In this paper the quality of service (QoS) constrained radio resource allocation problem at the downlink of a multiuser multicarrier system is investigated. We demonstrate and analyze the trade-off between energy consumption and transmit power within a cross physical and link layer system model, which jointly considers power allocation, adaptive modulation and coding and ARQ/HARQ retransmission protocols. A novel transmit power constrained energy minimization problem is formulated based on the competing nature of the two resources. Due to the combinatorial property of the problem, a suboptimal heuristic resource allocation algorithm is proposed, the accuracy of which is compared with the dual optimal value obtained as a byproduct of the algorithm. Simulation results also provide performance comparisons on ARQ and HARQ protocols, and their different impacts on the choice of the optimal modes of operations.