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Working Paper Series
Endogenous Financial Innovation and
the Demand for Money
WP 92-03
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Peter Ireland
Federal Reserve Bank of Richmond
This is a preprint of an article published in The Journal of Money, Credit and Banking, v. 27, iss. 1,
pp. 107-23, copyright 1995 by the Ohio State University Press. All rights reserved.
Reprinted with permission.
Working Paper 92-3
ENDOGENOUS FINANCIAL
AND THE DEMAND
INNOVATION
FOR MONEY
Peter N. Ireland*
Economist'
Research Department
Federal Reserve Bank of Richmond
PO Box 27622
Richmond, VA 23261-7622
804-697-8262
December 1992
Abstract
This paper embeds two key ideas about the nature of financial
innovation taken from the empirical literature into a familiar
equilibrium monetary model.
It provides formal support for several
alternative econometric specifications for money demand that attempt to
capture the effects of financial innovation and demonstrates that a
popular theoretical model of money demand, when suitably modified, can
account for some unusual monetary dynamics found in the data.
Thus, it
helps to establish both the theoretical relevance of recent empirical
work and the empirical relevance of recent theoretical work on the
demand for money.
I.
Introduction
Over the last two decades, an enormous body of literature has
documented the continuing instability of standard econometric money
demand specifications and attributed the instability to innovation in
the private financial sector.' In contrast, almost no theoretical work
has considered the possibility that financial innovation may have
n
important effects on the demand for money.L This paper seeks to fill
this gap on the theoretical side of monetary economics by embedding two
key ideas about the nature of financial innovation taken from the
empirical literature into a familiar equilibrium monetary model.
It
provides formal support for several alternative econometric
specifications for money demand that attempt to capture the effects of
financial innovation and demonstrates that a popular theoretical model
of money demand, when suitably modified, can account for some unusual
monetary dynamics found in the data.
Thus, it helps to establish both
the theoretical relevance of recent empirical work and the empirical
relevance of recent theoretical work on the demand for money.
As its starting point, this study takes two landmark pieces by
Stephen M. Goldfeld (1973, 1976). Goldfeld's earlier work (1973) finds
that a single-equation econometric model expressing the demand for real
Ml as a stable function of real GNP and nominal interest rates does a
remarkably good job of characterizing quarterly US data from 1952-1972,
as judged both by the accuracy of its forecasts and by the inability of
a Chow test to reject the hypothesis of parameter constancy across
subsamples.
In work published just three years later, however, Goldfeld
(19761 reports that by the same criteria of the accuracy of forecasts
and the results of Chow tests, the performance of his money demand
equation deteriorates markedly when the sample period is extended to
1976.
In fact, money demand regressions continue to be plagued by
instability when the sample runs through the present day, with their
forecasts systematically overpredicting actual real Ml figures for the
late 1970’s and underpredicting actual figures for the 1980’s (Goldfeld
and Sichel 1990).3
The years during which standard money demand equations broke down
also witnessed the proliferation of a number of assets that appear to be
very close substitutes for demand deposits, including NOW accounts and
security repurchase agreements, as well as the development of a variety
of new cash management techniques used by firms to economize on their
real balances.
As a result, Goldfeld’s findings launched an extensive
research program directed at repairing the conventional specification by
taking the effects of these financial innovations on the demand for
money into account. Lieberman (19771, for example, includes a time
trend in his money demand regression as a crude proxy for the
improvement in cash management techniques made possible by the
application of new technologies in the financial sector.
An alternative approach to modifying the standard equation, used
by Goldfeld (19761 himself, as well as by Enzler, Johnson, and Paulus
(19761, Simpson and Porter (19801, and Cagan (19841 includes a past
peak, or ratchet, interest rate as an additional independent variable
based upon an argument that can be traced back to Duesenberry (19631.
If the process of financial innovation involves significant initial
2
fixed costs because of the need for newly-trained personnel, newlydeveloped computer equipment, or a newly- established secondary market
for a new security, then the decision to innovate might not be made
unless the opportunity costs of continuing to hold higher money balances
instead--as measured by the nominal interest rate--exceed some threshold
level. Conversely, once these fixed costs have been incurred, the new
product might not be immediately abandoned should interest rates fall.
In addition, if the initial costs of bringing a new financial service on
line are quite high, there may be a lag between the decision to innovate
and the actual change in money demand as these costs are spread over
time.4 Thus, the current level of real balances will be found to depend
not only on how high nominal interest rates are today, but also on how
high they have been in the past.
Other studies employ more direct measures of financial innovation.
Kimball (19801 and Dotsey (1984, 19851 point out that since many cash
management procedures used by firms to economize on their demand deposit
balances involve the transfer of idle funds by wire into overnight
interest-bearing accounts, the number of electronic funds transfers is
likely to be highly correlated with the use of innovative financial
techniques. Dotsey (19841 notes that in contrast to a time trend, which
captures only changes in the costs of financial innovation from
technological change, and in contrast to the ratchet variable, which
proxies only for changes in the potential benefits of financial
innovation from peaks in nominal interest rates, the wire transfer
approach recognizes that the rate of innovation depends jointly on
changes in costs and benefits.
In equilibrium, the extent to which
3
resources are devoted to the process of financial innovation is
determined by agents who weigh the costs of computer and
telecommunications services against the benefits of recapturing the
interest income foregone by holding cash, just as the extent to which
resources are devoted to any other investment project is based on an
assessment of both costs and benefits. Dotsey (1984) reports that while
trend and ratchet variables both aid in explaining changes in the demand
for money, equations with the wire transfer variable perform best.
This paper takes two key ideas from the empirical work on
financial innovation. First, as in Dotsey (19841, the process of
financial innovation is regarded as an investment project.
