  For portfolio choice problems with proportional transaction costs, we discuss
whether or not there exists a "shadow price", i.e., a least favorable
frictionless market extension leading to the same optimal strategy and utility.
  By means of an explicit counter-example, we show that shadow prices may fail
to exist even in seemingly perfectly benign situations, i.e., for a
log-investor trading in an arbitrage-free market with bounded prices and
arbitrarily small transaction costs.
  We also clarify the connection between shadow prices and duality theory.
Whereas dual minimizers need not lead to shadow prices in the above "global"
sense, we show that they always correspond to a "local" version.
