The Review of Economic Studies, Ltd.
A Mathematical Formulation of the Ricardian System
Author(s): Luigi L. Pasinetti
Reviewed work(s):
Source: The Review of Economic Studies, Vol. 27, No. 2 (Feb., 1960), pp. 78-98
Published by: Oxford University Press
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A
MathematicalFormulation of
the
System*
Ricardian
Since his own time, David Ricardohas always occupieda privilegedplace among
economists, even in periods when economic analysis has been developingalong paths
very differentfrom the ones he pursued. It has neverbeen easy, however,for Ricardo's
many interpreters,to state his completesystemin a rigorousand concise form, and the
reasonlies in the peculiarityof some of the conceptshe used whicharenot alwaysdefined
in an unambiguousway. These concepts have encounteredstrong criticismsalmost at
any time, while-on the other hand-the bold analysesthey made possiblewere exerting
a sort of fascinatiDgattraction.
In this paper, criticismis left aside and the more constructiveapproachis taken of
stating explicitlythe assumptionsneeded in order to eliminatethe ambiguities. Then,
the Ricardiansystemis shownto be veryneat and even suitablefor a mathematicalformulation, with all the well-knownadvantagesof conciseness,rigourand clarity. The task-is
undertakenin sections4 to 9, which form the mainpart of the paper(partII). To avoid
digressionsand lengthy referencesthere,the difficultiesRicardowas faced with, and the
basic featuresof his theories,are brieflyreviewedin the firstthree introductorysections
(partI).
I
1. THEORYOF VALUE
The theoryof valuerepresentsthe most toilsomepart of Ricardo'stheoreticalsystem
and in our mathematicalformulationit will entailthe crudestassumptions.At the time it
was put forward,the theorysoon becamethe maintargetof the criticisms,whichRicardo
tried to answerby re-writingtwice (in the second and in the third edition) the chapter
' on value' of his Principles.' No fundamentalchangewas introduced,however,and the
threeversionsrepresentdifferentways of framing(in the light of the criticisms)a theoryof
value which remainsessentiallythe same2
* I am gratefulto Mr. Kaldor, Prof. Modiglianiand Mr. Sraffafor comments and criticism,and to
Dr. JamesMessagefor a most helpful suggestionin the third section of the appendix.
1David Ricardo, On thePrinciplesof PoliticalEconomyandTaxation(cit. as Principles). All references
to Ricardo'sworksin this paperreferto the editionpreparedby Piero Sraffa,The Worksand Correspondence
of David Ricardo,in 10 volumes, CambridgeUniversityPress, 1951, (cit. as Works).
2 This is a view to which recently Mr. Sraffa has given full support (Works, vol. I, Introduction,
pp. XXXVII and ff.). Fragmentsof an early version of the Ricardiantheory of value can be traced in
Ricardo'searly writingsand in some letters (see the evidencegiven by Mr. Sraffa, Works,p. XXXI). It
seems that Ricardo tried at the beginningto measurethe relevant variablesof his system in terms of a
main agriculturalcommodity,namely corn, claimingthat this commodityhas the propertyof being both
the capital and the product and, therefore,makes it possible to determinethe ratio of profit to capital in
physicalterms without any question of evaluation. This position was, however,very vulnerableand will
not be consideredin this paper, as Ricardo abandonedit long before writingthe Principles.
78
A MATHEMATICALFORMULATIONOF RICARDIAN SYSTEM
79
The theory is fundamentallybased on the cost of productionmeasuredin terms of
quantityof labour. Utility' is consideredto be absolutelyessentialto, but not a measure
of, exchangeablevalue. To commoditieswhichderivetheirvaluefrom " scarcityalone"2.
(e.g., rare paintings)only a few words are devoted-they are not consideredrelevantfor
economicanalysis; Ricardois concernedonly with commoditieswhich are the outcome
of a processof production. He begins by restatingAdam Smith'spropositionthat " in
the early stages of society, the exchangeable value of commodities . .. depends . . . on the
comparativequantityof labourexpendedon each ".3
Then, he takes a new and striking
step by asserting that the mentionedproposition is valid in general and not only in the
earlystagesof society, as Smithclaimed. His argumentmay be roughlyexpressedin the
followingway. Supposetwo commodities,A and B, the first of whichrequiresthe work
of one workerfor one year to be producedand the second the work of two workersfor
one year (the capital employedbeingjust the amount of wages to be anticipatedto the
workers). Whateverthe rate of profitmay be, either 10% or 20% or 30%, its amount
on the secondcommodityalwaysis twice as much as on the firstcommodity; hence the
relativepriceof the two goods alwayscomes out as equalto the ratio of the quantitiesof
labour requiredto obtain each of them.4 If a " commoditycould be found, whichnow
and at all times requiredpreciselythe same quantityof labour to produceit, that commodity would be of an unvarying value "5: it would be an invariable standard in
termsof whichthe value of all commoditiescould be expressed.
This formulationof the theory, of course, did not remain unchallenged. Strong
objectionswere immediatelyraised(by Malthus,McCulloch,Torrensand others)which
may be summarisedas follows. Let us suppose, returningto the mentionedexample,
that the productionof commodityB requiresthe workof one workerfor two yearsinstead
of the work of two workersfor one year. In this case, Ricardo'sprincipleno longer
appliesbecause,owing to the profitsbecomingthemselvescapital at the end of the first
year, a changein the rate of profit does imply a changein the relativeprice of the two
commodities,even though the relativequantitiesof labour requiredby them remainthe
same. Ricardocould not ignore these objectionsand alreadyin the first edition of the
Principleshe allowedfor some exceptionsto his generalrule. All exceptions-as he later
explainedin a letter-" come under one of time ",6 but he preferreddiscussingthem, in
the thirdeditionof the Principles,underthreegroups(i. differentproportionsof fixedand
circulatingcapital,ii. unequaldurabilityof fixed capital,iii. unequalrapiditywith which
the circulatingcapitalreturnsto its employer). However,while allowingfor exceptions,
Ricardo kept the fundamentalsof his theory and tried to overcomethe objectionsby
appealingto the orderof magnitudeof the deviationscausedby the exceptions,whichhe
consideredas responsibleonly for minordeparturesfromhis generalrule. In the previous
example,for instance,the modificationintroducedby the possibilitythat the samequantity
of labouron B might be employedin one year or in two differentyears amountssimply
to the effectscausedby the amountof profitto be calculatedon the wagesof the firstyear.
Ricardoholdsthat this is a differenceof minorimportance.7 Therefore,the conclusionis,
1 Needless to say, the term " utility " has for Ricardo, and in general for the Classics, a different
meaningthan for us to-day. It simply refers to the " value in use " of a commodity as opposed to its
" value in exchange". See Principles,p. 11.
2
Principles,p. 12.
3 Principles,p. 12.
4Principles,pp. 24 and ff.
6 Principles,version of editions 1 and 2, see p. 17, footnote 3.
6 Letterto McCulloch, Works,vol. VIII, p. 193.
7
Principles, pp. 36 and ff.
REVIEWOF ECONOMICSTUDIES
80
the theory of value as stated in terms of quantitiesof labour, and independentlyof the
distributionof commoditiesamong the classes of the society, does hold, if not exactly,
at least as a very good approximation.' With this premise,Ricardoconsidersas " the
principalproblemof Political Economy" that of determining" the laws which regulate
the distribution'"2
2.