The decision
to allocate resources to this investment project is made by agents who
balance its costs against its benefits, so that in equilibrium, the
level of financial innovation is endogenous. Second, as in the ratchet
variable literature, the process of financial innovation is assumed to
involve a significant initial fixed cost, the presence of which may
complicate the relationship between money demand and interest rates when
rates are high and volatile. These key ideas are embedded here into the
specification of a general equilibrium model of financial innovation.
The model extends Lucas and Stokey's (19831 interpretation of the
cash-in-advance framework to account for the effects of financial
innovation.' The model features a pure exchange economy. Hence, it
focuses on how financial innovation affects consumers's demand for
money, although its implications are compared to results from empirical
studies that aggregate household and firm behavior. Presumably, a more
elaborate version of the model with production opportunities would allow
4
innovations that facilitate firms's cash management activities to be
considered explicitly as well.
In the pure exchange economy, agents buy and sell a large number
of differentiated goods in a large number of spatially distinct markets.
Improvements over time in communications and record-keeping
technologies, brought about by irreversible investment in financial
capital, enable shoppers to purchase goods on credit in markets where
money was once required. Thus, the model's financial sector resembles a
credit card network; financial innovation allows credit cards to be used
in a wider range of transactions. White (19761, Garcia (19771, and
Dotsey (1984, 19851 all present evidence that increases in credit card
use have been associated with decreases in money demand in the United
States economy; the model's implications are consistent with this
evidence.
The model is specified at the level of preferences, endowments,
and technologies in the next section. Competitive equilibria for the
model economy are characterized analytically in section III and
numerically in section IV, both to provide theoretical support for the
empirical specifications surveyed above and to demonstrate that the
model is capable of generating artificial series that share some of the
features of the data uncovered by the empirical work.
Section V
concludes by pointing to some implications for model-building and for
policy-making.
II. A Model of Endogenous Financial Innovation and the Demand for Money
A discrete time, infinite horizon, perfect foresight economy is
imagined to consist of a continuum of markets arranged around the
boundary of a circle having unit circumference.6 By arbitrarily
selecting one of these markets as market 0, each is given a name is[O,l)
corresponding to its distance, moving clockwise around the circle, from
market 0.
A unique perishable consumption good is traded in each
market, so goods are also indexed by ie[O,l), corresponding to the
locations at which they are bought and sold.
this economy is then defined as L =&t
Lt
=
{c
t
The commodity space in
* where
:c t:[O,l)MJ? piecewise continuous I.
There is also a continuum of infinitely-lived households in the
economy, with names js[O,l). Household j's endowment of good i at time
t is denoted by e:(i), its consumption of good i at time t by cl(i).
Household j is imagined to inhabit a region on the boundary of the
circle including markets ie[j,j+e), where l>c>O, and is endowed at the
beginning of every period with positive amounts of each of the goods
traded in those markets.? For simplicity, it is assumed that household
j's endowment is distributed uniformly on [j,j+cI, so that
et>0 for ie[j,j+c)
e:(i) =
1
where et does not depend on j.
0 otherwise
Thus, for each t, the aggregate
endowment is a constant function et(i)=et.
Households have identical preferences defined on the consumption
O" J where
sets X' = TIIXt,
6
= {CJt :c’: t
[O,llH[O,oo) piecewise continuous ),
as represented by the additively time separable utility function
(1)
It is assumed that ~(-1 is strictly increasing, strictly concave, and
twice continuously differentiable with lim u'(c)=uJ;@(O,l)
C+O
is the
discount factor.
For all is[O,l) and for all trl, there is an uncountable number of
households having an endowment that includes positive amounts of good i
at time t.
Thus, it is assumed that all markets are competitive. In
addition, given the strong symmetry that has been imposed on preferences
and endowments, attention is confined to competitive equilibria in which
at each date, all goods trade for the same relative prices.
Opportunities and objectives are identical across households in such
equilibria, so that the behavior of a representative household with
endowment on [O,E) can be studied with the understanding that all other
households will behave symmetrically. Accordingly, the j superscripts
are now dropped and equilibrium conditions are expressed in terms of
quantities for the representative household.
The assumption lim u'(cl=w implies that although the
C+O
representative household is endowed with goods is[O,el only, it will in
general demand positive quantities of all goods ieIO,ll and must
therefore obtain goods I~[&,11 through trade with other households. To
describe the household's opportunities for trade it is assumed,
following Lucas and Stokey (19831, that each household consists of two
7
members: a buyer and a seller.
In each period, while the buyer travels
around the circle to purchase each of the different consumption goods,
the seller remains at home to sell the endowment to buyers from other
households. When visiting a market close to home, in the interval
IE,X), the representative household's buyer is known to the sellers
there and is able to make his purchases on credit. Farther from home,
in the interval [x,1), the buyer is not known to the sellers and must
pay for all purchases with government-issued noninterest-bearing money.
Symmetrically, the seller from the representative household is willing
to extend credit to buyers he knows, with names on (l-x,1-c], but
insists on receiving cash from everyone else.
The Lucas-Stokey interpretation of the cash-in-advance framework
is extended here by assuming that through a costly process of financial
innovation, it is possible for each buyer to become known in more
distant markets and thereby make purchases on credit where cash was once
required.
Formally, this process is modeled by indexing the variable x
defined above by time and allowing the representative household to
choose xt at each date t subject to the constraints
f(kt) 2
xt,
t=1,2,...,
(2)
where kt is its stock of financial capital at date t and where the
financial production function f:[O,co)H[O,lI is strictly increasing,
strictly concave, and twice continuously differentiable. The
representative household can increase its stock of financial capital
between periods t and t+l by choosing to invest, rather than consume or
sell, St(i) units of any good iE[O,c) with which it is endowed during
period t; the stock evolves according to
8
(l-61k
t
t=1,2,...,
+ b
t
(31
ki=O given,
where &[O,l]
is the depreciation rate for financial capital and where
bt is a technological parameter governing the rate of transformation
between consumption and investment. Increases in bt capture
technological progress exogenous to the financial sector, such as
improvements in computer and telecommunications technologies, which make
financial innovation less costly over time.8 The investment process is
assumed to be irreversible, so that st must be nonnegative for all t.
