)).
0
+
+
+
0
-
+
-
where the signs under the variable indicate the sign of the partial derivate of
currency with respect to that variable.
III.a
Developing the Empirical Tests.
The preceding section suggests a simultaneous equation systems approach for
estimating the model.
The two step maximization procedures yields equations
determining both the expenditure ratio and average currency balances which can be
expressed as
(7)
(8)
e - f(Y, a, ad,
T,
= g(eY, e, a, ad,
a(h), b(h), rs, rd
T,
rC, r , ud)
a(h), b(h), rs, rd, rc) rg, vt)
where ut and vt are stochastic disturbance terms.
Alternatively, one could substitute
(7) into (8) and represent C as a function of exogenous variables.
However, the
coefficients from such an equation would be a complicated combination of direct and
indirect effects on average currency holdings and therefore be far less revealing than
12
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13
the procedure outlined above.
Performing the estimation on the system (7) and (8)
allows one to discriminate between the effects a variable has on expenditure patterns
and any additional effects the variable has on the demand for currency.
Another important consideration in implementing the empirical tests, concerns the
derivation of measures which approximate the variables of the model.
A description of
these follows.
III.b
The Dependent Variables
The real per capita value of currency balances was used as the dependent variable
in the currency demand regressions.
Its calculation is straightforward. However, the
empirical measure of the expenditure ratio represents a fairly novel procedure.
Ideally, the expenditure ratio, e, is given by the ratio of goods and services bought
with currency to the total amount spent on goods and services using both currency and
demand deposits.
The volume of purchases made with currency is calculated by a method
devised by Laurent (1970), while debits to demand deposits (excluding New York City
debits) are used to measure expenditures made by check.
Using debits to demand deposits is appropriate only if purchases of goods and
services made by check is a fairly constant fraction of total debits.
Examining the
behavior of the ratio of debits to demand deposits to GNP in figure 2 indicates that
this is unlikely to be the case.
Beginning in the early sixties, debits to demand
deposits began to grow at a much faster rate than GNP.
The conjectured reason for
this behavior is that transactions of a purely financial nature have become an
increasingly important part of total demand deposit debits.
Because the volume of
demand deposit debits involving financial transactions are not a constant fraction of
total demand deposit debits, this series does not represent a consistent indicator of
the volume of goods and services purchased by check.
Therefore, the sample period
emphasized in the empirical work on the expenditure ratio covers the years 1920-1965,
with the belief that nonfinancial debits are a relatively constant fraction of total
debits over this period.
14
The measure of expenditures made with currency is obtained by using Laurent's
procedure. Laurent's methodology uses data on issuance and redemptions of currency by
denomination in order to calculate the value of transactions made with currency, given
that each bill performs a certain number of transfers before being retired.-/ The
actual number of transfers that the average bill makes is calculated by maximizing the
correlation between the sum of currency transfers and debits to demand deposits to
GNP.
Over the period 1920-1965 the number of transfers that an average bill made
before being retired was 85.
Using evidence from Laurent this would imply that a
one-dollar bill was involved in a transaction approximately every six days.-/
The real value of currency transfers, CT, obtained from this procedure is
depicted in figure 2.
War II.
This series is quite volatile and reaches a peak during World
The large wartime use of currency may have been associated to some degree
It could also have
with black market activity arising from the rationing of goods.
been partially due to an increased proportion of the population in military service
and to tax evasion (see Cagan (1958)).
The expenditure ratio is given by
(9)
e
= CT/(CT + DEBITS)
where DEBITS are the real value of debits to demand deposits.
As indicated by
the graph of the expenditure ratio (figure 3), currency has been an important
media of exchange accounting for approximately 60 percent of transfers in 1933
and 1934 and falling to roughly 20 percent of transfers in 1965.
It should be
noted that the figures obtained for CT and e represent an upper limit for the value
and percentage of the value of goods and services purchased with currency.
This is
because most transactions involving demand deposits are written for their exact
amount, while most transactions undertaken with currency involve total transfers which
are greater than the value of the purchase.
For example,
purchasing an $8.00 item
with a ten-dollar bill involves a currency transfer of $12.00. -
15
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16
IIc.
The Independent Variables
The next step is to examine the independent variables used as determinants of the
expenditure ratio and the demand for currency.
In Section II, it was shown that both
dependent variables should depend upon the transactions costs of cashing a check and
purchasing goods, on the rate yielded by demand deposits, currency, and goods, and on
the distribution of expenditure over goods.
In order to proxy for the fixed cost
component of transactions costs, the real wage rate, W, on average hourly earnings of
production workers in manufacturing was used.
are tried.
Two implicit rates on demand deposits
One, RD, is equal to one minus the maximum reserve requirement ratio on
demand deposit times the commercial paper rate.
This measure essentially assumes that
The
banks totally avoid Regulation Q, and remit services at a competitive rate.
other, RDI, is the average implicit rate derived by Startz (1979).
The inflation
rate, INF, is used to approximate the positive component of the return to goods
inventories.
Depreciation is therefore implicitly assumed to be constant.
Real per
capita consumption, CONS, is included to pick up changes in the types of goods
purchased caused by changes in wealth.
The percentage of consumption comprised by
durables, PDUR, is also used as a proxy for changes in expenditure patterns.
Since
durables are likely to intensively involve demand deposit use, a shift toward durable
goods purchases should lower the expenditure ratio.
However, once the effect of PDUR
and CONS are accounted for in currency transfers, they should have no additional
effect on average currency balances.-/
Three other variables, the percentage of bank failures, PFAIL, the marginal tax
on labor income, MTAX, derived by Barro and Sahasakul (1983),-/ and the rate paid on
savings, RS, are included.
The first is intended to capture the loss of liquidity (as
well as actual losses) suffered by deposit holders when banks are suspended.
This
loss of liquidity on demand deposits should make currency a relatively more attractive
medium of exchange.
Also, currency is a useful medium for avoiding income taxes since
no written records need to be kept.
