I don't think renting is better in CPH than in Stockholm. Right now I'm paying 10.000DKK (1500USD) for a one bedroom apartment, not in the center (NV). It was the only option that didn't involve any shady stuff like no-contract or not being able to register myself in the apartment. Most of my friends live doing one of those, or renting a room for around 5.000DKK (750USD).