Since all goods trade for the same price, st(*) can without loss of
generality be restricted to be a constant function st(il=st on iE[O,c)
and (31 simplifies to
(l-6)k
t
+
btstc h
kt+l,
t=1,2,...,
(41
klrO given.
Thus, as suggested by Dotsey (19841, the process of financial
innovation is modeled here as an investment project that involves paying
an initial cost at time t to purchase goods without money in more
distant markets beginning in period t+l. As suggested in the ratchet
variable literature, the initial cost is a fixed cost, since it is
independent of the dollar volume of goods purchased in each market and
since once incurred, it cannot be recovered should the fruits of
innovation no longer seem necessary.
The model specification is completed with a description of what
happens at the end of each period trl when, after consuming their
9
purchases as well as the fractions of their endowments that remain
unsold and uninvested, households convene in a centralized asset market
to settle outstanding debts and to accumulate the money balances needed
to make cash purchases in the following period.
The government
participates in this market by making a lump-sum transfer Ht of money to
each household (if Ht is negative, this is instead a lump-sum tax).
The
representative household leaves the asset market at the end of time t
with cash holdings denoted Mt+r.
Households are assumed to borrow and lend among themselves in the
end-of-period asset market by trading in one-period nominally
denominated discount bonds.
The representative household purchases
bonds paying Bt+l units of money in the time t+l asset market for
Bt+l/Rtunits of money in the time t asset market, where Rt is the gross
nominal interest rate between t and t+l.
The asset market is also open
in period 0, when each household receives an initial transfer Ho of
money from the government. Bonds are traded at this time as well; the
representative household's initial bond holdings are denoted Bo and the
prevailing interest rate is Ro. Since bonds are available in zero net
supply, Bt=O must hold in equilibrium for all tr0, as must the market
clearing condition Mt+r=MI+r,where the per-household money supply MI+1
is defined as
t
Ms+l =
c Hk
k=O
for all t20.
It is now possible to state formally the problem facing the
representative household and to define a competitive equilibrium for
10
this economy.
In the time 0 asset market, the representative household
faces the budget constraint
Bi
Bo
+
Ho
L
+
M1.
(5)
5
As sources of funds at time trl, the representative household has
the income from selling the fraction of its endowment that it chooses
not to either consume or invest, the money and bonds carried over from
the previous period, and the end-of-period government transfer. As uses
of funds, it has purchases of consumption goods as well as the money and
bonds to be carried into the next period.
It therefore faces the budget
constraints
&
Bt+Mt+Ht
+
s
0
Pt
[e,(i)-ct(il-st(illdi 2
1
I
&
ct(ildi +
M
t+1
%
+
B
t+1
,
ptRt
t=1,2,...,
where p, is the nominal price of every consumption good at time t.
Since et(il=et and st(il=st by assumption, these constraints may be
rewritten as
1
Bt+Mt+Ht
+
(et-st)c 5
Pt
s
0
ct(ildi +
M
t+1
Pt
+
B
t+1
,
ptRt
t=1,2,....
(61
The household's money balances at time t must be sufficient to
cover its purchases of the goods ie[max{f(ktl,e),l)that must be bought
with cash.
This requirement gives rise to the cash-in-advance
constraints
1
Mt
Pt
2
s
ct(ildi,
max{f(kt),E)
11
t=1,2,....
(71
Finally, no household is permitted to engage in Ponzi schemes
through which it can borrow more than it will ever be able to repay.
This requirement enters into the representative household's problem as a
nonnegativity constraint for each date t on the sum of the household's
current nominal asset position and the nominal value of its future
endowments and government transfers, all discounted back to date 0 using
the nominal interest rate:
L 1[
-1
wt
=
t-1
DoR,
Mt+r+Bt+/Rt ]
+
Jz~+~[;j;$l~JeJ’
+ ‘J]]
L
OS
(‘I
t=0,1,....
These no-Ponzi-game conditions guarantee that the period-by-period
budget constraints (5) and (6) may be combined to obtain an infinite
horizon budget constraint indicating that, as of time 0, the discounted
present value of the representative household's endowments and transfers
must be no less than the discounted present value of its consumption and
investment streams.
The representative household solves:
Problem:
Maximize by choice of {ctlyZIE X, nonnegative scalars
{St}yZl* {kt+I}~ZI~and {Mt+I}~Zo9and scalars (Bt+II~Cothe objective
function (1) subject to the constraints (4)-(g), taking Bo, kl, and the
sequences {p,)~=,, {Ht)yZo,and {Rt)TZoas given.
A competitive equilibrium is defined by:
Definition:
A competitive
equilibrium
consists of initial conditions
Bo=O and kirO and sequences of quantities ~ct,st,kt+,,Mt,M:,Bt)~=l,
12
prices {ptlysi,and interest rates {RtlyZosuch that:
t1;=I solve the representative
(a) The sequences {c s ,kt+r,Mt,B
t' t
household's problem given Bo, kr, {M:}y=r, {pt)y=r,and {Rt}yZo.
(b) Markets clear in every period:
.1
(i)
(et-st)c =
s
ct(ildi,
t=1,2,...,
0
(ii) M
(iii) B
t
t
=
MI,
=
0,
t=1,2,...,
t=1,2,....
