Existenceanduniquenessofthesolution

ofSraffa’sfirstequations

SaverioM.Fratini

April2019

TeachingMaterialn.1

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Existence and uniqueness of the solution of Sraffa’s first

equations.Anexamplewiththreecommodities

Weconsiderasystemofproduction forsubsistencewiththreecommodities.Theyare

denotedas“a”,“b”and“c”—insteadof“wheat”,“iron”and“pigs”,asinSraffa’sbook(1960,

p.4).

Systemofproduction

𝐴" ⊕𝐵" ⊕𝐶" → 𝐴

𝐴( ⊕𝐵( ⊕𝐶( → 𝐵

𝐴) ⊕𝐵) ⊕𝐶) → 𝐶

As in Sraffa,𝑋+ ≥ 0 is the quantity of commodity “x” employed in the production of

commodity“y”and𝑌 > 0isthequantityofcommodity“y”produced(with“x”and“y”=

“a”,“b”,“c”).

Forthissystemofproduction,commodityrelativepricesaredeterminedasthesolution

ofthefollowingsystemofpriceequations:

𝐴"𝑝" + 𝐵"𝑝( + 𝐶"𝑝) = 𝐴𝑝"

𝐴(𝑝" + 𝐵(𝑝( + 𝐶(𝑝) = 𝐵𝑝(

𝐴)𝑝" + 𝐵)𝑝( + 𝐶)𝑝) = 𝐶𝑝)

Assumptions

A1.Thesystemis inaself-replacingstate.That is tosay:𝐴" + 𝐴( + 𝐴) = 𝐴;𝐵" + 𝐵( +

𝐵) = 𝐵;𝐶" + 𝐶( + 𝐶) = 𝐶.

A2.Commodities“a”,“b”and“c”are“basiccommodities”,i.e.eachofthementersdirectly

orindirectlyintotheproductionofallthecommodities.

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Proposition

IfassumptionsA.1andA.2hold,thenthesystemofpriceequationsdeterminesoneand

onlyonesetofstrictlypositiverelativeprices.

Proof

Letusstartbyre-writingthesystemofpriceequations:

(𝐴 − 𝐴")𝑝" − 𝐵"𝑝( − 𝐶"𝑝) = 0

−𝐴(𝑝" + (𝐵 − 𝐵()𝑝( − 𝐶(𝑝) = 0

−𝐴)𝑝" − 𝐵)𝑝( + (𝐶 − 𝐶))𝑝) = 0

Now,itisclearthatthesystemisahomogeneouslinearsystem,whosetrivialsolutionis

𝑝" = 𝑝( = 𝑝) = 0.Therefore,inordertostudythepossibilityofnon-trivialsolutions,we

needtostudythepropertiesofthecoefficientmatrix:

𝐌 = 7𝐴 − 𝐴" −𝐵" −𝐶"−𝐴( 𝐵 − 𝐵( −𝐶(−𝐴) −𝐵) 𝐶 − 𝐶)

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Letusdenotebym1thefirstrowofmatrixM;bym2thesecondrow;andbym3the

third.AssumptionA.1impliesthat:

𝐦: +𝐦; +𝐦< = 𝟎

where0 is the null vector inℝ<. Accordingly,matrixM does nothave full rank1 and,

therefore,non-trivialsolutionsexist.

Moreover,wecanprovethatmatrixMisofranktwo.Inordertodothat,webegin

byprovingthattherearenottwoscalarsaandbsuchthat𝛼𝐦: + 𝛽𝐦; = 𝟎.Infact,ifa

and b have opposite signs, then 𝛼(𝐴 − 𝐴") + 𝛽(−𝐴() ≠ 0, because (𝐴 − 𝐴") > 0

(assumptionA.2)and(−𝐴() ≤ 0.If,instead,aandbhavethesamesign,then𝛼(−𝐶") +

1Therankofamatrixcorrespondstothemaximalnumberoflinearlyindependentrowsorcolumnsofamatrix.

