1. science
2. mathematics

Energeia
Vol. 7, no. 1, 2015, pp. 85–95
ISSN 1666-5732
MATHEMATIZATION IN ECONOMICS
A DOLFO G ARC´I A DE LA S IENRA
Instituto de Filosof´ıa
Universidad Veracruzana
[email protected]
RESUMEN : Para ejemplificar el inicio del proceso de matematizaci´
on de la teor´ıa econ´omica,
en este art´ıculo me propongo hacer una reconstrucci´on racional, desde un ventajoso si bien
anacr´onico punto de vista, de las condiciones que hubiera tenido que satisfacer Jevons para
producir una medici´on fundamental de la utilidad como e´ l la entend´ıa, mediante el tipo de
experimento que describe en los textos relevantes (Jevons 1866, 1871).
PALABRAS CLAVE : Experimento de Jevons ⋅ medici´
on de la utilidad ⋅ utilidad marginal ⋅
matematizaci´on ⋅ medici´on fundamental
ABSTRACT: In order to illustrate the beginning of mathematization of economic theory, in this
paper I propose to provide a rational reconstruction, from an anachronistic albeit advantageous point of view, of the conditions that Jevons would have had to satisfy in order to produce
a rigorous fundamental measurement of utility, by means of the experiment that he describes
in the relevant texts (Jevons 1866, 1871).
KEYWORDS : Jevons’ Experiment ⋅ Measurement of Utility ⋅ Marginal Utility ⋅ Mathematization ⋅ Fundamental Measurement
1. Introduction
What is the nature of a mathematical economic theory? By ‘mathematized economics’ it is understood, nowadays, mathematical economics roughly as defined, say,
by the series Handbook of Mathematical Economics edited by Arrow and Intriliga85
ADOLFO GARC´I A DE LA SIENRA
86
tor (1981). In the Historical Introduction to the series (which contains a brief, albeit
quite complete history of the mathematization of economics), these authors say:
Indeed, while the field of “mathematical economics” could be defined . . . as one that “includes various applications of mathematical concepts and techniques to economics, particularly economic theory”, an alternative approach to define the field would be to enumerate its parts. Pragmatically, our definition of the field in this sense is provided by the
Table of Contents of the Handbook. We recognize, however, that this definition is not truly
complete; considerations of space limitations and priorities have caused the omission of
some very active fields of mathematical economics (Arrow and Intriligator 1981, vol. 1, p.
1).
Standard textbooks on mathematical economics also provide a good idea of the
field. Outstanding among these are those of Takayama (1985) and Nikaido (1968).
The main mathematical techniques used in mathematical economics are, of course, linear algebra and calculus. But also convex analysis, mathematical programming (both linear and nonlinear), control theory, measure theory, probability and
statistics, and a whole array of fixed-point theorems. Beside the classical topic of
calculus-based marginalism, and the set-theoretic/ linear models of the period 1948–
1960, the main topics treated with these methods, enlisted by Arrow and Intriligator
(1981), are eleven:
(1) Uncertainty and information.
(2) Global analysis.
(3) Duality theory.
(4) Aggregate demand functions.
(5) Core of an economy and markets with a continuum of traders.
(6) Temporary equilibrium.
(7) Computation of equilibrium prices.
(8) Social choice theory.
(9) Optimal taxation.
(10) Optimal growth theory.
(11) Organization theory.
MATHEMATIZATION IN ECONOMICS
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Moreover, since the work of Leontief (1941), Georgescu-Roegen (1950), Burgess Cameron (1952), and Morishima and Seton (1961), the labor theory of value
has also been mathematized in terms as rigorous as those of the aforementioned
theories.1 Hence, the labor theory of value must be considered also as a province in
mathematical economics.
Sraffa’s economic theory, in contradistinction, has never received such a strict
mathematical formulation, and its followers are enmeshed in disputes about the use
of non-constructive mathematical methods in the demonstration of its main results.2
Other economic theories, like those of Keynes or the institutionalists, are far from
being formulated in a strict mathematical fashion, and so are far from being considered as parts of mathematical economics.
Mathematical economics gives rise to several philosophical questions. The first
one is whether mathematically formulated economic theories are the result of the
“mathematization” of a previously given, non-mathematical economic theory, or
whether they are innately mathematical; whether they are born with their mathematical structure “built in”. Another one is What conditions within the historical
development of mathematics have made possible the rise of mathematical economics? Has mathematical economics reached out to given, established mathematical
theories, or has she given rise to new branches of mathematics?
