A Companion to the History of Economic Thought - Chapter 33 potx

Whether or not there could be explanatory and predictive economic models that successfully resisted all attempts to represent them in mathematical form, most economic theorizing today is

C H A P T E R T H I R T Y - T H R E E Mathematical Modeling as an

Exegetical Tool:

Rational Reconstruction

A M C Waterman

33.1 TERMINOLOGY

33.1.1 Rational reconstruction

Rational reconstruction (hereinafter, RR) will be understood in this essay in the sense used by Imre Lakatos (1978, ch 2) in reference to the history of science

According to Lakatos, RR is equivalent to what he calls internal history: a putatively

diachronic account of what counts as “growth of knowledge” or “progress in science” – as “progress” is adjudicated by the particular normative methodology

favored by the historian Lakatos’s external history is confined to social and

eco-nomic conjuncture, the tastes, ideologies, and metaphysics of the scientists, and other circumstances that “explain the residual non-rational factors.” Thus

“external history is irrelevant for the understanding of science” (Lakatos, 1978,

pp 118, 102)

It is apparent that there must be at least as many rational reconstructions of any particular episode as there are methodologies Thus an historian who accepts the criteria of scientific progress proposed by “conventionalism” will offer a very

different internal history of some important scientific innovation from that of another historian whose criteria are those specified by Lakatos’s own methodo-logy of “scientific research programmes.” It also appears that those who accept

Thomas Kuhn’s account of The Structure of Scientific Revolutions (1962) must dis-qualify themselves from attempting any kind of RR For “In Kuhn’s view there can

be no logic, but only psychology of discovery ”: hence “scientific revolution is irrational,

a matter for mob psychology” (Lakatos, 1978, pp 90, 91; italics in original)

Some historians of economic thought (e.g., Blaug, 1990) have preferred Richard Rorty’s seemingly more open-ended usage of “rational reconstruction” (never clearly defined) to identify one of the four “genres” of the historiography of

philosophy; the others being “historical reconstruction” and “Geistesgeshichte”

(literally, a study of the “spirit” of the times) each of which, like RR, is legitimate and useful, and “doxography” (praise of dead philosophers), which is neither (Rorty, 1984) For Mark Blaug, RR corresponds to his own “economic theory in retrospect,” which he described as “absolutist” history in contrast with “relativist” history – the latter being “almost the same” as Rorty’s “historical reconstruction” (Blaug, 1997, pp 1–2, 7–8) But in fact it more closely resembles Lakatos’s

“external history.” Rorty himself, pretending to believe that modern philosophers know and understand some things that the greatest of their predecessors did not, defended RR as “self-consciously letting our own philosophical views dictate the terms in which to describe the dead” (Rorty, 1984, p 50) By doing this, Rorty claimed, we are “able to see the history of our race as a long conversational interchange”:

We need to think that, in philosophy as in science, the mighty mistaken dead look down from heaven at our recent successes, and are happy to find that their mistakes have been corrected (p 51)

It would appear from this that Rorty believes that there can be and is “progress”

in philosophy, that criteria exist to determine what counts as progress, and hence that we can reconstruct parts at least of the “long conversational interchange” in terms “dictated by our own philosophical views.” To this extent, Rorty’s fuzzier usage of RR is congruent with, if not identical to, Lakatos’s more rigorously specified definition

The reason for preferring Lakatos’s definition with reference to the history

of economic thought (hereinafter, HET) lies in an obvious difference between the “conversation” of economists and that of philosophers For Rorty was less than wholly serious in his claim that there can be “progress” in philosophy “We hesitate [to say that Aristotle or Leibniz or Descartes were ignorant of what now count as ‘facts’ in philosophy] because we have colleagues who are themselves ignorant of such facts, and whom we courteously describe not as ‘ignorant’, but

as ‘holding different philosophical views’ ” (Rorty, 1984, pp 49–50) It is at least

as plausible, therefore, to regard philosophy as a continual recycling of old ideas, and none the worse for that Although there is undoubtedly some element of this

