Books / Digital Text

II. Mundane Economics

7. “Mises and Hayek Mathematized”: Toward Mathematical Austrian Economics by Marek Hudík

In* his introduction to the second edition of Rothbard’s Man, Economy, and State, Professor Salerno (2004) argues that Rothbard’s purpose in writing his treatise was not to develop a heterodox school of economics and break with the prevailing body of thought. On the contrary, Rothbard examined contemporary literature and attempted to integrate this literature with his own views. As Salerno shows, Rothbard believed that his treatise could draw other economists to the ideas that used to be part of the mainstream in the not-so-distant-past. We now know that Rothbard did not succeed in this and that as of today, there still is a communication gap between the Austrians and the rest of economic profession. This paper argues that the gap could be narrowed if the Austrian economics becomes more mathematized.1

At a first glance, mathematization of Austrian economics may seem to be contradiction in terms. Yet, at a closer inspection, the idea turns out to be not paradoxical at all: note for instance, that the “literary” character of Austrian economics is typically not included among its defining characteristics (Machlup 1982; Leeson and Boettke 2006; O’Driscoll, Jr., and Rizzo 2002); in a similar vein, Vaughn (1998, p. 2) sees the Austrian aversion to mathematics as a “superficial identifying characteristic,” and Backhouse (2000, p. 40) points out that, to the best of his knowledge, no Austrian has “ever explained why mathematics cannot be used alongside natural-language explanations”; on top of that, Moorhouse (1993, p. 71) reviewing Mises’s views on mathematical economics concludes that there is “no major methodological gulf between praxeology and neoclassical mathematical economics.”

Admittedly, Rothbard, as well as some other Austrians, raised objections against mathematization; but his demonstrated preferences speak otherwise: he sometimes expresses his ideas formally or semi-formally (e.g., Rothbard 2004, pp. 120–121, 152–153, 234). In addition, there is a long line of authors whom we may count as Austrian or Austrian-inspired who occasionally use mathematics in their economic writings. These include Wicksteed (1910), Fetter (1915), Hayek (1941), Haberler (1950), Machlup (1939), Morgenstern (von Neumann and Morgenstern 1953), McCulloch (1977), Garrison (1978), Murphy (2005), Leeson (2010), etc.

Some of these authors even explicitly claim that mathematization of economics is, at least to a certain extent, methodologically acceptable or even desirable. For example, Hayek (1952, p. 214) sees mathematization as “absolutely indispensable to describe certain types of structural relationships”; Machlup (1991) roots for “polylinguistic scholarship” characterized by coexistence of mathematical and non-mathematical language; in Boettke’s (1996) view, formal models are “fine” when constrained by an understandability criterion; and according to Morgenstern (1963, p. 19), an outright supporter of mathematization of economics, the laws of society will be written in the language of mathematics, just like the laws of nature.

This paper acknowledges that mathematization has costs and benefits. At the same time, it admits that it is probably impossible to determine the range of levels of mathematization for which benefits outweigh costs. Given this limitation, the aim of this paper is thus rather modest: it merely attempts to show that the optimal level of mathematization is not zero. More specifically, this paper points out the benefits of mathematization that seem to have been overlooked by some Austrian authors and it shows that most of Austrian criticisms which supposedly challenge mathematization, in fact point to different issues.

Benefits of Mathematization

Mises (1996; 2003; 1977) and Rothbard (2004; 1997a; 1997b) claim that formalization adds nothing to our knowledge as it only involves translation of verbal statements into symbols.2 According to Rothbard (1997a, p. 61; 2004, p. 325), benefits of formalization are none, and therefore formalization should be cut through the principle of Occam’s razor.3 Mises (1996, p. 333) suggests that if there is any benefit to formalization at all, it is pedagogical: diagrammatic exposition can be helpful to students of economics. Mises thus indirectly admits that mathematics (in a diagrammatic form) contributes to clarity of exposition. But why restrict this benefit only to students? Should not economists always communicate with their colleagues in the clearest possible way, especially when presenting new ideas?

