Reserve Requirements and the Inflation Tax
Author(s): Philip L. Brock
Source: Journal of Money, Credit and Banking, Vol. 21, No. 1 (Feb., 1989), pp. 106-121
Published by: Blackwell Publishing
Stable URL: http://www.jstor.org/stable/1992581
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PHILIP L. BROCK
ReserveRequirementsand the InflationTax
ALTHOUGH
MOST
MODELS
OFINFLATIONARY
FINANCE
consider
only the inflation tax on currency (fiat money) governments typically levy the
inflationtax on non-interest-bearingrequiredreservesof the bankingsystemas well
as on currencyheld by the public. Two recent papers by Walsh (1984) and Romer
(1985) have provided generalequilibriumframeworksthat are potentially capable
of characterizinginflationary Elnanceon currency and bank deposits. Since the
reserve requirementacts as a tax on the intermediationservices performed by a
bank,any analysisof the reserverequirementmust specifythe specialcharacteristics
of banks' liabilities or assets that allow a positive reserve ratio to coexist with a
nonzero level of banking activity. Walsh places bank deposits and currencyinto a
cash-in-advanceconstraint,while Romer places currencyand deposits directlyinto
a representativeagent's utility function. Both formulationsthereforemotivate the
existence of the banking sector by stressingthe characteristicsof bank deposits that
differ from the characteristicsof liabilities of other nonbank Elnancialintermediaries.l
This paper incorporatesreserverequirementsinto a general equilibriumsetting
along the lines of Walsh and Romer. However, rather than specify a cash-inadvance constraintor place money and deposits into a representativeagentssutility
The author thanks Alvaro Donoso, Kent Kimbrough,and two anonymous refereesof this Journsl
for helpful comments on an earlierversion of this paper.Jinbang Kim and Ying Liu provided valuable
research assistance. This research has been partially supported by a research fellowship from the
FulbrightCommission and by a grantto the economics departmentof Duke Universityfrom the Sloan
Foundation.
I Williamson (1986) provides an alternative framework in which the general equilibriumimpact of
reserve requirementscan be assessed by stressing the role of the banking system in monitoring debt
contracts when the investment process is characterizedby asymmetric information.
PHILIP
L. BRocKis assistantprofessor of economics, Duke University.
Journal °+Money, Credit,and Banking,,Vol. 21 No. 1 (February 1989)
Copyright i' 1989by the Ohio State University Press
PHILIP L. BROCK : 107
function, this paper introduces banks into the model by specifyinga transactions
technology in which currencyand bank deposits permitagents to economize on the
amount of time spent on transactingin the goods market. The use of a 'shoppingtime" technology to motivate the demand for currencyhas been used recently by
Drazen (1979), Fischer (1983), McCallum (1983), and Kimbrough (1986). This
papergeneralizesthe currency-basedshopping technology by specifyinga transactions technology that is convex in currencyand demand deposits so that currency
and demand deposits are substitutesin allowing the representativeagent to economize on time spent transacting.Since many standardissues of inflationaryfinance
are precludedby the unitaryvelocity of money impliedby standardcash-in-advance
constraints, the use of a technology that conserves on transactingtime is a useful
way to motivate the demand for money without imposing a unitary velocity
restriction.The transactionstechnology of the paperis also consistentwith Romer's
money-andZeposits-in-the-utility-functionmodel, provided that all transactions
time comes out of leisure.In the moregeneralcase consideredin this paperthe use of
the inflation tax can also have output effects since an increase in time spent on
transactingwill generallycause the agent to reduce time spent producing.
As shown in Section 1, the transactionstechnology specifiedin this paperhas the
attractive property that utility-maximizingbehavior generates standard asset demand functions in which the demands for money and deposits depend on the
opportunity cost of holding each asset and on the level of consumption. Section 2
uses the assetdemandfunctionsto develop the characteristicsof inflationaryfinance
in the model. The analysis shows that the traditional use of the semielasticityof
demand for real money balances with respect to the nominal interest rate to
calculate the revenue-maximizingnominal interest rate may seriously underestimate the true revenue-maximizingrate. Section 3 develops the welfareconsequences of the use of the reserve ratio and nominal interest rate. The section demonstrates that a monetary authority that minimizes the welfare cost of inflationary
finance will generally alter both the reserveratio and the nominal interest rate in
response to changes in revenue requirementsfrom the inflation tax. Section 4
concludes.
1. THE MODEL
The economy is inhabited by a combined representativeagent and bank whose
preferencesover consumption (c) and leisure (t) are given by the concave utility
function U(c, t). Consumption and leisure are both assumed to be normal goods.
The agent produces a consumption good using a production technology that is
linear in labor inputs (n). In order to purchase the consumption good, the agent
must engage in shopping activity (s). The time spent shopping is reducedby the use
of a twiceZifferentiabletransactionstechnology that is convex in real balances of
currency (m) and demand deposits (d):s = ¢(m, d)c, where Xms Xd < 0; Xmm,
f dd Xmd >
Currencyand demand deposits are assumed to be normal goods,
o.2
2Thisform of the transactionstechnology implies that currencyand demand deposits are substitutes
in conducting transactionsand is based on the cash-and-deposits-in-advanceconstraint used by Walsh.
108 : MONEY,CREDIT,AND BANKING
< O. The total time spent
< 0 XmXdd-XdXmd
implying that XdXmm-XmXmd
producing, shopping, and enjoying leisure sums to one. Currency pays a zero
nominal interestrate while demand deposits pay the nominal interestrate id.
