This paper considers the broadcast channel with an energy harvesting transmitter equipped with an inefficient energy storage device. For this setting, the optimal offline power policy that maximizes the average weighted sum rate of the system is derived. It is observed that this policy has a double threshold structure, with piecewise constant thresholds determined by the energy harvesting process. The convexity of the capacity region for the energy harvesting broadcast channel for a finite deadline is established, showing that the weighted maximum sum-rate traces the boundary of the region. Next, the optimal online policy is found using dynamic programming, and it is observed that the solution has a double threshold structure as well with state dependent thresholds. Lastly, a double threshold policy is proposed with fixed thresholds that performs near optimal with reduced complexity.