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Dynamic tariffs such as time-of-use (TOU) were introduced to motivate consumers to change the operation of their behind-the-meter (BTM) resources to reduce their power consumption during peak demand times. Recently, export rates have been implemented that reimburse consumers for electricity exported to the grid at a rate less than the retail rate to reduce reverse power flow in distribution power systems with high photovoltaic penetrations. Export rates are expected to be employed more widely, and therefore it is important to understand their impact on consumers, the operation of BTM resources, and the distribution grid. This paper presents a home energy management system (HEMS) capable of optimally scheduling BTM resources under a tariff with export rates. The proposed HEMS is formulated as a multiobjective model predictive control problem. In the paper, we analyze the performance of the proposed HEMS under an export rate including the operation of each BTM resource and the impact on the cost and comfort of homeowners. We also compare the performance of the proposed HEMS under an export rate with the HEMS under a TOU rate. Simulation results are presented for a small power system with several homes that employ the proposed HEMS to respond to an export rate. The simulation results show that the estimated energy cost savings for houses with HEMS and energy storage under an export rate is 20% compared to the baseline scenario with no HEMS. The results also demonstrate that the proposed HEMS responding to an export rate increases the self-consumption of buildings more than HEMS responding to a TOU rate.