Continuum in Economics. On the Significance of the Realism of Assumptions
in Economic Theory
Juliusz Jabłecki1
Abstract
The present paper outlines the basic characteristics of the mathematical concept of a
continuum and shows its usage, and implications thereof, in modern neoclassical models.
It is argued that the continuum – i.e. an uncountable infinity – as a feature of certain
economic phenomena lacks any real-life interpretation, and thus implies that models
based on it are practically unrealizable. This is associated with a much broader problem
of whether scientific theories based on evidently unrealistic assumptions can be of any
use in understanding reality. The paper closes with some general remarks concerning
both the use of mathematics and the importance of realism of assumptions underlying
economic models.
1
The author wishes to thank Prof. Joseph Salerno, Dr. David Gordon and Mateusz Machaj for invaluable
comments and stimulating discussions during the process of writing this paper. The help of Ludwig von
Mises Institute, whose summer fellowship program facilitated research underlying this paper, is also
gratefully acknowledged.
1
In a recent paper Robert P. Murphy (2006) argued that the mathematical concept
of the continuum could be used to prove a fundamental mistake in Oskar Lange’s famous
market socialism approach. While it is immaterial here whether Murphy’s argument is
indeed correct,2 its unquestionable virtue consists exactly in drawing the attention of
Austrian scholars to the problem of using infinity in economic models.
It may come as a surprise to some economists that infinities – almost like T-shirts
– come in different sizes, i.e. there are “smaller” and “bigger” ones. Given that this is
intended to be an economics article, there is no need to go deep into set theory – which is
the branch of mathematics that deals primarily with that sort of problems – but since
infinities enter as standard assumptions into neoclassical models, it seems worthwhile to
concisely review the basic properties of the concept. In mundane speech “infinite” means
simply not finite or – as The New Webster Encyclopedic Dictionary of the English
Language puts it – “without limits, boundless”. This seems to be in accordance with
mathematical concept of infinity as long as we think of, say, the set of all positive
integers – it, too, is boundless, since if any number were to be its upper bound, it would
suffice to add 1 to it, thus obtaining a number that is even greater than the postulated
bound. Our intuition fails to square with mathematical theory, however, once we start to
think about the unit interval (0,1) – it is obviously infinite but in the same time bounded
from below by 0 and from above by 1. These simple examples are not meant to bore the
reader but simply to give a foretaste of how misleading it might be to apply rigorous
mathematical terms outside their natural theoretical, abstract domain.
Now, to make things slightly more complicated, let us note that even though both
the unit interval and the whole set of integers are infinite, the former can be meaningfully
said to be “larger” than the latter, in that there does not exist a one-to-one mapping of
(0,1) onto the set of all integers.3 Whenever there exists such a correspondence between a
given set S and the set of integers, we say that S is countably infinite. Infinite sets that are
not countable are uncountable. Referring to the example above, we may say that the
interval (0,1), or any interval – and in general the whole real line – are uncountable sets,
2
See on this Jablecki and Machaj (2007).
The proof, often referred to as Cantor’s diagonal argument, can be readily found in Murphy (2006).
However, readers interested in a much more detailed treatment of the basic set-theoretic concepts and
theorems should consult any book on set theory, e.g. Hrbacek and Jech (1999) or fairly advanced Jech
(2002).
3
2
whereas any infinite subset of integers as well as the whole set itself are obviously
countable. As a final terminological remark, we should note that the set of reals is often
called the continuum. And it is exactly in that latter sense that infinity is oftentimes
referred to in economic models.
The continuum in Economics
It could be argued that there is nothing particularly illuminating in the previous
statement. For doesn’t the continuum manifest itself in the assumption of marginal
analysis which assumes infinite divisibility of units of goods? Indeed it does, but Austrian
critics of mathematical method, following Leibnitz and Newton who first began using
calculus in science,4 typically understand by it merely a countably infinite process – i.e.
such a division of a given unit of good which at each step has finitely many predecessors
– yielding arbitrarily small parts. Their critique, therefore, is usually centered on the
problem of inapplicability of the methods of calculus to economics which, after all, is
first and foremost a study of human action.5 Thus, writes Rothbard (2004, p. 324 n.1)6:
… mathematics, particularly the calculus, rests in large part on assumptions of infinitely small
steps. Such assumptions may be perfectly legitimate in a field where behavior of unmotivated
matter is under study. But human action disregards infinitely small steps precisely because they
are infinitely small and therefore have no relevance to human beings. Hence, the action under
study in economics must always occur in finite, discrete steps. It is therefore incorrect to say that
such an assumption may just as well be made in the study of human action as in the study of
physical particles.
4
In fact, even Georg Cantor the famous mathematician who introduced the concept of the continuum into
mathematics, believed at first that the set of all reals is only countable, and thus not “greater” than the set of
integers. See on this Stillwell (2002).
5
An objection could also be raised that the continuum assumption manifests itself even in simple graphical
analysis, e.g. when supply and demand schedules are drawn as continuous functions defined on a real line.