Because the advantage of hiding income increases
17
with marginal tax rates, it is likely that currency will be used more intensively when
marginal tax rates are high.
Indeed Cagan (1958) argues that much of the increase in
currency holding during World War II was due to its increased use in order to avoid
taxes.
It is also possible that an increase in MTAX could cause individuals to hoard
some of their wealth in the form of currency. Therefore, MTAX could also affect the
demand for currency above and beyond its affect on expenditure patterns.
The rate paid on savings accounts are included to capture the effect of credit
usage.
This rate will also be a determinant of average currency holdings for
individuals that do not use a checking account.
The purchase of goods on credit
essentially allows an individual to use his savings account as a transaction medium.
That is, money can be left in the savings account until the bill becomes due.
Since
the model reasonably requires a trip to the bank each payments period,i/ no additional
costs are borne in transferring funds.
An increase in the rate paid on savings would
intensify the purchase of goods with credit.
To the extent that credit purchases
substitute for currency (since a credit purchase eventually causes a debit to demand
deposits), the expenditure ratio should be negatively related to this variable.
However, under the assumption that goods are purchased more frequently than the
payments period, credit use should have no direct effect on average currency balances
beyond the effects that are captured by the amount of currency transfers.
If, this
assumption is violated, the use of credit could have a substantial effect on average
currency holdings.
A direct measure of the intensity of credit is also used.
This measure (PCR) is
constructed as the ratio of installment retail credit, check credit, noninstallment
retail credit and service credit to consumption.
Optimally, one would like to
construct a variable that measured the amount of credit extended in relation to
purchases over a particular period.
However, no such measure is available over the
sample period and PCR uses the average amount of credit outstanding.
18
Before presenting the results, some potential difficulties should be
discussed.-O/ One difficulty is capturing the effects of illegal activities, besides
those of tax evasion.
No measure was derived for this type of behavior.
Also,
hoarding of currency by foreign residents could affect the data on issuance and
redemptions thus biasing the derivation of currency transfers.
found to account for this.
Again no measure was
A dummy variable taking on values of 1 in 1942-1945 was
tried in order to pick up the effects of rationing and black market activity during
World War II, but it was insignificant.
IVa
Empirical Results:
The Expenditure Ratio 1920-1965
The basic regression (run over the period 1920-1965) is depicted in equation
(10), where the notation "ln" signifies the natural logarithm of a variable.
(10)
ln(e
=
a
+ a ln(CONS ) + a 1n(RD ) + a 1n(RSAV ) + a 4 INF
+ a51n(Wt) +
+
+ a8PFAIL
+ ut
The results of this regression are reported in column 1 of table 3.
also run with a correction for serial correlation of order two.1 1 /
The regression is
The errors of the
corrected regression appear to be white noise.
Column 1 of table 3 represents the results of estimating equation (9) and will be
emphasized in the following discussion. Column 2 substitutes RDI for RD.
The
interpretation adopted here is that RD is an appropriate proxy for the implicit rate
earned on demand deposits, and that column 1 contains the most meaningful regression.
It is also important to report that the results of table 3 are not significantly
affected by altering the value of G, the number of times a bill is assumed to
circulate over its lifetime.
and 110.
Similar results were found for values of G equal to 50
Regressions were also run using the logarithm of the ratio of currency
transfers to demand debits with much the same qualitative results.
Examining the results of column 1, the coefficient on real per capita consumption
indicates that as wealth increases there is a significant change in the types of goods
19
purchased.
Specifically, an increase in the level of consumption alters the bundle of
goods purchased and implies that individuals purchase more of the types of goods
associated with checking account transactions. The coefficient on the percentage of
consumption comprised by durables reaffirms this conclusion. That is, as individual's
buying habits change, so does their method of purchase.
The negative coefficient on inflation establishes the condition that
(1/T)(Y/2) p(R*)(rd-rc) < (b(R*) - a(h*))ns(nd+O)n. Therefore, one would expect the
coefficients on the value of time as proxied by W, and on the implicit rate on demand
deposits to be negative, while variables that capture an implicit yield on currency,
say MTAX, to have a positive sign.
This is indeed the case.
on the real wage rate does seem rather large.
However, the coefficient
This could reflect the fact that as
time becomes more valuable individuals lump the purchases of goods together, and find
it more convenient to carry out this procedure with checks.
The assumption of the
model developed in section II is that agents purchase all goods during the same trip,
which is not a totally accurate assumption. As the value of time increases one would
expect a growing preference for stores that sold a large variety of merchandise,
allowing individuals to lump together purchases.
In this case it may be that using
checks becomes relatively more convenient, and that the wage rate variable is picking
up this effect as well.
The variable PFAIL also has the correct sign.
(The
coefficient is significant at the 11 percent level.)
The yield on the savings account is also significant and negative.
As mentioned
before, it may be proxying for the return on credit purchases, since credit in a sense
allows one to use the store of value as a means of exchange.
To the extent that
credit purchases substitute for cash, they would tend to lower the expenditure ratio.
A substitution of credit for a check would have little effect on the ratio since most
credit purchases are eventually paid off with a check.
(In some instances using
credit issued through a third party could result in two demand debits per purchase.
For example, American Express would pay the merchant and you would pay American
20
TABLE 3
REGRESSION RESULTS FOR THE EXPENDITURE
RATIO OVER THE PERIOD 1920-1965
Independent
Variables
Column 1
Column 2
CONSTANT
-2.95(-6.76)**
-2.57(-4.82)**
ln(CONS)
-.48(-5.56)**
-.61(-6.16)**
ln(RD)
-.057(-3.86)**
ln(RSAV)
-.18(-9.56)**
-.21(10.08)**
INF
-.50(-4.59)**
-.32(-2.56)*
ln(W)
-.76(-12.45)**
-.61(-10.43)**
ln(PDUR)
-.067(-2.38)*
-.10(- 3.25)**
ln(MTAX)
.046(2.60)*
.013(.62)
PFAIL
.120(1.64)
.14(1.49)
-.055(-1.21)
ln(RDl)
RHO(l)
.62(4.97)**
.54(4.04)**
RHO(2)
-.56(-4.84)**
-.41(-3.55)**
R2
.99
.99
S.E.R.
