On Computability of Equilibria in Markets with Production

Although production is an integral part of the Arrow-Debreu market model, most of the work in theoretical computer science has so far concentrated on markets without production, i.e., the exchange economy. This paper takes a significant step towards understanding computational aspects of markets with production. We first define the notion of separable, piecewise-linear concave (SPLC) production by analogy with SPLC utility functions. We then obtain a linear complementarity problem (LCP) formulation that captures exactly the set of equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production, and we give a complementary pivot algorithm for finding an equilibrium. This settles a question asked by Eaves in 1975 of extending his complementary pivot algorithm to markets with production. Since this is a path-following algorithm, we obtain a proof of membership of this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of existence of equilibrium (i.e., without using a fixed point theorem), rationality, and oddness of the number of equilibria. We further give a proof of PPAD-hardness for this problem and also for its restriction to markets with linear utilities and SPLC production. Experiments show that our algorithm runs fast on randomly chosen examples, and unlike previous approaches, it does not suffer from issues of numerical instability. Additionally, it is strongly polynomial when the number of goods or the number of agents and firms is constant. This extends the result of Devanur and Kannan (2008) to markets with production. Finally, we show that an LCP-based approach cannot be extended to PLC (non-separable) production, by constructing an example which has only irrational equilibria.Comment: An extended abstract will appear in SODA 201

A Primer on the Tools and Concepts of Computable Economics

Computability theory came into being as a result of Hilbert's attempts to meet Brouwer's challenges, from an intuitionistc and constructive standpoint, to formalism as a foundation for mathematical practice. Viewed this way, constructive mathematics should be one vision of computability theory. However, there are fundamental differences between computability theory and constructive mathematics: the Church-Turing thesis is a disciplining criterion in the former and not in the latter; and classical logic - particularly, the law of the excluded middle - is not accepted in the latter but freely invoked in the former, especially in proving universal negative propositions. In Computable Economic an eclectic approach is adopted where the main criterion is numerical content for economic entities. In this sense both the computable and the constructive traditions are freely and indiscriminately invoked and utilised in the formalization of economic entities. Some of the mathematical methods and concepts of computable economics are surveyed in a pedagogical mode. The context is that of a digital economy embedded in an information society

The Fundamental Theorems of Welfare Economics, DSGE and the Theory of Policy - Computable & Constructive Foundations

The genesis and the path towards what has come to be called the DSGE model is traced, from its origins in the Arrow-Debreu General Equilibrium model (ADGE), via Scarf's Computable General Equilibrium model (CGE) and its applied version as Applied Computable General Equilibrium model (ACGE), to its ostensible dynamization as a Recursive Competitive Equilibrium (RCE). It is shown that these transformations of the ADGE - including the fountainhead - are computably and constructively untenable. The policy implications of these (negative) results, via the Fundamental Theorems of Welfare Economics in particular, and against the backdrop of the mathematical theory of economic policy in general, are also discussed (again from computable and constructive points of view). Suggestions for going 'beyond DSGE' are, then, outlined on the basis of a framework that is underpinned - from the outset - by computability and constructivity considerationsComputable General Equilibrium, Dynamic Stochastic General Equilibrium, Computability, Constructivity, Fundamental Theorems of Welfare Economics, Theory of Policy, Coupled Nonlinear Dynamic

Rolf Mantel and the Computability of General Equilibria: On the Origins of the Sonnenschein-Mantel-Debreu Theorem

In this brief paper we revise the original motivations of Rolf Mantel to pursue a proof of Sonnenschein´s conjecture. We contend that his work on computational models of general equilibrium lead him to seek an alternative to the usual fixed point theorems used in proofs of existence. Confronted with a paper of Uzawa and his own experience in programming a national planning system he found that the use of theorems like Brouwer´s and Kakutani´s was unavoidable. To check out whether Uzawa was right he sought to find out whether the only properties required of excess demand functions to ensure the existence of equilibria in competitive markets were continuity, homogeneity and Walras´ law. In 1974, he found that this was actually the case. We will see that this result and his interpretation were informed by Mantel´s interest in economic development and planning.Fil: Tohmé, Fernando

