allais1966_a restatement of the quantity theory of money

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merican Economic ssociation

A Restatement of the Quantity Theory of MoneyAuthor(s): Maurice AllaisSource: The American Economic Review, Vol. 56, No. 5 (Dec., 1966), pp. 1123-1157Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/1815300

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A

RESTATEMENT

OF

THE

QUANTITY

THEORY

OF

MONEY

By

MAURICE

ALLAIS*

Up to

1950,

there

was

no

attempt to

derive a

formulation

of

the

demand

for

money.

The

Newcomb-Fisher

equation

of

exchange,

Walras'

formulation,

and

those

of

the

writers of

the

Cambridge

School-Mar-

shall,

Pigou,

and

Keynes-have had

little

value

other than

as

purely

formal

frameworks for

the

description of

the

facts.' A

new

formulation

which has an operational and general significance is proposed in this

paper:

the

hereditary,

relativistic,

and

logistic

formulation of

the

demand

for

money.

The

significance

of

this

formulation

can

be

fully

appreciated

only

by

considering

it

against

the

background

of

the

results

which

were

obtained

some

ten

years

earlier.

I.

Earlier

Results

During

1953-54,

Cagan

[10]

and

Allais

[3],

both

working

indepen-

dently of

each

other,

tried

to

find an

operational

formulation

of

the

quantity theory. They both reached the same formulation, although

their

starting

point

presentation and

terminology

were

different.23

This

formulation

was:

(1.1)

(a)

p

=

(u)

=

o-Ku

(Cagan)

(b)

-

=

b(u)

=

Oo(1

-

Ku)

(Allais

1954)

P

D

du

x(O)eX(t8)dO

(1.2) (a)

d=(x-u)

(b)

dt

1

dP ldD ldP

ldQ

(1.3)

(a)

x

=--

(Cagan)

(b)

x

=

=---+-

(Allais)

P

di

D dt

P

di

Q

di

*

The

author is

professor of

general

economics, Ecole

Nationale

Sup6rieure des

Mines; pro-

fessor of

theoretical

economics,

Statistical

Institute of the

University of

Paris and

research

director,

French

National

Center for

Scientific

Research.

I

For

an

accouint of

the

historical

development of the

quantity

theory see

[8, pp.

27-39 and

153-54].

2

For

clarity,

Cagan's

formulation is

given

here in the

notation used

in

this paper.

The cor

respondence between the

notations

is as

follows:

Cagan

(1954)

C

E

,

e7r

Allais

(1954)

x u K X @

3

As

far as

I

am

aware, the

formulation

(1.2a) was

suggested to

Cagan by

Milton

Friedman,

who later used

it

in

his

theory

of

permanent income

[12, pp.

142-45]. Cagan's research

was

brought

to

my

attention

by

Friedman

in a

discussion we had

in

July

1954 when

I

described to

him

the

interesting results

I

had

reached

using

the

formulation

(1.2b)

in

my

research

on the

theory

of the

business

cycle

[3].

This content downloaded from 195.220.72.138 on Sat, 19 Dec 2015 00:27:28 UTC

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1124

THE AMERICAN

ECONOMIC

REVIEW

where

D

is total

outlay,

P

the

price level, Q

the

activity index,

and

5o,

K, and x are

constants. Total

outlay

D

can

be

considered,

at least as

a

first

approximation,

proportional

to

national income

R

at current

prices, a property which is expressed by the relation

D

(1.4)

-

=

a

constant.

For hyperinflations,

it is approximately true

that

D

(1.5) - =

a

coistant.

If the case of hyperinflation is excluded, u is small and

(1.6)

e-Ku

1

-

Ku.

It follows from this

that the

formulations

(1.1) employed by

Cagan

and Allais were not

only

the

same, but

they also applied

in the

same

way to concrete economic data.

Allais

called

the

quantity X

the

psy-

chological

rate of

expansion; Cagan

called

it

the

coefficient

of

expecta-

tion.

Cagan

laid most

stress

on the

differential

equation

(1.2a); Allais

on the

equivalent

integrated expression

(1.2b).

For

Cagan,

future

expec-

tations were the important factor; for Allais, the memory of past events;

but these

were only two

facets of

one and

the

same mathematical

for-

mulation.

For both

Cagan and Allais, the only way

to calculate

the

co-

efficient u

was to consider the

approximation

(1.7)

l

n

+

fLi

1

+?

kPjcn

p

+

1

+

k

+

**+

kP-

+

with

(1.8)

= -

[i"Dn

1,D,1]4

p

where

p

is

the

length

of

the

interval

of

elementary

time

considered,

with

(1.9)

k

=

e-PX.

Relation

(1-7) indicates

that

the

coefficient

u

is

merely

the

weighted

average

of earlier

rates of

growth

of

total

outlay,

with

weighting

co-

efficients

which decline

exponentially

with distance in

time.

For

Allais,

however, this formulation was merely one component of a more gen-

eral

theory

of

monetary dynamics based

on

the relation

I

The notation

In

represents

the natural

logarithm.







ALLAIS:

QUANTITY

HEORYOF

MONEY

1125

(1.10)

dD-

x[M

-

MD]

dt

whereX s a constant and

MD

the desiredcashbalances. But becausethe

difference

M-MD

is

always

small,

Allais could

determine the

function

4

statistically

by

taking the ratio

M/D

and

assuming

(1.11)

MD

;

M.

Cagan's

formulation

assumed

implicitly that

(1.12)

M

MD.

Overall,

both

authors followed the

Walras-Cambridge

chool

formu-

lation. The terminologyemployed by Allais places him in the Walrasian

line

of

thought,

whereas

Cagan's

terminology

is

in

the

tradition

of the

Cambridge

school.

Fundamentally,

however,

since

Allais

considered

the

ratio

M

M

(1.13)

=

P

D

PQ

his

formulation

was

that of the

Cambridge

school,

whereas

Cagan,

con-

sidering the

ratio

M/P, was

using

the

Walrasian

formulation.

Leaving

aside differencesin terminology, the coefficient used by both authors

was the

weighted

average

of earlier

rates

of

growth

of

total

outlay,

with

weighting coefficients

declining

exponentially

with

distance

in

time.

Overall,

the

results

obtained

by Cagan

in

1954

were

excellent,

while

Allais'

results

(1954) for

France and

the United

States

were

quite re-

markable, and in

general

provided

a

satisfactory

fit

of

the

data. The

only

significant

exception

was the

period 1820-1870

for

France, but the

figures

used for

money

in

circulation

and

national income

were

highly

questionable.

However,

Cagan

had

only succeeded with his

formulation

in representingthe central period of the hyperinflationsstudied. It was

still

beyond

the

power

of

his formulation

to

deal

satisfactorily

with the

beginning

and

terminal

stages.

Furthermore,

he

had been no

more

suc-

cessful than

Allais in

finding

a

single

formulation

capable

of

dealing

with

all

the

cases

studied.5 The

main

results reached

by

the

Allais

(1954) and

Cagan

(1954)

formulations

are

presented

in

Table

1.

The

values found for

x and

K

are

very

different,

and

the

different

laws

cannot be

brought into

any consistent

relationship

with each

other.

But

two

remarks

may be made:

(1) the

values of

the product

Kx

are

remark-

ably steady and are all in the neighborhood of 0.75; (2) the rate of for-

getfulness

x

is

the

greater,

and

the

coefficient K the

smaller,

the

higher

I

Cagan

[10, pp.

43-46 and

55-571.







1126 THE AMERICAN ECONOMIC

REVIEW

TABLE

1-SUiMMAR

0o

CAGAN'S

AND

ALLAIS'

RESULTS:

1954

(Series

including

deposits)

Country

Period

Na

-b

|

1-p2

K

K

Average

u

1820-1848

29

0.520

0.730

0.005

211

1.05

France

1848-1867

20

0.320

0.900

0.006

169

0.98

1871-1913

43

0.970

0.060

0.022

44

0.92

Allais

(1954)

United

States

1871-1935

65

0.950

0.100

0.008

57.2

0.48

1918-1941

24

0.977

0.045

0.007

67.7

0.48

0.034

Average

0.747

0.367

0.010

109.8

0.78

Germany

Sept.

1920-July

1923

35

0.992

0.015

0.20

5.46

1.09

0.168

Austria

Jan.

1921-Aug.

1922

20

0.989

0.021

0.05

8.55

0.43

0.094

Greeceo

Jan.

1943-Aug.

1944

20

0.980

0.040

0.15

4.09

0.61

0.261

Hungary

I

July

1922-Feb.

1924

20

0.926

0.142

0.10

8.70

0.87

0.127

Cagan

Hungary

II

July

1945-Feb.

1946

8

0.998

0.004

0.15

3.63

0.54

0.485

(1954)

Poland

Apr.

1922-Nov.

1923

20

0.972

0.056

0.30

2.30

0.69

0.254

U.S.S.R.0

Jan.

1922-Feb.

1924

26

0.971

0.057

0.35

3.06

1.07

0.395

Average

0.976

0.048

0.20

5.11

0.76

0.255

General

average

0.880

0.181

0.112

48.73

0.77

N

=number

of

observations.

b

p

=coefficient

of

correlation

between

observed

and

calculated

values

of

MIR

(Allais)

or

ln[AI/PI

(Cagan).

a

Excluding

deposits.







