sraffa's mathematical economics – a constructive interpretation

31 Pages·2007·0.28 MB·English

UNIVERSITA' DEGLI STUDI DI TRENTO - DIPARTIMENTO DI ECONOMIA
_____________________________________________________________________________

SRAFFA’S MATHEMATICAL
ECONOMICS – A CONSTRUCTIVE
INTERPRETATION

Kumaraswamy Velupillai

_________________________________________________________

Discussion Paper No. 2, 2007
The Discussion Paper series provides a means for circulating preliminary research results by staff of or
visitors to the Department. Its purpose is to stimulate discussion prior to the publication of papers.

Requests for copies of Discussion Papers and address changes should be sent to:

Dott. Stefano Comino
Dipartimento di Economia
Università degli Studi
Via Inama 5
38100 TRENTO ITALIA
Sra⁄a(cid:146)s Mathematical Economics - A
Constructive Interpretation
(cid:3)
K. Vela Velupillai
y
Department of Economics
University of Trento
Trento
Italy
[email protected]
&
Girton College
Cambridge CB3 0JG
UK
[email protected]
February 7, 2007
Abstract
The claim in this paper is that Sra⁄a employed a rigorous logic of
mathematicalreasoninginhisbook,ProductionofCommoditiesbyMeans
of Commodities (PCC), in such a way that the existence proofs were
constructive. This is the kind of mathematics that was prevalent at the
beginning of the 19th century, which was dominated by the concrete, the
constructive and the algorithmic. It is, therefore, completely consistent
with the economics of the 19th century, which was the fulcrum around
which the economics of PCC was conceived.
KeyWords: ExistenceProofs,ConstructiveMathematics,Algorithmic
Mathematics, Mathematical Economics, Standard System.
JEL Classi(cid:133)cation Codes: B23, B24, B31, B41, B51
Constructive in the sense of constructive mathematics (say, Bishop-style constructivism,
(cid:3)
rather than Russian constructivism oreven classic Brouwerian constructivism).
Thispaperisabelateddedicationtothememoryofoneofthe(cid:133)nestscholars-particularly
y
of Sra⁄a and the Classical Economists - I have ever had the privilege of knowing: the late
Sukhamoy Chakravarty. Iam deeply indebted to my criticalfriends,Tom Boylan,Guglielmo
ChiodiandStefanoZambelliforyearsofinspirationonmattersdealtwithinthispaper. Alas,
they cannot be blamed forthe remaining infelicities and errors.
1
1 By Way of a Preamble
"Besicovitch insists that I publish [Production of Commodities
by Means of Commodities1, [35]]; the fact that I was able to foresee
interestingmathematicalresults showsthattheremustbesomething
in the theory."
Sra⁄a(cid:146)s Diary Entry, 31 May, 19582
Sra⁄a is nowhere, to the best of my knowledge, more speci(cid:133)c about these
(cid:145)interesting mathematical results(cid:146); nor is it made clear what he means by (cid:145)there
mustbesomethinginthetheory(cid:146)-i.e.,whatinwhichtheory. Perhapssomewhere
buried in Sra⁄a(cid:146)s voluminous unpublished writings and correspondences (espe-
cially with mathematicians and mathematically competent economists) these
thingsareclari(cid:133)ed.Itwouldbeironicalifthephrasemeant(cid:145)theremustbesome-
thing in the economic theory in PCC(cid:146)implied by the (cid:145)interesting mathematical
results(cid:146), given the almost uniform opinion in the economics profession about
PCC needing to be (cid:135)eshed out mathematically to make explicit the economic
originality of the book. Especially since the number of people who have made
careers out of recasting the economics of PCC into trivial propositions implied
by the mathematics of linear algebra and, occasionally, elementary topology, is
legion.
