Bridging concepts: gradient methods
Importantly, parallel to the general equilibrium theorizing neoclassical economists paid much attention to the development of mathematical programming. It was primarily Samuelson (1947 Samuelson, P. A. (1947). Foundations of economic analysis. Cambridge, MA: Harvard University Press.
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) who has to be given credit for formulating the economic problem in the spirit of Robbins by formalizing it as a constrained optimization problem (Rizvi, 2003 Rizvi, S. A. T. (2003). Postwar neoclassical microeconomics. In W. J.Samuels, J. E.Biddle, & J. B.Davis (Eds.), Blackwell companion to the history of economic thought (pp. 377–394). Oxford: Blackwell.
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). Indeed, general equilibrium models could be regarded as dealing with combinations of optimization problems and with coordinating individual consumption and production plans.
Along with formulating the models that defined the solutions of optimization problems economists and operations researchers8 8. Mathematical programming was institutionalized in the postwar American economics mainly as operations research implying multiple applications beyond the domain of economics proper.
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started to look for implementation mechanisms that could help achieve these solutions. In fact, what was proposed were calculative devices supporting theoretical solutions. This was quite clear in the frequently drawn analogy between the working of a price system and a computer (Lee, 2006 Lee, K. S. (2006). Mechanism design theory embodying an algorithm-centered vision of markets/organizations/institutions. History of Political Economy, 38(Suppl. 1), 283–304. 10.1215/00182702-2005-026.
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; Mirowski, 2002 Mirowski, Ph. (2002). Machine dreams: Economics becomes cyborg science. New York, NY: Cambridge University Press.
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). What we would like to claim is that in case of economics these calculative devices were, in fact, institutionally interpreted and transformed into something similar to what Callon (2007 Callon, M. (2007). What does it mean to say that economics is performative? In D.MacKenzie, F.Muniesa, & L.Siu (Eds.), Do economists make markets? On the performativity of economics (pp. 311–357). Princeton, NJ: Princeton University Press.
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) labeled as ‘sociotechnical arrangements’ – a socially organized techniques of implementation.9 9. Interestingly, Callon (2007 Callon, M. (2007). What does it mean to say that economics is performative? In D.MacKenzie, F.Muniesa, & L.Siu (Eds.), Do economists make markets? On the performativity of economics (pp. 311–357). Princeton, NJ: Princeton University Press.
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, p. 320f.) talks about adjustment – the term very frequently invoked by stability theorists and mechanism design scholars.
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This logic was quite well discernible in how the discipline of mathematical programming defined itself. In one important source, it was referred to as ‘the construction of a schedule of actions by means of which an economy, organization, or other complex of activities may move from one defined state to another, or from a defined state toward some specifically defined objective’ (Dantzig & Wood, 1951 Dantzig, G. B., & Wood, M. K. (1951). The programming of interdependent activities: general discussion. In T. C.Koopmans (Ed.), Activity analysis of production and allocation (pp. 15–18). New York, NY: Wiley.
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, p. 15, see also Kjeldsen, 2000 Kjeldsen, T. H. (2000). A contextualized historical analysis of the Kuhn–Tucker theorem in nonlinear programming: The impact of World War II. Historia Mathematica, 27, 331–361. 10.1006/hmat.2000.2289.
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). This algorithmic vision was tightly connected to the idea of adjusting reality toward a normatively given objective.
Hurwicz (1973 Hurwicz, L. (1973). The design of mechanisms for resource allocation. American Economic Review, 63, 1–30.
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) provides a helpful classification of these calculative procedures. For the linear case, the simplex method – introduced in 1947 by the mathematician Dantzig (1951 Dantzig, G. B. (1951). Maximization of a linear function of variables subject to linear inequalities. In T. C.Koopmans (Ed.), Activity analysis of production and allocation (pp. 339–347). New York, NY: Wiley.
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),1010. Dantzig (1951 Dantzig, G. B. (1951). Maximization of a linear function of variables subject to linear inequalities. In T. C.Koopmans (Ed.), Activity analysis of production and allocation (pp. 339–347). New York, NY: Wiley.
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, p. 339), in turn, acknowledges that ‘the general nature of the “simplex” approach… was stimulated by discussions with Leonid Hurwicz’.
