25
LEON WALRAS'S MATHEMATICAL ECONOMICS AND THE
MECHANICAL ANALOGIES
ALBERT JOLINK and JAN VAN DAAL*
Tinbergen Institute and Erasmus University Rotterdam
Order is what is needed: all the thoughts
that can come into the human mind
must be arranged in an order like the
natural order of numbers.
Rene Descartes
I. Introduction
In April 1909 Leon Walras (1834-1910) presented his last lecture before
the Societe Vaudoise des Sciences Naturelles at Lausanne. The subject dealt
with in this lecture was the analogy between the mathematics used in economics
and that employed in mechanics; it is reformulated in what turned out to be
one of Walras's last publications, entitled 'Economique et Mecanique.'1
The publication aroused hardly any reactions at the time of its appearance,
or even afterwards; yet for Walras himself the subject was of "...extreme
importance from the point of view of the diffusion of our method among
mathematicians... ." 2 The article appeared at the end of a long list of
publications by Walras3 and is in a way characteristic of his sequence of work.
From the very beginning an interest in the application of mathematics to
economics was one of the leading motives behind Walras's work. The strong
analogies between mathematical formulations in mechanics and those in
economics, as presented in 'Economique et Mecanique,' were for Walras the
ultimate justification of his efforts to found a system of mathematical economics.
We wish to express our gratitude to Mrs. Hefti and Mr. Stoll of the BCU at Lausanne for their
competent assistance during our stay at the University of Lausanne. We are grateful for critical
remarks made by Neil de Marchi (Duke University/University of Amsterdam). Andre Zylberberg
(University de Paris I) and participants in the British History of Economic Thought Conference.
Bristol. September 1988. We are indebted to the A.A. van Beek Fonds for a research grant. We
thank Donald Walker. Indiana University of Pennsylvania, for useful suggestions.
1. Walras (1909): the same year the article appeared as a brochure (Imprimeries Reunies. Lausanne).
2. Jaffe (1965). Letter 1666: L. Walras to A. Aupetit. 1 Dec. 1907. Walras's letters and those
of his correspondents all appear in Jaffe (1965). The letters will subsequently be indicated by
an 'L' and the number Jaffe assigned to them. Original test in French will, unless otherwise
indicated, appear in the translation of the authors. Text added to the original one has been
put between square brackets.
3. Walker (1987).
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26
HES Bulletin 11:1 (Spring, 1989)
Walras's article appeared in 1909 at the end of a life-long career in economics
during which his ideas on pure economics, applied economics and social
economics has become well established. His textbook on pure economics,
Elements d'economiepolitiquepure4 was founded on mathematical grounds,
and by its very nature the general economic equilibrium models presented in
it hardly needed a justification for the use of mathematics. The basic innovating
notion underlying Walras' explanation of the market behavior of individuals
was rarete. He used this term, which he borrowed from his father, to denote
marginal utility. In his theory it turned out that, if an individual maximizing
utility exchanges goods and services in a certain time period in equilibrium
the raretes of these goods and services would have the same ratio as their prices.
The assumption that for each individual and for each commodity there is a
rarete function in terms of the quantities of these commodities was thereby
sufficient to derive the Walrasian market supply and demand functions in terms
of the prices. The general criticism against this approach concentrated on the
subjective character of utility and rarete, which made it impossible to measure
them. Mathematics, so Walras was told, would therefore not be appropriate
in economics.
In this article we would like to present an interpretation of mathematics in
Walras's work. To this end, we would like to make a distinction between
mathematics used in economic theory and mathematics used in economic
practice. Once having established this distinction, we will suggest a possible
explanation for Walras's use of mechanical analogies in his theory. We will
see that the explanation of the use of mechanical analogies will depend on how
to look at the use of mathematics. Finally, we will summarize our point in
some concluding remarks.