THEORY OF DISTRIBUTION
The participantsin the process of productionare grouped by Ricardo in three
classes: landlordswho provideland, capitalists3who providecapital and workerswho
providelabour. Total productionis entirelydeterminedby technicalconditionsbut its
divisionamongthe threeclasses-under the form of rent,profitand wages-is determined
by the inter-actionof manytechnical,economicand demographicfactors. All Ricardo's
analysison this subjectrefersto what he calls the naturalpricesof rent,profitsand wages.
Divergenciesof marketpricesfrom their naturallevel are consideredonly as temporary
and unimportantdeviations.
Rent, namely" that portionof the produceof the earthwhichis paid to the landlords
for the use of the originaland indistructiblepowerof the soil "4 is determinedby technical
factors. The technicalpropertythat differentpieces of land have differentfertilityand
that successiveapplicationsof labour to the same quantity of land yield smaller and
smalleramounts of product (law of diminishingmarginalreturns)makes of rent a net
gain for the landlords. Therefore,rent does not enter Ricardo's theory of value-it
is a deductionfrom the total product. The value of commoditiesis determinedby the
quantityof labouremployedon the marginalportionof land-that portionof land which
yieldsno rent.
Wages are not relatedto the contributionof labour to the process of production,
as in the moderntheoriesthey normallyare. Like all economistsof his time, Ricardo
relatesthe level of wages to the physiologicalnecessityof workersand their familiesto
live and reproducethemselves. He is convincedthat in any particularstate of society
there exists a real wage-rate(so to speak, a certain basket of goods) which can be
consideredas the " natural price of labour". It need not necessarilybe at a strict
subsistence level5 (the minimum physiological necessities of life) ; but at that level which
in a givencountryand in a givenstate of society,besidesallowingworkersto live, induces
themto perpetuatethemselves" withouteitherincreaseor diminution".6 Whencapitalists
' With the acceptanceof criticismsbetween the first and the third edition of the Principles,also the
choice of a " standardof value " becamemore difficult. Ricardoreactedto the complicationby changing
his definitions. In the firstedition of the Principleshe regardedas " standard" a commoditywhich would
requireat any time the same amount of unassistedlabour (unassistedby capital); in the third edition he
mentionsa " commodityproducedwith such proportionsof the two kinds of capital(fixedand circulating)
as approach nearest to the average quantity employed in the production of most commodities".
(Principles,p. 63 and p. 45; see also Works,Introductionby Mr. Sraffa,vol. 1, p. XLII and ff.). Ricardo
consideredone year a good average and thought that perhaps gold could be the commodity that most
closely approachesthe requirementof an invariablestandard. (Principles,p. 45.)
2 Principles,p. 5.
"
" manufacturers" according as he refers to agri8 Ricardo calls them alternatively" farmers or
culturalor to industrialcapitalists.
4 Principles,p. 67.
5 " The naturalprice of labour-Ricardo says-varies at differenttimes, in the same country,and very
materiallydiffersin differentcountries . .. it essentiallydependson the habits and customs of the people
.
Many of the conveniences-Ricardo adds-now enjoyedin an Englishcottagewould have been thought
.u.
in an earlierperiod of our history". (Principles,pp. 96-97).
luxuries
6 Principles,p. 93.
A MATHEMATICALFORMULATIONOF RICARDIAN SYSTEM
81
accumulatecapital,demandfor labourincreasesand the marketwage-raterises above its
naturallevel. However,Ricardobelievesthat such a situationcannot be other than a
temporaryone because, as the conditions of workersbecome" flourishingand happy",
they " rear a healthyand numerousfamily"' and the growthof populationagain brings
back the real wage-rateto its naturallevel. It is very impressiveto notice how strongly
Ricardo is convinced of the operation of this mechanism. To be precise, he always
speaksof a processwhich will operate" ultimately"but the emphasison it is so strong
that his analysisis alwayscarriedon as if the responsewerealmostimmediate.
Profits,finally,representa residual. Rent being determinedby the produceof the
marginallandput into cultivation,andthe wagerateby non-economicfactors,whatremains
of the total productionis retained,underthe form of profit, by the capitalists,who are
the organizersof the process of production. The capitalistsare assumedto be always
intent on movingtheir capitaltowardsany sector of the economythat shows a tendency
to yield a rate of profit above the average. This behaviourensuresthe equalizationof
the rateof profit(afterrisk)all overthe economy.
OF ECONOmiC
3. THEORY
GROWTH
Economicgrowthis broughtabout essentiallyby the capitalists. The threeclasses in
which Ricardo divides the society have different peculiar characteristics. Landlords are
considered as an " unproductive class "2 of wealthy people who become richer and richer,
and consume almost all their incomes in luxury goods. Workers also consume everything
they get but in a differentkind of goods-" necessaries"-in orderto live. Capitalists,on
the other hand, are the entrepreneursof the system. They representthe " productive class "2
of the society. Verythrifty,they consumea smallamountof whatthey obtainand devote
their profitsto capitalaccumulation.
The processof transformingprofitsinto capital,however,cannot go on indefinitely.
Owingto the decreasingmarginalreturnsof new capital(and labour)appliedto the same
quantityof land, or to less fertile lands, rent increasesover time, in real and in money
terms,the moneywage-rateincreasestoo3, and consequentlythe profitrate continuously
falls.4 Whenthe rate of profithas fallento zero, capitalistsare preventedfrom accumulating any more; the growth process stops and the system reaches a stationaryst7te.
As a matterof fact-Ricardo adds-the stationarystatewill be reachedbeforethe extreme
point whereall profitshave disappearedbecause,at a certainminimumrate of profit,the
capitalistswill lose any inducementto accumulate. The final outcome (the stationary
state) is postponedin time by new inventionsand discoverieswhich increasethe productivityof labour,but it is Ricardo'sopinionthat it will eventuallybe attained.
II
IN A TWO-COMMODITIES
4. THE" NATURAL"EQUILIBRIUM
SYSTEM
It has been mentioned that Ricardo distinguishes two groups of commodities produced
in the economy: " necessaries "-or, we may call them wage-goods-and " luxuries ".
The most simple Ricardian system we can conceive of is, therefore, one where each of the
two groups is reduced to one commodity. Let us begin with this case and make the following assumptions:
1 Principles,p. 94.
2 Principles, p. 270.
9
How this happenswill appearvery clearlyin the mathematicaltreatmentof the following sections.
4 Principles,especiallychaptersVI and XXI.
REVIEWOF ECONOMICSTUDIES
82
(i) the systemproducesonly one type of wage-good,let us call it corn
(ii) to producecorn, it takes exactlyone year;
(iii) capital consists entirelyof the wage bill; in other words, it is only circulating
capital,whichtakes one year to be re-integrated;
(iv) theredoes exist an invariablestandardof value, namelya commodity,let us call
it gold-a luxury-good-,whichat any time and place alwaysrequiresthe same
quantityof labour to be produced. Its process of productionalso takes one
year. Prices are expressedin terms of such a commodityand the monetary
unit is that quantityof gold which is producedby the labourof one workerin
one year.
