Interactions between the Multiplier Analysis and the Principle of Acceleration
Author(s): Paul A. Samuelson
Source: The Review of Economics and Statistics, Vol. 21, No. 2 (May, 1939), pp. 75-78
Published by: The MIT Press
Stable URL: http://www.jstor.org/stable/1927758
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INTERACTIONS BETWEEN THE MULTIPLIER ANALYSIS
AND THE PRINCIPLE OF ACCELERATION
FEW economistswoulddenythatthe "multiplier"
analysis of the effects of governmental
deficit spending has thrown some light
upon this important problem. Nevertheless,
there would seem to be some ground for the
fear that this extremely simplified mechanism
is in dangerof hardeninginto a dogma,hindering
progress and obscuring important subsidiary
relations and processes. It is highly
desirable, therefore, that model sequences,
which operateundermore generalassumptions,
be investigated, possibly including the conventional
analysis as a special case.'
In particular, the "multiplier," using this
termin its usual sense, does not pretendto give
the relation between total national income induced
by governmentalspendingand the original
amountof moneyspent. This is clearlyseen
by a simple example. In an economy (not
necessarily our own) where any dollar of governmental
deficit spending would result in a
hundreddollarsless of private investment than
wouldotherwisehavebeenundertaken,the ratio
of total induced national income to the initial
expenditureis overwhelminglynegative, yet the
"multiplier"in the strict sense must be positive.
The answerto the puzzle is simple. What
the multiplierdoes give is the ratio of the total
increase in the national income to the total
amount of investment, governmental and private.
In other words, it does not tell us how
much is to be multiplied. The effects upon private
investment are often regardedas tertiary
influencesandreceivelittle systematicattention.
In order to remedy the situation in some
measure,ProfessorHansenhas developeda new
model sequencewhichingeniouslycombinesthe
multiplieranalysis with that of the acceleration
principle or relation. This is done by making
additionsto the nationalincomeconsistof three
components: (i) governmentaldeficitspending,
(2) private consumption expenditure induced
by previouspublicexpenditure,and (3) induced
private investment, assumed according to the
familiar acceleration principle to be proportional
to the time increaseof consumption. The
introductionof the last componentaccountsfor
the novelty of the conclusionsreachedand also
the increasedcomplexityof the analysis.
A numericalexamplemay be cited to illuminate
the assumptions made. We assume governmental
deficit spending of one dollar per
unit period, beginning at some initial time and
continuing thereafter. The marginal propensity
to consume, a, is taken to be one-half. This
is taken to mean that the consumptionof any
period is equal to one-half the national income
of the previous period. Our last assumptionis
that inducedprivate investmentis proportional
to the increase in consumption between the
previous and the currentperiod. This factor of
proportionality or relation, /3, is provisionally
taken to be equal to unity; i.e., a time increase
in consumptionof one dollar will result in one
dollar's worth of induced private investment.
In the initial period when the government
spends a dollar for the first time, there will be
no consumptioninducedfrompreviousperiods,
and hence the addition to the national income
will equal the one dollar spent. This will yield
fifty cents of consumption expenditure in the
second period, an increase of fifty cents over
the consumption of the first period, and so
accordingto the relationwe will have fifty cents
worth of induced private investment. Finally,
we must add the new dollar of expenditureby
the government. The national income of the
second period must thereforetotal two dollars.
Similarly, in the third period the national income
would be the sum of one dollar of
consumption,fifty cents inducedprivate investment,
and one dollar current governmentalexpenditure.
It is clear that given the values of
the marginalpropensity to consume,a, and the
relation, f, all succeeding national income levels
can be easily computed in succession. This is
done in detail in Table i and illustrated in
Chart i. It will be noted that the introduction
of the acceleration principle causes our series
to reach a peak at the 3rd year, a trough at
the 7th, a peak at the iith, etc. Such oscil'
The writer, who has made this study in connection with
his research as a member of the Society of Fellows at Harvard
University, wishes to express his indebtedness to Professor
Alvin H. Hansen of Harvard University at whose
suggestion the investigation was undertaken.
[75]
76 THE REVIEW OF ECONOMIC STATISTICS
latory behaviorcould not occur in the conventional
model sequences, as will soon become
evident.