II. Mathematics and Walras's Pure Economics
In Walras's day, the justification for the use of mathematical theorems and
techniques in the treatment of problems in economic theory was not at all
obvious. And indeed, Walras had great difficulty in convincing his
contemporaries of the necessity of the use of mathematics in economics. One
of the reasons, Walras claims, for the opposition to his mathematical approach
lies in the misunderstanding of the nature of mathematics. In the first place,
we should distinguish between mathematics, as such, and the application of
mathematics. In discussing the application of mathematics we should, according
to Walras, distinguish between economic theory and economic practice, as
explained in the article "Une Branche Nouvelle de la Mathematique." In our
opinion this latter article is one of the most interesting articles by Walras; it
remained until recently, however, unpublished in French.3 It contains Walras's
4. Walras (1874-1877).
5. Walras (1987, pp. 291-329). An Italian translation of this article appeared in the Giornali
degli Economisti (April 1876, vol. 3, pp. 1-40) as "Un Nuovo Ramo delta Matematica Dell'
Applicazione delle Matematiche all "Economica Politica."
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Jolink & Daal
Leon Walras's Mathematical Economics
27
explanation on how one should interpret his different uses of mathematics.
The mathematical language and method, according to Walras, would supply
a more rigorous analysis than any ordinary logic would make possible.
Walras's mathematics in economic theory has a strong Cartesian flavor. It
can be asserted that it is similar to Descartes' Mathesis Universalis (universal
mathematics). This Mathesis provides a process of discovery. Descartes states
that his order of study has been to start with the simplest and easiest of
disciplines and to master them before moving on. In this sense, "universal
mathematics" is a general introduction to 'higher disciplines':
.. .there must be some general science to explain everything which
can be asked concerning measure and order not predicated of any
special subject matter. This, I perceived, was called 'Universal
Mathematics'.6
In its application, universal mathematics is restricted to those matters "...in
which order or measure are investigated, and in that it makes no difference
whether it be numbers, figures, stars, sounds, or any other object that the
question of measurement arises." 7
It is in this light that Walras's introduction of mathematics can be seen. His
purpose, therefore, is not that of 'mathematizing' economics (whatever that
means), at least not at this stage. With the introduction of mathematics Walras
intends to present a method to discover 'universal truths.' Although Walras
does not seem to be very explicit about it, his manner of proceeding runs parallel
with Descartes: all the objects of rational knowledge somehow subsist in
deductive chains and are ordered from the simple to the complex, or from the
'absolute' to the 'relative.' This is prior to any application.
With the application, then, of mathematics to economic theory there is more
to it. In the case of application, Walras would have to show that economic
matters can be posed mathematically. As Walras writes:
All these operations: choice of variables and of functions; adoption
of notation that is both simple and expressive; use of geometry,
as Descartes says, to "consider particular relationships better" and
"to represent them more distinctly to the imagination and to the
intellect;" adoption of the algebraic mode to "retain them or to
understand several of them together;" use of the method of
reduction or of analysis, and use of the method of deduction or
of synthesis; all these are delicate operations. For them to be
successful it is less essential to possess some rather elementary
mathematical knowledge than to have economic data drawn always
carefully from experience. Be that as it may, what I have just said
truly indicates wherein consists the problem of the application of
mathematics to economic theory and to applied economics.8
6. Descartes, Regulae ad Directionem Ingenii, (quoted in Gaukroger (1980, p. 43)).
7. Idem.
8. Walras (1987, p. 314).
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28
HES Bulletin 11:1 (Spring. 1989)
At this stage one would expect Walras to find an identification between the
objects of mathematics and those of economics, as Descartes does for the case
of mathematical physics. Walras describes this identification in the Elements
as follows:
.. .the pure theory of economics ought to take over from experience
certain type concepts, like those of exchange, supply, demand,
market, capital, income, productive services and products. From
these real type concepts the pure science of economics should then
abstract and define ideal-type concepts in terms of which it carries
on its reasoning.9
In this sense, we could speak of Walras's mathematical economics, as contained
in his Elements of pure economics. In the case of pure economic theory we
study and explain facts, and the use of mathematics is, then, motivated by the
need of a method of analysis. The mathematical expressions are, and should
be, general, undetermined, non-numerical. Though dealing with magnitudes,
problems of measurement are beside the point. Or, as Walras writes:
...there are [certain elements], such as utility for example, which,
though being magnitudes, are not appreciable magnitudes. But what
does this imply for the analysis?10
Quite distinct is the practical application of mathematics. In economic
practice, according to Walras, mathematics is used to evaluate certain
consequences of an economic measure. The application of mathematics, in
this case, is motivated by the necessity of calculation. Consequently, the
parameters of the formulae need to be numerically determined, and the
expressions are specific for given circumstances. In the case of economic
practice, therefore, the magnitudes do have to be measured, which poses some
restrictions on the applicability of mathematics.