in economics too (Waterman, 1997), it is obvious that economists have a far more highly developed sense of “progress” or “growth of knowledge” in their discipline

than do philosophers Theories are formulated, models constructed, and hypo-theses tested in a way that closely resembles the method of the natural sciences Most economists believe that they can explain a wider range of social phenom-ena with modern theory than was possible for Smith and Ricardo, Walras and Edgeworth, or Wicksell and Keynes And because what motivates many is the desire to produce knowledge that is “useful” in that it can be applied to improve public policy and legislation, any serious doubt about that possibility – either among economists themselves or among those who pay for their services – would drastically reduce, if not eliminate, the profession Philosophy, however, thrives

on self-doubt

33.1.2 Mathematical modeling

Economic analysis is inconceivable without the use of models, either implicit

or explicit For example, the “mental experiment” at the heart of David Hume’s essay “Of the balance of trade” (1994 [1752] ) can only be conducted when aggregative concepts such as “money,” “the price of all labour and commodit-ies,” and “the art and industry of each nation” have been abstracted from the real world of commerce, implicitly quantified, and related to one another in the imaginary world of the analyst’s model Likewise, when Adam Smith states that

“the demand for men, like that for any other commodity, necessarily regulates the production of men” (1976 [1776], book I, p 98), he is manipulating abstractions and implying an imaginary causal nexus of other abstractions – the growth-rate

of “population” in relation to that of “capital,” and so on – some of which are

discussed in Wealth of Nations I.viii Therefore the term model will be used in this

essay to denote a formal arrangement of abstractions constructed to represent, emblematically or figuratively, some supposed system of cause and effect existing

in the real world of human societies By use of models, economists explain and predict social phenomena

Whether or not there could be explanatory and predictive economic models that successfully resisted all attempts to represent them in mathematical form, most economic theorizing today is in fact conducted in explicitly mathematical terms However, although exceptions exist as far back as the eighteenth century (Theocharis, 1961), this was not the case for most economic theorizing before

Walras’s Eléments (1954 [1874] ) And between the 1870s and the 1940s much

economic theory continued to be “literary.” It is a question for HET, therefore, whether the seemingly implicit mathematical reasoning in much literary economic theory invites explicit mathematical treatment; and, if so, whether the resulting

“translation” of literary theorizing into mathematics can tell us anything that the author did not reveal in his text, and which – were we able to bring him back to life – he might be brought to agree that he had really intended to say It is the purpose of this essay to address that question Meanwhile, it is sufficient to

define mathematical modeling (hereinafter, MM) in the historiography of economic

thought as the representation in mathematical terms of what seem to be the most important elements in literary economic theory The precise relation between

MM and RR will be considered below

33.1.3 Exegesis

The term, which is derived from the Greek verb εξηγοµαι = “I narrate, explain,” originated in connection with the study of sacred Scripture, and may be defined

as “the art of explaining a text.” In scriptural exegesis at any rate, “the

explana-tion may include a translaexplana-tion, paraphrase, or commentary on the meaning” (ODCC,

1977; italics added) There is no reason to exclude any of these possibilities from the exegesis of nonscriptural texts Now if translation be undertaken at all, any paraphrase will presumably be written in the language of translation And if the

MM of literary economic theorizing may indeed be regarded as paraphrase

writ-ten in the “language” of mathematics, then MM can properly be employed as an

“exegetical tool” in HET But since translation is prior, the question is whether translation into mathematics of an economic argument originally formulated in some natural language is possible; and whether, even if possible, there may be either or both a significant loss in content and a significant distortion of meaning This question raises further ones Is mathematics a “language” at all? Does eco-nomic theory possess any nonmathematical components, essential to its proper understanding, that resist translation? Are there any good criteria of “correct”

translation between any two languages?

