Heat pump water heaters (HPWHs) offer an efficient way to heat water using electricity, which aligns with efforts toward decarbonization and better utilization of renewable energy sources. Economic model predictive control (EMPC) provides an automated way to provide load flexibility of this new electric load by accounting for exogenous inputs like time-varying electric rates or marginal greenhouse gas emissions. Furthermore, a cloud-based supervisory controller implementation of EMPC enables retrofitting existing HPWHs with cloud-connection. In this work, the formulation of a supervisory multi-objective EMPC for HPWH is presented. The formulation of a supervisory multi-objective EMPC for HPWH is presented for equipment with a single heat pump and up to two backup resistive heating elements. A temperature setpoint is computed from the EMPC decisions using a logic-based setpoint calculator so the existing HPWH rule-based control (RBC) strategy activates the desired heat sources when deemed optimal by the EMPC. The performance of the simulated testing results under the supervisory EMPC is compared against the performance under an RBC strategy and under a regulatory EMPC that directly controls the HPWH. The simulation results demonstrate that the RBC in the proposed control architecture, operates the HPWH heat sources in the optimal manner computed by the EMPC, which indicates that the setpoint calculator can translate EMPC decisions to appropriate temperature setpoints. To minimize heat pump cycling, the effect of increasing the minimum on-time for the heat pump is also considered and the results show that increasing the minimum on-time increases the operating cost.