Fischer - Keynes Wicksell and Neoclassical Models of Money and Growth

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American Economic Association

Keynes-Wicksell and Neoclassical Models of Money and GrowthAuthor(s): Stanley FischerReviewed work(s):Source: The American Economic Review, Vol. 62, No. 5 (Dec., 1972), pp. 880-890Published by: American Economic Association

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Keynes-Wickselln d Neoclassical M o d e ls

o f M o n e y a n d G r o w t h

By STANLEY FISCHER*

The essential features of Keynes-Wicksell

(henceforth KW) monetary growth mod-

els, distinguishing them from neoclassical

models, are the specification of an inde-

pendent investment function and the

assumption that prices change only in

response to excess demand in the goodsmarket.! In neoclassical monetary growth

models, by contrast, there is no indepen-

dent 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 be-

fore 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 pricedetermination equation is incorporated

into a KW model. The standard compara-

tive 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 modelare then compared with those of neoclassi-

cal models. Essentially, familiar short-run

macro-economic conclusions emerge from

consideration of short-run behavior in the

modified model and neoclassical conclu-

sions 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.2Specifically, it is assumed in KW models

that

(1) r = X(D-S), O< X < oc

where 7r s the rate of inflation, D and S are

aggregate demand for and supply of goods,

each in real terms, and Xis a constant. It is

apparent that there cannot be inflation

without excess demand if equation (1)

determines the rate of inflation, and thusa steady state with inflation requires per-

sistent 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 GeorgeBorts, Rudiger Dornbusch, and Jerome Stein for theirhelpful comments on an earlier draft. Thanks for com-ments and discussion are due, too, to William Brock,Jacob Frenkel, Merton Miller, Michael Mussa, DouglasPurvis, and Richard Zecher.

1Jerome Stein-who is apparently responsible for theKW designation-has recently provided two very usefulexpositions of these models (1969, 1970). An earlierarticle of his (1966), using a KW model which is not sonamed, provides a full dynamic analysis for the typicalKW 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







FISCHER: MONEY AND GROWTH 881

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 un-appealing result and there are two possible

lines of attack on the problem: first, de-

mands could be expected to change in

response to such frustrations; alterna-

tively, 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 Walras-

ian answer is "the auctioneer"; another

frequent answer is "somewhat less than

competitive firms."

Consider the auctioneer explanation

first. In the standard single period ex-

change model, the auctioneer calls out

prices for each good sequentially on the

basis of the mechanism:

(2) Pi,j = Pi-l,j + Xj(pi-1)

where i is the iteration number of thecurrent 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 pro-

cess at the beginning and no further taton-

nement is required. If new information is

available in each period, as in models in-

cluding uncertainty, one supposes that

there is an "auction" each period. Thegoal 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 aggre-

gate price level. But it ignores the motive

of the auctioneer. If the auctioneer expects

the general price level at time t to be dif-

ferent 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 = * + 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 unsatisfactoryto discuss reasonable behavior for a non-

existent economic agent. Consider alterna-

tively the explanation in terms of the be-

havior 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 price-

setting 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 confineshimself to cases where the aggregate price

level is expected to remain constant. Sup-

pose now that all prices but the monopo-

list'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.







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 adjustingprices, 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. Aggregat-

ing 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 recon-

ciling 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) M/PN m =md

is sufficient to determine the price level. In

particular, a doubling of the stock of

money will double the price level but leave

the system otherwise unaffected.5

Is there any reason to regard this instan-

taneous 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 timeevery individual's nominal money bal-

ances 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 ac-

count 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 other-

wise the transfer payments by which the

money supply is expanded would be equiv-

alent to interest payments on money hold-

ings). 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 instan-

taneously. Hence the KW objection to this

neutrality has force.

Using (5)and

(6),the rate of inflation in

neoclassical models is given by

1(7) w=j,-X- [L,Dk + L2Dr*]

m

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 valueby capital accumulation and raised above

its steady-state value by increases in the

expected rate of inflation. Even leaving

aside the expectational factor, D7r*, equa-

tion (7) is not analogous to (4).

4 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

price level cannot be said to be determined by the re-

quirement of portfolio balance. It remains true that in

such models, "jumps" in the money stock affect only the

aggregate price level.








