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Keynes-Wicksell and Neoclassical Models of Money and Growth
Author(s): Stanley Fischer
Reviewed work(s):
Source: The American Economic Review, Vol. 62, No. 5 (Dec., 1972), pp. 880-890
Published by: American Economic Association
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Neoclassical
and
Keynes-Wicksell
of
and
Money
By
Models
Growth
STANLEY FISCHER*
The essential features of Keynes-Wicksell
(henceforth KW) monetary growth models, distinguishing them from neoclassical
models, are the specification of an independent investment function and the
assumption that prices change only in
response to excess demand in the goods
market.! In neoclassical monetary growth
models, by contrast, there is no independent investment function and all markets
are continuously in equilibrium.
In KW models a steady state of inflation
requires persistent excess demand in the
goods markets. This suggests that the
steady-state properties of such models are
unsatisfactory. In neoclassical models, an
instantaneous doubling of the quantity of
money, however the money is distributed,
produces an instantaneous doubling of the
price level so long as the expected growth
rate of the money supply is the same before and after the "blip" in the money
supply. This according to KW theorists
suggests that there is something amiss
in the short-run dynamics of the price level
in such models.
In this paper, the price dynamics of both
models are discussed, and a modified price
determination equation is incorporated
into a KW model. The standard comparative dynamic exercises for monetary
growth models are undertaken in this
modified model; the modification of the
price adjustment equation ensures steady
state equilibria rather than disequilibria.
The properties of the modified KW model
are then compared with those of neoclassical models. Essentially, familiar short-run
macro-economic conclusions emerge from
consideration of short-run behavior in the
modified model and neoclassical conclusions emerge from analysis of its long-run
behavior.
I. Price Dynamics
KW models use the Law of Supply and
Demand to determine the rate of inflation.2
Specifically, it is assumed in KW models
that
(1)
r = X(D-S),
O < X < oc
where 7ris the rate of inflation, D and S are
aggregate demand for and supply of goods,
each in real terms, and X is a constant. It is
apparent that there cannot be inflation
without excess demand if equation (1)
determines the rate of inflation, and thus
a steady state with inflation requires persistent excess demand. KW models can
accordingly have steady states in which
individuals are continuallv frustrated in
* Assistant professor, department of economics,
University of Chicago. I would like to thank George
Borts, Rudiger Dornbusch, and Jerome Stein for their
helpful comments on an earlier draft. Thanks for comments and discussion are due, too, to William Brock,
Jacob Frenkel, Merton Miller, Michael Mussa, Douglas
Purvis, and Richard Zecher.
1 Jerome Stein-who is apparently responsible for the
KW designation-has recently provided two very useful
expositions of these models (1969, 1970). An earlier
article of his (1966), using a KW model which is not so
named, provides a full dynamic analysis for the typical
KW model.
2 See Kenneth Arrow. It will be assumed that the
reader is familar with both types of monetary growth
models. A two-asset (money and capital), one-sector
model is used as the paradigm of neoclassical models
(see James Tobin and Miguel Sidrauski); places where
my conclusions would differ if some other neoclassical
model were used are footnoted. My paradigmatic KW
model is contained in Stein's 1969 article.
880
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FISCHER:
obtaining the goods they demand, even
though their demands are based on correct
expectations and perceptions of the price
level-and they are condemned to be so
frustrated forever after. This is an unappealing result and there are two possible
lines of attack on the problem: first, demands could be expected to change in
response to such frustrations; alternatively, the price determination equation
might be inadequate. I pursue the second
approach.
The question raised by (1) and similar
equations is: Whose behavior do such
equations describe? The standard Walrasian answer is "the auctioneer"; another
frequent answer is "somewhat less than
competitive firms."
Consider the auctioneer explanation
first. In the standard single period exchange model, the auctioneer calls out
prices for each good sequentially on the
basis of the mechanism:
(2)
881
MONEY AND GROWTH
Pi,j = Pi-l,j + Xj(pi-1)
where i is the iteration number of the
current call, j is the number of the good, p
is the vector of prices, and xi(pi-1) is an
increasing function of excess demand for
goodj at the previously called price vector.
In intertemporal models an equilibrium
price vector is obtained by the above process at the beginning and no further tatonnement is required. If new information is
available in each period, as in models including uncertainty, one supposes that
there is an "auction" each period. The
goal of the auctioneer in each period is to
establish market-clearing prices prices at
which demands are equal to supplies.
