La matematica nella storia dell’economia. Primo Workshop, Torino 16-17 ottobre 2003
History of Neoclassical Consumer Theory:
A Neo-Kantian Epistemological Perspective
Ivan Moscati*
ABSTRACT
This paper deals with the history of neoclassical consumer theory from 1871 to 1971, and with
the Neo-Kantian knowledge theory of the Marburg School, a philosophical movement whose
main exponents were H. Cohen, P. Natorp and E. Cassirer. The strictly historical investigation
focuses on two aspects somehow neglected in the previous literature: the acceptance in consumer theory of a general utility function in place of the original additive function, and the
comparison between the behavioral and the utilitarian formulation of the theory. In restating
these two aspects, I propose to show the usefulness of the Marburg School’s epistemology in
understanding the history of consumer theory and, more generally, the principles that rule the
process of knowing of neoclassical economics. I also suggest that the Marburg theory of knowledge provides a philosophical foundation to the cognitive logic of neoclassical economics.
Keywords:
Consumer Theory, Utility Theory, Revealed Preference Theory, Neo-Kantianism,
The Marburg School Epistemology
JEL Codes: B13, B21, B4, D11
*
Bocconi University, Milan, Department of Economics, and University of Florence, Ph.D.
in History of Economic Thought. E-mail: [email protected]
I am most grateful to R. Arena, P. Battigalli, N. Bellanca, L. Bruni, M. Dardi, N. De Vecchi, R. Faucci, A. Gay, N. Giocoli, G. Lunghini, F. Marzano, P. Mongin, A. Montesano,
S. Zamagni, C. Zappia for their comments on earlier drafts. All errors remain mine.
Neoclassical consumer theory emerged, developed and reached its current standard
form over the course of a century, from the time it first appeared in the works of Menger
and Jevons in 1871 up to the 1971 volume Preferences, Utility, and Demand edited by
Chipman et al., which fine-tuned it in many points. Although, up to now, there has been
no detailed overview of the century-long development of consumer theory, many of its
phases have been examined by numerous authors1. Some of them also try to draw certain epistemological conclusions about the criteria that seem to characterize the development of consumer (or demand) theory2. The history of this fundamental part of microeconomic neoclassical analysis and the logic that appears to guide its development
are also the subject of this paper.
Even I shall examine the history of consumption theory, focusing on two central aspects that have been somewhat neglected in the previous literature: the acceptance of a
general utility function in place of the original additive utility function, and the comparison between the behavioral and the utilitarian formulation of the theory3. This historical inquiry concludes that more realistic elements (such as general utility compared
with additive utility) are introduced into consumption theory only if they do not weaken
the systematic order of the conceptual representation of the phenomenon (consumption)
already achieved. If the inclusion of elements that correspond better to common-sense
evidence, empirical data or experimental verifications jeopardizes the determinateness
and the systematic unity of the theory, they are simply put aside. This does not mean
that the search for realism is irrelevant in neoclassical theorizing about consumption,
but just that the pursuit of conceptual integrity is stronger than that of realism. One can
say that neoclassical economics tries to maximize the realism of the theory under the
constraint of preserving its systematic view.
In this paper I argue that such a hierarchy between systematic character and realism
is not an idiosyncrasy of neoclassical consumer theory or of neoclassical economics in
general, but is rooted in a broader character of scientific understanding.
1
2
3
See Schultz 1935, Stigler 1950, Schumpeter 1954, Houthakker 1961, Hurwicz 1971, Samuelson
1974, Wong 1978, Blaug 1992 [1980], Dooley 1983, Mirowski 1989, Frambach 1993, Wade
Hands–Mirowski 1998, Hurwicz 1998, Mirowski–Wade Hands 1998, Weber 1999a, Weber
1999b, Mongin 2000, Bruni 2002, Mirowski 2002, Chipman–Lenfant 2002, Montesano 2003.
See in particular Stigler 1950: 392-396 and Mongin 2000: 1145-1149.
Throughout the paper, “utilitarianism” is meant to be related to individual choice theory, not to the
social choice and ethical doctrine of Bentham, Mill or Sidwick.
1
In setting out this thesis I rely on the epistemology of the Marburg School, a NeoKantian philosophical movement whose main exponents were H. Cohen (1842-1918),
P. Natorp (1854-1924), and E. Cassirer (1874-1945). In the literature on economic
methodology, this epistemological perspective is virtually unexplored, even if it seems
more powerful than others in grasping how neoclassical economics actually works4. In
particular, the Marburg Neo-Kantianism makes it possible to explain the reasons why
heterodox criticisms have had little effect on neoclassical theory. It also allows us to explain the formalist style of the neoclassical theory and proposes a philosophical foundation for the neoclassical method of successive approximations.
The aim of this paper is therefore threefold. First, I reconstruct the history of neoclassical consumer theory, trying to demonstrate that the Marburg School’s epistemological perspective is useful in understanding its development. Second, I try to show
how the Marburg epistemology makes it possible to grasp the principles that rule the
process of knowing of neoclassical economics and does so better than other epistemological theories. Third, I suggest that the Marburg theory of knowledge not only allows
us to comprehend the actual cognitive logic of neoclassical economics but also gives it a
philosophical foundation.
1. AN OVERVIEW OF THE EPISTEMOLOGY OF THE MARBURG SCHOOL
The Marburg School was part of the larger philosophical movement called NeoKantianism that, under the motto “Back to Kant!”, emerged in German philosophy in
the 1870s. Maintaining Kant’s critical method as the reference point, Neo-Kantianism
was opposed to both speculative idealism and positivism. During the late 19th and early
20th century, the movement reached a dominant position in German philosophy, which
it maintained until the 1920s when other philosophical trends (including phenomenology and logical positivism) put it on the fringe of the philosophical arena5.
4
The term “Neo-Kantian” is sometimes associated with Mises’ claim of deducing economic propositions from some synthetic, a priori statements regarding the nature of human action (cf. Mises
1949, Parsons 1990 and 1997). However, I do not use the term in the sense of Mises’ praxeology: I
refer to Neo-Kantianism for a theory of knowledge, not for a theory of action, and hence for an
epistemological theory and not, in the end, for an economic one. Moreover, as will be shown, the a
priori principles to which Marburg Neo-Kantianism appeals are the regulative principles of understanding, whereas Mises attempts to employ the category of action as the constitutive principle of
the economic domain. Regarding the flaws of such an attempt see Barrotta 1996.
5
For a general introduction to the Neo-Kantian movement see Ferrari 1997 and Ollig 1998.
2
Of the different trends included in Neo-Kantianism, the Marburg School is characterized by its logicist development of Kant’s theory of knowledge along the rationalist
Plato-Descartes-Leibniz line, and because it elaborates its epistemological view in connection with a thorough analysis of contemporary science. The main epistemological
tenets of the Marburg School can be outlined as follows:
1) Reality is never given in and of itself, but is always mind-correlative. Things are
always given within some kind of ‘sense’ of them: perception, common sense, scientific
knowledge, practical use, moral commitment, aesthetic experience etc. Of all these
forms of experience of things, scientific knowledge, however, holds a position of
prominence in the sense that ultimately something is ‘real’ exactly as science takes it to
be.
2) Because things are given within the knowledge of them, the a priori forms of understanding determine the things as given. Philosophy must therefore investigate the
principles and laws of understanding instead of the principles and the laws of things. In
line with the basic ideas of Kant, for the Marburg School, philosophy also becomes a
‘critique of reason’.
3) Since the a priori principles of understanding guide the development of scientific
knowledge, the Marburg School sees in the actual history of science an a posteriori
manifestation of these guiding principles. What Cohen calls “the fact of science” therefore becomes the starting point for the renewed Neo-Kantian critique of reason: “We
objectify […] reason in science. Critique of reason is a critique of knowledge or of science” (Cohen [1984] 1883: 6, my translation).
4) What are the principles that also regulate the process of scientific knowledge? According to the Marburg Neo-Kantianism, scientific understanding basically endeavors to
put in order a domain of phenomena in a systematic unity, i.e. attempts to connect the
diverse experiences in an ordered and stable system of dependencies. Scientific thought
aims to substitute the hazy given of perception with exact notions, it seeks to replace the
imprecise and uncertain relationships between pre-scientific things with concept connections that contain as much exactness and rational evidence as possible, it tries to
supplant variable and indecisive dependencies with constant and necessary ones. What
is “real and objective” in a properly scientific sense is gained by this synthesizing and
systematizing activity of understanding.
5) The goal of scientific knowledge is therefore not to attain a perfect copy of what is
presented in the kinds of experiences (perception, common sense, etc.) that precede sci-
3
entific knowledge. In its fundamental tendency to design a systematic unity of the
given, theoretical thought can lose the characteristics of things as they are immediately
offered in perception or common sense. This possible loss of immediacy and concreteness does not mean that the knowledge becomes unscientific or false. This sort of opinion misunderstands the essence of scientific understanding and thus makes it impossible
to account for the actual shape of science.
6) In particular, scientific concepts need not correspond to objects of perception or
common sense nor need they be accurate imitations of ‘something of the real world’ nor
abstractions maintaining the essential features of the ‘real world’. However, correspondence to common sense is not the right criterion to evaluate scientific concepts since
they are products of thought through which understanding reconstructs experience as a
system of dependencies thus making it intelligible.