The government finances lump-sum transfers (z) to the agent by levying the
+ Td, where T iS the legal reserve
inflation tax on the real monetary base, h-m
requirementon demand deposits. Legal reserves,like currency,pay a zero nominal
interestrate. The governmentwill be assumedto follow a time-consistentmonetary
policy that preventsprice leveljumps. As Auernheimer(1974) first showed, such a
policy impliesthat steady-staterevenuefrom the inflationtax will equal i(m + Td),
where i is the nominal interestrate.
Becausethereare no inherentdynamicissues connected with the agent'smaximization problem, the paper will consider only a steady-statecharacterizationof the
agent'sproblem. In the steady state the economy'srealinterestrateis determinedby
the agent's rate of time preferenceso that the monetary authority determinesthe
nominal interest rate. Bank loans pay the nominal rate of interest. By assuming
competitiveand costless banking,the nominal rate on deposits will equal one minus
the reserveratio times the nominal interestrate so that the differencebetween the
nominalinterestrateand the demanddeposit rateis the tax wedge i T on intermediation servicesof the bank.
Equation (1) gives the agent's maximization problem, where the agent's budget
constraint incorporates inflation tax payments and lump-sum transfersfrom the
government:
t
= Max{U(c, t) + A[1-t-+(m,
c,
d)c-c-i(m
+ Td) + z]} . (1)
t
m, d
The agent'smaximization problem given in (1) producesthe first-orderconditions
that are given in equations (2) through (4):
Uc(c, t) = [1 + +(m, d)]U(c, t)
-Xm(m,
d)c = i
-Xd(ms
d)c =
(2)
(3)
iT =
i -
i
.
(4)
Total differentiationof equations (3) and (4), while holding consumption constant,
generatesthe following asset demandequations for currencyand demanddeposits:
m =m(i,
c)
i-id,
-
+
+
(5)
d = d(i,
+
i-id,
-
C),
+
PHILIPL. BROCK : 109
dl = Xmd/
+ Xdfmd)/D;
=
¢>md/D, m3 = (-Xmfdd
m2
where ml = -Xdd/D,
C(¢mmdd-Wnd).
D
=
and
d3 = (-XdXmm + Xm¢md)/D
D, d2 =-Xmm/D
The asset demand functions show that the demands for currency and demand
deposits depend negatively on the opportunity cost of holding each asset and
positivelyon the opportunitycost of holding the otherasset. Asset demandsarealso
a positive function of the level of consumption.3
2. REVENUE FROM THE INFLATION TAX
Assume that the governmentwishes to calculatethe inflation rateand the reserve
ratio that will maximize the government'sinflation tax revenue:
R(i, T) = ah(i, r) = i(m + rd) .
(6)
The governmenthas two tax instruments,the nominal interestrate and the reserve
ratio, to determinethe opportunitycost of holding currency(i) and the opportunity
cost of holding demanddeposits (i-id). Revenuemaximizationwith respectto the
nominal interestrate, while holding the reserveratio constant, producesthe following Elrst-ordercondition:
i=-
m+Td
m, + m2r+ m3(d-1)+ r[dl + d2r+ d3(di)]
__
h
dh
(7
Equation (7) indicatesthat a revenue-maximizinggovernmentwill set the elasticity
of demand for the monetary base [-(i/ h)(dh/ di)] equal to one.4
Revenuemaximizationwith respectto the reserveratio is equivalentto maximizing the size of the monetarybase for any given interestrate, the condition for which
is shown in (8):
dr
m2i+ m3(d-T) +
Td2i + Td30
+
d= O
(8)
Equations(7) and (8) togetherdeterminethe revenue-maximizingcombinationof
the reserveratio and nominal interestrate. Using these two conditions, the revenuemaximizingelasticityconditions for the demandfor currency(m) and currencyplus
demand deposits (m 1) with respectto the nominal interestrate can be expressedas
in (9) 5
3Although consumption is taken as exogenous in the asset demandfunctions of (5), by making use of
all of the first-orderconditions for the agent's maximization problem, it can be shown that an increase
in the nominal interest rate or the reserveratio will lower consumption, given that consumption and
leisureare normal goods and providedthat shopping time is nondecreasingin the level of consumption;
i.e., that dsl dc = 8+(m, d)cl dc > O.
4Friedman(1971) mentions but does not analyze the revenue-maximizingcondition of equation (7)
while Calvo and Fernandez (1983) analyze the special case of (7) with currencyset equal to zero.
SThe elasticity expressions in (9) are based on the derivativesof the first two argumentsof the asset
110 : MONEY,CREDIT,AND BANKING
^9ma= 1 +
I
m [7RaT nac(72ci
- 71cr)]
h [I
where Rqmi=--(ml
h dr
n nhc =
+
h dc
T t7hT+ 77hc(71ci +
m2T)
h 71ct
=-
77ml,i =-ml(Ml
c di
t
?cr =-
r
a)]
+ m2T + dl + d2T), 6hr =
c d .Theelasticityexpressions
in (9) are gross elasticitiesthat take into account the total effect of a change in the
nominal interest rate on the demand for money, including the cost of financial
intermediation.The two expressionsin (9) indicatethat when the revenue-maximizing reserve ratio is chosen, the elasticity of demand for the monetary base with
respect to the reserveratio will equal zero [from equation (8)] so that the government will set the elasticities of demand for currency and currency plus demand
deposits (m 1) greaterthan or less than one dependingon the relativemagnitudesof
the elasticities of consumption with respect to the nominal interest rate and the
reserveratio.