.0239
.0279
D.W.
2.16
1.80
Column 1 represents the results for equation (6). RHO(l)
and RHO(2) are the coefficients for second order serial correlation. R2 is the corrected correlation coefficient, S.E.R. is
the standard error of the regression, and D.W. is the DurbinWatson statistic. Numbers in parentheses are t-statistics.
**, *, and t indicate significance at the 1%, 5%, and 10%
significance levels, respectively.
Column 2 is equation (6) with RD1 substituted for RD.
21
Express.)
The variable PCR, which measures the intensity of credit use was tried and
its coefficient was insignificantly different from zero, perhaps implying that the
interpretation given to RSAV is incorrect or that the credit variable was not a good
measure of credit usage.
The regression results of equation (4) were also checked for stability using the
cusum of squares test developed by Brown, Durbin, and Evans (1975).
Since the sample
period contains a number of very different economic environments, such tests are
certainly called for.
The maximum cusum of square statistics were .1131 for the
forward recursion and .1686 for the backward recursion.
Both values were well below
the 10 percent critical value of .2093, therefore rejecting instability.
Also, a
specification test suggested by Plosser, Schwert, and White (1982) which involves
overdifferencing of the equation failed to reject the level specification used in
estimation.
The test statistic for the regression given in column I of table 3 is
distributed X with 8 degrees of freedom and has a value of 11.48, which is below the
5 percent critical value of 15.51.
It is interesting to examine the behavior of the expenditure ratio that is
implied by the results of the regressions.
During the Depression the percent of
expenditures made with currency rose from .47 in 1930 to a high of approximately .63
in 1933, a 33 percent increase.
The largest part of this increase is explained by the
19 percent drop in real per capita consumption, while significant portions are
explained by the 25 percent fall in PDUR, the 21 percent fall in RSAV, the 53 percent
fall in RD, and the 39 percent increase in the marginal tax rate.
these effects only accounts for a 20 percent rise in e.
However, the sum of
The percentage of bank
failures rose by 400 percent between 1930 and 1933, which would account for an
additional rise in
e of 8 percent, thereby explaining most of the increase in the
expenditure ratio during the Depression.
During World War II, the continued high value of e can, to a large extent, be
attributed to a rapid increase in the marginal tax rate from 6.5 percent in 1940 to
22
26.2 percent in 1945, an increase of over 300 percent.
The decline in the percentage
of consumption composed by durables from 10.8 percent in 1940 to 6.7 percent in 1945,
a decline of 38 percent, was also significant in maintaining the relatively high
values of e.
The changes in these two variables largely offset the effects of a 23
percent rise in real per capita consumption and an 18 percent increase in real wages.
These results therefore support Cagan's contention that the rapid rise in tax rates
during World War II caused currency to be used more intensively than it otherwise
would have been.
After the war the expenditure ratio experienced a rather consistent decline.
This is largely attributable to the growth in per capita consumption, the increased
rate paid on both savings and demand deposits, and the rising real wage rate.
IVb
The Demand for Currency
The basic regression examining the demand for real currency balances is given by
(11)
dlnCt = b0 + bIdln(CT) + b 2 dln(et) + b 3 dln(RDt) + b 4 dln(RSAVd)
+ b 5 d(INFt) + b 6 dln(Wt) + b 7 dln(PDURt) + b 8dln(PCRt)
+ b 9 dln(MTAXt) + b 9 d(PFAILt) + Vt
where "d" indicates that all variables are first differenced,L3/ and estimation is
carried out using 2SLS.
Column 1 of table 4, reports the results of this regression.
The coefficient on the real per capita currency transfers, CT, is highly significant
as are the coefficients on PFAIL, and W.
The signs and size of these coefficients do
not violate any of the predictions of the inventory model.
The insignificance of PDUR
implies that currency is not intensively used in the purchase of durables, while the
insignificance of MTAX implies that currency was not hoarded in response to changes in
tax rates.
The insignificance of RSAV is consistent with the adjaceny property of the
inventory model if most individuals use a checking account and therefore do not use a
savings account as a temporary store of value for currency.
The effect of the
commercial paper rate was also investigated and its coefficient was insignificant
23
further confirming the adjacency property.
The insignificance of RD and INF are a
little puzzling. Regarding RD, it may only measure the marginal rate on corporate
balances while the marginal rate on individual's demand deposits is probably zero.
Since the behavior of corporations has a large effect on debits to demand deposits
this would account for the significance of this variable in the expenditure ratio
regressions. However, corporations only hold a small fraction of cash balances, which
could account for the lack of significance of this variable in explaining the behavior
of the stock of currency.
Therefore, with the exception of the results on INF and
PCR, the simple inventory model examined in this paper performs quite well.
The results concerning credit are indeed puzzling since these are not consistent
with the results reported for the expenditure ratio in which PCR was insignificant,
implying that credit usage did not directly substitute for currency.
The significance
of PCR in the currency regressions can only be accounted for if credit is a substitute
for currency and if the goods purchased are those that are bought less frequently than
the payments period. 4/
It is only in this case that credit can have an effect on
average currency balance beyond any affects already accounted for by changes in
expenditure patterns.
To see this, consider the following example.
Initially,
suppose currency is used to purchase two types of goods, one, good a, is purchased
more frequently than the payments period and the other, good b, is purchased every
other payments period, and that the individual does not meet the interior conditions
of the model regarding the use of demand deposits or a savings account.
In that case,
cash balances would accumulate in week one for expenditure on good b in week 2.
In
other words the one week's worth of interest is insufficient for this individual to
use a savings account.
But, if the individual could purchase good b with credit, he
could keep money in his savings account for a much longer period, since credit
payments usually involve a minimum of a one month lag.
This would increase the
likelihood that the individual would employ a savings account, with an accompanying
decline in his average cash balances.