7) It is worth noting that the Marburg scientific fictionalism is not to be confused
with instrumentalist fictionalism. Both epistemological views acknowledge that, in the
first instance, scientific theories do not aim to be exact copies of ‘reality’. Although instrumentalism regards a theory as a useful tool for practical ends (typically for prediction), Marburg Neo-Kantianism regards it as primarily useful for theoretical ends, that
is, in order to make experience intelligible.
8) It is also noteworthy that the Marburg epistemological view does not prescribe
what the ‘right’ theory for a particular field of phenomena should be. As a matter of
fact, the intellectual pursuit of a systematic synthesis of a given domain is just a formal
one, which actually and historically produces scientific systems with different contents.
Therefore, two divergent theories of a certain phenomenal domain could coexist, if they
are both able to connect the diverse elements of the domain in an evenly ordered and determined system of entailment relations. On the contrary, if the system of dependencies
set forth by one theory is more complete and exact than the one set forth by another,
scientific understanding will reject the latter and accept the former6.
6
The idea that there can be a multiplicity of theories for giving the manifold phenomena a synthetic
unity links Marburg to the conventionalism of Poincaré (1902) and especially of Duhem (1906). In
fact, the Marburg School shares with Duhem’s conventionalism the concept of scientific theories
as systematic wholes, whose parts cannot separately be subject to empirical verification, and the
subsequent idea that a theory can be evaluated just with reference to another theory and not with
reference to the observational data in themselves.
4
9) According to the Marburg School and particularly Cassirer, the relations among
the elements of a scientific system could be expressed in the most appropriate way
through mathematical relations, and in particular, as mathematical functions. This is due
to the fact that mathematics is regarded as a science of relations rather than as a science
of quantity: “The object of mathematics […] is not the compounding of magnitudes, but
the connection and reciprocal determination of relations” (Cassirer 1953 [1910]: 257).
Thus, the more the dependencies between the components of the theoretical system become exact, the more they can be expressed as mathematical functions. One could
therefore say that science intrinsically tends to mathematize its theories.
In the following Sections, I attempt to show how these epistemological tenets of the
Marburg School allow us to understand the historical development of neoclassical consumer theory and, more generally, to account for the epistemic (i.e. knowledge-relative)
logic of neoclassical economics. This does not mean that the economists that played a
part in developing consumer theory were somehow inspired by Marburg epistemology:
to my knowledge, none of them refer to it anywhere, nor do Cohen, Natorp, Cassirer refer to economic science in their writings. Therefore, I am not claiming that the Marburg
School’s epistemology is the common ‘muse-philosophy’ of neoclassical economists,
which has been concealed up to now. Neoclassical economists, moreover, have professed and profess different methodological views. Despite these differences, consumer
theory seems to have specific and stable principles that regulate its development. These
regulative principles, and not the possible, hidden roots of the methodological views of
neoclassical economists are the subject matter of the paper. The point I would like to
make is that the Marburg theory of knowledge explains such principles well. The following reconstruction of consumer theory is therefore not an investigation of hidden intellectual backgrounds, but rather a case for the general validity of Marburg School’s
theory of knowledge.
2. ADDITIVE UTILITY AND GENERAL UTILITY
Neoclassical consumer theory originates in the marginalist value theories of Menger
(1871), Jevons (1871) and Walras (1874), and becomes an autonomous body of doctrine
with Marshall’s Principles (1890). As is well known, according to the marginalists the
economic value of a commodity depends on the evaluation that the subjects give its
marginal unities. Such marginal evaluation is called “final degree of utility” by Jevons,
5
“Grenznutzen” by the Austrian School, “rareté” by Walras and then simply “marginal
utility”. Even though Jevons acknowledges that utility is not measurable, he first writes
down total utility as a cardinal function u (x) of the quantity of the good, and marginal
utility as the function φ (x) obtained by differentiating the former. In addition, Jevons assumes that marginal utility φ (x ) is a positive, decreasing and additive function. This last
feature presupposes that total and marginal utility of any one good are independent of
the quantities consumed of all others goods7. Under these assumption on φ (x ) and for
the two commodities/two agents case, Jevons obtains his famous “equations of exchange”, which state that, in order to maximize trader utility under the budget constraints, and with no trades at disequilibrium exchange ratios, it must hold that
φ1( x ) px φ 2 ( x )
=
=
φ1 ( y ) p y φ2 ( y )
where φ j (i ) is the marginal utility of the commodity i for the trader j, and p x p y is the
equilibrium exchange ratio between the commodities (Jevons 1871: 95 ff.).
Under the same assumptions as Jevons, in the Élements Walras systematizes the subjective value theory in a coherent and general price theory and first fixes the exact relationship between the marginal utility of a good and its demand (Walras 1874: 77 ff.). In
his Mathematical Psychics, Edgeworth stresses the need to introduce into the theory a
generalized (i.e. non additive) utility function, which would make it possible to understand the apparent phenomenon of the utility interdependence of goods (Edgeworth
1881: 20, 104). In order to analyze the exchange when utilities are interdependent,
Edgeworth constructs the indifference lines (or curves) which are to become an essential
conceptual tool for the subsequent whole consumption theory.
However, the introduction of a general utility function represents a tricky theoretical
problem. In fact, the assumptions of additive, at the margin positive, and decreasing
utility, make it possible to univocally determinate the relationships between the demand
for goods, their prices, and consumer income, and to ensure that the second order conditions for the constrained maximization of utility are certainly satisfied. As regards the
first issue, if the utilities of the goods are independent, the demand for a good definitely
decreases (increases) as its price rises (falls) (i.e. ∂xi ∂pi < 0 ). In other words, with addi7
In mathematical terms, Jevons assumes that u (x1 ,..., x n ) = u1 (x1 ) + ... + u n (x n ) and thus that
∂u ∂xi = φ i ( xi ) , with φi (xi ) > 0 and dφ i dxi < 0 . Additivity implies that ∂ 2 u ∂xi ∂x j ≡ dφi dx j = 0 .
6
tive utility the so-called “law of demand” holds, and demand curves are always downward sloping. With a general utility function this is no longer univocally true: demand
for a good might also increase as its price rises (the Giffen case). However, in order to
completely understand under what mathematical and economic conditions such a phenomenon can occur, we have to wait until the 1930s. The analytical expression of
∂xi ∂pi in the general utility case was already made by Pareto as early as 1893 (Pareto
1892-93, V: 306). However, it is only with the decomposition of ∂xi ∂pi in terms of income and substitution effect in Slutsky’s 1915 article (re-discovered in the Thirties) and
in Hicks-Allen’s 1934 paper, that the economic meaning of an upward sloping demand
curve becomes clear from a theoretical standpoint8.
Second, if the utilities of the goods are independent (and marginal utilities are usually positive and decreasing), it can be shown that demand for a good definitely increases (decreases) as consumer income I rises (falls). In mathematical terms ∂xi ∂I > 0 ,
which means that all goods are normal. With a general utility function, this statement is
no longer true. Even in this case, we need to wait for Slutsky and Hicks-Allen to have a
precise expression and an exact economic understanding of the ratio ∂xi ∂I 9.
As regards the utility maximization, the assumptions of additive, at the margin positive, and decreasing utility, ensure that the second order conditions for the constrained
maximization problem of the agent’s utility are satisfied. Therefore, if all commodities
are purchased, first order conditions are sufficient for a maximum. In geometrical terms,
Jevons’ assumptions imply that the agent’s indifference curves are strictly convex, so
that tangency conditions are sufficient to discover the optimal combination of commodities. With a general utility function, this is no longer true. Even with decreasing marginal utilities, plausible substitution relationships between commodities could cause indifference curves to be concave, so that a point that satisfies the tangency conditions can
also be of minimum, not maximum, utility (in the literature of the period such points are
called “unstable equilibria”). Moreover, if the indifference curves are convex in some
tracts and concave in others, there can be not one, but a multiplicity of tangency points,
8
9
On the relationship between Pareto’s expression and those of Slutsky and Hicks–Allen, see
Schultz 1935: 433-447, Dooley 1983 and Weber 1999b.
In Pareto, the analytical expression of ∂xi ∂I is not present. In Pareto’s general theory of exchange, in accordance with Walras’ approach, income is not regarded as a distinct exogenous variable, but is endogenously given as the value of the agent’s endowment at market prices. More on
this in Weber 1999b.
7
some of maximum and others of minimum utility. Edgeworth aims to introduce a general utility function, but without losing the convexity of indifference curves. Therefore,
the introduction of a general utility function raises two problems: that of providing the
analytical condition for convexity (i.e. the second order condition for the constrained
maximization problem), and that of justifying why such condition should hold. Edgeworth (1881: 34 ff., 108 ff.) provides the condition at issue for the two-commodities
case, and claims that it is, by and large, satisfied, nonetheless with an utterly deceptive
argument.
In conclusion, although Edgeworth draws attention to the unrealistic nature of the
additive utility assumption, he is unable to solve the problems raised by the adoption of
general utility. In this state of affairs, the introduction of a general utility function in the
theory would have caused the loss of the determinateness of the dependencies between
demand, prices, and income obtained by Jevons and Walras. With a general utility function, the demand curve can be downward or upward sloping (as well as horizontal) and
indifference curves can be convex or concave, but if one is unable to understand and to
exactly determine under which theoretical circumstances each case occurs, introducing
general utility just means introducing unpredictability and indetermination into the picture. In this situation, according to the epistemological theory of the Marburg School,
economics is expected to hold to the systematic theory based on the unrealistic additive
utility assumption. This is what, in fact, happens.