It is this distinction, theoretical application versus practical application of
mathematics, which is of importance for the understanding, and in a sense
the justification, of the use of mathematics in Walras's theory. Since Walras
spent most of his time on theory, it can be argued that in most cases, Walras's
mathematics should be regarded as a method of analysis in abstract terms,
something his opponents failed to see. As Walras states in "Une Branche
Nouvelle de la Mathematique":
Of these two modes, the theoretical, abstract, and analytical
application, and the practical, concrete, and numerical application,
the people to whom one speaks of applying mathematics to
economics can conceive of only the second... . Our adversaries
absolutely neglect the first mode of application of mathematics to
9. Walras (1984, p. 71).
10. Walras (1987, p. 311-312).
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Jolink & Daal
Leon Walras's Mathematical Economics
29
economics, that which consists in speaking and reasoning about
value in exchange, about demand and supply, about utility and
quantity, which are magnitudes, in the language of magnitudes,
and which consists in following the method of the science of
magnitudes. They neglect the first mode that consists in doing those
things and in treating the magnitudes as functions of others, thus
making the knowledge of the properties of functions contribute to
the study of economic phenomena."
ID. Mechanical Analogies in Walras's Economics
The objections of Walras's contemporaries to the use of mathematics in
economics can be summarized in four points.12 These points are:
i) mathematics has no heuristic role to play in economics;
ii) the complexities and subtleties of economic reality can not be
reduced to mathematical formulae of any practical significance;
iii) mathematics is too difficult to understand for economists who are
not mathematicians;
iv) mathematical economics does not provide unambiguous results.
Persisting in the use of mathematics, as a method of analysis, in economic
theory, Walras spent quite some time answering the objections. To the objection
on the minor heuristic role of mathematics, Walras replied that his own work
had proved the contrary. It was only through the mathematical method that
Walras had been able to obtain the results in the Elements. The inability to
reduce the complexities and subtleties of economic reality to mathematical
formulae of any practical significance was, according to Walras, an objection
valid for any generalization; mathematics, as such, has nothing to do with it.
Mathematics could, however, be of practical relevance by formulating general
rules. The third objection, mathematics being too difficult for economists, is
overruled by formal education and therefore hardly an argument against the
use of mathematics. Walras did admit the objection that mathematical economics
does not provide unambiguous results. In reply he states that it was never a
question of finding the absolute truth but, rather, whether mathematics could
be of any help in clearing up matters.
It would have been quite legitimate if Walras had restricted the defense of
his approach to arguments that lie entirely within the field of economics. An
alternative line, and, as it proved to be, his last, culminated in 1909 in the
article "Economique et Mecanique." The object of the article is explained
in a letter to George Renard:
11. Walras (1987. p. 306-307).
12. These objections are reviewed by Jafft in letter 1509 of the Correspondence.
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30
HES Bulletin 11:1 (Spring, 1989)
But it is not enough to popularize the mathematical method in
political economics by elementary education; it also has to be
justified in the eyes of competent men. This is what I have tried
to do in a Memoir: "Economique et Mecanique"... 13
The central theme in this article was to show the analogy between mathematical
formulations in economics and those in mechanics. For, as Descartes had
asserted, any science dealing with magnitudes can be subjected to the analytical,
mathematical method, thus being a mathematical science. This of course, could
apply to any science dealing with magnitudes. They all have one thing in
common: the mathematical method.