While this example can account for the effect
24
of credit in equation (11) it can not jointly explain the results on the expenditure
ratio and the currency equation. As mentioned earlier, the variable PCR is a somewhat
crude measure and it is clear that more work needs to be done regarding the effect of
credit.
Finally, the expenditure ratio does not appear to be significant beyond its
effect on CT (recall CT = eY).
This is somewhat of a blessing in disguise since
e is the only variable that can not be accurately measured over the entire sample
1921-80.
Indeed as indicated in column 2 of table 2, omitting e
substantially alter the regression results.
does not
Therefore, a regression omitting e is
examined over the entire sample period in column 3.
The only major change is that the
marginal tax rate is now significant at the 10 percent level, implying that increases
in tax rates lead to hoarding of currency.
The stability of the regressions in columns 2 and 3 of table 4 were checked using
two procedures developed for simultaneous equation systems.
One developed by Erlat
(1982) is essentially an extension of the usual Chow test and the other developed by
Giles (1981) extends the use of the Brown-Durbin-Evans cusum of squares test to
simultaneous equations.
Over the period 1921-1965, with the sample broken at 1944
which is the date that the Quandt log likelihood ratio takes on its maximum value, the
test statistic which is distributed F1 takes on a value of .31.
12
the 5 percent critical value.
This is well below
The maximum cusum of squares statistic takes on a value
of .2637 which is below the 5 percent critical value of .2857.-15
Regarding the
1921-1980 period, the sample was broken at 1968 with the test statistic being
15
= .268 which is again well below the 5 percent critical value.
distributed F
34
The
maximum cusum of squares statistic is .2932 which is above the 5 percent critical
value of .244 but below the 1 percent critical value of .2972.
Therefore, the tests
broadly suggest that the demand for currency is stable over the period.
As depicted in figure 4, the major changes in real per capita currency balances
occurs during the Depression and during World War II.
After WWII, currency declines
25
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26
sharply and then grows at a relatively steady rate.
During 1931 real per capital
currency balances grew by 22 percent and in 1932 they grew by 25 percent.
This growth
is largely but not totally explained by the increase in the use of currency and the
large number of bank failures. The increased use of currency occurs, even though
income was falling during this period.
In fact, CT grew at a rate of 8.8 percent in
1931 and 10.4 percent in 1932.
The largest growth in currency occurs during the war.
Over that period real per
capita balances grew by 15.5 percent in 1941, 22.9 percent in 1942, 29.8 percent in
This rapid growth was also sharply reversed
1943 and 23.2 percent in 1944.
immediately after the war.
The rapid rise was partially accounted for by the
increased use of currency during the war, especially in 1943 and 1944 when CT grew by
at an annual rate of roughly 13 percent.
A look at figure 2 shows that CT reached a
peak during the war and thereafter experienced a somewhat uneven pattern of decline.
The behavior of credit also helps explain movements in currency balances.
Credit fell
substantially during the war, but began to grow quite strongly after the war.
The
growth in wage rates during the war also helps explain the increase in currency.
Wage
rates also fell immediately after the war, and then grew fairly consistently through
the rest of the sample.
As mentioned earlier, marginal tax rates also grew
substantially during the war.
However, the coefficient on the marginal tax rate is
only significant when the sample is extended through 1980.
This provides some
evidence that currency hoarding is associated with high tax rates, although the
evidence is far from conclusive.
An examination of the residuals of the currency regressions indicate that the
large swings in currency holdings that occurred over the sample are captured quite
well by the regressions.
somewhat large.
Only the residuals for the years 1931, 1935, and 1949 are
The above results therefore indicate that currency behaves in a
systematic fashion and that this behavior is consistent with the predictions of an
inventory model of money demand.
Indeed the inventory model is especially useful in
27
indicating that the proper transactions scale variable involves a measure of the
actual expenditures made with currency.
As will be shown in the next section, simply
employing real per capita GNP in a reduced form regression yields poor results and
casts doubt on previous studies of the demand for currency.
V.
A Critical Examination of the Literature
The empirical methodology of section IV was used to estimate the structural
parameters of equations (10) and (11).
Based on the parameter estimates obtained for
these structural equations, the reduced form regression for currency consistent with
the structural equations would be
(12)
C = .71dln(CT + DEBITS) - .34dln(CONS) -
.17dln(RSAV) + .08dln(MTAX) -
+ .54dlnPFAIL -
.23 dln(RD)
.047 dln(PDUR) - .45dln(PCR)
.36dINF + .08dln(W)
where CT + DEBITS are in real per capita terms.
period 1921-1965 is given in column 1 of table 5.
The estimated reduced form over the
The estimated coefficients, with
the exception of the coefficient on CONS, are all extremely close to the values given
in (12), and all the coefficients are within one standard error of the coefficients in
(12).
If one, instead runs a regression using real per capita GNP, gnp, (or real per
capita consumption) as the transactions scale variable the results are markedly
different. As seen in column 2 table 5, the coefficient on dlngnp is insignificantly
different from zero.
A similar result is found for the entire sample period, which
make spurious correlation a strong possibility with regard to a number of results
reported in the literature.
For example, Becker (1975) uses the level of retail trade as a scale variable on
quarterly regressions over the period 1953-1971. Although he obtains a significant
positive coefficient on this variable, the coefficient on lagged currency is .98, and
may indicate that this coefficient is picking up a good deal of serial correlation.
28
Garcia and Pak (1976) find that the log of real GNP is significant and positive in
quarterly regressions over the period 1952:2-1967:4 and 1952:2-1977:2. The
coefficients on lagged currency are between .80 and .86 while the first order serial
correlation coefficients are .70 and .53 respectively. Ochs and Rush (1983) also find
that the log of real GNP is significantly positive, but obtain a very high coefficient
on lagged currency.
In all these studies using first differencing without lagged
dependent variables appears to be a needed specification check.
VI.
Conclusions
Building on an inventory model of currency demand, the empirical work in this
paper indicates that the demand for currency has behaved systematically over a sample
period that includes very diverse conditions. The inventory model indicates the
importance of correctly specifying the transactions variable, and this importance is
confirmed in the empirical work.