3. BETWEEN EDGEWORTH AND PARETO
Marshall basically rejects Edgeworth’s suggestion of a general utility function and
never utilizes indifference lines, which are the typical analytical tool that makes it possible to take into account the interdependence of commodity utilities. In all the editions
of the Principles (1890-1920), Marshall’s consumption theory relies on the additive
utility assumption in a crucial, even though subtle, way. Without this assumption “the
one universal rule to which the demand curve conforms [i.e.] that it is inclined negatively throughout the whole of its length” (Marshall 1961: 99, note) breaks down. However, this means that Marshall’s whole partial equilibrium analysis, which is based on
the intersection of a downward sloping demand curve and of an upward sloping supply
curve, totters. As regards the admission of the interdependence of utilities, Marshall’s
attitude is therefore rather inconsistent. Although in the first edition of the Principles
(1890), he explicitly deals with rival commodities, in the second edition (1891) he tries
8
to defend the additive utility assumption against Edgeworth (Marshall 1961: 845). Subsequently, in the third edition (1895), he openly admits that “we cannot say that the total
utility of [two commodities which contribute to the same purpose] is equal to the sum of
the total utilities of each separately”, and afterwards introduces the Giffen goods case,
which can be explained only if the additive utility assumption is removed, but luckily
such a case is “rare” (Marshall 1961: 131 f.). These are all surface adjustments which do
not modify the analytical groundwork of Marshall’s demand theory which continues to
be implicitly based on an additive utility function until the final 1920 edition of the
Principles10.
Apart from the question of additive utility, Marshall is to be remembered because he
introduces a way of dealing with the consumption theory, different from that of Jevons,
Walras and Edgeworth. First, Marshall considers purchasing by an isolated agent rather
than the exchange among many agents. Second, the consumer prices are fixed by the
market and are not the result of the exchange process. Third, the household does not
have an endowment of commodities to trade but a money endowment with which to
purchase the commodities. In this way, Marshall limits consumption analysis to the current boundaries and separates exchange theory from consumption theory.
In the 1890s, in addition to admitting the unrealistic nature of the additive utility assumption, another serious limitation of the classical utilitarian theory is highlighted, that
is, the absence of a measure of utility, even if barely theoretical. It is not at all clear
what the values that the function u (x) associates to the commodity quantities x mean,
how these values could be measured, and, foremost, what the unit of measure of utility
is. In his Mathematical Investigation, Irving Fisher tries to find an empirical measure of
utility based on the economic choices of agents. Nevertheless, Fisher’s attempt also collides against utility non-additivity: he himself recognizes that his method of measurement works only if commodity utilities are independent (Fisher 1926 [1892]: 64-67).
Gustav Cassel also attempts to overcome the problems related to utility by adopting an
alternative demand theory independent from the utility notion, but obtains no satisfactory results (Cassel 1899: 413 ff.).
Therefore, at the end of the 19th century the unrealistic nature of the classical – additive, cardinal – utility is quite clear. Classical utility is unrealistic in at least three ways:
10
Regarding the methodological reason of why Marshall rejected Edgeworth’s suggestion of a general utility function, see Dardi 1991: 96-101.
9
i) as an assumption in itself, since utilities of commodities appear to be interdependent;
ii) for its empirical implications which conflict with Giffen and inferior goods; iii) in the
sense that it does not correspond to anything actually measurable. However, despite the
recognized unrealistic character of classical utility, neoclassical economists hold to it.
The succeeding editions of Walras’ Élements (1889-1900) follow the first on this critical point. As already mentioned, not even Marshall changes the groundwork of his theory in the subsequent editions of the Principles. In their works, Wicksteed (1888),
Wicksell (1893), Barone (1894) and, although in a non mathematically explicit form,
Pantaleoni (1889 and 1898), Wieser (1889), and Clark (1899) maintain the cardinal additive utility assumption. In his first cardinalist phase, Pareto builds up a general utility
analysis more than any other without applying his own results. In the five-part article,
Considerazioni, he recognizes the soundness of Edgeworth’s generalization, states the
exchange analysis in general terms, but then develops it within the additive framework.
He subsequently resumes the general utility analysis. He first provides the exact expression of ∂xi ∂pi in the general utility case, but then continues by using the additive utility
which he declares “approximately true” (Pareto 1892-93, V: 306-307). Similarly, in the
Cours, Pareto develops the entire analysis with additive utility functions and discusses
the general case only in a footnote (Pareto: 1896-97: 332-334)11.
4. THE METHOD OF SUCCESSIVE APPROXIMATIONS
The Marburg explanation for this state of affairs is that neoclassical economists hold
themselves to the unrealistic classical utility since it allows them to construct a coherent
and determinate theory of demand. In this case, the commitment to realism conflicts
with the commitment to conceptual intergrity, and the latter prevails. This does not
mean that grasping the observed phenomena ceases to be the task of neoclassical economic inquiry. The point is that this understanding must be a theoretical one, that is, the
phenomena at issue (e.g. those related to the demand of commodities) have to be understood in a structured system of clear entailment relations. If capturing empirical evidence requires the introduction of assumptions that, however, make the theory indeterminate, then they are temporarily put aside. The method of successive approximations
basically consists in this effort of ‘approximating’ as much as possible the available
11
As regards the use of the additive utility assumption among leading neoclassical economist at the
end of the 19th century and at the beginning of the 20th, see Stigler 1950: 326-327.
10
evidence by means of successive theories based on progressively more plausible assumptions which must not, however, blur the theoretical system. Neoclassical economists – at least from Menger’s defense of theoretical economics against Schmoller’s
historical school in the Methodenstreit – have repeatedly upheld such a method in response to the criticisms of the lack of realism and empirical inconsistency often raised
by the different heterodox schools.
What I assert is that the Marburg theory of knowledge suggests a transcendental (i.e.
relative to the laws of thought) foundation for the method of successive approximations
and, accordingly, a vindication of neoclassical economists’ methodological common
sense. As a matter of fact, it explains the introduction and the maintaining of unrealistic
hypotheses, and it recognizes that even those hypotheses, whose implications conflict
with observational evidence, are not rejected until an alternative competitive theoretical
system is proposed. More decisively, the Marburg epistemology legitimizes the neoclassical condition that in the approximation sequence the model must remain determinate by tracing it back to the regulative principles of scientific understanding.
5.
PARETO THE BEHAVIORIST, PARETO THE UTILITARIAN
The second stage of neoclassical consumer theory began in 1900, with the publication
of Pareto’s Sunto (1900), and ended in the middle of the 20th century, with Samuelson’s
last momentous contributions to revealed preference theory. This phase is characterized
by the opposition of two approaches in consumption analysis: the behaviorist and the
utilitarian-ordinalist one. The behaviorist approach, in line with Fisher’s suggestion, intends to overcome every hint of internal and unobservable utility, by referring back to
the effective, observable consumer choice behavior. The utilitarian-ordinalist one instead, continues to base consumer theory on the utility concept and on the utility function, which, however, are regarded as having just an ordinal, not a cardinal meaning.
Pareto’s work, rich as well as ambiguous, represents the starting point of both approaches.
As is well known, after having embraced the classical utility theory, between 1898
and 1900 Pareto changes his perspective and considers it possible to construct a demand
theory just by analyzing consumer acts of choice, thus making the reference to utility
superfluous. Nonetheless, as observed by many authors from the 1930s on, Paretian
11
demand theory actually implies many implicit references to utility, not only to ordinal
but also to cardinal utility12.
In particular, utilitarian elements are evident in the way Pareto states the features of
indifference curves. Edgeworth’s indifference curve is indeed the analytical tool that Pareto brings into play in order to obtain the equations of pure economics from the “pure
and naked” fact of choice (Pareto 1900: 217). For Edgeworth the bundles on the same
indifference line are those that give the same utility to the subject, whereas for Pareto
they are simply those the subject is unable to choose from, and in this sense, is
“indifferent”. However, in order to reconstruct a complete demand theory, Pareto requires that the indifference curves resulting from choices be convex, since this ensures
(as seen above) that the first order conditions are sufficient to determine the optimal
combination of commodities. Like Edgeworth, Pareto therefore faces the problem of
justifying convexity but he should solve it just by referring to the characteristics of the
acts of choice. Pareto instead refers surreptitiously to psychological-utilitarian motives
and bases convexity implicitly in the assumptions of decreasing marginal utility of
goods and in disregarding their interdependence relations. He also assumes that the indifference curves are decreasing. This feature is also justified on the basis of psychological motives, namely tracing back the downward slope to the hedonistic principle of
non-satiation (cf. Pareto 1906: 258-259)13.
Moreover, it is noteworthy that decreasing marginal utility is not only a utilitarian
concept but a cardinal one, since this property is not invariant to an arbitrary, strictly increasing transformation of the utility function14. Similarly, Pareto’s notions of complements and substitutes in terms of second-order cross-partial derivatives of the utility
function, are cardinal. His idea that subjects can order not only bundles but also preference differences among bundles is also cardinal15.
12
13
14
For the 1930s see e.g. Hicks–Allen 1934: 52 ff. For a defense of Pareto’s utilitarian ambiguities
from a methodological standpoint see Bruni-Guala 2001.
On these issues see Ranchetti 1998.
In fact, even if
∂ 2 u ∂ 2 x ≡ u xx < 0 ,
the second-order derivative of
f (u (x) ) ,
∂ 2 f (u ( x ) ) ∂ 2 x ≡ f xx = f u′ xx + f ′u′ x u x , could be positive if f ′′u xu x is large ‘enough’.