ALLAIS: QUANTITY

THEORY OF MONEY 1127

the

average value

fu

of the rate of expansion during the

period consid-

ered. Further, since

the

value

of

relative

desired cash balances ?b s the

smaller

the

higher

the value

of u,

it

may be said

that

the coefficient of

forgetfulness is the higher the lower the value of relative desired cash

balances, or

the

greater the velocity of

circulation.6

II. The

Hereditary,

Relativistic,

and

Logistic

Formulation of the

Demand for Money

A.

The H.R.L. Formulation

The

H.R.L.

formulation

of

the demand for

money,

the

fruit of fifteen

years

of

research,7

is

based

essentially

on

the

following three ideas.

1.

Monetary theory requires a

formulation based on

a

general theory

which can bring all monetary phenomena into the same logical frame-

work,

whether

they

relate to

equilibrium situations,

to

cyclical

fluctua-

tions,

or to

hyperinflations.

2. At any given moment, the

relative demand for money-in other

words,

the demand

for

money

in

relation

to the

national income-

depends only

on the

historical

development

of the nominal

value

of

national income.

There is thus a "hereditary" effect,

with the influence

of

past

events

becoming

attenuated

as

they fall farther back

in

time.

3.

Provided

that a psychological

time scale is considered such that

the rate of forgetfulness per unit of time is constant, the "hereditary"

expression

of

the demand for

money is

the

same at all places

and

times.

The

postulates of the H.R.L.

formulation were

derived from the re-

sults of the

previous researches only

after a long prior

analysis.8

This

paper is limited to their

enunciation, and the deduction of the conse-

quences flowing

from

them.

B.

Relativistic Postulate

(Postulate I)

In relation to

physical time,

it

is assumed that an

instanteous

coeffi-

cient of forgetfulness,

x(t),

can be defined such that the effect of attri-

tion of the

memory

of

past

events between r

and

t

can

be

represented by

the

coefficient

rt

(2.1) exp

-

x(u)du

which has

an

exponential

form. A

psychological

time scale

t'

can

then

6

For

more

details see

[8,

pp.

159-80].

7

Allais [1] to [8].

8

The

starting

point

of the

formulation

below was

my

books

Economie el intergi

[1]

and

Les

fondements

comptables

de la

macro-gconomique

4]

and

the research

which I

undertook be-

tween 1952 and

1955,

which is

described

in

three

successive

papers [2]

[3] [5];

for more details

see

[6]

[8]

[9].







1128

WHE

MERICAN

CONOMIC EVIEW

be defined

such that

(2.2)

x(t)dt

=

x'dt'.

wherex' is a constant; i.e., when measured in relation to

psychological

time,

forgetfulness per unit of

time is constant.

In

other

words, by

reference to this

psychological time scale,

memory

is

invariant.9

This leads to the relation

(2.3)

1'

=

hk)

linking

psychological and physical time.

Writing

D(t)

for

total

outlay per

unit

of

physical

time

by

all eco-

nomic agents,

(2.4)

D(t)

=

E

pi(t)q.(l)

- P(t)Q(t)

where

pi

and

qi represent the

price

and

volume of the

various basic

transactions i, and where

P

and

Q

are

appropriate indices

of

price

and

of

economic activity. When

referred to the

psychological

time

scale,

total

outlay per

unit

of time is

D'('), whence

(2.5)

D'(t')dt'

=

D(t)dt.

However, it is assumed that

the macroeconomic

quantity taken into

consideration

by

economic

agents

in the

process

of

monetary decision-

making

is total

outlay D(t),

whatever the

time-scale referred to. In

other

words,

it is assumed that

D(t) is

an

invariant psychological

datum,

independent of any time-scale

reference.10

his hypothesis was

initially

developed on

the

basis of

psychological

considerations,

and its

retention

is

attributable to the

success of its

application

in

the

analysis of the

em-

pirical series.

When

expressed

in

terms

of

the

psychological

time

scale,

the rate

1

dDQt')

(2.6)

X'(d')D= - -

D(t')

dt'

will

be referredto

as

the instantaneous

rate of

increase

of

total

outlay.

Where

physical

time units are

employed,

this

rate is

given

as"

1

dD(t)

(2.7)

x(t)

=

3-

D(t)

di

I

This

assumption

had

already

been

formulated

in

my

1955

paper

[5, ?26,

pp.

273-74,

rela-

tion

481.

10

See

note (11) below.







ALLAIS: QUANTITY THEORY

OF MAONEY

1129

and of

course,

from

(2.6)

and

(2.7)

(2.8)

x'dt'

=

xdt.

The first postulate thus signifies that there exists a psychological time

t'

which

differs from physical time

t. Measured against this

psycholog-

ical

tim.e scale, the rate of forgetfulness per

unit of time is

assumed

to

be

independent of the

period under consideration. This condition

defines

psychological time. Furthermore, the

quantity of money M and

total

outlay

per unit of physical time D(t), are

taken to be accepted

by eco-

nomic agents as invariant

psychological data, independent of

the time

scale

used for

reference.12

C. Hereditary Postutlate (Postulate II)

Decisions by economic agents are assumed

to be taken in

relation to

past

changes

in

total

outlay, represented by an index which is

termed

"the rate of

psychological expansion."

In

relation to the psychological

time

scale,

the

"psychological rate of

expansion" is assumed to be a

weighted

average

of all

preceding rates of

expansion and is

defined by

the

relation

x

(,r')e-x

"-

)d,r

(2.9)

z

t'

re-x1(?#-')drT

where

X'

is a

constant. This relation

specifies the "hereditary"

nature of

the

link

that

exists between the

psychological rate z'(t') and the

instan-

taneous rates

x'(T').

The

weighting

coefficients

e-x It'W')

decline exponentially with time when referred to the psychological time

scale. It follows from

relation

(2.9) that

dz'

(2.10)

di

IX

-

z'].

And

not, it

cannot be too

strongly

stressed

xIQ')

= 1 dD'(t)

D'(t)

dt'

To consider D and not D' means that psychologically it is D which constitutes an invariant

concept

from the

point of view of

appreciation of

the

economic situation.

On this

point see

[8, pp.

80-81].

12

On

relativistic

effects

in

the

social

sciences,

see

[8, pp.

23-25].

13

On

hereditary

effects

in

the

social

sciences,

see

[8, pp.

21-22].







1130 THE

AMERICAN ECONOMICREVIEW

Postulate

II

is thus

equivalent to the assumption that the formula-

tion

(1.2)

used

by

Cagan and Allais

in

1954

is valid, but that it is valid

only when considered

in

relation to the

psychological

time scale.

The

parameter z' is therefore a magnitude which may be compared with the

parameter

u

of Section

I.

Similarly

X'

is comparable

to the

coefficient

%of that section.

Using

the definition

of z in

the

physical

time scale

(2.11)

z(t)

=z'(t);

then

from

the

relation

(2.3)

and

(2.8),

and using physical time,

we

have

dz

Fx1

(2.12) d= XLX-,Zj

The behavior

of

economic agents insofar

as their desire

to hold

money

is concerned

can be measured by

MD

(2.13)

OD

-

D

D

i.e.,

by

the

ratio

between

desired

money

balances

MD

and

total

outlay

per

unit

of

physical

time, D.

This

ratio is referred

to

throughout

as

"rela-

tive desired money balances"; it can also be designated as the relative

demand

for

money."4

It is assumed

that relative desired

money

balances

PD

are

a

function

of z:

(2.14)

OD

=

?(Z))

Thus,

Postulate

II

signifies

that the

monetary

behavior of economic

operators,

when referred

to a

psychological

time

scale,

is

a function

of

a

psychological

rate

of

expansion.

This rate

is found

as

the

weighted

average

of

the instantaneous

rates of increase

x' of total

outlay D,

mea-

sured against the psychological time scale, with the weighting coeffi-

cients

declining exponentially

with

the

passage

of

time,

and the

coeffi-

cient

of

forgetfulness

x'

being

constant.

D. The

Postulate

of

Invariance

of

the Function

of

Relative

Desired

Money

Balances When

Referred

to

the

Psychological

Time Scale

(Postulate III)

The constant

fo

and the function

sp(z)

are

defined

by

the relation

(2.15)

Oo

=

FD(0)

14

In earlier papers, [11to [7], I used the term "real value of wanted cash balances," but this

suggested

that

OD

was derived

by dividing

M by P, whereas in fact it is the quotient

resulting

from division by

the product PQ. This

was a possible

source of confusion

and I

have altered

the

terminology.

16 It follows

from

this that

0'0

s not the initial

value of ', but the value

of

' for

z=

O.







ALLAIS:

QUANTITY

THEORY OF

MONEY

1131

and

(216)

p(z)

---

It is

assumed that

'p

is an invariant function of z,

i.e.,

that it is inde-

pendent

both of

circumstances

and of

the

institutional framework.

This

function

is

designated

as the

"function

of

desired money balances."

Thus,

Postulate II

signifies

that,

subject

to

application of

a

given

coefficient,

relative desired

money

balances

are

a

function

of

the psy-

chological

rate of

expansion

which is

independent of

time

and

place.

E.

