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In the present paper we study endogenous price leadership in the context of a homogeneous product Bertrand duopoly model in which the firms have different, strictly convex cost functions. In such a framework it is well known that a simultaneous move price choice game does not have an equilibrium in pure strategies, but it has an equilibrium in mixed strategies. In the Stackelberg games with an exogenous price leader, we show that a pure strategy subgame perfect Nash equilibrium (SPNE) always exists. Although the SPNE might not be unique, the payoffs are the same across all SPNE. Finally, we analyze the issue of endogenous price leadership using the continuous version of the Robson (1990) timing game. The result is unexpected. One would expect the more efficient firm to emerge as the endogenous price leader. This is not always true. In most cases the endogenous leader is the firm with the highest "threshold" price. However, we also provide conditions under which the more efficient firm emerges as the leader. Our paper essentially complements Yano (2001), which is based on the Hamilton and Slutsky (1990) framework.