The Ricardiansystemcan now be statedin terms of equations. Takingthe quantityof
land in existenceas given and supposingthat its technicalcharacteristics(fertility and
possibilitiesof intensiveexploitation)are known,the productionof corn can be expressed
by a technical productionfunction, which we may assume to be continuously differentiable:
(1) X1
f (N1)
where: X=
N1
physical quantity of corn producedin one year;
=
number of workers employed in
the corn production;
with the followingproperties:
(la) f(0) > 0
(lb) f' (0) >
(lc) f"(NI)
x
where: x=
natural wage-rate in terms of
corn;
< 0.
The firstinequalitymeansthat when no labouris employed,land is supposedto produce
either somethingor nothing at all (negativeproductionis excluded). The meaningof
(lb) is that, at leastwhenthe economicsystembeginsto operateand workersare employed
on the most fertilepiece of land, they must producemore than what is strictlynecessary
for their support,otherwisethe whole economicsystemwould nevercome into existence.
Finally,(ic) expressesthe law of decreasingmarginalreturns.
The productionfunctionfor gold is much simpler:
(2) X2 = a N2
where:
physical quantity of gold producedin one year;
X2
N2
=
=
number of workers employed in
the productionof gold;
physicalquantityof gold produced
by one workerin one year(o > 0).
A MATHEMATICALFORMULATIONOF RICARDIAN SYSTEM
83
The following equations are self-explanatory:1
where: N = total numberof workers;
N1 + N2
(3) N
N,
N2
(4)
agriculturalworkers;
workersin the gold industry;
W = total wage-bill, in terms of physical
Nx
W
=
=
units of corn;
(5)
(6)
(7)
K
R
P1
W
f(N1)
XI -
-
x
K
R
PI
Nlf '(N1)
R - Nlx
=
=
=
=
real wage-rate (corn);
physical stock of capital (corn);
yearly rent, in real terms (corn);
yearly total profits, in real terms
(corn), in the corn producing
sector.
All variablesintroducedso far are in physicalterms. Turningnow to the determination
of values,we have
(8) p,X, - p1 R
(9)
P2X2
N,
where: p,
price of corn;
P2
price of gold.
N2
=
Equatiolns(8) and (9) are very importantin the Ricardiansystem. They state that the
valueof the yearlyproduct,afterdeductionof rent,is determinedby the quantityof labour
requiredto produceit. In our case, owingto the definitionof the monetaryunit, the value
of the product,after paying rent, is exactlyequal to the numberof workersemployed.
From(1), (2) and (6), equations(8) and (9) may be also written
1
N,
(8a)
(8a)
Pi_
Pi
-
(9a)
P2
-
f ' (N1)
Xl-R
Profitsin the gold industryand total profitsin the economyemergeas
profits,in terms of physicalunits
where: P2
(10) P2 P2 =P2 X2- N2 pIx
of gold, in the gold industry;
7 = total profits, in terms of the
(11) 7 :- piX+ p2X2-plR
-p,W
standardof value.
After substitutingfrom (1)-(10),equation(11) may be also written
(11a)
7
=
(N1 + N2) (1-xpl).
1 Equation(6) may not appearso evidentas the otherequations. Let me state, therefore,an alternative
way of writingit. As explainedin section2, rentrepresentsfor Ricardoa surplus,a net gain for the owners
of the more fertile lands with respect to the owners of the marginalland (the land which yields no rent).
Therefore,when N1 workersare employedon land, the resultingtotal rent can be expressedas a sum of all
the net gains of the non-marginalland-owners. In analytical terms
rN,
(6a)
R = f (O) +
f ' (y) - f ' (N1)]dy
wheref (O),from (la), is the producethat the land-ownerscan get from land without rentingit, i.e. without any labour being employed. By solving the integralappearingin (6a), we obtain
R = f (O) + f (N1) - f (O) - N1f ' (N1)
which is exactlyequation(6).
REVIEWOF ECONOMICSTUDIES
84
At this point, the equationscontaina theoryof value and a theoryof distributionbut not
yet a theory of expenditure. Since Ricardo assumesthat all incomes are spent (Say's
law), to determinethe compositionof total expenditureonly one equationis necessary
in the presentmodel, specifyingthe productionof one of the two commodities. Then
the quantityproducedof the other commodityturns out to be implicitlydetermined,as
total productionhas alreadybeen functionallyspecified. The Ricardiantheory is very
primitiveon this point. Workersare supposedto spendtheirincomeon necessities(corn,
in our case)capitalistson capital-accumulation(corn again,in our case) and land-owners
on luxuries. Hence the determiningequationis
(12)
p 1R.
P2 X2
Let us also write
(13)
Pi x
w
(14) r
=
pK
where: w
r
monetary wage-rate;
=
rateof profit.
Pi, P2, w, r,
So far 16 variableshave appeared: Xl, X2, N1, N2, N, W, x, K, R, P1, P2, wc,
but only 14 equations. Two moreequationsare neededin orderto determinethe system.
In a situation which Ricardo considers as natural, the following two data have to be added:
(15)
x =
> 0
(16)
K =
> 0
where: x.
- natural real wage-rate,definedas
that wage-ratewhich keeps population constant;
=
given stock of capital at the begin-
ning of the year.
The system is now completeand determinate.2It can be easily demonstrated(see the
appendix)that properties(la), (lb), (lc) and the inequalitiesput on (15)-(16)are sufficient
conditionsto ensure the existenceand uniquenessof non-negativesolutions. We may
consider, therefore, the system of equations (I)-(16) as defining the natural equilibrium
of the Ricardiansystem.3
OF THE RICARDIANSYSTEM
5. SOMECHARACTERISTICS
Alreadyat this stage, the system of the previoussection clearlyshows some of the
most peculiarcharacteristicsof the Ricardianmodel. First of all, it contains a theory
of valuewhichis completelyand(owingto our explicitassumptions)rigorouslyindependent
1 To be precise,we should allow for a minimumof necessitiesto be bought by the land-owners. This
minimum,however, introducesonly a constant into the analysis without modifying its essential features.
For simplicity,therefore,the procedureis followedof neglectingthe constant,whichamounts to considering
the minimumas negligibleand supposingthat the whole rent is spent on luxuries. Similarly,a minimum
of luxuriesmight be allowed to be bought by the capitalists. This minimumalso will be consideredas
negligible.
2 It may be interestingto notice that equations(1), (4), (5), (6), (7), (15) and (16), taken by themselves,
form an extremely simplifiedbut determinedRicardian system expressed in terms of corn, where any
question of evaluationhas not yet arisen, corn being the single commodityproduced. This is the system
which has been used by Mr. Kaldor in his article " Alternative Theories of Distribution," Review of
EconomicStudies,1955-56.
8 To justify the terminology,let me mention that in his article " On the Notion of Equilibriumand
Disequilibrium" (Reviewof EconomicStudies, 1935-36),ProfessorRagnar Frisch distinguishestwo types
of equilibria: stationaryand moving. The naturalequilibriumof the Ricardiansystem is not a stationary
one, as will be seen in a moment; it belongsto the movingtype. ProfessorFrisch,in that article,describes
a somewhat similarsituation for the Wickselliannormalrate of interest.
A MATHEMATICALFORMULATIONOF RICARDIAN SYSTEM
85
of distribution. From equations(8a) and (9a), it appearsthat the value of commodities
dependsexclusivelyon technicalfactors(the quantityof labourrequiredto producethem)
and on nothingelse.