The Postulate of

a

Constant

Velocity of Circulation of Desired Money

Balances when Referred to a Psychological TimneScale (Postulate IV)

When

related

to

a

psychological

time

scale,

the

velocity

of

circulation

tD

of desired

money

balances

can

be

defined

by

the relation

(2.17)

D'

=

JID

I'D1

where

D'

represents total

outlay per unit of psychological time. It is

assumed

that,

as

for the

coefficient

of

forgetfulness x', the velocity V'

is

constant.

When the physical time-scale is referred to, we have

(2.18)

D

=

MDVD

where

VD

is

the

velocity

of circulation of

desired

money balances MD

measured

in

relation to

physical

time,

and

from

(2.13)

(2.19)

VD

-

O5D

The

velocity of circulation

of

desired

money

balances thus appears as

the reciprocal of relative desired money balances. From relations (2.2),

(2.5,

(2.14), (2.17), (2.18)

and

(2.19), we have

(2.20)

x

_

1 1

x

V'D

4(Z)

The

instantaneous coefficient of

forgetfulness x thus appears as a func-

tion of

z. We see that the coefficient

of

forgetfulness is inversely pro-

portional

to

the level

of

relative desired

money balances.

Postulate IV thus

signifies

that the

velocity

of

circulation of desired

16

MD is

invariant,

.e.

MID

=

MD

where

M'Drepresentshe value of desired

money balanceswith

reference o the psychological

time

scale.







1132

THE

AMERICAN

ECONOMIC

REVIEW

money balances, when referred to

a

psychological

time

scale,

is

assumed

to be a constant,

independent of time. It follows from this and the pre-

ceding postulates that the

rate of forgetfulness is inversely

proportional

to relative desired money balances, i.e., proportional to the velocity of

circulation of

desired cash

balances.

It is natural

to

assume

that for

(2.21)

z=

z

=

0

the instantaneous scale of

psychological time is identical

to

the physical

time

scale. This implies that

(2.22)

Xo

=

x(z

=

O) =x'

17

From (2.14), (2.15) and (2.20)

x'

(2.23)

Xo

=

-

-

i.e.,

from

(2.22)

(2.24)

VD

(PO

so that from (2.20)

x f

(2.25)

x'

____

and

also,

from

(2.16)

x

1

(2.26)

,

and

whence, from (2.12)

and (2.22)

dz

F

z

(2.27) --_=

xoL

s

F. The Hereditary

Relativistic Formiulation

To facilitate

the

econometric analysis, it is advantageous to consider

the

index

Z

defined by the

relation

(2.28)

Z(t)=

z(t)

xo0

17

It

follows fromii

his

assumnption

hat in

a stationary process.

in

Ohich total outlay

D)

is

constant,

the

psychological time scale is identical

to the physical time scale.

In

this case,

.(t)-n,

so

that

Z(t)=0,

whence

X=Xo

and, from (2.2), di=dt'.







ALLAIS:

QUANTITYTHEORY OF

MONEY

1133

Z has

no

dimension with

reference to

time,

and

it

can for

this reason

be

denoted

as the

"intrinsic

psychological

rate

of

expansion." Putting

(2.29) s(z)

from the set of

Postulates I,

II,

III, and

IV, we can

write

(relations

2.13, 2.14,

2.16, 2.29,

2.27, and

2.28)

MD

(2.30)

OD=-

(PD

(2.31)

-

=--(Z)

dZ xoz

(2.32)

-

-

__-

where

4'

is a

determinate function of

Z.

Tnis formulation

can

be

thought

of as

being

hereditary

and

rela-

tivistic;

hereditary

because

of

relation

(2.9),

relativistic

because

of

rela-

tion

(2.2).

For

simplicity,

it

may

be

designated

as the

"H.R.

formula-

tion."

In

fact,

this

formulation leads to the

following

assumptions:

(a) that the

velocity

of

circulation of

desired

money

balances

(2.33)

VD(t)

=

D(t)

MD(t)

is

a

function of time

only through

the

intermediary

agency

of

the

psy-

chological

rate

of

expansion

Z(t),

which

summarizes

the

past

history

of

the rate of

growth

x(t)

of nominal total

outlay

;18

and

(b)

that

if

the

time scale is

modified in

such a

way

as to render the

instantaneous

rates

of

forgetfulness

constant-in

other words

if the

the

passage

of

time

is

so

regulated

that the

memory

of

past

events fades

at a

constant rate-

the velocity of circulation of desired money balances is constant when

measured

against

this

new reference scale. This

is

equivalent

to

assum-

ing

that

in

physical time the

velocity

of

circulation of

desired

money

balances

VD(t)

is

proportional

to the

instantaneous rate

of

forgetfulness

x(t).

The H.R. formulation does not

determine the function

41(Z);

this

can

only

be

done

by

the

introduction

of

further

postulates.'9

G.

The

Logistic

Postulate

(Postulate

V)

The

following

postulate

has been

derived

by

inductive

reasoning:

The relative change in relative desired money balances is proportional

18

Relations

(2.9), (2.11),

and

(2.28).

19

M

and

MD

of course

are

quantities

without

any

time dimension.

D(t),

x(t), x(t), z(t), and

VD(t)

are

quantities

whose

dimension

s inverted ime.

?(t)

and

so(t)

have a

time

dimension.







1134

THE

AMERICAN

ECONOMIC

REVIEW

CHART 1.

+(Z)

. z0

Z

to

the

change

in

the

psychological

rate of

expansion, the

coefficient

of

proportionality

being proportional

to the

relative

gap between

relative

desired

money

balances and

their

maximum,

which is

assumed to be

finite.

This

property

is translated

by

the

relation

1

d

tM-A

dZ20

(2.34)

-

d-

=M-da

ql

d

OM

di

in

which

{lM designates

the

maximum

value

of

V/,

whence, by

integra-

tion,

1 + b

(2.35)

41(Z)

1= ba

where a and b are two constants and

(2.36)

{AM

=

1

+

b.

Postulate

V

thus

signifies

that the

function

A1

s a

logistic

function

of

Z.

The

shape

of

the

curve

of

the function

Af(Z)

is

shown

in

Chart

1.

H. H.R.L. Formulation

From relations

(2.29), (2.30),

(2.35), (2.9),

(2.11), (2.8),

(2.2), (2.28),

(2.26),

and

(2.22),

the

hereditary,

relativistic,

and

logistic

formulation,

20

The

assumption

of

proportionality

f

(1/#)(d4/dt)

and

dZ/dt

is

quite

natural,

but

if

q'

is

assumed

o

have

AM

as a

maximum

valuLe,

he

coefficient

f

proportionality

hould

cancel out

for

;

=#M.

The

simplest

assumption, hen,

is

to assume

hat the

coefficient f

proportionality

s

linear

n

,6.







ALLAIS:

QUANTITY

THEORY

OF

MONEY

1135

which may

for brevity

be referred

to as the H.R.L. formulation,

and

which

follows

from Postulates

I to V,

is

summarized

by the

following

relationships:

MD

(2.37)

OD-=D

D

OD

(2.38)

-

=

4(Z)

1 +b

(2.39) A(Z)=

1

+

berz

(2.40)

Z(t)

=

f

x(r) exp

[-

f

X(u)du]dr

x_

1

(2.41)

X

1

XO

A(Z)

or,

in differential form21

(2.42)

--

=

+

]

dZ X0

(2.43)

-

=

X Z

di

A/(Z)

The

parameters

a

and

b

have

no dimension. They specify

the form of

the function

VJ(Z).

The coefficient

Xo

specifies

the

rate

of

forgetfulness

of

past

events

when

the value

of Z is zero.

I.

Postulates

Concerning

the

Constants

The constants a, b, and

Xo

of the H.R.L. formulation can be deter-

mined from

three

postulates:

the

asymptotic

postulate,

the

postulate

of

conjunctural

symmetry,22

and the

postulate

of

temporal

symmetry.

The Asymptotic

Postulate

(Postulate

VI)

When

the

relative

demand

for

money

is

very

small,

so that

changes

of its

square

can

be

neglected,

although

the

changes

themselves are

not

necessarily

negligible,

the rate

of

psychological

expansion

is

equivalent

to the

rate

of increase of

the

circulation

of

money

referred

to the

psychological

time

scale.

Mathe-

matically,

this

postulate

can

be written

21

Relations(2.34)and (2.32).

22

In

French

"Postulat

de

symetrie

conjoncturelle."

n

French,

the

phrase

"appreciation

de la

conjoncture"

elates

to

the assessment

of the economic ituation

and its

development

taking

both

terms

n

a

broad

sense.







1136

THE AMERICAN

ECONOMICREVIEW

1

dMD

d,iP

(2.44) z

- -*0

when

0

a0nd

----*0

MD

dt

dt

even if

-

does

not tend to zero.

dt

In

this case, it can

be shown

that it follows

necessarily that

(2.45)

c=

1.

It can in

fact be

shown that the

H.R.L.

formulation

implies2

(1-a +

-)

I dMfD 1

+

b {7v

(2.46)

-

dM (

z

-

I

MID

dt'

f

(t

1 +

b

whence

it follows

immediately that Postulate

VI

implies (2.45).