If thereare no real output effects of the inflation tax that is, if all shopping time
comes out of leisure then the usual resultholds at the point of revenuemaximization: the monetary authority will set the elasticity of demand for currencyor for
currencyplus demand deposits with respect to the nominal interest rate equal to
one.6 If, on the other hand, the monetary authority is constrained to require a
reserveratio less than the revenue-maximizingratio, the elasticityof demandfor the
monetarybase with respectto the reserveratio will be positive, so that a constrained
revenue-maximizingmonetary authority will set the elasticity of demand for currencywith respectto the nominal interestrategreaterthan one at the same time that
the elasticity of demand for currency plus demand deposits with respect to the
nominal interestrate is set less than one.
The intuition behind the constraineduse of the nominal interestrate to generate
inflation tax revenueis straightforward.With a reserve-ratioconstraint,the monetaryauthorityattemptsto set two tax rates(on currencyand demand deposits)with
one tax instrument. Without the use of the reserveratio, the monetary authority
may wish to raise the inflation rate to a level greaterthan the revenue-maximizing
ratefor currencyif the loss of revenueis less than the incrementalrevenuegenerated
from the inflation tax on requiredreserves.Full control over both the reserveratio
demand functions with respect to the nominal interest rate. This convention was adopted because
empiricalestimates of the revenue-maximizinginflation ratealways express the demand for money as a
function of the nominal interestrate (or expected inflation rate)and consumption (or output), without
taking into account the additional effect of the nominal interestrate on consumption. To determinethe
total revenue-maximizingelasticities in (9), the expressions T1meT1cland T1ml, cT1czshould be added to the
right-handsides of the elasticity conditions.
6Siegel (1981) arrived at the same condition in a partial equilibriumsetting. Siegel's results correspond (in this paper'smodel) to a specificationof a separabletransactionstechnology (so that jtt)md = O)
with transactionstime coming entirely out of leisure.
PHILIPL. BROCK : 111
and nominal interestrate eliminatesthe need to tax currencyat a rate greaterthan
the revenue-maximizingrate, since the asset substitutionaway from currencyand
toward demand deposits that accompanies a higher nominal interest rate can be
offset by the use of a higher reserveratio.7
Many empirical studies on hyperinflations,beginning with Cagan (1956), have
reliedon estimates of the demand for currencyor currencyplus demanddeposits to
calculate revenue-maximizinginflation rates. The estimated revenue-maximizing
inflation rates are uniformly lower than observed inflation rates. The elasticity
expressionsin (9) of the total effect of an increasein the nominal interestrate on the
demandfor money suggestthat even accurateempiricalestimatesof the demandfor
money will not generally produce accurate estimates of the revenue-maximizing
inflation rate. For example, let the transactions cost function take the following
form:
+ ;3 (!n d-,80-1)
¢(m, d) = m (!n m-alO-1)
+ K'
(10)
where O < m, d < 1, and K is a positive constant that is large enough to make
¢(m, d) positive. By makinguse of the asset demandfunctions given in equation (5)
it is easy to show that (10) impliesthat asset demandscan be writtenas tn m = ogOHence, the semielasticityof the demand for curai and ln d = /30-,8(i-id).
rencywith respectto the nominal interestrate is equal to ceand the semielasticityof
the demand for deposits with respect to the opportunity cost of holding deposits
(i-id) is equal to ,8.
With the transactionstechnology given by (10), the set of points deElnedby the
Elrst-orderrevenue-maximizingcondition with respectto the reserveratio given in
(8) becomes the condition ir = 1/ ,8. The set of points defined by the first-order
revenue-maximizingcondition with respectto the nominal interestrate given in (9)
becomes the condition
7The observation that constrained revenue maximization results in different elasticity conditions
than unconstrainedrevenuemaximization can be extended to regimesin which a reserverequirementis
also placed on time deposits, perhapsbecause the liquidity of time deposits allows them to enter into the
transactions technology via arrangements such as the automatic transfer of savings into checking
accounts. Setting aside output effects of the inflation tax, the relevant set of elasticity conditions for
revenue maximization with two reserve requirements is the following, where m 1 is currency plus
demand deposits, m2 is m 1 plus time deposits, 8 is the reserve ratio on time deposits, and h is the
monetary base that is composed of currency plus required reserves on demand deposits and time
deposits:
= 1+ m
71ml
n X = 1+
r1m2,, =
1
[tthr + T1h@]
h
[- I
h rl-r
m2 L
T t1h + T1hG]
1-8
nhr +
49
1
tS1@2
These elasticity expressions suggest that the calculation or estimation of the revenue-maximizing
nominal interest rate is very sensitive to the assumption that the government is also jointly using
requiredreserve ratios on different classes of bank deposits to maximize revenue.
112 : MONEY,CREDIT,AND BANKING
I =-+_
1
pTd
-1
;
-
.1
IT] .
(11)
When the reserveratio is zero, the revenue-maximizingnominal interest rate will
equal 1/a. At the point of unconstrainedrevenuemaximization(where iT = 1 / wf)
the revenue-maximizingnominal interestrate will also equal 1/ cxand the revenuemaximizingreserveratio will equal a/,B. If the monetaryauthorityis constrainedto
use the reserveratio at a level less than a/,B, constrainedrevenuemaximizationwill
cause the monetary authority to set the nominal interestrate greaterthan 1/ CY.
