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A o ?dcnf>fVnP
A Series of Occasional Papers in Draft Form Prepared by Members
of the Research Department for Review and Comment.
76-1
Money, Prices, and Interest Rates
in Stable Monetary Growth Models
Thomas A. Gittings
Federal Reserve Bank of Chicago
ji
t
i t tu
> *♦ ♦ ♦ ♦ ♦ ♦ *< z i i x i z i a
♦♦I
Research Paper No. 76-1
MONEY,
PRICES,
STABLE
AND
INTEREST
MONETARY
RATES
IN
GROWTH MODELS
By
Thomas
A.
Department
Federal
The
views
and
do n ot
Reserve
material
necessarily
to s t i m u l a t e
permission
of
of
is
of
discussion,
the
are
or
the
Research
Bank
of
solely
represent
Chicago
contained
of
Reserve
expressed herein
Bank
Gittings
the
Chicago
those
views
of
of
the
the
authors
Federal
Federal
Reserve
System.
a preliminary
nature,
is
and
authors.
is
not
to b e
The
circulated
quoted without
I. INTRODUCTION*
The purpose of this paper is to analyze a series of theoretical macroeconomic models that describe the possible dynamic relationships among money,
a price index, and an interest rate.
Having been subject to Occam's razor,
the models are presented in their simplest form and from a macroscopic point
of view.
All behavioral equations are assigned specific functional forms so
that the models can be solved analytically or simulated numerically.
This en
ables one to study both the qualitative and quantitative properties of the
alternative models.
Each of the models is formulated as a system of ordinary differential
equations.
The first three models are presented as "equilibrium" models
where the real quantity of money is inversely related to the rate of inflation
in the "long run."
Two of these three models have been analyzed extensively
in the literature; the third is a subtle extension.
After discussing the
dynamic relationship between the rate of growth of the quantity of money and
the rate of inflation, we extend the third model by including an interest rate
equation and a simple unemployment relationship.
Finally, a corresponding
"disequilibrium" model is derived to show the mathematical equivalence be
tween equilibrium and disequilibrium growth models.
This equivalency is based
on the concept that a system of differential equations is a system of differen
tial equations.
In the spirit of mathematical growth models, we make a series of heroic
assumptions.
The models represent a closed economy that produces and consumes
a constant amount of a single homogeneous nondurable good.
There is no in
vestment, depreciation, technological change, or inventories.
The population,
2
level
of
initial
are
employment,
state
normalized
that
any
is
changes
in
models.
rate,
ing
second
the
test
path
test
not
the
rate
initial
to
of
parameter
We
values
begin
by
hyperinflation
the
primary
cost
difference
as
bonds,
in
of
crease
the
of money.
in
the
third
cash
test
of
be
level
a
the
constant.
are
type
of
In
this
constant
fiat
and
money
economy without
its
and
a period
inflation
zero,
is
causing
state
on
assign
the
the
output
is
the
rate
is
destabilizing
Marshall
[9,
p.
we
some
shall
in
his
constant
change
by
the
force
48]
is
growth
he
rate
of
the
the
The
of
in
de
quantity
succinctly
states:
with
such
balances
summarized
where
and
in prices.
dominated
cash
on
compatible
reserves,
of
use
study
of
rate
rule
models.
fairly
real
that
infla
feedback
model,
A
initial
of
forms
hyperinflation,
the
but
This
rate
reasonably
of
correspond
model.
following
Cagan
remains
the
path.
alternative
by
constant
constant
simple
first
alternative
exceeds
a
a
the
expected
the m o d e l
formulated
durables,
of
an
by
alternative
at
observe
steady
of
in
and
the
grow
remains
to
in
of
generated
shock or
is
some
of m o n e y
hyperinflation
balances
on
supply
simulate
Real
consumer
of
that
exception
models.
work
off
and
amount
money
two mo d e l s
on money
of
to
into
period
stability
period
potentially
theoretical
A
ensure
or
remain
price
conducted
exogenous
the
our
such
rate
This
an
With
holding
equities,
During
as
of
time
position
reviewing
return
flation.
a
be
nominal
money
A
injected
inflation
of
which
[2].
assumptions
of
reflect
disturbance.
and
assumed
the n o minal
the
zero.
is
will
initial
that
growth
is
money
rate
distribution.
rate
from
could
equal
and
have
the
the
begins
displacement
to
of
Money
experiments
from
for
one.
income
is
utilization
quantity
produce
assumes
economy
tion
to
the
beginning
time
equal
basic
One
capital
nominal
to
costless
Several
for
the
and
3
The
total
fore
value
cannot
increase
in
repeated,
is
to
the
to
this
lower
is
fundamental
analysis.
of
of
such
a
by
paper
increasing
which
the v a l u e
stresses
the
rapid
confidence
relationship
long-run
Given
inconvertible
seems
currency
its
there
quantity;
likely
to
an
be
of
each
unit
more
that
this
type
of
than
in
increase.
[8]
because
inverse
an
quantity,
the
p u b l i c ’s l o s s
that
a
its
Reform Keynes
self-defeating
due
of
increased
will
proportion
In M o n e t a r y
be
between
property
consensus,
of
increase
in
the
the
let
the
us
in
the
inflationary
velocity
currency.
rate
of
of
Friedman
inflation
and
taxation
circulation
[5]
argues
real
balances
theoretical
framework
now
the math e m a t i c a l
look
at
for m o n e t a r y
impli
cations.