Moreover,the system shows that wage-goodsand luxury-goodsplay two different
roles in the system. The productionfunction for the wage commodityturns out to be
of fundamentalimportance,while the conditions of production of the luxury-goods,
expressedby a, havein the systema verylimitedinfluence.As can be easilyfoundout (see
also the appendix),the solutionsfor all variables,exceptP2, dependon the functionf (NL)
or on its firstderivative,whilethe constanta only entersthe solutionsfor X2andp2. As a
consequence,the rate of profitand the moneywage-rateare determinedby the conditions
of productionof wage-goodsand are entirelyindependentof the conditionsof production
of luxurygoods.' It follows,for example-to mentionone problemof concernto Ricardothat a tax on wage-goodswould affect all the participantsin the process of production
by changingboth the money wage-rateand the rate of profit (as can be inferredfrom
equations(13a) and (14a)),while a tax on luxury-goodswould affect only the purchasers
of these goods becauseit leavesthe rates of profitand of wagesentirelyunaffected.2
OF THE " NATURAL" EQUILIBRIUM
AND THE ATTAINMENT
6. THE MARKETSOLUTIONS
Ricardoadmitsthat the marketoutcomesmay not necessarilycoincidewith those of
his " natural" equilibrium,but he considerstwo types of mechanismswhich make the
formerconvergetowardsthe latter. First, he mentionsthe behaviourof the capitalists,
whose readinessto move theircapitaltowardsthe most profitablesectorsof the economy
alwayscause the rates of profitto equalizein all sectors. Secondly,he considersthe increase of the workingpopulationin responseto increasesin wages. About the first of
thesetwo processes,Ricardodoes not reallysay muchmore than what is said above. He
does not find it useful to enterinto complicateddetails(and in this case they would have
been very complicatedindeed for him, who did not possess a demandtheory). Simply
he allowsfor the processandcarrieson his analysis(thesystem(I)-(16)) on the assumption
that the equalizationof the rates of profit has alreadybeen permanentlyachieved. On
the otherhand,his analysisis moreexplicit,and can be clearlyformulated,on the second
type of mechanism.
At the beginningof our hypotheticalyear, what is really given (besidescapital) is
not the wage-ratebut the numberof workers. Therefore,the solutions determinedby
the market(supposingthe ratesof profitalreadyequalized)aregivenby the system(I)-(14),
(16) plus the followingequality(replacing(15)):
1 This can be seen more clearly by re-writing(13) and (14) after substitution from (4), (8a), (I Ia)
(15). We obtain:
r = f (N1) - 1.
w - f(N)
(14a)
(13a)
2 The independenceof the rate of profit from the conditions of production of luxury-goods is a
propertyof all the theoreticalmodels which use the distinctionbetweenwage- and luxury-goods. In plain
words, it is due to the peculiaritythat wage-goodsare necessaryto produceany type of goods, while luxurygoods are not. Mr. Sraffapointed out to me that the property was first discoveredby Ladislaus von
theoretischenKonstruktionvon Marx im drittenBandde.q
Bortkiewicz(Zur Berichtigungder Grundlegenden
' Kapital', in " Jahrbuicher
fur Nationalokonomieund Statistik,"July 1907. An English translationcan
be found as an appendixto the volume Karl Marx and the Close of his System, by E. B6hm-Bawerk and
Criticismof Marx, by R. Hilferding,translatedand edited by P. M. Sweezy, New York
BAhm-Bawerk's
1949). From our mathematicalformulation,the propertycomes out very simply and clearly.
86
(15a)
REVIEWOF ECONOMICSTUDIES
N
where:
=
numberof workersat the beginning of the year.
The systemis againcompleteand determinatebut the wage-ratehas now becomea variable
and has a solution(the marketsolution). Ricardois firmlyconvincedthat this solution
can only be a temporaryand unstableone because,if it comes out differentfrom the
naturalwage-rate(x), the populationwill adjustitselfin sucha wayas to bringthe two rates
together. Analytically,the mechanismmay be expressedas follows:
(17)
(17) =ddt-
~=F(x
F(x - t)
where: t denotestime and x the wage-rateresulting fromthe system(1)-(14),(15), (16),
with the properties':
(17a)
=
FF(O)
0
F' > 0
(7)
which mean that populationis stablewhen x
x> x(or x <x).
x, and it increases(or decreases)when
The differentialequation(17) with the properties(17a) is of a type which has been
extensivelystudiedby ProfessorSamuelsonin connectionwith what he calls the correspondenceprinciplebetweencomparativestaticsanddynamics.2 In ourcase,it can be easily
demonstrated3
that the dynamicmovementfor x(t) generatedby (17) is convergenttowards
x (the naturalwage-rate),providedthat dN < 0, a condition which the system fulfils.3
Hence,for x, only the naturalsolutionx = x is a stablesolution.
7. THE EQUILIBRIUM OF THE STATIONARY STATE
The naturalequilibriumexaminedin the previoussectionsis still not a stablestate of
affairs. Two other types of change are in operationin a Ricardiansystemas time goes
on: (i) the improvementswhich take place in the technicalconditionsof productionin our terms,the shiftsin timeof the productionfunctionf (N1)-, and(ii) the accumulation
of capitalby the capitalists,who add eachyeara substantialpartof theirprofitsto capital.
Here again, Ricardodoes not considerthe firsttype of change-technical progress-in a
systematicway (a characteristicwhich only to-day can be found in models of economic
growth). He only points out that improvementsin the technicalconditionspostponein
time the effects of the changesof type (ii). Since he thinks that these changes(capital
accumulation)are-in order of magnitude-the more relevantones, he concentrateshis
analysison them,with the qualificationthat the effectshe showsmight be delayed,though
not modified,by technicalprogress.
In analyticalterms,capitalaccumulationrepiesentsanotherdynamicmechanism,in
operationon the systemalreadydescribed,of the followingtype:
I The functionF and the similarfunction0 of the followingsection are supposed to be continuously
differentiable.
2
P. A. Samuelson," The Stabilityof Equilibrium: ComparativeStaticsand Dynamics,"Econometrica,
April 1941, also Foundationsof EconomicAnalysis,HarvardUniversityPress, 1948, especiallyChanterTX.
3 A proof is given in the appendix,
A MATHEMATICAL FORMULATION OF RICARDIAN SYSTEM
(18)
dt
(D
dK
(
dt
P
87
)\
or, from (8a) and (1la),
(19)
with the properties :1
(19a)
1D(0)
NEf '(NI)- x
)
0
-
>0.
4V
The differentialequation (19) is of the same type as (17) and has now to be considered
jointlywithit. Frommereinspectionof the two equations,it can be seenthatthe solutions
of the system at which the two dynamic mechanismscease to operate (the stationary
solutions) emerge when x
=
x and
i-
- 0. Therefore, for any given state of technical
knowledge,representedby the technicalfunctionf (N1),the stationaryequilibriumis given
by equations(1)-(14)plus the followingtwo:
(15)
x-x>
O
=
0.
(16a)
In order to ensurethe existenceof non-negativesolutions for this system, a somewhat
strongerconditionthan (lb) is required,namely
where: f'(ao) = limf' (N1).