The revenue-maximizationproblemcan be portrayedgraphicallyby drawingthe
two Elrst-orderrevenue-maximizingconditions as functions of the two revenuegeneratingpolicy instruments.The transactionstechnology speciEledin (10) implies
that the set of points formed by the first-orderrevenue-maximizingcondition with
respect to the reserveratio is a hyperbola and that the set of points formed by the
Elrst-order
revenue-maximizingcondition with respectto the nominalinterestrateis
concave with respect to the reserveratio.
The Appendix shows that this characterizationof the Elrst-orderconditions for
revenuemaximization also holds wheneverthe transactionscost function is quadraticand transactionstime comes out of leisure.8In the more generalcase derivedin
the Appendix, the verticalasymptote for the Elrst-order
revenue-maximizingcondition with respect to the reserveratio is the line defined by T = ¢>md/ Xmm. Figure 1
presentsthe graphicalsolution to the revenuemaximizationproblemas the point of
intersection R* of the curves defined by the two first-orderconditions dR / di = O
and dR/dT = 0.9
Bombergerand Makinen(1983)have alreadyverballymade much the same point
as is made by Figure 1 in theiranalysis of the Hungarianhyperinflationof 194546.
They provide evidence that it was the existence of partiallyindexed bank deposits
(tax pengo accounts) that producedthe extremeseverityof the hyperinflation.Since
indexing of tax pengo accounts was lagged from one to two days, the inflation rate
was raised to a level much greaterthan the revenue-maximizingrate for currency
alone in order to tax the partiallyindexed bank accounts. Revenue maximization
that is constrainedby partialindexation of bank accounts is analyticallyequivalent
to revenuemaximizationwith a less-than-revenue-maximizing
reserverequirement.
In both cases the constrainedmaximization of revenuewill resultin the elasticityof
the demand for currency set at a value that is greater than the unconstrained
revenue-maximizingelasticity.
Bombergerand Makinen(1983)indicatedthat the constraintimposed on revenue
8The assumption that all transactions time comes of leisure simplifies the analytical derivation of
Figure 1. Includingthe negative output effect of the inflation tax servesprimarilyto shift the dRI di = O
schedule downward and the dRI dT = Ocurve inward.
9Although the transactionstechnology given by (10) implies that the revenue-maximizingnominal
interest rate is equal to 11cYboth at a zero reserve ratio and at the reserveratio T*, Figure 1 shows the
more general case in which the semielasticity of the demand for currency with respect to the nominal
interestratediffersat the two reserveratios. In Figure 1the dRI di = Oschedulereachesa peak between
a zero reserveratio and the revenue-maximizingreserveratio (T*). This resultholds for both the semilog
asset demand example and for the general quadraticcase discussed in the Appendix but need not hold
for all transactions cost functions.
\
PHILIPL. BROCK : 113
xtr
l
\
:
bR=o
\<
I
\8T
I
\
I
\
m'0
f md
fmm
T
T
FIG. 1. Revenuefromthe InflationTaxas a Functionof the ReserveRatioandNominalInterestRate
maximizationfrom the inflation tax in Hungarywas relatedto governmentresistance to making large reparations payments imposed by the Soviet Union after
WorldWar II. In other countriesit may be the case that monetaryauthoritiesface
more bindinglegal constraintsin the discretionaryuse of reserverequirementsthan
in varying the rate of growth of the monetary base. Such legal constraints in the
United States currentlylimit the Fed to an 18 percent ceiling on reserve requirements on checkabledeposits and to a 9 percentceiling on time and savingsdeposits.
Tamagna(1965) has also documented the widespreadincorporationof upper legal
limits on reserverequirementsin a number of Latin-Americancountries. In some
cases, the constraintthat the reserveratio cannot exceed one may also be binding,as
will be the case if a exceeds ,8 for the transactions technology given in (10). If an
administration is constrained in its use of legal reserve requirementson bank
deposits, this section'sresultsemphasizethat calculatedrevenue-maximizinginflation ratesthat relyon estimatesof the demandfor currencyor currencyplus demand
deposits as a function of expected inflation or the nominal interestrate may poorly
approximatethe constrainedrevenue-maximizinginflation rates.
114 : MONEY,CREDIT,AND BANKING
3. CONSTRAINED
WELFAREMAXIMIZATION
In general,governmentsthat rely on the inflationtax do not attemptto maximize
revenue. Given that a governmentmust raise a certainamount of revenuewith the
inflation tax, this section derivesthe optimal use of the reserverequirementand the
inflation rate. If the level of lump-sum transfersis less than the maximum revenue
that can be generatedfrom the inflationtax, the governmentwill be able to generate
the required revenue with a wide range of combinations of inflation rates and
reserveratios. For any revenuerequirementthat is less than the maximum revenue,
the locus of points found by setting dR = (8R/di)di + (8R/8T)dT
= O will
define an iso-revenue curve. In addition, for any given level of utility of the
representativeagent, the locus of points found by setting dU=(dU/di)di+
(d U/ AT)dT= 0 will define an iso-welfarecurve facing the government.
Optimal combinations of the reserve ratio and the inflation rate will occur at
points of tangency between the iso-revenue and the iso-welfare curves. At these
points, the Appendix shows that the elasticity of the demand for currency with
respectto the nominal interestrate is the following:
m [(
where 71ui=--U
i
8 U
ai
t7Ur) t1hT
i t/Ur =--U
T
t7hc(tlci
8
U
ar t and
rlCT)]
t/Ui
rlur
*
(
1 +
12)
XmXddXdXmd
T(+dXmm-XmXmd)
> 1 by the assumption that money and demand deposits are normal goods. A
comparison of (12) and (9) shows that a welfare-maximizinggovernment that is
unconstrainedin its use of the reserveratio and nominal interestratewill always set
the gross elasticityof demand for currencywith respectto the interestrateto a value
that is less than one, even though the revenue-maximizingelasticitymay exceed one
at the chosen reserveratio. Equations (9) and (12) together also indicate that the
only condition underwhich a welfare-maximizinggovernmentwill set the elasticity
of demandfor currencygreaterthan one is if the governmentis constrainedin its use
of the reserverequirement.