Paul Samuelson (e.g., 1947, 1952) has argued strongly for the equivalence of mathematical and literary theorizing, and for the greater efficiency of the former

in economics George Stigler (1949, p 45) assumed the possibility of translation, which, he asserted, “is absolutely necessary, not merely desirable.” Contributors

to the symposium on mathematical economics in the Review of Economics and

Statistics (1954, pp 357–86) took the possibility of translation for granted Whether they were justified in so doing was later contested (Dennis, 1982) Meanwhile,

W V Quine (1960) showed that without some generally agreed set of rules for translation, it is possible that two equally “good” translations from any one language into any other may be mutually incompatible In this essay, it will simply be assumed that translation of literary into mathematical economic theory

is usually sufficiently reliable for the attempt to be worthwhile; and hence that a paraphrase of the former in the language of mathematics might be a useful

“exegetical tool.” Quine’s objection holds against all forms of translation and therefore against all use of translation and paraphrase in exegesis: including the attempt to translate the original Hebrew and Greek of the scriptural texts first into Latin and then into modern languages In this respect at any rate, exegesis of

Wealth of Nations beginning with a “translation” into mathematics is on all fours with exegesis of the Book of Genesis beginning with a translation into English

33.2 MATHEMATICAL MODELING, RATIONAL RECONSTRUCTION,

AND HISTORY

It is obvious that MM is not RR as Lakatos understands the latter, for MM is synchronic and RR diachronic Consider the first example of MM in Takashi

Negishi’s History of Economic Theory (1989), which addresses John Locke’s

con-cept of “vent” and its function in Locke’s value theory A simple model of supply

is constructed which takes account of limited information, search for buyers and sellers, and both buying and selling costs Negishi uses the model to show that John Law (1966 [1705] ) and all who followed him were wrong to interpret Locke’s

“vent” as “demand”; that Karen Vaughn (1980) was correct to interpret Locke’s price theory based on “vent” as the microeconomic foundation of his version of the Quantity Theory; and that Locke’s results are sensitive to the range of certain parameters There is, however, no connection between this piece of MM and anything that follows We have not been offered an account of a “progressive problem shift” (supposing that we have chosen to understand growth of know-ledge in terms of Lakatos’s methodology): simply a snapshot – or X-ray photo-graph – of what the Quantity Theory “scientific research programme” looked like

in 1691

Yet it would seem that MM is well suited to be a tool of RR Provided that we can model the theory of a dead economist by means of the symbols and functions that we had used to model that of his predecessors, we can show exactly what theoretical value added – if any – was contributed by the former Consider an extremely simple example Let us model Keynesian macroeconomic equilibrium, following J R Hicks (1937), as:

where i = r + d/dt(Pe) and expected price level, Pe = P, the actual price level assumed to be constant M and ∆ are shift parameters Let – ∞ ≤ ∂L/∂i ≤ 0 It is then obvious that equilibrium Y will be invariant with respect to ∆ if ∂L/∂i = 0.

Hicks regarded this as a “classical assumption,” and so was able to illustrate part

of Keynes’s claim to have “generalized” classical macroeconomics For the novel doctrine of “liquidity preference” makes the demand for money a decreasing

function of i, the bond yield (that is, ∂L/∂i < 0) In general, therefore, equilibrium

Y would be responsive to any change in the shift parameter ∆ In this example,

MM formulates precisely what Lakatos (1970, p 118) called a “theoretically pro-gressive problem shift,” and is therefore part of the rational reconstruction of a putative step forward in macroeconomic theory

Negishi’s MM of Locke’s value theory shows that MM can be a tool of exegesis.

We understand Locke’s text better as a result of Negishi’s model, and we can use the model to criticize and appraise the merely literary exegesis of other authors The Hicksian MM of Keynes’s macroeconomic theory shows that MM can be a

tool of RR We understand more clearly why Keynes claimed to have provided a

“general” theory that exhibited the results of what he called “classical” theory as

a special case It is possible to imagine an “internal” history of economic theoriz-ing based on a temporal sequence of related and commensurable mathematical models But before going further it is important to consider the relation of MM

and RR to history.

