There shouldn't be a minimum wage without a minimum productivity.
We already have that. With at-will employment, any employer can decide what level that is and replace the worker if they don't meet that level. So higher min. wage would help the worker out, cause the employer already can get the minimum productivity they want, or fire the person.
Interestingly, I have seen an example of pay/productivity value in construction jobs.
Contractor 1 has been winning most bids for the past several years. He pays his craftspeople $20-$25/hr. I've always thought that his jobs are slow - seems to take much longer than it should.
Contractor 2 came on the scene about a year ago, significantly underbidding Contractor 1 (25% or more) - Does he squeeze his employees to cut costs and make bids? Nope, he pays his craftspeople $40-$45/hr.
Quality of the job is at least as good, but his jobs go MUCH faster. Contractor 2 completes the job in half the time. The craftspeople are paid almost twice as much, and are twice as productive. Where does his savings come in? Overhead. This is a pretty capital-intensive specialty - remote work. By being twice as fast, he cuts overhead in half (Example: Industrial compressor @ $8k/month, probably the same in diesel cost to run it). He also saves half the per diem/travel cost for his workers since they are on the jobsite half as long. His managers/QC/foremen don't make double what Contractor 1 pays, but somewhat better. Probably saves 60+% there on a per-job basis. One other difference - he doesn't cheap out on equipment, so he doesn't have all the breakdowns/downtime/productivity losses that Contractor 1 seems to have. Maybe his equipment overhead savings is only 45% instead of 50%.
The real kicker? He is using some of the exact same craftspeople that Contractor 1 used to hire.