  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 $N\theta<1$.