Economics is more akin to physics than it is to philosophy (and more like ecology than either) But it does resemble philosophy, especially political philo-sophy, in one important respect Its central ideas refer to human consciousness

in human society, and are formulated in a language of discourse generated by the entire range of “humane” studies available to its authors: theology, philosophy,

history, and the arts and letters In order to write a satisfactory intellectual

history (hereinafter, IH) of the production and significance of economic ideas, we must imagine ourselves eavesdropping upon a bygone conversation: inward with the language and literature, religion and politics, tastes and morals, of those

we are observing We put ourselves in the position of our subjects and look in the same direction, and with the same eyes, as they The fact that this is strictly impossible does not absolve the historian from the obligation to attempt it (Blaug, 1997, p 8) This sense of IH was classically enounced by Quentin Skinner (1969) However, Lakatos (1978, p 102, n 1) reported that in his discipline

“internal history” is usually defined as “intellectual history”; “external history” as

“social history.” In his own, “unorthodox, new demarcation,” what was meant

by “internal history” is simply RR as explained above But Lakatos’s “external history” is a mere residual and is not to be taken as IH in the Skinnerian sense necessary for a truly “historical” HET

It has lately been suggested (Waterman, 1998a) that HET be regarded as synonymous with “history of political economy,” and as a catch-all for any and all “historical” studies of economic thought Then IH (of economic thought) is a subset of HET (or, more generally, the intersection of IH in general with HET) concerned with the past “as it really was.” It includes both “historical

reconstruc-tion” and “Geistesgeschichte” in Rorty’s senses Each of these in turn includes, but

is not exhausted by, Lakatos’s “external history.” What is known as history of

economic analysis (hereinafter, HEA) is then what Blaug correctly describes as

Economic Theory in Retrospect (1997), which is “internal history” in Lakatos’s

“unorthodox, new” sense, or simply RR Its purpose is to trace the lines of descent

of leading analytic themes in economics, and account for their articulation HEA is like a topological diagram of the kind that helps us get from Uxbridge to Charing Cross on the London Underground But IH is like a genuine, scale map

of London We cannot superimpose the former on the latter without distorting geographical truth

It is important to note that just as a topological diagram can be a cartographic tool, so HEA can be a tool of IH For by attending to the internal logic of the conversation we are observing we enrich our understanding of what is being said and why Since it has been argued above, therefore, that MM can be a tool of

RR [= HEA], it follows that MM may have a part to play in IH as a component of

the relevant HEA And since it has also been shown that even when MM is not

used in RR it can be an exegetical tool, it may also have an “autonomous” use in

IH The complex relations among (1) IH, (2) HET, (3) HEA [= RR = “internal

history”], (4) “historical reconstruction,” (5) “Geistesgeschichte,” (6) MM, and (7)

“external history” may be illustrated in the Venn diagram of figure 33.1

Set (1) is IH in general, which of course includes much more (10) than simply the IH of economic thought (12) Set (2) is HET, some of which is IH (12 and its

Figure 33.1 The relations among mathematical modeling (MM), rational reconstruction (RR), intellectual history (IH), and the history of economic thought (HET) (1) Intellectual history (IH); (2) history of economic thought (HET); (3) rational reconstruction (RR) or history of economic analysis (HEA);

(4) historical reconstruction; (5) “Geistesgeschichte”; (6) mathematical modeling

(MM); (7) “external history”

(2)

20

(3)

23 236

26 (6)

(1)

(5)

(4)

12346

1246

(7)

1257 127 1247

subsets) and some not (20, 23, and 236) Both “historical reconstruction” (124)

and “Geistesgeschichte” (125) belong to the IH of HET (12) Set (3), which is HEA

= RR, intersects in part with IH (123, 1236, 1246, and 12346) and in part does not (23 and 236) Set (7) is Lakatos’s “external history” and belongs wholly in IH,

intersecting with “historical reconstruction” (1247), with “Geistesgeschichte” (1257), and with neither (127) Mathematical modeling is represented as set (6) It may be

merely an exegetical tool of HET (26); a tool of RR (236); an exegetical tool of

“historical reconstruction” (1246); a tool of RR used in the service of IH (1236); or either or both an exegetical tool and a tool of RR used as a tool of “historical reconstruction,” itself a component of the intellectual history of economics (12346)