For other chosen values of a and A similar
model sequences can be developed. In Table 2
national income totals are given for various selected
values of these coefficients. In the first
column, for example, the marginal propensity
to consume is assumed to be one-half, and the
relation to be equal to zero. This is of special
interest because it shows the conventionalmultiplier
sequencesto be special cases of the more
generalHansen analysis. For this case no oscillations
are possible. In the second column the
oscillationsin the nationalincomeareundamped
and regular. In column three things are still
worse; the oscillations are explosive, becoming
largerand largerbut always fluctuatingaround
an "averagevalue." In the fourth column the
behavioris no longeroscillatorybut is explosive
upward approachinga compoundinterest rate
of growth.
By this time the investigator is inclined to
feel somewhatdisorganized. A variety of qualitatively
differentresults emerge in a seemingly
capricious manner from minor changes in hypotheses.
Worse than this, how can we be sure
that for still different selected values of our
coefficientsnew and strongertypes of behavior
will not emerge? Is it not even possible that
if Table 2 were extendedto cover moreperiods,
new types of behavior might result for these
selected coefficients?
Fortunately, these questions can be given a
definitenegative answer. Arithmeticalmethods
cannot do so since we cannot try all possible
values of the coefficientsnor compute the endless
termsof each sequence. Nevertheless, comparatively
simple algebraic analysis can be
appliedwhich will yield all possible qualitative
types of behavior and enable us to unify our
results.
The national income at time t, Yt, can be
written as the sum of three components: (i)
governmentalexpenditure,gt, (2) consumption
expenditure, Ct, and (3) induced private investment,
It.
Yt- gt+Ct+It.
But accordingto the Hansen assumptions
Ct=aYt-1
It=/3[Ct-Ct-i] =a/3Yt-i-a/3Yt-2
and
gt= I.
Therefore,ournationalincomecan be rewritten
Yt= I +a [ I+#] Yt-1-aflyt-2CHART
I.-GRAPHIC REPRESENTATION OF DATA IN
TABLE I
(Unit: one dollar)
Goyernmentexpenditure
Consumption
Privateinvestment
2 -
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14
TABLE I.-THE DEVELOPMENT OF NATIONAL INCOME
AS A RESULT OF A CONTINUOUS LEVEL OF GOVERNMENTAL
EXPENDITURE WHEN THE MARGINAL PROPENSITY
TO CONSUME EQUALS ONE-HALF AND THE
RELATION EQUALS UNITY
(Unit: one dollar)
Current Current
Current consump- private
govern- tion investment Total
Period mental induced proportional national
expendi- by to time income
ture previous increase in
expenditure consumption
I ....... I.00 0.00 0.00 I.00
2 ...... I.00 0.50 0.50 2.00
3 ...... I.00 I.00 0.50 2.50
4 ....... I .00 I.25 0.25 2.50
5 .00 .00 I.25 0.00 2.25
6 ...... I.00 I.I25 -O.I25* 2.00
7 ...... 1.00 I.00 -O.I25 I.875
8 ...... I .00 0.-9375 -o.o625 I.875
9 ...... I .00 0.9375 0.00 I*9375
I 0 ....... I.00 0.96875 0.03I25 2.00
I I . 1..... I.00 I.00 0.03I25 2.03I25
I2 ......I...00 I.OI5625 O.OI5625 2.03I25
I3 ......I...00 IOI5625 0.00 2.0I5625
I4 ...... I .00 I.0078I25 -0.0078I25 2.00
~~~~~~~. . . . . . . . . . ,.. . . . . . . . . . . . ........
*Negative induced private investment is interpreted to
mean that for the system as a whole there is less investment
in this period than there otherwise would have been. Since
this is a marginal analysis, superimposed implicitly upon a
going state of affairs, this concept causes no difficulty.
MULTIPLIER ANALYSIS-PRINCIPLE OF ACCELERATION 77
In words, if we know the national income for
two periods, the national incomefor the following
period can be simply derived by taking a
weighted sum. The weights depend, of course,
upon the values chosen for the marginal propensity
to consumeand for the relation.
This is one of the simplesttypes of difference
equations, having constant coefficientsand being
of the second order. The mathematicaldetails
of its solution need not be entered upon
here. Sufficeit to say that its solution depends
upon the roots-which in turn depend upon
the coefficientsa and,8- of a certainequation.'
It can be easily shown that the whole field of
possible values of a and , can be divided
into four regions, each of which gives qualitatively
differenttypes of behavior. In Chart 2
these regions are plotted. Each point in this
diagramrepresentsaselection of values for the
marginal propensity to consume and the relation.
Correspondingto each point there will be
a model sequence of national income through
time. The qualitative properties of this sequence
depend upon whether the point is in
Region A, B, C, or D.2 The propertiesof each
region can be briefly summarized.