Graphical analysis, however, performs a purely descriptive role in that it does not serve as a primary tool of
theoretical analysis but rather as an illustration of prior reasoning. Moreover, graphical analysis can still be
carried out once the assumptions of continuity and differentiability are dropped, though it loses then some
of its expository power.
6
See on this also Machaj (2007).
3
In fact, however, there seems to be more to the critique of the application of
calculus to economics than just that, since the use of even the most elementary methods
of calculus, like differentiation or integration, implicitly presupposes uncountability of
the set of elements on which those mathematical operations are carried out. In other
words, modern economics assumes conventionally not only that any unit of any good is
divisible through finitely many steps into arbitrarily small units – which would mean that
the infinity is only potential in that although we approach it, we never really reach it – but
that the set of divisions is actually infinite, meaning e.g. that all the infinitesimally small
units cannot even be enumerated (or listed).7 To put it even more bluntly, economic
models typically rest on the assumption that somehow a good “approximation of reality”
can be found not in the natural, intuitively clear concept of infinity of positive integers,
but in an infinitely greater infinity – clearly, neither of which exists in the real, physical
world around us.8
It is perhaps easier to excuse mainstream economists for resorting to infinitesimal
calculus whenever really small changes are under consideration. After all, one is tempted
to say, some goods, say gold, are divisible up to such tiny units that the use of
differentiation according to the popular explanation of that technique (“the change in a
given function caused by a very little change of its argument”) prima facie does not seem
utterly nonsensical, especially if one refers to the analogy of physics. Such a view,
though as noted above completely mistaken, seems at least intuitively plausible.
Mathematical economists, however, inasmuch as their truth-seeking has been gradually
replaced by theorem-seeking,9 have gone much further than that. For example, in a very
influential paper on perfect competition, Robert Aumann (1964) makes a case for the use
of the continuum in microeconomics as a magnitude describing the number of market
agents. According to him the most natural model for the purpose of developing the notion
of perfect competition “contains a continuum of participants, similar to the continuum of
7
On the distinction between potential and actual infinity see Lorenzen (1987, pp. 195-202).
This was understood even by Aristotle who in his Book 8 of the Physics writes: “In that which is
continuous there are indeed infinitely many points as parts, but these are not in reality but rather merely as
a possibility. For it merely appears that a line has infinitely many points as parts, while by its nature and in
its essential being it is otherwise”; quoted in Lorenzen (1987, p. 201); a more recent author, the
mathematician Donald Greenspan (1990, p. 59), confirms this conviction: “Indeed, even today, atomic and
molecular theory maintain that everything in the material world is of finite character.”
9
Cf. Ward (1972, p. 49).
8
4
points on a line or the continuum of particles in a fluid” (Aumann 1964, p. 39). What
immediately strikes one as completely mistaken is the conviction that the particles in a
given volume of a fluid form an uncountable set. It has been brilliantly observed by
David Hilbert, one of the greatest mathematicians of the 20th century, that:
If we have a piece of metal or a volume of liquid, the idea impresses itself upon us that it is
divisible without limit, that any part of it, however small, would again have the same properties.
But, wherever the methods of research in the physics of matter were refined sufficiently, limits to
divisibility were reached that are not due to the inadequacy of our experiments but to the nature of
the subject matter… The infinite divisibility of a continuum is an operation that is present only in
our thoughts; it is merely an idea which is refuted by our observations of nature and by the
experience gained in physics and chemistry. (Hilbert 1967, pp. 371, 392)
Interestingly enough, in another paragraph of his paper Aumann explicitly admits that
“we are intellectually convinced that a fluid contains only finitely many particles”, but
nevertheless clings to his assumption as he considers it an “approximation to the ‘true’
situation” and believes, furthermore, that finite methods would be hopeless in this regard
(Aumann 1964, p. 41).10
In order to properly assess Aumann’s contribution, it is crucial to ask what exactly
does he mean when he postulates that the number of market agents should be
uncountably infinite. As has been noted above, nowhere in the real, physical world does
there exist anything even remotely resembling the continuum. “An approximation” of the
way things really are on a market where individual agents have little influence on price –
an approximation still exaggerated but at least intuitively imaginable – would perhaps be
to assume that traders are like integers, i.e. that there is so many of them that any given
quantity can be shown to be insufficient to include them all, but nevertheless that one can
10
Incidentally it might be argued that the real motivation behind developing the model of perfect
competition is revealed by Aumann himself when he writes (1964, p. 41):
“Though we are intellectually convinced that a fluid contains only finitely many particles, to the naked eye
it still looks quite continuous. The economic structure of a shopping center, on the other hand, does not
look continuous at all. But, for the economic policymaker in Washington … there is no such difference.
He works with figures that are summarized for geographic regions, different industries, and so on; the
individual consumer (or merchant) is as anonymous to him as the individual molecule to the physicist”
[emphasis added].
It seems therefore, that Aumann must have been fully aware of the nature of the real world. His model,
however, as it is clear from the perspective that he consciously chooses, was not meant to help identify the
mechanisms of a market economy, but to provide a useful tool for policymakers.
