  We revisit the problem of maximizing expected logarithmic utility from
consumption over an infinite horizon in the Black-Scholes model with
proportional transaction costs, as studied in the seminal paper of Davis and
Norman [Math. Operation Research, 15, 1990]. Similarly to Kallsen and
Muhle-Karbe [Ann. Appl. Probab., 20, 2010], we tackle this problem by
determining a shadow price, that is, a frictionless price process with values
in the bid-ask spread which leads to the same optimization problem. However, we
use a different parametrization, which facilitates computation and
verification. Moreover, for small transaction costs, we determine fractional
Taylor expansions of arbitrary order for the boundaries of the no-trade region
and the value function. This extends work of Janecek and Shreve [Finance
Stoch., 8, 2004], who determined the leading terms of these power series.