As is typical in general equilibrium environments, the prices and
quantities consistent with a competitive equilibrium as defined above
will also be consistent with competitive equilibria obtaining under a
variety of different market arrangements. It could be assumed, for
instance, that competitive firms rather than households have access to
the financial technology. These firms, acting as financial
intermediaries, rent capital to produce and sell financial services to
households during each period trl. An argument similar to those in
Stokey and Lucas with Prescott (1989, Sec. 2.3) shows that equilibrium
prices and quantities under this alternative market structure are
identical to those under the original market structure assumed above.
Therefore, although the sections to follow characterize competitive
equilibria for this model by taking households as the only type of
economic agent, the equilibrium outcomes may always be thought of as
being generated in an economy in which firms, acting as private
financial intermediaries, supply households with financial services at
cost.
13
III. Analytic Results
The task of characterizing competitive equilibria as defined in
section II becomes considerably easier when it is assumed first that
preferences are logarithmic, so that
u({ctl~~*)
(91
=
and second that the government permits the nominal money supply to vary
over time as necessary to target a sequence {Rt)~=oof nominal interest
rates with Rt>l for all tr0, so that bonds will always dominate money in
rate of return and the cash-in-advance constraint will always bind.g
Under these additional assumptions, the first order conditions for
the representative household’s problem are given by
m*+v,
ho
I
(10)
=
Pl
’
(111
et(i) =
= (ht+ptl-‘.
cp
eb
t t
5
At
-
%
5
=
ie[max{f(kt),c).l),Vtrlp
(131
(14)
with equality if st>O, Vtzl,
A
t’
(3(1-SIC3
(121
ie[O,max(f(kt).e)l,Vtrl,
h;l,
t+1
+ 13%
t+lpt+lCt+l if (kt+l ) I f’ (kt+l ) ’
tlt=1,
(15)
13(At+1 +pt+1 1
=
.
(161
vtr1,
Pt+1
14
at
m t+*
ptRt
Pt+*
-=-(
(17)
vtz1,
where Ao' {~t);=l, h
Ia0
t t=*'
and ~etI~=lare the nonnegative Lagrange
multipliers on (51, (61, (71, and (41, respectively, and the indicator
functions xt+* are defined for all trl by
1 if f(kt+J=s
x t+*
0 if f(kt+ll<c.
=
There are, in addition, three transversality conditions among the
necessary conditions for the household's problem; these are given by
lim BTOTkT+*= lim BTXT(MT+l/pT+l)
= lim /3ThT(BT+l/pT+l)
=
T+W
T+OJ
0.
(181
T+CO
Equations (lo)-(13) and (16)-(17) imply that for all trl, the
optimal et(i) is a step function:
C
et(i) =
C
1t
2t
= a;*
for ie[O,max{f(kt),c))
(191
= GtRt *I-* for ie[max~f(kt),c).lI.
As in more conventional versions of the cash-in-advance model, a
positive nominal interest rate drives a wedge between the representative
household's marginal utility of consuming goods that are bought on
credit and its marginal utility of consuming goods that must be
purchased with cash.
Equation (14) says that the marginal utility of financial capital
will be equated to the marginal utility of consumption only when
investment in financ .a1 innovation is positive.
It retells the ratchet
variable story by indicating that when the interest rate falls from a
peak, so that investment in new financial innovation ceases, the
household will continue to have access to the fruits of past innovation,
15
which will no longer be as valuable as they were under higher rates.
Equation (15) is analytically similar to a capital asset pricing
formula, indicating that the shadow price of financial capital at time t
is equal to the discounted sum of the price at time t+l (after
accounting for depreciation) and the time t+l dividend: the value of the
capital in relaxing the cash-in-advance constraint.
If Xt+*
=l, equations (16), (17), and (19) imply that
cct+lct+l[f(kt+l)l=
(Rt-1)At+*c2t+1
=
(Rt-1)/R ,
t
(20)
which when substituted into the asset pricing equation (15) yields"
et
=
fi(l-a)et+l+ Bxt+l[(Rt-l)/Rtlf'(kt+I)
=
@c
[B(l-S)lJ~t+J+l[(Rt+J-l)/Rt+JIf'
(kt+'+*).
(21)
J=O
Using the market clearing condition for goods as well as equations (14),
(19), and (21), total investment stc is found to be
StE
=
etE
-
J
ct(i)di
0
max{f(kt),cI
=
ec
t
1-max{f(kt),c)
at
=
ec
t
-
htRt-l
+
j\tRt-l
(l-Rt-l)max{f(kt),c}-l
=
max
et& +
8 0
btetRt-l
I
=
max {s:,O),
(22)
where
16
[(l-Rt_l)/btRt_llmax{f(kt),E)-l
s: =
etc
.
+
(23)
w
J=O
Equations (22) and (231, indicating that investment depends on
current and future values of the nominal interest rate, is a nonlinear
version of the investment function emerging from the linear-quadratic
environment studied by Sargent (1987, pp. 399-401). These equations
show how financial innovation is a response to the joint presence of
improved technology and high interest rates; neither alone is likely to
be sufficient. Holding all else constant, s: becomes negative (so that
st equals zero) as bt approaches zero. Thus, for any fixed path {Rt}yZo
of interest rates, financial innovation will not occur at time t if bt
is too small. On the other hand, s; may also become negative if, with
bt held constant, future interest rates are low enough to make the
discounted sum in (23) sufficiently small.
In this sense, financial
innovation will not occur if interest rates too low.
Since by assumption the cash-in-advance constraint is always
binding, the demand for real money balances may be written
1
Mt
-
=
Pt
J
ct(i)di
max(f(kt),E)
=
[l-max{f(kt),cIlc
2t
l-max{f(ktl,c)
=
htRt-l
17
[l-max~f(ktl,e~l(et-stle
=
(241
(Rtl-llmax{f(ktl,~~+l
'
using (19) to substitute for et(i) and the third line of (22) to
substitute for htRt *.