Sra⁄a(cid:146)scontributiontoeconomictheoryisawelldocumentedchapterinthe
history of economic thought. Sra⁄a may or may not have found it (cid:145)immoral to
write more than one page per month(cid:146)([32],p.43), but every one of those pages
aredistilledessencesofpureeconomictheoreticalelixir(cid:150)atleastinthiswriter(cid:146)s
opinion. I want to suggest that there were also, particularly in PCC, but not
exclusively in that elegant work, mathematical and methodological elixirs still
to be discovered.
From a purely mathematical point of view, PCC lacks nothing. In these
times of increasing formal mathematization of economic theory, the point I
wish to raise is that there are alternative ways of mathematising that may not
lose sight of the economic underpinning of a theory. The concerns in PCC
are the solvability of equation systems and, whenever existence or uniqueness
proofs are considered, they are either spelled out in completeness, albeit from
a non-formal, non-classical, point of view or detailed hints are given, usually in
the form of examples, to complete the necessary proofs in required generalities.
Standard economic theory, on the other hand, is naturally formalized in terms
of inequalities. A case can even be made that this is so that (cid:133)x-point theorems
caneasilybeappliedtoprovetheexistenceofequilibria. Acasemadeelegantly
by Steve Smale:
"We return to the subject of equilibrium theory. The existence
theory of the static approach is deeply rooted to the use of the
1Henceforth referred to as PCC in this paper.
2"Besicovitchinsistecheiopubblichi[;]ilfattochehopotutoprevedererisultatimatematici
interessantimostra che c(cid:146)Ł qualcosa nella teoria" (cited and translated in [20],p.193).
2
mathematics of (cid:133)xed point theory. Thusonestepinthelibera-
tion from the static point of view would be to use a mathematics
of a di⁄erent kind. Furthermore, proofs of (cid:133)xed point theorems
traditionally use di¢ cult ideas of algebraic topology, and this has
obscured the economic phenomena underlying the existence of equi-
libria. Also the economic equilibrium problem presents itself most
directly and with the most tradition not as a (cid:133)xed point problem,
but as an equation, supply equals demand. Mathematical econo-
mists have translated the problem of solving this equation
into a (cid:133)xed point problem.
I think it is fair to say that for the main existence problems
in the theory of economic equilibrium, one can now bypass the
(cid:133)xed point approach and attack the equations directly to
give existence of solutions, with a simpler kind of mathe-
matics and even mathematics with dynamic and algorithmic
overtones."
[33], p.290; bold emphasis added.
Sra⁄a, in PCC, (cid:145)bypassed the (cid:133)xed point approach and attacked the equa-
tionsdirectlytogiveexistenceofsolutions,withasimplerkindofmathematics,
mathematics with [an] algorithmic overtone(cid:146). That is the main point I try to
make in this paper.
In this brief paper, then, in addition to the above issue, I also want to em-
phasisetwoissues thathavecausedunusualandunfortunatemisunderstandings
inthereadingandunderstandingofPCC,byso-calledmathematicaleconomists
and economists with a mathematical bent, almost all of whom are hopelessly
incompetentinmathematicalphilosophyandalmostequallyhopelesslyignorant
about the existence (sic!) of alternatives to classical mathematics3. The two is-
sues are rigour and proof in PCC. The elementary misunderstandings by these
so-called mathematical economists have led to quite incredible assertions about
the mathematical content and validity of the formal propositions in PCC, and
their proofs, particularly the existence proofs. I should like to add that even
some of the mathematically competent - albeit only in classical mathematics
and its underpinning mathematical logic4 - economists who are widely known
to be sympathetic to their visions of Sra¢ an Economics, have made incorrect
assertions and unnecessary formalizations that have had the e⁄ect of diverting
attention from the more basic economics in PCC.
3By(cid:145)classicalmathematics(cid:146)Iamreferringtothemathematicsofrealanalysisunderpinned
by set theory plus the axiom ofchoice.
