  Aldous and Bandyopadhyay have shown that each solution to a recursive
distributional equation (RDE) gives rise to recursive tree process (RTP), which
is a sort of Markov chain in which time has a tree-like structure and in which
the state of each vertex is a random function of its descendants. If the state
at the root is measurable with respect to the sigma field generated by the
random functions attached to all vertices, then the RTP is said to be
endogenous. For RTPs defined by continuous maps, Aldous and Bandyopadhyay
showed that endogeny is equivalent to bivariate uniqueness, and they asked if
the continuity hypothesis can be removed. We introduce a higher-level RDE that
through its $n$-th moment measures contains all $n$-variate RDEs. We show that
this higher-level RDE has minimal and maximal fixed points with respect to the
convex order, and that these coincide if and only if the corresponding RTP is
endogenous. As a side result, this allows us to answer the question of Aldous
and Bandyopadhyay positively.
