🌀 nonlinear sciences

Revising price coordination in the classical and neoclassical economics based on elementary cellular automata

This paper utilizes elementary cellular automata, Shannon entropy, and Monte Carlo simulations to argue that classical economics offers a consistent, interaction-based framework for price coordination, whereas the neoclassical approach fails to provide a mechanism for such coordination, portraying individuals as passive spectators.

Original authors: Igor Lugo, Martha G. Alatriste-Contreras

Published 2026-07-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Igor Lugo, Martha G. Alatriste-Contreras

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the economy as a giant, bustling marketplace where millions of people are constantly buying and selling things. The big mystery economists have tried to solve for centuries is: How do all these prices magically line up so that everyone gets what they need without chaos?

This paper suggests that the old ways of explaining this (using complex math equations) might be missing the point. Instead, the authors propose using a simple computer game called Elementary Cellular Automata (ECA) to understand how prices actually coordinate.

Here is the breakdown of their ideas using simple analogies:

1. The Game Board: A Row of Light Switches

Think of the economy as a long row of light switches.

The Switches: Each switch represents the price of a specific item.
The States: A switch is either ON (1) meaning the price went up, or OFF (0) meaning the price went down.
The Neighbors: Just like in real life, a price doesn't exist in a vacuum. It reacts to the items next to it.
- The neighbor on the Left is a "Complementary Good" (like a hot dog bun). If the price of buns goes up, the price of hot dogs might change.
- The neighbor on the Right is a "Substitute Good" (like a burger). If burgers get expensive, people might buy more hot dogs, changing the price.

The computer runs this game step-by-step. It looks at a switch and its two neighbors, applies a simple rule, and decides what the switch will be in the next moment.

2. The Two Schools of Thought: The Architect vs. The Spectator

The authors used this game to test two famous ways of thinking about economics: Classical and Neoclassical.

The Classical Approach: The Active Architect

The Metaphor: Imagine a group of people building a wall together. They talk, they look at the bricks, and they make logical choices based on what they see. If a brick is too heavy, they adjust.
In the Paper: This approach assumes people are active participants. They use logic and real data (like how much labor it took to make a product) to decide prices.
The Result in the Game: When the authors programmed the computer to act like this "logical" person, the prices settled into stable, repeating patterns (like a neat, rhythmic wave). The system found a balance because the "architects" were actively coordinating.
The Takeaway: The Classical view is consistent. It shows that when people make logical choices based on facts, prices naturally find a stable order.

The Neoclassical Approach: The Passive Spectator

The Metaphor: Imagine a person standing on the sidelines watching a storm. They can't stop the rain or change the wind; they just see the rain falling and say, "Oh, it's raining now." They don't know why the rain started, they just react to it.
In the Paper: This approach assumes the "market" is a mysterious force that magically balances itself. The individual is just a spectator who sees the final price and reacts to it. They don't know the rules of how the price got there.
The Result in the Game: When the authors programmed the computer to act like this "spectator," the system only looked for simple, repetitive patterns (like a solid block of color or a simple repeating stripe).
The Takeaway: The Neoclassical view is limited. It ignores the complex, messy, and chaotic ways prices actually interact. It assumes the market is always in a perfect, simple balance, ignoring the fact that real life is often messy and unpredictable.

3. The "Classes" of Chaos

The paper mentions that these computer games can produce four types of results (Classes):

Class 1 & 2 (Simple/Stable): Everything settles down quickly. (This is what Neoclassical economics likes to see).
Class 3 & 4 (Complex/Chaotic): The patterns are wild, messy, and unpredictable, like snowflakes or traffic jams. (This is what the Classical approach actually describes when you let people interact logically).

The authors argue that because old computers were too slow to handle the "messy" Class 3 and 4 patterns, early economists were forced to only look at the "neat" Class 1 and 2 patterns. They built their theories on the idea that the world is simple and orderly, just because they couldn't compute the complex stuff.

4. The Big Conclusion

The paper concludes that:

Classical Economics is closer to the truth because it acknowledges that people interact, make choices, and create order out of chaos.
Neoclassical Economics is a bit of a fantasy. It treats people as if they are just watching a magic show, assuming the prices will magically balance themselves without anyone actually doing the work of coordinating them.

In short: The authors used a simple computer game to show that the economy isn't a perfectly tuned machine (as Neoclassical theory suggests); it's more like a complex dance where people are actively leading and following each other (as Classical theory suggests). By using these simple rules, we can finally see the "facts" behind the "beliefs" of economic theory.