The

significance of this

postulate

becomes clear if

it is assumed

that

(as

experience

confirms) the

difference (M-

MD)

always

remnains ela-

tively

small,14 .e.,

that

in

practice

I

dM

D

I

dM

(2.47) _ M 1dM

3MD

dt'

M

dt'

The

practical

interpretation

of Postulate

VI is that at the

end of

a hy-

perinflation

the

psychological rate

of

monetary expansion

z

is

equivalent

to

the rate

of

growth of the

supply of

money

referredto the

psycholog-

ical

time scale,

which

appears to be a

quite reasonable

property to

assume.

It

is

interesting to note

that from

(2.46) this

property is

always valid

whatever a

if the

assumption is made

that

dil/dt

tends to zero

at the

same time as ,6.But in point of fact, for none of the hyperinflations to

date is it

possible to

consider

d/./dt

as

negligible by

comparison

with

the

rate

z=XOZ2s

The

Postuiate of

Psychological

Conjunctural

Synmetry (Postulate

VII).

Starting from

stability (Z

=

0), the

intensity of the

differentialbehavior

of the

economic

agents is

assumed to

be

the same,

whether

expansion

or

recession

is in

progress.

This

condition

implies

that the

derivative

dq,/dZ

of relative

desired

money balances

with

respect to

Z

is a

paired

function of Z in

the

neighborhood of Z=O.

It

can

then be deduced

immediately that

23Relations

(2.2),

(2.6),

(2.10),

(2.11),

(2.13),

(2.16),

(2.29),

and

(2.42).

See

[8, pp.

82-841.

24

See

below,

condition

(3.1).

Is

As

will be

seen from

the

fittings

obtained.







ALLAIS:QUANTITYTHEORY

OF MONEY

1137

(2.48) b

=

1.

The

Postulate of

Psychological ''emporal

Sytmmetry (Postulate

VIII).

Economic agents are assumed

to take the

past into consideration

as

they do

the future. This

postulate implies

that

(2.49)

x

=

i

where i is

the pure rate of

interest.

It follows that

Xo=io

for

Z

=

0.

For the

period

1880-1956, the pure rate of

interest

io

can be taken

as

having been 5 per cent

per annum

in

the United

States, for which

country satisfactory estinmates are

available.26

Thus we

can take

io

0.004

per

montlh

so

that for all countries for which

the rate

io

can

be

considered as prac-

tically equal

to

its

value

in

the

United States

(2.50)

x,

=

0.004

where the time

unIit

is

the month.

J.

The Hlereditar

,

Relativistic and

Logistic

La-w

(a

=

1, b

=

1,

Xo

=

0.004)

In

sum,

the

following formulation follows from

the eight postulates

which have

been

given

earlier:27

2

(2.51)

(z)

1+ ez

dZ

(2.52) -

=

x

-

0.002(1 +

eZ)Z.

This formulation follows from

principles

which

are,

fromi both

an eco-

nomic and

psychological

standpoint,

extremely simple and quite appeal-

ing.

III. The

Confrontation

of

the Theoryand

EmnpiricalData

A.

Money Balances Desired

and

Held

"Desired Money

Balances"'

MD

is

a psychological

concept and

of

course no

statistical

data

to

nmeasure

t

are

available.

However,

at

any

given moment, economic agents are in a

position

to adjust their money

26

The

exact fig-ure s giveii by

Allais "The

Induence of the

Capital-Output Ratio on Real

National

Incomne," Econometrica,

Oct. 1962, 30, p.

714) as 4.87 per cent.

27

Relations

(2.39), (2.43),

(2.45), (2.48),

and

(2.50)







1138

THE

AMERICANECONOMIC

REVIEW

balances

M

from

existing

towards desired

levels

MD

either by

spend-

ing more

or

buying

less.28

Naturally,

this

adjustment

is

never

perfect,

but

it

can

reasonably

be

suggested that the discrepancy between the actual and the desired value

of

money holdings is

always

relatively

small. It then

follows

that

it

can

be

assumed

at

least as

a first

approximation

that

(3.1) M

-MD

MD

where

e

is a

small

quantity.

It further

follows that it

is possible to

write as

a

first

approximation

(3.2) MD - M

(3.3)

OD

4)

with

M

(3.4)

D

Likewise as a

first

approximation,

it

can

be

assumed

that

D

(3.5)

R-

a constant

where

R

is national

income at current

prices. Thus we can

take

M

(3.6)

OD

R

R

l

dR

(3.7)

X

:Z::x

R

-

R

dt

During periods of hyperinflation, the index of activity Q does not in

general

change

appreciably

and the relation

R

(3.8)

- a

constant

where

P

is

the

price

index,

can be

assumed.

It

follows that

for

these

periods,

it is

possible

to take

M

(3.9)

PD= p

28

This

is

naturallynly

an

assumption,

ut as will

be

seen,

t

is

onewhichs

justified

y

its

consequences.







ALLAIS:

QUANTITY

THEORY

OF

MONEY

1139

1

dP

(3.10)

X

P-_

B. The Available Statistical Data

Discrete

weekly,

monthly,

quarterly, or

annual

series are

available

directly

or

may be

calculated

for the

values

Mn,

Rn

or

Mn,

P,b

(hyperinflations)

from

which series we

can derive

the

values of

1

Rn

1

Pn.

(3.11)

X=-1n

(or

-In

Pn)

p

Rn_

P

p

where

p

is

equal

to

0.2308, 1,

3,

or

12,

according

to

whether

the

series

considered

is

weekly,

monthly,

quarterly,

or

annual,

the time

unit

being

the

month.

Xn represents

the

average rate of

growth

of

total

outlay

between

t,_1

and

tn.

Fifteen

series have

been

analyzed

for

nine

countries. The

data used

were

those

of M.

Friedman

and M.

Allais for

the

United

States,

Great

Britain, and France, while P. Cagan's figures were used for the seven

hyperinflations

considered.29

C.

Fitting of the H.R.L.

Formulation

From

(3.2)

and

(3.6) it is

possible

to use

(3.12)

M*=

RD

as an

estimate of

M.

Then, from

(2.38)

(3.13) M*

=

OPoRVI*(Z)

where

4f*(Z)

represents the

theoretical value of

i1, given by

(2.51),

with

Z

determined by

the

differential

equation

(2.52). Thus30

2'00

(3.14)

Al*-

R

1

+

eZ

dZ

(3.15)

-

=

x

-

0.002(1 +

eZ)Z

with

29

For

a

discussion

of the data

used and

their

characteristics,

see [8, pp.

45-63] and

[9].

30

Relations

(2.51),

(2.52) and

(2.2).







1140 THE

AMERICAN

ECONOMIC

REVIEW

1 dR

(3.16)

x(t) -

-

If the R, are taken as given so that the xn are also given,

Zn

can be cal-

culated by integration of

(3.12) from the values of xn and the

initial

value

Z,

of Z. If a value of

P0

s then taken, M* can be calculated from

equation (3.14). For hyperinflations, of course, the index of

national

income R is replaced by the

price index P.

In

each case, the values

of the two constants

Z,

and

%o

ave

been

fixed by the condition that

1

n=-N*

(3.17)

e2

-

-

[In

Mn

-

In

AI,]2

Nn~l

be a

minimum,

i.e.,

by the

conditioin

that the sum of squares of the

devi-

ations of the calculated from

the observed values of the logarithm of

M

is a minimum. This condition is considered as the criterion of

the best

possible fit. The minimum

E2

of e2 can be considered as a coefficient which

represents the goodness of

the fit.

Practically, rather than consider the initial value

Z,

of Z, it is more

convenient

to

consider the

coefficient k defined by the relations

(3.18)

II

1

+

b

t4>o

1 +

bea(z1)

where

Z,

is the value of Z at the initial instant

ti

considered,

and

(ZA)

represents what the value

of

Z,

would be if the empirical point corre-

sponding to instant t1 were exactly on

the

fitted curve. For illustrative

purposes,

the

coefficient of correlation

p

between

In

MIf

and

ln

Mn*

has

been calculated for each

series fitted.

Once

the

two

values

of k

and

Sb0

re determined,

the

calculation

of

AM*(t) rom the values of

R(t) is completely independent of the

values

of

M(t). These two

coefficients

oo

and

iz appear

as constants of

integra-

tion."3

D. Results Obtained for

thte

Nominal Value of Desired Money

Balances

The

calculations

which have so far

been made for the nine countries

considered at

different times show that the

course

of

events

is

exactly

as if social

psychological attitudes towards

the

relative value

of desired

money balances ,6 were practically invariant over time and between

different countries and that this

invariance

is characterized

by

the

three

universal constants.

31

For details on the

computations iinvolved in fitting the data, see [8,

pp

96-1011.







ALLAIS: QUANTITY

THEORY OF MONEY 1141

(3.19)

a

=

1

b-1

Xo-0.004

Three

countries were studied

for the whole of the

period 1900-

1960: France, the United Kingdom, and the United States.32 Six

more

countries were studied

during

periods

in

which their econo-

mies were dominated

by hyperinflationary

conditions:

Germany,

Austria, Greece, Hungary, Poland,

and Soviet Russia. Table

2

gives

the

principal results corresponding to

the fitting of the

same HI.R.L.

law

(a= 1, b= 1,

Xo=0.004)

to the

relevant

series.

