We study the problem of computing the lower hedging price of an American contingent claim in a finite-state discrete-time market setting under proportional transaction costs. We derive a new mixed-integer linear programming formulation for calculating the lower hedging price. The linear programming relaxation of the formulation is exact in frictionless markets. Our results imply that it might be optimal for the holder of several identical American claims to exercise portions of the portfolio at different time points in the presence of proportional transaction costs while this incentive disappears in their absence.