A number of other properties of the inventory model
are consistent with the data, among which are the insignificance of the commercial
paper rate in the currency demand regressions and the sign patterns on the
coefficients in the expenditure ratio equations. The systematic and largely
explainable behavior of currency over the sample lends support to the view that
currency control may be a viable alternative to current monetary policy.
29
FOOTNOTES
-/Implicit in this argument is the notion that the demand for currency would not
radically change under a regime of targeting currency.
Given the wide disparity of
the economic environments examined in the empirical work, my guess is that the demand
for currency would, if anything, exhibit more systematic behavior under a regime of
constant growth.
2/Introducing the intertemporal problems associated with credit would severely
complicate the model.
-/The qualitative results of this model are not sensitive to a variety
of alternative specifications. Similar results were found for a model containing a
continuum of goods with no fixed transactions cost associated with the purchase. of
goods and for a model that contained a discrete number of goods, the purchase of each
being associated with a different fixed cost.
For a detailed discussion of the latter
model, see Dotsey (1983).
4/If goods are equally weighted in the budget, then goods can be ordered in such
a way that purchasing behavior would accord with this assumption. If goods were
ordered so that (b(i) - a(i)) > (b(i+c) - a(i+e)), then buying the first goods with
cash will be optimal.
For unequally weighted goods I have not been able to derive a
similar ordering.
5/The volume of transactions negotiated with coins is not considered.
6
/The correlation coefficient series generated by this procedure is
extremely flat making it difficult to state a maximal value with a high degree of
statistical certainty.
However, the results of the regressions presented in section
IV are not very sensitive to changes in G.
Similar results were obtained for values
of G = 50 and G = 110.
The method of expenditure could have a direct effect on average currency
balances if goods purchased with currency are bought less frequently than the payments
period (see Grossman and Policano (1975)).
An increase in the proportion of goods
30
that are bought less frequently than the payments period ((behavior which is likely to
be associated with durables) would cause average cash balances to rise.
However, if
durables are not generally purchased with currency, an increase in the proportion of
expenditure made on durables would only affect currency through its indirect effect on
the expenditure ratio.
-/Barro and Sahasakul's derivation is an improvement over earlier work done on
marginal tax rates.
In particular, they calculate a rate derived from the actual tax
schedule and therefore avoid downward biases caused by deductions to income.
They
also include the effects of social security taxes.
9/A 1979 survey by the Federal Reserve Bank of Atlanta indicates that individuals
make on average 3 deposits per month to their demand deposit accounts.
-O/These difficulties are potentially severe since the latest Federal Reserve
Study could account for less than 15 percent of the U.S. currency stock.
"l'The best fit for the errors from an uncorrected regression was ARIMA (2,0,0)
with the AR parameters equalling .51 with a standard error of (.15) and -.49 with a
standard error of (-.15).
L2/A maintained hypothesis of the Brown-Durbin-Evans test is white noise errors.
The test was therefore also run after the variables were filtered using an AR2 filter.
The maximum cusum of squares statistics were .1780 for the forward recursion and .2009
for the backward recursion, which are below the 10 percent critical value of .2140.
L3/The results of the specification test suggested by Plosser, Schwert, and White
can be extended to simultaneous systems and indicate that the level regression is an
inapproriate specification.
'4/For a detailed discussion of the effects of the relative frequency of goods
purchases on money balances, see Grossman and Policano (1975).
15/In the stability tests a maintained hypothesis is that the errors of the
regression are white noise.
Therefore, the variables are filtered before running the
tests so that the resulting errors are indeed white noise.
31
LITERATURE CITED
1.
Barger, Harold.
Outlay and Income in the United States:
1921-1938. Nat. Bur.
Econ. Res., New York, 1942.
2.
Barro, Robert J.
"Integral Constraints and Aggregation in an Inventory Theory of
Money Demand." J. Finance 31 (March 1976), 77-87.
3.
Barro, Robert J., and Sahasakul, Chaipat.
"Average Marginal-Tax Rates from
Social Security and the Individual Income Tax."
NBER Working Paper 1214,
October 1983.
4.
Barro, Robert J., and Santomero, Anthony M.
Demand Deposit Rate."
5.
Baumol, W. J.
"Household Money Holdings and the
J.M.C.B. 4 (May 1972), 397-413.
"The Transactions Demand for Cash--An Inventory-Theoretic
Approach." Q.J.E. 66 (November 1952), 545-56.
6.
Becker, William E.
Ratio."
7.
"Determinants of the United States Currency-Demand Deposit
J. Finance 30 No. 1 (March 1975), 57-74.
Brown, R. L.; Durbin, J.; and Evans, J. M.
of Regression Relationships Over Time."
"Techniques for Testing the Constancy
J. of the Royal Statistical Society,
Series B (1975), 149-63.
8.
Cagan, Phillip. "Demand for Currency Relative to the Total Money Supply." J.P.E.
66 (August 1958): 303-29.
Reprinted in Nat. Bur. Econ. Res.
Occasional
Paper 62.
9.
Dotsey, Michael.
"A Study of the Relative Use of Currency and Demand Deposits."
Ph.D. dissertation (unpublished), Department of Economics, University of
Rochester (1983).
10. Dutton, Dean S., and Graham, William P.
the Demand for Money."
11. Erlat, Haluk.
"Transactions Costs, the Wage Rate and
A.E.R. 63 (September 1973), 652-65.
"A Note on Testing for Structural Change in a Single Equation
Belonging to a Simultaneous System." Ec. Letters 13 (1983), 185-189.
32
12. Fama, Eugene F. "Banking in the Theory of Finance," JME 5 (April 1980): 39-57.
_
"Financial Intermediation and Price Level Control."
JME Vol. 12
No. 1 (July 1983): 7-28.
13. Feige, Edgar, and Parkin, M.
"The Optimal Quantity of Money, Bonds,
Inventories and Capital."
14. Garcia, G. and Pak S.
States."
Commodity
A.E.R. 61 (June 1971), 335-49.