Clarity of exposition achieved through diagrammatic representation is but one (and perhaps even not the most important) benefit of the use of mathematics in economics. I propose that mathematics offers also the following three benefits: First, mathematics is nowadays a common language of most economists and other researchers across disciplines — it is thus necessary to communicate ideas; Second, mathematics is less ambiguous than verbal language as it forces one to define precisely the meanings of concepts; and Third, mathematics is generally more efficient than verbal language, both for “producers” and “consumers” of economic ideas. These three benefits of mathematization are now discussed in turn.

Mathematics as a Common Language

If the great majority of economists use mathematics, it pays for each individual economist to use mathematics too; this is simply a coordination problem. The use of verbal language may lead to misunderstanding by the rest of the profession. When an Austrian and another economist speak of marginal utility or time preference, for example, do they in fact mean the same things?4

There are numerous examples in the history of economic thought when translation into the language of mathematics helped to clarify the differences between competing approaches.5 For instance, Marshall’s (1982) translation of Ricardo’s theory of price formation into mathematics allowed for distinguishing between the classical and marginalist theories and facilitated the latter’s acceptance. Similarly, mathematics in the hands of Hicks (1937) and some others helped to detect the differences between “Keynes and the classics” on macroeconomic issues and contributed to the creation of the “neoclassical synthesis.” According to one observer:

Keynes was impressed by the help given by mathematics when numerous economists (Harrod, Hicks, Samuelson, Bryce) cleared up confusions in his General Theory and also presented his system neatly with the help of mathematics. (Harris 1954, p. 384)

Several decades later, formalized language of mathematics revealed that the dispute between “monetarists” and “Keynesians” was not about a general theoretical framework but about different empirical assessment of the value of parameters of the same model (e.g., Modigliani 1977; Mayer 1995). To plunge into more heterodox waters, Roemer (1982; 1988) is one of several economists who formalized Marxian economics and thus helped readers to compare the similarities and differences between Marxism and other mathematized approaches.

With respect to Austrian economics it is interesting to note that according to Chipman (1954, p. 364), “it is hard to find in mathematical economics any discussions more abstruse and difficult to follow than the great verbal debates between the Austrian and American schools on capital theory.” Fortunately for Chipman and others, several attempts to formalize Böhm-Bawerk’s theory have emerged (e.g., Dorfman 1959; 1995; 2001; Potužák 2014) and helped to clarify the debate. Very helpful in this respect is also Garrison’s (1978; 2000) partly formalized treatment of Austrian macroeconomics.

Mathematization is of course not the only way of dealing with the “language-coordination problem.” For instance, one may ignore the majority of economists and choose to “play the game” only with those who use his (i.e., verbal) language. However, this would in effect amount to creating a closed school of thought whose members are able to communicate only with each other but would not be able to interact with the rest of the discipline.6 Closed schools of thought are analogous to closed economies: they protect their cherished ideas from competition. As in the case of trade, such a state of affairs benefits “producers” of ideas but hurts the “consumers” who receive products of inferior quality. Rothbard (1987) seems to have been aware of these adverse effects of isolated groups and perhaps that is also why he chose to communicate with the mainstream.7

Another possibility of approaching the “language-coordination problem” is to stick to verbal language with the proselytizing aim of persuading the rest of the profession to use it, too. In other words, one may be trying to change the language convention, and achieve a switch from a “mathematized equilibrium” to a “verbal equilibrium” of the “language coordination game.” Nevertheless, success of such an attempt seems unlikely, all the more for the fact that the “mathematized equilibrium” is — as I argue below — superior.