The

parameter p corresponds to the

number of months between two

successive

observations,

and n the number

of

observations.

p

is the

coefficient

of

correlation between the

observed

and

calculated values of

In

Mn

and In

M*,

so that

I_-p2

represents the unexplained part of the

variance of

In

AM,.

It can be seen that the

coefficients

C2

n general have

very low values.

If

the coefficients of correlation are

examined, it will further be ob-

served

that out of

fifteen

coefficients eleven

exceed 0.99;

eight are equal

to or

greater

than

0.995,

while there

are

two whose

value exceeds

0.999.33,34,35

By way

of

illustration, the values of

hInand

M*

are

plotted against

time in Charts 2, 3,

4,

5, 6 for

France

(1919-1938), United States (1918-

1941), Germany (December 1919-October 1923), Hungary (July 1945-

July 1946),

U.S.S.R.

(January

1922-February 1924).

The

German

fit-

ting

is

particularly

excellent.

The

values of

(An,n,

A/*)

are

plotted

in

Chart

7

as

a

function of

Z,,

for

France

(1947-1962),

the

United States

(1918-1941),

Germany (Dec. 1919-Oct. 1923),

corresponding

in

all

to

87

pairs

of values of

M

and

R.

The

coefficients of correlation

between

the 87

pairs

of values

of

Vt',

and

41*

and

ln

41V,

nd

In

4/*

are

respectively

0.9983 and

O.9959).38

Chart

8

shows

the

values of

4/,n

and

iA*

as

a

func-

22

Excluding wartime

periods. During such periods,

the

assumption that is translated by

relation (3.1) cannot be considered as valid.

33

These last two results are

practically equal

to those

obtained

in

simulation runs on arti-

ficial series whose structure

corresponded exactly

to

the

H.R.L.

law.

For

example, for the

20-term artificial series

n-IA,

the coefficient

of

correlation

betwseen

V&

nd

V/*

was

0.99976,

whereas for

the Ger-man

figures (excluding deposits)

over

the

period March

1920-October

1923,

a total of

43

termris,

he coefficient of correlation between

(In

Al

and

ln

M*)

was

0.99987.

"

Naturally,

if shorter series

only

are

fitted,

it

is

generally possible

to obtain

better fits.

Lack of space precludes their

presentation

here.

See

[6].

35

t is interesting

to

note that

if the

differential

equation (2.43)

in

Z is integrated using the

observed

valtues

p(Z)

=

(Z)/Oo

instead of

the

theoretical values

V*(Z)

as

calculated

by (2.51),

the results

are

less

good.

For

example,

for

Germany (15th

Dec.

1919-16th

Oct. 1923,

includ-

ing deposits),

the lowest

value

of e2

corresponding

to the

integration using

the

qt(Z)

is 0.019,

whereast is 0.0032

using

the

VI,*(Z).

16

The

respective regression equations

are

=

=

0.9872,p6

-

0.0037

ln

V/

=

0.9991

ln

V/*

0.0030







1142

THE

AMERICAN

ECONOMIC

REVIEW

TABLE

2-THE

NOMIN

VALUE

OF

DESIR

MONE

BALAN

FITTED

BY

TEE

COMMO

H.R.L.

LAW

a

=

1

b=

1

X=

0.004

(Series

including

deposits)

Index

of

National

Income

Correlation

Country

Period

Chart

Series

p

N

(or

Prices)

|

b0

k

e2

1

(nM,

InM*)

Start

of

Period

End

of

Period

|p

i

-p2

1898-1913

1

J

12

16

0.73

1.00

1.2

1.00

0.00028

0.992

0.015

France

1919-1938

2

2

C

12

20

2.06

7.53

1.4

1.80

0.0104

0.974

0.051

1947-1962

4

12

16

63.23

587.8

1.8

0.95

0.0019

0.997

0.0051

Great

Britain

1925-1940

i

C

12

16

0.95

1.46

1.4

1.10

0.00090

0.973

0.053

1952-1962

1

E

12

11

3.04

5.27

0.95

0.80

0.00088

0.973

0.053

1895-191

2C

12

18

0.40

1.09

1.65

1

0.0012

0.997

0.0061

United

States

1918-1941

3

1

12

24

1.67

2.84

1.8

1

0.00089

0.990

0.021

1946-1958

4

A

3

52

4.65

9.61

1.5

0.95

0.00067

0.978

0.043

Average

(or

total)

173

1.08

0.0021

0.984

0.031

Germany

Dec.

1919-Oct.

1923

4

3

bis

1

47

8.03

1.09X

109

1.35

1.0

0.0032

0.9987

0.0026

Austria

Jan.

1921-Aug.

1922

3

1

20

65.91

1.07X

104

2.1

1.1

0.017

0.994

0.013

Greeceb

June

1941-Oct.

1944

2B

bis

1

41

1.00

9.95X

108

3.5

1.0

0.050

0.9982

0.0039

Hungary

Ib

July

1921-Feb.

1924

2

quarter

1

32

1.00

438.00

5.5

1.0

0.018

0.9960

0.0079

Hungary

II

July

1945-July

1946

5

3

ter

1

13

105

4.00X102

1.2

1.0

0.429

0.9991

0.0019

Polandb

Jan.

1921-Jan.

1924

2

C

bis

1

37

1.00

9.63X

103

14.5

1.0

0.022

0.9992

0.0015

U.S.S.R.b

Jan.

1922-Feb.

1924

6

2

1

26

2.88X

105

1.71X

1010

0.700

1.0

0.051

0.995

0.011

Average

for

hy-

erinflations

(or

total)

216

1.01

0.0270

0.9971

0.0060

General

average

(or

total)

_

-

I

|

|

389

|

_

-

|

1.05

0.013o

0.990

0.019

a

e2=Average

of

[In

A,,-In

M.*]2.

b

Excluding

deposits

(series

including

deposits

are

not

available

for

the

periods

considered).

e

Excluding

Hungary

II.







,

~~1.51,.

-DSRED

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BALANCE$ _.''_

rx_,.....I:...--:--1--'--

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3

m

$

l..lA..l^9X9Sz.l...l...:l,l....l.,,.,.,,..l,L.li': l

~~~~~~~~~~~~~~~

I' 4' ' 1 i' l ';: '4i '

1 1

I:i:''4'

1'''-''' ''-''1"

:i,

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,,,,,

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11.

.

...

1 $

1 ,'

rr.t.tit4C'1i5 -I.li

1:

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.&

w

-

l ;l l .

2 l

- t

'-

--'-

I j

l -1 t @''''

l''

,:''|'''X '''','t D rll;

l l l l

l: l:, ? Fi l l -

l ll- l: 's: ;''i

' i

ii -Ili

1

1 ;

t

L

F lt 1 - l - 1 ' |

' '1' ' ' |'

;|' :' |' ' ' |' ' ' |" ' '|' ' ' |

:| . ^ | " |

~~j

1:

. f

j

lo,

P.

915

6 7 8

9 920 1

Z

3 4 6

7

6 9 193

5

2 5 4

5 6 1'

e

941940

RART 3

~

i

;

4







1144

THE AMERICAN

ECONOMIC

REVIEW

L;, __

--l-

r

-i4-t-i-+-A-.=-4

-

_--H------rh--

-

n

___/DRED

ION4CY

SAhANCCS4-

FT"

10~~~~~~~~~t'Lt11Th-f

T lJ l l r T

r l

rr- i

I

l0

oo000X0glVD'1,..ll

FL

T

5~~~47eo

='=

;; 11 ?IT

vs A r

t0d~~

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ST-'lllikJli

K

SJl

_7'I

2

10---

- -

I

]

1

aI0,

1

, I I > .,p, 1 1 1 I i

I~

~~~

I

? i 1 1 1 1 1 ;

I

1

i

1 1 1 1 1 1 1 1 1 1

1i'

,

r

10

4~~~'4

T

HARTF4

to:

IllllX

Iiii

1i1Ifl11

m

fofOrTlltSt lilGlllllSi

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i||.j1 -1.

:0

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.

A

so DJ,,

A

JJASONDJtnA"

2

~ ~ ~ ~ ~ ~ ~~HB 4







ALLAIS:

QUANTITY

THEORY OF MONEY

1145

DIC

IIED

/IOA/-Y SALA

A111V-'1

I

1'

i0a

30

F -

11|1

SIE S

ion

to

1o

10

-~~~~~~~0

100

I

I I I I

I

I

10

10

I10

ND

J

f

M J

J.

A

S O

N

0

J f

M

P

J

A

S

N

,

J

F-

1

A M

J

*

0

J

F

K

A

J J

N

CHART







1146

THE

AMERICAN

ECONOMIC

REVIEW

_

OE3/RD

MONEY

8AL ANCE6S

-_

if

f

I

I I

,, ~~~~~~)

AA

:0077

000

11

0

Iz

I I

L

I

J I

i

-1W-1

I

I

.

