Gilles & Pesca Theorem Links Production Viability to Coherence

Robert Gilles and Marialaura Pesce have released a revised paper (v2, June 2026) on arXiv that formalizes a new equilibrium framework for structured production systems. The framework distinguishes between consumption goods traded in competitive markets and intermediate goods exchanged through bilateral relationships. Their central concept is viability—the requirement that all producers earn positive incomes—which they posit as a foundational prerequisite for equilibrium.

The main theorem proves that a structured production system is viable if and only if it is coherent, meaning it admits no circular conversion processes that yield no net output. This coherence condition is equivalent to the non-singularity of the system's matrix representation. The authors further investigate completely viable systems, where viable prices exist for all consumption good price vectors. They show that complete viability sits between two input restrictions: guaranteed when no consumption good is an input in production, and requiring that no consumption good is used as an input in producing another consumption good. The analysis reveals fundamental relationships between system architecture and economic sustainability, with contributions to the Hawkins-Simon condition in input-output analysis.