15
According to Pareto’s notion of complements and substitute, two goods are complementary if
u xy > 0 , and substitute if u xy < 0 . Hicks and Allen (1934: 59-60) highlight that the sign of
f xy = f u′ xy + f ′u′ x u y can be different from that of u xy . Regarding Pareto’s idea that subjects can order preference differences, see Pareto 1906: 252-253. Lange 1934 shows that this is tantamount to
assuming cardinal measurability of utility.
12
Pareto’s utilitarian ambiguities are therefore not only of the ordinal sort but also of a
cardinal nature and exist not solely in the interpretations of the notions but also in the
theoretical analysis of demand. As a result, such ambiguities make Pareto’s behavioral
project of reconstructing demand theory on the basis of the “naked fact” of choice,
flawed.
6.
THE ROLE OF THOUGHT EXPERIMENTS
Realist critics of Pareto’s theory also draw attention to the fact that the experiment
through which Pareto expects to determine the indifference lines is a fictitious one (see
e.g. Mayer 1994 [1932]). Pareto, in fact, imagines putting a subject – a sort of Buridan’s
donkey – between two bundles x1 and x2, then changing the composition of x2 up to the
point where the subject is no longer able to choose between x1 and x2, thus determining
that x2 belongs to the same indifference line as x1. This experiment is to be repeated infinitely many times in order to discover all the bundles indifferent to x1, and then repeated all over again to determine the entire indifference map of the subject. However,
in this case it is not a matter of factual experiments: there is actually no subject, no bundle, and Pareto’s test is just a thought experiment16. From the empiricist standpoint, Pareto’s behavioral project of reconstructing demand theory on the basis of the observable
fact of choice is to be regarded not only as flawed, but simply as groundless.
However, from a theoretical standpoint, the problem is not that Pareto’s experiment
is just a mental one, but that it is somehow inconsistent. First of all, Pareto takes for
granted that the subject’s indecisiveness in the face of two alternatives is tantamount to
indifference, although this interpretation is legitimate only under what is today termed a
completeness assumption. Subsequently, in determining the indifference map, Pareto
presumes that if bundle x1 is chosen against bundle x2, every other x1-indifferent bundle
will be chosen against every other x2-indifferent bundle. However this inference is valid
only under the currently termed transitivity assumption. Lastly, as already seen, indifference curves can be assumed as downward sloping only under the hypothesis of nonsatiation, and convex under other specific postulates, such as preference for variety.
How is the development of neoclassical consumer theory affected by these two kinds
of criticisms of Pareto’s choice analysis – the realist one and the theoretical one? As
16
The first actual empirical attempt to determine the indifference curves is that of Thurstone 1931.
13
will be shown, the subsequent contributions of Slutsky, Hicks-Allen, Hicks, and Samuelson all aim to improve the new Paretian conceptual framework by overcoming its
logical inconsistencies and not its lack of realism. Even in this case, the line of development of neoclassical economics is skillfully rationalized by the Marburg School’s
epistemological tenets. Cassirer particularly recognizes the importance of thought experiments as fundamental tools through which understanding reconstructs and represents to itself the phenomena under consideration (cf. Cassirer 1911: 328 ff.). The fact
that thought experiments just occur ‘in the laboratory of the mind’ is certainly not a
problem for Marburg’s epistemological view, nor for neoclassical economics. Instead
problems arise if a conceptual representation proves to be inconsistent or incomplete
(e.g. as a consequence of the discovery of a paradox) since this makes it difficult for the
understanding to acquire an integral logical picture of the phenomenon. Removing these
inconsistencies therefore becomes an urgent task for scientific research and the same
sense of urgency seems to explain the efforts of neoclassical economists to correct the
inconsistencies of Pareto’s new and promising conceptual framework.
7. HICKS, ALLEN AND THE MARGINAL RATE OF SUBSTITUTION
In 1934, Hicks and Allen publish a famous paper which aims to work out a theory of
consumption choices along a Paretian line but “free of the inconsistencies detected in
Pareto” (Hicks–Allen 1934: 55). Even though a widespread interpretation puts the
Hicks–Allen paper at the origin of utilitarian ordinalism, it belongs to the behaviorist
camp: in fact, they start by admitting the cardinal immeasurability of utility and note
that “if total utility is not quantitatively definable, neither is marginal utility” (Hicks–
Allen 1934: 55)17. In the place of total and marginal utility, Hicks and Allen do not put
ordinal utility but the marginal rate of substitution (MRS). The MRS, defined as the
amount of good 2 which substitutes a marginal unit of good 1 for the individual, is a
quantitatively definable entity which can be empirically observed, and whose notion is
independent from the notion of utility. Starting from the MRS, Hicks and Allen CAN 1)
determine the relationships between the demand for goods, their price, and the consumer income in elasticity terms; 2) decompose the effect of a price change on demand,
17
The first derivative of a strictly monotonic transformation of u(x) , f x = f ′u x , only keeps the sign
of u x , not its magnitude. More importantly, the second-order derivative, f xx = f ′u xx + f ′u′ x u x , does
not even keep the sign of u xx .
14
in what Hicks will later call income and substitution effect; 3) provide a new definition
of complements and substitutes which replaces the Edgeworth-Pareto cardinal notion
relying on utility second-order cross-partial derivatives.
The problem with the Hicks-Allen construction is that all their results rely on two
crucial assumptions: first, that the MRS is negative (which means decreasing indifference curves) and second, that the MRS is decreasing (which means convex indifference
curves). The assumptions in hand are the same critical ones, which Pareto found difficult to justify. As regards the first item, Hicks and Allen postulate the MRS negativity
without any particular explanation. This statement can however seem natural only on
the basis of an unspoken endorsement of the hedonistic non-satiation principle. As regards the decreasing MRS assumption, Hicks and Allen only assert that it is not falsified
by experience, without putting forward any positive justification for it. Therefore, as
Samuelson (1938a: 61-62) will promptly underline, the convexity hypothesis once more
seems to ultimately rely on psychological-utilitarian considerations.
In the end, this second attempt (after that of Pareto) to overcome every reference to
utility does not succeed. The Hicks-Allen restatement of consumer theory in terms of
MRS fails to become the standard neoclassical one for two reasons. From a methodological standpoint, the fact of overcoming the utility notion is to a great extent just illusory since the crucial assumptions of the MRS rely on postulations that are utilitarian in
nature. In spite of this, the Hicks-Allen paper presents the most advanced results on the
relationships between price, income and demand from a theoretical viewpoint. However, with the rediscovery of Slutsky’s article and with the publication of Value and
Capital by Hicks in 1939, it becomes clear that the same results, obtained on the basis
of the MRS, can be obtained through ordinal utilities in a much simpler way. Therefore
there are no methodological or theoretical reasons to adopt the Hick-Allen approach
which is hence abandoned.
8. SLUTSKY, THE FINDER REDISCOVERED
In 1915 Slutsky published his article on consumer theory in Italian. It remained basically ignored until Dominedò (1933) in Italy and later in international journals Schultz
(1935) and Allen (1936) call attention to it18. In his article, Slutsky provides the follow-
18
For a detailed analysis of the rediscovery of Slutsky’s article see Chipman–Lenfant 2002.
15
ing results for any number of goods and for a general utility function (cf. Slutsky 1915:
4-19):
1) The second order condition for the constrained maximization of utility. As already
seen, such a condition coincides with that of convexity of indifference curves, which
however, slutsky does not bring into play.
2) The exact mathematical expressions of ∂xi ∂I , ∂xi ∂p j and ∂xi ∂pi .
3) By introducing the notion of “compensating variation”19, Slutsky decomposes the
effect of a price change on the demand for goods in the part due to the substitution effect and in the one due to the income effect thereby obtaining what will be called “The
Slutsky Equation”:
∂ˆ x
∂x
∂x
∂xi
∂ˆ x
∂x
= i − xi* i , which in the particular case j = i becomes i = i − xi* i
∂p j ∂ˆ p j
∂I
∂I
∂pi ∂ˆ pi
ˆ ˆ
where ∂xi ∂p j is the compensated variation of the demand for i when the price j varies
(the cross-price substitution effect), ∂ˆ xi ∂ˆ pi is the compensated variation of the demand
for i when the price of i itself changes (the direct substitution effect, or simply substitution effect), and xi* is the quantity of good i contained in the initial consumption bundle.
Slutsky notes that ∂ˆ xi ∂ˆ p j and ∂ˆ xi ∂ˆ pi are, at least ideally, empirically observable quantities.
4) Slutsky demonstrates that, if the second order condition is satisfied, the substitution
effect is always non-positive, i.e. ∂ˆ x i ∂ˆ p i ≤ 0 . That means that for the compensated demand, the “law of demand” holds.
5) Slutsky draws attention to the fact that the cross-price substitution effect is symmetric, i.e. ∂ˆ xi ∂ˆ p j = ∂ˆ x j ∂ˆ pi . If we call “substitution matrix” the matrix of the terms
∂ˆ xi ∂ˆ p j
, this means that the substitution matrix of the demand function is symmetric.
6) Even if Slutsky does not demonstrate, if the second order condition holds, the
substitution matrix is negative semi-definite20.
19
20
Slutsky’ compensating variation is a change in consumer income accompanying a change in prices
which makes the consumer’s initial consumption bundle just affordable at the new prices.
This result was proved first by Samuelson 1938a: 69.