The theoretical money demand equation (24) can
be used to interpret the performance of the empirical money demand
equations discussed in section I.
If economic conditions make financial
innovation impossible or unnecessary in the model, then kt=k, st=O, and
hence
Mt
[I i-
[l-max{f(k),c)lete
In
Pt
=
In
=
(Rtl-llmax{f(kl,c)+l
I, + ylln(et) - r2Rtwl
where ~o=~2+ln{[l-max{f(kl,e~lc~,ir,=l,and T2=max{f(kl,e), so that as
discovered by Goldfeld (19731, money demand will be a stable function of
real income and the nominal interest rate.
When innovation is taking place, however, (24) implies that money
demand changes over time with the stock of financial capital. Both
Lieberman's (19771 time trend and Kimball (1980) and Dotsey's (1984,
1985) electronic funds transfer variable might be thought of as proxies
for kt in (24). Since equation (41 implies that
kt = (l-6)-k + c bJSJc
t-1
1
J=l
kt depends on {s,}ii:,each element of which in turn depends on
‘R$;zJ-l’
Thus, if a proxy for kt is not included in the equation, the
demand for money may be found to depend on all past and future interest
rates as well as the contemporaneous rate.
To the extent that a past
peak interest rate summarizes the entire history of interest rate
behavior, a ratchet variable specification for money demand will be
18
appropriate. More generally, however, both leads and lags of the
interest rate may be needed to account for the role of past and future
interest rates in determining current money demand.
In fact, equation (241 indicates that introducing a technology for
financial innovation will allow the cash-in-advance model to account for
a variety of unusual monetary dynamics. Since little can be said
analytically about the properties of (24) under arbitrary patterns of
interest rate behavior, numerical methods are applied in the next
section to study the behavior of money demand in this model economy in
more detail.
IV. Numerical Results
A.
Computing Equilibrium Dynamics
Substituting (22)-(23) into (41 and solving the asset pricing
equation (21) for et+* yields a system of two nonlinear first order
difference equations in k and 8,
(l-Rt-llmax{f(ktl,e)-l
k
t+*
= Cl-6)kt + btmax
et& +
8
=
pt1-S)
I
~,+,(l-R~)f'(k~+~l
et
t+*
(25)
, 0
btetRt-l
(261
+
Rt(l-61
'
Along with the boundary conditions klzO given and lim BT9TkT+1=0,(25)
T+Q
and (261 completely describe the dynamic behavior of the model economy
given the sequences (Rt}~zo, {et)~zl,and {bt)y=l.
19
TO
solve the system (25)-(26) numerically,
it is
necessary
to
specify a functional form for the financial technology f(*l and to
assign values to the parameters ,!3,
6, and E.
In all of the examples
discussed below, ft.1 is specialized to
f(kt) =
kt
,
1 + kt
which, as required, maps [O,wl into [O,ll and is strictly increasing,
strictly concave, and twice continuously differentiable. The discount
factor /3 is chosen to be 0.99, so that a period in the model represents
one quarter in real time.** The depreciation rate 6 is set equal to
zero, since financial capital is imagined to consist primarily of
disembodied knowledge, computer hardware, and computer software, which
depreciate slowly if at all.
Finally, the interval [O,E) is chosen to
be quite small, with e=O.OOl, so as to make the range of goods that the
representative household must acquire through trade as large as
possible.
Below, a variety of patterns for the time varying parameters
are fed through the model and the effects
{Rt)Fd, {et}FSl,and {bt)"
t=*
on the income velocity of money, which using (24) is computed as
V
t
etc[(Rt l-l)maxif(kt),c)+ll
etE
=
(271
[l-max{f(kt),s)lc
2t
are traced out.
=
[I-max(f(kt),c}l(et-stlc '
The first two numerical examples demonstrate how the
model economy behaves under the simplest conditions. The next example
shows how an upward spike in the nominal interest rate produces a
ratchet effect on money demand.
The fourth and fifth examples examine
the effects of real economic growth and exogenous technological change
20
on the demand for money.
A final example generates artificial series
that are compared to actual US time series data.
B.
Convergence
to a Steady State
When Rt 1=R, et=e, and bt=b for all trl, (25)-(261 becomes a timeinvariant system.
is a fixed point of (25)In this case, if (k*,tJ*I
(261, then lim kt=kf and lim vt=v* whenever kl is sufficiently close to
t-JO
t.30
k*.
As a first experiment, a constant endowment level e=lOOO is chosen
so that with Rt 1=1.05 and bt=l for all tzl and with kl=0.05, velocity
(figure 1) converges to a stationary value of 4.41,12 about what the
income velocity of the US monetary aggregate Ml-A (currency plus demand
deposits) was when nominal rates were around 5% in the mid-1960's (see
figure 7),13
Figure 2 reveals that similar dynamics are associated with a
permanent increase in the nominal rate of interest. Here, the model
economy at time 0 is assumed to be in the steady state reached in
example 1.
When R increases permanently from 1.05 to 1.15, new
financial innovations help to gradually push velocity up to a new
stationary value of approximately 7.6.
In both examples 1 and 2, the
costs of the financial innovations that permit velocity to increase are
spread out over several years.
C.
The Ratchet Effect
Example 3 is identical to example 2 except that the increase in
interest rates is only temporary; after rising to 1.15 for five years,
21
R t l returns to 1.05 for all 020.
Figure 3 shows that velocity
increases in response to higher rates, but does not reach the levels
seen when the change in rates is permanent. Velocity remains higher
after the interest rate returns to its previous level. Comparing the
behavior of velocity in examples 2 and 3, therefore, demonstrates that
the demand for money in any given period depends nontrivially on the
entire sequence of nominal interest rates.
