Your example, although meant to show that the point was silly, can still serve to illustrate it. Imagine that for the vast majority of people, widgets are worth $3 each, but for one person they are worth $10 each (because he really likes them). Further, suppose that a widget takes an hour to make. With a minimum wage, one person can make $10/hour serving the one person who want widgets that badly, but without one it becomes profitable to open up an entire new market sector of mass-produced widgets for the common man.

You can change the numbers and the details, but the general principles of microeconomics will prefer the free market every time.

To carry that example further, you have many people that are willing to pay $3 per widget. But there are barriers to opening a widget factory, such that there is only one widget factory in town. And the owner of that factory only wants to pay worker 50 cents per hour, selling the widgets produced for $3.00. Now total money paid to workers is much lower than if minimum wage is set to $3.00

So the moral of the story is minimum wage should be raised to the point where total employment is still at an acceptable level.