In what follows, some attempt will be made to identify in terms of this taxonomy the historiographic function of the examples of MM reported and appraised

33.3 EXAMPLES OF MATHEMATICAL MODELING IN THE

HISTORY OF ECONOMIC THOUGHT

The earliest example of an author’s using MM in HET is that of William Whewell (1971b [1831] ), who produced a “mathematical exposition” of “some of the lead-ing doctrines” in Ricardo (1851 [1817] ) It was Whewell’s belief, expressed in an earlier lecture on mathematical economics, that:

Some parts of this science of Political Economy may be presented in a more systematic and connected form, and I would add, more simply and clearly, by the use of mathematical language than without such help; and moreover to those accus-tomed to this language, they may be rendered far more intelligible and accessible than they are without it (Whewell, 1971a [1829], p 1)

It is remarkable that Whewell identifies, in this passage, the most important claims advanced by Samuelson (1947, 1952) with respect to the use of mathem-atics in economics: (a) mathemmathem-atics is a “language”; (b) it is possible to translate

“some parts of Political Economy” (i.e., “the elements and axioms which are its materials,” Whewell, 1971a [1829], p 3) into “mathematical language”; (c) such translation may achieve the greater rigor of “a more systematic and connected form”; and (d) it presents economic theory “more simply and clearly,” so render-ing it “far more intelligible and accessible.”

It was Whewell’s purpose in the two lectures which comprised this article “to trace the consequences” of “the principles which form the basis of Mr Ricardo’s system,” and which Whewell himself believed to be without justification (1971b [1831], pp 2, 3) In this case, MM was simply an exegetical tool of HET (26) Although his mathematics was “awkward and sometimes simply ‘incorrect,’ ” Whewell is judged by a modern commentator to have discovered, through his mathematical reconstruction of Ricardo, several important analytic concepts usually attributed to the “marginal revolution” of the 1870s (Cochrane, 1970) A few years later, one of Whewell’s Cambridge colleagues, John Edward Tozer, published two papers on mathematical economics (Tozer, 1838, 1841 [1840] ), the

first of which applied MM to analytic work of Barton, Sismondi, M’Culloch, and Ricardo, with the object of criticizing their arguments (Gehrke, 2000)

Alhough important work in mathematical economics was published in the first half of the nineteenth century by Heinrich von Thünen (1966 [1826, 1850, 1863] and Augustin Cournot (1838), there appears to have been no further attempt at the MM of other authors’ work after Tozer’s until Alfred Marshall busied himself

in 1867 with translating “Ricardo’s reasoning into mathematics” (Keynes, 1972 [1933], p 181) Not only did Marshall never publish this work, however: he ex-erted his powerful influence to discourage mathematical methods in economics except as a preliminary and private, ground-clearing exercise For “ 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, 1952 [1890], p ix) Three generations of English-speaking economists, enthralled by Marshall’s doctrine, spent their time instead in the “laborious liter-ary working over of essentially simple mathematical concepts” (Samuelson, 1947,

p 6) Sophisticated and far-reaching developments in mathematical economics (e.g., Ramsey, 1928; Leontief, 1936; Neumann and Morgenstern, 1944; Neumann, 1945–6 [1938] ) went largely unnoticed by the profession Even the (elementary algebraic) general equilibrium theory of Walras (1954 [1874] ) was widely ignored until its popularization by J R Hicks (1939) And such HET as then existed (e.g., Roll, 1938; Gide and Rist, 1944 [1909] ) was entirely literary All this was changed

quite suddenly by the publication of Samuelson’s Foundations of Economic

Ana-lysis (1947)

From the parochial standpoint of this essay, the most important thing about

Foundations is that it contains – in passing and merely to illustrate a mathematical idea – what may be the first published example of genuine MM in HET since

