  The Merton investment-consumption problem is fundamental, both in the field
of finance, and in stochastic control. An important extension of the problem
adds transaction costs, which is highly relevant from a financial perspective
but also challenging from a control perspective because the solution now
involves singular control. A further significant extension takes us from
additive utility to stochastic differential utility (SDU), which allows time
preferences and risk preferences to be disentangled.
  In this paper, we study this extended version of the Merton problem with
proportional transaction costs and Epstein-Zin SDU. We fully characterise all
parameter combinations for which the problem is well posed (which may depend on
the level of transaction costs) and provide a full verification argument that
relies on no additional technical assumptions and uses primal methods only. The
case with SDU requires new mathematical techniques as duality methods break
down.
  Even in the special case of (additive) power utility, our arguments are
significantly simpler, more elegant and more far-reaching than the ones in the
extant literature. This means that we can easily analyse aspects of the problem
which previously have been very challenging, including comparative statics,
boundary cases which heretofore have required separate treatment and the
situation beyond the small transaction cost regime. A key and novel idea is to
parametrise consumption and the value function in terms of the shadow fraction
of wealth, which may be of much wider applicability.
