Life is cheap everywhere in China.  I suppose, you could buy someone for less than 6k and force them to teach you Mandarin.  But at 6k they wouldn't be very useful at anything other than menial labor.

If you are talking about cost-of-living, very standard Mandarin, living on a budget, if you can endure the climate, then Harbin might be the answer.  Mandarin is the "lingua franca" of Harbin and its "Mandarin" is quite favorably to that of Beijing, sometimes more so. Actually there is no real "Harbin talk" as they say around it, because Harbin was created artificially. Things in Harbin are cheaper than other places as it is considered believe it or not to be an economically distressed area.

Beijing is outrageously expensive, except for the buses and subways, and Shanghai is wonderful but Shanghai Mandarin is so influence by the local Wu dialect that even people from Beijing wince at it sometimes.

Guangzhou is a great city but the Mandarin there is not standard at all, given the almost usage of Cantonese by the locals.  The migrants there speak Mandarin but they tend to be the day-workers and peasants from the countryside and the qualiy of their Mandarin is hardly worth mentioning, sad to say.

Just my thoughts.