II.
(1)
where
cash
The
first
M/P
»
M
tt
is
the
nominal
balances,
base
inflation
tt
of
is
quantity
the
Cambridge
of money,
instantaneous
logarithms,
operator
Taking
spect
is
defined
the n a t u r a l
This
a modified
cash-balance
equation,
and
ot i s
P
rate
a
is
of
the
price
positive
price
index,
inflation,
constant.
The
M/P
exp
is
is
rate
real
the
of
as
= DP/P = DlnP,
the
tt
is
I
e x p (-air),
is
natural
where
model
MODEL
=
model
to
the
3DlnP
3InP
is
is
interpreted
logarithm
the
(InP
D
-
of
logarithms
the
of
as
price
equation
the
right-hand
time
derivative
and
InP
with
re
index.
(1)
yields
lnM)/a.
unstable
logarithm
since
of
the
prices
derivative
is
of
positive.
the
rate
of
inflation
4
When
the
above
nominal
its
equilibrium
inflation.
In o t h e r
generating
al.
This
is
level,
words,
the
is
constant
the m o del
when
when
there
there
antithesis
this
heavenly model
stabilizing monetary
greater
than
of m o n e y
y
money
the
initial
states
that
is n o t
enough money
is
of
and
too
the
much
there
money
Marshall,
will
one
price
be
level
is
self-generating
one
has
has
self-generating
Keynes,
self
Friedman,
Cagan,
thesis.
In
the
of
inflation, and
deflation.
et.
quantity
and
the
rate
where
policy
of
there
is
to
inflation.
are
perfect
increase
Let
y be
foresight
nominal
the
money
rate
of
and
no
frictions,—
balances
growth
of
at
the
a
rate
quantity
n be the logar i t h m of real cash balances:
= DM/M « DlnM,
n = In(M/P) - InM - InP.
Assume
that
y
is
positively
related
to
tt so t h a t
y — 0tt.
Since
Dn = y -
tt
= (0-1)tt
and
n = -CX7T,
therefore
Dn
This
If
=
(l-6)n/a»
model
is
3Dn
=
3n
(1-0) / a
heaven
is
stable
if
<
and
only
if 0
stable,
(2)
M/P
alternative
=
greater
than
one
so
that
0.
St.
Peter
must
conduct
MODEL
An
is
2/
model—
exp(-f3Tr*),
used
by
such
an
aggressive
II
Cagan
is
the
following:
monetary
policy.
5
(3)
D tt* -
Y ("“ * * ) »
ic
where
it
is
the
expected
rate
of
inflation,
p
and
Y
are
positive
constants.
This model is stable if the reaction index (By ) is less than unity.
To see why,
take the logarithm of equation (2) to get
(4)
n = - B it*.
Differentiating with
(5)
Dn
=
y -
tt
=
respect
to
time
yields
B D tt*.
Use equations (2)-(5) to obtain the following relationships:
(6)
n* -
(7)
n *
Dn =
( ( B y - l ) v ; + v ) / (By) ,
((l-gy)TT-u)/Y,
Y(n+3u)/(6Y-l)*
When By is less than one, the model is stable and real cash balances are posi
tively related to the rate of inflation and negatively related to the rate of
change of nominal money balances.
In the long run, which corresponds to the
particular solution of a differential equation model, the rate of inflation
equals the rate of growth of the quantity of money and is inversely related to
real cash balances.
Notice in equation (6) that the expected rate of inflation
is negatively related to the actual rate of inflation and serves as a counter
balance.
Figure 1 plots the dynamic responses of these two models when the quantity
of money is increasing at a constant rate y .
Notice that the rate of inflation
asymptotically approaches its long-run level from above, so that real cash
balances instantaneously fall whenever the nominal quantity of money is increased.
This may be a reasonable description of a theoretical economy where new fiat
money is introduced by having the monetary authority bid directly for goods and
services.
When money is introduced via transfers or a bond market, one would
expect that real money balances initially increase, reach a maximum, and then
6
Model 2.
n = -3ir*
Dir* *
y
(rr— it*)
3 * .5
y = .5
&
*
Dir = C (7T— 7T )
C = ,5
Dn = a(nd-n)
a = 1
7
decline to its new and lower long-run position.
In other words, the rate of
inflation is initially below its new long-run value.
In order for real
cash balances to fall eventually, the rate of inflation must at some time rise
above its long-run value and then settle onto this new equilibrium position.
The minimum mathematical requirement for a model to generate this cyclical
process is that it be a second-order differential equation model.
The previous
two models considered are first-order differential equations linear in the
logarithm of real cash balances(n).
Let us now analyze some linear second-
order models.
III.
SECOND-ORDER MODELS
Assume that the dynamic relationship between money and prices can be re
duced to the following linear second-order nonhomogeneous differential
equation:
(8)
D2n +
where D
2
pDn
+
an
* -xp
+
<f)Dp,
denotes the second-order time derivative; p, a, x, and <f> are
constants.