Equation (1) is an attempt to use (2) in
a temporal context so that the i subscript
becomes a t, and to apply (2) to the aggregate price level. But it ignores the motive
of the auctioneer. If the auctioneer expects
the general price level at time t to be different from that at t- 1, then he might use
as his rule of thumb
-e
(3)
Ptj = pt-l,
+ xj(pi)
where fi is the general price level expected
to prevail at t, and pt-, is the general price
level at t- 1. Aggregating over goods, and
in continuous time, an analogue of (3) is:
(4)
l
=r*
+ X(D-S)
where 7r is the actual rate of inflation, and
7r* is the expected rate of inflation.
The auctioneer is not present in most
markets and it is somewhat unsatisfactory
to discuss reasonable behavior for a nonexistent economic agent. Consider alternatively the explanation in terms of the behavior of price-setting firms. As suggested
by Arrow, and developed by Robert Barro
in a recent and interesting paper, since the
existence of disequilibrium is inconsistent
with certain assumptions of the perfectly
competitive model,3 we may expect pricesetting by firms even in industries for
which the competitive model is adequate
for comparative static analysis.
Barro analyzes optimal price-setting
behavior for a monopolistic firm faced
with uncertain demand and a fixed cost of
adjusting its selling price; the optimal
policy is to adjust price only when excess
demand or supply reaches certain barriers.
He then shows that, by aggregating over
firms, the average price may be expected
to behave according to (1). Barro confines
himself to cases where the aggregate price
level is expected to remain constant. Suppose now that all prices but the monopolist's price were expected to increase at the
rate 7r*; then costs would be expected to
rise at the rate 7r*(since the cost function
I In particular, in disequilibrium it cannot be true
that each firm can sell as much as it wants at the going
price and each consumer can purchase as much as he
wants at the going price.
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882
THE AMERICAN ECONOMIC REVIEW
is homogeneous of degree one in prices),
and as of any given price fixed over an
interval by the monopolist, the relative
price of the monopolist's output would be
falling at the rate 7r*. Then, in adjusting
prices, the monopolist could be expected
to include an adjustment for the trend in
prices over the period for which he expects
to keep his own price constant. Aggregating over firms, one would expect to reach
an equation similar to (4).
Thus, on either score, an equation such
as (4) is a more adequate representation of
price adjustment than is (1). Accordingly,
I proceed in Section II to an analysis of a
KW model incorporating equation (4).
Stein (1970) has in fact suggested that an
equation like (4) might be useful in reconciling KW and neoclassical models. Similar
equations may be found to describe wage
and price adjustment in the literature.4
Before presenting the modified KW
model, it is necessary to discuss the price
dynamics implicit in the usual neoclassical
model. The per capita demand for real
balances (md) is a function of the per capita
capital stock (k) and the expected rate of
inflation (r*):
(5)
md
=
L(k, r*,
Li > O,L2 < O
At any instant of time the capital stock
(we omit "per capita" where no confusion
is likely to result) and the expected rate of
inflation are given, as is the nominal money
stock and population. Then, adding to (5)
the neoclassical specification that the
money market is always in equilibrium
(6)
is sufficient
particular,
money will
the system
M/PN
m =md
to determine the price level. In
a doubling of the stock of
double the price level but leave
otherwise unaffected.5
See, for example, Edmund Phelps.
I In two-sector neoclassical models (e.g., Duncan
Foley and Miguel Sidrauski) determination of the price
level requires also commodity market clearing, and the
4
Is there any reason to regard this instantaneous neutrality with suspicion? There
are circumstances under which it might be
regarded as reasonable: for instance, if it
was announced that at some point of time
every i ndividual's nominal money balances would be doubled, then, given some
sophistication by economic agents, it
might be realized that this action was
analogous to creation of a new unit of account and the price level might simply
double. It is, however, a basic assumption
of neoclassical models that injections of
money are not distributed on the basis of
existing holdings of money (since otherwise the transfer payments by which the
money supply is expanded would be equivalent to interest payments on money holdings). Given this assumption, increases in
the nominal balances of some individuals
in the economy can be expected to produce
their effects on prices gradually, through
real balance effects, rather than instantaneously. Hence the KW objection to this
neutrality has force.