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 func-

tions 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) f' > 0, f " < 0

where, for convenience, it is assumed the

Inada conditions hold and that real bal-

ances do not enterthe

production function.It is also assumed that the labor force,

growing at the rate n, is supplied inelasti-

cally and that full employment is main-

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 substi-tutes. The variable y, output, enters to

represent the transactions demand for

money. Per capita wealth, a=(k+m),

(9) md = L(y, a, f'(k) + r*, p) L1 > O,1 > L2 > O, L3 < O, L4 < 0

(10) bI = H(y, a,f'(k) + r*,p) H1 < O H, > O,H3 < O,H4 > 0

(11) kd = J(y,a,f'(k) + w*,p) Ji < O, 1 > J2 > O,J3 > O,J.j < 0

enters as the stock budget constraint.8

Bonds and capital are not perfect substi-

tutes so thatf'(k)+7r*, the expected nomi-

nal return on capital, may differ from p,

the nominal interest rate. The three de-mand functions are dependent since the

sum of the demands for assets is con-

strained by wealth at each instant.

Per capita savings is a function of dis-

posable income and wealth:

(12) s = S(ye, a), 1 > sl > (, s2 < 0

Expected disposable income, ye consists of

factor payments, f(k), plus transfer pay-

ments 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 per-

ceived), 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'1which are desired additions,

per capita, to real balances, bonds andcapital, 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< o

Thus far we have outlined a fairly stan-

6 For a KW model with variable employment, seeKeizo Nagatani.

I The demand functions for assets differ from thoseused in Foley and Sidrauski only in that the price ofcapital does not enter. It is assumed that productionalways takes place away from corners of the productionpossibility frontier so that the relative price of capitaland consumption goods remains fixed. I note, quoting

David Levhari and Don Patinkin, "that it would bemore consistent with general considerations of economictheorv if . . [the demands for assets] . . . were repre-sented as depending upon disposable income . .. This,however, would greatlv complicate the . .. analvsiswhich follows...." (p. 720).

8 Since there are no outside bonds in the model, the

net per capita value of bonds is zero.








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 oftime these are functions of the capital

stock and expected rate of inflation. Then

assume that consumption demand is al-

ways 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 equi-

libria, 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 de-

manded at the current levels of wealth and

current rates of return and income. The

basic justification for (15) lies in the exis-

tence 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

investment functions in Stein's KW mod-

els that an increase in the difference be-

tween the expected nominal return on

capital, f'(k) +7r*, and the nominal interest

rate, p, increases investment demand. 'I'hisis 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 de-

mands for consumption and investment;

the supply is simply full employment

output.

Third, it is specified that the bond mar-

ket be continuously in equilibrium, so that

(18) b = bd = 0

This is an assumption of conveniencerather 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 ex-

cess demand; in particular, planned invest-

ment 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 con-

I See Robert Eisner and Robert Strotz for the deriva-tion of an investment demand function such as (15);see also Marc Nerlove for critical comments on this andsubsequent developments. Note that although adjust-ment costs are invoked in explaining (15), thev are notexplicitly incorporated in the model.

10The "Keynes" part lies in the specification of an

independent investment demand function; other Kev-nesian features, such as unemployment, can be captured

in KW models with variable employment.

11 n Stein (1966), for instance, it is assumed that the

money market is in equilibrium.








stant; while (19) is very much a deus exmachina, theories of allocation under dis-equilibrium are not well developed andthere is no formulation which is obviously

theoretically superior at this stage. Notethat (19) is equivalent, through (17), to

(20) x = x" -( *

Before proceeding to an exposition of theshort- and long-run properties of the modi-fied model, we use the assumption that thebond 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 ofinflation. Given k, m, and 7r*, here is, from(10) and (18), only one nominal interestrate which equilibrates the bond market.