Mathematics as a More Precise Language

One of the benefits of mathematization is that it forces us to formulate our ideas precisely (e.g., Klein 1954; Tinbergen 1954; Chiang 1984; Clower 1995). It is sometimes correctly argued that verbal language can be made as precise as the language of mathematics (e.g., Menger 1973; Beed and Kane 1991). In reality, however, this opportunity very often goes unexploited: unless one is forced to express ideas formally, one is perhaps not even aware that the language is ambiguous. Perhaps the best example of increased clarity due to formalization is the creation of the supply and demand model. As Schumpeter (1994, p. 602) points out:

the sponsors of supply and demand [of the 19th century], again with the unnoticed exception of Cournot (and very few others, such as C. Ellet and D. Lardner), even experienced difficulty in setting on its feet the very supply-and-demand apparatus, the claims of which to a place in economic theory they tried to assert. They talked of desires or desires backed by purchasing power, of “extent” of demand and “intensity” of demand, of quantities and prices, and did not quite know how to relate these things to one another. The concepts, so familiar to every beginner of our own days, of demand schedules or curves of willingness to buy (under certain general conditions) specified quantities of a commodity at specified prices, and of supply schedules or curves of willingness to sell (under certain general conditions) specified quantities of a commodity at specified prices, proved unbelievably hard to discover and to distinguish from the concepts—quantity demanded and quantity supplied.

Precision of mathematics also helps to derive implications of one’s assumptions and to demonstrate possible inconsistencies (e.g., Dorfman 1954; Clower 1995). For instance, Samuelson (1957), by formulating Marxian model of wages and interest discovered an error in Marx’s theory that went unnoticed for 90 years (Brems 1975). Mathematics may also help to discover inconsistencies in the Austrian economics: Austrian economists work with preference scales; at the same time, they sometimes criticize the transitivity assumption used by other economists (Block and Barnett 2012). Yet, it is straightforward to show formally that an ability to rank alternatives on a single scale corresponds to the assumptions of completeness and transitivity of the preference relation. In other words, whenever a preference scale is introduced, completeness and transitivity of preferences are implicitly assumed (Hudík 2012). To use a different example, with the help of some simple mathematics it can be demonstrated that, contrary to Rothbard’s (2004, p. 240) claim, the principle of diminishing marginal utility does not necessarily imply a downward-sloping demand curve (Hudík 2011a).

Interestingly, Rothbard sees the ambiguity of the verbal language as an advantage. He quotes Bruno Leoni and Eugenio Frola:

the lack of mathematical precision in ordinary language reflects precisely the behavior of individual human beings in the real world. ... We might suspect that translation into mathematical language by itself implies a suggested transformation of human economic operators into virtual robots. (Rothbard 1997a, p. 62)

This argument is unpersuasive on several grounds: First, it is not at all clear why researchers should use imprecise language just because their researched subjects are imprecise; one can (and, indeed, should) talk precisely even about imprecision. Second, Leoni and Frola’s argument seems to imply that economists should not describe human behavior by concepts which are not used by the acting individuals themselves. However, this requirement imposes unnecessary constraint on economic theories. For instance, economists would be barred from referring to the law of marginal utility merely because people are generally unaware of this law. Finally, Leoni and Frola neglect the fact that economics mostly deals with an order which emerges as an unintended consequence of human actions (Hudík 2011b) where their argument is inapplicable. Consider, for example, activities of speculators which inadvertently contribute to efficient allocation of resources. I assume that we want to be able to describe these consequences even though speculators themselves are unaware of them.

