Algorithmic foundations of computable general equilibrium theory

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Abstract

A constructive and recursion theoretic analysis of the standard computable general equilibrium (CGE) model of economic theory is undertaken. It is shown, contrary to widely expressed views and textbook versions of the CGE model, that the standard CGE model is neither computable nor constructive in the strict mathematical senses.

Introduction

One of the great achievements of mathematical economics in the 20th century was the Walrasian economic equilibrium existence proof of Arrow and Debreu [2]. It is listed as the 7th of 10 significant1 achievements in applied mathematics in Piergiorgio Odifreddi’s overall list of the 30 great solved problems of The Mathematical Century [13]. Its extension to dynamics is listed as the 8th of 18 problems for the 21st century—in ‘Hilbertian mode’—by Steve Smale [19]. Given its undoubted and acknowledged significance in the intellectual canvas of 20th century mathematical economics, economic theory and applied mathematics, it is not surprising that attempts have been made, most notably by Herbert Scarf, to devise algorithmic methods to compute Arrow–Debreu equilibria. These attempts have resulted in the development of the (almost) independent discipline of computable general equilibrium (CGE) theory. It will not be an exaggeration to claim that, till Scarf’s pioneering work on CGE theory and modelling, the Arrow–Debreu achievements remained in the realm of pure-theory – whether of economics or mathematics; after Scarf, it is, surely, also a significant chapter in applied mathematics.2

The economic foundations of CGE models lie in Uzawa’s equivalence theorem ([22], [7], p. 719, ff); the mathematical foundations are underpinned by topological fix point theorems (Brouwer, Kakutani, etc.). The claim that such models are computable or constructive rests on mathematical foundations of an algorithmic nature: i.e., on recursion theory or some variety of constructive mathematics. It is a widely held belief that CGE models are both constructive and computable.3 That the latter property is held to be true of CGE models is evident even from the generic name given to this class of models; that the former characterization is a feature of such models is claimed in standard expositions and applications of CGE models. For example in the well known, and pedagogically elegant, textbook by two of the more prominent advocates of applied CGE modelling in policy contexts, John Shoven and John Whalley [18], the following explicit claim is made:

“The major result of postwar mathematical general equilibrium theory has been to demonstrate the existence of such an equilibrium by showing the applicability of mathematical fixed point theorems to economic models. …Since applying general equilibrium models to policy issues involves computing equilibria, these fixed point theorems are important: It is essential to know that an equilibrium exists for a given model before attempting to compute that equilibrium.

The weakness of such applications is twofold. First, they provide non-constructive rather than constructive proofs of the existence of equilibrium; that is, they show that equilibria exist but do not provide techniques by which equilibria can actually be determined. Second, existence per se has no policy significance. … Thus, fixed point theorems are only relevant in testing the logical consistency of models prior to the models’ use in comparative static policy analysis; such theorems do not provide insights as to how economic behavior will actually change when policies change. They can only be employed in this way if they can be made constructive (i.e., be used to find actual equilibria). The extension of the Brouwer and Kakutani fixed point theorems in this direction is what underlies the work of Scarf … on fixed point algorithms….” [16, p. 12, 20-1; italics added]

However, in Scarf’s classic book of 1973 there is the following characteristically careful caveat to any unqualified claims to constructivity of the algorithm he had devised:

“In applying the algorithm it is, in general, impossible to select an ever finer sequence of grids and a convergent sequence of subsimplices. An algorithm for a digital computer must be basically finite and cannot involve an infinite sequence of successive refinements. ……. The passage to the limit is the nonconstructive aspect of Brouwer’s theorem, and we have no assurance that the subsimplex determined by a fine grid of vectors on S contains or is even close to a true fixed point of the mapping.” [16, p. 52; italics added]

The main goal4 in this paper is to sort out this and other ambiguities by clarifying the precise roles played by computability and constructivity in the theory of CGE models. The paper is organized as follows. In Section 2 the Uzawa equivalence theorem is analysed from the point of view of computability theory. In Section 3, the (non-) constructive content of the combinatorial proof of the Brouwer fix point theorem is made explicit and other, related, issues on constructivity in proofs of this theorem are also discussed. Following this, in the brief concluding section, further clarifying remarks on the mathematical and algorithmic foundations of the theory of CGE models are made, together with some suggestions on going beyond reliance on topological fix point theorems in the proof of equilibrium existence.

Section snippets

Uncomputability and undecidability in the Uzawa equivalence theorem

The Uzawa equivalence theorem is the fulcrum around which the theory of CGE modelling revolves. This key theorem5 provides the theoretical justification for relying on the use of the algorithms that have been devised for determining general economic

Notes on non-constructivity in CGE theory and modelling

An algorithm, by definition, is a finite object, consisting of a finite sequence of instructions. However, such a finite object is perfectly compatible with ‘an infinite sequence of successive refinements’ [16, p. 52], provided a stopping rule associated with a clearly specified and verifiable approximation value is part of the sequence of instructions that characterize the algorithm. Moreover, it is not ‘the passage to the limit [that] is the nonconstructive aspect of Brouwer’s [fix point]

Concluding notes

In his characteristically perceptive review of the important papers by Pour-El and Richards [14], Kreisel [11, p. 900] observed:

“The [papers by Pour-El and Richards] add to the long lists of operations μ in analysis with some recursive ‘input’ I for which no output in μ(I) is recursive  Familiar examples are provided by: (i) Brouwer’s fixed point theorem in dimension >1  where I ranges over (i) continuous maps of the unit circle into itself … and where μ(I) is the set of (i) fixed points….”

References (22)

  • Morris W. Hirsch, A proof of the non-retractability of a cell onto its boundary, in: Proceedings of the American...
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    By ‘algorithmic foundations’ I refer to constructive and recursion theoretic underpinnings, bearing in mind the basic difference between the two approaches to the notion of algorithms. The former does not accept the phenomenological discipline of the Church–Turing thesis; the latter does. However, there are varieties of constructive mathematics where algorithmic constructions do occur within the discipline of the Church–Turing thesis (cf. [3]). A simpler and less detailed version of the results reported in this paper were presented during my lectures at the ‘Winter School’ in Experimental and Computable Economics held in the department of economics at the University of Trento on 4/5 February, 2004.

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