A logical question that arises is What is the logical structure of a mathematical
economic theory? Is it a set of sentences formulated in first-order logic? Is it really
possible to formulate mathematical economic theories in first-order logic? If the
answer is negative, is there a “canonical way” of describing their logical structure?
What is, indeed, the conception of the axiomatic method behind the work of the
mathematical economists in formulating their theories?
Other questions are epistemological. What is the structure of explanation in mathematical economics? What empirical claims can be made using a mathematical
economic theory? Does the explanatory power of a previously non-mathematical
theory increases once it receives a mathematical formulation? Why is it precisely
that it has been deemed necessary or useful to mathematize economics? In responding to these questions, the role of fundamental measurement (in the sense of Krantz
et al. 2007) must be taken into account.
1
2
For a history of the mathematization of the labor theory of value, see Garc´ıa de la Sienra 1992, §2.3.
Cf., for instance, Vela Velupillai 2008.
ADOLFO GARC´I A DE LA SIENRA
88
2. The Rise of Mathematical Economics
It is fair to say that mathematical economics arose with the introduction of calculus
in economics, due mainly to William Stanley Jevons (1871), Carl Menger (1871) and
Leon Walras (1874). This marks the appearance of the idea of utility as a numerical
magnitude and to the related idea of marginal utility, which gave the new theory
its name: marginalism. Marginalism provides the most important historical example
of a mathematical economic theory. In a paper read before the Royal Statistical
Society in 1862, Jevons makes clear that economics “has of necessity always been
mathematical” because it is “concerned with quantities”, but this does not mean that
the theory has received a “strict and general statement”:
The following paper briefly describes the nature of a Theory of Economy which will
reduce the main problem of this science to a mathematical form. Economy, indeed, being
concerned with quantities, has always of necessity been mathematical in its subject, but
the strict and general statement, and the easy comprehension of its quantitative laws has
been prevented by a neglect of those powerful methods of expression which have been
applied to most other sciences with so much success. (Jevons 1866, §1)
This suggests that economics is inherently mathematical but that not all economic
theories receive a rigorous mathematical formulation. I shall briefly try to show now
how the concept of utility was the result of a certain “mathematization” carried out
by Jevons.
To mathematize an economic magnitude means to provide a metrization of it,
in the sense of Garc´ıa de la Sienra (2013). Metrization gives rise to some mathematical object, usually a function of a certain sort. Jevons’ theory begins with the
metrization of certain types of sensations of pleasure and pain, namely those that
“arise periodically from the ordinary wants and desires of body or mind, and from
the painful exertion we are continually prompted to undergo that we may satisfy our
wants” (Jevons 1866, §2).
The very starting point of marginalism is the metrization of a magnitude capable
of more or less, of a certain comparison relation:
We always treat feelings as being capable of more or less, and I now hold that they are
quantities capable of scientific treatment. [. . . ] Our estimation of the comparative amounts
of feeling is performed in the act of choice or volition. Our choice of one course out of
two or more proves that, in our estimation, this course promises the greatest balance of
pleasure. When there is a large overbalancing force on one side, indeed, the estimation of
the amount of this balance is no doubt very rude; but all the critical points of the theory
MATHEMATIZATION IN ECONOMICS
89
will depend on that nice estimation of the opposing motives which we make when these
are nearly equal, and we hesitate between them. (Jevons 1866, §3)
In the previous paragraph Jevons introduces a relation of comparison among
feelings, and also anticipates a sort of “revealed preference”, but in the next one
he is quite accurate regarding the magnitudes he wants to measure:
feelings have two dimensions, intension and duration. A pleasure or a pain may be either
weak or intense in any indivisible moment; it may also last a long or a short time. If
the intensity remain uniform, the quantity of feeling generated is found by multiplying the
units of intensity into the units of duration. But if the intensity, as is usually the case, varies
as some function of the time, the quantity of feeling is got by infinitesimal summation or
integration.
