In this paper, we consider the sum $α$-fair utility maximization problem for joint downlink (DL) and uplink (UL) transmissions of a wireless powered communication network (WPCN) via time and power allocation. In the DL, the users with energy harvesting receiver architecture decode information and harvest energy based on simultaneous wireless information and power transfer. While in the UL, the users utilize the harvested energy for information transmission, and harvest energy when other users transmit UL information. We show that the general sum $α$-fair utility maximization problem can be transformed into an equivalent convex one. Tradeoffs between sum rate and user fairness can be balanced via adjusting the value of $α$. In particular, for zero fairness, i.e., $α=0$, the optimal allocated time for both DL and UL is proportional to the overall available transmission power. Tradeoffs between sum rate and user fairness are presented through simulations.