Whewell and Tozer Samuelson (1947, p 297) noted that in his Essay on Population

(1798), “Malthus implicitly and explicitly assumed the law of diminishing (per capita) returns.” Hence we may write:

where f ≡ F/N is per capita real income, F is the total production of “the means

of subsistence” or “food,” and N is the total population Now equation (33.3) implies the (diminishing marginal returns) aggregate production function of “food”

in the agricultural economy assumed by Malthus and Ricardo:

F = F(N), F′ > 0, F″ < 0. (33.4) Equation (33.3) is a simple example of MM as an exegetical tool: a straight-forward translation into mathematical language of Malthus’s assumption (1798,

pp 25–6) that the ratio of “population” to “the means of subsistence” increases

as the former rises

But equation (33.4) may be regarded as the RR of a theoretically progressive problem-shift For it was a short step from Malthus’s original argument of 1798

to the implicit formulation in 1815 and 1817 of a diminishing-returns aggregate

production function (Malthus, 1815; Ricardo, 1815, 1951 [1817] ) It is also the starting point of a series of mathematical models of the “classical” theory of growth and distribution, beginning with Peacock (1952) One of the most inter-esting is that of Stigler (1952), in which it is shown that Malthus’s “ratios” imply

a production function of the form F = L ln N, where L is a shift parameter that

captures land availability, capital–labor ratios, and technique The concept of

an aggregate production function of the general form (33.4) has been much employed since the 1950s in MM of the “classical” authors, in particular Malthus and Ricardo L L Pasinetti’s (1960) influential formulation of “the Ricardian System” begins with (33.4) Samuelson’s (1959) two-part “Modern treatment” is chiefly concerned to employ linear-programming ideas to explore Ricardo’s value theory, but diminishing returns in agriculture enter the picture Samuelson (1959)

is genuine RR in that its object is to show that “Poor as our knowledge and insights are, they are way ahead of those of our predecessors” (1959, p 231) The most famous of all such production-function models, however, is Samuelson’s

“Canonical classical model of political economy” (1978) Inputs into the

pro-ductive process are “doses” of V, “made up of balanced proportions of L and K applied to a fixed vector of lands” (1978, p 1418); where L is labor (represented

as N in (33.4) above) and K capital By means of this model, we can see that

“Adam Smith, David Ricardo, Thomas Robert Malthus, and John Stuart Mill shared in common essentially one dynamic model of equilibrium, growth and distribution” (Samuelson, 1978, p 1415) The “canonical classical model” is clearly

a case of RR, for “within every classical economist there is to be discerned a modern economist trying to be born” (Samuelson, 1978, p 1415)

The production-function approach allows macroeconomic growth modeling, originally devised for neoclassical theory (Solow, 1956), to be extended to the classical authors by adding a fixed factor (land) and by making population growth endogenous (Swan, 1956, pp 340–2) This line of inquiry was pursued by Walter Eltis in a series of papers between 1972 and 1981, incorporated in his monograph (Eltis, 1984), which includes MM of the growth theory of Quesnay, Smith, Malthus, Ricardo, and Marx Eltis’s work inspired an important MM of “Malthus’s theory

of wages and growth” (Costabile and Rowthorn, 1985)

The concept of a diminishing-returns aggregate production function, so illu-minating when applied to the Malthus–Ricardo doctrine of rent, is less helpful

in the MM of other aspects of classical political economy In the first place, despite Samuelson’s (1978) assumption, not only Marx but also Adam Smith and his contemporaries ignored “the limitation of land and natural resources” in their analyses (Hollander, 1980; Waterman, 1999, 2001) Secondly, much light can be thrown on classical and pre-classical value theory and growth theory by the modeling assumption of many goods, produced interdependently by processes with constant inputs Thirdly, so far as Adam Smith at any rate is concerned, neglect of increasing returns to scale must seriously impair the exegetical cred-entials of any “Smithian” model

Few attempts, if any, have been made to deal rigorously with the third topic Smith’s analysis seems to assume the persistence of competition, which perva-sive IRS must undermine (however, see Negishi, 1989, pp 89–95) Only informal,