This model is stable if and only if p and a are positive since p
is the determinant of the corresponding Jacobian matrix of partial derivatives
and o is the negative of the trace of the Jacobian.
The thesis that real cash
balances are inversely related to the rate of inflation in the long run implies
that x is positive.
The form of the solution depends upon whether the roots
of the characteristic equation are real and distinct, real and equal, or complex.
An interesting exercise is to derive the corresponding cash balance
equation in logarithmic form.
Dn
*
D2n =
p Dp
tt
-
9
D tr.
By definition the following equations exist:
8
Substituting
following
When
rate
negatively
do
+
the m o d e l
the
not
equations
equation
n ■ (ptt
(9)
to
these
of
to
the
D-tt
-
is
stable,
(p +
t
an
to
the
and
the
the
rate
of
third
demand
some models
portional
the
to
expected
(10)
n =
model
assumes
a weighted
rate
of
real
of
for
and
rearranging
yields
the
balances:
that
change
of
the
of money.
MODEL
is
rate
Since
balances
positively
with
assume
of
inflation
economists
this
related
form,
and
is
usually
it
such
an
cash
balances
is
in
equation.
III
logarithm
between
The
balances
implicitly
the
difference
real
real
that
inflation.
of
growth
IV.
The
(8)
(<J>-l)Dp/a.
quantity
rate
explicit
analyze
equation
l o g a r i t h m of
) p ) / cj +
inflation
related
specify
structive
for
into
the
equations
of
real
actual
of
rate
of
3/
the m o d e l —
is
inflation
are
as
pro
and
follows:
(7T-g7T*)/a,
Dir* »
a and Y are assumed to be positive, and
3
is assumed to be greater than one so
that in the long run real balances and the rate of anticipated inflation are
inversely related.
Since
the
as
the
quantity
the
(11)
rate
of
growth
of m o ney minus
following
s y s t e m of
of
the
real
rate
linear
balances
of
equals
inflation,
differential
the
this
rate
model
of
can
growth
of
be
written
simple
substi
equations:
Dn * -an - 3 tt* + p ,
Dtt* = ayn + (g-l)Y,
n‘*«
By
taking
tutions
to
expressed
the
time
derivative
eliminate
as
a
linear
the
of
equation
expected
second-order
rate
of
(11)
inflation
differential
D 2n + (a+(l-3)Y)Dn + ayn “ ( 1 - 3 ) y p + Dp.
and
using
term,
equation:
some
this
model
can
be
9
The stability condition therefore is that
a + (l-g)y > 0.
Equation (9) gives the functional form which relates real cash balances to the
relevant observable variables.
It is interesting to compare this model with one Cagan proposed that in
cludes a lag in forming expectations and a lag in adjusting actual levels
of real cash balances to their desired levels.
variables:
He uses two unobservable
an expected rate of inflation and the logarithm of ndesiredM real
balances or nd -
The three basic equations of his model are as follows:
nd = -bTT*,
D7T* « c(7r-7T*),
Dn = a(nd-n),
where a, b, and c are positive constants.
In a manner similar to our previous
analysis, this model can be reduced to a system of two simultaneous differential
equations.
After substituting to remove the desired real cash balances and the
actual rate of inflation terms, the model can be written as follows:
(12) Dn = -an - abir*,
Dtt = acn + (ab-l)cTT + cp.
The corresponding second-order differential equation is
D2n + (a+c - abc)Dn + acn = -abcp.
This model is locally stable if and only if
a + c - abe > 0.
By using equation (12) and the definition of Dn> one can obtain the following
equation, which relates the logarithm of real balances to the expected rate
of inflation:
n = (Tr-u)/a - bit*.
The easiest way to get a feel for the difference between this model of
1
10
U,tt,T
Dir
(ir-gir*)/a
a « 1.
Y (ir-n * )
Y * 10 - 4-1^6
(ir-8ir*)/o
a - 1.
Y (ir-ir*)
Y - -5
6 * 1 .5
e - 1 .5
11
Dir* = c(n-n*)
c = .5
Dn = a(nd-n)
a = 1.
FIGURE 6.
n
Du *
Synthesis Second-order Differential Equation Model
(ir-gTr*)/a
o-l.
Y - -5
6 = 1.5
12
Cagan and of this paper is to compare their response paths following some dis
turbance.
Figures 2-4 plot the time paths of the actual and expected rates
of inflation and real cash balances after the nominal quantity of money be
gins to increase at some constant rate.
In these simulations the initial
expected rate of inflation is assumed to be equal to zero.
Figures 5-6
plot the time paths generated when the quantity of money remains constant and
the initial expected rate of inflation is equal to some positive value.
Since the present model can have real or complex roots, two sets of parameters
are simulated and plotted in the first experiment.
Caganfs model only can
have real roots when the constants are assumed to be positive real numbers.
V. INTEREST RATES
Up to this point we have concentrated on the dynamic relationship be
tween the quantity of money and a price index.
The model will not be extended
via a combination of traditional arguments to include a macroscopic analysis
of an interest rate.
A nominal market rate of interest (R) is assumed to be
equal to the sum of a natural rate of interest (R*) and a reaction index (r):
R = R* + r,
where the natural rate of interest is constant and reflects an average rate of
time preference.
