Abstract We consider an energy harvesting transmitter sending messages to two users over parallel and fading Gaussian broadcast channels. Energy required for communication arrives (is harvested) at the transmitter and a finite-capacity battery stores it before being consumed for transmission. Under off-line knowledge of energy arrival and channel fading variations, we obtain the trade-off between the performances of the users by characterizing the maximum departure region in a given interval. We first analyze the transmission with an energy harvesting transmitter over parallel broadcast channels. We show that the optimal total transmit power policy that achieves the boundary of the maximum departure region is the same as the optimal policy for the non-fading broadcast channel, which does not depend on the priorities of the users, and therefore is the same as the optimal policy for the non-fading scalar single-user channel. The optimal total transmit power can be found by a directional water-filling algorithm. The optimal splitting of the power among the parallel channels is performed in each epoch separately. Next, we consider fading broadcast channels and obtain the transmission policies that achieve the boundary of the maximum departure region. The optimal total transmit power allocation policy is found using a specific directional water-filling algorithm for fading broadcast channels. The optimal power allocation depends on the priorities of the users unlike in the case of parallel broadcast channels. Finally, we provide numerical illustrations of the optimal policies and maximum departure regions for both parallel and fading broadcast channels.