Existence of Equilibria in Large Competitive Markets with Bads, Production and Comprehensive Externalities

Technical Summary: Existence of Equilibria in Large Competitive Markets with Bads, Production, and Comprehensive Externalities

Problem Statement
The paper addresses a fundamental gap in Measure-theoretic General Equilibrium (MGE) theory: the failure of equilibrium existence when modeling "bads" (undesirable commodities) alongside production and externalities. While the MGE framework, pioneered by Aumann (1964) and Vind (1964), serves as the canonical benchmark for perfect competition, Hara (2005) demonstrated that standard MGE exchange models fail to guarantee equilibrium existence in the presence of bads. Specifically, Hara showed that candidate equilibrium allocations may involve non-integrable consumption of bads, rendering the equilibrium undefined.

The authors argue that existing literature often relies on assumptions that are economically problematic when bads are central to the model. These include:

1. Monotonic Preferences: These force non-negative prices, eliminating the negative prices necessary to incentivize the control of bads.
2. Free Disposal in Production: This assumption implies that byproducts (e.g., emissions) can be discarded without cost, making them disappear from accounting and precluding the pricing mechanisms needed for environmental regulation.
3. Free Disposal in Consumption: This allows agents to purchase bads to generate income (via negative prices) and then discard them, leading to empty demand sets.
4. Limited Externalities: Previous models often restrict externalities to the consumption choices of others, failing to capture harms arising from production plans that are not consumed (e.g., emissions).

Methodology
The paper employs nonstandard analysis as its primary technical tool. Rather than using nonstandard analysis merely to derive asymptotic results for large finite economies, the authors utilize it as a systematic bridge to extend existence theorems from finite economies to continuum (measure-theoretic) economies. The proof strategy proceeds in five distinct steps:

1. Finite Foundation: The authors first establish the existence of equilibrium for weighted finite production economies with externalities (Theorem 1), adapting results from Florenzano (2003).
2. Hyperfinite Embedding: A standard measure-theoretic economy is embedded into a hyperfinite economy via a "lifting" process. This involves constructing a hyperfinite partition of the consumer space where agents within the same partition set share similar characteristics (endowments, preferences, consumption sets).
3. Transfer Principle: The existence theorem for finite economies is transferred to the hyperfinite economy, yielding a hyperfinite equilibrium.
4. Loeb Construction: The hyperfinite equilibrium is pushed down to a Loeb measure economy. The authors prove that under specific regularity conditions (integrable bounds on bads and weak continuity of preferences), the hyperfinite allocation is S-integrable and near-standard, allowing for the construction of a Loeb equilibrium.
5. Standard Part Push-down: Finally, the Loeb equilibrium is pushed down to the original standard measure-theoretic economy. This step relies on the weak topology on $L^1$ spaces and the convexity of quasi-demand sets to ensure the resulting allocation is a valid equilibrium in the original model.

Key Contributions and Model Features
The paper introduces an MGE production model with eight key economic ingredients that distinguish it from prior literature:

• Categorization of Commodities: Commodities are explicitly divided into goods and bads, with preferences not required to be monotonic over bads.
• Non-Free-Disposal Equilibrium: The model explicitly rules out free disposal in both production and consumption. This is crucial for allowing negative prices, which provide incentives for firms to eliminate bads and consumers to absorb them.
• Comprehensive Externalities: Consumer preferences depend not only on other consumers' allocations but also on firms' production plans and equilibrium prices. This captures externalities arising from emissions (production byproducts not consumed) rather than just consumption externalities.
• Integrably Bounded Consumption Sets: To resolve the non-integrability issue identified by Hara (2005), the model imposes an integrable bound on the consumption of bads. This reflects the physical limitation of agents (e.g., land capacity for waste) rather than a preference restriction.
• Individualistic Approach: The model uses a measure space of individual consumers rather than a distributional approach.
• Convex Preferences: Unlike pure exchange economies where convexity is not strictly necessary, the presence of comprehensive externalities necessitates convex preferences to ensure the existence of equilibrium.

Main Results

• Theorem 1: Establishes the existence of equilibrium in weighted finite production economies with general externalities.
• Theorem 2: The paper's main result, proving the existence of a non-free-disposal equilibrium in a measure-theoretic production economy with bads and comprehensive externalities. This theorem overcomes the nonexistence example of Hara (2005) by incorporating production, non-free disposal, and integrable bounds on bad consumption.
• Theorem 3 (Supplementary Material): Extends the existence result to "quota equilibrium," a generalization of cap-and-trade systems where disposal is allowed up to a regulator-specified cap.

Significance and Claims
The authors claim this work provides the first existence theorem for MGE models that simultaneously incorporate bads, production, and comprehensive externalities. The paper asserts that by formulating the model in an "economically natural way"—specifically by rejecting free disposal in production and consumption and acknowledging physical limits on bad absorption—equilibrium existence can be restored.

The significance lies in providing a rigorous theoretical benchmark for analyzing production-generated bads (such as CO2 emissions) and their external effects in large competitive markets. The paper notes that while the proof is technically demanding due to the use of nonstandard analysis, the economic statement is standard. The result validates the use of MGE models for environmental economics and policy analysis, where negative prices and externalities are central, overcoming the theoretical barriers previously identified in the literature. The authors dedicate the work to Peter A. Loeb, whose measure-theoretic foundations are central to the methodology employed.