Using (5) and (6), the rate of inflation in
neoclassical models is given by
(7)
1
w=j,-X-
m
[L,Dk + L2Dr*]
where ,u is the (assumed constant) rate of
expansion of the nominal money supply, n
is the rate of population growth, and D
denotes the time derivative. In the steady
state 7rw- n; thus the rate of inflation
will be reduced below its steady-state value
by capital accumulation and raised above
its steady-state value by increases in the
expected rate of inflation. Even leaving
aside the expectational factor, D7r*, equation (7) is not analogous to (4).
price level cannot be said to be determined by the requirement of portfolio balance. It remains true that in
such models, "jumps" in the money stock affect only the
aggregate price level.
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FISCHER:
11. The Modified KW Model
In outlining this KW model I shall point
to its departures from neoclassical analysis.
Both types of model have in common a
production function, stock demand functions for assets, a savings function, and
an expectations function. I shall specify
forms of these functions which could be
usecl in either type of model.
The per capita output of goods is
(8)
y = f(k)
tained.'
There are three assets: money, private
bonds, and physical capital. Stock demand
functions for real balances, real bonds (the
excess demand function, since it is assumed
there are no outside bonds), and capital are
given by (9), (10), and (11), respectively.7
The assets are assumed to be gross substitutes. The variable y, output, enters to
represent the transactions demand for
money. Per capita wealth, a=(k+m),
6 For a KW model with variable employment, see
Keizo Nagatani.
I The demand functions for assets differ from those
used in Foley and Sidrauski only in that the price of
capital does not enter. It is assumed that production
always takes place away from corners of the production
possibility frontier so that the relative price of capital
and consumption goods remains fixed. I note, quoting
David Levhari and Don Patinkin, "that it would be
more consistent with general considerations of economic
theorv if . . [the demands for assets] . . . were represented as depending upon disposable income . .. This,
however, would greatlv complicate the . .. analvsis
which follows...."
(p. 720).
md
=
L(y, a, f'(k) + r*, p)
(10)
bI = H(y, a,f'(k) + r*, p)
(11)
kd = J(y,a,f'(k)
+ w*,p)
enters as the stock budget constraint.8
Bonds and capital are not perfect substitutes so thatf'(k)+7r*, the expected nominal return on capital, may differ from p,
the nominal interest rate. The three demand functions are dependent since the
sum of the demands for assets is constrained by wealth at each instant.
Per capita savings is a function of disposable income and wealth:
(12) s
f' > 0, f " < 0
where, for convenience, it is assumed the
Inada conditions hold and that real balances do not enter the production function.
It is also assumed that the labor force,
growing at the rate n, is supplied inelastically and that full employment is main-
(9)
883
MONEY AND GROWTH
=
1>
S(ye, a),
sl > (, s2 < 0
Expected disposable income, ye consists of
factor payments, f(k), plus transfer payments ym, where y is the constant and
preannounced rate of expansion of the
nominal money supply (it is assumed that
the current price level is correctly perceived), minus expected capital losses on
money holdings, 7r*m. Thus
ye =
f(k) + ( - 7r*)m
Saving is definitionally equal to desired
additions to asset holdings; it is the sum of
Id, h'd, and x'1 which are desired additions,
per capita, to real balances, bonds and
capital, respectively. Consumption demand
and savings demand are constrained by
dlisposable income:
(13)
ye =
Cd +
s
It is well known that the stability of
dynamic models is heavily dependent on
the expectations function. We assume
here adaptive expectations:
(14)
7r*= f(7r-7r*),
0
< d <0o
Thus far we have outlined a fairly stan8 Since there are no outside bonds in the model, the
net per capita value of bonds is zero.
L1 > O, 1 > L2 > O, L3
H1 < O H, > O, H3
<
<
O, L4
O,H4
> 0
Ji < O, 1 > J2 > O,J3 > O,J.j
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< 0
<
0
884
THE AMERICAN ECONOMIC REVIEW
dard neoclassical model. A neoclassical
analysis would proceed as follows: assume
asset market equilibrium and use any two
of (9)-(1 1) to determine the price level and
the nominal interest rate at each instant of
time these are functions of the capital
stock and expected rate of inflation. Then
assume that consumption demand is always satisfied and obtain the rate of
capital accumulation as the residual of
output minus consumption.
The scene is then set for determining
"next instant's" short-run equilibrium; the
economy proceeds through these equilibria, and if it is stable, ultimately reaches
a steady state in which the capital stock
and expected rate of inflation are constant.
In fact, the model we have set up is very
similar to Levhari and Patinkin's "Money
as a Consumer Good" model.