The reaction index is assumed to be a function of the ex
pected rate of inflation and real cash balances.
It is a proxy variable used
to synthesize the arguments of Keynes [7] and Fisher [3].— ^
According to Keynes1 liquidity-preference function, real balances and a
composite interest rate are inversely related.
Since we have normalized the
initial quantity of real balances to be equal to one, a simple liquiditypreference function can be expressed as follows:
*
r
= -Cn,
13
where r* is a Keynesian reaction index.
If there is a sustained period of a
constant rate of inflation, one would expect this reaction index to equal the
rate of inflation.
C «
In terms of our third model this implies that
o/(B-l).
Fisher's reaction index in a continuous time model is the expected rate of inflation
&
“nr .
The synthesis reaction index is assumed to be a weighted average
of these two alternative indices:
r =
6 tt* +
(l-6)r*,
where 6 is a positive constant.
Figure 7 plots the actual and expected rate of inflation, the Keynesian
reaction index, and the synthesis index in a simulation where the nominal
quantity of money begins to increase at some constant rate.
The difference
between the actual rate of inflation and the synthesis index is plotted to
show the time path of the "real” rate of interest.
There is an initial and
short-lived decrease in the nominal market rate of interest.
The real rate
of interest decreases for a longer duration before it eventually increases,
overshoots, and then returns to its long-run position.
In this simple experiment it is easy to see how unexpected inflation
can redistribute income.
When the real rate of interest is negative, lenders
are subsidizing borrowers and when it is positive, borrowers are subsidizing
lenders.
If such a monetary policy were conducted, the persons with better
foresight would initially borrow and later lend money so as to maximize their
effective subsidy.
The fiscal analogue of this monetary policy is a tax on
those persons with "poorer" foresight and a transfer payment to those persons
who have better foresight or "inside" information.
VI. AN ALTERNATIVE INTEREST RATE
An implicit assumption of the previous method of introducing an interest
rate variable is that the fundamental relationship between money and a price
14
FIGURE 7.
n
Dir*
r*
r
Synthesis Second-order Differential
Equation Model
(Tr-gir*)/a
a = i.
Y (iT—TT*)
3 = 1.5
an/ (1-3)
Y * 6 =
<5tt* + (1-6) r*
? ■ r -
tt
15
index is unaltered when borrowing and lending is allowed.
The coefficients
of such an extended model may change, but the dynamic function between money
and prices is assumed to remain a linear second-order differential equation.
An alternative approach is to allow this fundamental relationship to change
after interest rates are introduced.
This change may be of the form of
introducing a nonlinear relationship and/or a different order differential
equation.
An example of the latter possibility is the following model:
n = (TT-BTT*-xr)/a,
Dir* = y (tt- tt*) ,
r =
tt*
Dr =
- 4>Dn,
ui(r-r),
where ip and w are positive constants.
r is a reaction index of an actual or a
hypothetical interest rate that is instantaneously decreased by an increase
in the rate of growth of money.
If money is introduced via an open-market
operation, this interest rate may be a proxy for the federal funds rate,
the reaction index for an interest rate of a longer-term bond.
r is
This model can
be rearranged into the following system of three linear differential equations
DlnP
=
-alnP
+
Bn* +
x* +
otlnM,
Dir* =
-a y ln P 4- y ( B - 1 ) tt* +
Dr =
-ai/>u)lnP 4- a)(14-B^)ir* 4- a ) ( x ^ l ) r 4- i|>u)(alnM-DlnM).
yxr
4* aylnM,
Representative simulations of this type of model are presented in the sections
on variable output and disequilibrium models.— ^
V I I . AUTOREGRESSIVE FORM
When the rate of growth of money is constant, it is possible to derive
the corresponding second-order autoregressive equation where the current value
of the logarithm of real balances is expressed as a function of real balances
16
in two previous and equally spaced time periods.
Suppose that our initial
model has real and distinct roots so that the solution equation is the
following form:
n(t) = c exp(m t) + c exp(m t) 4* (l-3)y/a.
1
c
m
1
2
2
and c^ are constants of integration and depend upon the initial conditions.
1
and m
2
are the roots of the characteristic equation
m2 4* (a4-(l-B)y)m 4- ay = 0.
Therefore
m ][ = P 4- q,
= P -
q3
where
P “ -(a+(l-e)y)/2,
q = /p2 - ay.
The values of n in the time periods t-0
and t-20
are given by the following
equations:
n(t-0) - c exp(m (t-0)) 4- c exp (m (t-0)) 4- (l-3)u/a,
1
1
2
2
n(t-20) = c exp(m (t-20)) 4- c exp (m (t-20)) 4* (l-3)p/a.
1
1
2
2
The autoregressive equation is
n(t) = <f> 4- <p n(t-0) 4- cp n(t-20),
0
1
2
where
(13) 1 = <f> exp (-m 0) 4- <j> exp (-2m 0),
1
1
2
1
(14) 1 88 <P exp (-m 0) 4* <j> exp (-2m 0),
v J
1
2
2
2
(p = (1-4) ~<P )(l-3)p/a.
