Abstract
Philosophers of science regard the Lotka–Volterra model as an exemplar of model transfer across disciplines. This article traces three cases of the Lotka–Volterra model transfer to economics during the 1960–70s. Each model represents a different kind of methodological attitude towards model transfer. After detailing the historical case studies where the Lotka–Volterra model was transferred to economics and how the economists actually adopted it into their model constructions, the following philosophical discussions on interpretation and justification suggest that formal template or structure alone does not produce successful transdomain model transfer; rather it is subject to the model builders’ intentions and interpretation. In addition to interdisciplinary transfer, intradisciplinarity is also crucial to how models are constructed in order to be resituated in their economic context.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
See May (1973, Ch. 2) for the notions and the mathematical treatments of stability in biology.
As stated in a letter from Le Corbeiller to Goodwin in 1958 (reproduced in Velupillai, 1990, pp. 20–23).
Another detailed narrative of mechanism was given in Goodwin et al. (1984): “[t]he share of wages is determined by the reserve army of labour, or, more precisely, by the employment ratio. The pace of capital accumulation determines the demand for labour. If the rate of accumulation is rising sufficiently, so does the employment ratio. Beyond a certain value of the employment ratio (i.e., in the neighborhood of full employment) the vigorous accumulation of capital leads to a rising real wage and a rising wage share. This process goes on until the rise in the wage share is sufficient to reduce the rate of profit to the point where the accumulation rate slows down and unemployment begins to rise. The replenishment of the reserve army of labour yields a falling wage share, therefore a rising rate of profit and eventually an upturn in the rate of accumulation” (Goodwin et al., 1984, pp. ix–x).
There is a third interpretation of the Lotka–Volterra model transfer to the Goodwin model in the philosophy of science literature. Weisberg (2013), without realizing the actual history of the Goodwin model, focuses on interpreting the model in the new domain—“construal” in his own words. Weisberg argues that there are three possible ways of interpreting how Goodwin saw the similarity between Lotka–Volterra and his own models, which Goodwin himself had characterized as “purely formal, but not entirely so”. Goodwin recognized the similarity in mathematical structures, realizing that the construals are different therefore the models are not identical, and understanding that the two models share a “higher-order similarity”, indicating the analogy between prey-predator and worker-capitalist relationships. Weisberg concludes that the Goodwin model is an “especially clear illustration of how the same [mathematical] structure can become a different model with a different construal”, while at the same time the model is an “unusual, extreme case” in which “a mathematical structure is completely reinterpreted in a new domain” (ibid.).
Samuelson termed it the “Verhulst–Pearl logistic law of population growth” (1972b [1967], p. 480) and the “Malthus-logistic law” (1972a [1965], p. 435), to underline that Malthus had already presented the idea of diminishing returns, even though his population theory has been characterized (as it is currently) as the exponential curve.
That is, the logistic curve is treated as a special case of the Lotka–Volterra model. Whereas the former describes the growth of a population that is limited by finite resources, the latter describes the competition between two populations of predators and prey.
This is parallel to Humphreys’ (2004, pp. 77–79) problems of “template correction”, and Knuutila and Morgan’s (2019) account of deidealization. Humphreys is concerned with the assumptions of template construction (“construction assumptions”) comprising four categories: ontology, idealization, abstraction and constraints. Similarly, Morgan and Knuuttila consider four categories: recomposition, reformulation, concretization, and situation.
Solow attended Goodwin’s economics course at Harvard in 1940 as an undergraduate. He credited Goodwin for being the inspiration for his future economic studies.
However, Humphreys and Suppes are different in that Suppes’ position was a model-theoretic structuralist position that explicitly argued that the specific syntactic representation used in a model needed to be abstracted from. Humphreys explicitly argued for a central role for the mathematical syntax in understanding computational templates and their transfer (Humphreys, 2004, pp. 59, 95–100). I thank the referee for pointing this out to me.
Knuuttila and Loettgers’ (2016b) usage of inter- and intradisciplinary transfer is restricted in indicating template transfer across and within the disciplines, respectively. For example, the interdisciplinary model transfer of the Ising model is instantiated by the transfer from physics to Thomas Schelling’s (1969) segregation model in economics. In contrast, intradisciplinary transfer indicates the application of the same mathematical structure within the same domain, such as the transfer of the Ising model to the Lattice–Gas model (Kunnttila and Loettgers, 2016b, p. 389).
One possible way to provide a clear picture of how a model is constructed is to use Boumans’ (1999) receipt account, in which a model is represented by a diagram containing all its equations and identifying the sources of all its ingredients. Of course such a work requires extensive knowledge of the history and methodology of the specific model, and the work of identification gets harder as the model becomes more complicated.
See, for example, Hendry (2011) for the methodology of empirical economic modeling.
However, Goodwin then expressed skepticism, citing Le Corbeiller and arguing that the model yields only a conservative system which is not a proper explanation of the persistence of cycles (Goodwin, 1982, p. ix). On this view, Goodwin was in agreement with Samuelson’s early discovery.