"Economique et Mecanique" shows that both economics and mechanics deal
with magnitudes, though very distinct ones. In the case of economics, according
to Walras, we are dealing with 'intime' phenomena, which happen within us
and are therefore individual and subjective. On the other hand, mechanics deals
with 'exterior' phenomena, which happen outside of us and can be observed
by everyone. The fact, then, that both deal with observable phenomena makes
in both cases a mathematical method applicable. In doing so, some analogies
in mathematical expressions could appear between economics and mechanics,
due to their common method of analysis. Walras shows in the article
"Economique et Mdcanique" that for some particular cases this analogy can
be found. Among Walras's papers preserved at the University of Lausanne
we found the following chart which clarifies his point:
Chart of Analogies14
Falling bodies
2
e = '/2g.t distance moved e = f (t)
v = g.t
velocity
v = F'(t)
a = g
acceleration
a = f"(t)
Exchange
Utility
u = 0 (q)
rarete'
r = 0'(q)
(? Inflexion)
= 0"(q)
convexity or
concavity
This chart above shows that the analogy between economics and mechanics
is contained in the general, undertermined mathematical expressions, here
represented by the process of derivation.
This is also as far as the analogy goes. There is not, in Walras's opinion,
an analogy in ideas or concepts between economics and mechanics. Arguments
in favor of stressing the analogy any further than the methodological one would
13. Jaffe(1965); L. 1722, L. Walras to G. Renard, 21 January 1909 (emphasis added).
14. Note presently in the 'Fonds Walras,' Lausanne; original in French (our translation). This
note is part of a collection of notes from which it becomes clear that Walras intended to extend
the article "Economique et Mecanique' with a section on dynamics. Eventually he decided
to withdraw this extension. This is discussed in Van Daal and Jolink (1989).
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Jolink & Daal
Leon Walras's Mathematical Economics
31
imply that the objects found in both sciences would be comparable. Walras
does not go as far as that. In a letter to Boninsegni he writes:
I consider these magnitudes [utility and rarete] as psychical
magnitudes, susceptible of being appreciated and measured only
by the consumer who experiences them and, in that sense, distinct
from physical magnitudes susceptible of being appreciated and
measured by all the spectators who observe them. In addition, I
attempt only to demonstrate that we proceed in economics, with
regard to these [psychical magnitudes], exactly as the
mathematicians [proceed] with regard to the physical magnitudes
in mechanics and astronomy..."
The differences in magnitudes that can be encountered in either economics
or mechanics, also have their implications on the practical application of
mathematics. In the case of psychical magnitudes, e.g., utility, and the
impossibility of measuring them, a practical (numerical) application of
mathematics is excluded. A practical application of mathematics in the case
of measurable physical magnitudes would not encounter these problems.
Therefore, the analogy between mathematical expressions in both economics
and mechanics is restricted to analogies in mathematical method.
FV. Concluding remarks
In this article we have tried to show that Walras's conception of mathematics
can be attributed to Descartes' Mathesis Universalis. In this sense, mathematics
is a method of analysis, rather than an instrument for numerical calculation.
Although this might seem quite obvious at present, it was not at all clear to
his opposing contemporaries.
In an attempt to justify the mathematical method in economics, Walras
compared the analytical results with those in the exemplary mathematical
science of mechanics. By illustrating his ideas with mechanical analogies in
economics, however, Walras ran the risk of being misunderstood. His
contemporaries as well as present-day authors16 tend to see the physical idea
as the original, after which the economic analogy is found, instead of the other
way around. In our opinion this was never Walras's intention, due to an
unbridgeable difference between observed magnitudes; 'intime' versus
'exterior' phenomena. Therefore, Walras's conception does not allow an
analogy in mathematical method between economics and physics. Thus a correct
understanding of Walras's mathematical economics explains the presence of
mechanical analogies through his work.
15. Jaffe (1965); L. 1705, L. Walras to P. Boninsegni, 10 Sept. 1908, Walras's italics.
16. See, for instance, Mirowski (1984) and Mirowski and Cook (1989).
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HES Bulletin 11:1 (Spring, 1989)
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