5
list or number them, e.g. by saying that Mr. X is the first, Mr. Y second, Mr. Z third, and
so on. No such thing can be done with an uncountable set, however, which reveals the
absurdity of Aumann’s assumption, as has been rightly recognized by Paul Ormerod who
writes:
To take just one example, the phrase ‘assume a continuum of traders’ will be encountered in many
theoretical papers on the idealised market economy… But what does this phrase ‘continuum’
actually mean? It sounds quite innocuous, yet spelt out in words it might lead people to query the
realism of any academic paper based on this assumption, or even begin to doubt whether the
article was worth writing in the first place. For the phrase means that the number of people,
whether as individuals or as firms, carrying out trade in this theoretical economy is not just large
but quite literally infinite. In fact, to be strictly accurate, it even means rather more than this. If one
were to start to count the whole numbers—one, two, three and so on—one could go on for ever.
There is an infinite number of them. . . . But mathematicians have the apparently bizarre but
nevertheless logical concept of infinities which are even bigger than this infinity! A continuum is
exactly one of these. In other words, it is assumed that there is not just an infinite number of
traders, in the sense that the set of whole numbers is infinite, but there is an even bigger number of
them than this” (Ormerod 1995, pp. 43-44).
What Ormerod does not say, but what is essential to understand in this context, is that if
agents were to have three dimensions and some minimum “size”, then – by an elementary
set-theoretic argument – there could only be countably many of them in a 3-dimensional
space. Aumann, however, wants his traders to be located on the real line, so that his
assumption essentially boils down to thinking of people as coordinates in a 1-dimensional
space, merely points, with no bodies, mass or any physical characteristics. Yet the
flirtation with infinity does not end there. As has been noted by an eminent
mathematician and economist Nicholas Georgescu-Roegen (1979, p. 318):
T.C. Koopmans, perhaps the greatest defender of the use of the mathematical tool in economics,
countered the criticism of the exaggeration of mathematical symbolism by claiming that critics
have not come forward with specific complaints. The occasion was a symposium held in 1954
around a protest of David Novick. But, by an irony of fate, some twenty years later one of the
most incriminating corpora delicti of empty mathematization got into print with the direct help of
none other than Koopmans.… In 1972, Koopmans presented to the National Academy of Sciences
a paper by Donald Brown and Abraham Robinson for publication in its official periodical. The
authors assumed that there are more traders even than the elements of the continuum. Now,
6
since the authors of those papers and Koopmans are well versed in mathematics, they must have
known the result proved long ago by Georg Cantor, namely, that even an infinite space can
accommodate at most a denumerable [i.e. countable] infinity of three-dimensional objects (as the
traders must necessarily be) [emphasis added].
Following Aumann’s example, economists started to adopt the continuum
assumption to describe not only the number of market agents but also other parameters of
their models. For example, Rudiger Dornbusch et al. (1977) in a very influential paper set
the standard for research in international economics by extending the famous Ricardian
trade and payments theory from two commodities – as Ricardo himself, or many as e.g.
Gottfried Haberler (1937) – to a continuum of goods.11 Now, it ought to be stressed that –
to borrow an expression from Kenneth Boulding – “generalizations” in economic theory
often tend to run into a law of diminishing returns, in that moving from simpler to more
difficult cases might give the argument a certain fancy elegance but it seldom affects the
fundamental conceptual framework. And this is certainly the case with the work of
Dornbusch et al., who might have perhaps added a mathematical ornament to Ricardo’s
original theory, yet the whole edifice is still his. Indeed, commenting on Samuelson’s use
of n-dimensional analysis – which is a different but certainly related problem – Boulding
(1948, p. 192) states:
So we find that the n-dimensional analyses of Samuelson and his confreres add much to the
aesthetics of economics but surprisingly little to its substance.
More fundamentally, however, to form a proper judgment on the paper by
Dornbusch et al., we should ask what is the significance and real meaning of the
assumption that there is a continuum of goods in the economy. Note that if the stock of
goods in an economy is assumed to be uncountable, then – in mathematical terms – this
means that uncountably many goods can be subtracted from it leaving still no less than
11
The result obtained in 1977 was subsequently developed in a later article (Dornbusch et al. 1980)
structured in a very similar manner as the previous one – an old theory, this time the Heckscher-Ohlin
model, was evoked and “generalized” by the assumption of an uncountable number of goods.
7
uncountably many!12 What’s more, this procedure can be repeated over and over again
without significantly decreasing the “size” of the original stock of goods. Therefore, in
economic terms, the fact that a single individual consumes a unit of good does not reduce
the total available supply of goods, which seems to imply that the postulated model
simply assumes away the most fundamental fact of economic inquiry, namely scarcity.
This, however, means that the whole model is fallacious, since as Mises (1998, p. 93)
explained:
Means are necessarily always limited, i.e., scarce with regard to the services for which man wants
to use them. If this were not the case, there would not be any action with regard to them. Where
man is not restrained by the insufficient quantity of things available, there is no need for any
action.