0
1 2
Solving equations (13) and (14) and substituting in the functions for the
roots yields the following equations for the two autoregressive parameters:
(p
= 2 cosh(q0)exp (-p0) ,
1
<p = -exp(2p0) ,
2
17
where
cosh
first
autoregressive
<f>
=
is
the
hyperbolic
cosine.
When
the
roots
are
real
and
equal,
the
term becomes
2 exp(-pG),
1
and w h e n
the
(J)^ =
Notice
of
2
that
nominal
A
lates
roots
complex,
cos(q0)exp(-p0).
the
intercept
term
exercise
current
value
is
to
of
real
a lagged
real
balances
term.
creasing
at
constant
rate
position.
rates
a
In
this
included
coefficient
model
has
initial
and
and
beginning
can
be
the
and
to
derived
=
<j>
o
determine
and
to
the
constant
roots.
increase
of
at
rate
When
<}> r ( t ) .
+
example
of
=
-20/(l-(/6-4)0),
=
exp((2
is
initially
of
growth
rate
zero,
y,
the
re
interest
and
balances
in
equation with
interest
for
the
the
balances
and
off
are
long-run
Consider
real
of
which
its
equations
equal
rate
nominal
differential
initial
constant
+
i
current
equation
that
model
simple
inflation
a
the
coefficient.
A):
<j> n ( t - e )
the
such
rate
to
corresponding
assume
second-order
interest
rates
we
that
generate
equal
the
balances
(see A p p e n d i x
numerical
<f>
proportional
Again
the
not
expected
are
A
case
does
real
n(t)
is
balances.
similar
the
are
first
case
where
equal
nominal
lag
one,
the
the
balances
corresponding
equation
2
these
coefficients
yields
the
following
equations
2
<f>
/ 6 ) 0 ) / ( 1 - ( ./S -
-
4)0),
<f>0 = — ((1— 2)/2+4>2 )U »
when
With
for
a =
the
the
1.0,
3 =
1.5,
appropriate
cases
where
Y =
amount
there
10-4^6,
of
are
and
6 =
cranking,
real
and
.75 .
these
distinct
coefficients
roots
or
can
complex
be
determined
roots
and
18
where
the
these
relationships
change
area
coefficients
of
for
the
when
quantity
further
and
initial
stochastic
of m o n e y
is
of
constant
relaxed
by
our
[4]
Real
unemployment.
the
inflation.
the
will
tion will
change
the
Phelps
to
the
output
actual
rate
real
is
was
in n o m i n a l
are
different.
introduced
to v a r y
could
What
and
be
an
happens
the
rate
to
of
interesting
OUTPUT
that
the
level
balances.
This
accelerationist
theory
[10].
of
The
assumed
this
between
real
assumption
unemployment
unemployment
the
output
actual
and
can
as
be
developed
assumed
expected
rate
of
is
above
(below)
its
long-run
level
inflation
is
above
(below)
the
expected
rate
a
is
zero,
smaller
assumed
then
amount
a
that
to
given
an
be
the
be
output
is
to
to
inversely
relationship
related
is
remains
be
inflation
by
rate
of
of
to
real
of
output
decrease
possible
of
VARIABLE
difference
rate
Furthermore
increase
allowed
assumptions
Therefore,
expected
One
and
related
inflation.
whenever
a
introducing
Friedman
inversely
If
initial
following
by
elements
are
study.
VIII.
One
conditions
rate
of
of
6/
nonlinear.—
rate
equal
of
inflation
rate
of
defla
it.
equation— ^ that
characterizes
this
relationship
is
the
following:
lnq
where
Q
index
is
s ln(Q/V0 ) =
is
the
level
constant,
constants.
The
q
i (k ~ 1- (k +
of
real
is
the
cash-balance
tt- tt*
) " 1) ,
output,
ratio
of
equation
VQ
is
the
Q divided
is n o w
form:
n
=
ln(M/P)
■
ln(Q/Vc ) +
(7r-37T*-xr)/a,
or
n = i (k ” 1-( k+ tt- tt*)” 1) + (7r-B7r*-xr)/a
income
by
velocity
VQ .
assumed
i and
to
have
when
the
price
< are positive
the
following
19
This
model
has
which
can be
bined
with
the
expressed
combination
rate,
of
Figures
with
the
is
at
variable
primary
to
rate
output.
By
purposes
for
the
the
of
of
alternative
impact
for
relatively
when
at
and
our
the
of
then
begins
of
the
converge
run when
third-order
figures
output
rate
of
inflation
equation
of
is
inflation
numerically,
In
the
rates,
output,
of
the
rate
7 and
to v a r y
Notice
in
com
and
using
a
begin
to
time.
actual
of
Next,
and
downwards
onto
one
observes
such
a
prescribed
of
its
9
the
the
to
the
rate
real
of
is
rates
turning
reaches
of
interest
points
inflation.
The
rates
a peak
inflation
its m a x i m u m
long-run
manner
plotted
interest
output
attains
that
synthesis
first
reduce
expected
inflation
equation model
equation model
and
is
balances
8,
figure
timing
balances
nominal
differential
differential
nominal
period
between
expected
rate
this
approximated
adjustment.
interest
Finally,
to
the
second-order
short
difference
a maximum.
in
comparing
increasing
be
simulation
allowing
period
can
the
When
the
8 /
comparison.
first
a
of
for
form.
for
R u n g e - K u t t a ’s a l g o r i t h m s . —
a
rate
lengthen
for
solution
9 plot
equation
implicit
equations
constant
effect
equation
the
in
N e w t o n ’s a n d
8 and
a
differential
only
differential
interest
increase
a nonlinear
is
level
value.
