samuelson restat 1939

7/30/2019 Samuelson RESTAT 1939

http://slidepdf.com/reader/full/samuelson-restat-1939 1/5

Interactions between the Multiplier Analysis and the Principle of

Acceleration

Paul A. Samueison

The Review ofEconomie Statisties, Val. 21, No.2 (May, 1939),75-78.

1IIìiiiil"1IìiII®

Stable URL:

http://links.jstor.org/sici?sici=0034-6535%28193905%2921%3A2%3C75%3A1BTMAA%3E2.0.CO%3B2-D

The Review ofEconomie Statistics is current1y published by The MIT Presso

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at

http://www.jstor.org/about/terrns.html. JSTOR's Terms and Conditions ofUse provides, in part, that unless you

have obtained prior permission, you may not download an entire issue of ajoumal or multiple copies of articles, and

you may use content in the JSTOR archive only for your personal, non-commerciai use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at

http://www.jstor.org/joumals/mitpress.html.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or

printed page of such transmission.

JSTOR is an independent not-for-profit organization dedicated to creating and preserving a digitaI archive of

scholarly joumals. For more information regarding JSTOR, please contact [email protected].

http://www.jstor.org/

Mon Oct 1707:58:03 2005



INTERACTIONS BETWEEN TRE MULTIPLIER ANALYSIS

AND TRE PRINCIPLE OF ACCELERATION

FEW economists would deny that the "multiplier" analysis of the effects of govern

mental deficit spending has thrown some lightupon this important problem. Nevertheless,there would seem to be some ground for thefear that Ithis extremely simplified mechanismis in danger of hardening into a dogma, hindering progress and obscuring important subsidiary relations and processes. It is highly

desirable, therefore, that model sequences,which operate under more generaI assumptions,be investigated, possibly including the conventional analysis as a special case.1

In particular, the "multiplier," using thisterm in its usual sense, does not pretend to givethe relation between total national income induced by governmental spending and the originaI amount of money spente This is clearly seenby a simple example. In an economy (notnecessarily our own) where any dollar of governmental deficit spending would result in ahundred dollars less of private investment thanwould otherwise have been undertaken, the ratioof total induced national income to the initialexpenditure is overwhelmingly negative, yet the"multiplier" in the strict sense must be positive. The answer to the puzzle is simple. What

the multiplier does give is the ratio of the totalincrease in the national income to the totalamount of investment, governmental and private. In other words, it does not tell us howmuch is to be multiplied. The effects upon private investment are often regarded as tertiaryinfiuences and receive little systematic attention.

In order to remedy the situation in somemeasure, Professor Hansen has developed a new

model sequence which ingeniously combines themultiplier analysis with that of the acceleration

principle or relation. This' is done by makingadditions to the national income consist of threecomponents: (I) governmental deficit spending,(2) private consumption expenditure inducedby previous public expenditure, and (3) induced

1 The wri ter, who has made this study in connection with

bis research as a member of the Society of Fellows a t Har-

vard University, wishes to express his indebtedness to Pro-

fessor Alvin H. Hansen of Harvard University at whosesuggestion the investigation was undertaken.

private investment, assumed according to thefamiliar acceleration principle to be propor

tional to the time increase of consumption. Theintroduction of the last component accounts forthe novelty of the conclusions reached and alsothe increased complexity of the analysis.A numerical example may be cited to illumi

nate the assumptions made. We assume governmental deficit spending of one dollar perunit period, beginning at some initial time andcontinuing thereafter. The marginaI propensity to consume, a, is taken to be one-half. Thisis taken to mean that the consumption of any

period is equal to one-half the national incomeof the previous period. Our last assumption isthat induced private investment is proportionalto the increase in consumption between theprevious and the current period. This factor ofproportionality or relation; {3, is provisionallytaken to be equal to unity; Le., a time increasein consumption of one dollar will result in onedollar 's worth of induced private investment.

In the initial period when the goveinmentspends a dollar for the first time, there will beno consumption induced from previous periods,and hence the addition to the national incomewill equal the one dollar spente This will yieldfifty cents of consumption expehditure in thesecond period, an increase of fifty cents overthe consumption of the first period, and so

according to the relation we will have fifty centsworth of induced private investment. Finally,we must add the new dollar of expenditure by

the government. The national income of thesecond period must therefore total two dollars.Similarly, in the third period the national in

come would be the sum of one dollar ofconsumption, fifty cents induced private investment, and one dollar current governmental expenditure. It is clear that given the values ofthe marginaI propensity to consume, a, and therelation, f3, alI succeeding national income levelscan be easily computed in successione This isdone in detail in Table I and illustrated inChart I . It will be noted that the introductionof the acceleration principle causes our seriesto reach a peak at the 3rd year, a trough at

the 7th, a peak at the 11th, etc. Such oscil-[ 75]