Specifically

(21) p A(k, m, 7r*)

where

-1A1=- [Hif' + H2 +Hf"] < O

H1

- H,

H-

- H3A3= - > 0

H4

Ihe only ambiguity in (21) concerns theeffects of an increase in the capital stockon the nominal rate: there is, in adldition

to the substitution effect (IIff") andwealth effect (11.), an income effect, (11f');we assume that the substitution andwealth effects dominate and that the re-duced real rental on capital resulting froman increase in k leads to a decrease in thenominal rate as of any given r*. Thus, weassume that increases in the capital stockten(l to reduce the nominal rate; our earlierassumptions imply that increases in realbalances tend to reduce the nominal rate

while increases in the expected rate of

inflation tend to increase the nominal rateof interest.

III. The Short and Long Run in theModified Model

We now discuss the behavior of this KWmodel in the short and long run. Giventhe assumption that the adjustment coeffi-

cient, X, in (17) is finite, the price level is

given at any instant-that is, it is inher-ited from the past. Accordingly, m, realbalances per capita, is determined exoge-nously, for M, nominal balances, is a policyvariable. TIhe capital stock and the ex-pected rate of inflation are also inheritedfrom 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 nominalrate of interest and thence, through the

goods market, the rate of inflation. Giventhe rate of inflation, and k, m, and 7r*,therate of capital accumulation is determinedfrom (19), and the rate of change of the

expected rate of inflation from (14). Thestage is then set to determine the capital

stock, real balances, and the expected rate

of inflation at the next "instant"; the econ-

omy proceeds in this way through time,

reaching a steady state if. the system is

stable. The remainder of this section con-sists of a more detailed examination of this

process.12

Given k, m, and 7r*,the predetermined

variables, the nominal interest rate is de-

termined through the requirement of bondmarket equilibrium, and is given by (21).TI at nominal rate in turn, together withthe predetermined variables, determinesthe demands for consumption and invest-

ment and the consequent rate of inflation.

12The verbal description we give of the dynamic pro-cess of this economv corresponds more closely to a dif-ference e(quationsystemn han to the differential equationsystem contained in the formal analysis; this is simp)lya matter of convenience.








Using (17) and the flow budget constraint,

the rate of inflation is

(17') 7r= 7r*+X(xd+ (Au-7r*)m-S(ye, a))

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/ i-) -+n-sif'?S2)

G2=XQ -\dm

+ 7r*)1Sl) - S2) >O0

/ dJG3=1+--X' -m(1-sSi) > 0

GiA=Xm(l-si)>0

The derivatives of the J function are writ-

ten 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 capi-

tal13 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

n-sif'>O and the term (n-s1f'-s2) will

be positive. Thus the sign of G1 is am-

biguous.

Increases in real balances are inflation-

ary; they increase both consumption and

investment demand. Increases in the ex-

pected rate of inflation have a direct effect

on actual inflation throughout the expec-

tations effect-they also increase invest-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 pay-

ments and is inflationary.

The "short-run" position of the econ-

omy is determined by (21) and (22). Itsbehavior 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 expecta-

tions equation (14). For convenience we

rewrite and renumber these equations

here:

(23) Dk = 44J(y, a,f'(k) + r*, A( ))-k]

- -7 [G(k,m, 7r*, ) -r

(24) Dm = - n - G(k, m,m, ,,)]m

(25) D7r* = O[G(k, m, 7r*, ) -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 the13 This may be shown by computing the derivative

dJ/dk- 1 and using the stock budget constraint.








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), theexcess 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 ref ormula-

tion of the price adjustment equation is

thus sufficient to remove the unsatisfac-

tory feature of previously published KW

models the persistence of excess demand

in the steady state.

IV. Changes in the Stock of Money andin 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 sys-

tem is stable, it will return to the samesteady state. However, this economy, un-

like 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 de-mand is increased. The increase in the rate

of inflation increases the expected rate of

inflation and begins to reduce real bal-

ances. The effects of the increase in the

money stock on capital accumulation are

ambiguous: the demand for both invest-

ment goods and consumption goods is

increased, and investment is more likely

to increase the relatively greater are real

balance effects on investment demand and

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 bondmarket and result in an increase in invest-

ment 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 Key-

nesian for the increase in both consump-

tion 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 prop-

erties, 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 sta-

bility 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 sensitivityof the de-

mand for capital to the expected rate of

inflation. It is also shown that slow adjust-

ment of expectations as in the neoclassi-

cal model and rapid adjustment of prices

to eliminate excess demands are conducive

to stability. However, the conditions for

the rapid adjustment of expectations a

large 0 to produce instability are less

stringent than they are in neoclassical

models.