IX. DISEQUILIBRIUM APPROACH
A parallel way of looking at the dynamics of a monetary growth model
is the disequilibrium apprach, where the rate of inflation is a function
of the excess demand for goods and the expected rate of inflation.
This
excess demand is the difference between "planned*1 investment and "planned"
savings.
Actual investment usually is assumed to be a weighted average
of planned investment and savings.
These hypothetical variables are
assumed to be functions of real balances, an interest rate and/or the ex
pected rate of inflation when the real output and quantities of capital
20
*
TT,TT ,D
U.lnq
Time
FIGURE
8.
Synthesis
Model
Third-order
with
Variable
Differential
Equation
Output
lnq + (ir-8ir*-xr) fa
a
lnq
i (ic-1-(K+ir-ir*)-1)
3 = 1.5
D tt*
Y (it- tt*)
Y = S - ip - w =
*
IT - l^rDn
l =s .015
u>(r-r)
K = .15
n
r
Dr
1.0
X -
0.
V
•1
=
21
Time
Synthesis Third-order Differential Equation
Model with Variable Output
n * lnq + (n-Sir*-xr)/“
a = 1.0
lnq = l(le
e = 1.5
Dtt* ss y (tt-■if*\)
*
r = TT ■ ipDi)
y = 6 =
Dr = aj(r- r)
f = r - ir
r* » an/(l-e)
r" = 6n* + (l-(S)r*
i = .015
K = .15
X
0.
u
.1
to
22
and labor are constant.
References to these growth models can be found in
the works by J. Stein, [11], Burmeister and Dobell [1] , Urrutia [12], and
L. Johnson [6].
To see how this method is equivalent to the previous approach, one need
only derive the corresponding disequilibrium model.
Suppose the cash-
balances equation model is the following:
(15)
n = In (M/P) = (ir-8ir*-xr)/o.
The market rate of interest is assumed to equal a natural rate plus a reaction
index:
(16)
R = R* + r.
These equations can be combined into the following differential equation for
the logarithm of the price index:
it
= DlnP ■ alnM - alnP + $tt* +
- xR*
or
7T =
an +
3 tt* +
x**.
A representative disequilibrium form for this differential equation is
7T = X (I*(* )“S*(* )) + A
where A is a positive parameter.
I*(«) and S*(») are the unobservable functions
for planned investment and savings, respectively.
The actual level of in
vestment is assumed to equal zero and is a weighted average of the planned
levels of investment and savings:
I = al*(0 + (l-a)S*(.) = 0,
where a is a nonnegative constant that is less than one.
Simple substitutions
yield the following equations:
(17) S* = -a(an + (8-1)it* + XR - xR*)A,
(18) I* =
(l-a)(an + (8-l)ir* + XR - xR*)A.
The usual assumption that the partial derivative of planned savings with
respect to the interest rate is positive and the partial derivative of
23
planned investment with respect to the interest rate is negative implies
that X is less than zero.
This assumption means that the partial deriva
tive of real balances with respect to the interest rate is positive.
In
x affects the dynamic properties
order to get an idea of how the sign of
of such a model, several simulations were run using the third-order model
with a constant level of output.
One simulation has a negative value for
X; the other simulation uses a positive value.
The coefficient of the
price expectations variable is adjusted so that the simulations have the
same long-run, or particular, solutions.
The dynamic paths of the rate of
inflation and the reaction index for the rate of interest are plotted in
Figure 10.
While these simulations show the sign of the interest rate coefficient
is not necessarily of particular importance in determining the dynamic
response of a model, most disequilibrium models specify that planned
savings is positively related to the interest rate, while real balances
and planned investment are negatively related to the interest rate.
In
order for all these conditions to be satisfied, it is necessary to impose
an explicit relationship between real balances and the rate of interest.
This means that the partial derivative of planned savings and investment
with respect to the interest rate cannot treat real balances and the in
terest rate as independent variables.
For example, the partial deriva
tive of planned savings in equation becomes
41^
= -a(oiH + x)/A.
The explicit relationship between real balances and the interest rate
can be used to express planned savings and investment as functions of only
24
Dir
=
r =
y (tt—TT*)
tt*
y m u M .5
- ljiDn
Dr = w(r-r)
Run i.
6 = 1.0
X = .5
Run 2.
6 = 2.0
X = -.5
25
the actual and expected rates of inflation.
Substituting equations (15)
and (16) into equations (17) and (18) gives the following equations for
planned savings and investment:
S* « -a(TT-TT*)/X,
I* - (l-a)(ir-ir*)/A,
This makes the disequilibrium form for the differential equation of the
price index a tautology.
Let us now consider the more general disequilibrium model where we be
gin by specifying the following functions for planned savings and investment:
S* = f(M/P, R,
tt*),
I* = g(M/P, R, TT*),
where the level of output is assumed to remain constant.
