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Existenceanduniquenessofthesolution
ofSraffa’sfirstequations
SaverioM.Fratini
April2019
TeachingMaterialn.1
1
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.
2
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.
3
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.