In this paper, we explore whether there is a potential for shifting load between different times of the days, i.e., between peak and off-peak hours. In particular, we explore whether the fact that electricity is a necessity to modern life puts restrictions on the possibilities for load shifting. To do that we provide a structural framework for peak and off-peak electricity demand, where households are assumed to have Stone-Geary utility functions with subsistence levels for electricity demand that varies within the day, and that depends on household characteristics and temperature. As an empirical illustration, we fit our model to Swedish data on residential electricity usage at the sub-daily level. Our results indicate that the potential to shift load from peak to off peak is limited. One reason for this is that the subsistence levels are larger during peak than off-peak, implying that households assign a high value on electricity during peak time, relative to off-peak time. Overall, the results have important policy implications, not the least with respect to effects of real time pricing, as it suggests that there are limits to households' price responsiveness.