Paradoxically, even Samuelson in his own textbook admits that “economics is the study
of how people choose to use scarce or limited productive resources” (Samuelson 1980, p.
2). It is left to the reader to assess to what extent the extension of Ricardo’s model to a
case of a continuum of goods, co-authored by Samuelson, satisfies that criterion.
There is at least one more respect in which the two examples mentioned above
constitute – to use Nicholas Georgescu-Roegen’s wonderful expression – an “empty
exercise with symbols” (Georgescu-Roegen 1979, p. 318). It is the fact that in many
neoclassical models fundamentally different results are obtained depending on whether
one considers a finite or uncountable set of agents/goods.13 Indeed, Aumann (1964, p. 41)
explicitly states that:
It should be emphasized that our consideration of a continuum of traders is not merely a
mathematical exercise; it is an expression of an economic idea. This is underscored by the fact that
12
Indeed, there are uncountably many points on the real line, but also uncountably many in each unit
interval. Hence, uncountably many points can be subtracted from the continuum still leaving no less than
uncountably many.
13
An example might be the famous Coase Conjecture stating that a durable goods monopolist has no
market power and, in effect, sells his good at a price equal marginal cost. Once this subject is treated
mathematically, however, it turns out that if there are finitely many consumers, then the seller can perfectly
price discriminate (Bagnoli et al. 1989), but when the number of buyers increases to continuum the good is
sold immediately at the seller’s reservation price (Fundenberg et al. 1985). Levine et al. (1995, p. 1160)
note that the divergence of outcomes occurs also in mathematical analyses of “corporate takeovers, and
time consistency of optimal government policy.”
8
the chief result holds only for a continuum of traders – it is false for any finite number [emphasis
added].
This questioned has been pondered in a somewhat more general case by David Levine et
al. (1995) who in an article with a most telling title “When are Agents Negligible?” write:
…the continuum-of-agents model is widely used either explicitly or implicitly in applied
economic situations ranging from competitive markets to public finance and political economy.
The rationale for using the continuum-of-agents model is that it is a useful idealization of a
situation with a large finite number of agents, but if equilibria in the continuum model are
radically different from equilibria in the model with a finite number of agents, then this
idealization makes little sense.
Unfortunately, the rest of the paper is rather disappointing, since having noted the
significance of that “paradox”, the authors do not analyze its implications further, rather
they set out to provide mathematical conditions under which models with a large but
finite number of agents may have similar equilibria to those in the continuum limit. The
fact that such purely mathematical conditions can be presented clearly does not invalidate
the conclusion only hinted at by Levine et al., namely that the assumption of a continuum
of agents is an idealization that makes very little sense.
One could object – as mathematical economists probably would – that the
foregoing analysis resembles rather childish efforts to catch an esteemed novelist on
misspelling a name. This accusation would be to some extent correct. So far we did point
out only the rather minor, or even petty, problems in the whole edifice of mathematical
economics, namely the problems of interpreting the continuum assumptions. It should be
stressed, however, that our deliberations – suggesting e.g. that uncountability of the stock
of goods essentially brings an end to scarcity – are provoked by the fact that human mind,
as limited and imperfect as it is, simply cannot imagine what an actual (uncountable)
infinity looks like, just as it cannot envision a 150-dimensional space,14 though both of
those concepts have a perfectly legitimate and rigorous mathematical meaning.15 Thus, it
14
Prescott and Lucas (1972), for example, consider not merely a 150- but infinitely-dimensional space.
It is somewhat similar in philosophy were efforts have been made to construct systems of alternative,
many-valued logics, which although seem perfectly sound “on paper”, have no operative meaning and thus
15
9
seems that Mark Blaug is perfectly right when he describes present-day economics as a
kind of “social mathematics”, which, to be sure, uses words such as “price”, “market” or
“commodity”, but gives them purely mathematical meaning with no likeness to real
world observations (Blaug 1998).16
The realism of assumptions in economic theory
We have argued so far that the assumptions of either a continuum of agents or a
continuum of goods are not only unrealistic, but simply unrealizable, which raises a more
fundamental question that cannot be so easily dismissed by mathematical economists,
namely that of the relevance of the realism of assumptions for economic theories. And
indeed, this problem – inextricably linked, as it is, with the use of mathematics in
economics – has received considerable attention even in the mainstream. For example,
the famous Cambridge economist Nicholas Kaldor was very critical of the
overrepresentation of abstract, anti-realistic models in economics. In 1972 he wrote:
The whole progress of mathematical economics in the last thirty to fifty years lay in clarifying the
minimum [equilibrium] requirements … without any attempt at verifying the realism of those
assumptions, and without any investigation of whether the resulting theory of “equilibrium prices”
has any explanatory power or relevance in relation to actual prices (Kaldor 1972, p. 1237-1238).
