So lets work with the idea that pay does not represent the productivity/hr in dollars. I would think that your view is that the fast food worker is making 10/hr, but is actually producing a value of 16+/hr.
First, do you have any data to show that people are producing 16+/hr, or is that a belief without data? I would ask that you provide that, because it would be interesting to me. The question then becomes, how are companies able to get around wage competition? Assuming that there are enough jobs to employ everyone, why are wages not higher?
If firm A has 10 jobs that if each filled, would make 20/hr each in profits, and firm B has the same number of jobs that make the same profit, and there are only 10 workers, you would see competitive pressure to raise wages. This is because each firm wants to make the most money possible. If Firm A were to offer 15/hr, and Firm B were to offer 16/hr, then Firm A would make 0 profits, and Firm B would make 3/hr per employee. Firm A would then have to decide if they are willing to go above 16/hr to try and capture the 3/hr in profits. This is a simple situation and reality is a bit more complex.
So assuming people are producing far more than the wages they are collecting, must be due to downward pressure on wages. What most economists would say, is there are likely 2 main reasons for this. First, its possible that due to unequal information, people are not able to negotiate a higher wage, because they dont know what their worth is. That seems strange, because you would think that if companies were willing to pay more, it would be worth their time to lure those underpaid employees away. The other main idea, is that we have far more people fighting for low paying jobs, than there are low paying jobs. In that 2nd case, the competition on wages is on the workers, who do the same thing as I was describing before. Either they get a job for 10/hr, or they dont work at all.
I will assume that your argument is, that if its the 2nd idea, that companies should be forced to pay more to make up for labor competition. The problem with that is net job loss, at the expense of higher wages.
I said a lot though, so before I go into that part, I will leave this open to allow you to respond if you wish.