We study the consequences of modeling asymmetric bargaining power in two-person bargaining problems. Comparing application of an asymmetric version of a bargaining solution to an upfront modification of the disagreement point, the resulting distortion crucially depends on the bargaining solution concept. While for the Kalai-Smorodinsky solution weaker players benefit from modifying the disagreement point, the situation is reversed for the Nash bargaining solution. There, weaker players are better off in the asymmetric bargaining solution. When comparing application of the asymmetric versions of the Nash and the Kalai-Smorodinsky solutions, we demonstrate that there is an upper bound for the weight of a player, so that she is better off with the Nash bargaining solution. This threshold is ultimately determined by the relative utilitarian bargaining solution. From a mechanism design perspective, our results provide valuable information for a social planner, when implementing a bargaining solution for unequally powerful players.