4Mathematical logic, as distinct from the logic of mathematics, is generally understood,
these days, to consist of set theory, proof theory, model theory and recursion theory. I do
not know of a single mathematical economist or an economist with (cid:145)o¢ cial(cid:146)competence
in mathematics (cid:150)whether sympathetic or hostile to the message in PCC - who has tried
to read the mathematical reasoning in PCC in any other way than in terms of classical
mathematics and set theory (plus the axiom of choice) - i.e., ZFC, as it is routinely referred
to in mathematical circles. ZFC, in turn, refers to the Zermelo-Fraenkel axiom system of set
theory plus the axiom ofchoice.
3
I shall, however, refrain from making any comments or interpretations on
theeconomics ofPCC.Theeconomicimplicationsofthemethodsofproofused
by Sra⁄a in PCC for the economics is quite a separate issue, with which I hope
to struggle on another occasion. One point that bears on the economics, as an
implication of the (cid:145)equation systems(cid:146)approach must be mentioned, however
brie(cid:135)y. A natural generalization, in the direction of economic reality, would
be to consider general diophantine systems of equations. Anyone familiar with
computabilitytheorywillknowthatsuchsystemsformparadigmaticunsolvable
systems. But I shall not enter this weird and wonderful world in this paper. If
the case I try to make in this paper proves sensible, I may undertake the more
general task at a later date.
The paper is, therefore, structured as follows. In the next section I discuss
a representative sample of the largely negative assertions made about rigour,
proof and the mathematics of Sra⁄a in PCC. However, mercifully, there have
beendistinguishedeconomists, notquiteknowntobemathematicaleconomists
or even mathematically minded ones, who gleaned immediately that PCC was
impeccable in its mathematical rigour and, to a lesser extent, also aware - how-
ever dimly - that the methods of proof employed by Sra⁄a were sound, even
if (cid:145)unconventional(cid:146). The foremost examples of the former class are Richard
Quandt([28]), EdwinBurmeister([3])andFrankHahn([18]); agoodsampleof
thelattersetconsists,atleast,ofPeterNewman,HarryJohnson,MelvinReder
and, above all, Sukhamoy Chakravarty. Somewhere in between are some of the
mathematically competent so-called (cid:145)Neo-Ricardians(cid:146).
I go on,then, in §3 to suggest an alternative mathematical reading of PCC
whichexposestheerrorsofrepresentationdiscussedintheprevioussectionand,
hopefully,makescleartheimpeccablerigourwithwhichthepropositionsofPCC
have been demonstrated by Sra⁄a, particularly the existence proofs.
Finally, in §4, I try to derive broad methodological precepts, particularly
from the point of view of a mathematical philosophy, for economic theory, from
the exercises attempted in §2 and §3.
Iwouldliketoaddabriefexplanatorynote,informedbyreadingtwobrilliant
works of unusual nature, on a single topic. One by Richard Feynman and the
other by Subhramnayan Chandrasekhar, both on re-reading and re-doing the
mathematics of Newton(cid:146)s Principia. Richard Feynman, when asked to give a
guestlecturetotheFreshmanclassatCaltech,inMarch,1964,decidedtoprove
Kepler(cid:146)s law of ellipses (cid:145)using no mathematics more advanced than elementary
geometry(cid:146)([13], p. 18; italics added):
"Why did Feynman undertake to prove Kepler(cid:146)s law of ellipses
using only plane geometry? The job is more easily done using more
the powerful techniques of more advanced mathematics. Feynman
was evidently intrigued by the fact that Isaac Newton, who had
inventedsomeofthosemoreadvancedtechniqueshimself, neverthe-
less presented his own proof of Kepler(cid:146)s law in the Principia using
only plane geometry. Feynman tried to follow Newton(cid:146)s proof, but
he couldn(cid:146)t get past a certain point, because Newton made use of
4
arcane properties of conic sections (a hot topic in Newton(cid:146)s time)
that Feynman didn(cid:146)t know. So,.... Feynman cooked up a proof of
his own.
.....