Next, one makes
the initial assumption that real balances and the interest rate are inde
pendent variables and that the partial derivatives have the following signs:
as*
-< 0,
a(M/p)
It
3R
r *
31'
> 0,
3(M/P)
> 0.
9I* < 0.
3R
By the implicit-function rule these assumptions imply that
IMP!
3R
U#
In order to avoid this conclusion, we impose the restriction that real balances
and the interest rate are dependent and inversely related.
A representative
restriction is the following explicit function for the rate of interest:
R
=
h(M/P),
where the derivative is assumed to be negative, or
dR
d(M/P)
< °*
The differential equation for the rate of inflation can now be expressed
as
TT =
A(g(M/P,
h(M/P),
TT*) - f ( M / P , h ( M / P ) ,
it*))
+
tt*
26
or
■n
A
= F(M/P,
tt*)
semi-logarithmic
approximation
of
this
general
function
is
n = ln(M/P) ■ (tt— 3 tt*)/a,
which is the cash-balance equation (10) of the third model.
The semi-
logarithmic approximation of the interest rate equation is the Keynesian
version of the synthesis second-order differential equation model.
X. CONCLUSION
This paper has analyzed a series of dynamic macroeconomic models designed
to reflect an inverse relationship between the rate of inflation and real
balances in the long run.
differential equations.
These models were presented as systems of ordinary
The particular solutions of differential equation
models correspond to the economic concept of the long run.
Each of the
alternative models has been simulated using parameter values that ensure
stability.
This enables one to study the subtle differences in the quantita
tive responses of dynamic models that have the same qualitative properties.
The synthesis models proposed are shown to be compatible with a broad spectrum
of economic models.
Some possible areas for future studies would be to
relax some of the initial and very restrictive assumptions, to develop a
microeconomic framework for the synthesis models with an interest rate in
cluded, and to analyze the stochastic behavior of the models using different
monetary policies.
27
FOOTNOTES
*1 am grateful for comments and suggestions offered by my colleagues
at the Federal Reserve Bank of Chicago and by participants at a talk given
to the Special Studies Section of the Board of Governors. Bob Laurent,
Vince Snowberger, and my brother Mike provided challenging discussions that
influenced my research. The motivation for this paper stems from some of
the difficult and fundamental questions raised by my former teacher
Nicholas Schrock. All biases and errors are, of course, mine.
i^An alternative model of such a perfectly perfect world is
n = -ap.
For a discussion of this type of model, see Stein [11].
— ^A variation of this model is
n - -B p *,
Du* = y(u-u*)
where B and Y are assumed to be positive. P is the expected rate of change
of nominal balances and is used as a proxy for the expected rate of inflation.
In a manner similar to that used in analyzing Cagan’s model, real balances
can be expressed as a function of the actual rate of inflation and the rate
of change of nominal balances.
n = (tt-( i+ b y )p )/y .
This model is stable when Y is positive.
—3/ An extension of this model is the following cash-balances equation:
n -
(u - g T r* - 3 * u ) / a ,
where B* is a positive constant. This model implies that an increase in the
rate of growth of nominal balances instantaneously causes an increase in the
rate of inflation.
— ^It is often difficult, if not impossible, to represent the logical
patterns in the arguments of Keynes and Fisher by elementary functions. The
equations used should be viewed as stylized constructs of broader theories.
— ^When testing feedback rules where the rate of growth of money is a
function of the rates of inflation and/or the interest rates, a fourth dif
ferential equation must be added to this type of model. The values of the
parameters in the feedback rule should be selected so as to maintain the
stability of the model.
Another way of extending this model is to specify a production function,
where the level of real output is related to the capital stock, and an in
vestment function that is the differential equation for capital. Care should
28
be exercised when formulating
the implicit assumption about
"neutral" model is considered
is independent of the rate of
the investment function since it will include
the neutrality or nonneutrality of money. A
to be one in which the long-run capital stock
growth of nominal balances.
6 /
— A linear relationship between real output and the difference between
the actual and expected rates of inflation may be simulated by just changing
the parameters of the second-order model. If
lnq
e
ln(Q/VQ) = i (tt- tt*)
and
n = ln(M/P) = lnq + (7r-07r*)/a,
then
n “ ((l+ia)7r - (B+ia)7r*)/a,
or
n = (ir-8*ir*)/a*
where
a* = a/ (l+i a) ,
0* = (0+ia)/(l+ia).
— ^This equation was derived from a translated rectangular hyperbola.
Assume that the nonlinear relationship between lnq and the difference be
tween the actual and expected rates of inflation is the following:
(k+ tt-tt ) (lnq - Q) = -i,
where - k and Q are the asymptotes. The model is normalized so that lnq
equals zero when v equals tf*. Therefore,
Q=
\/<>
and
lnq =
x Ck -1
- (k + tt- tt*)"1)
8 /
— The implicit differential equation is evaluated using Newton’s method
for solving a nonlinear equation. The system of simultaneous differential
equations is numerically integrated using Gill’s modification of a fourthorder Runge-Kutta’s algorithm. Variable step integration was used with an
absolute error criterion of exp(-8). The subroutines used were XCNLSB and
XCRKGM in the CDC Library of Mathematical Subprograms (publication number
86614900).
