There aren't enough jobs with that number of hours because companies have per-employee overhead and will choose someone who can work full time over two part-time employees. So then you get stuck with the full time job you hate that pays more than you need instead of the full time job that you enjoy that pays exactly what you need.
Moreover, even if you could find a job working only three hours a day, you're still commuting two hours a day to get there when the lower-paying job was closer, so now you're still spending almost as much time and ending up with even less money because you don't get paid for commuting and you have to pay for transportation.
But you can't end up with less money since that's not enough, so now you have to work more hours. You need the full time job in order to pay for the transportation to get to it. Which means all you've done is add two hours of your own time commuting while trading a job you enjoyed for one you don't.
> And for the person getting the training, that still doesn't affect their cost of living. Training isn't cash in hand.
Schools don't pay cost of living for their students. If you don't have the money you're expected to do some other paid work to come up with it.
The employer offering the training could offer you more paid hours, but that's not going to result in exceeding a minimum wage when averaged in with the negative $/hour training time. Especially if the extra hours are counted as overtime even though they wouldn't be if the employer and the school were separate entities.
> I'd personally target the living expenses for a single person, living in a studio or 1 bedroom apartment, with the expected average commute for the area, and who is generally healthy.
Which is completely arbitrary, since that is neither the average person nor would you want it to be. In fact, people making minimum wage tend to have longer commutes, because they have a low skill level and you prohibited them from trading lower wages for a shorter commute.
It also doesn't make any sense. If you're anyone other than that person, you may not need as much (e.g. you have a roommate or live with parents), so why shouldn't you be able to take a job that you prefer for non-wage reasons over one that pays slightly more? Or on the other side, if this is regarded as necessary (i.e. we're not going to solve the problem in a way that would make this unnecessary, such as a UBI), you'll be screwing over anyone who doesn't qualify for your arbitrarily structured assistance programs. And the assistance programs are frequently poverty traps because overlapping means-tested assistance programs cause their recipients to be subject to excessive marginal tax rates -- sometimes exceeding 100% of their marginal income, not that 50+% is anything reasonable in the alternative.
The issue is that there is no such uniform amount.
If your spouse makes 85% as much as the two of you need to live with dignity, you don't need to make as much as they do and may prefer not to because the lower-paying job is less difficult or more satisfying. For someone else it could be 95%, or 99%, or 250% and the second job is pure supplemental income, so what's the minimum meant to be here?
The minimum wage doesn't account for costs. A job across the street from you means you don't need a car, don't have to pay for fuel or maintenance or transit tickets and don't have to spend hours in traffic (which you could have spent working and getting paid). For a long commute this difference in value can be more than the entire minimum wage.
It's possible for jobs to provide training, which is attractive for entry-level workers or anyone trying to change occupations or learn new skills, because school otherwise costs money. But if you're spending extra hours "at work" getting trained while the value of the training comes out of your paycheck, there isn't any sensible "minimum wage" for this. You could be receiving $50,000/year worth of training while "getting paid" only $5000/year and you're the one coming out ahead because you don't otherwise qualify for a job that pays $55,000/year. You might still be ahead if you were paying them $5000/year.
So how are you supposed to set a minimum wage if the "minimum amount required to live peacefully and with dignity" is different for everyone, and setting it below that threshold deprives people of valuable alternatives?