(20) f' (0) > x > f' (oo)
N1* 00oo
The meaningis that there must be a certainpoint, as populationincreases,at which the
productof the last workerput to work descendsbelow the naturalwage-rate(a condition
whichis implicitin Ricardo'sarguments). If this conditionwerenot satisfied,the system
would expandindefinitelyand the stationarystate would never be reached. When (20)
is satisfied,it is shownin the appendixthat two types of solutionsexist-one of themcorrespondsto the equalityf' (N1) = and the otherto the equalityN = 0. The solutions
of the secondtype, however,so called trivial(they meanthat thereis no economicsystem
at all) are uninterestingand moreoverthey are unstable. On the otherhand,the solutions
correspondingto f' (N1) = x are unique and perfectlystable. Therefore,the system
necessarilyconvergetowardsthem. When the situation they representis attained,all
dynamicmechanismscome to a standstill. The wage-rateis at its naturallevel (no longer
disturbedby capitalaccumulation)and the rate of profithas fallento zero. The system
has reacheda stable equilibrium-the Ricardianequilibriumof the stationarystate.
OFECONOMIC
GROWTH
8. THEPROCESS
It has been shownin the foregoingsectionsthat the Ricardiansystemcontainsmany
dynamicprocesses,althoughsome of themare not systematicallyanalysed. The two dynamic processeswhich are explicitlytaken into considerationare convergentand lead to a
stationaryand stable state. Ricardo,however,investigatesthe propertiesof his system
at a very particularstage of the whole movement,which he considersthe relevantone.
Most of his analysisis carriedon as if the demographicmechanismhas alreadyfullyworked
through,while the capital accumulationprocess has not yet been completed. In other
1 If a minimumrate oI profit(let us call it r) is considerednecessaryin orderto inducecapital accumuHowever, the conclusions
lation, equation
(18) has to be modifiedas follows dK = (D
$
~~~~~~~tPi _K)
and stable point of conthe
the
modification
stationary
that
drawnin the text remainthe same, with
single
vergencyof the systeminsteadof being at 7 = 0, is at 7 = Fp1K.
88
REVIEW OF ECONOMIC STUDIES
words, he concentrates on describing the changing characteristics of his system in terms
of natural behaviour of the variables in a process of capital accumulation.
In mathematical notations, this task becomes very easy. It is enough to consider the
system (1)-(16) (in which the natural wage-rate has been permanently achieved) and to
take the derivatives of each variable with respect to capital, which represents the datumin the natural equilibrium-whose variation in time brings about economic growth. A
substantial part of the Ricardian analysis is simply expressedby the signs of these derivatives.
Let us consider them:
(21)
(21)0
(22)
(22)
dN
1
== > 0
dK
x>
dN
-l~
I1
f
1 \_(N)f"(Nj)
__
[fI(N1)]2
'(N)f (NO
f"
(23)
^-dN2~ 1
dK
(24)
dK
f
(N1) dK-
(25)
(26)
dK
dX2
a
dK2
dK >
(28)
(29)
(30)
(31)
dpl
_
_
-f"(N1)
dK-
[f (N)]2
dp2
0
>0
> 0
J
->
0
> °
0
d
*
dK >
dw .
dp1
> 0
dK dK-dr
N ,
daN <
dr = f"(N).
d
0
dK
dK
The derivativeshavebeenobtainedfromthe system(1)-(16)and the inequalitysignsfollow
from(lb), (lc), (15), (16), and from otherpreviousinequalitiesamong(21)-(31)themselves.
Theireconomicmeaningmaybe statedas follows. The numberof workers(employment),
all physicalproductions,the wage bill, total rent, the price of corn and the naturalmoney
wage-rate: all increaseas long as the processof capitalaccumulationis going on. As an
effectof the same process,the rate of profitconstantlydecreases. For Ricardo,it took,
of course,a much longerprocessto show what here is demonstratedmerelyby the sign
of a derivative.
Another variable whose response to capital accumulation particularly requires a
long analysis in the Principles1is total profit. For this variable too, let us consider its
derivative with respect to capital. From (1 la) we obtain
1 Principles,chapterVI.
A MATHEMATICAL FORMULATION OF RICARDIAN SYSTEM
(32)
I
dK _
d-K f'(Ni)
[f K(f
(N1)
x
1 +
+
89
" (N)) dN1
'dK '
f(NO
> 0.
Now, from (lc) and (22), f" (N1) < 0 and dNMoreover1
(N) > x as long
as the stationary state has not yet been attained. (At the stationary state f' (N1) = x).
Therefore, the sign of (32), unlike all the others, is not independent of the amount of K.
At the beginning of the process of capital accumulation, where K = 0, the third term into
brackets vanishes and therefore d > 0. At the stationary state, where f' (N1) -= x, the
first two terms into brackets cancel out, and the third is negative, so that dK < 0.
between, there must be at least one point of maximum total profits (wheref (N
In
=
f" (N1) dN1
. f, (N1. dK
at which (32) changes its sign from positive to negative as capital
accumulates.1 Hence
dK > 0
according as to whether f' (Nl)
1
-K.f "(N1) dN1
d Kf'(N)'dK
9'<
'
- 1 Ewhich may also be written
(NK)]
X
EK '
<
where the first member of the inequality represents the rate of profit
(see equation (14a)) and the second member represents the elasticity
of the marginal product from land with respect to capital.
Analytically, the possibility cannot be excluded of more than one point of maximum, in
the sense that :t might alternativelyincrease and decrease many times as capital accumulates.
For such a possibility to realize, however, the third derivative of f(Nx) must behave in a
very peculiar way. Ricardo, of course, did not consider these complications; he explained
the process by a long numerical and obviously non-rigorous example which allowed him
to consider the normal case in which, as capital accumulation goes on, total profits increase
up to a certain point and then decrease.23
-K
1 The readermay easily verifythat, as capital accumulates,profitsin sector 1 and in sector 2 (namely
the variablesP1 and P2) behave exactly in the same way as total profit (7r).
2
pp. 110 and ff.
8 Principles,
While correctingthe proofs, I was pointed out a recent paper by H. Barkai where a very simplified
one-commodityRicardianmodel is worked out in mathematicalterms in order to analyse the movements
of relativesharesas capital accumulates(H. Barkai, " Ricardo on Factor Prices and Income Distribution
in a Growing Economy," Economica,August 1959). Dr. Barkai uses a procedurewhich has some similaritieswith the one I have adopted in this section. His analysis,however, seems to me ratherinaccurate.
Without going into details here (among other things, his conclusionsabout the behaviourof the share of
profits are not altogethercorrect)I shall only mention that Dr. Barkai'smain contentionis that the relative
shareof total wages in total productincreasesas an effectof capitalaccumulation(whichis obvious, as the
real wage-rateis constant and the productionfunction is at diminishingreturns)and that this resultcontradicts what Ricardo said on page 112 of his Principles,namely that as capital accumulates," the labourer's
... real sharewill be diminished" (Barkai'squotation). But where is the contradiction? Dr. Barkai'sproof
refersto the relativeshareof total wagesin total product,while Ricardois talkingaboutthe single labourer's
real share, which is evidentlya differentthing.
This is a case wherethe interpretationof PJcardois quite straightforward.However,as it is alwaysso
easy to be misled by particularpassagesin Ricardo'swritings,let me recall the advice of Alfred Marshall:
"If we seek to understandhim [Ricardo]rightly,we must interprethim generously,more generouslythan
he himselfinterpretedAdam Smith. Whenhis wordsare ambiguous,we must give them that interpretation
which otherpassagesin his writingsindicatethat he would have wishedus to give them. If we do this with
the desireto ascertainwhat he reallymeant,his doctrines,though far from complete,are free from many of
the errorsthat are commonlyattributedto them ". (Alfred Marshall,Principlesof Economics,Macmillan
London, 8th ed. reset, page 670.)