Kaldor is very much to the point here, and ironically enough, one of the most
prominent exponents of the mathematical method, Gerard Debreu, would probably fully
embrace that observation. After all, it was Debreu himself who described the postWalrasian general equilibrium theory – which owes its existence largely to him – as
amount to no more than mere symbolic formalism. As the philosopher Brand Blanshard notes, those
propositions consist essentially in introducing many gradations of the word “not”, and none of them is
incompatible with the “black-and-white” logic which is the only system natural to humans (Blanshard
1962, p. 273-274).
16
Blaug (1998) notes also: “It is not just that economics has become technical; it is that economics prizes
technicalities above everything else and that is why I call it formalism. Formalism is the tendency to
worship the form rather than the content of the argument. That is the kind of subject it has become. We care
only about the form in which an economic theory or hypothesis is presented, and we care almost nothing
about the actual content of the hypothesis” [emphasis added].
10
“logically entirely disconnected from its interpretation.”17 Let it be noted as an aside, that
the difficulty of determining the economic content of assumptions underlying
mathematical analysis seems to be a general shortcoming of mathematical economics,
and not only of general equilibrium theory (Bodenhorn 1956, p. 26).18
The problem is, as another renowned econometrician, Nobel Prize Laureate in
Economics Wassily Leontief suggests, that “uncritical enthusiasm for mathematical
formulation tends often to conceal the ephemeral substantive content of the argument
behind the formidable front of algebraic signs” (Leontief 1971, p.1-2). This, in turn, leads
to a situation in which attention is only centered on the mathematical derivation of formal
properties of a given model, leaving completely aside a thorough discussion of the
reasonableness of assumptions on which the model has been based. As Leontief stresses,
however, “it is precisely the empirical validity of these assumptions on which the
usefulness of the entire exercise depends” (ibid, p. 2).
Leontief’s view contrasts sharply with Milton Friedman’s program of “positive
economics” which did not care much for assumptions as long as the theory developed
with their help had any predictive power.19 Thus, writes Friedman:
17
Quoted by Kaldor op. cit., p. 1. In fact, Debreu has come up with an ingenious method of selecting
papers to be published in the prestigious Econometrica or the Review of Economic Studies. His so called
acid test consisted in removing all economic interpretations from an article and checking whether the
remaining mathematical infrastructure would stand on its own as a contribution in the field of theoretical
mathematics (Debreu 1991, p. 3). In his address to the American Economic Association Debreau notes with
considerable disappointment that not many of the articles published in those journals until the late 1930’s
would now be accepted for publication.
18
Bodenhorn (1956, p. 32) proposed the following way out of the formalistic trap:
“Many mathematical economists have suggested that communication between literary and mathematical
economists would be greatly facilitated if literary economists would learn more mathematics…. However,
it is also possible that communication might be improved if mathematical economists would learn more
economics … Moreover, communication might be improved if mathematical economists would state their
economic assumptions clearly in literary form and discuss fully the economic implications of the
mathematical model which they are employing.”
19
Though Leontief does not explicitly address Friedman, he blames the program of “positive economics”
for pushing economics into a state of “splendid isolation”. He writes: “True advance can be achieved only
through an iterative process in which improved theoretical formulations raises new empirical questions and
the answers to these questions, in their turn, lead to new theoretical insights. The ‘givens’ of today become
the ‘unknowns’ of tomorrow. This, incidentally, makes untenable the admittedly convenient
methodological position according to which a theorist does not need to verify directly the factual
assumptions on which he chooses to base his deductive arguments, provided his empirical conclusions
seem to be correct. The prevalence of such a point of view is, to a large extent, responsible for the state of
splendid isolation in which our discipline nowadays finds itself” (Leontief 1972, p. 5).
11
Truly important and significant hypotheses will be found to have “assumptions” that are wildly
inaccurate descriptive representations of reality, and, in general, the more significant the theory,
the more unrealistic the assumptions (Friedman 1953, p.14).
Therefore,
the relevant question to ask about the “assumptions” of a theory is not whether they are
descriptively “realistic”, for they never are, but … whether the theory works, which means
whether it yields sufficiently accurate predictions (ibid. p. 15).
Probably the most accurate response to Friedman has been given by Roderick
Long (2006) who argues that abstraction – that is the very mental process by which we
arrive at good explanatory theories – does not necessarily have to be unrealistic. Drawing
inspiration from Aristotle, Long proposes to distinguish between two different types of
abstraction: precisive and non-precisive. The former being the one in which certain
characteristics of the object under consideration are specified as absent, whereas in the
latter certain characteristics are merely absent from specification. Writes Long (2006, p.
7):
We may consider the horse as not having a determinate color, or else we may consider the horse
not as having a determinate color. To consider the horse as not having a determinate color is to
hold, or attempt to hold, as the object of our thought a horse that simply has no determinate color –
a creature never encountered in physical reality, and having its home either in Platonic heaven or
nowhere. This sort of abstraction falsifies and contradicts the concretes on which it is based. But
to consider the horse not as having a determinate color is simply to consider the horse as a horse
without considering its color one way or the other; and here no falsification is involved.