Assuming that the agent has perfect memory, so that he can compare any two
feelings as to their intension, let F be the set of feelings and ≿ the comparison
relation among them, so that x ≿ x′ iff the agent finds feeling x more intense
ADOLFO GARC´I A DE LA SIENRA
90
(pleasant) than feeling x′ . Now, a necessary condition for the metrization of relation
≿ is that the structure ⟨F, ≿⟩ be a weak order. The condition is also sufficient if F is
at most countable, but the metrization might not exist if F is uncountable. Moreover,
any metrization g∶F → R need not be continuous, something which Jevons needs
in order to map F into the set of real numbers.3 Jevons wants to consider a time
interval T = [0, τ ] during which the agent is enjoying some good; for instance,
drinking water. The intensity of the sensation may be constant along this interval,
but it may also vary in many ways. Actually, Jevons requires another function u
relating each instant t ∈ [0, τ ] with a certain level of intensity; i.e. with a number in
R. This is just the composition u = g ○ f , as depicted in figure 2. Moreover, in order
to define the concept of quantity of feeling, Jevons needs u to be continuous almost
everywhere, as this amount at time t is defined as the integral
U (t) = ∫
t
u(z) dz.
0
Clearly, while u(t) ≥ 0, U (t) increases with time. The marginalist idea, of course,
is that U grows with time up to a point but: if the agent keeps consuming the good,
there will be a satiation point t∗ in which u becomes negative and forces U to start
decreasing. Clearly, at point t∗ , U reaches a maximum, which is precisely when u
crosses the horizontal axis, that is when u(t) = 0. Thus, u is nothing but the marginal
utility, and it is seen that the condition
dU ∗
(t ) = u(t∗ ) = 0
dt
is exactly right.
As Jevons puts it,
3 The existence of all types of metrizations in science, especially economics, is the topic of Krantz et al.
2007.
MATHEMATIZATION IN ECONOMICS
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Amount of utility [the level of U ] corresponds to amount of pleasure produced. But the
continued uniform application of a useful object to the senses or the desires, will not
commonly produce uniform amounts of pleasure. Every appetite or sense is more or less
rapidly satiated. A certain quantity of an object received, a further quantity is indifferent
to us, or may even excite disgust. Every successive application will commonly excite
the feelings less intensely than the previous application. The utility of the last supply of
an object, then, usually decreases in some proportion, or as some function of the whole
quantity received. This variation theoretically existing even in the smallest quantities, we
must recede to infinitesimals, and what we shall call the coefficient of utility, is the ratio
between the last increment or infinitely small supply of the object, and the increment of
pleasure which it occasions, both, of course, estimated in their appropriate units. (§8)
If “time” is interpreted as amounts of the good consumed (or as the time during
which the good is consumed), u(t) can be interpreted as the coefficient of utility at
t. Then the fundamental law of Jevon’s theory can be formulated thus:
The coefficient of utility [our u] is, then, some generally diminishing function of the whole
quantity of the object consumed. Here is the most important law of the whole theory. (§9)
This rational reconstruction of Jevons’ ideas gives us a glimpse of the role of
metrization in mathematical economics but does not tell us anything about the way
these theories can be applied (if they can be applied at all) to make empirical claims
and make explanations. This is the topic of the final section, but this topic cannot
be dealt with unless a disquisition about the logical structure of such theories is
introduced before.
3. The Logical Structure of Mathematical Economic Theories
Perhaps the clearest example of a mathematical economic theory is the theory of
value as presented by Gerard Debreu in his classic Theory of Value (1959). Along
the lines of Bourbaki’s mathematics, based mainly on the set-theoretic concept of
structure (cf. Weintraub 2002, ch. 4), Debreu formulated his theory of value following a logical methodology which was going to be further elaborated at Stanford
by J. C. C. McKinsey, and mainly by Patrick Suppes, to produce one of the most
influential views in the philosophy of science. The application of this methodology
in economics was pursued at the Institute for Mathematical Studies in the Social
Sciences (IMSSS) of Stanford University.
Instead of identifying scientific theories as sets of sentences formulated in some
formal language (as Rudolf Carnap among others proposed to do), Suppes proposes
to identify the theory through an extrinsic characterization of its models, using the
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language of informal set theory. Here the relevant notion of model is precisely the
one that Bourbaki (1968) defined under the label ‘mathematical structure’. These
structures are introduced by Suppes through the definition of a set-theoretic predicate, like ‘A is a topological space’ or ‘A is a game’.4 In order to illustrate this method,
let us try to reconstruct Jevons’ theory as formulated in the previous section.