Mathematics as a More Efficient Language

Mathematics is often more efficient than verbal language for both “producers” and “consumers” of economic ideas. From the perspective of the “producers”, mathematics economizes on effort: laborious thought processes are “embodied” in simple rules for manipulation of mathematical symbols (Whitehead 1911, p. 41). In this context Duesenberry (1954) understands mathematics as a “capital good” increasing productivity of economist’s “labor.” On the one hand, Duesenberry admits that it may be true that one cannot do anything with mathematics which cannot be done with verbal language; on the other hand, however, he claims that verbal language is much less efficient; according to his analogy, “[o]ne probably cannot do anything with power shovels that cannot be done with picks and hand shovels” (Duesenberry 1954, p. 361). Analogously, Chiang (1984, p. 5) thinks of mathematics as a “mode of transportation.”8

Chiang (1984, p. 4) mentions another aspect of the efficiency of mathematization of economics: there exists a large number of mathematical theorems at economists’ disposal. Consequently, we do not have to rediscover these theorems whenever they arise in a new context (Dorfman 1954, p. 376). Thus, for instance, in order to prove his theorem of the existence of (“Nash”) equilibrium in strategic games, Nash applied first Brouwer’s and later Kakutani’s fixed point theorems (Kuhn and Nasar 2002). Half a century before Nash, Euler’s theorem was applied to address the “adding-up problem” in the theory of distribution (Stigler 1994).9

As for “consumers” of economic ideas, mathematics often allows them to economize on their time and attention: as Klein (1954, p. 360) puts it, “[t]here is a real merit in condensing wordy volumes or manuscripts into a few understandable pages.” Nash may again be used as an example here: his famous dissertation thesis that earned him the Nobel Prize has only twenty seven pages; his paper on the existence of Nash equilibrium takes up only one page (Nash 1950a), while his ground-breaking paper on the bargaining problem is eight pages long (Nash 1950b). It is safe to assume that without formalization Nash’s papers would have to be considerably longer.10

Costs of Mathematization

Mathematization does, naturally, have its costs. As pointed out by Morgenstern (1963, p. 2), when evaluating costs of mathematization, one has to distinguish among (i) criticism of inappropriate use of mathematics, (ii) criticism of the underlying economic model which happens to be analyzed mathematically, and (iii) criticism of mathematization.

In the first category we find criticisms of Bourbakism in economics (McCloskey 1994), of the use of calculus (Boulding 1948; Rothbard 1977), or of applying the mathematics of nineteenth-century mechanics to economics in general (Mirowski 1989). Likewise, criticisms of failed attempts to mathematize phenomena which seem to be impossible to address with known mathematics belong to this category (Beed and Kane 1991; Wutscher et al. 2010), as do also criticisms of misinterpreting quantitative economics (Mises 1996, pp. 55–56)11 and measurement (Rothbard 1977). None of these or similar criticisms, justifiable or not, represent arguments against the use of mathematics in economics as such.

Type (ii) criticisms are also not arguments against mathematization. They include criticism of unrealistic assumptions (e.g., Keynes 1964; Leontief 1971; Beed and Kane 1991; Wutscher et al. 2010) or criticism of particular concepts that happen to be used by mathematical economics, such as equilibrium (Wutscher et al. 2010). It is important to repeat that most mathematization is simply a translation of verbal statements into symbols; hence, the problem must be with theories themselves, not mathematics (Backhouse 1998; 2000). One may interject that the use of certain branches of mathematics (e.g., calculus) requires some additional assumptions such as continuity and differentiability (Menger 1973); but again, this criticism concerns only the application of a particular branch of mathematics to particular economic problem and is consequently not a general argument against mathematization. Furthermore, technical assumptions used by mathematical economics are often harmless: for instance, it is well-known that all important conclusions of standard demand theory can be obtained without the assumption of continuous and differentiable utility functions. Yet, continuous and differentiable functions are often used for the sake of convenience.

Actual costs of mathematization are identified by type (iii) criticisms. What are these costs? I identify three: first, tendency to downplay factors which are difficult to formalize; second, tendency to lose touch with reality; third, decrease of intelligibility for lay people. Note, that the first two costs are not inherent to mathematization per se; they are rather incidental to it and can perhaps be avoided. More importantly, though, none of these costs constitutes by its nature an argument for avoiding the use of mathematics altogether.

