There is clearly a counter-argument that the minimum wage is a market intervention and can cause inefficiencies (it may result in some folks who just can't be trained losing their $3 per hour jobs).
That's an inefficiency, but it seems to me that a far more central one is embedded in the assumptions of your toy model: how many unskilled jobs ($3) funge against skilled or semi-skilled ones ($10). In practice, it seems to me that the kind of jobs an employer can offer are often narrowly constrained by business requirements.
A factory owner, for example, might be able to retrain unskilled line workers (fitting Subwidget A to Subwidget B) to do semi-skilled work (operating a widget-fitting machine) for higher total productivity; that's consistent with your model's assumptions. But if you run, say, a hardware store, someone's got to stack shelves, mop the floors, and run the registers, all of which take roughly the same level of training, and there's only so many places you can squeeze out more per-body productivity by investing more. Anyone you have to fire because of minimum-wage laws there represents an economic loss: they aren't getting paid, and you aren't running as efficient a business as you could be.
OK, a fair criticism of the "toy" model, which was simplified to make the point. There are always multiple choices of productivity and wage level, and big moves (more than doubling employee productivity, while simultaneously quadrupling the cost of labour) usually can't happen quickly.
Back in the real world, I did a quick look at the economic evidence, and was surprised. The latest evidence base is that the minimum wage has surprisingly little effect on anything. It seems to have no discernible effect on employment levels - see here - but it has ...
Note: Originally posted in Discussion, edited to take comments there into account.
Yes, politics, boo hiss. In my defense, the topic of this post cuts across usual tribal affiliations (I write it as a liberal criticizing other liberals), and has a couple strong tie-ins with main LessWrong topics:
The issue is this: recently, I've seen a meme going around to the effect that companies like Walmart that have a large number of employees on government benefits are the "real welfare queens" or somesuch, and with the implied message that all companies have a moral obligation to pay their employees enough that they don't need government benefits. (I say mention Walmart because it's the most frequently mentioned villain in this meme, but others, like McDonalds, get mentioned.)
My initial awareness of this meme came from it being all over my Facebook feed, but when I went to Google to track down examples, I found it coming out of the mouths of some fairly prominent congresscritters. For example Alan Grayson:
Or Bernie Sanders:
Now here's why this is weird: consider Grayson's claim that each Walmart employee costs the taxpayers on average $1,000. In what sense is that true? If Walmart fired those employees, it wouldn't save the taxpayers money: if anything, it would increase the strain on public services. Conversely, it's unlikely that cutting benefits would force Walmart to pay higher wages: if anything, it would make people more desperate and willing to work for low wages. (Cf. this this excellent critique of the anti-Walmart meme).
Or consider Sanders' claim that it would be better to raise the minimum wage and spend less on government benefits. He emphasizes that Walmart could take a hit in profits to pay its employees more. It's unclear to what degree that's true (see again previous link), and unclear if there's a practical way for the government to force Walmart to do that, but ignore those issues, it's worth pointing out that you could also just raise taxes on rich people generally to increase benefits for low-wage workers. The idea seems to be that morally, Walmart employees should be primarily Walmart's moral responsibility, and not so much the moral responsibility of the (the more well-off segment of) the population in general.
But the idea that employing someone gives you a general responsibility for their welfare (beyond, say, not tricking them into working for less pay or under worse conditions than you initially promised) is also very odd. It suggests that if you want to be virtuous, you should avoid hiring people, so as to keep your hands clean and avoid the moral contagion that comes with employing low wage workers. Yet such a policy doesn't actually help the people who might want jobs from you. This is not to deny that, plausibly, wealthy onwers of Walmart stock have a moral responsibility to the poor. What's implausible is that non-Walmart stock owners have significantly less responsibility to the poor.
This meme also worries me because I lean towards thinking that the minimum wage isn't a terrible policy but we'd be better off replacing it with guaranteed basic income (or an otherwise more lavish welfare state). And guaranteed basic income could be a really important policy to have as more and more jobs are replaced by automation (again see gwern if that seems crazy to you). I worry that this anti-Walmart meme could lead to an odd left-wing resistance to GBI/more lavish welfare state, since the policy would be branded as a subsidy to Walmart.