Academic wages and pyramid schemes: a mathematical model

This paper analyzes a steady state matching model interrelating the education and labor sectors. In this model, a heterogeneous population of students match with teachers to enhance their cognitive skills. As adults, they then choose to become workers, managers, or teachers, who match in the labor or educational market to earn wages by producing output. We study the competitive equilibrium which results from the steady state requirement that the educational process replicate the same endogenous distribution of cognitive skills among adults in each generation (assuming the same distribution of student skills). We show such an equilibrium can be found by solving an infinite-dimensional linear program and its dual. We analyze the structure of our solutions, and give sufficient conditions for them to be unique. Whether or not the educational matching is positive assortative turns out to depend on convexity of the equilibrium wages as a function of ability, suitably parameterized; we identity conditions which imply this convexity. Moreover, due to the recursive nature of the education market, it is a priori conceivable that a pyramid scheme leads to greater and greater discrepancies in the wages of the most talented teachers at the top of the market. Assuming each teacher teaches $N$ students, and contributes a fraction $\theta \in]0,1[$ to their cognitive skill, we show a phase transition occurs at $N\theta=1$, which determines whether or not the wage gradients of these teachers remain bounded as market size grows, and make a quantitative prediction for their asymptotic behaviour in both regimes: $N\theta \ge 1$ and $N\theta<1$.