29
BIBLIOGRAPHY
[1]
Burmeister, E. and A. R. Dobell, Mathematical Theories of Economic
Growth, New York: Macmillan Company, 1970.
[2]
Cagan, P., "The Monetary Dynamics of Hyperinflation," in Studies in the
Quantity Theory of Money, edited by M. Friedman, Chicago: University
of Chicago Press, 1956.
[3]
Fisher, I., The Theory of Interest, New York:
1954.
[4]
Friedman, M., "The Role of Monetary Policy," American Economic Review,
58 (March 1958), 1-17.
[5]
Friedman, M . , "A Theoretical Framework for Monetary Analysis," Journal
of Political Economy, 78 (March/April 1970), 193-238.
[6 ]
Johnson, L., "Portfolio Adjustment and Monetary Growth," Review of
Economic Studies, (forthcoming).
[7]
Keynes, J. M., The General Theory of Employment, Interest, and Money,
New York: Harcourt, Brace & World, Inc., 1965.
[8]
Keynes, J. M., Monetary Reform, New York:
1924.
[9]
Marshall, A., Money Credit and Commerce, London:
Kelley & Millman, Inc.,
Harcourt, Brace and Company,
Macmillan & Co., 1923.
[10] Phelps, E. S., "Phillips Curves, Expectations of Inflation and Optimal
Unemployment over Time," Economica, 34 (August 1967), 254-81.
[11] Stein, J. L., Money and Capacity Growth, New York:
Press, 1971.
Columbia University
[12] Urrutia, J. E., "Optimality in a Growing Monetary Economy," Ph.D. dis
sertation, University of Colorado, 1972.
30
APPENDIX A
Structural equations of the model:
(1 )
n = (TT-eTT*)/ct,
(2)
(3)
(4)
Dtt* ■ y ( tt- tt* ) ,
r* = an/(l-S),
r *» 6tt* + (1-6) r ,
where
n =
it =
y =
R =
ln(M/P),
DlnP,
DlnM,
R* + r.
a, 6, y, 6, R*, and u are assumed to be constant.
Initial conditions:
n(0) =
tt*(0)
= r*(0) = r(0) = 0.
By the general rules of differentiation:
Dn = y
-
it,
D 2n * Du - D u = - D tt.
Model as a system of differential equations:
&
Dn « -an - $7r + p ,
Dtt* ■ ayn + (B-I)ytt*.
Two corresponding second-order differential equations:
D2n + aDn + bn = (l-B)yu,
D2tt* + aDTT* 4- bTr* = ayy,
where
a - a + (l-B)y,
b = ay.
Stability condition when a , B , and y are positive:
a > 0.
CASE I:
ROOTS REAL AND EQUAL
When roots are real and equal,
a2 » 4b,
p = -a/2 ,
31
where p is the common root.
Solutions of the nonhomogeneous second-order differential equations:
(5)
n(t) « A exp(pt) + A t exp(pt) + (l-B)y/ot,
1
(6)
2
7r*(t) * B^exp(pt) +
t exp(pt) + y.
Constants of integration given the initial conditions:
Ai = (B-l)y/a,
A2 « y - pAj,
Bi * -y,
B2 ■ py.
From equations (1), (5), and (6):
7T = an + Btt*
(7)
tt(t)
- C} exp (pt) + C2t exp(pt) + y,
where
C
- -y,
1
C
- (a+p)y.
2
From equations (1), (3), and (4)
(8)
r =s eiT + (l-e)ir*,
where
e = (1-6)/ (1-3).
Solution for interest rate’s reaction index from equations (6), (7), and (8)
r(t) ■ D exp(pt) + D 0t exp(pt) 4- y,
l
*
where
Di = -y,
D2 = (ae+p)y.
Lagged value of n:
(9)
n(t-0) = E 1exp(pt) + E21 exp(pt) + (l-B)y/a,
where
E 1 “ (A r 6A2)exp(-p0),
E2 = A2exp(-p6).
n(t) as a function of r(t) and l(t-0):
n(t) =
<f>0 +
<}>J n (t— 0) +
4>2r(t) ,
32
where
(10)
Aj -
+ ♦jjD j ,
(11)
A2 - ♦ jE j + 4>2D2,
+0 » ((1-3)(1-*1)/o -*2)p .
Solve equations (10) and (11) using Cramer's rule:
Let A = E 2D2 - D xE2
= ((X-e(l-pX))(ae+p) + (l-pX))y2exp(-pe),
where
X = (e-D/a,
4>1 * (A1D2-d 1a 2)/^
- ((0-l)e+l)u2/A,
*2
■ ff.A j-A jE jjM
“ -0(l-pX)2y2exp(-p0)/A.
When <*=1, 3=1.5, 6=.75 and the roots are real and equal
Y “ 10-4>^6,
e = -.5,
p ■ 2-/6,
X - .5,
A -
75((^6-4)0-l)w2exp(-p0) ,
d>2 “ =1.50p2exp(-p0)/A,
- 20/((^-4)0-l),
♦j - .75 P2/A
» -exp(p0)/((v^6-4)0-l)
" -<t>2exp(2-v^60)/26,
♦0 - -((l-^^/a+^jw.
@FedFRASER