Finally, we consider the comparativesteady-state properties of the modified KW

model. An increase in the growth rate of

the nominal money supply ultimately in-

creases the expected rate of inflation by the

same amount as the increase in the mone-

tary growth rate. T he higher expected rate

of inflation increases the demand for capi-

tal and reduces the demand for real bal-

ances; one of the factors determining the

new stea(ly state is thus the asset demand








functions and the fact that there will be noexcess demands in the long run; the otherfactor determining the new steady state is

savings behavior. Working with our full

system of differential equations, we obtain

dk* Om'VX dJ(26) - = - *- - (n(1-S1) -S2)

d,u Z3 dr*

and

dm* fmc'X dJ(27) -- = __ * (n-Slf'-S2)

d,u Z3 dr*

where Z3 is the determinant of the matrixin 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 ina stable steady state, increases in the rateof growth of money unambiguously in-increase the equilibrium capital intensity;and if that steady state is near the goldenrule capital stock, increases in the rate ofgrowth 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* andreduce 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 neoclassicalmodel of Section JI, which was set up to be

very similar to earlier neoclassical mone-

tary 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 statein terms of asset market equilibrium andthe requirement that savings be just suf-ficient to maintain real per capita assetsconstant.

It is, incidentally, interesting to use (20)

to examine the impact effect on invest-

ment of an increase in the growth rate

of the money supply. The demand for

investment goods Xd is unaffected by in-creases 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 in-

creases. 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 re-

moves 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 un-

satisfactory features of KW models is the

price adjustment equation, and argu-

ments have been presented for using an

alternative adjustment equation in which

prices may change because they are ex-

pected to change, as well as because there

is excess demand. The potential of KWmodels for a useful theory of short-run

dynamics, emphasized by others, has been

demonstrated in the context of the modi-

fied model. It has also been shown that

there is no inherent reason for the long-

run properties of KW and neoclassical

models to differ, so long as the KW in-

vestment demand function is consistent

with the neoclassical stock demand func-

tion for capital.

14 In particular, our use of output rather than dispos-ab)le income in the asset demand functions, and theomission of imputed interest on real balances enable usto avoid several pitfalls.








-/dJ \ 1-z y dJ 1-y dJ 1-y

(Al)~~b

i)-

l-G1

K1'd

---

G24td*--

(31

-Gim - G2m - G3m

gcG13G2 O(G3 1) _

APPENDIX

Stability Conditions

The matrix involved in determining thelocal 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 condi-

tions for local stability are

z1 < 0

(A2) Z3 < 0

Z1Z2 - Z3 < 0

A necessary condition implied by (A2) is that

Z2 be positive.Now,

(A3) Z, = -Om(G3-1) Gs[ --1)

1-- -yG < 0

From the derivatives given in (22), we know

that G2 is positive; it follows that the productof

(G3 - 1) and

[ (dJ 1 -y 1

must be positive. These are, respectively, the

terms a(Dr*)/a7r* and &(Dk)1ak. Consider

first &(Dk)/ k which is

a(Dk) /dJA4) - = yq/ - - 19k \dk

- (1 - Y)(n - Slf - 2)

The first term in parentheses is negative by

virtue of the gross substitute assumption,

and if (n-s1lf-s2)>0, the whole expression

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 thedynamics 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 in-

crease in the expected rate of inflation in the

goods market be negative-this requires that

the adjustment coefficient in the investment

equation, V', be sufficiently small and/or

that dJ/d7r* be small.

Second

dJ(A5) Z3= -flC/m G1-

_dm

/dJ \-G2 - -1 < O

dk /

dJ--AX4'm _- (n-sif'-s2)

_dm

-(- -1) (n( 1- SI) -S2)]

Given the assumption n-s1f'- s2>0, this isnegative.

The value of Z2 iS

/dJ\Z29=mt3G2+I3KP' (G3-1) (k -1

(A6)

-Gi- -->O

The sign of the bracketed term is ambiguous:

after substitution the term becomes





Fischer - Keynes Wicksell and Neoclassical Models of Money and Growth

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