Since the theory intends to describe the behavior or actions of a consumer, we
need first a conceptual apparatus describing such behavior. It is usual to adopt a
choice correspondence η for this effect. In the present case, the correspondence or
function must describe the amount of the good selected by the agent over the set
of all the possible amounts; i.e., η is a function from the family of all subintervals
S = [0, t] of T into T . For any such set S, η(S) is the amount actually chosen by the
agent. The “most important law of the theory” is actually a description of the utility
function: It says that u is strictly quasi-concave (or ≿ strictly convex). The law is in
fact the assertion that utility begins to decrease after a certain amount of the good
has been consumed. An implicit presupposition of Jevons is the agent is “rational”;
i.e. that she chooses that point in S that maximizes her utility. More precisely:
∀S
U [η(S)] = m´ax U (t).
t∈S
In particular, η(T ) is the point where U reaches its maximum; i.e. where u(t) = 0.
Hence, we can formulate the theory as follows.
D EFINITION 1 J is a Jevonian consumer iff there exist T , 7, η and u such that
(0) J = ⟨T, 7, η, u⟩.
(1) T is a closed interval [0, τ ] of real numbers.
(2) 7 is the set of all closed subintervals of T of the form [0, t], 0 ≤ t ≤ τ .
(3) η∶7 → T is a function that assigns to each element S ∈ 7 a point η(S) of
S.
(5) u∶T → R is a quasi-concave, real-valued continuous function.
(6) ∀S
U [η(S)] = m´axt∈S U (t) (fundamental law).
4 For a rigorous formulation of Suppes’ notion of set-theoretic predicate, see Da Costa and Chuaqui
(1988).
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4. Explanation Through the Application of Mathematical Economic Theories
To explain in economics is to “apply” successfully the conceptual apparatus of some
economic theory to empirical data; i.e. to find specific functions, out of empirical
data about the object of the application (e.g. a consumer), that not only satisfy
the characterizations of the theory, but above all the fundamental law. Yet, as I
try to explain in Garc´ıa de la Sienra (2009), the production of empirical claims
by means of mathematical economic theories is not straightforward, but requires
several mediating steps.
First of all, it requires a pre-theoretical identification of the phenomenon to
be explained. This explanandum must be described in appropriate terms, the nontheoretical terms of the theory. In the present example, T , 7, and η, in order to get a
partial potential model C = ⟨T, 7, η⟩, specifying the time interval T , and the choice
function η; all these out of empirical observations.
To explain C consists of finding a u of the appropriate sort that satisfies the
fundamental law. The utility function obtained by integration out of u
is peculiar to each kind of object, and more or less to each individual. Thus, the appetite
for dry bread is much more rapidly satisfied than that for wine, for clothes, for handsome
furniture, for works of art, or, finally, for money. And every one has his own peculiar tastes
in which he is nearly insatiable. (§10)
Hence, C is to be specified depending upon the agent and the kind of object under
consideration. Since u is a Jevons-theoretical function, its determination will make
essential use of the fundamental law. The key is to find the specific form of u and
to infer out of it the behavior of the agent, inducing a “theoretical” behavior in the
form of a model C′ of the same similarity type as C. The explanation (prediction
or retrodiction) is “perfect” if C′ is identical to C, but usually, they will be only
“similar”, in the best case. This means that the distance between η and η ′ will not be
“too large”.
5. Conclusions
The first conclusion is that, usually, mathematically formulated economic theories are the result of the “mathematization” if not of a previously given, nonmathematical economic theory, at least of some crucial ideas (like marginal utility
in the case of Jevons). Mathematical economics was made possible in the nineteenth
century thanks to the availability of calculus. Twentieth century theories were made
possible by the rise of set-theory, topology, the development of convex analysis and
94
ADOLFO GARC´I A DE LA SIENRA
fixed-point theorems, and both linear and nonlinear programming. It is fair to say,
I think, that mathematical economics has reached out to given, established mathematical theories, and I really do not recall any case in which it gave rise to a new
mathematical method or theory.
It is clear that no real-life scientific theory can be formulated as a set of sentences
formulated in first-order logic. I have tried to make a case for a “canonical way” of
describing their logical structure. The conception of the axiomatic method behind
the work of the mathematical economists in formulating their theories is basically
the one developed by Suppes at Stanford University.
Once the theory has been rationally reconstructed following Stanford’s methodology, it is possible to raise in precise terms the question of its empirical claim. I
have explained the structure of explanation in mathematical economics basically in
terms of an imbedding of empirically-determined structures into the non-theoretical
structures stemming from the theoretical part. There is no doubt that the explanatory
power of a previously non-mathematical theory increases once it receives a mathematical formulation, if only because it becomes more precise. It has been deemed
necessary or useful to mathematize economics because the topic itself requires it (as
Jevons claimed), mainly because some propositions cannot be even formulated, or
some concepts defined, in terms of ordinary language alone.
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