IsaacNewtoninventedthedi⁄erentialandintegralcalculus. There
is little doubt that he used these powerful analytical tools to make
his great discoveries. ..... [However] the Principia is presented in
the classical languages of Latin and Euclidean geometry. Thereason
is obvious enough. Newton had to speak to his contemporaries in a
language they would understand."
ibid; p. 19 & 44; last two sets of italics, added.
Sra⁄awishedtospeaktothemajorityofhiseconomiccontemporaries,many
of whom were not versed in the advanced mathematics underpinning (cid:145)Perron-
Frobeniustheorems(cid:146). Hepresentedhisworkinthe(cid:145)classicallanguagesofEnglish
and the Higher Arithmetic(cid:146). It is a pity that the mathematical economists had
become immune to the aesthetic elegance of good prose and the deep beauty of
the Higher Arithmetic.
Chandrasekhar5, in his monumental re-reading of the Principia ([6]), with-
out any recourse to secondary sources, undertook to redo Newton(cid:146)s proofs with
modern mathematics, but acknowledged that (cid:145)the manner of his study of the
Principia was to(cid:146):
"[R]ead the enunciations of the di⁄erent propositions, construct
proofs for them independently ab initio, and then carefully follow
Newton(cid:146)s own demonstrations. In the presentation of the propo-
sitions, the proofs that I constructed (which cannot substantially
di⁄er from what any other serious student can construct) often pre-
cedeNewton(cid:146)sproofsarrangedinalinearsequenceofequationsand
arguments, avoiding the need to unravel the necessarily convoluted
style that Newton had to adopt in writing his geometrical relations
and mathematical equations in connected prose. With the impedi-
mentsoflanguageandofsyntaxthuseliminated,thephysicalinsight
and mathematical craftsmanship that invariably illuminate Newton(cid:146)s
proofs come sharply into focus."
ibid, p. xxiii (Prologue); italics added.
(cid:145)The economic insight and mathematical craftsmanship that invariably illu-
minateSra⁄a(cid:146)sproofs(cid:146),ontheotherhand,havebeendistortedandobscuredby
5Lesttheunlikelymathematicaleconomistreaderofthispapergetscarriedawayandreads
intoChandrasekhar(cid:146)smathematicalapproachajusti(cid:133)cationforwhatheorshehasdonewith
PCC,let me also add the following wonderfulcaveat in [6],p.44 (italics added):
"This simple notational device (suggested by Tristan Needham) allows us (cid:146)to
draw on the intuitive power of in(cid:133)nitesimal geometry while continuing to pay
lip service to the tyrannical legacy of Cauchy and Weierstrass(cid:146). "
5
the attempts to re-mathematise PCC by economists without the deep knowl-
edge and wide vision one needs to have of economics and mathematics. Chan-
drasekhar had these, for physics and mathematics, together with a deep under-
standing of the mode of reasoning of the classics, not only in physics.
2 Clarity and Confusion in Interpreting the Math-
ematical Underpinnings of PCC
"Hardy was right after all: mathematicians are out to debunk
thefakerythatliesconcealedunderneatheverylogicallycorrectproof.
But they will not admit that their task is one of debunking; they
will instead pretend that they are busily proving new theorems and
statingnewconjecturesincompliancewiththecanonsofpresent-day
logic. ....
[M]athematical proofs come in di⁄erent kinds, that need to be
classi(cid:133)ed. Thenotionofunderstanding,thatisusedininformaldis-
cussion but quashed in formal presentation, will have to be given a
place in the sun; what is worse, our logic will have to be accommo-
dated to admit degrees of understanding.