90
REVIEW OF ECONOMIC STUDIES
9. MULTI-COMMODITY PRODUCTION
We are now in a position to drop the two-commodityassumptionand extend our
production.Asfar as the wagesystemof equationsto the generalcase of multi-commodity
goods are concerned, the extension does not present particulardifficulties,although
it does emphasizethe crudenessof Ricardo'sassumptions. The economictheory of demand had not yet been developed,at that time, and there is no questionof substitution
among wage-goodsin the Ricardianmodel. The naturalwage-rateis representedby a
fixed basketof goods, to be acceptedas given by factors lying outside economicinvestigation. With this specification,the introductionin our systemof any wage-goodi, besides
corn, introduces8 morevariables: Xi, Ni, Wi,Ri, Pi, Ki,pi, xi, but also 8 moreequations
of the types (1), (4), (5), (6), (7), (8), (15), (16). The system is again determinateand
maintainsits basicfeaturesalreadyanalysedin the previoussections. As a matterof fact,
when the naturalwage-rateis acceptedas a fixedbasket of goods, thereis no gain at all,
from an analyticalpoint of view, in extendingthe systemto includeany numberof wagegoods morethan one. The whole structuralcharacterof the modelis alreadygivenby the
systemof equations(1)-(16),providedthat our interpretationof the singlewage-commodity
is modifiedin the sense of consideringit as a compositecommodity,made up of a fixed
of the modelalso remainunchanged
mixtureof wage-goods.' The dynamiccharacteristics
as they dependexclusivelyon the wage-goodspart of the economy.
The problembecomesmuchmorecomplicatedwhenthe extensionto multi-commodity
productionis made for luxury-goods. Here, the introductionof each commodity 4j,
besides the one which is used as a standard,introduces4 more variables: XlK,N11,py
P11,but only 3 moreequationsof the types(2), (9) and (10). Moreover,it changesequation
(12) into the followingone:
p+..
nXinX pw R
*plj Xl +
(12a) Pli Xll + P12 X12 +
wherethe subscriptw stands for the compositewage-commodityand the subscriptsl;'s
standfor the luxury-goods.
Hence, for each luxury-commodityintroducedbesides the first, one more relation
is neededin orderto keepthe systemdeterminate. Ricardodoesnot providethis relation.
Again the difficultyis that he does not have a theory of demand. The assumptionof a
naturalwage-ratesolvesthe problemfor theworkers(andby consequencefor the capitalists)
but leavesit still open for the landlords,whosepossibilitiesof substitutingone luxury-good
for anotherand whose changesof tastescannot be ruledout; and Ricardodoes not rule
them out. Have we to conclude,therefore,that the Ricardiansystem is indeterminate
with respectto the luxury goods? It certainlyis, but-interestingly enough-only for
the particularvariablesXl1's.N,j's, Plj's,whichare not reallyof much interestto an economist like Ricardo, once their totals are determined.
1 ProfessorSamuelson,in his recent " Modern Treatmentof the RicardianEconomy, (The Quarterly
Journalof Economics,Februaryand May, 1959),has been unableto graspthese propertiesof the Ricardian
model. The reason seems to me that he has treated a Ricardianeconomy with a production function
of the neo-classicaltype (see especially his appendix),which is inappropriateand is responsiblefor the
conclusions he then criticises. Professor Samuelson argues that the classificationof lands in order of
fertility-namely, in our terms, the technicalfunctionf(N) - is not an unambiguouslydeterminedone
because, accordingto the type of produce which is considered,the classification,i.e. the functionf(N1),
may be different. This argumentis valid in a neo-classicaltheoreticalframework,wheresubstitutionamong
goods (in consumptionand in production)is the main featureof the theory,but is irrelevantin a Ricardian
type of analysis, which excludes substitution. When the proportion of the differentproduces is fixed.
the classificationof lands in order of fertilityis a perfectlydeterminedone.
A MATHEMATICAL FORMULATION OF RICARDIAN SYSTEM
91
To see this surprisingpropertyof the Ricardiansystem,let us supposethat n luxurygoods are pioduced.
Then 4(n-1) new variables of the types Xl,, Nij, P11,ply,and 3(n-1)
new equationsof the types(2), (9) and (10) are introducedin the alreadyanalysedsystem.
Provisionally,let us writedown n demandequationsfor the luxurygoods:
X12 =
p... Pln, R)
P2(Pw, PIDP12 ..... Pin, R)
Xin
(on(Pw,Pll,P2 ......
Xil
91 (Pw, Pll, Pl2.
(33)
pin, R).
Equation(12a), which representsSay's law (namely,landlordsspend all their incomeno more and no less-on luxurygoods), puts a restrictionon the (33), so that one of the
equationsmay be dropped. We are left with (n-1) equations,whichis the numbernecessary to determinethe system. It is very interestingto notice now that the solutionsfor
all the variablesof the system,exceptthe Xlj's,Pu;'s, Nuj's,are independentof the (33).
In other words, the (33) are requiredonly to determinethe single physicalproductions,
employmentsand profitsin each particularluxury-goodssectorbut not to determineall
other variables. Whateverthe demandequationsforluxury-goodsmay be, i.e., independently
of them, all the variablesreferringto the wage-goodspart of the economy,all prices,the
rate of profit,and all the macro-economicvariablesof the system-like total employment,
nationalincome,total profits,total rent,total wages,total capital-are alreadydetermined
by the system.
This is perhapsthe most interestingoutcomeof the whole mathematicalformulation
attemptedin this paper and it will be useful to remindthe readerof the assumptions
underwhichit has been reached: (i) perfectmobilityof capital; (ii) Say'slaw ; (iii) the
assumptionof circulatingcapitalonly, and of a one-yearperiodfor all processesof production. The last assumption,so stated, is too restrictive. As a matterof fact, it may be
droppedandfixedcapitalintroducedinto the analysiswithoutaffectingthe alreadyattained
conclusions,providedthat the somewhatmore generalrestrictionis kept of supposing
that all the sectorsof the economyuse fixedand circulatingcapitalof the samedurability
and in the same proportions. This is indeed the crucial assumption: the determinateness
of the whole Ricardiansystemitself dependson it, in an essentialway.
Ricardohimselfbecameawareof this limitationof his theoreticalmodelin connection
with the problemof the determinationof total employmentin the economy. He was
disturbedby the discoveryand, as a result,in the thirdeditionof the Principles,he added
the well-knownchapter " on machinery". The problemis that, when the mentioned
crucial assumptionholds, total employmentin the economy, for any given amount of
capital,is determinedindependentlyof the (33). But when the conditionsof the assumption are not realized,total employmentcomes out differentaccordingto the way in which
demand(and thereforecapital)is distributedamong the luxurygoods sectors. Having
realizedthis, Ricardo declaredexplicitly,in the added chapter,that he was mistaken
earlierwhenhe extendedto the introductionof machinery(i.e., to the case wherethe proportionsof fixedandcirculatingcapitalchange)his generalconclusionsabouttotal employment dependingon total capitalalone and not on how and wherethiscapitalis employed.