The constant-revenuetrade-offbetweenthe reserveratioand the nominalinterest
rate is shown in Figure 1 by the five downward-slopingiso-revenuecurves labeled
Ro through R4. The curves are vertical where they cross the dR/di = 0 schedule
and horizontal wherethey cross the dR/ aT = 0 schedule.The government'stradeoff between the nominal interestrate and reserveratio combinationsthat maintain
the agent at a constant level of welfare is representedby the downward-sloping
curves labeled Uothrough U4. As the Appendix shows, these curveshave the slope
i(¢dXmm - XmXmd) / [T(+dXmm - XmXmd) + (¢mXdd - XdXmd)] < O by the
normalityassumption on currencyand demand deposits.
The set of welfare-maximizingcombinations of the reserveratio and the nominal
interestrate, given the government'srevenueconstant, connect the horizontalaxis
and the revenue-maximizingpoint R* via the tangency points eOthrough e4. The
tangency points are the graphicalrepresentationof equation (12). The slope of the
PHILIPL. BROCK : 115
locus of tangency points will depend on the functional form of the transactions
technology. With the transactionstechnology given by equation (10), it is straightforward to show that the locus of tangency points is vertical. In an economy
characterizedby such a transactionstechnology,a monetaryauthoritythat attempts
to minimize the welfarecost of inflationaryElnancewill choose the unique reserve
ratio T = 0g/: and vary the inflation rate as revenue requirementschange. An
alternativetransactionstechnology is the following: +(m, d) = m + d V.With
such a technology the slope of the set of tangency points between the iso-revenue
and the iso-welfarecurveswill equal i v / (,u-v) at the point R *. Consequently,the
upward-slopinglocus of points shown in Figure 1 would correspondto the condition , > v for this transactionstechnology.
If monetary authoritiesdo attempt to minimize the welfarecosts of inflationary
Elnancefor any given revenuerequirement,the analysis of this section suggeststhat
there may be an observable systematic correlation between movements in the
nominal interestrate and movements in the reserveratio for countries that rely on
the inflation tax. Thereare some regionsof the world wheregovernmentsdo relyon
inflationtax revenueand wheremonetarypolicy is characterizedby an active use of
the reserveratio. For instance,in LatinAmericavirtuallyall governmentsmodified
their central bank chartersduring and after the Depression to permit flexibility in
the use of reserverequirements.By the early 1970smany countriesin Latin America
had set legal reserve requirementson demand deposits at levels that exceeded 30
percent and frequently modiEledthe reserve requirementsin response to Elscal
Elnancing requirernents.l°
Colombia, for example, altered the legal reserve ratio on demand deposits
seventy-fourtimes between 1952 and 1974, gradually raising the base level of the
reserveratio from 14 percentat the start of the period to 41 percentby mid-1974.
The basic legal reserveratio on demand deposits was frequentlysupplementedby
marginalreserveratios for new deposits that ranged between 40 and 100 percent.
Substantialincreasesin the reserveratio were associated with exchange rate crises
(from 14 to 23 percentduring 1957-58, from 18 to 27 percentduring 1962-63, and
from 23 to 34 percent during 1967-68) as well as with a period of expansionary
governmentexpenditureprograms(from 31 to 41 percentduring 1973-74).1l
To determine whether the use of reserverequirementsin Latin America and in
other parts of the world is systematically related to the inflation rate, Table 1
computes correlations between the average reserveratio on bank deposits and the
inflation rate for forty-one countries during the period 1960-84.12lThe IMFs
l°Tamagna (1965) contains a detailed description of the legislation in each country that created
flexible reserverequirements,as well as a thorough discussion of the uses to which reserverequirements
were put.
l l The information on changes in the legal reserveratio can be found in the Colombian centralbank's
Revista del Banco de la Republica, various issues. Diaz Alejandro (1976) is a good source for a
description of the exchange rate crises and fiscal policies.
I2Inflationrateswere used becauseof the absence of informationon nominal interestrates.Thereis a
potential pool of 134 countries in the IMF that could have been studied. To reduce the number to a
manageable size, countries were chosen that possessed at least twenty years of data and had at least ten
million people. The two criteria are arbitrary and would need to be relaxed in a more thorough
investigation of the link between fiscal variables and the reserve ratio.
*
c-
z
.
...
..
n
t
.
.
TABLE 1
INTERNATIONAL CORRELATIONS OF THE INFLATION RATE AND RESERVE RATIO
Country
Argentina (60-84)
Brazil (60-84)
Chile (60-84)
Colombia (60-84)
Mexico (60-84)
Peru (60-84)
Venezuela(60-84)
Country
Egypt (60-84)
Ethiopia (61-83)
Ghana (60-83)
Madagascar(65-84)
Morocco (60-84)
Nigeria (60-84)
South Africa (60-84)
Sudan (60-83)
Zaire (64-84)
Latin America
Average Reserve Ratio
Inflation
Mean
(Std. Dev.)
Mean
(Std. Dev.)