Rota ([31], p.195; italics added)
This paper is dedicated to the memory of Sukhamoy Chakravarty for many
reasons, but primarily because I had some of my most fruitful and enlightening
discussions on Sra⁄a(cid:146)s contribution to economic theory with that erudite man,
notlongbeforehisuntimelydeath. WhenwebegantalkingaboutSra⁄a,mostly
aboutthemethodsofproofusedinPCC,Chakravarty(cid:146)sviewonitwasexpressed
in his early Arthanithi review:
"We come now to the (cid:145)piece de resistance(cid:146)of the book: the con-
structionoftheso-calledstandardsystemandtheproofofitsunique-
ness. Herewhiletheskilloftheliteraryexpositionistobeadmired,
nonetheless the roundaboutness of proofs (if we can call such dis-
cussion proof), is hardly a factor conducive to clear understanding.
RestatementofMrSra⁄a(cid:146)sproblemintermsofinter-industryanaly-
sis shows how the proof of the existence and uniqueness of such a
"standard system" follows from the well-known theorem of Perron
and Frobenius6 in connection with non-negative square matrices."
6In view of what I think is the (cid:145)insidious(cid:146)role this theorem has played in distorting the
mathematicsofPCC,Iwouldliketotakethisopportunitytocorrectanegregiousmistakeon
the origins of the use of this theorem in economics. It was introduced to the mathematical
economicsliteraturebyRichardGoodwinintheearly50sduringaslightlyacrimoniousdebate
with John Chipman in the pages of the Economic Journal ([14]). In spite of Goodwin(cid:146)s own
clear statements (cf., [16]) on the source for his own knowledge of this theorem (which was
G(cid:246)ran Ohlin,whohadbeenastudentinoneofGoodwin(cid:146)sclassesatHarvardin1949),Desai
and Ormerod state ([12],p.1433),incorrectly:
"This ([14]) was also the article which (cid:133)rst cited the Frobenius Theorem, (cid:133)rst
6
Chakravarty ([4], p.87; italics added)
In 1980 and 1981 I had long discussions with Sukhamoy Chakravarty on
the nature of proofs in PCC. I pointed out to him, referring to the irrelevance
and, indeed, the dangers of formalizing the propositions of the (cid:133)rst part of
PCC using (cid:145)Perron-Frobenius Theorems(cid:146), that Sra⁄a(cid:146)s proofs were invariably
(cid:145)constructive(cid:146). Chakravarty listened patiently to the case I was making and
promised to re-think his interpretation of the method of proofs used in PCC. A
quarterofacenturylater,inareviewofyetanother(cid:145)Sra¢ an(cid:146)book,Chakravarty
wentatleasthalf-waytowardsmyinterpretationofthenatureofproofsinPCC:
"Sra⁄a(cid:146)s austere prose of Production of Commodities by means of
Commodities canprovemoredauntingtomoststudentsofeconomics
thantheuseofmatrixalgebra. Inrecentyears,anincreasingnumber
of textbooks have, therefore, made liberal use of the basic tools of
linear algebra, including some results on non-negative square matri-
cestoderivetheanalyticalresultswhichSra⁄a largely demonstrates
constructively with the help of English prose."
Chakravarty ([5], p.122; second set of italics, added)
Apart from my own interpretation of the nature of proofs in PCC as con-
structive, this is the only other mention of such a possibility in the entire lit-
erature on that elegant and rigorous piece of (cid:145)austere prose(cid:146)(with another,
well-meaning, albeit technically incorrect exception in recent years, to be men-
tioned below), that I am aware of. For the rest, the profession simply recast
the economics of PCC in the mathematics of linear algebra and proceeded to
assure itself, as in the gratuitous words of a leading exponent of this genre,
Frank Hahn:
"Sra⁄a(cid:146)s book contains no formal propositions which I consider to
be wrong ....."8
suggested to him, as Goodwin acknowledged, by a young student called Robert
Solow."
7I am in the privileged position of having received a reprint of this rare review from
Chakravarty himself. However, the pagination in the reprint is obviously not that of the
published version.