The four KW features of the model
follow. First, there is the specification of
an investment demand function, xd. We
assume a stock adjustment demand for
investment.
(15)
xd = nzk + D(kd-k),
'> O
The flow demand for capital consists of the
replacement demand, nk, plus a term
which depends on the divergence between
the actual capital stock and that demanded at the current levels of wealth and
current rates of return and income. The
basic justification for (15) lies in the existence of adjustment costs in changing the
capital stock: the greater the divergence
between actual and desired capital stocks,
the greater the costs that can profitablv be
incurred in changing the capital stock.9
The investment demand function (15)
has the property which is the basis for
I See Robert Eisner and Robert Strotz for the
derivation of an investment demand function such as (15);
see also Marc Nerlove for critical comments on this and
subsequent developments. Note that although adjustment costs are invoked in explaining (15), thev are not
explicitly incorporated in the model.
investment functions in Stein's KW models that an increase in the difference between the expected nominal return on
capital, f'(k) +7r*, and the nominal interest
rate, p, increases investment demand. 'I'his
is the "Wicksell" feature of KW models for
(16) f'(k) + * - p = f'(k) - (p -*);
the first term on the right-hand side of (16)
is the natural rate and the second is the
real rate, and differences between these
two rates affect investment demand.10
Second, there is the price adjustment
equation, in which it remains to specify
aggregate demand and supply.
(17)
7r = 7r* + X(cd
+
Xd
-f(k))
The demand for goods consists of the demands for consumption and investment;
the supply is simply full employment
output.
Third, it is specified that the bond market be continuously in equilibrium, so that
b = bd = 0
(18)
This is an assumption of convenience
rather than necessity.1"
Fourth, there is the question of the
allocation of output in periods of excess
demand or supply. Here it is assumed that
both consumption and investment plans
are partially frustrated when there is excess demand; in particular, planned investment is reduced by some positive fraction
(1 -y) of excess demand to give the actual
rate of investment.
x
-
(1
-
y) [cd +
xd
f(k)j,
In general -y could be expected to be an
endogenous variable rather than a con10 The "Keynes" part lies in the specification of an
independent investment demand function; other Kevnesian features, such as unemployment, can be captured
in KW models with variable employment.
11In Stein (1966), for instance, it is assumed that the
money market is in equilibrium.
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FISCHER: MONEY AND GROWTH
stant; while (19) is very much a deus ex
machina, theories of allocation under disequilibrium are not well developed and
there is no formulation which is obviously
theoretically superior at this stage. Note
that (19) is equivalent, through (17), to
(20)
x=
-(
x"
*
Before proceeding to an exposition of the
short- and long-run properties of the modified model, we use the assumption that the
bond market is always in equilibrium (18),
to derive the implied relationship between
the nominal rate and the capital stock,
real balances, and the expected rate of
inflation. Given k, m, and 7r*,there is, from
(10) and (18), only one nominal interest
rate which equilibrates the bond market.
Specifically
(21)
p
A (k, m, 7r*)
where
A1=-
-1
H1
[Hif' +
H2
+Hf"]
<
O
- H,
HA3= -
-
H3
H4
>0
Ihe only ambiguity in (21) concerns the
effects of an increase in the capital stock
on the nominal rate: there is, in adldition
to the substitution effect (IIff") and
wealth effect (11.), an income effect, (11f');
we assume that the substitution and
wealth effects dominate and that the reduced real rental on capital resulting from
an increase in k leads to a decrease in the
nominal rate as of any given r*. T hus, we
assume that increases in the capital stock
ten(l to reduce the nominal rate; our earlier
assumptions imply that increases in real
balances tend to reduce the nominal rate
885
while increases in the expected rate of
inflation tend to increase the nominal rate
of interest.
III. The Short and Long Run in the
Modified Model
We now discuss the behavior of this KW
model in the short and long run. Given
the assumption that the adjustment coefficient, X, in (17) is finite, the price level is
given at any instant-that
is, it is inherited from the past. Accordingly, m, real
balances per capita, is determined exogenously, for M, nominal balances, is a policy
variable. TIhe capital stock and the expected rate of inflation are also inherited
from the past. Tlhus, at an instant of time,
k, m, and r* are predetermined.
