Theory versus application: does complexity crowd out evidence?

AuthorCoelho, Philip R.P.

JEL Classification: All

  1. Introduction

    Winnowing theories by appeals to evidence is a practice that dates to the beginnings of modern science. Here we test the hypothesis of Donald F. Gordon (1955) that complex mathematical statements are less operational than other economic statements. Operationalism means that non-selfreferential evidence has a dominant role in the assessment of theories. Mathematical "'proofs" of lemmas and theorems are self-referential and are generally nonoperational; (1) the 'proven" theorems may or may not be operational. Gordon, echoing concerns raised in 1920 by Alfred Marshall (1964) (2) about the use of mathematics in economics, argued that

    ... the essential point is the difference between theories using a large number of functions and those using one or two, since formal and mathematical reasoning is normally required when the number of relationships simultaneously being considered becomes large. As we have seen, even though each may be quite plausible, a combination of very many will rarely be so. Consequently, it happens that the cases in which formal and mathematical reasoning is most likely to be required are precisely the cases in which, for other reasons, the validity of any conclusions is likely to be conjectural. It is frustrating but nevertheless true that, where mathematics is most likely to be useful, the theory is least likely to be valid, while, where the theory is most likely to be true, complex deduction is generally not needed (p. 160). Gordon argued that the realms of mathematics and real-world economic behavior are not identical. Operational propositions are less likely to arise from protracted mathematical formalism. He used an example of a theory relating three variables x, y, and z to illustrate:

    Again, the relationship between x and y may be stable long enough for a shift along that function but not stable long enough for a shift along that function plus a subsequent shift along another [z] (p. 155). Expressed more formally, let the relationship between x and y be expressed as y = f(x), and that between y and z be z = g(y). Substituting f(x) into the second expression, a composite function, z = g[f(x)], is obtained. Differentiating the composite yields

    (1) dz/dx = (dz/dt) x (dy/dx).

    Equation 1 expresses the impact of a change in x upon z as an indirect effect; a change in x leads to a change in y, and the change in y then leads to a change in z. Mathematical conventions assume that the indirectness of the effect of x upon z is irrelevant. It is irrelevant because units of measure such as historic time do not exist in pure mathematics. But if a mathematical technique is used to represent a real-world situation in which it takes time for a change in one variable to affect another, then the functional relationship may be devoid of practical application. Gordon emphasized that economic phenomena are time dependent; the more functions that were linked in a theory, the more likely it is that the passage of time will materially affect the relationships in ways that are inherently unpredictable. Gordon saw the timelessness implicit in mathematics as an impediment to operationalizing complex relationships between and among variables in economic models. (3)

    Our analysis of the Gordon hypothesis extends the literature that has brought the content of published journal articles and citation data to bear on issues in the history of economic thought. In a classic article, George J. Stigler (1969, p. 229-30) concluded

    Economics ... has a useful past, a past that is useful in dealing with the future. Many useful commodities and services are not produced in society because they are worth less than they cost: it remains the unfulfilled task of the historians of economics to show that their subject is worth its cost. Since Stigler, the literature has provided practical reasons for studying the history of economics. In an analysis of the citations of "great" economists, Gary Anderson, David Levy, and Robert Tollison (1989, p. 182) showed that although "a considerable number" of the listed economists had little connection to the "living" literature, "a fair number of pre-twentieth century economists have impressive citation counts ... What Ricardo, Marx, and Smith, et al. may not have been able to solve may be what is most important about their work for contemporary economists." In another article, David Laband and Robert Tollison (2000) quantified aspects of intellectual collaboration in economics; one example was a positive relationship between the probability of coauthorship and the frequency of "'equations, tables, figures, and appendices" (p. 641). Finally, Laband, Tollison, and Karahan (2002) conducted a content analyses for The American Economic' Review publications that produced insights into editorial quality control, the decline of commentary, and rent seeking by authors.

  2. Evidence on Gordon's Hypothesis

    The Gordon hypothesis is that complex mathematical statements are less likely to be operational relative to other economic statements: We offer evidence on this proposition. (4) We use data from the JSTOR (Journal Storage) archive for 1963 through 1996. (5) We collected data on four general interest economic journals in the archive: The American Economic Review (AER), The Economic Journal (EJ), The Journal of Political Economy (JPE), and The Quarterly Journal of Economics (QJE), as well as The Journal of Economic History (JEH). We included the JEH because it is empirically oriented: we wanted to observe how a journal that emphasizes real-world applications compared to the general interest journals. The Social Sciences Citation Index provided data for our citation analysis. We used EViews3 and EViews5 software.

    Trends in Theoretical Complexity

    We examine the trends of mathematical complexity in the literature to assess the importance of the Gordon hypothesis. If trends in the publication of mathematically complex papers are constant or declining, then the Gordon hypothesis is relatively less important than if the trends turned out to be increasing. Toward an assessment of trends in complexity, we conducted an annual full-text search in our sample journals for either of the terms "multiple equilibria," or "lemma." Articles found to contain either (or both) of these temps were viewed as being more mathematically complex than those that did not contain them.

    The terms "lemma" and "'multiple equilibria" were selected as proxies because they are indicative of mathematical complexity, and because their initial usages in the journals in JSTOR were in 1910 and 1934, respectively. These terms were used infrequently prior to 1963, but they were in usage. (6) We did not "clean" the data...

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