Houkes and Zwart (2019) would call it creative interpretation.
See Lin (2022) for a discussion on the travel of justification (what Lin calls a spillover) with an incoming formalism.
References
Akerlof, G. A. (1967). Stability, marginal products, putty and clay. In K. Shell (Ed.), Essays in the theory of optimal economic growth (pp. 281–294). MIT Press.
Akerlof, G. A., & Stiglitz, J. E. (1969). Capital, wages and structural unemployment. Economic Journal, 314, 269–281.
Boumans, M. (1992). A case of limited physics transfer: Jan Tinbergen’s resources for re-shaping economics. PhD dissertation, University of Amsterdam.
Boumans, M. (1993). Paul Ehrenfest and Jan Tinbergen: A case of limited physics transfer. In N. De Marchi (Ed.), Non-natural social science: Reflecting on the enterprise of More Heat than Light (pp. 131–156). Duke University Press.
Boumans, M. (1999). Built-in justification. In M. S. Morgan & M. Morrison (Eds.), Models as mediators: Perspectives on Natural and social science (pp. 66–96). Cambridge University Press.
Boumans, M., & Davis, J. B. (2015). Economic methodology: Understanding economics as a science. Macmillan.
Di Matteo, M. (1988). Goodwin and the evolution of a capitalistic economy: An afterthought. In G. Ricci & K. Velupillai (Eds.), Growth cycles and multisectoral economics: The Goodwin tradition (pp. 93–101). Springer.
Di Matteo, M., Filippi, F., & Sordi, S. (2006). The confessions of an unrepentant model builder’: Rummaging in Goodwin’s Archive. Structural Change and Economic Dynamics, 17, 400–414.
Gandolfo, G. (2008). Giuseppe Palomba and the Lotka–Volterra equations. Rendiconti Lincei, 19, 347–357.
Goodwin, R. M. (1967). A growth cycle. In C. H. Feinstein (Ed.), Capitalism and economic growth: Essays presented to Maurice Dobb (pp. 54–58). Cambridge University Press.
Goodwin, R. M. (1972). A growth cycle. In E. K. Hunt & J. G. Schwartz (Eds.), A critique of economic theory (pp. 442–449). Penguin Education.
Goodwin, R. M. (1982). Preface. In R. M. Goodwin (Ed.), Essays in economic dynamics (pp. vii–x). Macmillan.
Goodwin, R. M. (1986). The M-K-S system: The functioning and evolution of capitalism. In H. J. Wagener & J. W. Drukker (Eds.), The economic law of motion of modern society: A Marx-Keynes-Schumpeter Centennial (pp. 14–21). Cambridge University Press.
Goodwin, R. M., Krüger, M., & Vercelli, A. (1984). Introduction. In R. M. Goodwin, M. Krüger, & A. Vercelli (Eds.), Nonlinear models of fluctuating growth (pp. vii–xvii). Springer.
Gordon, S. (1989). Darwin and political economy. Journal of the History of Biology, 22(3), 437–459.
Harcourt, G. C. (1985). A twentieth-century eclectic: Richard Goodwin. Journal of Post Keynesian Economics, 7(3), 410–421.
Hendry, D. F. (2011). Empirical economic model discovery and theory evaluation. Rationality Markets and Morals, 2, 115–145.
Houkes, W., & Zwart, S. D. (2019). Transfer and templates in scientific modelling. Studies in History and Philosophy of Science, 77, 93–100. https://doi.org/10.1016/j.shpsa.2017.11.003.
Humphreys, P. (2004). Extending ourselves: computational science, empiricism, and scientific method. Oxford University Press.
Humphreys, P. (2019). Knowledge transfer across scientific disciplines. Studies in History and Philosophy of Science, 77, 112–119.
Kaldor, N. (1955). Alternative theories of distribution. The Review of Economic Studies, 23(2), 83–100. https://doi.org/10.2307/2296292.
Kingsland, S. E. (1985). Modeling nature: episodes in the history of population ecology. University of Chicago Press.
Knuuttila, T., & Loettgers, A. (2012). The productive tension: Mechanisms versus templates in modeling the phenomena. In P. Humphreys & C. Imbert (Eds.), Models, simulations and representation (pp. 3–24). Routledge.
Knuuttila, T., & Loettgers, A. (2014). Magnets, spins, and neurons: The dissemination of model templates across disciplines. The Monist, 97(3), 280–300. https://doi.org/10.5840/monist201497319.
Knuuttila, T., & Loettgers, A. (2016a). Contrasting cases: The Lotka–Volterra model times three. In T. Sauer & R. Scholl (Eds.), The philosophy of historical case studies (pp. 151–178). Springer. https://doi.org/10.1007/978-3-319-30229-4_8
Knuuttila, T., & Loettgers, A. (2016b). Model templates within and between disciplines: From magnets to gases – and socio-economic systems. European Journal for Philosophy of Science, 6, 377–400. https://doi.org/10.1007/s13194-016-0145-1
Knuuttila, T., & Loettgers, A. (2017). Modelling as indirect representation? The Lotka–Volterra model revisited. British Journal for the Philosophy of Science, 68, 1007–1036.