Tlhe behavioral relations of the model
determine, in the short run, the nominal
rate of interest and thence, through the
goods market, the rate of inflation. Given
the rate of inflation, and k, m, and 7r*, the
rate of capital accumulation is determined
from (19), and the rate of change of the
expected rate of inflation from (14). The
stage is then set to determine the capital
stock, real balances, and the expected rate
of inflation at the next "instant"; the economy proceeds in this way through time,
reaching a steady state if. the system is
stable. The remainder of this section consists of a more detailed examination of this
process.12
Given k, m, and 7r*, the predetermined
variables, the nominal interest rate is determined through the requirement of bond
market equilibrium, and is given by (21).
TIhat nominal rate in turn, together with
the predetermined variables, determines
the demands for consumption and investment and the consequent rate of inflation.
12The verbal description we give of the dynamic process of this economv corresponds more closely to a difference e(quationsystemnthan to the differential equation
system contained in the formal analysis; this is simp)ly
a matter of convenience.
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886
THE AMERICAN ECONOMIC REVIEW
Using (17) and the flow budget constraint,
the rate of inflation is
(17')
7r= 7r*+X(xd+
a))
(Au-7r*)m-S(ye,
Consider now the effects of changes in k,
m, and 7r* on the rate of inflation. The
effects of changes in k and m occur only
insofar as excess demand is affected (recall
that xd is a function of the nominal rate, so
that effects working through the bond
market must also be considered) while a
change in r* has an expectational effect on
the rate of inflation in addition to excess
demand effects. We obtain
(22)
7r=
G(k, m, *,
)
where
G1=XQi/
G2=XQ
i-) -+n-sif'?S2)
(i
-
\dm
+
7r*)1Sl)
/
G3=1+--X'
dJ
- S2)
-m(1-sSi)
>O0
>0
GiA=Xm(l-si)>0
The derivatives of the J function are written as total derivatives to indicate that
bond market effects are to be included.
Increases in the capital stock have an
uncertain effect on excess demand; they
reduce the stock excess demand for capital13 but may either increase or decrease
savings since the income and wealth effects
on savings work in opposite directions. If
the system is near the golden rule, then
and the term (n-s1f'-s2)
n-sif'>O
will
be positive. Thus the sign of G1 is ambiguous.
Increases in real balances are inflation13 This may be shown by computing the derivative
dJ/dk- 1 and using the stock budget constraint.
ary; they increase both consumption and
investment demand. Increases in the expected rate of inflation have a direct effect
on actual inflation throughout the expecalso increase investtations effect-they
ment demand but reduce consumption
demand by reducing the value of expected
transfer payments. Thus, whether the
actual rate of inflation increases by more
or less than the expected rate depends on
whether increases in the expected rate
produce an excess supply or excess demand
for goods; in other words, on whether the
reduction in consumption demand is
greater than or less than the increase in
investment demand. It later turns out that
this is an important factor in determining
the stability of the system, and it may be
seen that the smaller is 4V' the more
slowly is the capital stock adjusted the
more likely is (G3- 1) to be negative.
Finally, an increase in the rate of growth of
the money stock increases transfer payments and is inflationary.
The "short-run" position of the economy is determined by (21) and (22). Its
behavior through time is determined by
the capital accumulation equation (20),
the rate of change of real balances equation
which can be derived by differentiating
m with respect to time, and the expectations equation (14). For convenience we
rewrite and renumber these equations
here:
(23) Dk = 44J(y, a,f'(k) +
(24)
Dm =
-
A( ))-k]
[G(k, m, 7r*, )
-7
-
r*,
n
-
m,
G(k, 7m,
(25) D7r* = O[G(k, m, 7r*,,)
-r
,,)]m
-7r*]
Consider now the steady state for this
economy. In the steady state, Dwr*= 0, and
so the actual rate of inflation is equal to the
expected rate, and, from (24), each is equal
to (,u-n). From (23), the demand for the
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FISCHER: MONEY AND GROWTH
capital stock is equal to the existing capital
stock, and there is no excess demand for
capital; from the stock budget constraint
it follows that there is no excess demand
for real balances either. From (17), the
excess demand for goods is also zero and
since investment demand is satisfied, so is
consumption demand.
As in the neoclassical model, there are
no unsatisfied demands in the steady state
of the modified KW model. The reformulation of the price adjustment equation is
thus sufficient to remove the unsatisfactory feature of previously published KW
models the persistence of excess demand
in the steady state.