In particular, as suggested
in the ratchet variable literature, a past peak in rates has lasting
effects on money demand.
D.
Economic Growth and Technological
Change
Equation (24) indicates that in the absence of financial
innovation, the income elasticity of money demand is unity.
Thus, when
st=O and kt=k for all trl, equation (27) implies that velocity depends
only on the contemporaneous nominal rate of interest. Series on
velocity and income generated by the model may be consistent with the
presence of economies of scale in money demand, however, because of the
possibility for endogenous financial innovation.
In example 4, the nominal interest rate is held constant over
time, with Rt l=1.05 for all t=O, and the parameter bt is held constant
at unity.
Real economic growth is captured by increasing the endowment
level over time according to
e
t+1
= (l.Olle
t'
t=1,
el=lOOO.
The model economy is again assumed to be in the steady state from
example 1 as of time 0.
22
The investment function described by (22) and (23) indicates that,
given the sequences {Rt)y=oand {bt)~=i,and given an initial stock of
financial capital ki, financial innovation will take place only after
the endowment exceeds some threshold level. Velocity in figure 4
remains constant until et exceeds this threshold level. As long as
innovation continues, velocity rises along with income, so that there
appear to be economies of scale in the demand for money. Decreasing
returns, however, imply that innovation ceases once the stock of
financial capital is sufficiently large; hence, velocity eventually
levels off again even as income continues to rise.
An econometrician using the artificial series {vt,et,Rtilyzl from
example 4 to deduce the income elasticity of money demand without
accounting for changes in kt would find evidence of economies of scale
in some subsamples but not in others.
If kt were included along with
income and interest rates in a regression equation, however, the income
elasticity would be found to be constant at unity.
Similarly, using
actual data Laidler (19711 and Cagan and Schwartz (1975) conclude from
regression equations that do not attempt to account for the effects of
financial innovation that the income elasticity of money demand has
varied considerably over time in the United States, while Dotsey (1984)
reports that estimates of the scale elasticity of money demand increase
from 0.31 to approximately 0.90 once various proxies for financial
innovation are added to his regression equation.
Since the preferences represented by the logarithmic utility
function (9) are homothetic, the ratio cat/e, is invariant to increases
in et and hence by (27) velocity depends on the sequences {e,}Tzland
23
t 1 only through their effects on the growth rate of capital.
{btI"=
Moreover, since
(l-Rt_i)max{f(kt),&}-l
btSt
=
btmax
et& +
, 0
btetRt-l
(l-Rt-i)max{f(kt),c)-l
=
max
I
btete +
$0
etRt-l
*
I
the growth rate of financial capital depends on the sequences {e,}yzl
and {bt)E1 only through the evolution of the product btet. Two sets of
A A
sequences {et,bt)y-land {et,bt}rB1such that etbt=etbtfor all trl,
therefore, generate exactly the same time paths for velocity. This
result is confirmed by example 5, which is identical to example 4 except
that instead of holding bt constant and increasing et by 1% per period,
et is held constant and bt is increased by 1% per period; the time path
for velocity is the same in figure 5 as in figure 4.
Using US time series data, Lucas (1988, p. 146) notes that it is
extremely difficult to distinguish the effects of income growth on
velocity from those of technological change. Examples 4 and 5 show that
these effects can be indistinguishable in theory as well.
E.
Comparison with US Data
Money in this model economy, as in most cash-in-advance economies,
is used exclusively as a means of exchange and does not bear interest.
Its closest analog in the US data, therefore, is Ml-A, which includes
currency and demand deposits but excludes the interest-bearing checkable
24
deposit component of the broader aggregate Ml.
Figure 7 compares the
behavior of Ml-A velocity to that of the six-month commercial paper rate
from 1961 through 1991. Nominal rates have peaked on three occasions:
in 1969, 1974, and 1981. Hester (1981) notes that periods of rapid
innovation in US financial markets coincide with each of these peaks.
Following each peak, velocity remained higher even as interest rates
returned to levels seen previously; in fact, velocity marched steadily
upward as rates became higher and more volatile.
In example 6, a pattern of interest rates stylized after that
experienced by the US economy during the past 30 years is fed through
the model economy. Rates are assumed to reach ever increasing peaks
during periods 1 through 84 (the first 21 years) before declining
erratically. The parameters et and bt are both assumed to grow at a
rate of 1% per period, The economy begins time 0 in the steady state
that would obtain if Rt 1, et, and bt were to remain constant at their
time 1 levels for all trl.
Figure 6 shows that velocity in the model economy, like velocity
in the US data, trends steadily upward, apparently responding very
little to contemporaneous movements in the nominal interest rate.
An
econometrician using the artificial series generated in this example
would report on an unstable relationship between velocity, income, and
interest rates. As velocity remains permanently higher after each peak
in rates, the series would be found to be consistent with ratchet
variable specifications for money demand; if data (such as the series
for ktl were available to proxy for the rate of financial innovation,
the proxy would be significant in a money demand regression. All of the
25
unusual monetary dynamics associated with financial innovation in the
empirical literature are captured by the model in this example.
V.
Conclusions and Implications
Conventional versions of the cash-in-advance model have recently
been criticized (e.g., Christian0 19911 for failing to account for all
but the simplest kinds of monetary dynamics.
In fact, since as Lucas
(19881 and McCallum and Goodfriend (1987) demonstrate, standard
theoretical models imply that the demand for money can be expressed as a
stable function of income and interest rates, these models cannot be
used to understand why standard money demand functions are not found to
be stable when estimated with data from the past thirty years.
