Economics has a few basic rules that can't be evaded. A bank can't give money away and remain in business. It's like the law of gravity: if you jump off a tall building you are going to land hard and break bones.
There are other economic laws that some people will tell you are engraved on stone tablets. The minimum wage is supposed to be one of those. It's supposed to be part of the rules of the "free market".
It's not necessarily true.
Some laws are more unchangeable than others.
There are laws of electronics that say you cannot turn off a DC motor.
If you have a lot of electricity flowing through a big coil (like you would find in an electrical motor), that electrical current does not want to suddenly stop.
If you throw open one of the old style switches, the current wants to continue flowing so badly that it will jump the gap like a small lightening bolt. When motors were first being used, a lot of people were badly burned openinging that kind of switch.
You can make a chart of the math that describes the current and voltage going through that motor. And if you look at the chart and consider the voltage and current, the equations will absolutely prove that you can't safely turn off the motor.
An analogy is a large grindstone. If you get a grindstone moving, you can't stop it instantly. It wants to keep rotating in the same way that the electric current wants to keep flowing.
Fortunately engineers are smarter, or at least more creative, than economists.
When you tell an engineer that turning off a big motor causes a dangerous spark, the engineer looks at what he wants to happen and figures out a way to make it work. It turns out that adding a capacitor and/or a resistor gives the current a path to travel that doesn't do any harm.
If you had to do something similar with a grindstone, you could build a mechanical gadget with a brake and some kind of windup spring that would absorb the energy of the turning stone.
We have a few "laws" of economics that are supposed to the the same as the ten commandments. They are supposed to be part of the gospel of "free enterprise".
You might notice that when a bunch of people got their hands burned by electrical switches, no one advocated the gospel of "free electricity". They hired some engineers and told them to figure out how to build motors and switches that served human needs.
If the law of supply and demand is used to set wages, this means that some of the time, (when you have enough people available), the value of a person's labour might be less than what it takes to keep him alive.
The charts and the equations would show that some part of the population has to die of starvation. It's not the same thing as a spark jumping a gap and burning someone, but it is a fair analogy. Human beings constructed an economy -- something that is supposed to make life better. But some of the time some of us get burned.
The job of economists is to describe what is going on. In general economists are not engineers.
This does not mean that the laws of supply and demand are nonsense and it doesn't mean that the rules can be changed at will. It's possible to have a minimum wage rule that works, but, in general, you can't make it work unless you can make it apply equally to all employers.
And even then, there are limits. You can't have a minimum wage of $100 per hour in todays money. No one knows exactly what would happen if you tried, but it's easy to figure out that whatever happened would not be good.
Most of the predicted bad consequences of rent control actually happened in the real world. Vast amounts of New York real estate became worthless. Rent control made it impossible to fix up a property without throwing away money. Landlords simply walked away and abandoned the buildings.