IV. Changes in the Stock of Money and
in the Rate of Growth of Money Stock
Suppose the economy is in the steady
state and there is an increase in the money
stock, but no change in the rate of growth
of the money supply. T hen since , is the
only exogenous variable in the system in
the long run, it is apparent that if the system is stable, it will return to the same
steady state. However, this economy, unlike our earlier neoclassical system, will be
forced out of equilibrium by the increase
in the money stock, and will take time to
return to its steady state. T he steady-state
neutrality is of course neoclassical but
the dynamics is not.
Consider now the impact effects of an
increase in the money stock. The nominal
interest rate is reduced, and the rate of
inflation is increased because excess demand is increased. The increase in the rate
of inflation increases the expected rate of
inflation and begins to reduce real balances. The effects of the increase in the
money stock on capital accumulation are
ambiguous: the demand for both investment goods and consumption goods is
increased, and investment is more likely
to increase the relatively greater are real
balance effects on investment demand and
887
the more fully are investment plans, rather
than consumption plans, realized. This
short-run story is very Keynesian insofar
as the effects of the change in the money
stock manifest themselves in the bond
market and result in an increase in investment demand through the lowering of the
nominal rate. If we had been dealing with
a model with unemployed resources, the
story would have been even more Keynesian for the increase in both consumption and investment demand could have
called forth more output, rather than
resulting in inflation.
The path followed by the economy
thereafter depends on its stability properties, which are analyzed in the Appendix.
It is shown in the Appendix that if the
steady state is near the golden rule capital
stock, then a necessary condition for stability is that increases in the expected rate
of inflation reduce excess demand this,
as discussed above, is helped by the slow
adjustment of investment demand to
changes in the desired capital stock, and
damaged by a great sensitivity of the demand for capital to the expected rate of
inflation. It is also shown that slow adjustment of expectations as in the neoclassical model and rapid adjustment of prices
to eliminate excess demands are conducive
to stability. However, the conditions for
a
the rapid adjustment of expectations
large 0 to produce instability are less
stringent than they are in neoclassical
models.
Finally, we consider the comparative
steady-state properties of the modified KW
model. An increase in the growth rate of
the nominal money supply ultimately increases the expected rate of inflation by the
same amount as the increase in the monetary growth rate. T he higher expected rate
of inflation increases the demand for capital and reduces the demand for real balances; one of the factors determining the
new stea(ly state is thus the asset demand
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888
THE AMERICAN ECONOMIC REVIEW
functions and the fact that there will be no
excess demands in the long run; the other
factor determining the new steady state is
savings behavior. Working with our full
system of differential equations, we obtain
dk*
(26)
-
=
d,u
-
Om'VX dJ
Z3
*-
-
dr*
(n(1-S1)
-S2)
and
(27)
dm*
--
d,u
fmc'X
=
__
Z3
dJ
* (n-Slf'-S2)
dr*
where Z3 is the determinant of the matrix
in the Appendix which has to be negative
for stability. This negativity is assured if
(n-slf'-s2)
> 0
Thus we can say that if the system is in
a stable steady state, increases in the rate
of growth of money unambiguously inincrease the equilibrium capital intensity;
and if that steady state is near the golden
rule capital stock, increases in the rate of
growth of money reduce equilibrium real
balances. In any event, if increases in the
capital stock reduce savings, so s1f'-s2
<0, then increases in ,u increase k* and
reduce m*.
These results are familiar and early
comparative steady state neoclassical
propositions. We obtain them, of course,
because this KW system has the same
steady-state properties as our neoclassical
model of Section JI, which was set up to be
very similar to earlier neoclassical monetary growth models."4 Although we chose
to represent our steady state by using
(23)-(25) we could equally well have been
neoclassical and described the steady state
in terms of asset market equilibrium and
the requirement that savings be just sufficient to maintain real per capita assets
constant.
14 In particular, our use of output rather than disposab)le income in the asset demand functions, and the
omission of imputed interest on real balances enable us
to avoid several pitfalls.
It is, incidentally, interesting to use (20)
to examine the impact effect on investment of an increase in the growth rate
of the money supply. The demand for
investment goods Xd is unaffected by increases in ,. Thus the impact effect of a
change in , depends only on its effect on
the rate of inflation. The rate of inflation
increases with ,, so that the actual rate
of investment falls when , is increased.
The increase in , increases consumption
demand but not investment demand and
so some investment is displaced. Thus,
initially the capital stock falls when the
rate of growth of money is increased,
though ultimately the capital stock increases. This is similar to the behavior of
the capital stock following an increase in
, in Sidrauski.