This proposition,in the light of our formulation,appearsquite obvious, but it has not
appearedso to many of Ricardo's interpreters. Indeed, because of the assertionsit
contains,which seem to be in contradictionwith the generalconclusionsfollowingfrom
the whole previousanalysis,the chapter" on machinery"has alwayspuzzledRicardo's
92
REVIEWOF ECONOMICSTUDIES
readers. The mathematicalformulationof the presentpaper helps to clarify the issue.
It showsthatthe entireRicardianmodelstandson the assumptionof a uniformcomposition
of capital all over the economy. The problem of introductionof machineryexactly
hypothesizesa violation of this assumption. Therefore,the generalconclusionscannot
be extendedto this case. Lookedat in theseterms,the chapter" on machinery" appears,
ratherthan a contradiction,an honest acknowledgementby Ricardo of the limitations
of his theory.
10. CONCLUDING REMARKS
A few remarksmay be drawnas a way of conclusion.
Ricardo'smodel is built on very crude assumptions. The most crucial of them is
that all sectorsof the economyuse-we mightsay in moremodernterms-the sameperiod
of production. This was just the point againstwhichhis contemporarycritics(especially
Malthus)threw their most violent attacks. In their function as critics, they were right.
The limits entailedby the assumptionare relevantnot only for the Ricardiantheory of
itself of the whole
value-as has alwaysbeen thought-but also for the determinateness
system,as soon as the simplecase of two-commodityproductionis depaitedfrom.
On the other hand, once the assumptionsunderlyingthe whole analysis have been
explicitlydefined,the systemappearsto be logicallyconsistentand determinatein all its
macro-economicfeatures and even in its sectoral details, except for some particular
sectoralvariablesin which Ricardowas not interested. A mathematicalformulationof
the model is possible, claiifies many issues-among others, those connected with the
controversialchapter" on machinery"-and permitsa representationof the Ricardian
dynamicprocesses-in particularthe processof economicgrowth-in a few rigorousand
concisenotations. The solutionsof the naturalsystemRicardowas dealingwithare shown
to existand to be uniquebut not stable. They reacha perfectstabilityonly in the equilibriumof the stationarystate.
The whole model, in its crudenessand simplicity,appearsremarkablycompleteand
synthetic. Ricardo is always looking for fundamentals. Detailed relations are dealt
with only in the light of basictendencies-when they becometoo complicatedand lead to
difficulties,those relationswhich are thought to be less importantare frozen by crude
assumptions. Whetherthis is a fruitfulmethodologicalline to pursueis open to controversy. Later, neo-classicaleconomistspreferreda radicallydifferentline of approach.
They abandonedtoo ambitiousdynamicoutlooks and instead startedto analyse, in a
at least a more simplified(static)
completeway and in all its functionalinterrelationships,
versionof economicreality. The step which was supposedto follow, however,-that of
passingto a dynamicanalysis-has not comeout as easyand spontaneousas was expected,
and, in recentyears,it has not beeninfrequentfor economists,facedwith urgentproblems
of economic development,to have second thoughts on the subject. In this light, the
Ricardiananalysis,with all the naiveteand the limits of its particulartheories,appears
less primitivenow-a-daysthan it appearedsome decadesago.
Harvard Universityand Cambridge University.
LUIGIL.
PASINETTI
A MATHEMATICAL FORMULATION OF RICARDIAN SYSTEM
93
APPENDIX
EXISTENCE
ANDUNIQUENESS
OFSTABLE
SOLUTIONS
It has been a widespreadconcern among mathematicaleconomistsin the last few
decades not to be satisfiedany longer (as economistsused to be) with mere counting
the numberof equationsand unknownsof their theoreticalsystemsand to enquiremore
rigorouslyinto the conditionsfor the existence,uniquenessand stabilityof the solutions.
The task has not provedto be an easy one, as it normallyentails mathematicalnotions
and manipulationsof a fairly highly sophisticatednature. In our case, fortunately,
the proofs can be given in a relativelyelementaryway, except perhapsfor the stability
conditions.
1. Existence and uniqueness of non-negative, non-trivial solutions. The Ricardian
system contains one single functionalrelation, the f(N1). Therefore,the fundamental
step to solvingit is to findthe value of N1 whichsatisfiesthe restrictionsput by the system
on the f(N1). In the system (1)-(15),(16a), we may start by taking ( lla), substituteit
into (16a) and obtain
N[f'(N1)
-
0.
This equationis satisfiedeitherbyf' (N1) = x or by N = 0. The lattersolutionmeans
that there is no economicsystemat all. Any theoreticalrepresentationof an economic
systemhas this solution, but it is an uninterestingone-it representsthe so-calledtrivial
case. Evidently,the relevantsolution is the other one. Let us prove therefore:
(i) that N;-defined as the solutionof the equationf' (N1)
negative;
- - existsand is non-
(ii) that N; is unique; and finally,
(iii) thatf (N;) > N; f ' (N;).
(Thereasonfor this proof will appearin a moment.)
Proof (i). From (20),.f' (0) exists and is greaterthan x; f' (xo) also exists and is
smallerthan Z Since x is a positiveconstant,there must be a value 0 < N; < oo, at
which f' (N) = x. Hence N; exists and is non-negative.
Proof (ii). From (Ic), f " (N1) < 0, namelyf' (N1)is a monotonicfunction. Since
x is a constant,then (by a straightforward
applicationof Rolle's theorem)N; is unique.
Proof (iii). Call G
=
f(N1)
-
N1 f' (N1). Then dG=
dN1
-N1
f" (N1) >
0,
namelyG is a monotonicallyincreasingfunction. Sincef(0) > 0 and N1 > 0, from (la)
and (i), then G is never negative,or f (N1) > N1 f ' (N1) and, in particular,f (N) >
N; f ' (N;).
By substitutingnow N, into the system of equations(1)-(15), (16a), the solutions
come out as:
REVIEW OF ECONOMIC STUDIES
94
(Al)
X1 = f(N)
1
[f(N*')
(A2) X2. =
f (N)
1
(A9) p
[if'(N)
-N
]
(A10) P2 =1
f (N*)
(A3) N = f(N)
(All) w =
(A4) N1 =
N;
(A12) P =- 0
(A5) N2
f(NI *)
N;
(N)
(A13) P2
=
(A14) r
= 0
(A15) x
= x
W = x (N)
f (N*)
(A7) K = x f N
(A6)
(A8)
R =
(N;)
-
N f' (N)
1f(N
0
(A16) rt = 0.
It follows that, if N 1exists is unique and non-negative and, moreover, iff(N ) > N f (N ),
then all (Al)-(A16), namely the non-trivial solutions of the system, exist, are unique and
are non-negative. The proofs have been given so far with reference to the equations
(1)-(15), (16a). A fortiori, the solutions of any other system of equations (1)-(16), defined
by a given . between 0 and K*, exist, are unique and non-negative. For the system (1)-(16)
the trivial solutions are even excluded by hypothesis as 1 > 0. (The stars * are taken to
denote the non-trivial solutions of the stationary equilibrium).
2. The stability of the stationary equilibrium. The stationary equilibrium is defined
by the solutions of the system of equations (1)-(15), (16a). In order to find out whether
it is stable, an investigation has to be made into the dynamic behaviour of the system
when displaced from the equilibrium solutions. That behaviour is represented by the
two differentialequations (17) and (18). For a rigorous proof of stability, the two equations
have to be considered jointly. Such a proof is given below but, as it entails a rather
sophisticated mathematical treatment, it may be useful to give first a more simple proof
which, although less rigorous, is intuitively easier to grasp and perhaps also more pertinent
to the Ricardian logic.