Downplaying Factors Not Amenable To Formalization

A tendency to neglect everything that cannot be easily formalized is a drawback of mathematization acknowledged by mathematical economists themselves (e.g., Debreu 1986). For instance, Krugman (1996; quoted in Backhouse 1998) argues that economists ignored important models for spatial economics just because these models could not be formalized.

Sometimes economists go so far as to demand that theories must refer only to quantifiable magnitudes. In his Nobel lecture Hayek (1975, p. 434) points out that

while in the physical sciences the investigator will be able to measure what, on the basis of a prima facie theory, he thinks important, in the social sciences often that is treated important which happens to be accessible to measurement.

He gives an example of quantifiable relationship between aggregate demand and total unemployment on one hand, and relationship between unemployment and the structure of relative prices and wages on the other. The former is accepted as “scientific,” while the latter is neglected as not testable because we never know what the equilibrium prices and wages are.

Other phenomena that are difficult to treat mathematically and are often mentioned by the Austrians are subjectivism and Knightian uncertainty. Again, these can be argued to receive insufficient attention by economists.12 Still, one may wonder if perhaps the limits of mathematization, whether in this particular case or in general, do not often coincide with the limits of scientific investigation: are currently non-mathematizable phenomena amenable to science at all?

I suggest that the way to deal with the phenomena which are currently difficult to mathematize is not only a careful use of known mathematic tools but also development of new tools. For example, before von Neumann and Morgenstern (1953) mathematical economics (and, as a matter of fact, any branch of economics) was unable to deal with strategic decision problems. Hence, von Neumann and Morgenstern constructed a completely new branch of mathematics to deal with strategic issues. As this example illustrates, the limits of mathematization are not given but constantly evolve.

Losing Touch with Reality

It is often argued that mathematization leads to a loss of contact with reality (e.g., Boulding 1948; Champernowne 1954; Novick 1954; Šímová and Šíma 2012).13 This can have several reasons: In Debreu’s (1986, p. 1268) view, the power of mathematics is such that the “seductiveness of [mathematical] form becomes almost irresistible” and researchers thus tend to forget economic content. Still, Debreu argues that separation of models and reality can sometimes be an advantage. For instance, it is said to bring economics closer to the ideology-free ideal (see Düppe 2010).14

According to Duesenberry (1954, p. 362), loss of touch with the real world is simply given by the job description of an economic theorist: the aim of the theorist is not to explain a particular set of observations but to show general consequences of a set of premises. To this argument we may add that a theorist also aims at universalization: she also attempts to show that two or several seemingly separate theories are merely different manifestations of the same principle. Hence, theoretical research is necessarily often disconnected from reality as it focuses on logical consistency of theories. From this perspective, criticism of the separation of mathematical models from reality could be interpreted as a criticism of theoretical research as such and as a plea for focusing on applied research. I hasten to add that the debate on optimal allocation of resources between theoretical and applied research is extremely important (see e.g., Šťastný 2010); yet, it is a different debate than the one on costs and benefits of mathematization.


It is probably true that the more formalized a model is, the less intelligible it is to lay people. Should economists worry about this trade-off? On the affirmative side stands the consideration that economic literacy is low which in turn has substantial negative externalities as citizens and voters are called upon to form opinions on many economic issues (e.g., Becker 2000; Šťastný 2010). On the other side stands the argument that, as in any other science, researchers should write primarily for other researchers and educating lay people should be left to popularizers: as individual economists differ in their skills and talents, there are benefits from specialization.15 Trading off benefits of formalization for intelligibility of academic writing to the general public thus seems inefficient. A different question is whether economists have sufficient incentives to be popularizers; but that is again for another debate.


Examination of benefits and costs of mathematization suggests that the issue is not whether to use mathematics in economics or not; instead, the issue is what kind of mathematics is appropriate and how it should be used (cf. Backhouse 2000; Rosser 2003). It should be stressed that mathematization by no means is in conflict with the Austrian methodology, although some aspects of Austrian economics may be difficult to formalize at the present state of knowledge. This limitation, however, does not imply that we should give up on pushing the limits of mathematization further. Given that spreading ideas among the bulk of modern economists requires the use of mathematical language, one may only hope to see more and more mathematized Austrian economics in the future. For as they say: b(m) - c(m) > 0, for some m > 0.


