Training in the Age of AI: A Theory of Apprenticeship Viability

Apprenticeships let juniors pay for training by doing menial work. AI now performs an increasing share of that work, putting the bargain at risk. We introduce AI into a dynamic apprenticeship model with an automation threshold and possible complementarity for experts. A single statistic—the expertise leverage ratio, measuring the AI-augmented value of a graduate relative to AI's standalone output—governs the impact of AI. Our central result is that apprenticeships are guaranteed viable, in the sense that they are at least as profitable as they were before the arrival of AI, when this ratio is above a critical threshold, specifically Euler's number e; in this case, training has a fixed duration and the apprenticeship is not at risk. Below the threshold, training compresses as the master's saleable knowledge shrinks; in this case, advances in AI threaten wholesale apprenticeship collapse.