The numerical work performed in section IV shows, however, that
just as conventional econometric models for money demand have been
modified to account for the effects of financial innovation, the
conventional cash-in-advance model of money demand can be modified to
capture the dynamics associated with financial innovation. The
necessary theoretical modifications are suggested by the empirical
literature and, in turn, provide formal support for alternative
econometric models.
These results suggest that introducing a
transactions technology such as the one used here, which recognizes that
households and firms have access to a variety of means for circumventing
the use of noninterest-bearing assets in exchange, may be a useful step
in developing a general equilibrium model that is consistent with enough
26
data to be of use in evaluating policy experiments.
Certainly, acknowledging that possibilities for financial
innovation exist is critical if the presence of a stable money demand
function is to be relied on in policy-making. As equation (24) and the
numerical work make clear, simple money demand relationships will break
down when interest rates are high and volatile; instabilities will
persist even after rates have settled down.
27
Notes
Thanks go to participants in the macroeconomics seminar at the
University of Michigan and in the Federal Reserve System Committee
Conference on Financial Analysis at the Federal Reserve Bank of Boston
as well as to Michael Dotsey, Joseph Haslag, Jeff Lacker, Milton
Marquis, Kevin Reffett, Stacey Schreft, and two anonymous referees for
extremely helpful comments and suggestions. The opinions expressed
herein are those of the author and do not necessarily represent those of
the above-mentioned individuals, the Federal Reserve Bank of Richmond,
or the Federal Reserve System.
1
Early empirical studies attributing money demand instability to
financial innovation include Enzler, Johnson, and Paulus (1976) and
Goldfeld (1976). Judd and Scadding (1982) and Goldfeld and Sichel
(1990) survey the subsequent literature. Hester (1981) and Dotsey
(1984) describe the major innovations to have occurred in US financial
markets during the past 30 years.
2
Two exceptions are Simpson and Porter (1980) and Dotsey (19851, which
present versions of the classic inventory model of money demand modified
to allow for endogenous changes in the intensity of agents's cash
management efforts.
See note 5 below.
3
Most recently, the empirical money demand literature has focused on
unusual weakness in the broader aggregate M2.
28
Just like the earlier
episodes of instability in Ml demand, this recent episode of weakness in
M2 has been associated with changes in the private financial sector,
including the growth of bond mutual funds and the closing of insolvent
savings and loan institutions; see Carlson and Parrott (1991) and Duca
(1992).
4
The potential magnitude of the fixed costs associated with a particular
financial innovation are documented by Iida (19911, who reports that the
startup cost to a commercial bank of installing software to monitor its
daylight overdraft position might run as high as $700,000. Similarly,
Tufano (19891 cites costs ranging from $50,000 to $5 million associated
with the development of new financial instruments.
5
Thus, Lucas and Stokey's cash-in-advance model is extended here just as
the inventory model is extended in Simpson and Porter (19801 and Dotsey
(19851.
6
An economic environment similar to this one was originally described by
Schreft (19921.
7
Here and below it is assumed that j*cIl. When j+c>l, the interval
[j,J+&) should be replaced by [j,j+c-1).
8
Thus, it is assumed here, as in Solow (19691, that technological
progress must be embodied in newly-installed capital.
29
9
The well-known indeterminacy of nominal quantities under interest rate
targeting policies (Olivera 1970) may be eliminated here, of course, by
allowing the government to choose Ho as well as the sequence {RtIyzo;
given the choice of Ho, the remaining transfers {Ht)yzl are then
supplied so as to clear markets at the given interest rates {RtIy-o.
"Since
strO for all thl, kT+lzll-GITkIand hence
0
=
[f3(1-611Tt3T
=
j!3T6TkT+I/kI.
The transversality condition lim @TBTkT+l= 0 therefore implies that
T+O
lim [8(1-S)l’CBT= 0 and hence that the asset pricing equation may be
T+O
solved forward to obtain (21).
11
One period in the model is both the holding period for money and the
gestation period for investment in financial capital. The holding
period for money suggests that one model period ought to be identified
with, perhaps, one month in real time. On the other hand, the gestation
period for investment suggests a longer period length, perhaps one year.
One quarter is chosen, therefore, as a compromise between these two
interpretations.
12
Here and below, as well as in the figures, the quarterly interest rate
and velocity series are expressed in annual terms. That is, R=l.OS
means that a quarterly interest rate of approximately 1.012 is fed
through the model.
Similarly, v=4.4 translates into a quarterly
velocity of 1.1. Reporting the artificial series in this way makes them
30
comparable to US data as they are most frequently reported (e.g., in
figure 7).
13A11 data presented in figure 7 are taken from the DRI/McGraw-Hill
database. Ml-A is Ml less the OCD component. Velocity is defined as
nominal GDP divided by Ml-A.
31
References
Cagan, Phillip. "Monetary Policy and Subduing Inflation." In Essays in
Contemporary Economic Problems:
Disinflation,
William Fellner,
project director, pp. 21-53. Washington: American Enterprise
Institute, 1984.
Cagan, Phillip and Anna J. Schwartz. "Has the Growth of Money
Substitutes Hindered Monetary Policy?" Journal of Money, Credit,
and Banking 7 (May 19751: 137-159.
Carlson, John B. and Sharon E. Parrott. "The Demand for M2, Opportunity
Cost, and Financial Change." Federal Reserve Bank of Cleveland
Economic Review 27 (1991 Quarter 21: 2-11.
Christiano, Lawrence J. "Modeling the Liquidity Effect of a Money
Shock." Federal Reserve Bank of Minneapolis Quarterly
Review 15
(Winter 19911: 3-34.
Dotsey, Michael. "An Investigation of Cash Management Practices and
Their Effects on the Demand for Money." Federal Reserve Bank of
Richmond Economic Review 70 (September/October 1984): 3-12.