Problem Statement
The paper addresses the complexity of price coordination in economics, noting a lack of scientific consensus regarding the simplest mechanisms that explain how prices clear markets. While classical and neoclassical economic theories offer different formulations, the authors argue that these approaches have failed to provide simple, predictive mechanisms for understanding price coordination. The classical approach relies on logical choices based on objective data and human interaction, whereas the neoclassical approach often treats the individual as a passive spectator reacting to exogenous price signals without explaining the underlying coordination mechanisms. The study posits that these theories were historically constrained by a lack of accessible computational power, leading to a reliance on reductionist models that seek uniform or repetitive patterns (stable equilibria) rather than exploring complex, dynamic behaviors.

Methodology
The authors employ elementary cellular automata (CA) as a computational framework to model and analyze the foundational principles of classical and neoclassical economics.

Model Construction: Using Python and standard libraries (NumPy, Matplotlib), the authors simulate one-dimensional elementary CAs with a lattice length of 100 and wraparound (torus) boundary conditions.
Rule Definition: Price interactions are modeled using binary states (0 for decreased price, 1 for increased price). The update rules for each cell are derived from the economic concepts of substitute and complementary goods. The study defines specific rules where a cell's next state depends on its current state and the states of its left (complementary) and right (substitute) neighbors.
Analysis Techniques:
- Shannon Entropy: Used to quantify the uncertainty and order/disorder of the generated price patterns. Lower entropy indicates high order (stable patterns), while higher entropy suggests complexity or randomness.
- Model Validation: A conceptual model validation is performed using Monte Carlo simulations. The deterministic outputs of the CA rules are compared against null models based on binomial distributions (with success probabilities $p=0.3, 0.5, 0.7$ ) using Spearman correlation coefficients to determine statistical significance against random chance.
Scenarios: The study tests both random initial conditions (uniform distribution of 0s and 1s) and simple initial conditions (a single '1' in the center).

Key Contributions
The primary contribution of the paper is the translation of economic theories into computational rules to reveal the "facts and beliefs" underlying classical and neoclassical economics.

Mapping Theory to CA Classes: The study attempts to map economic approaches to Wolfram's four classes of elementary CAs. It hypothesizes that historical limitations forced economists to focus on Class 1 (uniform states) and Class 2 (repetitive structures), which represent stable equilibria, while potentially overlooking the complexity of Class 3 (chaotic/complex) and Class 4 (localized structures).
Differentiation of Approaches: The paper distinguishes between the two schools by their treatment of price coordination: the classical approach is modeled as endogenous coordination driven by rational interaction, while the neoclassical approach is modeled as exogenous, where coordination is assumed rather than mechanistically described.

Results

Classical Economics: The study identifies specific CA rules (e.g., Rule 23, Rule 85, Rule 51) that align with the classical emphasis on rational choice and objective data. These rules generate stable or complex patterns (Classes 2, 3, and 4) with high statistical significance compared to random null models. The results suggest that the classical approach supports mechanisms where individual interactions (substitutes and complements) drive collective coordination, capable of producing both stable and complex patterns.
Neoclassical Economics: The neoclassical approach is found to be exclusively interested in Class 1 and Class 2 patterns (stable or repetitive large-scale structures). The study notes that the neoclassical framework avoids Class 3 and 4 patterns because they represent non-equilibrium or unstable prices, which are analytically intractable within the traditional mathematical formalism. The authors argue that under the neoclassical view, price coordination is treated as an unpredictable event or a "spectator" phenomenon, lacking a mechanism to describe the generation of complex or unstable patterns.
Statistical Validation: The deterministic rules derived for both approaches showed statistically significant differences from random binomial variables, confirming that the modeled economic theories generate structured, non-random patterns.

Significance and Claims
The paper claims that modeling economic theory through computational concepts allows for a re-evaluation of the assumptions behind classical and neoclassical economics.

Revealing Assumptions: By using CA, the authors reveal that the classical approach provides a consistent mechanism for understanding price coordination based on logical human interactions, whereas the neoclassical approach lacks a descriptive mechanism, treating coordination as an external, unpredictable event.
Complementarity: The study suggests that while the two approaches differ in formulation, they complement each other. The classical approach explores the potential for complex coordination (Classes 3 and 4), while the neoclassical approach focuses on identifying stable equilibria (Classes 1 and 2).
Educational Implication: The authors propose that students and researchers should recognize the distinction between "facts" (mechanisms that can be modeled) and "beliefs" (assumptions about equilibrium) within economic theory. They argue that the economy should be recognized as a complex system, and that computational tools are necessary to explore coordination problems that traditional analytical methods cannot fully capture.
Limitations: The authors modestly acknowledge that the study is limited to identifying direct theoretical fundamentals and does not fully map all 256 CA rules to specific social theories, suggesting this as a direction for future research.