.50
.35
.33
.29
.35
.44
.23
(.53)
(.08)
(.16)
(.12)
(.19)
(.15)
(-03)
1.16
.56
.84
.17
.18
.31
.06
(1.52)
(.44)
(1.32)
(.09)
(.24)
(.33)
(.06)
Africa
Average Reserve Ratio
, Inflation
Mean
(Std. Dev.)
Mean
(Std. Dev.)
.27
.23
.32
.05
.06
.17
.07
.32
.44
(.07)
(.12)
(.13)
(.05)
(.02)
(-09)
(.02)
(.11)
(.14)
.08
.06
.26
.10
.06
.12
.08
.12
.38
(.06)
(.07)
(.30)
(.09)
(.05)
(.11)
(.05)
(.12)
(.28)
Correlation
(Signif. Prob.)
.42*
-.30
.41*
.47*
.57**
.53**
-.18
(.0363)
(.1418)
(.0444)
(.0191)
(.0028)
(.0067)
(.3763)
Correlation
(Signif. Prob.)
.04
.45*
.73**
.61**
-.46*
54**
-.68**
.41*
-.05
(.8460)
(.0315)
(.0001)
(.0043)
(.0218)
(.0051)
(.0002)
(.0464)
(.8427)
Correlation
(Signif. Prob.)
.66**
.16
.32
.02
.10
-.21
.21
.43*
-.20
-40
(.0005)
(.4314)
(.1498)
(.9271)
(.6461)
(.3257)
(.3057)
(.0325)
(.3401)
(.0551)
Correlation
(Signif. Prob.)
.02
-.50**
.35
.58**
.30
(.9299)
(.0107)
(.0911)
(.0023)
(.1436)
Correlation
(Signif. Prob.)
-.59**
-.33
-.66**
.05
.10
.78**
.60**
-.48**
-.25
-.49**
(.0017)
(.1087)
(.0004)
(.8004)
(.6288)
(.0001)
(.0017)
(.0146)
(.2279)
(.0125)
Asia
Country
Burma (60-83)
India (60-84)
Indonesia (62-84)
Korea (60-84)
Malaysia (61-84)
Nepal (60-83)
Pakistan (60-84)
Philippines (60-84)
Sri Lanka (60-84)
Thailand (60-83)
Country
Greece (60-84)
Portugal (60-84)
Spain (60-84)
Turkey (60-84)
Yugoslavia (60-84)
Country
Australia (60-84)
Belgium (60-84)
Canada (60-84)
France (60-84)
Germany (60-84)
Italy (60-84)
Japan (60-84)
Netherlands (60-84)
United Kingdom (60-84)
United States (60-84)
Average Reserve Ratio
Mean
(Std. Dev.)
.42
.09
.33
.20
.10
.31
.12
.13
.17
.10
(.40)
(.02)
(.17)
(.08)
(.02)
(-13)
(.02)
(.03)
(.03)
( 04)
Mean
.07
.08
1.06
.14
.04
.07
.08
.11
.07
.06
Inflation
(Std. Dev.)
(.11)
(.07)
(2.53)
(.08)
(.04)
(.08)
(.06)
(.11)
(-07)
(.06)
Middle-Income Europe
Average Reserve Ratio
Inflation
Mean
(Std. Dev.)
Mean
(Std. Dev.)
.12
.15
.08
.29
.29
(.04)
(-05)
(.06)
(.08)
(.09)
.10
.13
.11
.21
.19
(-09)
(.09)
(-06)
(.25)
(.13)
IndustrializedCountries
Average Reserve Ratio
Inflation
Mean
(Std. Dev.)
Mean
(Std. Dev.)
.10
.02
.06
.04
.1 1
.12
.03
.01
.07
.08
(.03)
(.01)
(.02)
(.02)
(.02)
(.03)
(.01)
(.01)
(.03)
(.02)
.07
.05
.06
.07
.04
.10
.07
.06
.08
.05
(.05)
(.03)
(.04)
(.04)
(.02)
(.06)
(-°S)
(.02)
(.06)
(.04)
*Significant at the .05 Ievel.
**Significantat the .05 Ievel.
Source: International Finance Statistics Yearbook, 1985, International Financial Statistics, April 1986. Inflation rates are given at the
beginningof the IFS Yearbookand measurethe proportionalchange in the consumerpriceindex from midyearto midyear. Reserveratios
are calculated by thefollowing formula: ( 14-14a)/(34 + 35-14a), wherethe numbersrefertothe lFSline numbersforthe mo eta y
base ( 14), currency outside of banks ( 14a), money (34), and quasi money (35). Reserve ratios are end-of-year figures.
n r
PHILIPL. BROCK : 117
International Financial Statistics allows the computation of the average reserve
ratio on demand and time deposits but does not contain information on legal
reserveratios. For the computed averagereserveratios, standardmoney multiplier
theory predictsthat fixed legal reserveratioson demandand time depositswill cause
the average reserve ratio to decline with an increase in the interest rate. Positive
correlationsbetween the average reserveratio and the inflation rate will therefore
reflect monetary policies that actively raise reserve requirementswhen inflation
rates rise.l3
Table 1 reportsinformation on the inflation rate, average reserveratio, and the
correlation between the two variables over the period from 1960-1984. Table 1
shows that statisticallysignificantpositivecorrelationsbetweenthe reserveratioand
the inflation rate occur in five of the seven Latin-Americancountries,in five of the
nine African countries, in two of the ten Asian countries (Burma and the Philippines), in one of the five middle-incomeEuropeancountries(Turkey),and in two of
the ten industrializedcountries (Japan and Italy). Statisticallysignificantnegative
correlationsoccurredin none of the Latin-Americancountries,in two of the African
countries(South Africaand Morocco), in none of the Asian countries,in one of the
middle-income European countries (Portugal), and in four of the industrialized
countries.
