A General Equilibrium Theory of Orchestrated AI Agent Systems

Abstract

We establish a general equilibrium theory for systems of large language model (LLM) agents operating under centralized orchestration. The framework is a production economy in the sense of Arrow-Debreu (1954), extended to infinite-dimensional commodity spaces following Bewley (1972). Each LLM agent is modeled as a firm whose production set Y a $\subset$ H = L 2 ([0, T ], R R ) represents the feasible metric trajectories determined by its frozen model weights. The orchestrator is the consumer, choosing a routing policy over the agent DAG to maximize system welfare subject to a budget constraint evaluated at functional prices p $\in$ H A . These prices-elements of the Hilbert dual of the commodity space-assign a shadow value to each metric of each agent at each instant. We prove, via Brouwer's theorem applied to a finitedimensional approximation V K $\subset$ H, that every such economy admits at least one general equilibrium (p * , y * , $\pi$ * ). A functional Walras' law holds as a theorem: the value of functional excess demand is zero for all prices, as a consequence of the consumer's budget constraint-not by construction. We further establish Pareto optimality (First Welfare Theorem), decentralizability of Pareto optima (Second Welfare Theorem), and uniqueness with geometric convergence under a contraction condition (Banach). The orchestration dynamics constitute a Walrasian t{\^a}tonnement that converges globally under the contraction condition, unlike classical t{\^a}tonnement (Scarf, 1960). The framework admits a DSGE interpretation with SLO parameters as policy rates.