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an advisor of US Air Force and consultant of the Research and Development (RAND) corporation, – is the most important technique. It redefines a solution of a linear programming problem as a search of extreme points on a polytope and involves moving along its edges in the direction determined by the function to be optimized. Dantzig's method was enthusiastically received in the Cowles commission as a technique that would finally allow to circumvent the non-computability issue that had been haunting the whole business of solving concrete allocation problems in the 1940s (Erickson et al., 2013 Erickson, P., KleinJ. L., Daston, L., Lemov, R., Sturm, Th., Gordin, M. D. (2013). How reason almost lost its mind. The strange career of Cold War rationality. Chicago, IL: University of Chicago Press.
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).
Nonlinear unconstrained optimization required similar implementation technologies, and it was Arrow and Hurwicz who were fulfilling the task. Note that it is not that common in the histories of general equilibrium theory to refer to the other work of Arrow and Hurwicz, done before their joint papers on stability and somehow underlying it. This earlier technical work was devoted to the so-called gradient methods in mathematical programming. According to Hurwicz's recollection, ‘it was natural to interpret the dynamics of programming as a certain kind of mechanism for resource allocation’ (Feiwel, 1987 Feiwel, G. (Ed.). (1987). Arrow and the ascent of modern economic theory. Houndmills: Macmillan.
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, p. 259).
Gradient method refers to the way of finding a solution of an optimization problem, or, in other words, the technology of implementing an optimum. In the simplest formulation, it involves moving toward the optimum in the steepest way, which means to move along the gradient of the function optimized.
What is striking in the whole business of applying gradient methods is its affinity to the socialist way of issuing commands and providing concrete algorithms for action.1111. To be sure, the pervasiveness of simple rules and algorithms can be also seen as a basic element of the Cold War rationality, as it is shown by Erickson et al. (2013 Erickson, P., KleinJ. L., Daston, L., Lemov, R., Sturm, Th., Gordin, M. D. (2013). How reason almost lost its mind. The strange career of Cold War rationality. Chicago, IL: University of Chicago Press.
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). They also make the case for the rules being relevant both as description and prescription, thus rendering the very idea of rationality inherently normative.
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Thus, Arrow and Hurwicz (1957 Arrow, K. J., & Hurwicz, L. (1957). Gradient methods for constrained maxima. Operations Research, 5, 258–265. 10.1287/opre.5.2.258.
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) provide economic interpretations of the gradient systems by invoking a firm, which changes the scale of one of its activities and manipulates various parameters in order to achieve the optimal state. Usual analysis of the conditions of convergence is also given. At that time, Arrow and Hurwicz (1960a Arrow, K. J., & Hurwicz, L. (1960a). Stability of the gradient process in n-person games. Journal of the Society for Industrial and Applied Mathematics, 8, 280–294. 10.1137/0108016.
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) still believed that gradient method could also become a simple and universal computational technique. In general, ‘the focus was… on the parallelism between market processes and their stability on one hand, and the convergence of iterative computational procedures on the other’ (Feiwel, 1987 Feiwel, G. (Ed.). (1987). Arrow and the ascent of modern economic theory. Houndmills: Macmillan.
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, p. 272).1212. Arrow's (1974 Arrow, K. J. (1974). Limited knowledge and economic analysis. American Economic Review, 64, 1–10.
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) enthusiasm persisted in the 1970s as well: ‘[W]ith the development of mathematical programming and high-speed computers, the centralized alternative no longer appears preposterous. After all, it would appear that one could mimic the workings of a decentralized system by an appropriately chosen centralized algorithm’ (p. 5).
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But since Samuelson (1947 Samuelson, P. A. (1947). Foundations of economic analysis. Cambridge, MA: Harvard University Press.
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) a constrained optimization technique has also gained significance as a tool for economic analysis. Again, Hurwicz (1973 Hurwicz, L. (1973). The design of mechanisms for resource allocation. American Economic Review, 63, 1–30.
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) notes that the Kuhn–Tucker theorem that associated a constrained maximum with a saddle point of the corresponding Lagrangian can be seen as a result inviting to apply gradient methods to finding this solution as well. |