16
7) Slutsky demonstrates that, although the substitution effect is always non-positive,
the income effect − xi*
∂xi
can be either negative or positive and the Giffen case is due to
∂I
a positive income effect which is greater in absolute value than the substitution effect.
8) Lastly, Slutsky demonstrates that with additive utility functions and under the assumption of decreasing marginal utility: i) the second order condition is certainly satisfied (i.e. indifference curves are convex), ii) the income effect is always negative, iii)
when the price of a good rises its demand falls and vice versa, i.e. there are no Giffen
goods and the law of demand holds.
ˆ ˆ
Slutsky draws the analytical expressions of ∂xi ∂I , ∂xi ∂p j and ∂xi ∂p j starting from
2
the derivatives ∂ 2u (x ) ∂ 2 xi and ∂ u(x ) ∂xi ∂x j of a cardinal utility function. The values of
ˆ ˆ
∂xi ∂I , ∂xi ∂p j and ∂xi ∂p j are, however, empirically observable so that it becomes
theoretically possible to go back from them to the values of ∂ 2 u (x ) ∂ 2 xi and
∂ 2u( x ) ∂xi ∂x j , which would thus cease to be merely psychological. This is the initial
purpose of Slutsky’s article. However, in the final part of the paper, Slutsky himself
shows that unfortunately, such a going-back is not possible. Devoid of this piece of the
initial project, Slutsky’s contribution remains cardinal in nature: i) he is apparently unaware that second-order and cross-partial derivatives of utility make sense only within a
cardinal view of utility; ii) there is no attempt to verify whether the results obtained in
terms of such utility derivatives are, in any case, independent from increasing transformations of u (x ) ; iii) he finally maintains the cardinal notions of complementarity and
2
substitutability in terms of the sign of ∂ u(x ) ∂xi ∂x j (see Slutsky 1915: 23-25)21.
However, in quite a providential way, Slutsky’s results are independent from cardinal
measurability assumptions on utility and can be entirely obtained within an ordinalist
framework, as Allen stresses in his 1936 paper devoted to Slutsky’s results. In fact, if
the function u (x) employed by Slutsky is replaced with an arbitrary f (u (x)) with f ′ > 0 ,
all the derivatives of f cancel out, so that the second order condition, the expression of
∂x i ∂I , ∂x i ∂pi , ∂xi ∂p j and all the other results obtained by Slutsky do not change (Al-
len: 1936: 191-192). This opens the way to the ordinalist restatement of Slutsky’s find21
As regards this issue, I claim that the interpretation of Slutsky’s contribution as ordinalist in nature, e.g. by Samuelson 1974: 1282 or by Chipman–Lenfant 2002: 562-563 ff., is inexact.
17
ings and to the formulation of the standard ordinalist neoclassical consumer theory
which will be worked out by Hicks.
9. HICKS THE ORDERER
Although for a while, Allen (1938: 344 ff.) continues to maintain consumer choice
analysis in terms of MRS as a primary concept, Hicks already in 1937 sets forth consumer theory in terms of ordinal utility (Hicks 1937) and fine-tunes his formulation in
Value and Capital (1939). In this work, the MRS ceases to be the basic element of the
model and is regarded as the opposite of the ratio between the partial derivatives of the
ordinal utility function. In any case, the required features for the MRS are the same as in
the paper written together with Allen: the MRS has to be negative and decreasing.
Negativity is still upheld on the basis of the hedonistic principle of nonsatiation which
makes marginal utilities positive22. The case for the decreasing MRS is more complex
than in the article written with Allen but remains basically the same: the assumption of a
decreasing MRS (i.e. of indifference curve convexity) is required for the determinateness of the theory and does not seem to be disproven by experience (Hicks 1939: 13 ff.).
In the end, Hicks contributes to re-presenting Slutsky’s results in a systematic and
mathematically clear way and demonstrating, more thoroughly than Allen and Schultz,
how these findings are invariant to strictly increasing transformations of a general utility
function.
10. GENERAL UTILITY AS THE RULE
The epilogue of this part of the story concurs perfectly with the Marburg epistemological view. From the mid-1930s, the use of a general utility function in consumption theory ceases to be a knotty problem for neoclassical utilitarian theory. On the contrary,
general utility becomes the standard whereas additive utility becomes a special case. In
fact, the works of Hicks, Allen, Schultz and especially Slutsky show that the introduction of general utility – whose realism was acknowledged by Edgeworth 55 years earlier
– does not imply introducing unpredictability into the theoretical picture. In fact, it becomes clear that not only with additive utility but also with general utility is it possible
22
If u i > 0 , then the MRS = −(ui u j ) < 0 . Note that the sign of ∂f (u ( x )) ∂xi = f ′ui is the same of
ui since f ′ > 0 .
18
to determine exactly the conditions for the maximization of consumer utility and the interrelations between demand, prices, and income, as well as to set univocally the
substitution and complementary relations between goods23.
The establishment of the Slutsky-Hicks theory as the new orthodox one is not hindered by the fact that it is difficult to apply it empirically, as some scholars of the University of Chicago pointed out. Schultz (1935, 1938) tries to test the empirical implications of Slutsky’s analysis by comparing them with the data from the statistical demand
curves of some commodities. Although Schultz’s findings do not refute Slutsky’s results, they certainly do not corroborate them24. Other Chicago scholars, such as Stigler
(1939) and Wallis and Friedman (1942), subsequently point out the difficulty of reconciling the demand curves of the ordinal Slutsky-Hicks theory with the statistical demand
curves. In particular, Wallis and Friedman question the possibility of empirically deriving the indifference function, either through psychophysical experiments, like that of
Thurstone (1931) or through a statistical estimation. This makes the Slutsky-Hicks
framework useless for empirical evaluations and for predicting the effect of changes in
economic conditions on the consumption of various goods (Wallis–Friedman 1942: 177
ff.).
However, once again this kind of empirical criticisms has no effect. As a matter of
fact, neoclassical consumer theory does not take it into account and instead develops in
two different directions with divergent aims but a similar approach. On the one hand,
Samuelson strives to obtain analytical results comparable to those of Slutsky-Hicks, but
on the basis of a set of postulates different from the utilitarian ones. On the other hand,
first Wold and then Debreu, maintain the reference to utility but seek to rigorously
ground the ordinal utility function and its features starting from an axiomatic theory of
consumer preference.
23
24
According to the new definition, which is also the current one, two goods are substitutes if their
compensated cross-price effect is positive, and are complements if their compensated cross-price
effect is negative; see Schultz 1935: 459, Schultz 1938: 620-628, Hicks 1939: 311.
More on the problems related to Schultz’s findings in Wade Hands–Mirowski 1998. Wade Hands
and Mirowski propose an alternative history of the American neoclassical demand theory of the
Thirties centered on Schultz’s statistical findings and on Hotelling’s different model of demand
(Hotelling 1932, 1935). Although Wade Hands–Mirowski’s reconstruction is full of new and interesting information, in my opinion, it is untenable, as Hurwicz 1998 convincingly points out.
19
11. SAMUELSON AND THE BEHAVIORIST-ORDINALIST EQUIVALENCE
In an often quoted passage of his 1938 note on consumer theory, Samuelson (1938a: 71)
says that he seeks to “develop the theory of consumer behaviour freed from any vestigial traces of the utility concept”. This does not mean that Samuelson wants to preclude
the introduction of the utility notion or to contradict the results obtained by using related
constructs (as shown by his work of the late 1930s on welfare economics with Bergson),
but merely that he thinks that consumer behavior analysis “can be carried on more directly […] from a different set of postulates” (Samuelson 1938a: 62)25. According to
Samuelson, the purpose of consumer theory is to place some restrictions on the features
of the empirical demand functions x = f ( p, I ) , that is, at least on the ideally observable
values of ∂xi ∂I , ∂xi ∂p j , ∂ˆ xi ∂ˆ p j . He claims that such restrictions can be obtained starting from a simple coherence assumption on consumer behavior which will be later
termed the Weak Axiom of Revealed Preference. As is well known, the Weak Axiom
states that if at the price vector p1 a bundle x1 is chosen when another bundle x 2 is also
affordable (and in this sense x1 is “revealed to be preferred” to x 2 ), then there can be no
budget set containing both bundles for which x 2 is chosen and x1 is not.
What are the implications of the Weak Axiom for consumer demand? Samuelson
(1938a, 1947, 1953) proves that, if consumer purchasing choices fulfil the Weak Axiom:
1) The sum of price changes multiplied by the changes in compensated demands
proves to be non-positive:
n
∑ ∆p i ∆xˆ i
i =1
≤ 0 . If the consumer demand function x = f ( p, I ) is
differentiable, for infinitesimal price changes this inequality becomes
n
∑ dp i dxˆ i
i =1
≤ 0 , so
that the substitution matrix of the demand function is negative semi-definite, as in the
utility maximization problem.
2) In particular, when only the price k changes, the last expression reduces to
dpk dxˆk ≤ 0 . This means that the substitution effect is always non-positive (negative if
dxˆ k ≠ 0 ) as in correspondence of a bundle of maximum utility, and that for the compen-
sated demand the “law of demand” holds.
25
Mongin 2000: 1135-1139 convincingly demonstrates that, already in 1938 and not only later,
Samuelson was not interested in eliminating utility from microeconomic theory.
20
3) The Weak Axiom has no implications on the sign of the income effect. This is perfectly consistent with the utilitarian analysis which contemplates goods both with negative income effect (normal goods) and with positive income effect (inferior goods).