One conclusion that emerges from Table 1 is that systematic increases in the
reserveratio accompany increasesin the inflation ratein Latin Americaand Africa.
The reserve ratio appears to be less actively used in Asia and in industrialized
countries,with the observednegativecorrelationsbetweenthe inflationrateand the
averagereserveratio partlyreflectingchanges in the money multiplier.The empirically observedcorrelationsbetweenthe reserveratio and the inflationratein Table 1
may reflect attempts by monetary authorities to minimize the welfare cost of
inflationaryfinance, given that a certain amount of revenuemust be raised by the
inflation tax. However, it is worth noting that many developing countries do not
possesscompetitivebankingsystemsof the sort modeled in this paper.To the extent
that an oligopolistic banking structure or government monopoly control of the
'3The average reserveratio can be expressed as the following, where T iS the legal reserveratio on
demand deposits, 0 is the legal reserveratio on time deposits, t is the ratio of time deposits to demand
deposits, and e is the ratio of excess reservesto demand deposits:
T + at + e
Tivg =
1 + t
Differentiation of the average reserve ratio with respect to the nominal interest rate shows that the
correlation between the reserveratio and the interest rate depends on the three terms shown below:
dT<fvg
= de/di + (0 T-e) di
di
(1 +t)
(1 +t)2
di + t d°
(1 +t)
The first term in the above expression will be negative under the standardassumption that the demand
for excess reservesby banks is negatively related to the opportunity cost of holding them. The second
term,also, will be negative underthe usual assumptionthat the legal reserveratio on time deposits is less
than the legal reserve ratio on demand deposits and that portfolio switching out of currency and
demand deposits into time deposits in response to higher nominal interestrates raises the ratio of time
deposits to demand deposits. The general presumption,then, is that optimizing behavior by banks and
the public to higher nominal interest rates will tend to lower the average reserveratio. The third term
captures the response of the central bank to increasesin the nominal interest rate.
118 : MONEY,CREDIT,AND BANKING
banking system is important in a country, the analyticalresults of this paper may
need modification.14 Whetherthe observed correlationsbetween the reserveratio
and the inflation rate in Table 1 are, in fact, the correlationsthat would correspond
to the slopes of the sets of tangency points between iso-revenue and iso-welfare
curves must be the subject of future research.
4. CONCLUSION
This paper has developed the analysis of inflationaryfinance for a production
economy in which the government uses the reserve ratio in conjunction with the
nominal interestrateto tax currencyand demanddeposits. Essentiallythreeconclusions emergefrom the model. First,the semielasticityof the demandfor money with
respectto the directand indirecteffects of changesin the nominal interestraterarely
gives an accuratemeasureof the revenue-maximizingnominalinterestrateand may
even seriously underestimatethe revenue-maximizingnominal interest rate if the
government is constrained in its use of the reserve requirement (including the
constraintthat the reserveratio cannot exceed 100percent).In addition, even for an
unconstrainedgovernmentthe semielasticityof demandfor money will only give an
accurate estimate of the revenue-maximizingnominal interest rate if there are no
output effects associated with the use of the inflation tax (so that on the marginall
time spent on transactionscomes out of leisure).
Second, when governments rely on inflationary finance at levels less than the
revenue-maximizingamount, the paper shows that the minimization of welfare
costs will resultin the choice of a combinationof a reserveratioand nominalinterest
rate that lies on a locus of tangencypoints formed by the government'siso-revenue
curves and the iso-welfarecurves that maintain the representativeagent at a given
level of welfare.Third,the examination of empiricalcorrelationsof the reserveratio
and the inflation rate for forty-one countries over the period from 1960 to 1984
shows that a number of governments do actively alter the reserve ratio with a
tendency for increasesin the reserveratio to accompany increasesin the inflation
rate in Latin America and Africa. Although the paper'sempiricalevidence is not
sufficientto determinewhethergovernmentsdo attemptto use reserverequirements
to minimize the welfare cost of inflationaryfinance, the evidence does provide a
starting point for a more thorough investigation of the interactionbetween fiscal
deficits, the inflation rate, and the reserveratio in economies that rely on revenue
from the inflation tax.
APPENDIX
The first elasticity expression in (9) is derived by noting that the first-order
condition for revenuemaximizationwith respectto the nominal interestratecan be
writtenas
'4See McKinnon and Mathieson (1981) for a discussion of the use of reserve requirements in
developing countries. Fry (1981) examines revenuegenerationfrom the inflation tax when the government is a monopoly supplier of currencyand deposits.
PHILIPL. BROCK : 119
nmi--m
(ml
+
m2T)
=
1
+
m
(A1)
di + Tdl + T2d2+ Td3 ,.] .
+-[m3
Thederivationis completedby addingand subtracting
m + m
m qhT-
[iTm2
+
Tm3
d +
iT
d2 +
T
d3 21
fromthe right-handsideof (A1)notingthatm2 = dl. Theotherelasticityexpressionsin (9)andfootnote7 arederivedsimilarly.Equation(12)is derivedfromthe
condition(8R/ Ar)(dU/ di) = (8R/ di)(d U/ aT) thatholdsat thepointsof tangency
of the iso-revenueand iso-welfarecurves.Since dR/8T = i(dh/8T), multiplying
as
bothsidesof thetangencyconditionby T/ h Uallowstheconditionto berewritten
h
m
nh
r/Ui =
t/Ur
1 dR = _ t1mi + 1 + Td
m
m di
+-[m3
di + Tdl + T2d2+ Td3t1
(A2)
Equation( 12)resultsfrom(A2)byaddingandsubtractingh 7yhr , as was done in
(A1)in deriving(9).