Thus, while building a theory of, say, consumer behavior, it is perfectly legitimate to
leave aside a whole array of consumer’s features – such as e.g. eye and hair color, height,
weight, number of family members, and so on – which according to Friedman would
make it “too realistic” and completely unintelligible.20 We cannot, however, specify those
20
In fact, Friedman has once been asked in an interview about the development of New Keynesian
economics which – unlike his monetarism – focuses on wage and price rigidities. Defending his own
position, Friedman stated: “I don’t have any doubt that there are wage rigidities because obviously there
12
characteristics as absent, as our theory would then only be valid for “consumers” who,
accordingly, have no eyes, bodies or families.
To fully grasp the significance of Long’s point, let us consider a geometrical
example. Suppose one wanted to prove some general theorem concerning triangles – say
that all three altitudes lie within a triangle or constitute either of its sides. Suppose further
that the proof would start with a following specification: “for the purpose of simplicity,
let us consider an acute triangle.” Clearly, then, should the “theorem” turn out to be true –
as it really is for acute and right-angled triangles – it would still have to be shown that the
result holds also for obtuse triangles, which most certainly is not the case.
Coming back to economics, the important point which follows from Long’s
discussion of abstraction is that if one builds a theory intended to explain a certain
economic phenomenon by specifying the absence of some of its characteristic features,
one does not obtain by this procedure an explanation of the phenomenon in all of its
generality. Rather, if successful, the reasoning explains only the specified – and
nonexistent – particular case, much like the aforementioned mathematical “proof” was
valid only for “special” types of triangles (namely the acute and right-angled ones). It still
remains to be shown, however, that what we have come up with applies also to the
general class of phenomena under consideration found in everyday life.
Assuming a continuum of traders in an economic model amounts to – as has been
explained above – specifying e.g. the absence of bodies or any physical features that
market agents might have. Whatever the end result of the analysis, then, the conclusions
ascertained apply only to those disembodied “economic zombies”, and it still has to be
rigorously demonstrated that the theory explains the behavior of real humans.
The reason we have so ardently kept emphasizing the word “explain” in the
foregoing discussion, is that Long’s critique of Friedman, as brilliant and insightful as it
is, seems to some extent to miss its target. It is not that Long’s remarks on abstraction are
incorrect – for, as far as the present author is concerned, they are correct, and indeed
constitute a major development of Austrian methodology – it is just that they do not
are; it’s a fact of life, it’s hard to deny it. The question is whether they are important or not, in what ways
they are important and in what kind of phenomena are they important. As I said before, the essence of a
successful theory is that it extracts the key elements from the whole host of attendant circumstances”
(Snowdon et al. 2005, p. 210).
13
apply to Friedman who has a distinctly different understanding of economics. To be sure,
Long is perfectly aware of that, and in fact starts his very investigations by stressing that
difference (Long 2006, p. 4), but then sets out to prove that Friedman’s conception of
abstraction is erroneous.
It seems, however, that this way of formulating the problem hinders a proper
comprehension of differences between the Austrians and mathematical economists, and –
perhaps more importantly in this context – it does not really help in understanding the
proper role of the realism of assumptions in scientific theories. This is most clearly seen
if we consider what Long writes about physics. In his view physical laws, properly stated,
are non-precisive abstractions, similar to counterfactual economic laws. Thus, he writes
(Long 2006, pp.12-13):
The law of gravity, too, is not a precisive abstraction applicable only to motion in a vacuum, but
rather a non-precisive abstraction that omits reference to other forces but does not thereby regard
them as nonexistent. The trajectory of a falling object is the resultant of all the forces acting on it;
if during a given period an object would fall five feet if gravity were the only force acting on it,
then whatever other forces may be acting on the object, we can still predict that the object will end
up five feet further downward than it would have if gravity had not been acting on it.
This is all very well, but there seems to be more to the laws of physics than just
that. For the law of falling bodies, referred to above, has a precise mathematical
formulation s=½(gt2), where s stands for the distance traveled by a falling body, t stands
for time, and g is a constant expressing the acceleration of a body dropped in a vacuum.
Now, it is obvious that the formal exposition of that, as well as many other physical laws,
requires calculus. In fact, the whole Newtonian mechanics is based on calculus, meaning
that it is based on precisive abstraction and manifestly false assumptions, such as the
infinite divisibility of time and space etc. Clearly, however, no one would reject the
whole body of modern physics because of its unrealistic, or in fact, unrealizable,
assumptions. Therefore, contrary to what Long seems to believe, Friedman is exactly on
the right track when he notes that false assumptions are useful in physics.21 So why
21
See Friedman (1953, pp. 16-19). As a matter of fact this was also Rothbard’s contention – see e.g.
Rothbard (1997, p. 102).
14
exactly is it – one could ask at this point – that unrealizable, demonstrably false
assumptions are accepted and yield good results in physics but not in economics? As
Rothabrd (1997, p. 61) explains, it is because:
In physics the axioms and therefore the deductions are in themselves purely formal and only
acquire meaning “operationally” insofar as they can explain and predict given facts.