4) The Weak Axiom has neither implications on the way gross demand for a good
changes when its price changes, i.e. on the sign of ∂xi ∂pi . This is again quite in agreement with the utilitarian analysis: when the price of a commodity rises the consumer
can either purchase less (ordinary good) or more (Giffen good).
5) Moreover, the Weak Axiom implies that any good whose demand increases when
the income rises, shrinks in demand when its price rises, which corresponds to Slutsky’s
utilitarian demonstration that Giffen goods are a subset of inferior goods.
6) Lastly, the Weak Axiom implies that the demand functions are homogeneous of order zero, i.e. a change in prices and income in the same proportion leaves all demanded
quantities unchanged. This restriction upon the demand functions is also embodied in
the solutions of the utility maximization problem.
The only restriction on demand functions that derives from the utility maximization
problem but that is not involved in the Weak Axiom, is the symmetry of the cross-price
substitution effects, i.e. the symmetry of the substitution matrix. The importance of this
property of demand functions lies in its being a necessary condition (with also the negative semi-definiteness of the substitution matrix) to go back from a demand function
x = f ( p, I ) to an ordinal utility function u ( x ) which generates the demand function itself,
in the sense that the constrained maximization of u (x) brings the consumer to the consumption choices expressed by x = f ( p, I ) . The problem of defining exactly under which
necessary and sufficient conditions a generating utility function for a given demand
function exists, is the so-called “integrability problem”, which was first studied by Antonelli (1886) and which has been conclusively solved in some of the works collected in
Chipman–Hurwicz–Richter–Sonnenschein (1971)26.
It is beyond the scope of this paper to give a detailed account of the integrability
problem. Suffice it to say that in 1950 Houthakker shows that a stronger axiom on consumer behavior, together with appropriate continuity assumptions on the demand function (Lipschitz continuity) is sufficient for the integrability of the demand function.
Houthakker’s Strong Axiom is an iteration of Samuelson’s Weak Axiom, which rules
26
A reconstruction of the history of the integrability problem can be found in Hurwicz 1971.
21
out the possibility of cyclical consumer choices: in a chain of commodity bundles, each
of which is shown to be preferred to its successor in the sense of the Weak Axiom, the
last element cannot be shown to be preferred to the first.
On the basis of Houthakker’s contribution, Samuelson re-examines the problem of
integrability and demonstrates that the symmetry and negative semidefiniteness of Slutsky’s substitution matrix are not only necessary but also sufficient conditions for the integrability of Lipschitz continuous demand functions (Samuelson 1950: 376-385)27.
Hence, the Strong Axiom on consumer choices implies the symmetry and negative
semidefiniteness of the substitution matrix, which are in turn sufficient for the existence
of an ordinal utility function, whose constrained maximization generates those consumption choices. On the other hand, the constrained maximization of an ordinal utility
function leads to consumption choices which satisfy the Strong Axiom. Therefore, it
turns out that the Slutsky-Hicks framework and the Samuelson-Houthakker one lead to
identical restrictions on demand functions. This realizes Samuelson’s program of carrying out consumption theory from a non-utilitarian set of postulates, and at the same time
makes the opposition between the ordinalist and the behaviorist approach vanish.
12. THE UTILITY FRAMEWORK AS THE STANDARD ONE
Despite these equivalence results, Samuelson’s restatement of consumption analysis
does not replace the ordinal utility apparatus as the standard framework of neoclassical
demand theory. What are the reasons for this outcome?
From a methodological standpoint, it is important to point out that Samuelson’s revealed preference theory is not more empirically based than the utilitarian one. As in
Pareto’s case, the compelling power of the Weak Axiom does not stem from a factual
experiment on real consumer behavior but from a thought experiment on an imaginary
consumer confronted with fictional prices. In principle, the Weak Axiom can be tested
through individual experiments or on the basis of statistical demand functions. As regards the first suggestion, we must bear in mind that the Weak Axiom, by its very nature, concerns hypothetical alternative choices, not actual successive choices. This
makes it difficult to verify or disprove it through experimental sequences of choices. On
the second point, it must be remembered that since consumer theory axioms place re-
27
Samuelson’s demonstration will be perfected by Hurwicz-Uzawa 1971.
22
strictions on individual demand, and only under specific auxiliary assumptions on aggregate demand, it is difficult to test the Weak Axiom starting from the statistical demand functions. Therefore, as regards the actual testability of the basic postulates, there
is no genuine advantage of Samuelson’s framework over Slutsky-Hicks’ one, even if
one embraces the epistemological canons of positivism or of operationalism.
From the standpoint of the Marburg theory of knowledge, there is instead a crucial
advantage of the latter approach over the former: the preference/indifference apparatus
makes it possible to analyze some issues that, within the Samuelson scheme, are not directly workable and which require, instead, the recovery of the preference/indifference
device. As shown above, if the consumption choices of the individual, expressed by his
demand function, satisfy the Strong Axiom, there exists an ordinal utility function
which generates those choices and which can be determined by integrating the demand
function itself. Therefore, even the results of consumer theory that cannot be deduced
by directly using the Strong Axiom can be obtained indirectly through the integration
procedure. In particular, such a device is necessary for the welfare analysis of economic
changes: without the preference/indifference apparatus, for example, it is impossible to
evaluate the impact on the consumer’s welfare of a prices change, due e.g. to the introduction of a tax28. Thus the preference-based approach proves to be more powerful for
the theoretical understanding of some valuable issues of consumer demand and this can
therefore explain its dominance over the behaviorist approach.
Lastly, preferences and utility functions are basic analytical tools in other fundamental areas of neoclassical microeconomics which start to intensely develop at the end of
the 1930s: welfare economics, general equilibrium theory, social choice analysis and the
theory of choice under uncertainty. Therefore, it appears that employing the ordinal utility framework makes it possible to connect demand theory to these other areas in a systematic analytical picture of the microeconomic field. For the Marburg knowledge theory, this is another important advantage of the Slutsky-Hicks theory over the Samuelson
one which rationalizes the prevalence of the former.
28
Passing from consumer theory to exchange theory or to general equilibrium theory, the preference/indifference apparatus is required to evaluate the efficiency of an allocation. On this issue cf.
Montesano 1996.
23
13. DEBREU AND THE AXIOMATIC FOUNDATION OF SLUTSKY-HICKS’ THEORY
From the 1940s on, utilitarian consumer theory strives to find a sounder and more rigorous basis for its fundamental analytical tools, preferences and ordinal utility functions.
In this connection, the pioneer attempt was made by Frisch (1926), but it is Wold who,
in 1943-44, gives the first axiomatic treatment of the utilitarian theory of demand. In
particular, Wold states axiomatically the general properties of the preference/indifference relations that allow us to represent “every indifference map […] as the level map
of a continuous, non decreasing function” (Wold 1943-44, II: 223). Furthermore Wold
proposes a method to construct such a function which is ordinal in nature. Unfortunately, Wold’s treatment is not completely rigorous and correct. In 1951 Arrow introduces in economics the handling of preferences as binary relations whose properties are
stated axiomatically. Preference is conceived as a formal relation R between two generic
alternatives (not just commodity bundles), in which they stand when one alternative is
preferred to the other (Arrow 1951: 11 ff.). However, the subject matter of Arrow’s
book is not to investigate the problems related to the representation of a preference ordering by a numerical function. In 1954, the so-called “paradox of lexicographic preferences” introduced by Georgescu-Roegen (1954) makes it apparent that it is not always
possible to define a real valued order-preserving function on a set of alternatives ordered by the preferences of some agent. In the same year, Debreu specifies assumptions
under which the representation of a preference ordering, by means of a real-valued continuous function, is possible and demonstrates the consequential representation theorem
(Debreu 1954). In his classic monograph, Theory of Value (1959), Debreu carries out
the axiomatic analysis of economic equilibrium, determining when it is possible to derive the specific utility function required by the Slutsky-Hicks theory.
Debreu (1959: 52 ff.) defines in mathematical terms the features of both the consumption set Xi and the binary relation “preference”. In particular, he deals with the
weak preference relation (x can be either strictly preferred or indifferent to y), which is
labeled with the symbol ≿. Debreu demonstrates that if: 1. Xi is connected, 2. ≿ is complete, 3. ≿ is reflexive, 4. ≿ is transitive; 5. ≿ is continuous (in an appropriate technical
sense), then it is possible to construct a function that associates to each indifference
class induced by ≿ a real number, in such a way that, if a class is preferred to another,
the number of the first is greater than the number of the second. This function is called a
utility function and can be chosen to be continuous. Moreover, if it is assumed that: 6.
24
Xi is closed, 7. ≿ is convex, and the hypothesis 1. is reinforced becoming: 1'. Xi is convex (convexity implies connectedness), every utility function that represents the given
preferences is also quasi-concave29.
Yet, a continuous quasi-concave utility function is exactly the ordinal utility function
that the Slutsky-Hicks consumer theory needs, since such a function ensures that the
second order conditions for the constrained maximization of utility are satisfied, i.e. that
the indifference curves are convex. In this way, it is possible to recover all the results
already obtained by ordinal utility theory, but now starting from a basis fully specified
in axiomatic terms. Debreu therefore provides the Slutsky-Hicks theory with a sound
axiomatic foundation and brings neoclassical consumer theory to its current standard
form.
