A population threshold for dedicated teaching.

Teachers hold a prominent place in modern societies, particularly where education is compulsory and widely institutionalized. This ubiquity obscures an underlying puzzle: Why do societies assign some individuals solely to the instruction of others? This question is especially enigmatic for dedicated teachers, who invest their labor in cultivating others' skills but do not themselves participate in the productive activities for which their students are being trained. To address this question, we develop a simple, mathematically tractable model of teaching and learning in a population that shares a common productive aim. We identify a tradeoff between the size of the workforce and its collective level of expertise; and we analyze the optimal proportion of a population that should serve as teachers, across diverse scenarios. We find that a population must exceed a critical size before it is beneficial to allocate anyone as a dedicated teacher at all. Subsequently, the peak demand for teachers is achieved at an intermediate population size and never exceeds one half of the population. For more complicated tasks, our analysis predicts the optimal allocation of teachers to instruction across different levels of expertise. Although our model is not evolutionary per se, this population-level account lays a foundation for understanding the adaptive advantage of dedicated teachers in both human and nonhuman populations.