| Abstract: |
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. |