.
"The Use of Electronic Funds Transfers to Capture the Effects
of Cash Management Practices on the Demand for Demand Deposits: A
Note.
” Journal
of Finance 40 (December 19851: 1493-1503.
32
Duca, John V. "The Case of the Missing M2." Federal Reserve Bank of
Dallas Economic Review (Second Quarter 1992): l-24.
Duesenberry, James S. "The Portfolio Approach to the Demand for Money
and Other Assets." Review of Economics and Statistics 45 (February
1963, supplement): 9-24.
Enzler, Jared, Lewis Johnson, and John Paulus. "Some Problems of Money
Demand." Brookings Papers on Economic Activity
(1976):
261-280.
Garcia, Gillian. "A Note on Bank Credit Cards' Impact on Household Money
Holdings." Journal of Economics and Business 29 (Winter 1977):
152-154.
Goldfeld, Stephen M. "The Demand for Money Revisited." Brookings Papers
on Economic Activity
(1973): 141-189.
. "The Case of the Missing Money." Brookings Papers on Economic
Activity
(1976):
683-730.
Goldfeld, Stephen M. and Daniel E. Sichel. "The Demand for Money." In
Handbook of Monetary Economics, edited by Benjamin M. Friedman and
Frank H. Hahn, pp. 299-356. Amsterdam: North-Holland, 1990.
33
Hester, Donald D. "Innovations and Monetary Control." Brookings Papers
on Economic Activity
(1981): 141-189.
Iida, Jeanne. "Fed Nears Decision on Plan to Charge Overdraft Fees."
American Banker 156 (5 November 19911: 1 and 3.
Judd, John P. and John L. Scadding. "The Search for a Stable Money
Demand Function: A Survey of the Post-1973 Literature." Journal of
Economic Literature
20 (September 1982): 993-1023.
Kimball, Ralph C. "Wire Transfer and the Demand for Money." New England
Economic Review (March/April 19801: 5-22.
Laidler, David E.W. "The Influence of Money on Economic Activity--A
Survey of Some Current Problems." In Monetary Theory and Monetary
Policy
in the 1970s, edited by G. Clayton, J.C. Gilbert, and R.
Sedgwick, pp. 75-135. London: Oxford University Press, 1971.
Lieberman, Charles. "The Transactions Demand for Money and Technological
Change." Review of Economics and Statistics
59 (August 19771: 307-
317.
Lucas, Robert E., Jr. "Money Demand in the United States: A Quantitative
Review." Carnegie-Rochester
Conference
(Autumn 1988): 137-168.
34
Series
on Public
Policy
29
Lucas, Robert E., Jr. and Nancy L. Stokey. "Optimal Fiscal and Monetary
Policy in an Economy Without Capital." Journal of Monetary
Economics 12 (July 1983): 55-93.
McCallum, Bennett T. and Marvin S. Goodfriend. "Demand for Money:
Theoretical Studies." In The New Palgrave Dictionary of Economics,
edited by John Eatwell, Murray Milgate, and Peter Newman, v. 1,
pp. 775-781. London: The Macmillan Press Limited, 1987.
Olivera, Julio H.G. "On Passive Money." Journal of Political Economy 78
(July/August 19701: 805-814.
Sargent, Thomas J. Macroeconomic Theory, 2nd ed. New York: Academic
Press, 1987.
Schreft, S.L. "Transaction Costs and the Use of Cash and Credit."
Economic Theory 2 (April 19921: 283-296.
Simpson, Thomas D. and Richard D. Porter. "Some Issues Involving the
Definition and Interpretation of the Monetary Aggregates." Federal
Reserve
Bank of Boston Conference
Series 23 (October 1980): 161-
234.
Solow, Robert M. "Investment and Technical Progress." In Readings in the
Modern Theory of Economic Growth, edited by Joseph E. Stiglitz
and Hirofumi Uzawa, pp. 156-171. Cambridge: MIT Press, 1969.
35
Stokey, Nancy L. and Robert E. Lucas, Jr. with Edward C. Prescott.
Recursive
Methods in Economic Dynamics. Cambridge: Harvard
University Press, 1989.
Tufano, Peter. "Financial Innovation and First-Mover Advantages."
Journal of Financial
Economics 25 (December 1989): 213-240.
White, Kenneth J. "The Effect of Bank Credit Cards On the Household
Transactions Demand for Money." Journal of Money, Credit,
Banking 8 (February 1976): 51-61.
36
and
Fig. 1.
R(t11
Example 1, Convergence to a Steady State
1)
1
Fig. 2.
Example 2, Permanent Increase in Interest
Rates
R(t-1)
1 16
v(t)
e(t)
78r-----
Fig. 3.
R(t‘=Y
Example 3, Temporary Increase in Interest Rates
b(t)
1)
1
51
r
1
Fig. 4.
Example 4, Real Economic Growth
R(t-1)
11
66
C"
t
__
6
58I
Fig. 5.
R(t-
Example 5, Technological Change
b(t)
1)
88
I
-__
e(t)
v(t)
74,
I
Fig. 6.
Example 6, Comparison with US Data
b(t)
Fig. 7.
COMMERCIAL
US Data
PAPER
RATE
1.17
1.16
1.15
1.14
1.13
1.12
1.11
1.1
1.09
1.08
1.07
1.06
1.05
1.04 1.03 1.02 '
I
1
1
I
I
I
I
1961:l
1966:l
1971:l
1976:l
1981:l
1986:l
1991:l
VELOCITY
OF
Ml -A
12
11 -
8-
6-
3-
'
1961:l
1
I
I
I
I
I
1966:l
1971:l
1976:l
1981:l
1986:l
1991:l
@FedFRASER