:

1~~~~~~1

1&

t 4 ; . ;

fIllllkl'llldlllT[l ll'~~~~~~~~~~~~~~~~~~~Iii

10

-

-~~~~~~~~~~~-.-*i

0

STITCk9L

4

X

S +

X

X S2 |

lE

2 l |

IN

D

F

M

A

M

J

J

A

I I

0

i

1M

A

1

J 0 N

1

1

M

A-

1

-A

S

0

N

D

J

F

M

A

M

J

J

A S

I I

CHART6







ALLAIS:

QUANTITY

THEORY

OF

MONEY

1147

\FUNC

TION

OF

0'S1/REO

MONEY

8ALAWCCJ

9

AS

A

.

FUNC7TON OF

THE

COJAINCTURAL INDEX Z

.

-\---|-

OVRALL

P/T

Or

7THT

DAA

_vsl-

/ sI

.

He

ISAIC

A

...AW

--5_

~AAArC

(ws~ t$s2

?/A,'77f0

.

* cu

Deposts ,

i

. .

\

......................AXILES'

0ASS

ew F

AGSZ.USC....

._

--

; -\

.-

-7rH4-

-F

___PA_Z_OF

ZLCG

1

I-

--~~~~~

~ ~

,,

,,38~3.(

,I ''' --' - ' " ''. '

.001 ____

I

*i.

O7~~~~

_______~~~~~~~~~

he

o)re4

c

alj

.c9ve1

,

.

IX ?*

voJiuji;,OJ 10

z

} l

....*

* -'-5*.i...t--. ,.-',*-

Vai

I'- 7,2*oS ........... ._.,te.fnuLo

_id...._

N-l47?

..

.......

.r

e ; . ..........

,

{

0 | * i l

.

...I

N

.

.......

O..... ._ _

____ _____.

IN.

_

A

.

_ _

__;

1

.......

- '

.1

--1- . _ N4 i

l |

K.-~ ~~~T

1f947

T

-i

'germnany'

.(Dc.

.99

..Oct. .?3

So.:

.rce.:

Ta

l

f

.................

0 -a

-1 0

I

o 7 J

CHART

7







1148

THE AMERICAN

ECONOMIC

REVIEW

AMMCTON

OF

O1C./RZO

MIONEY

BALANCCJ

AS

A

AUNcTrION

OP

THE'

CONJUNCTURAL

INVDEXZ

"T-7.

~ ~ ~ ~ ~

OA~

I 2fl

~~L~~aA14Ea M~~4O~f

,(nIR~

-A.

71~~~~~~~~~~~6n4'Y~frsa

4

I ~ ~ ~ ~

~ ~ ~

Lc~~~~(fs~~~~~a

~~~S4)

77

7

-.4~~~w.~.b(i

i4d34~9(

wim~~~4

LIAND

..

..A.4ICT

oo {~~

XPAN.5/M'A

........C.....O..LA.

V_. As_._

-0004IV

'0.75

~~ ~ ~ ~ ~

~~~.......

. ..

r;+~~~~GE'

j t

.

AL

Ware .

0.25 . ... ...

4* Ezd~~~~~,d~~~,.9

.p..~~~~~~4aar

Greec.e...

....N

o

1

2 3~~~~....5

...

7. 8 9

CHART

...

8

tion of

Z~~~orhe fifteen numerical

series

in Table

2 in total 389

airs~~~~~~~~~~t7

of vaues

f M nd R

ThecoeficieTs-fcreain

ewe

h

8

and025."

~~ heespective

regression~.quations.ar

=

96A

001

lnJ.

.07n~~-007







ALLAIS:

QUANTI'TY

THEORY

OF MONEY

1149

E.

The

Significance

of the Results

As has been seen,

the fitting of the time series AI,

R for a given

country over a given period using the same formulation with the same

constants

ot=1

b

=1

Xi,

=

0.004t

depends only on the two constants of

integration

qo

and

Z1, or

fo

and

k.

Overall,

the

agreement between the values

forecast by the

theory

and

the

observed data is

surprising.

It

goes far beyond what could reason-

ably have been

hoped for on a priori grounds. The

values

Of

_2

and

I-p2 are so low

that the degree of explanation can be

considered as

complete. The fact that the sauie curve could be fitted to all empirical

data

is the

more remarkable when

it

is

considered that,

for

example,

the

highest and lowest values of the index

of national income in the United

States between 1918 and 1941 were

1.08 and 2.84,

whereas between

December 1919 and October 1923,

the index of prices

in Germany

varied

from 15 to

1.09X 1tO, while for

Hungary, between July

1945

and

J-uly 1946, a base

value of 105 rose to the

astroinomical figure of 4X

1029.38

Once the

integration.

constants

k and

5o

have

been

chosen,

the

numer-

ical

integration of

equation (3.15)

dZ

(3.20) --

=

x

-

0.002(1 + ez)Z

di

and

the calculation

of the

M*

using relation (3.13)

(3.21)

M* --

R

1

+

ez

with

1 dR 1 dP \

(3.22)

x

-

-= (or --- for the

liyperinflations1

depends only

on

the value

of

R or P.

The value

of M

does

not

enter

into

the

calculation

in

any way. Thus, once

the integration

constants

k and

?o

are

established,

the

calculated

values, MI,

are

wholly independent

of

the

observed

values of 21.39,40

38

See

Charts 2, 3,

and

4.

39

But,

of

course,

the criterion

that

the

arbitrary coefficients

00

and

k which

represent

the

constants of

integration should

be so

chosen as

to

minimize

e2

means that both

of

these

param-

eters depend on the observed values of

M,,.

40

Moreover,

it can

easily be shown that the

influence of the initial value

Z,

of

Z, i.e.,

of the

coefficient

k,

fades

progressively,

and

that it has

an

asymptotic

value of 0. It follows cor-

respondingly

that the values of

A*

for

the end-period observations are

practically independent

of

the selected value of

k,

whose

influence

is

restricted to

the beginning-of-period

values

of,.







1150

THE AMERICAN

ECONOMICREVIEW

The

results

that

have been

obtained are so extraordinarily close

that

they seem too good

to be true; they lead one to

wonder whether

they

are

not

necessarily

so close, i.e., whether there may

not be some implicit

circularity. The answer is simple. As the relationships are defined, each

curve

fitted

to

the data

depends only on two arbitrary

parameters.

This

means that on the

graph of

M

and

M*

it is possible to arrange for the

terminal points of

the curves that

represent

M

and M* to coincide. But

when

this

condition

has been fixed

there are

no

longer any degrees of

freedom, and there

is

no

a priori reason for the

MI

and

M*

curves to

remain near to each

other, or indeed to be

practically identical to each

other,

as

they

are for

Germany (Chart 3) over the whole

range of varia-

tion between the

two extreme

points. The correspondence that is ob-

served is thus a reflection of the nature of things, and it must be con-

cluded that the

observed dependence

is indeed

a

real one.41

It

should

further be stressed that

the parameters

a, m, and

Xo

have

exactly

the

same values for each

set of data

fitted, and that better

results

could still

be obtained for each country if, as

seems

reasonable,

each parameter

were allowed

a

certain

range

(01, CX2),

(b17

b2),

(XO,1,X0,2)

of

variation

about the

central values

a= 1,

b=

1,

Xo=0.004.

Such

a

pos-

sibility can improve the results, but without greatly affecting their con-

ceptual

and

economic

significance.

The values

a=

1, b=1,

x=0.004

are in any event

only orders of

magnitude which

it

seems

perfectly

reasonable to

assume. Thus,

in

the same way as the pure rate

of

interest

may vary

over

a

certain

range,

it is

reasonable to

envisage

that the rate

xo

may

fluctuate about

a certain

average

value.42

From the discussion

above,

the

value X0=0.004

can be

validly

re-

tained

only

in

cases

in

which

the

equilibrium

level of

the

pure

rate of

interest is in the

region of 0.4 per cent per month,

i.e.,

5

per

cent

per

annum. If this condition is not met, other values of Xoshould be used.

It

should be added that the

better

the

statistical

quality

of the time

series

used,

the better

the

fit

obtained.

In

fact,

the curve

fittings

that

are

relatively speaking the least

satisfactory concern

time series whose sta-

tistical quality can

be

seriously questioned.43

In

any event,

the

slight

41

On

the

testing

of

the

results

see

[8,

p.

143,

note

120].

42

As

a

matter of

fact,

if

it is assumed that the

parameters c, b,

and

xo

can take

any

value

whatsoever for each series studied, better results still can be obtained, with reductions of the

values of

62

and

1

-R2 of the order of 2 to

1

or more in certain cases. When considering rela-

tively long periods in which structural changes have occurred, the results can be further

im-

proved by introducing

an

exponential trend of

the

form

+-='PePo(Z).

These results will be described in a separate paper.

43

It may also be remarked that if the three postulates VI, VII and VIII concerning the

con-







ALLAIS: QUANTITY

THEORY OF MONEY

1151

differences

between

M

and

M*

are

merely

the

reflection

of the gaps

(3.23) M-MD-M-M*

between actual and desired cash balances. The theory here presented

carries

no

implication

that

the

gap

is

zero; merely

that

it

is

small.

In

fact, the author has shown

elsewhere that the differenceM- MD (which

may

be termed the inflationary or deflationary gap, as the case may be)

plays a decisive role in monetary

developments, and that at least as a

first approximation we can write the relation

(1.10).44

The

determina-

tion of the inflationary gap

which is rendered possible by

the present

theory can clearly

be

of great

importance for the implementation of

monetary policy.

