Scale and Staging in Competitive Automation: Demand Externalities with State-Dependent Reabsorption

Abstract Falk and Tsoukalas show that competitive firms may choose an excessive automation endpoint because each firm captures its own labour-cost saving while internalising only part of the product-demand loss generated by worker displacement. This paper studies a second choice margin: the temporal staging of displacement. We first establish a full-horizon null. Under stationary exogenous recovery, all finite displacement paths reaching the same automation endpoint generate identical complete-lifetime unresolved exposure, despite potentially different peaks. Timing becomes consequential only when the transition changes subsequent recovery. We introduce state-dependent reabsorption and show that, under an increasing, concave total service flow with a stable residual process, an undivided displacement batch weakly maximises complete-tail exposure; genuine splitting is strictly beneficial under strict residual convexity. A declining individual recovery hazard alone is insufficient. At a fixed endpoint, partial private internalisation produces a signed staging distortion: excessive front-loading where marginal common timing cost is positive and excessive delay where it is negative. We then solve the joint endpoint-staging problem, establish unique private and cooperative targets, and derive a target-specific two-rate correction with a controlled return to the source model. The analysis is theoretical and uncalibrated; it does not establish that current AI adoption is excessive or prescribe numerical policy rates. Research status Working paper, version 0.1 (September 2026). Not peer reviewed. Independent technical review and empirical validation have not been completed. AI assistance and research transparency OpenAI ChatGPT was used substantively during the research and manuscript-development process, including theoretical exploration, mathematical formulation, proof and counterexample analysis, literature exploration, drafting and editing, and the development and execution of computational verification code. The computational tests examine theoretical model configurations and do not constitute empirical observations or experimental evidence. AI-assisted mathematical arguments, references, and code remain subject to the author's responsibility and independent review. The named human author assumes responsibility for the content and integrity of the deposited work.