Region A (relatively small values of the re-
lation)
If there is a constant level of governmental
expenditurethrough time, the national income
will approachasymptoticallya value times
I-a
the constant level of governmentalexpenditure.
A single impulseof expenditure,or any amount
of expenditure followed by a complete cessation,
will result in a gradual approach to the
original zero level of national income. (It will
be noted that the asymptoteapproachedis identically
that given by the Keynes-Kahn-Clark
formula. Their analysis applies to points along
the a axis and is subsumedunderthe moregeneral
Hansen analysis.) Perfectly periodic net
governmentalexpenditurewill result eventually
in perfectly periodic fluctuations in national
income.
RegionB
A constant continuinglevel of governmental
expenditure will result in damped oscillatory
movements of national income, gradually approaching
the asymptote times the con-
I-a
stant level of government expenditure. (Cf.
Table i.) Governmentalexpenditurein a single
orfinitenumberof periodswill resulteventually
in damped oscillations around the level of income
zero. Perfectly regular periodic fluctuations
in government expenditure will result
eventuallyin fluctuationsof incomeof the same
period.
Region C
A constant level of governmentalexpenditure
will result in explosive, ever increasingoscillations
aroundan asymptote computedas above.
(Cf. column3 of Table 2.) A single impulseof
expenditure or a finite number of expenditure
impulseswill resulteventuallyin explosiveoscillations
aroundthe level zero.
Region D (large values of the marginalpropensity
to consume and the relation)
A constantlevel of governmentalexpenditure
will resultin an ever increasingnationalincome,
eventually approaching a compound interest
rate of growth. (Cf. column4 of Table 2.) A
TABLE 2.-MODEL SEQUENCES OF NATIONAL INCOME FOR
SELECTED VALUES OF MARGINAL PROPENSITY TO CONSUME
AND RELATION
(Unit: one dollar)
Period a= .5 a= .5 a= .6 a= .8
3=0 8=2 8=2 8 =4
I ... I.00 I.00 I.00 I.00
2 . 1I..50 2.50 2.80 5.00
3 ...... .75 3-75 4.84 I7.80
4 ........ I.875 4.I25 6.352 56.20
5 ... .I9375 3-4375 6.6256 I69.84
6 ........ I.9688* 2.03I3 5.3037 500.52
7 . 1...... I-9844 .9I4I 2.5959 I,459.592
8 ........ I.9922 - JII72 - .69I8 4,227.704
9 ........ I-996I .2 I48 -3.3603 I2,24II.2I6
. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .
*Table is correct to four decimal Dlaces.
Actually, the solution can be written in the form
I
Yt=-~+aj[xi]t+aJ[x2V t
I-a
where Xi and X2 are roots of the quadratic equation
x2-a[i+,P]x+aq=o,
and a, and a2 are constants dependent upon the a's and 6's
chosen.
2Mathematically, the regions are demarcated by the conditions
that the roots of the equation referred to in the previous
footnote be real or complex, greater or less than unity
in absolute value.
78 THE REVIEW OF ECONOMIC STATISTICS
single impulse of net investment will likewise
send the system up to infinity at a compound
interest rate of growth. On the other hand, a
single infinitesimal unit of disinvestment will
send the systemever downwardat an increasing
rate. This is a highly unstable situation, but
correspondsmost closely to the pure case of
pump-priming, where the total increase in
national income bears no finite ratio to the
original stimulus.
The limitations inherent in so simplified a
picture as that presented here should not be
overlooked.' In Darticular.it assumes that the
marginal propensity to consume and the relation
are constants; actually these will change
with the level of income,so that this representation
is strictly a marginalanalysis to be applied
to the study of small oscillations. Nevertheless,
it is more generalthan the usual analysis. Contrary
to the impressioncommonlyheld, mathematical
methods properly employed, far from
makingeconomictheorymoreabstract,actually
serve as a powerful liberating device enabling
the entertainment and analysis of ever more
realistic and complicatedhypotheses.
CHART 2.-DIAGRAM SHOWING BOUNDARIES OF REGIONS YIELDING DIFFERENT
QUALITATIVE BEHAVIOR OF NATIONAL INCOME
OL.
12 -
Ot.
0.8
02
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.
,- _ _._ _ _ _ _ .................. ,.A
'It may be mentioned in passing that the formal structure
of our problem is identical with the model sequences of
Lundberg, and the dynamic theories of Tinbergen. The
present problem is so simple that it provides a useful introduction
to the mathematical theory of the latter's work.
PAUL A. SAMUELSON
HARvARDUNivERsiTY