Finally, to the extent that the formulation obtained can be considered

valid, it becomes possible

to make meaningful comparisons between

situations that are a

priori

incomparable,since the intensity

of a hyper-

inflationarysituation can

be measured by the rate Z, which

has intrinsic

significance. Thus

the Polish

hyperinflation

of

November

1923 may

be

compared with the German hyperinflation of July 1923,

the

two

cor-

responding values of

Z

being 4.60 and 4.56 respectively.

IV.

The Economic

Significance of

the Formulation

A. The H.R.L. Formulation

The

aim

of

the

formulation proposed here is

to

give operational

con-

tent to

the

quantity theory

of money, by showing that the velocity of

circulation

of

money

at

any given

moment can be

very

closely

described

by

an

invariant

functional of

all the

earlier rates of growth

x of total

outlay

defined

by (2.7).

It shows that this

function

can itself be ex-

pressed very simply

as a function of

a rate

Z, designated

as

the

psy-

chological

rate

of

expansion,

and

defined

by

the

differential

equation

(3.15). The evidence provided

by the series studied to date suggests

that the interdependenceof V (or 4) and Z appearsto be the same at all

times and

in all

places. Thus

the formulation

proposed

in

this

paper

constitutes a

key

to the

placing

of all the

known

facts

within

a

single

framework.

Desired money balances

MD

are defined as

the

Droduct

of

relative

stants

in

?18 were not

retained,

but the

assumption

of

a

common law were

maintained

never-

theless, the common values of

the

parameters a,

b, and

Xo

would no longer be determinate and

could then be chosen

in

such

a

way

as to render the overall fitting of

the curves as close as

possible to

the

data.

In this

case,

the H.R.L. law (a= 1, b

=

1,

Xo=0.004)

would no longer be

the best common law

conceivable, although it would doubtless represent

a good approxima-

tion to the best common

law.

In

other words, from this point of view,

excellent as the results

appear to be, they could

probably be further improved.

The

degree

of

improvement,

however,

would of

course be

less

than

the

possible improvement,

under the

assumption

that a,

b,

and

Xo

are not necessarily identical for all the series considered

(note 42 above).

44

Allais, 1954, A non-linear model

of

cyclical

fluctuations [3, ?8,

relation (7), p. 186].







1152

TFHE

AMERICAN ECONOMIC

REVIEWNt

desired cash balances

4

by

total

outlay.

Relative

desired

money

bal-

ances

are

moreover inothing

other than the

coefficient

K

of the Cam-

bridge school. The proposed formulation

is

hereditary in the sense

that

it establishes a functional dependence between the demand for money

as recordedat any

given moment

and

the previously observed values of

the rate of growth of

total outlay. It is relativistic in the sense

that

this

dependence can be given a

simple formulation

in

terms

of a

psycholog-

ical time-scale. Finally, it is logistic, in the sense that it leads to a logis-

tic curve when the

expression of relative desired money balances is con-

sidered as a function of the

psychological conjunctural index.

The

eight postulates

on

which

the

H.R.L. formulation is

based are

not indissolubly linked

to each other. While the hereditary and rela-

tivistic postulates do seem to form a unit, the logistic postulate, despite

its appeal, need not

necessarily be retained.

In

the same way, Postulate

VIII on

psychological temporal symmetry

is

completely independent

of

the

remaining two postulates VI

and

VII

on

the

constants

a

and

b.

B. The Economtic

Interpretation of the Results

The

results obtained can be interpreted

in

either of two

ways, accord-

ing to whether the expression for relative desired money balances or

the

interrelationship

of

the price

level

and

the

quantity

of

money

is

being considered. These two interpretations, however, correspond to

one, and one only,

underlying reality.

Relative desired

money

balances which differ little from the

recipro-

cal

of the

velocity

of

circulation

may

be taken first.

Their level

is ex-

pressed

as a determinate function

(4.1)

PD

=

Po0b(Z)

of

an

index

of

psychological expansion,

which

in

its turn is

a

determinate

function of the historical development of total expenditure.45

If instead of

considering

relative

desired

money balances,

attention

is focussed on V, the

velocity of circulation,

then

1

46

(4.2)

V

---

Since

(4.3)

OD

=

MD/D

=

MD/PQ

so

that

45

Or,

in

hyperinflations,

of

prices.

4'

From

(2.4), (3.1),

and

(3.4).







ALLAIS:

QUANTITY

THEORY

OF

MONEY

1153

(4.4)

MD

=

PQPD

whence it

follows that, at

any given

moment,

desired

money

balances

are proportional to the product of the price level and the level of eco-

nomic

activity.

Turning

to

the

link

between

prices

and

the quantity

of

money,

ex-

pressed by the

relationship

1

M

i7

(4.5)

--

4

_

sb0)f4Z)

Q

it can

be seen

that, at any given

time, there

is a proportional

relation-

ship between the

price level

and

the

quantity of

money. But the coeffi-

cient

of

proportionality is not

constant; its value at

each moment

depends on the past historical development of total outlay. Thus, the

quantity

theory

of

money appears to be

fundamentally correct,

pro-

vided that

it

is redefined on the

basis of the hereditary

and

relativistic

formulation

which has been

presented

above. The velocity of

circula-

tion

which

is

inversely

proportional to

relative desired

money balances

is not a

constant,

but

appears as

an

invariant

function

of

past

develop-

ments.

The

link of

proportionality

at

any given time

between the

quantity of

money and the

product of the price

level and

the level of activity

(rela-

tion 4.5) is a relationship of interdependence, and from the dynamic

standpoint of causality can

obviously be

interpreted

in either

of

two

ways.

It

may

mean

that the

price level is

proportional

to

the ratio of

the

quantity of money to

the level

of

activity;

or it may

mean

that

the

quantity

of

money

is

proportional

to

the

product

of the

price

level and

the

level of

economic activity.

The

examination of this double

interpre-

tation

is

beyond the

scope

of

this

study.

In

general,

both

viewpoints

are

valid, for

in

changing economic

situations, effects

influence causes,

and

different weights must be

given

to the one or the

other according to

circumstances. In some cases, it may be the first interpretation which

applies,48

whereas at

other times the second

may provide the

explana-

tion

of

what

constitutes

the

motive

force.49

The

theory

proposed

here

makes

possible

a

new, operational reformu-

lation

of the

quantity

theory

of

money,

and

enables a

synthesis to be

made

of

points

of view

which hitherto

have

been

considered

as

com-

pletely

opposed. The

standpoint

of the

quantity

theorist is validated in

as

much as there is a

proportional

relation between

prices and the

quantity

of

money.

But

so

is

the

anti-quantity theory view

since

the

coefficient of proportionality is not constant. Clearly, considered in

itself the

formulation of the

demand for money does not and cannot

47

Which follows from

(2.4),

(2.13),

(2.31), (3.3),

and

(3.4).

48

This

is so, for

example, of

inflations reflecting

the growth of the means

payment

(infla-

tion in the

16th

century,

hyperinflations, etc.).

49

E.g., wage

inflation (as in

France in

1936 and 1945,

etc.).







1154

THE

AMERICAN

ECONOMIC

REVIEW

constitute a

theory of

monetary

dynamics. It

is

but one

element

of a

much

greater

whole;50 but

it

is

obviously an

element

of

considerable

importance.

The only

aim

of

the

H.R.L.

formulation

is

to

specify

how,

at any given moment, the demand for money and the velocity of circu-

lation

are

determined.

C.

The

H.R.L.

Formulation

and

Previous

Researches

It

may

be of

help to

show

very

briefly

the

relation of

the

present

H.R.L.

formulation

to

earlier

formulations.

Fundamentally,

"desired

money

balances"

MD

in

this

paper is

none

other than

Walras'

"desired

money

balances," but

the

concept of

relative

desired

money

balances

4D

corresponds

to

the

coefficient K in

the

Cambridge

formulation,

and

in fact is little different from the reciprocal, 1/V, of the velocity of

circulation,

V,

in

Newcomb

and

Fisher's

formulation.5'

Cagan's 1954

formulation

has

been

compared with

Allais'

1954

formulation

in the

introduction

of

this

paper.

The

present

Allais

formulation

(1963)

differs from

the 1954 Allais

formulation in

that

the

two

equations

(1.ib)

and

(1.2a) are

replaced

by

the

two

equations

(4.6)

OPD

+b

1 + be-z

dZ

1

+beaz

(4.7)

d

-x-x

-z

By contrast

with all

earlier

formulations, the

theory

underlying both

Allais'

formulations

(1954

and

1963)

is

based on

the

existence of

a

nonzero

difference

M-MD

between

the

supply

of

and

the demand

for

money.

This

difference

determines

changes in

total

outlay

D

by the

equation

(1.10).

The

fact that

this

difference

remains small

enables

empirical

study

of

relative

desired

money balances to be founded on the

assumption that

for

practical

purposes

the two

quantities are

equiva-

lent

(condition

3.2).

The

essential

differences

between

the

H.R.L.

formulation

of 1963

and

Allais'

and

Cagan's 1954

formulations are:

(a) the

coefficient

x

of for-

getfulness is no

longer a

constant,

but

a

function

of the

index

I6(Z)

which

satisfies the

condition

(2.41);

(b)

the

function

i1,

instead of

being

exponential

(Cagan)

or

linear

(Allais

1954)

in

u,

is

a

logistic

function of

Z.