THE REVIEW OF ECONOMIC STATISTICS

TABLE I . - TRE DEVELOPMENT OF NATIONAL IN COME

AS A RESULT OF A CONTINUOUS LEVEL OF GOVERN

MENTAL EXPENDITURE WHEN THE MARGINAL PRO

PENSITY TO CONSUME EQUALS ONE-HALF AND THE

RELATION EQUALS UNITY

(Unit: one doUar)

Current CurrentCurrent consump- privategovern- tion investment Total

Period mental induced proportional nationalexpendi- by to time incometu re previous increase in

expenditure consumption

I . . . . . . . 1.00 0.00 0.00 1.00

2 . . . . . . 1.00 0·50 0·50 2.00

3 . . . . . . 1.00 1.00 0·50 2·50

4 . . . . . . . 1.00 1. 2 5 0.25 2·50

5 . . . . . . . 1.00 1. 2 5 0.00 2.25

6 . . . . . . 1.00 1.12 5 -0 .12 5 * 2.00

7 . . . . . . i.oo 1.00 -0.12 5 1.875

8 . . . . . . 1.00 0·9375 -0.062 5 1.8 75

9. . . . .

.1.00

0·93750.00

1.9375IO . . . . . . . 1.00 0. 9 68 75 0.03 12 5 2.00

I I . . . . . . . 1.00 1.00 0.03 12 5 2.03 12 5

12 . . . . . . 1.00 1.01 56 2 5 0.0 1 56 2 5 2.03 12 5

13 . _ .... 1.00 1.01 56 2 5 0.00 2. 0 1 56 2 5

14 . . . . . . 1.00 1.0078125 - 0 . 007 812 5 2.00

. . . " . . . . ' . .. . . . . . . . . . . . . . . . . . . . . . . .*Negative induced private investment is interpreted to

mean that for the system as a whoIe there is less investmentin this period than there otherwise wouId have been. Sincethis is a ,marginaI anaIysis, superimposed implicit1y upon agoing state of affairs, this concept causes no difficuIty.

Iatory behavior could not occur in the conventional model sequences, as will soon becomeevidenteFor other chosen values of a and f3 similar

model sequences can be developed. In Table 2

national income totaIs are given for various seIected values of these coefficients. In the firstcolumn, for example, the marginaI propensityto consume is assumed to be one-half, and therelation to be equal to zero. This is of specialinterest because it shows the conventional muItiplier sequences to be special cases of the more

generaI Hansen analysis. For this case no oscilIations are possible. In the second column theoscillations in the national income are undampedand regular. In column three things are stilIworse; the oscil1ations are explosive, becomingIarger and Iarger but always fluctuating aroundan "average value." In the fourth column thebehavior is no Ionger oscillatory but is explosiveupward approaching a compound interest rateof growth.By this time the investigator is inclined to

feeI somewhat disorganized. A variety of quali-

tatively different results emerge in a seeminglycapricious manner from minor changes in hypotheses. Worse than this, how can we be surethat for stilI different selected values of ourcoefficients new and stronger types of behaviorwiII not emerge? Is it not even possible thatif Table 2 were extended to cover more periods,new types of behavior might result for theseselected coefficients?Fortunately, these questions can be given a

definite negative answer. Arithmetical methodscannot do so since we cannot try alI possiblevalues of the coefficients nor compute the endIess terms of each sequence. Nevertheless, comparatively simple algebraic analysis can beapplied which will yield alI possible qualitativetypes 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)governmental expenditure, gt, (2) consumptionexpenditure, Ct, and (3) induced private investment, I t •

Yt=gt+Ct+lt.

But according to the Hansen assumptions

Ct=aYt-l

It={3[Ct-Ct-l] =a{3Yt-l-a{3Yt-2

and

,gt= I.Therefore, our national income can be rewritten

Y t= l+a[I+!3] Yt- l - a!3Yt-2.

CHART 1 . -GRAPHIC REPRESENTATION OF DATA IN

TABLE I

(Unit: one doUar)

r:=1 Goyernment expenditure~ Consumpfion

3- Private investment

2

o

2 3 4 5 6 7 8 9 IO 11 12 13 14



MULTIPLIER ANALYSIS - PRINCIPLE OF ACCELERATION 77

In words, if we know the national income fortwo periods, the national income for the following period can be simply derived by taking aweighted sum. The weights depend, of course,upon the values chosen for the marginaI propensity to consume and for the relation.

This is one of the simplest types of differenceequations, having constant coefficients and being of the second order. The mathematical details of its solution need not be entered uponhere. Suffice it to say that its solution dependsupon the roots - which in turn depend uponthe coefficients a and p - of a certain equation.1

TABLE 2.-MoDEL SEQUENCES OF NATIONAL INCOME FOR

SELECTED VALUES OF MARGINAL PROPENSITY TO CON

SUME AND RELATION

(Unit: one dollar)

Period a= ·5 a= .5 a= .6 a= .8

8=0 8=2 8=2 8='4

I . . . . . . . . 1.00 1.00 1.00 1.00

2 . . . . . . . . . . 1.50 2·50 2.80 5·00

3 . . . . . . . 1.75 3·75 4.84 17.80

4 . . . . . . . . . . . 1.875 4. 125 6.352 56.20

5 . . . . . . . . . . 1.9375 3·4375 6.6 256 169.846 . . . . . . . . . . 1.9688 * 2.0313 5.3037 500.52

7 . . . . . . . . . . . 1.9844 .9141 2·5959 1,459.592

8 ............ 1.9922 - .117 2 - .69 18 4,227.704

9 . . . . . . . . . 1.9961 .21 48 -3.3 603 12,241.1216

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

*TabIe is correct to four decimaI places.