"The Ratio of Currency to Demand Deposits in the United
J. Finance 34 (June 1986), 703-715.
15. Giles, David E.A. "Testing for Parameter Stability in Structural Econometric
Relationships." Ec. Letters 7 (1981), 323-326.
16. Grossman, Herschel I., and Policano, Andrew.
"Money Balances, Commodity
Inventories, and Inflationary Expectations.
J.P.E. 83, no. 6 (1975),
1093-1112.
17. Karni, E. "The Value of Time and the Demand for Money."
J.M.C.B. 6
(February 1974) 45-64.
18. Kuznets, S.
National Income and Its Composition. Nat. Bur. Econ. Res.,
New
York, 1941.
19. Laidler, David E. W.
The Demand for Money Theories and Evidence. 2nd ed. New
York: Dun-Donnelley, 1977.
20. Laurent, Robert D.
"Currency Transfer by Denomination." Ph.D. dissertation
(unpublished), University of Chicago (1970).
21. Ochs, Jack and Rush, Mark.
demand for Money."
22. Patinkin, Don.
"The Persistence of Interest Rate Effects on the
JMCB 15 (November 1983), 499-505.
"Financial Intermediaries and the Logical Structure of Monetary
Theory," AER 51 (March 1960), 95-115.
23. Plosser, Charles I.; Schwert, G. William; and White, Halbert.
"Differencing as a
Test of Specification." I.E.R. 23 (October 1982): 535-52.
24. Santomero, Anthony J.
"Optimal Transactions Behavior and the Demand for Money."
Ph.D. dissertation, Brown University (1971).
-
33
_
"The Role of Transactions Costs and Rates of Return on the Demand
Deposit Decision." J. Monetary Economics 5 (1979), 343-64.
25. Startz, Richard.
"Implicit Interest on Demand Deposits." J. Monetary Economics 5
(1979), 515-34.
26. Tobin, James.
"The Interest-Elasticity of Transactions Demand for Cash." R.E.S.
38 (August 1956), 241-47.
34
APPENDIX A
Given the structure of the model, the various inventories of goods and
assets can be expressed in terms of ns, nd, n, and the proportion of goods
bought with currency, e.
Recall, that in the expressions below it is assumed
for simplicity that all goods lie along the continuum from 0 to 1 and that
the goods whose indices are an element of (0, h) are purchased with currency,
while those whose indices are an element of (h, 1) are purchased by check.
The average holdings of savings, demand deposits, currency, and goods are
given by equation (Ala)-(Ald), where p(i) is the proportion of the budget
spent on good i, T is the length of the payments period and Y is the flow of
transactions income over the period.
p(i)YT/(Number of shopping trips in each period).]
by z(i)
(Ala)
[Hence z(i) is related to p(i) and YT,
S
1/2YT[l - 1/nSI
(Alb)
[
2n 8
e
d
nd+l
(Alc)
C
(Ald)
C = 2n (n +T)n fO p(i)di =2n (nel)n
nnS~
0
s d
where e
e
2
|O
n(nd+l) [e -
1
1
n(nd+l) fi p(i)di]
nnd1
-J
p(i)di]
p(i)di represents the proportion of the budget spent using
currency and is therefore called the expenditure ratio.
An individual maximizes the profits, Tr, of managing his transactions
with respect to his choice variables.
Specifically he maximizes
35
r= r TS + r TD + r TC + r TG - a n -adn snd
ss
g
d
c
s
(A2)
(T +
-
A(h) + B(h))nS(nd+l)n
Using expressions (Ala)-(Ald),
a(i)di and B(h) = fh b(i)di.
where A(h) =
equation (2) may be rewritten in the following form.
YT
YT
s
= 1l/2r
(A2')
2 l/2YT~~2
(r-r)
n
s dd
2
1/2YT__
Y
2
1
1 /2YT
nS(nd+l)n (-r r p(i)di +
h d
nsndln
(a -ad)n - adcns(nd+l)
-
e(r-r
h
r p(i)di- r)
g
0 c
-
(T
+ A(h) + B(h))n (nd+l)n
The optimal values of the choice variables are derived in two steps.
First, for any given h, the optimal values of ns, nd+l, and n are expressed
as functions of h and the parameters of the model.
These solutions are then
substituted back into (A2'), resulting in an expression which depends on h.
The optimal value of h is then implied by the first order condition for
profit maximization. The solutions for n5 , nd+l, and n are given by
equations (A3a)-(A3c).
(A3a)
ns
T
(A3b)
nd+l
(A3c)
n
r(r -rd)
2(as- ad)
1
]
[e(rd-rc)(as-ad)
ad rc) -rd)]
[(e)(rd rg) + e(rC-rg))ad
e(T + A(h) + B(h))(rd-rc
36
As can be seen from the above solutions, the number of trips to the bank
is independent of h and therefore e.
This independence occurs because the
optimal average savings balance only depends on the relative yields between
savings and demand deposits and on the incremental cost of going to the bank
over the cost of cashing a check.
These considerations are removed from the
optimal manner in which goods are purchased. The number of check cashings is
sensitive to the manner in which goods are purchased, since the checking
account is used to control average currency holdings.
If a large portion of
income is spent using cash, then it will be more worthwhile to economize on
currency holdings by cashing more checks. Similarly the number of trips to
the store given by ns(nd+l)n =
[ (l--e)Y(rd-rg) + ieY(rc-rg) 1½
2(T + A(h) + B(h))
I
depends on h,
but it is not immediately clear in which direction the dependence lies.
It
can be shown that both the numerator and denominator are decreasing functions
of h.
Intuitively, since demand deposits earn a greater return than currency
th
one would only buy a particular good, say the i , with currency if a(i) < b(i).
Therefore in the neighborhood of the optimal h, as h increases A(h) increases
but by less than B(h) -decreases, so that the denominator falls.
It is clear
that the numerator is declining in e and therefore h.
For a given h, the transactions behavior of an individual with respect
to interest rates and transactions costs is straightforward. The greater
the relative yield on an asset, the higher will be the optimal holdings of
that asset.
