In this paper we study the bandwidth efficiency (throughput) and energy efficiency of relay networks considering both the physical layer and the medium access control (MAC) layer. Due to the wireless signal power attenuation with transmission distance, adopting a relay for packet transmissions leads to more energy efficient wireless networking at the expense of requiring multi-hop transmissions. To understand benefits of using a relay, the energy-throughput tradeoff needs to be analyzed. In a decentralized wireless network, not only the physical layer but also the MAC layer should be considered because the physical layer affects the MAC layer and the MAC layer overhead is not negligible. However, the energy-throughput tradeoffs for relay networks considering both the physical layer and the MAC layer has not been studied yet. We analyze the energy consumption and the throughput of relay networks with transmit power control. We determine the conditions in which wireless communication using a relay has better energy efficiency or bandwidth efficiency than direct transmission.