This brings us to the very crux of the matter, namely that most physical laws are
not intended to explain but to predict – the former and the latter being, of course, two
related but nevertheless completely different things. In other words, physics as such,
following the methodological program of Mach, is not interested in causal but in
functional relations,22 and as far as the formulation of the latter is concerned, unrealistic,
unrealizable, and manifestly false assumptions do not pose a serious problem.
The foregoing remarks might seem slightly off the mark but they enable us to
understand clearly why Long’s critique of Friedman seems to miss the target. It is, as we
have already suggested, because Long and Friedman have a completely different
understanding of economics. The former describes his own approach by quoting
Lachmann who explains that the sole purpose of economics is to seek to “make the world
around us intelligible in terms of human action and the pursuit of plans” (Long 2006, p.
4). The latter, on the other hand, considers economics to be “a body of tentatively
accepted generalizations about economic phenomena that can be used to predict the
consequences of changes in circumstances” [emphasis added] (Friedman 1953, p. 39). To
make this difference even more apparent let us consider the following example. It is well
known that people eat ice-cream in the summer. It is also obvious that it is in the summer
that the number of drowning incidents reaches its peak. Friedman, therefore, would have
to believe that ice-cream consumption “explains” the number of drowning incidents, in
that including the former in an econometric model would – due to a high correlation
between those two phenomena – undoubtedly help predict, or estimate, the extent of the
latter. Austrians, however, would never say that ice-cream consumption is a cause of
people drowning. It is only if one mechanistically regards cause to be a factor which
22
See Menger (1973, p. 54).
15
improves prediction – as Friedman does – that one ends up with such superficial
“explanations”.23
The crucial difference between Austrians and mathematical economists has been
brilliantly perceived by Karl Menger (1973, p. 54):
[the] methodological difference between some Austrian and some mathematical economists stems
from the fact that the former look for causal explanations of some phenomena, whereas the latter
wish to confine themselves to the study of functional relations.
The Austrians, in other words, look for the very essence of economic phenomena, while
mathematical economists satisfy themselves with finding functional, statistical relations
between economic variables.24 Long (2006, p. 4) quotes the philosopher Peter Winch
who suggests that the essential difference between the two approaches is analogous to
trying to actually learn Chinese as opposed to studying the probability of occurrences of
various sounds in that language. Whatever we may think of the latter approach, it should
be understood that precisive abstraction is a perfectly legitimate tool of research there.
A good way to illustrate this point is to consider the rational expectations
hypothesis. As developed by Muth (1961) and further elaborated most notably by Lucas
(1972), the hypothesis asserts that economic agents’ subjective probability distribution of
outcomes tend to coincide with the true, or objective, conditional distribution of
23
As a matter of fact, this is precisely the econometrical definition of causality, sometimes referred to as
Granger’s causality. As Greene (1993, p. 553) explains: “Causality in the sense defined by Granger and
Sims is inferred when lagged values of a variable, say xt, have explanatory power in a regression of a
variable yt on lagged values of yt and xt.” In other words, x is said to be a cause of y if an autoregressive
model can be built in which x is a statistically significant predictor of y. Greene is fully aware of the
problems with such a formulation of a causal relation: “As such, the causality tests are predicted on a model
which may, in fact be missing either intervening variables or additional lagged effects that should be
present but are not. For the first of these, the problem is that a finding of causal effects might equally well
result from the omission of a variable that is correlated with both (or all) of the left-hand-side variables”
(ibid, p. 553). Note that this is exactly what happened in our ice-cream and drowning example. It seems,
however, that econometricians, more so than regular economists, are well aware of the limitations of their
method. For example, Adrian C. Darnell, the author of a dictionary of econometrics writes explicitly: “Such
a test of ‘causality’ does not allow any particular philosophical position to be adopted regarding the causal
structure of the Y, X relationship” (1994, p. 42), and suggests even that for the purpose of clarity,
“causation” should be replaced in econometrics by “precedence” (1994, p. 43).
24
Indeed, this approach has been advocated by Joseph A. Schumpeter who in his Wesen und Hauptinhalt
der theoretischen Nationalökonomie proposed to “avoid the concepts of ‘cause and ‘effect’ and replace
them by a more perfect concept of the function”, since “the latter, carefully developed by mathematics, has
a content which is clear and unambiguous, while the concept of causation has not” (quoted in Machlup
1951).