14. FORMALISM AS NEOCLASSICAL METHODOLOGICAL AWARWNESS
The axiomatic method mainly aims to pursue the logical rigor of the theory and not its
realism. In an oft-quoted passage of the Theory of Value, Debreu peremptorily states
that: “Allegiance to rigor dictates the axiomatic form of the analysis” (Debreu 1959: x).
In another text, Debreu observes that the aim of the axiomatization of a certain part of
economic theory is the “full specification of the assumptions under which any one of its
conclusions is asserted”, and that “the complete specification of assumptions, the exact
statement of conclusions, and the rigor of the deductions of an axiomatized study provide a secure foundation on which the construction of economic theory can proceed”
(Debreu 1983 [1977]: 5-6).
Within the formalist attitude of the axiomatic method, the time-honored question of
overcoming every hint of psychological conjectures (nonsatiation principle, decreasing
marginal utility or preference for variety) in the statement of some basic assumptions of
consumer theory (downward slope and convexity of indifference curves) fades away.
From the formalist standpoint, such assumptions do not, in fact, aim to grasp something
real, whether psychological or not, but have a primarily logical-systematic function:
they in fact ensure the determinateness and the connectedness of the entire theoretical
construction. Therefore, the interpretation e.g. of the convexity of the indifference
29
Quasi-concave is a function u(x ) for which u(x3 ) ≥ min{u(x1 ), u (x2 )} for each x3 = λx1 + (1 − λ )x2 ,
with λ ∈ [0, 1] . Quasi-concavity is an ordinal feature, i.e. a feature which is invariant to a monotonic transformation of u(x ) .
25
curves in the psychological terms of preference for variety cannot put the theory at risk.
On the contrary, if such interpretations make some axiom appear more plausible, they
are even welcome. However, if an axiom seems indisputably false, the task is to either
remove it (‘to relax the assumptions’) without letting the edifice of the theory collapse,
or to replace the entire theoretical construction with another one which ought to be
comparably complete, determined and exact.
However, as I have tried to show, this formalist allegiance to conceptual rigor, and
the priority it has over both the pursuit of a descriptive accuracy of the assumptions and
the search for factual relevance of the empirical implications, are the same that have tacitly guided the development of neoclassical consumer theory from its very beginnings.
Instead it is sometimes claimed that after the Second World War there was a “formalist
revolution” in neoclassical economics which involved a change in the very aims of the
theory30. The driving element of this revolution would have been a much greater use of
mathematics which was to have important methodological consequences: economic
theory became more and more directed at achieving internal consistency to the detriment of factual relevance, the empirical element became distorted or vanished altogether in economic model-building, and a permissive attitude to unreal assumptions
pervaded economic theorizing. In short, with the formalist revolution realism was sacrificed to tractability. However, from the historical reconstruction I propose, no formalist
revolution took place since the sacrifice of realism on behalf of tractability has been the
rule from the origin of neoclassical theory. On the contrary, it seems that, according to
the formalist methodological statements expressed by Debreu, neoclassical economics
becomes aware of those which already were its internal epistemic laws.
In line with a Lakatosian methodology it might be suggested that “allegiance to
rigor” is to be included in a set of “hard core propositions” which could characterize the
neoclassical research program from its very beginnings. However, in line with the Marburg theory of knowledge, I argue that “allegiance to rigor” is not a hard core proposition among a possible set of propositions included in a research program among many
others. Rather, the pursuit of rigor depends on a more fundamental tendency of the understanding to achieve a systematic picture of the given. It is for this reason, that “allegiance to rigor” is helpful in explaining not only the internal development of consumer
30
See Ward 1972, Hutchison 2000.
26
neoclassical theory, but also why the neoclassical research program (i.e. the research
program that endorsed such an allegiance from its beginnings), displaced and superseded the others, and became the prevailing one.
15. MARBURG AND THE OTHER TRENDS IN ECONOMIC METHODOLOGY
After having tried to illustrate the explanatory power of the Marburg epistemology, I
conclude by trying to roughly determine its position in relation to some of the other
principal trends in economic methodology. First of all, the starting point of the Marburg
Neo-Kantianism is “the fact of science” which means that the Marburg epistemology
aims to understand economics as it is, without trying to dispense recipes for economic
inquiry, without attempting to decide whether economics meets some alleged ‘right’
methodological rule, without dealing with demarcation issues (“Is economics really a
science?”) of a Popperian nature. The Marburg epistemology aims to be a positive philosophical theory of scientific knowledge, not a normative doctrine. In the recent literature, the epistemological position that is closest to the Marburg one is that of Rappaport.
Referring to a suggestion made by Gibbard and Varian (1978), Rappaport calls attention
to the fact that much economic theorizing is based on the construction of models and
their use in various cognitive activities. He claims that economists put forward models
for the purpose of resolving conceptual and normative problems. In doing so, they “do
not put forward these models as true descriptions of reality” (Rappaport 1998: 138). Although Rappaport’s study on the nature, uses, and types of models is certainly useful, in
my opinion, it fails to investigate why these models, even if ‘false’, can play the fundamental role in economic knowledge that Rappaport himself highlights, and which for
the Marburg School is related to the tendency of the understanding to an exact reconstruction of experience.
As pointed out in Section 1, with regard to the non descriptive nature of scientific
theories, there is some affinity between the Marburg cognitive fictionalism and instrumentalist fictionalism which in economics is usually associated with the methodological
positions of Friedman and Machlup. Friedman’s position is the more distant from the
Marburg one since he quite univocally stresses the idea that scientific theories are tools
used to make predictions, and sometimes claims that “in general, the more significant
[is] the theory, the more unrealistic [are its] assumption” (Friedman 1953: 14). This thesis upon the irrealism of the assumptions is alien to the Marburg School, which just affirms that assumptions can be unrealistic. Machlup is less distant from Marburg Neo-
27
Kantianism since he does not even mention the irrealism of assumption thesis and
points out that theories not only serve as an instrument to predict but also “as an instrument of explanation” of phenomena (Machlup 1955: 12).
With respect to the methodological trends that try to understand economic science
and explain its historical developments on the basis of extra-scientific factors – discourse analysis, as in the rhetorical approach; social factors, as in the sociology of scientific knowledge (SSK); self-interest, cost-benefit calculation, marketplace of ideas etc.
as in the economics of scientific knowledge (ESK) – the Marburg epistemology is in
flagrant contrast31. Obviously (economic) science is a human activity, and as such, is influenced by the following factors: communication skills and persuasion strategies, quest
for wealth and status, customs and fashions, good or bad functioning of selection
mechanisms and institutions, ability to raise funds or create network relationships,
youthful creativity or attachment to old ideas, opportunities and chance. However, if
one considers at least 20-30-year time spans, the success or the abandonment of a research program appear to depend foremost on theoretical factors, which characterize
that particular human activity which is scientific knowledge. I did not discuss here the
general question of whether the internal rather than the external factors are, in the end,
the crucial ones in determining what goes on in (economic) science. I simply contend to
have shown that an ‘old-fashioned, internal and ultra-Whiggish’ reconstruction, like the
Neo-Kantian-inspired one presented here, can still be a sufficiently powerful resource to
account for the history and the logic of neoclassical theory without resorting to extrinsic
explanations.
REFERENCES
ALLEN R.G.D. 1936: “Professor Slutsky’s Theory of Consumers’ Choice”, Review of Economic Studies,
3: 120-129.
— 1938: Mathematical Analysis for Economists, Macmillan, London.
ANTONELLI G.B. 1886: Sulla teoria matematica dell’economia politica, Tipografia del Folchetto, Pisa.
ARROW K.J. 1951: Social Choice and Individual Values, Wiley, New York.
BACKHOUSE R. et alii (eds.) 1998: Economics and Methodology, Macmillan, London–New York.
BARONE E. 1894: “A proposito delle indagini del Fisher”, Giornale degli Economisti [2], 9: 413-439.
BARROTTA P. 1996: “A Neo-Kantian Critique of von Mises’s Epistemology”, Economics and Philosophy, 22: 51-66.
BLAUG M. 1992 [1980]: The Methodology of Economics, Cambridge University Press, Cambridge.
BRUNI L. 2002:Vilfredo Pareto and the Birth of Modern Microeconomics, Elgar, Cheltenham.
BRUNI L.–GUALA F. 2001: “Vilfredo Pareto and the Epistemological Foundations of Choice Theory”, History of Political Economy, 33: 21-49.
31
For a detailed assessment of these trends see Wade Hands 2001.
28
CASSEL G. 1899: “Grundriss einer elementaren Preislehre”, Zeitschrift für die gesamte Staatswissenschaft, 55: 395-458.
CASSIRER E. 1911: Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren Zeit, Bd. 1,
B. Cassirer, Berlin.
— 1953 [1910]: Substance and Function, Dover, London–Chicago.
CLARK J.B. 1899: The Distribution of Wealth, Macmillan, London–New York.
CHIPMAN J.S.–HURWICZ L.–RICHTER M.C.–SONNENSCHEIN H.F. (eds.) 1971: Preference, Utility and Demand, Harcourt Brace Jovanovich, New York.
CHIPMAN J.S.–LENFANT J.-S. 2002: “Slutsky’s 1915 Article: How It Came to be Found and Interpreted”,
History of Political Economy, 34: 553-597.
COHEN H. 1984 [1883]: Das Prinzip der Infinitesimal-Methode und seine Geschichte; in: Werke, Bd. 5,
Olms, Hildesheim.
