To clarify, badges allow Pokemon you receive in trades, transfers, and certain gifts (Like the Zorua in B2/W2 or the event Beldum in OR/AS) to obey you at increasingly high levels. This is obviously to prevent the player from sweeping the game with a powerful, high-level Pokemon from another save.
All the Pokemon who fall under this do so because of their Trainer ID, which is different for every save. If a Pokemon has a Trainer ID that differs in any way from that of your current save's ID, badges are required to have them obey. If a Pokemon has a Trainer ID that is completely identical to yours, they will obey you no matter what. Pokemon you have personally caught (and the majority of gift Pokemon) have the ID of your save.
So, to answer your question, yes. Your starter will obey you no matter what. You could level it to Level 100 without any badges and it will still always obey.