Elysium and its government
The government was a democracy with a caste system built in. There were people of power, they were the citizens of Elysium. The caste system was obvious when the people that lived there had generations of family on the floating city. The people that live on Elysium stay on Elysium. The same goes for the poor who don't have enough money or power to get to Elysium.
Purpose and Goals of the Government
The Social Contract
Executive, Judicial, Legislative
The judicial branch really applies to the secretary of defense and the robot police. Delacourt interpreted the law and the robot police are the ones who act on them.
The legislative branch wasn't really well represented. John Carlyle could be considered to have "made" the legislative branch. He created Elysium and all the rules/programs that Elysium runs on.