DARDI M. 1991: “The Concept and the Role of the Individual in Marshallian Economics”, Quaderni di
Storia dell’Economia Politica, 9: 89-114.
DEBREU G. 1954: “Representation of a Preference Ordering by a Numerical Function”, in: R.M. Thrall et
alii (eds.): Decision Processes, Wiley, New York: 159-165.
— 1959: Theory of Value, Wiley, New York.
— 1983 [1977]: “The Axiomatization of Economic Theory”, excerpts in: Mathematical Economics:
Twenty papers of Gerard Debreu, Cambridge University Press, Cambridge: 5-6.
DOMINEDÒ V. 1933: «Considerazioni intorno alla teoria della domanda», Giornale degli Economisti e Rivista di Statistica, [4], 73: 30-48, 765-807.
DOOLEY P.C. 1983: “Slutsky’s Equations Is Pareto’s Solution”, History of Political Economy, 15: 513517.
DUHEM P. 1906: La thèorie physique, Chevalier et Rivière, Paris.
EDGEWORTH F.Y. 1881: Mathematical Physhics, Kegan Paul, London.
FERRARI M. 1997: Introduzione a il Neocriticismo, Laterza, Roma-Bari.
FISHER I. 1926 [1892]: Mathematical Investigations in the Theory of Value and Prices, Yale University
Press, New Haven.
FRAMBACH H.A. 1993: Die Evolution moderner ökonomischer Kategorien, Dunkler und Humblot, Berlin.
FRIEDMAN M. 1953: “The Methodology of Positive Economics”, in: Essays in Positive Economics, University of Chicago Press, Chicago: 3-43.
FRISCH R. 1926: “Sur un problème d’economie pure”, Norsk Matematisk Forenings Skrifter [1], 16: 1-40.
GEORGESCU-ROEGEN N. 1954: “Choice, Expectations, and Measurability”, Quarterly Journal of Economics, 68: 503-534.
GIBBARD A.–VARIAN H. 1978: “Economic Models”, Journal of Philosophy, 75: 664-677.
HICKS J.R. 1937: Théorie mathématique de la valeur, Hermann & Cie, Paris.
— 1939: Value and Capital, Claredon Press, Oxford.
HICKS J.R.–ALLEN R.G.D. 1934: “A Reconsideration of the Theory of Value”, Economica [N.S.], 1: 5276, 196-219.
HOTELLING H. 1932: “Edgeworth’s Taxation Paradox and the Nature of Demand and Supply Functions”,
Journal of Political Economy, 40: 577-616.
— 1935: “Demand Functions with Limited Budgets”, Econometrica, 3: 66-78.
HOUTHAKKER H.S. 1950: “Revealed Preference and Utility Function”, Economica [N.S.], 17: 159-174.
— 1961: “The Present State of Consumption Theory. A Survey Article”, Econometrica, 29: 704-740.
HURWICZ L. 1971: “On the Problem of Integrability of Demand Functions”, in: Chipman et alii 1971:
174-214.
— 1998: “Comment”, in: Backhouse et alii 1998: 399-421.
HURWICZ L.–UZAWA H. 1971: “On the Integrability of Demand Functions”, in: Chipman et alii 1971:
114-148.
HUTCHISON T. 2000: On the Methodology of Economics and the Formalist Revolution, Elgar, Cheltenham.
JEVONS W.S. 1871: The Theory of Political Economy, Macmillan, London–New York.
LANGE O. 1934: “The Determinateness of the Utility Function”, Review of Economic Studies, 1: 218-225.
MACHLUP F. 1955: “The Problem of Verification in Economics”, Southern Economic Journal, 22: 1-21.
MARSHALL A. 1961: Principles of Economics. Ninth (Variorum) Edition, Macmillan, London 1961.
MAYER H. 1994 [1932]: “The Cognitive Value of Functional Theories of Price”, in: I.M. Kirzner (ed.),
Classics in Austrian Economics, Vol. 2, William Pickering, London: 55-168.
MENGER C.1871: Grundsätze der Volkswirtschaftslehre, Braumüller, Wien.
MIROWSKI P. 1989: More Heat than Light, Cambridge University Press, Cambridge.
— 2002: Machine Dreams, Cambridge University Press, Cambridge.
29
MIROWSKI P.–WADE HANDS D. 1998: “A Paradox of Budgets”, in: M.S. Morgan–M. Rutherford (eds.),
From Interwar Pluralism to Postwar Neoclassicism, Duke University Press, Durham: 260-292.
MISES L. VON 1949: Human Action, Yale University Press, New Haven.
MONGIN P. 2000: “Les préférences révelées et la formation de la théorie du consommateur”, Revue économique, 51: 1125-1152.
MONTESANO A. 1996: “Introduzione”, in: G. Sabattini (a cura di), Abraham Wald e il programma di
ricerca sull’equilibrio, Angeli, Milano: 11-31.
— 2003: “Umberto Ricci, l’utilità marginale e la teoria della domanda”, Università Bocconi, Milano.
OLLIG H.-L. 1998: “Neo-Kantianism”, in: Routledge Encyclopedia of Philosophy, vol. 6: 776-792.
PANTALEONI M. 1889: Principii di economia pura, Barbera, Firenze.
— 1898: Pure Economics, Macmillan, London.
PARETO V. 1892-93: “Considerazioni sui principii fondamentali dell’economia politica pura”, Giornale
degli Economisti [2], I: 4: 389-420; II: 42: 485-512; III: 5: 119-157; IV: 6: 1-37; V: 7: 279-321.
— 1900: “Sunto di alcuni capitoli di un nuovo trattato di economia pura del prof. Pareto”, Giornale degli
Economisti [2], 20: 216-235; 511-549.
— 1896-97: Cours d’économie politique, Rouge, Losanne.
— 1906: Manuale di Economia Politica, Società Editrice Libraria, Milano.
PARSONS S.D. 1990: “The Philosophical Roots of Modern Austrian Economics: Past Problems and Future Prospects”, History of Political Economy, 22: 295-319.
— 1997: “Mises, The A Priori, and the Foundations of Economics”, Economics and Philosophy, 13: 175196.
POINCARE H. 1902: La science et l’hypothèse, Flammarion, Paris.
RANCHETTI F. 1998: “Choice without Utility?”, in: M. Bianchi (ed.), The Active Consumer, Routledge,
London: 21-45.
RAPPAPORT S. 1998: Models and Reality in Economics, Elgar, Cheltenham.
SAMUELSON P. 1938a: “A Note on the Pure Theory of Consumer’s Behaviour”, Economica [N.S.], 5: 6171, 353-354.
— 1947: Foundations of Economic Analysis, Harvard University Press, Cambridge (Mass.).
— 1950: “The Problem of Integrability in Utility Theory”, Economica [N.S.], 17: 355-385.
— 1953: “Consumption Theorems in Terms of Over-Compensation Rather Than Indifference Comparisons”, Economica [N.S.], 20: 1-9.
— 1974: “Complementary”, Journal of Economic Literature, 12: 1255-1289.
SCHILPP P.A. (ed.) 1949: The Philosophy of Ernst Cassirer, Open Court, La Salle (Ill.).
SCHULTZ H. 1935: “Interrelations of Demand, Price, and Income”, Journal of Political Economy, 41:
433-481.
— 1938: The Theory and Measurement of Demand, University of Chicago Press, Chicago.
SCHUMPETER J.A. 1954: History of Economic Analysis, Oxford University Press, New York.
SLUTSKY E. 1915: “Sulla teoria del bilancio del consumatore”, Giornale degli Economisti [3], 51: 1-26.
STIGLER G.J. 1939: “The Limitations of Statistical Demand Curves”, Journal of the American Statistical
Association, 34: 469-481.
— 1950: “The Development of Utility Theory”, Journal of Political Economy, 58: 307-327, 373-396.
THURSTONE L.L. 1931: “The Indifference Function”, Journal of Social Psychology, 2: 139-167.
WADE HANDS D. 2001: Reflection without Rules, Cambridge University Press, Cambridge.
WADE HANDS D.–MIROWSKI P. 1998: “Harold Hotelling and the Neoclassical Dream”, in: Backhouse et
alii (eds.) 1998: 322-397.
WALLIS W.A.–FRIEDMAN M. 1942: “The Empirical Derivation of Indifference Functions”, in: O. Lange
et alii (eds.), Studies in Mathematical Economics and Econometrics, University of Chicago Press,
Chicago: 175-189.
WALRAS L. 1874: Eléments d’économie politique pure, Corbaz & C., Lausanne.
WARD B. 1972: What’s Wrong with Economics?, Macmillan, New York.
WEBER C.E. 1999a: “Slutsky and Additive Utility Functions, 1947-1972”, History of Political Economy,
31: 393-416.
— 1999b: “More on Slutsky’s Equation as Pareto Solution”, History of Political Economy, 31: 575-586.
WICKSELL K. 1893: Über Wert, Kapital und Rente, Fisher, Jena.
WICKSTEED P.H. 1888: The Alphabet of Economic Science, Macmillan, London.
WIESER F.F. VON 1889: Über die Ursprung und die Hauptgesetzte des wirtschaftslichen Werthes, Hölder,
Wien.
WOLD H. 1943-44: “A Synthesis of Pure Demand Analysis”, Skandinavisk Aktuarietidskrift, I: 26, 1943:
85-118; II: 26, 1943: 220-263; III: 27, 1944: 69-120.
WONG S. 1978:The Foundations of Paul Samuelson’s Revealed Preference Theory, Routlege, London.
30