In Figure 1 the dR/dT = O scheduleis definedby the conditionthat i =
costfunctionsthatarequadFortransactions
(T¢mm-¢)md)*
d (¢dd¢>mm-Xmd)Cl
raticthe dR/ aT = Oscheduleis boundedon theleftbya verticalasymptoteat T =
Xmd/(4mm and from below by the horizontalaxis. The dR/dr = Oscheduleis
definedby the conditionthat i(dm/ di + Tdd/di) + m + Td = O,whichcan be
rewrittenas
i(-Xdd
+
2Th4md-
T (4mm) +C(¢dd¢>mm
-
Xmd)(M
+
Td) =
°
(A3)
by makinguse of the derivativesof the assetdemandfunctions.For transactions
of (A3)showsthatalongthe
costfunctionsthatarequadratic,totaldifferentiation
+ (1/2)dh/8T]/
dR/di = O schedulethe slope (di/dT) is equal to-[(i)8d/di
(8h/ di) so thattheschedulereachesitsmaximumat a reserveratiothatliesbetween
the verticalasymptoteof the dR/ aT = O scheduleand the revenue-maximizing
reserveratio(wheredh/ aT = O).
Theslopeof theiso-welfarecurvesandtheslopeof the locusof tangencypoints
nuf. Witha fixed levelof consumption(andwork
dependon the expression7Rui/
d)c. Therefore,
effort),leisureis givenas follows:t = 1-n-¢(m,
120 : MONEY,CREDIT,AND BANKING
dU
ai
=
_UtC[(4m(
(4dd +
T(4md) + (4d(¢md
(4mmh4dd
nd
T¢>mm)]
(A4)
and
aT
=
Ulc [iXmXmd iXdXmm]
(4mmh4dd-
(AS)
nd
Expressions(A4) and (AS) can be combined to show that
. n Ui =
Sf/UT
1 +
(4m (4dd
T( ¢>d¢>mm
(4d (4md
>
1
(A6)
(4m (4md)
Equation (A6) can be used to show that the slope of the iso-welfare curves is
negative:
di
dT | U
_ i
r/UT =_
T nUi
i((4dh4mm-(4mh4md)
T((4dh4mm-(4mh4md)
<
O.
(A7)
+ ((4mh4dd-(4dh4md)
LITERATURE CITED
Auernheimer, Leonardo. "The Honest Government's Guide to the Revenue from the
Creation of Money." Journal of Political Economy 82 (May/June 1974), 598-606.
Bomberger,William A., and Gail E. Makinen. "The HungarianHyperinflationand Stabilization of 1945-1946."Journal of Political Economy 91 (October 1983), 801-24.
Cagan, Phillip. "The Monetary Dynamics of Hyper-Inflation."In *tudies of the 0wantity
Theoryof Money, edited by M. Friedman, pp. 25-117. Chicago: University of Chicago
Press, 1956.
Calvo, Guillermo, and Roque B. Fernandez. "CompetitiveBanks and the Inflation Tax."
Economics Letters 12 ( 1983), 313- 17.
Diaz Alejandro, Carlos R. Foreign TradeRegimesand Economic Development: Colombia.
New York, Columbia University Press, 1976.
Drazen, Allan. "TheOptimal Rate of Inflation Revisited."Journal of Monetary Economics
5 (April 1979), 231 48.
Fischer, Stanley. "A Frameworkfor Monetary and Banking Analysis."Economic Journal
(Conference Papers, 1983), 1-16.
Friedman, Milton. "GovernmentRevenue from Inflation."Journal of Political Economy
79 (July/ August 1971), 846-56.
Fry, Maxwell J. "GovernmentRevenue from Monopoly Supply of Currencyand Deposits." Journal of Monetary Economics 8 (September 1981), 261-70.
Kimbrough, Kent P. "Inflation,Employment,and Welfarein the Presenceof Transactions
Costs." Journal of Money, Credit,and Banking 18 (May 1986), 127-40.
McCallum, Bennett T. "The Role of Overlapping-GenerationsModels in Monetary Economics." Carnegie-RochesterConference*eries on Public Policy 18(Spring 1983),9-44.
McKinnon, Ronald I., and Donald J. Mathieson. "How to Manage a Repressed Economy." Essays in International Finance Number 145, Princeton University, December
1981.
PHILIPL. BROCK : 121
Romer, David. "Financial Intermediation, Reserve Requirements, and Inside Money: A
General EquilibriumAnalysis." Journal of Monetary Economics 16 (September 1985),
175-94.
Siegel, Jeremy J. "Inflation, Bank Profits, and Government Seignorage."American Economic Review 71 (May 1981), 352-55.
Tamagna, Frank. Central Banking in Latin America. Mexico City: Centro de Estudios
Monetarios Latinoamericanos, 1965
Walsh, Carl E. "OptimalTaxation by the Monetary Authority." WorkingPaper Number
1375, National Bureau of Economic Research, Inc., June 1984.
Williamson, Stephen D. "Costly Monitoring, Financial Intermediation,and Equilibrium
Credit Rationing."Journal of Monetary Economics 18 (September 1986), 159-80.