I agree with you that a minimum wage law is effectively a ban on pay for very unproductive work, but I disagree with you on what that implies.

If someone can't support themselves, then I believe that the wider community should help.

The two groups of people which I have in mind are: 1) those who can't produce enough value to support themselves, and 2) those who can produce enough value to support themselves, but who are not paid a large enough fraction of that value to actually do so.

My understanding is that X is set high enough so that people in group 2 will get a large enough share to support themselves (so it moves people from the poverty-stricken group to the somewhat-poor group at the cost of company profits) while still being low enough such that anyone who produces less would fall into group 1 anyway, and would therefore need other more direct help, independent of any minimum wage laws.

Is your position that for any X, some people will be forced into group 1 who would otherwise consider themselves self-sufficient, and that the cost to them outweighs the benefit to those in group 2?

> What about the people who are unable to produce at least $X of value per hour? They don't matter?

They get fired as soon as statistics show they don't meet their targets. We already have tons of those jobs in existence and the rise of technology only oversimplified this further. Case in point: a call center. Say the goal is you need to deal with X customers in an hour. That doesn't mean their problems are solved. And even if the problem is (seemingly) solved that doesn't mean their problems are solved in the best way possible. And yet, this is the basis of how call centers operate these days.

What happens when people get fired? Well, in some countries you get social benefits and in some you don't but there's no country where those benefits are going to pay your bills AND allow you to enjoy any form of luxury. So what happens either way? They look for a new job. Preferably legal job, but if they can't, they resort to shady businesses. Grey areas. You know, like Uber*?

What if they can't provide more value than enough to get food and shelter?

Then they should still work at $4/h or whatever and the government should top them off to $10/h with basic income or welfare.

It's more efficient and humane than basically telling them they're worthless and they should just stay unemployed.