Time and symmetry in models of economic markets (0902.4274v1)

Abstract: These notes discuss several topics in neoclassical economics and alternatives, with an aim of reviewing fundamental issues in modeling economic markets. I start with a brief, non-rigorous summary of the basic Arrow-Debreu model of general equilibrium, as well as its extensions to include time and contingency. I then argue that symmetries due to similarly endowed individuals and similar products are generically broken by the constraints of scarcity, leading to the existence of multiple equilibria. This is followed by an evaluation of the strengths and weaknesses of the model generally. Several of the weaknesses are concerned with the treatments of time and contingency. To address these we discuss a class of agent based models. Another set of issues has to do with the fundamental meaning of prices and the related question of what the observables of a non-equilibrium, dynamic model of an economic market should be. We argue that these issues are addressed by formulating economics in the language of a gauge theory, as proposed originally by Malaney and Weinstein. We review some of their work and provide a sketch of how gauge invariance can be incorporated into the formulation of agent based models.

• The paper critiques the Arrow-Debreu model by exposing its unrealistic treatment of time and equilibrium assumptions.
• It employs gauge theory to reinterpret economic invariances and demonstrates how scarcity-induced symmetry breaking leads to multiple equilibria.
• Additionally, the paper advocates for agent-based models to capture decentralized decision-making and non-equilibrium market dynamics.

Time and Symmetry in Models of Economic Markets: An Overview

This paper, authored by Lee Smolin, addresses foundational issues in economic market modeling, focusing on time and symmetry within neoclassical economics and alternatives. Smolin begins by summarizing the canonical Arrow-Debreu model of general equilibrium, highlighting its fundamental assumptions and mathematical formulations. He notes that while the model establishes economics as a mathematical science comparable to physics, it falls short in its treatment of time and contingency. This critique provides a basis for exploring agent-based models and the application of gauge theory to economics.

Major Themes and Concepts

Firstly, the Arrow-Debreu model is dissected to reveal its strengths and limitations. The model is critiqued for treating time like space, where goods are distinguished only by their temporal and spatial labels, neglecting the asymmetry and irreversibility of real-world economics. It assumes perfect foresight and rationality, which leads to unrealistic market dynamics where all contingencies are pre-known by agents who can simultaneously optimize their entire lifetime's utility—a notion largely disconnected from practical decision-making processes.

Symmetry and Equilibrium

An insightful examination of symmetry within economic systems is provided. Smolin proposes that symmetries, such as those arising from similarly endowed individuals or products, are typically broken by scarcity constraints, leading to multiple equilibria. This multiplicity echoes the findings of the Sonnenschein-Mantel-Debreu Theorem, suggesting that a general economy does not lead to a unique set of equilibrium prices. Smolin emphasizes that, like in physics, spontaneous symmetry breaking plays a crucial role in understanding which of many possible equilibria is realized in a market.

Gauge Theory and Economics

Smolin introduces gauge theory as a potential framework for addressing some of the shortcomings in current economic models, particularly with regard to time and contingency. Building on the work of Malaney and Weinstein, he suggests that gauge invariance underlies economic dynamics just as it does in physical systems. This insight addresses the fundamental meaning of prices out of equilibrium and the role of arbitrage. By employing gauge invariance, economics could potentially develop an operational framework that ensures decisions remain consistent across changes in observed price units, akin to curvatures and connections in gauge physics.

Agent-Based Models

Agent-based models are explored as a means to incorporate more realism into economic modeling. These models simulate economic activities at a micro-level, capturing the behavior of various agents acting on limited information—a stark contrast to the global optimization assumptions of Arrow-Debreu. They offer a platform for examining how macroeconomic phenomena, such as money emergence or price stabilization, can evolve from decentralized interactions. Smolin suggests that these models, if framed within a gauge-theoretic approach, could offer significant insights into non-equilibrium economics.

Implications and Future Directions

The implications of Smolin's exploration are profound. Practically, adopting a gauge-theoretic perspective can lead to more robust models for financial institutions, better accommodating market shifts and regulatory interventions. Theoretically, this approach challenges the deterministic predictions of classical models, steering towards a description of the economy that accommodates indeterminism and complexity found in real markets.

While the paper does not offer finalized theories, it marks a crucial step toward integrating non-equilibrium conditions into the economic modeling landscape. The suggestion that gauge invariance can drive novel economic insights, similar to its role in modern physics, is a compelling narrative for future research, potentially leading to a new understanding of economic systems driven by local interactions and global invariances. This paveway illuminates the path forward for a statistical economics analog to statistical mechanics in physics, opening possibilities for future breakthroughs in understanding economic dynamics.