Shadow price in the power utility case

We consider the problem of maximizing expected power utility from consumption over an infinite horizon in the Black.Scholes model with proportional transaction costs, as studied in Shreve and Soner [ Ann. Appl. Probab. 4 (1994) 609.692]. Similar to Kallsen and Muhle-Karbe [ Ann. Appl. Probab. 20 (2010) 1341.1358], we derive a shadow price, that is, a frictionless price process with values in the bid-ask spread which leads to the same optimal policy.