This result is accomplished by making more trips to the asset.
Conversely, a higher transactions cost of using an asset will imply fewer
trips.
For example, a rise in rd (ord a fall in a~~~~d
) implies more check cashings.
Also, to further increase the holding of demand deposits more trips to the
bank and the store will be made.
With respect to goods, a rise in r
or an
increase in the transactions costs of purchasing goods will result in a rise
37
in desired goods inventories.
This is accomplished by purchasing goods
in larger amounts and, therefore, by making fewer trips to the store.
The necessary conditions for an individual to adhere to the above
inventory model are that ns > 2, nd+l > 1, and n > 1.
These imply the
following relationships.
(a -ad)
(A4a)
1/2YT2(r -r) >
(A4b)
e(rd-rc)(a -ad) > ad(rSr d)
(A4c)
[(l-e)(rd-rg) + e(rc-rg)]ad > e(T + A(h) + B(h))(rd-rc)
4
Equation (A4a) states that in order for one to use a savings account the
interest rate earnings on savings relative to demand deposits must outweigh the
increased transactions costs of making a trip to the bank relative to cashing
(A4b) and (A4c) have similar interpretations regarding the use of the
a check.
checking account as a store of value and the holding of positive quantities
of C and D respectively.
To examine the meaning of equations (A4a)-(A4b) more fully, a numerical
example is presented. For'a payments period of 2 weeks, a salary of $1000
every 2 weeks, and rs = .05 per year and rd ' .02 per year, (A4a) will only
hold if a
5
-
a
d
< 14.5 cents.
It may therefore be true that many individuals
use demand deposits as a store of value for transactions income.
however, would not be substantially changed.
(Ald), and the solution for nd+l = T [
in the model in the text.
ad
The model,
In this case ns = 1 in (Ala)]c
,
which is merely n (nd+l)
The solution for n is unchanged.
Finally, if there
doesn't exist a store of value, the optimal n equals the expressions for
ns(nd+l)n in the text.
The only consequence for the comparative statics would
be that more parameters would have no direct effect on the optimal value of h.
38
Substituting (A3a)-(A3c) into (Ala)-(Ald) yields expressions for average
asset holdings in terms of h.
These are given in (Ala')-(Ald').
[Y:§:r
(Ala')
S(h)
(Alb')
D(h) = 1/2 Y[[( Ys)
:
2
-r~
T ]½~ S-e [
d)r-c
=
1/2YT{1
T
_ _ ¶YF0
2
]½}
1
ad
[
(Alb'Ly(r2-rdc)]
(Ald')
G(h)
2(T+
= 1 /2Y
]J
- (l-.e)[((2!(T+A(h)
+ B(h))
(le(r-g
~Cr) ]½}
2(T + A(h) + B(h))
LY((l-e)(rr-re)
dgdc +e(r
cg
-r
9c
A(h) + B(h))
Y((l e)(rd-rg) + e(rC-rg))j
Also, substituting (A3a)-(A3c) into (A2') gives an expression for profits
which depends on h and the exogenous variables of the model.
(A5)
7Ch) = 1/2YT2 rs
s
T[2Y(rs-rd )(as-a)]
d
- T[(2 Y(l-e)(fd-rg) +
2
-
T[I2Ye(rd-red)a
Ye(rc-rg))(T + A(h) + B(h))]
This expression is then maximized with respect to h, and the resulting first
order necessary conditions yield an implicit function h.
This is given by
equation (A6)
(A6)
-l/2T[2Ye(rd-rc)a]
dcd
2
-
+ 1/2T
~
2Y[( + A(h) + B(h))
L(1-e)(r d-rg) + e(rc rg)
(1-e)Y(rd-r ) + 2eY(r -r )cg
T + A(h) + B(h)
-
- 1/2Td
L
½ k(h)
½
r9rph
(a(h) - b(h)) =0
J))
39
More intuitively (A61 can be expressed as,
(A7)
T (aD) rd+ T(-) rc +T (G-)r
ah/
\ah'
a
g
h-
d
a(n (nd+l)n)
sadh)
(T + A(h) + B(h)) - ns (nd+l)n(a(h) - b(h)) = 0
Equation (A7) states that the change in the interest rate earnings and
transactions cost is exactly balanced at the optimal h.
Using condition (A4c) it can be shown that the sum of the first two terms
in (A6) is negative.
This implies that a(h*) < b(h*) for profit maximization,
where h* denotes the optimal value of h.
to hold
V-
< 0)
,
Also, for the second order conditions
it is necessary for the derivative of b(h*), bt(h*) to
be sufficiently less than a'(h*) (see Appendix B).
Comparative Statics
Equation (A6) will now be used to examine how changes in transactions costs,
income, and interest rates affect h* and the expenditure ratio, j0 p(i)di.
Since
the relationship between h and the expenditure ratio is monotonic, the signs of
the effects will be the same.
Therefore, only changes in h* are analyzed.
It
is assumed that the conditions of the implicit function theorem are met so that
a function h = f(x), where x is a vector of parameters, exists in the neighborhood of h*.
Bxa
,will
The derivative of h with respect to any particular parameter,
be of the same sign as
axi
(h) evaluated at h*.
Therefore, the
latter expression is all that is derived.
First it is easy to see that changes in as, rs, and Y do not affect the
optimal method of purchase. However, this is only true to the extent that the
consumption bundle remains unchanged, or more explicitly that the distribution
of p(i) is invariant to movements in Y, a8 , and r
5
.
While this may be a
40
reasonable assumption regarding r5 , it is unlikely to be true for changes in
transactions income.
The qualitative effect of a change in the cost of cashing a check is
given by equation (A8).
/
a
d
j
-(T/2)(Y/2) [
)
ch
p(h*) <
0
Therefore, a decrease in the cost of cashing a check implies that more goods
will be bought with currency.
The comparative static effects for changes in the various interest rates
cannot unambiguously be signed, since they depend on the relative values of
interest rates and transactions costs.