It can be easily shown that the whole field ofpossible values of a and f3 can be dividedinto four regions, each of which gives qualitatively different types of behavior. In Chart 2these regions are plotted. Each point in thisdiagram represents selection of values for themarginaI propensity to consume and the relation. Corresponding to each point there will bea model sequence of national income through

time. The qualitative properties of this sequence depend upon whether the point is inRegion A, B, C, or p.2 The properties of eachregion can be briefiy summarized.

1 Actually, the solut ion can be written in the form

Y t =_ I_+a l [X l ] t +a2[x2] t

I-a

where Xl and X2 are roots of the quadrati c equa tionx2 - a[I +,8]x+a,8= 0,

and al and a2 are constants dependent upon the a's and ,8's

chosen.2 Mathematically, the regions are demarcated by the con

ditions that the roots of the equat ion referred to in the pre-

Region A (relatively small values of the relation) -

If there is a constant level of governmentalexpenditure through time, the national income

will approach asymptotically a value _I- timesI-a

the constant level of governmental expenditure.A single impulse of expenditure, or any amountof expenditure followed by a complete cessation, will result in a graduaI approach to theoriginaI zero level of national income. (It willbe noted that the asymptote approached is identically that given by the Keynes-Kahn-Clarkformula. Their analysis applies to points alongthe a axis and is subsumed under the more generaI Hansen analysis.) Perfectly periodic netgovernmental expenditure will result eventually

in perfectly periodic fiuctuations in nationalincome.

Region B

A constant continuing level of governmentalexpenditure will result in damped oscillatorymovements of national income, gradually ap-

proaching the asymptote _ I- times the con-I-a

stant level of government expenditure. (Cf.Table I. ) Governmental expenditure in a single

or finite number of periods will result eventuallyin damped oscillations around the level of income zero. Perfectly regular periodic fiuctuations in government expenditure will resulteventually in fiuctuations of income of the sameperiod.

Region C

A constant level of governmental expenditurewill result in explosive, ever increasing oscillations around an asymptote computed as above.(Cf. column 3 of Table 2.) A single impulse of

expenditure or a finite number of expenditureimpulses will result eventually in explosive oscillations around the level zero.

Region D (large values of the marginaI propensity to consume and the relation)

A constant level of governmental expenditurewill result in an ever increasing national income,eventually approaching a compound interestrate of growth. (Cf. column 4 of Table 2.) A

vious footnote be real or compIex, greater or Iess than unity

in absolute value.



THE REVIEW OF ECONOMIC STATISTICS

CHART 2.-DIAGRAM SHOWING BOUNDARIES OF REGIONS YIELDING DIFFERENT

QUALITATlVE BEHAVIOR OF NATIONAL IN COME

5.0.5.0.5.0.5~

2.0.5.0.5

cl

.2::

/0(.= ~ '/

-0( .= . ~ ~V V

.00/ /

/....- y- ~ /

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

A / \ ~ ~ ....... D•8::

/ ~ ........... ~ ~---I " --...........

--...........

"

,...

------ ---

/"Ì'-. ..

' - ' --- ----...

BI

, ~ ~./11I~I

~--...........

--...........

- I-

----i ' ---- r---

j ---.

----_.

I

0.2

o

o.

o

single impulse of net investment wiII Iikewisesend the system up to infinity at a compoundinterest rate of growth. On the other hand, asingle infinitesimal unit of disinvestment wiII

send the system ever downward at an increasingrate. This is a highly unstable situation, butcorresponds most closely to the pure case ofpump-priming, where the total increase innational income bears no finite ratio to theoriginaI stimulus.The Iimitations inherent in so simplified a

picture as that presented bere should not beoverlooked.1 In particular, it assumes that the

1It may be mentioned in passing that the formaI structureof our problem is identical wi th the model sequences of

marginaI propensity to consume and the relation are constants; actually these wiII changewith the Ievel of income, so that this representation is strictly a marginal analysis to be appliedto the study of smaII oscillations. Nevertheless,it is more generaI than the usual analysis. Contrary to the impression commonly held, mathematical methods properly employed, far frommaking economic theory more abstract, actuallyserve as a powerful Iiberating device enablingthe entertainment and analysis of ever more

realistic and complicated hypotheses.

Lundberg, and the dynamic theories of Tinbergen . Thepresent problem is so simple that it provides a useful introduction to the mathematica l theory of the lat te r' s work.

HARVARD UNIVERSITYPAUL A. SAMUELSON

samuelson restat 1939

Documents