The function (17) depends on the deviation of x from x and the function (18) on the
deviation off' (N1) from x. The two dynamic mechanisms are, so to speak, one on the
top of the other. We may begin, therefore, by proving first, for a given R, the convergency
of the first dynamic process towards x and then substitute this stable solution into the second
process and carry on a similar investigation on it, for a given x.
Let us take equation (17) and expand it in a Taylor series around a value of N defined
as N+ =d(N-
N+) = F(O) + (N-
N+) F' (0) dx + (N- N+)2 [F
(0) dx )2
F'(0) d]
+
+
A MATHEMATICALFORMULATIONOF RICARDIAN SYSTEM
95
Neglectingthe terms of higher order than the first and recallingthat F(O) = 0, the
equationbecomes
dt
N+) * F'(O) * dx
(N-
=
This is a simple differentialequationand its solution is'
N(t)
=
N+ + [N(0)
-
N+] exp
[' (0) d
F'
dN
]
where
N (0)
is the value of
Nat time
zero.
N
Since F' (0) > 0 from (17a), a necessaryconditionfor N(t) to convergetowardsN+ (and
thereforefor x(t) to convergetowardsx) is d <0. Now, fromthe system(1)-(14),(15a),
dx
(16), we have d-N
is stable.
K
=---
< 0.
The condition is fulfilled. Hence, the solution x
=
x
By substitutingnow x = x into (19) and developingthe same type of analysis,the
necessaryconditionfor K to convergetowardsK*-defined as the stationaryequilibrium
solutionfor K-is
d /1
<0.
-V(1) In <0
Now, from (1la) we can write
d
-1
Nf "(N1)dK
Sincef" (N,) is negativeandf' (N,) is greateror equalto x accordingas to whetherN, <
N* or N, = N*, then conditiond-i-7- ) < 0 is not satisfiedwhen N =0 , while it is
dK P2/
satisfied when f ' (N,) = . Hence, the solutions of the system correspondingto f ' (N,)
= x are stable, while the trivial solutions are unstable-the system necessarily converges
towardsthe first ones. As a conclusion,the system (1)-(15), (16a) has stable solutions.
Such stable solutionsare also unique.
3. A more rigorousproof of stability. Consider equations (17) and (18), representing
the variationsin time of N and of K. SinceN, is a monotonicallyincreasingfunctionof
N, the equationsmay be equallyexpressedin terms of N, (namelyin termsof the wagegoods sector):
(A17)
(A18)
dNt
dt
dK
dt
g(x -);
p(Pi),
g(O)
0;
where K,=N,x;
g > 0;
cp(O)=O;
cp'> 0.
1 See any elementarytreatiseon differentialequationsor also R. G. D. Allen, MathematicalEconomics,
London 1956, chapter5.
REVIEW OF ECONOMIC STUDIES
96
Our purpose is now to investigate the dynamic behaviour of the system in the vicinity
of the stationary solutions x = x and f' (N1) = x. Let us expand (A17) in a Taylor series
around the value x. Neglecting the terms of higher order than the first the equation
may be written
Ndt
(A17a)
(x-
=-
x)g' ().
Equation (A18) is more complex. Let us first write it in terms of the same vaiiables entering (A17),
dK
d(N x)
x
(N)
By expanding also this equation in a Taylor series and neglecting the terms of higher order
than the first we obtain
N, dtx +
(A19)
dN
9' (0) * Nl[f' (N) -
x]
Let us now express the variables in terms of deviations from their stationary solutions and
utilize Taylor's theorem for the f' (N1). We have
-
(N,-Nj)]d(x
(N --N;)]
[N; +Nj±
d
dt
y'(0)
fc)
d(Nl --N)
+
+ (x - -fC
[ +dt
)
[N* + (N -
N
[(N -N*') 'f"(N)-x+f'(N;)]}.
N)]
The squares of (N1 - N;) and of (x - x), and their products, represent magnitudes of
second order and we may neglect them, re-writing the whole expression as
* d(x --
)
dt
d(Nl-N)N
dt
-
N
'(O)
'
o {(x-x)2;
[(N
--N)
(N,
N)2;
f" (N)-x
(x
x) (N1-
N)}
where the last term denotes the order of magnitude of the neglected products. Multiplying now (A17a) by x and subtractingit from (A20) we can at last write down our equations
in a suitable form for an immediate solution
d(Nl-N)
dt
(dt(
=
(x-)
) = (N1-N*)
. g'(O) + 0 ({(x
* '(0) *f" (N)-
)2}
x) * [
(x
0 {(x
-
)2;
g'(O)+
(N - N
'(O)] +
)2; (x -
) (N - N;)}
The solutions of this system of equations-apart from the neglected terms-take
form'
the
1 See, for example,A. R. Forsyth, A Treatiseon DifferentialEquations,London, 1921,pp. 342 and ff.
In order to make the procedureeasier to follow, I shall take here the same steps as ProfessorSamuelson
in his alreadymentionedarticlein Econometrica.
A MATHEMATICAL FORMULATION OF RICARDIAN SYSTEM
N1 (t) = N + kll eXlt
x (t) =
+ k2l exit
97
+ k12 eat
+k22ex2t
wherethe k's dependon the values of N1 and x at time zero and the X'sare the roots of
the characteristic
equation
I
0-X
g'(0)
*
q'(0) f" (Ni)
-
x
* g'(0) -
'() -
=
0
=
0.
X
or
(A21)
X2
* g'(0)] - g'(0)*'(0)
+ X [q(0) +
f" (N;)
For the equilibrium to be stable the real part of X must be necessarily negative,
i.e.,
R(X) < 0.
(A22)
Now, since q'(0) > 0, g'(0) > 0, andf " (N1) < 0, (A21) can be written as
X2 + 2 mX + n2 = 0
where:
(A23)
m = ½[qp'(0)+ N
n
=
]
'(0) f" (N
V- g'(0)
Hence:
X=
-
m
(m2 -
n2)i
from which it appears that the real part of X is always negative, namely that condition
(A22) is satisfied. Therefore, the stationary equilibriumdefined by the couple of solutions
x = x andf' (N1) x= is stable.
A proof of the instability of the trivial solutions, characterizedby N1 = 0, can be given
in an easier way because in this case the products involving N1 itself-besides those involving
(x - x)-are of second order of smallness and equation (A18) may be considered in
isolation, as appears by re-writing (A19) as
N1
dt
+ (x-±
+ x)-
=p '(0) Nl[f' (0)-x
+
-
x].
and then, neglecting the squares of N1 and of (x - x) and their products,
1
1 dN1
'(0) [f' (0) - x] + 0 {N;
N dt =
(x - X;
N(x -)
}.
This is a simple differential equation whose solution-apart from the neglected termstake the form
(A24)
logn N1 = k t + logn C
REVIEW OF ECONOMIC STUDIES
98
namely
(A25)
N1(t)
C ekt
where : k =
and
C=
1
1-
p'(0) [f' (0) -
],
N1 (0).
1
[f' (0) - x] > 0, then the solution (A25) is explosive, which means that
x 1p'(0)
the stationary equilibriumdefined by the solution N1 = 0 is unstable.
Since