ABSTRACT
Complexity confounds our ability to comprehend completely the multiple layers of material and organic structures. We do not understand how mind emerges from this intricate evolution or how mind participates as creator in that evolution – except to give the source of novel thought and action a name: the intuition. But once armed with the concepts of number, measurement, causality and logic, reason can advance. To derive existence and stability properties of equilibria, economists usually ground theoretical work on the real number system. We implicitly assume that ‘real’ theory is close enough to the truth. In the rationals the task is more difficult, sometimes without answers at all. Still, it is a fact that all monetary and physical calculations in an actual market economy exploit the rationals or the integers (depending, for example, on whether you use a penny or a dollar as numeraire), a fact that most theorists ignore without a qualm. An exception occurred more than half a century ago when optimization theory was extended to the case of integers in order to solve the transportation problem. While considerable applied work was subsequently undertaken in this vein, few economists took note. In the meantime the construction of computer representations of empirical economic processes has gone on apace, living as it were in the rational world of digital numbers. Now a theorist of note has taken up the fundamental issue implicit in this distinction between ‘real’ economic theory and ‘rational’ computational economics. Kumaraswamy Velupillai, in a series of challenging articles and books, calls for a reconstruction of the discipline directly on a computational, numeric basis. Velupillai is a rare, philosophically oriented economist, concerned with the fundamental meaning of theory and the appropriateness of its methods. Central to his analysis is the distinction among the alternative methods of mathematical proof, in particular between constructive proof and proof by contradiction. The central implication of his argument is important: if you have ‘proved’ that something ‘exists’, say ‘A’, but you have no way of representing ‘A’, then inferences based on ‘A’ have no practical relevance. Consider the economist’s representation of economizing as the solution of a constrained optimizing problem, that is, as the choice or decision that provides the most preferred alternative among those that satisfy financial and physical
constraints. The choice alternatives include contemplated activities such as buying, selling, consuming and producing, while the constraints may involve cash, demand deposits, stocks of durable goods, legal restrictions and so forth. Consumers in a modern economy have a vast array of potential alternatives at any one time and a vast array of potentially relevant information to consider. In various ways they drastically reduce the complexity of the problem by concentrating on a few alternatives at any one time among all those that will arise over a lifetime. Indeed, the same is true for producers, however sophisticated. The more complex their business, the more drastically they must decompose their economizing activity into an interactive structure of individual managerial functions presented over time. In short, real world economizing involves relatively simple decision making and managerial activities that over time arrive at realized decisions to consume and produce. However simple or complex, alternative ways of thinking and acting arise. The ideas of feasible and best ways of acting emerge in each given situation which guides the agent through time. But termination of the process for arriving at action is required because a decision must be made. Common classes of mathematical programming models possess algorithmic sequences of arithmetic computations designed to produce or converge to an optimal solution. Such algorithms search – in practice – always among the rational numbers. In some cases a best, real-valued solution can be determined as ‘existing’ theoretically. This implies in effect that a solution is worth pursuing. But in many cases a best solution can only be approached in the sense that a better solution can be found but only ones for which improvement is negligible from a practical point of view. Especially, when significant nonlinearities and or large numbers of variables and constraints are involved, computation must be terminated short of a ‘best’ solution. If one accounts for the time and cost of reaching a ‘best’ decision, however, so-called ‘suboptimal decisions’ are actually optimal. In any case, real solutions can only be approximated. Indeed, real numbers only ‘exist’ as a limiting concept. The economic theorist is not usually concerned with solving particular individual consumption or production problems. His problem is to characterize their solution for the purpose of further theorizing: to explain how the actions of individual households and firms add up to market aggregates on the basis of which the operation of the various sectors or the economy as a whole can be understood. For this, it is the outcome of algorithmic behaviour, not that computational behaviour itself that is his building block. Any of his results carried out in the real domain are convenient abstract characterizations only of the actual phenomena of interest that of necessity exist practically only in the rational numbers. Let us turn to the analysis of the market economy as a whole, that is, to the Walrasian formulation of general equilibrium in the form based directly on the commodity supply and demand functions. These functions characterize the potential outcomes of the collective of individual households’ and firms’ algorithmic decisions given a system of commodity prices. They shift attention from
the behaviour of individuals to the existence of market clearing prices. The algorithmic behaviour of the participants is subsumed. Velupillai, here again, makes an important point. The equilibrium existence theory à la Arrow-Debreu is not algorithmic. The proofs are all carried out in the real number domain based on non-constructive, fixpoint arguments. Here, however, we should also recognize that Walras did also specify an algorithmic representation of market clearing. It is governed by a ‘tâtonnement process’: a sequence of prices each one in the sequence chosen as an adjustment over its predecessor by reducing its value if its corresponding product was in a surplus and increasing it for those for which demand exceeded surplus. Most midtwentieth century mathematical economists analysed this simple algorithmic scheme. As Goodwin somewhere pointed out, Walras thought of it as a way to prove existence. Here again, most theorists are not interested in the algorithmic modes by which households and firms arrive at their respective decisions, but rather the properties of the final outcome of those decisions. They implicitly assume that the final outcome can be characterized as the solution to a constrained optimization problem – without worrying about the manner in which that problem has been solved. But suppose now that the tâtonnement process does not converge in principle or, even if it does, that it cannot converge in finite time, so that the economy is out-of-equilibrium all the time? Then some commodities will be in short supply, others exhibiting a surplus. The process of adjustment will still continue to function out of equilibrium, just as it does in reality all the time. From this point of view it is the tâtonnement process itself that must be of greatest interest, not its potential equilibrium. Much of what has been discussed above is a kind of defence of standard ‘real’ but potentially computational economic theory, classic ideas from which we learn something useful: that it contains some algorithmic content and in itself provides a coherent partial understanding of how – in principle – a decentralized private ownership economy is coordinated. One must remember the fundamental problem faced by Adam Smith and that continued to be posed again and again in the succeeding centuries with ever greater symbolic formality: ‘Can a system of prices coordinate the exchange of consumption and production goods among a completely decentralized collection of households and firms?’ General equilibrium theory finally provided the first completely rigorous answer to this question, one that is purely formal to be sure and not algorithmic. It does not describe how an economy can potentially solve this problem. Walras’ tâtonnement does solve it in the sense of producing a sequence of adjustments that – given the right regularity conditions and given the stationarity of the technology and preferences of the market participants – could approximate a competitive equilibrium ever more closely – and that is the only sense in which real numbers exist. It is important to understand that the convergence of a dynamic process in general is a limiting concept. Consider the simplest Keynesian model, y = αy + j + g where α is the marginal propensity to consume, 0 < α < 1, y is income, αy is consumption, j is investment, and g is government expenditure; all in per capita
terms and all rational. The solution in terms of income yields the familiar multiplier equation for equilibrium income,
y~ = 1_____1 – α ( j + y), (12.1)
which will be rational. However, consider the algorithmic dynamic process,
yt+1 = αyt + j + g.