V. Conclusions
The purpose of this paper has been to
modify a KW model in a way which removes the feature of steady-state excess
demand in such models and to compare the
resulting model with a neoclassical model
based on the same demand functions for
assets and savings. The paper has made it
clear that the element producing the unsatisfactory features of KW models is the
price adjustment equation, and arguments have been presented for using an
alternative adjustment equation in which
prices may change because they are expected to change, as well as because there
is excess demand. The potential of KW
models for a useful theory of short-run
dynamics, emphasized by others, has been
demonstrated in the context of the modified model. It has also been shown that
there is no inherent reason for the longrun properties of KW and neoclassical
models to differ, so long as the KW investment demand function is consistent
with the neoclassical stock demand function for capital.
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\
-/dJ
~b
(Al)~
i
-d
G1
K1'
l- -Gim
G2
---
1-y
dJ
1-y
dJ
1-z y
)-
889
MONEY AND GROWTH
FISCHER:
- G2m
(31
4td*--
- G3m
gcG1
i3G2
APPENDIX
Stability Conditions
The matrix involved in determining the
local stability of the system (23)-(25) is
shown in (Al) above.
Let Z1 be the trace of Z, Z2 the sum of its
second-order principal minors, and Z3 its
determinant. Necessary and sufficient conditions for local stability are
z1 < 0
Z3 < 0
(A2)
Z1Z2
Z3
-
< 0
A necessary condition implied by (A2) is that
Z2
be positive.
Now,
(A3)
Z, = -Om(G3-1) Gs[
--1)
O(G3- 1) _
virtue of the gross substitute assumption,
the whole expression
and if (n-s1lf-s2)>0,
will be negative. Now, at low levels of the
capital stock, f' is very large and the above
expression may be negative unless y is close
to unity; for higher levels of the capital
stock, and certainly when it is near the
golden rule, we are assured that 3(Dk) 8k is
negative. We shall assume that the steady
state about which we are examining the
dynamics is such that nt-s1f'- s2>O and
hence &(Dk)ak< O.
Given this, it is necessary that (G3-1) be
negative, or that the direct effects of an increase in the expected rate of inflation in the
requires that
goods market be negative-this
the adjustment coefficient in the investment
equation, V', be sufficiently small and/or
that dJ/d7r* be small.
Second
(A5)
1- -yG
<0
-G2
From the derivatives given in (22), we know
that G2 is positive; it follows that the product
of
(G3 -
1)
/dJ \
- -1
/
dk
--AX4'm
<
<O
dJ
_-
_dm
(n-sif'-s2)
and
[
-(-
(dJ
1
(1
-
Y)(n
(n( 1-
SI)
-S2)]
Given the assumption n-s1f'negative.
The value of Z2 iS
Z29=mt3G2+I3KP'
s2>0,
(G3-1)
this is
/dJ\
(kk-1
(A6)
a(Dk) =
- - 1
yq/ /dJ
\dk
9k
-
-1)
1
-y
must be positive. These are, respectively, the
terms a(Dr*)/a7r* and &(Dk)1ak. Consider
first &(Dk)/ k which is
(A4)
dJ
G1_dm
Z3= -flC/m
-Gi- Slf
-
-->O
2)
The first term in parentheses is negative by
The sign of the bracketed term is ambiguous:
after substitution the term becomes
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THE AMERICAN ECONOMIC REVIEW
890
-f3AV>
L
-
i)(
si)m
(-
(!J)
+--
(n
-
dir*J
slf'
-
S2)
While the first term within the brackets is
negative, the second is positive and potentially destabilizing. Note that I have already
discussed the size of (IJ/(1dr*,for if this term is
large, there may be trouble with price level
stability (see the discussion after equation
(22) and above (A5)). Note also that the
more slowly do expectations
smaller is d-the
more likely is this stability conadapt-the
dition (A6) to be met.
Finally,
Z1Z2-Z3
(
ZFm)G .m3G2+V
((G3-1
(-(1k
-/ )]Z3
[ l+Zl
<0O
-G,
Evidently, the larger is Z1, in absolute value,
the more likely is the system to be stable prothe greater (in absolute value)
vided Z2>0;
are O(Dk)/Ok, OG am and (dOGC1r*- 1), the
more likely is the system to be stable. The
last two of these derivatives are increasing
functions of X, and thus the faster does the
price level adjust in response to excess
demand, the more likely is stability.
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