Knuuttila, T., & Morgan, M. S. (2019). Deidealization: No easy reversals. Philosophy of Science, 86(4), 641–661. https://doi.org/10.1086/704975.
Le Corbeiller, P. (1933). Les systemes autoentretenus et les oscillations de relaxation. Econometrica, 1(3), 328–332. https://doi.org/10.2307/1907044.
Lin, C. H. (2022). Knowledge transfer, templates, and the spillovers. European Journal for Philosophy of Science. https://doi.org/10.1007/s13194-021-00426-w
Lotka, A. J. (1925). Elements of physical biology. Williams and Wilkins.
May, R. M. (1973). Stability and complexity in model ecosystems. Prinction University Press.
Mirowski, P. (1984). The role of conservation principles in twentieth-century economic theory. Philosophy of the Social Sciences, 14, 461–473.
Mirowski, P. (1989). More heat than light: Economics as social physics, physicas as nature’s economics. Cambridge University Press.
Morgan, M. S. (2011). Travelling facts. In P. Howlett & M. S. Morgan (Eds.), How well do facts travel? The dissemination of reliable knowledge (pp. 3–39). Cambridge University Press.
Morgan, M. S. (2014). Resituating knowledge: Generic strategies and case studies. Philosophy of Science, 81(5), 1012–1024. https://doi.org/10.1086/677888.
Phillips, A. W. (1958). The relation between unemployment and the rate of change of money wage rates in the United Kingdom, 1861–1957. Economica, 25(100), 283–299. https://doi.org/10.2307/2550759.
Samuelson, P. A. (1947). Foundations of economic analysis. Harvard University Press.
Samuelson, P. A. (1971). Generalized predator-prey oscillations in ecological and economic equilibrium. Proceedings of the National Academy of Sciences of the United States of America, 68(5), 980–983.
Samuelson, P. A. (1972a). Some notions on causality and teleology in econonics. In R. C. Merton (Ed.), The collected scientific papers of Paul A. Samuelson (Vol. III, pp. 428–472). MIT Press.
Samuelson, Paul A. (1972b). A universal cycle? In R. C. Merton (Ed.), The collected scientific papers of Paul A. Samuelson (Vol. III, pp. 473–486). MIT Press.
Samuelson, P. A. (1998). How foundations came to be. Journal of Economic Literature, 36(3), 1375–1386.
Samuelson, P. A., & Solow, R. M. (1960). Analytical aspects of anti-inflation policy. The American Economic Review, 50(2), 177–194.
Solow, R. M. (1962). Substitution and fixed proportions in the theory of capital. The Review of Economic Studies, 29(3), 207–218. https://doi.org/10.2307/2295955.
Solow, R. M. (1990). Goodwin’s growth cycle: reminiscence and rumination. In K. Velupillai (Ed.), Nonlinear and multisectoral macroeconomics (pp. 31–41). Macmillan.
Solow, R. M., & Stiglitz, J. E. (1968). Output, employment, and wages in the short run. The Quarterly Journal of Economics, 82(4), 537–560. https://doi.org/10.2307/1879599.
Stiglitz, J. E. (2002). Biographical. Retrieved March 8, 2022, from https://www.nobelprize.org/prizes/economic-sciences/2001/stiglitz/biographical/.
Suppes, P. (2002). Representation and invariance of scientific structures. CSLI Pub.
Swoyer, C. (1991). Structural representation and surrogative reasoning. Synthese, 87(3), 449–508. https://doi.org/10.1007/BF00499820.
Velupillai, K. (1990). Nonlinear and multisectoral macroeconomics. Macmillan.
Velupillai, K. (2015). Richard Goodwin: The indian connection. Economic and Political Weekly, 50(15), 80–84.
Vercelli, A. (1988). Goodwin and the M-K-S system. In G. Ricci & K. Velupillai (Eds.), Growth cycles and multisectoral economics: The Goodwin tradition (pp. 102–111). Springer.
Weisberg, M. (2013). Simulation and similarity: Using models to understand the world. Oxford University Press.
Acknowledgements
I am grateful to Chia-Hua Lin for the invitation to contribute to the special issue “Transdisciplinary Model and Template Transfer”. I would also like to thank the participants of the workshop “Transdisciplinary Model Transfer and its Interfaces”, especially the late Paul Humphreys, and anonymous reviewers for their constructive comments.
Funding
Funding was supported by Ministry of Science and Technology, Taiwan (Grant No. 108-2420-H-007 -012 -MY5).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares no conflicts of interest.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Chao, HK. Three kinds of the Lotka–Volterra model transfer from biology to economics. Synthese 202, 124 (2023). https://doi.org/10.1007/s11229-023-04341-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11229-023-04341-w