  This paper studies continuous-time optimal contracting in a hierarchy problem
which generalises the model of Sung (2015). The hierarchy is modeled by a
series of interlinked principal-agent problems, leading to a sequence of
Stackelberg equilibria. More precisely, the principal can contract with the
managers to incentivise them to act in her best interest, despite only
observing the net benefits of the total hierarchy. Managers in turn subcontract
with the agents below them. Both agents and managers independently control in
continuous time a stochastic process representing their outcome. First, we show
through a continuous-time adaptation of Sung's model that, even if the agents
only control the drift of their outcome, their manager controls the volatility
of their continuation utility. This first simple example justifies the use of
recent results on optimal contracting for drift and volatility control, and
therefore the theory of second-order backward stochastic differential
equations, developed in the theoretical part of this paper, dedicated to a more
general model. The comprehensive approach we outline highlights the benefits of
considering a continuous-time model and opens the way to obtain comparative
statics. We also explain how the model can be extended to a large-scale
principal-agent hierarchy. Since the principal's problem can be reduced to only
an $m$-dimensional state space and a $2m$-dimensional control set, where $m$ is
the number of managers immediately below her, and is therefore independent of
the size of the hierarchy below these managers, the dimension of the problem
does not explode.