8The completion of the sentence reads: (cid:145)...although here and there it contains remarks
which I think to be false(cid:146). (ibid, p.353). This is, in my opinion, a statement that is not easy
to substantiate about (cid:145)remarks(cid:146)in a rigorous book, where there is not a single categorical
statement-asremarksorinanyotherformwhatsoever-withoutrocksolidlogicalunderpin-
nings. There are, of course, suggestions, with impeccable caveats - the prime example being
the famous one to end the penultimate paragraph ofp.33 in PCC:
"Therateofpro(cid:133)t,asaratio,hasasigni(cid:133)cancewhichisindependentofanyprices,
and can well be (cid:145)given(cid:146)before the prices are (cid:133)xed. It is accordingly susceptible
of being determined from outside the system of production, in particular by the
levelofthe money rates ofinterest." (italics added)
7
Thesimplestofexamplesofhowheandlegionsofotherssatis(cid:133)edthemselves
thatPCC (cid:145)containsnoformalpropositions[thatthey]considertobewrong(cid:146)can
be given by taking one of Hahn(cid:146)s own renderings of a (cid:145)formal proposition(cid:146), os-
tensibly from PCC. According to Hahn(cid:146)s reading of PCC, Sra⁄a in PCC, when
constructing thestandardsystem,islookingforapositivevectorx anda(posi-
(cid:3)
tive)scalarG suchthatthefollowingvector-matrixequationissatis(cid:133)ed(op.cit,
(cid:3)
p.355)9:
x =G Ax (1)
(cid:3) (cid:3) (cid:3)
where n n matrix A consists of elements a >0;i;j =1;:::;n
ij
(cid:2)
It is at this point that the usual (cid:145)distortion(cid:146)and misreading of PCC enters
the fray. Having formulated the problem of the construction of the standard
system as one of (cid:133)nding particular eigenvalues and eigenvectors of a system of
linear equations, Hahn goes on to claim, with almost dismissive disdain (ibid):
"We now have a purely mathematical problem for which there is a
standardmathematicalresult. ... The[vectorx ]isapureconstruct
(cid:3)
as of course is [1] used in its derivation."
Heevenhelpsthereaderbyreferringtotheappendixinhisownbook(writ-
ten jointly with Arrow, but he refers to the wrong appendix) for the (cid:145)standard
mathematical result(cid:146). He does not, of course, tell us in the article or in the
appendix of the book with Arrow, what assumptions were needed to prove the
mathematical result he invokes. Nor does he add any caveat on the care with
which PCC avoids any matrix formalizations. Above all, he does not warn the
reader that (1) is not used in the derivation of the construction of x in PCC.
(cid:3)
Tobemoreprecise,wearenotinformed,eitherintheabovearticlebyHahn
or in the book with Arrow to which he refers for (cid:146)the mathematical result(cid:146), of
the assumptions, frameworks and the methods of proof used in the derivation
ofthoseresults. Perhapstheywerederivedbyhand-waving,appealtoESPs,or
undecidable disjunctions! In fact, the Perron-Frobenius theorems are generally
proved by an appeal to the Brouwer (cid:133)xed point theorem (although there are
other ways to prove them, too) where, at a crucial stage of its proof, appeal
is made to the Bolzano-Weierstrass theorem, which is provably impossible to
constructivise. WhetherSra⁄awasawareofthisparticularinfelicityinderiving
the(cid:145)mathematicalresult(cid:146)whichHahnandotherswavewitha(cid:135)ourishwhenever
they mention the standard system and its construction is not the issue. The
point really is that uncritical appeal to standard mathematical results means
the mathematical and logical baggage underpinning it comes with it and could
makea mockeryof the economic rationaleforthe resultand, mostimportantly,
for the way its validity is demonstrated - i.e., proved.
Richard Quandt(cid:146)review of PCC (op.cit), is slightly more explicit about
appealing to the Brouwer (cid:133)xed-point theorem - so beloved of the mathematical
9NotalloftheassumptionsinHahn(cid:146)srenderingarefaithfultotheeconomicsofPCC;but
let that pass.
8