Uncomputability and Undecidability in Economic Theory

Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory ([28]) and Algorithmic Statistics ([13]) are frontier research subjects. All of them, each in its own way, are underpinned by (classical) recursion theory - and its applied branches, say computational complexity theory or algorithmic information theory - and, occasionally, proof theory. These research paradigms have posed new mathematical and metamathematical questions and, inadvertently, undermined the traditional mathematical foundations of economic theory. A concise, but partial, pathway into these new frontiers is the subject matter of this paper. Interpreting the core of mathematical economic theory to be defined by General Equilibrium Theory and Game Theory, a general - but concise - analysis of the computable and decidable content of the implications of these two areas are discussed. Issues at the frontiers of macroeconomics, now dominated by Recursive Macroeconomic Theory, are also tackled, albeit ultra briefly. The point of view adopted is that of classical recursion theory and varieties of constructive mathematics.General Equilibrium Theory, Game Theory, Recursive Macro-economics, (Un)computability, (Un)decidability, Constructivity

Computability and Evolutionary Complexity: Markets As Complex Adaptive Systems (CAS)

The purpose of this Feature is to critically examine and to contribute to the burgeoning multi disciplinary literature on markets as complex adaptive systems (CAS). Three economists, Robert Axtell, Steven Durlauf and Arthur Robson who have distinguished themselves as pioneers in different aspects of how the thesis of evolutionary complexity pertains to market environments have contributed to this special issue. Axtell is concerned about the procedural aspects of attaining market equilibria in a decentralized setting and argues that principles on the complexity of feasible computation should rule in or out widely held models such as the Walrasian one. Robson puts forward the hypothesis called the Red Queen principle, well known from evolutionary biology, as a possible explanation for the evolution of complexity itself. Durlauf examines some of the claims that have been made in the name of complex systems theory to see whether these present testable hypothesis for economic models. My overview aims to use the wider literature on complex systems to provide a conceptual framework within which to discuss the issues raised for Economics in the above contributions and elsewhere. In particular, some assessment will be made on the extent to which modern complex systems theory and its application to markets as CAS constitutes a paradigm shift from more mainstream economic analysis

Computation in Economics

This is an attempt at a succinct survey, from methodological and epistemological perspectives, of the burgeoning, apparently unstructured, field of what is often – misleadingly – referred to as computational economics. We identify and characterise four frontier research fields, encompassing both micro and macro aspects of economic theory, where machine computation play crucial roles in formal modelling exercises: algorithmic behavioural economics, computable general equilibrium theory, agent based computational economics and computable economics. In some senses these four research frontiers raise, without resolving, many interesting methodological and epistemological issues in economic theorising in (alternative) mathematical modesClassical Behavioural Economics, Computable General Equilibrium theory, Agent Based Economics, Computable Economics, Computability, Constructivity, Numerical Analysis

Behavioural Economics: Classical and Modern

In this paper, the origins and development of behavioural economics, beginning with the pioneering works of Herbert Simon (1953) and Ward Edwards (1954), is traced, described and (critically) discussed, in some detail. Two kinds of behavioural economics – classical and modern – are attributed, respectively, to the two pioneers. The mathematical foundations of classical behavioural economics is identified, largely, to be in the theory of computation and computational complexity; the corresponding mathematical basis for modern behavioural economics is, on the other hand, claimed to be a notion of subjective probability (at least at its origins in the works of Ward Edwards). The economic theories of behavior, challenging various aspects of 'orthodox' theory, were decisively influenced by these two mathematical underpinnings of the two theoriesClassical Behavioural Economics, Modern Behavioural Economics, Subjective Probability, Model of Computation, Computational Complexity. Subjective Expected Utility

Variations on the Theme of Conning in Mathematical Economics

The mathematization of economics is almost exclusively in terms of the mathematics of real analysis which, in turn, is founded on set theory (and the axiom of choice) and orthodox mathematical logic. In this paper I try to point out that this kind of mathematization is replete with economic infelicities. The attempt to extract these infelicities is in terms of three main examples: dynamics, policy and rational expectations and learning. The focus is on the role and reliance on standard xed point theorems in orthodox mathematical economics