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𝛽(−𝐶() ≠ 0,because(−𝐶") ≤ 0and(−𝐶() ≤ 0,butatleastoneofthemmustbesurely

negativeduetoassumptionA.2.Inboththecaseswecannotobtainthenullvectorasa

linearcombinationofthefirsttworowsofmatrixM.Withanalogousargument,itcanbe

provedthathoweverwetaketworowsofmatrixM,theirlinearcombinationcannotbe

thenullvector.

Therefore,all thenon-trivialsolutionsof thesystemareonthesameray.More

precisely,let𝐩 = [𝑝", 𝑝(, 𝑝)]beanon-trivialsolutionofthesystem,then𝐩′ = [𝑝"′, 𝑝(′, 𝑝)′]

isasolutionofthesystemifandonlyifthereisascalarlsuchthat𝐩 = 𝜆𝐩′.Thismeans

thattherelativeprices𝑝( 𝑝"⁄ and𝑝) 𝑝"⁄ areunivocallydetermined.2Havingestablished

this, for the non-trivial solution of system tobe economicallymeaningful, the relative

pricesmustallbestrictlypositive.

Therelativeprices𝑝( 𝑝"⁄ and𝑝) 𝑝"⁄ arestrictlypositiveiftheprices𝑝", 𝑝( and𝑝)

areeitherallstrictlypositive,orallnegative.Inotherwords,thenon-trivialsolutionof

systemwould not be economicallymeaningful if someprices are strictly positive and

someothernegative.Inthelattercasetherewouldcertainlybenegativerelativeprices.

Withtheaimofprovingthatthenon-trivialsolutionofthesystemiseconomically

meaningful,letusstartbyassumingthatthepriceofcommodity“a”isnegative,thatis,

letusassume𝑝" < 0.

Fromthefirstequationofthesystemwehave:(𝐴 − 𝐴")𝑝" = 𝐵"𝑝( + 𝐶"𝑝) .Since

(𝐴 − 𝐴") > 0(assumptionA.2),if𝑝" < 0,thentheLHSissurelynegativeand,accordingly,

atleastoneoftheothertwopricesmustbenegativeaswell.Letussayitis𝑝( .

Fromthesumof the firstandthesecondequationwehave:(𝐴 − 𝐴" − 𝐴()𝑝" +

(𝐵 − 𝐵" − 𝐵()𝑝( = (𝐶" + 𝐶()𝑝) . If𝑝" < 0 and𝑝( < 0, then the LHS is surely negative3

and,accordingly,alsotheprice𝑝) mustbenegative.

Inconclusion,eithertheprice𝑝", 𝑝(and𝑝) arestrictlypositive,ortheymustall

benegative.Itisnotpossibleforthemtohaveopposingsigns.Therefore,thenon-trivial

solution univocally determines economic meaningful – i.e. strictly positive – relative

prices𝑝( 𝑝"⁄ and𝑝) 𝑝"⁄ .

QED

2Infact:𝑝𝑏′ 𝑝𝑎′⁄ = 𝜆𝑝𝑏 𝜆𝑝𝑎⁄ = 𝑝𝑏 𝑝𝑎⁄ andsimilarly𝑝𝑐′ 𝑝𝑎′⁄ = 𝑝𝑐 𝑝𝑎⁄ .3Weknowthat(𝐴 − 𝐴" − 𝐴() ≥ 0and(𝐵 − 𝐵" − 𝐵() ≥ 0,butatleastoneofthemmustbestrictlypositivebecause,otherwise,commodities“a”and“b”wouldnotbebasiccommodities,violatingassumptionA.2.

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References

Maroscia,P.(2008)SomeMathematicalRemarksonSraffa’sChapterI.In:G.Chiodiand

L.Ditta(eds)SraffaoranAlternativeEconomics.PalgraveMacmillan:Basingstoke

andNewYork.

Sraffa, P. (1960) Production of Commodities by Means of Commodities. Cambridge

UniversityPress:Cambridge.