For instance, equation (A9) gives the
results for a change in rg (inflation).
(A9)
a-=
;h-*
ar9
(Y){I
(rdCr)p(h*)
ITrdrcpC
2(2)
-
T
II
(b(h*) - a h*)
where for notational simplicity TC = T + A(h*) + B(h*) and
I1= (l-e*)(rd-rg) + e*(rcrA).
This expression is positive if
p(h*)(rd-rc ) > (b(h*) - a(h*))
for nd(n+1)n
TC .Using
sn the
h expression
xrsinfrn(i+~
d
yields equation (3) in the text.
Regarding a change in the own rate on currency, there are two effects
which support each other.
(AbO)
(AlO)a~r
a
( _ )
arc
a-h*
This can be observed by examining (Al0).
TY
( 2)
2Tj)
-
{2 [-
a-1
d
]
e(rdrc)I
c
-ie*p
(h*)(rd-r)
I
TC½
p~h*)
-
TCFp(h*)
pj*r..)Jph*h
-12
e*(b(h*)
+ ½T;*( h)-a(*
2 TCI
41
Using the first order condition (A6), to substitute for the first term in
brackets, the sign of equation (A10) can be reduced to the sign of the
following expression.
bh*) - a(ih*))
-
]1 +
P(W*)(rd-rc
dIc)]
An increase in r causes more trips for goods to be made.
If currency is
relatively cheap at the margin to transact with when compared to its weighted
opportunity cost, individuals will use more currency.
The response of h* to a change in rd also reflects two avenues.
On the
one hand a change in rd should have an opposite effect to that of rg, while
on the other it should have an opposite effect with respect to r , since D
and C are substitutes.
It can be shown that the substitution effect between
D and C dominates and that h* rises with rd when
p(h*)(rd-rc) > (b(h*) - a(h*)) TC
The outcome of a change in the fixed cost of going to the store,
T,
is similar to the results encountered for changes in various interest
rates.
Specifically an increase in T will cause h* to rise if
p(h*)(r -r ) > (b(h*) - a(h*))L .
d c
TC
This is the same condition encountered
for a change in r9.
Regarding C(h), comparative static effects will involve a number of
considerations. First, there will be direct effects on currency independent
of changes in transactions patterns.
Secondly, since transactions patterns
change when exogenous variables change, there will be an additional effect
on currency balances.
The change in currency holdings due to a change in
transactions patterns will depend on whether more or less goods are bought
with currency and whether currency balances are increasing or decreasing
42
around the optimal transactions pattern.
The shape of the currency function
C(h) can be ascertained by examining aC(h)/ah. This is given by
(All)
aC(h)
=
=h
p(h) C(h)
1/2
27 C-ih)c
It is easy to see that at h
Y I)
TC p~h) + e(a(h) - b(h))
e TC P(h)(rd-r)1
2T
0 this derivative is positive. Also, using the
F.O.C. from (A6), the derivative evaluated at h* can be expressed as
(A12)
aCh)
I
I
1/2
ah
)
(Y-q½
2-TC
b~h*) -ra(h*)
L
-
-TC
p1h)1
d
I
j
I
--
which is positive if p(h*)(rd-r) < I(b(h*) - a(h*)) or if
(l/T)(Y/2)½p(h*)(rd-rc) < (b(h*)
-
aGh*))ns(nd+l)n.
This turns out to be the empirically relevant case in the text and therefore
in the neighborhood of the optimal expenditure ratio an increase in the
proportion of goods purchased with currency will cause average currency
balances to rise.
The comparative static effects caused by changes in the exogenous
variables on C(h*) are
to show that
aC(h*)/aad
>
dC(h*)/dx =
%
c(h*) +C(h*) ah
~~ax
+ a**
ax
.
It is easy
aC(h*)/arc > 0, 3C~h*)/ard < 0, ac(h*)/arg < 0,
0, and
aC(h*)/aTC > 0.
The empirical results on e, which imply that column 2 of table 2 is
relevant, also imply that all indirect effects reinforce the direct
effects that changes in exogenous variables have on currency holdings.
.4
43
APPENDIX B
Derivation of 2nd order condition.
Then 2nd order condition is expressed in equation (Bi), where a prime over a
variable indicates its derivative with respect to h.
For notational simplicity,
the symbol TC = t + A(h) + B(h), and the symbol I = (l-p)(rd-rg) + p(rc-rg).
a2T
T(Y~k
2
(Bl)
a-h2
rd
LjJ
p(h)(rd-rc)(b(h)
-
l/27e
a(h))
(I )(TC)
dd
pCh)2 [ad(rd-r)I½;
(TC) p(h)(rdrc)
(I)
Pr
r)[ TC 1
p(h)(rd-r)(b(h) - a(h))
- p'(h)(rd-rc) L i
+ 1/2 -(I)(TC)½
-
1
/
2
(b(h) - a(h))2
[C] (b'(h) - a (h))]
(Bi) will be less than zero if the expression in brackets is greater than zero.
Using the first order condition to eliminate the first term in brackets, and
performing the necessary algebra reduces the bracketed term to the following
expression.
44
p(h) 2ad(rd-rc)]
+ 1/2
+
[
pCh) L
-)
~p
Fb~~~i
a(h)I
b(h)-
TC ~~~
r
2.,F
½ p'(h)1
(b(h) - a(h)) -
-1/2e
-
p h)
(b'(h)
41h)
-a'(h))]
)-½ (h)(rd
-ri
b(h) - a(h)
c-c p(hi)(r r )TC½(l
JTC
J
The last term is composed of two terms of different sign and is therefore
negative as is the first term.
be sufficiently positive, or
sufficiently positive.
For profit maximization the second term must
b(h) - a(h)
-
Ph) (b'(h)
p' (h)
-
a'(h)) must be
This reduces to (b(h) - a(h))P'(h) > b'(h) - a'(h),
p(h)
which is certainly true if b'(h) < a'(h).
If goods are of the same weight
around the optimal h*, then b'(h*) < a'(h*) is required.
@FedFRASER