16
outcomes (Muth 1961, p. 316). In mathematical terms, this is equivalent to the statement
that ∆tPe=∆tP+εt, where ∆tPe stands for expected rate of, say, inflation from t to t+1; ∆tP
is the actual rate of inflation from t to t+1, and εt is a random error term which is
supposed to be normally distributed. In plain English this simply amounts to saying that
expectations of economic variables formed by market agents will on average be correct.25
It should be fairly obvious that the rational expectations hypothesis is an unrealistic
precisive abstraction in that it necessarily assumes that all economic phenomena can be
grouped into sets of repetitive events (over an infinite time span), thus reducing every
case of what Austrians would call case probability to class probability, i.e. changing
unmeasurable probability into a measurable one.26 As such, therefore, the hypothesis
cannot claim to explain any real phenomena, but it may – as we have argued in the
general case above – try to help predict their occurrence. And indeed, it is exactly what
its proponents expect from it, since as Muth (1961, p. 330) notes, the only real test of
theories involving rationality, is whether they fit the data any better than alternative
theories.27 By the same token, one may also very well build a model which rests on the
assumption that the number of goods is uncountable and try to “explain” with its help –
as Dornbusch et al. (1977; 1980) do – the production patterns in different countries
depending on the corresponding factor endowments. It is essential to grasp, however, that
in both of those cases one does not improve one’s understanding of the causal relations
between the economic phenomena under investigation – rather, one simply learns
25
By now, a variety of interpretations of the rational expectations hypothesis has emerged, but we refer
here to Muth’s (1961) original formulation of the concept.
26
Lucas himself notes that the rational expectations hypothesis can only be applicable to situations “of risk
in Knight’s terminology”, i.e. in cases where distribution of the outcome is known through calculations or
from statistics. He notes furthermore, that whenever the distribution of outcome isn’t known a priori
“economic reasoning will be of no value” (Lucas 1983, pp. 223-224).
27
It seems that the rational expectations hypothesis, both in its scope and method, draws heavily on
Friedman’s example concerning the prediction of density of leaves around a tree (Friedman 1953, pp. 1920). Friedman holds that one may very well put forth an assumption that leaves are thinking entities and
that each of them seeks to maximize the amount of sunlight it receives, as long as this improves our
predictions of where the density of leaves will be greater. Friedman argues that since it really turns out to
be so that leaves are in general denser on the south than on the north side of trees, our theory has “great
plausibility”, even though its assumptions are manifestly absurd. It is somewhat similar with the original
formulation of rational expectations hypothesis which starts off with blatantly untrue assumptions, but its
only role is to form predictions as to consequences of monetary policies.
17
something about nonexistent, abstract entities which may or may not be helpful in
predicting certain real life conditions.28
Conclusion
Neoclassical economic models often rest on the assumption that the number of
either goods or market agents is not only infinite in the popular meaning assigned to the
term, but also uncountable – i.e. so unimaginably large that one cannot even list them in
order. Such unrealizable assumptions cause obvious problems with interpretation, since it
is in practice impossible to determine with certainty what is meant by saying that, e.g.,
the number of goods available in an economy is uncountable. A rigorous mathematical
meaning of the term continuum implies that the only possible economic interpretation of
the aforementioned assumption is the elimination of scarcity in a model, which obviously
amounts to contradicting the very purpose of economic theory, as described even by the
proponents of mathematical models.29
This raises a more fundamental question of the importance of the “realism” of
assumptions in scientific theories, in particular in economics. The correct answer to that
question has been suggested already by Long (2006) who put forth a proposition to
distinguish between two different types of abstraction in scientific reasoning. As long as
one wishes to say something true about real things, the only legitimate abstraction
consists in abstaining from specifying certain features of the phenomena under
investigation. On the other hand, if in order to simplify analysis, one specifies the
absence of certain aspects of the analyzed process, then the conclusions thus obtained
apply only to the individual case specified at the outset.
This somewhat abstract argument provides background for understanding the
crucial difference between Austrian economics and mathematical economics and makes
28
In an interesting recent paper Salerno (2006) analyzes the famous monetarist Quantity Equation MV=PQ
and reaches the conclusion that it mixes the real cause and effect, i.e. it is not the flow of spending that
determines the price level, but rather the money prices are the causal determinants of the spending flow
(ibid, p. 51). Thus, the Equation of Exchange seems to be an excellent example of the mainstream
functional approach to economic theory – it does not capture the real economic cause and effect
relationship, but focuses rather on building a statistical, functional relation between the price level and the
flow of spending.
29
See e.g. Samuelson (1980, p. 2).
18
possible a deeper understanding of the role of realism of assumptions in economic theory.
The former approach focuses on causal-realist analysis and through the use of
counterfactual laws and non-precisive abstraction seeks to explain real life economic
phenomena. In stark contrast, the latter approach, as argued in the first part of the paper,
rests on necessarily false and unrealizable assumptions, and thus resorts to precisive
abstraction, with the sole purpose of improving the prediction of consequences of certain
changes (Friedman 1953, p. 39). It seems that depending on circumstances, both methods
may be resorted to, provided that one is constantly aware of their respective scopes and
purposes. Therefore, there is nothing a priori wrong in starting one’s analysis with
assuming that, say, there is a continuum of market agents – whatever that means. One has
to be aware, however, that any conclusions that follow from such an analysis can – to use
Peter Winch’s30 expression – shed some light only on the statistical probabilities for the
occurrence of the various words in the Chinese language, a not on the understanding of
the true meaning of those words.
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