Final answer:
The structure described with a hereditary elite class topped by a king and a royal family pertains to both chiefdoms, when expanded, and monarchical states.
Step-by-step explanation:
The power structure you're describing is characteristic of both chiefdoms and states, but the very specific hierarchy with a hereditary elite class and a king or royal family as the highest authority is particularly indicative of a monarchical state. Chiefdoms, while they can grow into states, typically have a single chief who holds the economic, political, religious, and military power. As chiefdoms expand, they might encompass multiple towns and ethnic groups, and the chief might then be referred to as a king, which aligns with the description given.