Badges allow Pokemon obey to a certain level. Once you get all badges, any traded Pokemon will obey at any level.
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Yes, because whoever owns the egg when it hatches is considered the original trainer.
in Pokemon pearl / diamond you need 2 badges for any Pokemon up to level 20 to obey you need 4 badges for any Pokemon up to level 40 to obey you need 6 badges for any Pokemon up to level 60 to obey you need all 8 for any Pokemon (even traded) to obey in heart gold / soul silver its exsacly the same :)
As far as I know, no. If you want your Pokemon to obey you, youll have to do it the old fashioned way: beat the gyms. Each gym badge will extend the limit before Pokemon will disobey you. All 8 badges means all of them will listen.
It depends in which Pokemon game, but in the newest games (Diamond/Pearl) You will need to earn badges to get a higher level for them to obey. if you have all 8, all Pokemon will obey you.
Pokemon whose levels forty and under will only obey you when you obtain the 3rd gym badge.