Further,

in

the

context

of

analysis

of

the

current

situation,

there

is

the

essential

point that

Allais'

formulations

of

1954 and

1963

differ from

Cagan's

formulation,

in

that

Allais

defines x

by

(1.3b),

whereas

in

Cagan's

formulation it

is

defined

by

(1.3a).

In

fact,

defining

x

by

(a)

50

In

my

papers

of 1953

[2],

1954

[3], and

1955

[5] I

have

tried to

trace the

outlines of such

a

theory; see

in

particular

relation

(1.10)

above.

61

See

Allais

[8, pp.

33-34 and

153-54].







ALLAIS:

QUANTITY

THEORY OF MONEY

1155

results

in

a

less

satisfactory fit

of the data

than when

(b)

is

used.

This is

because definition

(b)

takes account of

both

changes in the price

level

and

changes in

the level of

economic

activity,

and

correspondingly

pro-

vides a better image of the development of the economic situation.52

The

superiority of the

H.R.L. formulation

over

Cagan's

formulation

can be

assessed

by comparingthe

periods

covered, and

the values

of N

and

1-

p2,

as

between Tables 1

and 2. The main

advances represented

by the

H.R.L. (a= 1, b=

1,

Xo=O.004)

formulation

by comparison

with

Cagan

formulation are

a

better

fitting

of

short

series,53the

possi-

bility of

fitting long

series, the

applicability of the

same

formulation to

different

situations. The IH.R.L.

common

law generally

offers a better

description of

the facts

than the earlier

Cagan

or Allais

formulations,

though the characteristic parameters in the latter varied from one

series

to

the

next54

and

though

the

number

of

arbitrary

parameters is

two

instead of three.

Accordingto

Friedman's

and

Cagan's

view,

the

coefficient u (or

E)55

appears

as

a

"coefficient

of

expectation"

and it

is

quite

natural to

as-

sume

that

the

expected

rate of

change

du/dt

of

the

rate of

expectation

u

is

determined

in

each

period of

time

in

proportion

to the

difference

be-

tween the actual rate of

change

x

of total

outlay

D

and

the

rate of

change

u

that

had been

expected

(differential equation

1.2a).

Accord-

ing to Allais' view in 1954, u is related to the past and not to the future.

This rate

reflects the

memory

of the

past

retained

by

economic

agents,

and

the

farther

back

in

the

past

that events

are

recalled,

the

more

it

fades. When the coefficient

of

forgetfulness

x

is

constant the

two formu-

lations

(1.2) are

mathematically

the

same,

but

their

philosophical

start-

ing points

are

different.

Since

Cagan's

and

Allais'

1954

views

are

merely

two different

interpretations

of

the

same

mathematical formulation

there is

no

point

in

suggesting

that either

is

superior.

The new

H.R.L.

formulation

which

is

proposed

in

this

paper

differs

from Allais' earlierformulation and the Cagan and Friedmanformula-

tions

in

that it takes

the

psychological

time

scale

into

consideration.

Here,

from the

differential

equation

(2.27)

it

is

no

longer possible

to in-

terpret

the

rate of

expansion

z

as

being proportional

to

the

difference

x-z. On

the

contrary,

the new

formulation

appears

to be

a

quite

natural

generalization of Allais' earlier

conception.56

To

the

extent

that

formulation

gives

better

results,

Allais' view

appears

to

correspond

bet-

ter to the real nature of the facts.

52

For further details on the comparison of the present formulation and those which have

been put forward

in the

past,

see

[8,

Table

VI, p.

154].

53

Allais [8,

Tables

IX

and

X, pp. 178-79].

"4For further details see

[8, Appendix I].

6"

See Introduction

and footnote 2.

56

Since from postulate

of

II.c.,

relation

(1.2) holds when

referred

to

a

psychological time

scale II.c., relation

(2.10).







1156

THE

AMERICAN ECONOMICREVIEW

D. The Proposed Formulatwon

The

results

show

that

human

societies placed

in

very

different cir-

cumstances, rangingfrom normal situations to hyperinflations, in capi-

talistic or communistic

regimes, today or

a

half-century ago,

react ac-

cording

to one

and the same

law,

one

which

moreover

might

validly

have been deduced

from a priori reasoning. This similarity of

behavior

should

be

interpreted

as corresponding o

the

invariance

of human psy-

chology

in

space

and

over

time,

at

least in its

collective aspect.

In

fact,

the

proposed

formulation

can be applied

in

numerous other

fields. The index of psychological expansion,

which

can

easily

be

calcu-

lated

at

any time,

provides

a

quantitative

measure of the

economy's ap-

preciation of the conjunctural situation.57 It is therefore very probable,

if

not certain,

that the introduction

of

this

index into

economic fore-

casting

and

analytical

calculations

would

greatly improve

both the

analysis

of

past

developments

and

the

methods employed

for

making

projections.

The results obtained

in this

study

of

monetary phenomena

are

no

more than an illustration of the importance

of

hereditary

and rela-

tivistic effects

in

the behavioral sciences.

There

is no

doubt that these

phenomena are merely one particular

example

of

many

cases

in

which

the methods and principleswouldbe valid over the whole rangeof social

phenomena

to

the

extent that these phenomena

are

linked

by

relation-

ships

of

a

hereditary type,

and

to the extent

that

time

scales are

subject

to

relativistic

effects.58

The

proposed

formulation may

at

first

sight appear

to be

complex.

But it is verified

by the available empirical data,

at

least

in

the

cases

studied

up

to

now.

So

long

as

fresh data

do

not invalidate

it,

it

must be

considered

as

corresponding

o the

nature of

things.

Whether

the heredi-

tary, relativistic, and logistic

formulation

(a= 1,

b=

1,

Xo

0.004)

is

confirmedor invalidated as a general law by the analysis of further

numerical series,50

he numerical

regularities

which

have been

brought

out will stand.

As

with

other

theories,

so

with

this

one.

Its

logical

formu-

lation

may

be maintained

or

modified,

but the

real

relationships

between

observable quantities which

it

has brought

out

will

have

been

estab-

lished once

and for

all,

and this is certainly

what

must be

considered as

the essential

contribution

of this

study.

67

In

French

"Situation conjoncturelle."

See footnote 22 above.

58

From

relations (2.2), (2.26),

and (2.29)

the

function

A/(Z)

represents

the

value

of

the unit

of psychological

time for society

as a whole

as a function of the intrinsic

rate

of

expansion

Z.

69 The author

of

this

paper would be grateful

to receive

time series

(M.,

Rl) or (AI,,,

P,,)

relating to any

country and covering

any

period for purposes

of analysis.

Monthly values

are

preferable,

if they can

be supplied.







ALLAIS:

QUANTITY THEORY

OF MONEY 1157

REFERENCES

1.

M. F. C. ALLAIS,

Economzy nd

Interest

(Economie et

interet).

Librairie

de Medicis (in deposit). Paris 1947.

2.

,

Illustration

of the Theory of

Economic

Cycles by Non-linear

Models (Illustration

de la theorie

des

cycles

econorniques

par

des

modeles non-lineaires).

Paper

submitted

to the European

Congress

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the Econometric

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August 31, 1953.

3.

,

Explanation

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Economic

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Model with

Lagged Relations

(Explication

des cycles economiques

par

un

modele

non-lineaire

a

regulation retardee).

Paper submitted

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European

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August

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1954.

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1954.

S.

- -,

Explanation

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modele

non-lineaire

a regulation

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Paper

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International

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Published

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Dynamic

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Collection

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Proceedings

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International Colloquia

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169-308.

4. ,

Hereditary Effects

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T'imten the Social Sci-

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Application

to

the

Theory of

MIonetary

Phenomena (Des

effets hereditaires et

de la relativite

du temps

dans les sciences sociales

--Application 'a

la

theorie

des phenomenes

monetaires).

Paper sub-

mitted to the Centre

National de

la Recherche

Scientifique,

Decem-

ber 31,

1963.

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, The Hereditary, Relativistic

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Logistic

Formulation

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the

Demand of

Money.

Lecture delivered

at the University

of

Chlicago,

May 11,

1964.

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,

A Restatementof the Quantity Theory of Money (Reformulation

de

la

theorie

quantitative

de la monnaie),

Bull.

SEDEIS,

Sept. 1965,

No. 928, Supp.

9.

-,

A Restatement

of

the

Qutantity

Theory

of

Money (Reformulation

de

la

theorie

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de

la

monnaie),

in

preparation

for

publication

by

Hermann,

Paris.

(This

volume

will

present

the

full framework

of

the

study

dealt

with

in

the

present

paper,

which

has

been

limited to

giving

a

summary

of the

results obtained.)

10. P. D.

CAGAN,

"The Monetary

Dynamics

of

Hyperinflation,"

in

M.

Friedman,

ed., Studies

in the Quantity

of Money,

Chicago 1956, pp.

25-

117. (The essential results had been given earlier in Cagan's doctoral

dissertation, 1954.)

11.

MILTON FRIEDMAN,

"The Quantity

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of

Money-A

Restate-

ment,"

in

Studies

in the Quantity

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,

A Theory of

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Documents