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  • *. Marek Hudík is postdoctoral fellow at the Center for Theoretical Study at Charles University in Prague, Prague, Czech Republic. I was a summer research fellow at the Mises Institute in 2009. Throughout the fellowship, I greatly benefited from Professor Salerno’s kind help and constant encouragement.
  • 1. By “mathematization of economics” I mean the “use of mathematical techniques … in economic arguments” (Backhouse 1998, p. 1848). An alternative definition of the term can be found in Beed and Kane (1991, p. 581), who understand it as the “increasing emphasis given to mathematical economics.” For a discussion of the concepts of mathematization, formalization, axiomatization, and abstraction, see e.g., Weintraub (1998) and Backhouse (1998).
  • 2. This claim seems uncontroversial: it is put forward by both critics of mathematization (e.g., Novick 1954) and its advocates (e.g., Samuelson 1952). However, see Dennis (1982a; 1982b) for criticism of this view; see also Weintraub (1998, p. 1844) who posits the view of mathematics as an engine of discovery as an alternative to mathematics as a language.
  • 3. This Rothbard’s claim is problematic: if true, how would we explain that mathematics itself (or any other discipline) became formalized? Indeed, until the Renaissance, there was basically only “literary mathematics”: for instance the symbols “+” and “—” first appeared in the late fifteenth century and “=” was introduced only in the early sixteenth century (Cooke 2005, p. 432).
  • 4. For a discussion of different definitions of marginal utility, see Hudík (2014a). On the ambiguity of time preference definition, see Potužák (2014).
  • 5. Admittedly, there are also instances when mathematization contributed to ambiguity of economic concepts (Stigler 1950). In this context, it should also be noted that there usually is more than one way of formalizing a theory and this further complicates the issue (Beed and Kane 1991).
  • 6. Interestingly, until the first half of the twentieth century, i.e., before mathematical methods spread through the discipline, mathematical economics was considered to constitute such closed group. See e.g., Clark (1947).
  • 7. Similar attitude was adopted by many Austrians before and after Rothbard, including Böhm-Bawerk, Mises, and Hayek.
  • 8. This metaphor seems to have been used for the first time by Fisher (2007); for similar metaphors, see e.g., Pareto (1897), Champernowne (1954), Tinbergen (1954), Menger (1973) and McCloskey (1994).
  • 9. For more examples of mathematical theorems that were directly applied in economics, see Debreu (1984).
  • 10. As usual, there is a dissenting view, this time it is Marshall’s: The chief use of pure mathematics in economic questions seems to be in helping a person to write down quickly, shortly and exactly, some of his thoughts for his own use … It seems doubtful whether anyone spends his time well in reading lengthy translations of economic doctrines into mathematics, that have not been made by himself. (Marshall 1982, p. ix)
  • 11. It should be added that Mises criticized the use of quantitative methods to test theories; there is no argument in Mises’s writings against using quantitative methods in applied research. See also Leeson and Boettke (2006).
  • 12. For the debate on formalization of Knightian uncertainty, see Caplan (1999) and Wutscher et al. (2010); for an attempt to formalize subjectivism in games, see Hudík (2014b).
  • 13. On the other hand, Brems (1975) provides the following counter-example of verbal treatment leading to focus on imaginary problems: investment in the Keynesian theory was considered a function of the rate of interest instead of the change of the rate of interest, only because verbal economics was unable to handle difference or differential equations.
  • 14. Morgenstern praised mathematical economics for exactly the same reason. See Leonard (2010).
  • 15. Steven Levitt is an exception that may in fact prove the rule: his pop-economics books are co-authored with the journalist Stephen Dubner.
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