======== "If you raise the price of something–anything–it lowers demand"
I hate to break it to you, but this is completely wrong, I was being polite before, I'm being more direct now.
I know what you are trying to say, but you're using the wrong language, which is maybe an indication of the lack of understanding of the the underlying micro econ in this statement.
Changing the price of a good does not 'decrease demand'. It changes the supply curve - and given an identical 'demand curve' (aka demand does not change), there may end up being less consumption of that good.
But it also depends entirely on other factors, which I explained.
======== "If you're earning your employer $10/h and the minimum wage rises to $11, you will not have a job. Either right away if the employer realizes it, or eventually when they go out of business if they don't."
No - again, this is even worse than wrong, I don't know what to say at this point.
To put it in simple terms, wage changes happen all of the time, and even minimum wage is increased without necessarily affecting jobs.
In short, if minimum wage at Subway sandwiches go up from $14 to $15 - more than likely we'll see prices go up a bit, profits go down a bit, and probably not a change of employment.
(And again, the price of wages does not affect demand, it affects supply)
========= "Profits have nothing to do with it. "
"If a business has employees at some wage and the minimum wage rises above that wage, they will, at least eventually, replace the low-wage employees with new ones"
Ok, now either you're trolling, or you really are having a hard time understanding the basics of finance, economics or business here.
Yes, if an input cost is raised, then a business will adjust. You are correct to suggest they will likely 'do something'.
But what are their options: 1) 'Replacement' - no, they can't replace with cheaper workers - because of minimum wage 2) 'Automation' - ok, but what 'automation'? And at what cost? Do you think there are magic robots waiting to replace workers at Subway? Now, there definitely are a lot of new technologies, but, Subway is looking at that new tech irrespective of what 'minimum wage' is.
And profits have everything to do with it. A wage increase means the money has to come from somewhere: higher prices, lower output to suppliers, or profits. Subway sandwiches, faced with this dilemma - has to make a choice. It may be more rational to simply have less profits.
Now, as you hint, there may be no room for manoeuvre - maybe there is 'no way out' and Subway sandwiches is going to go bankrupt. That could happen, but in most cases, not.
========= "Otherwise, prices may rise. Not without an associated drop in demand."
No - again - a change in the price of an item does not change demand - it may reduce the amount of units sold. And in the case of Subway, yes, you're right, probably they will sell a few less sandwiches. But how many fewer is the key question, and how does it impact their business. It depends.
========= "Assuming you are talking about the US, "
I'm talking about everywhere.
Everywhere in the world, healthcare is treated differently from most other business sectors. Everything - drugs, materials, pricing, publications, communications (what you can say in an ad), what you can advertise, how you are allowed to bill - even in some cases that you must provide service to some people even if they cannot pay (!) - this is due to the asymetry in the system where providers have the power of life and death over customers.
In many countries, including the US, Hospitals cannot turn away patients and must treat in some cases, no matter what. Think about it: imagine if the US Gov told Ford and GM that they have to 'give a car' to people if they camp outside the dealership.
FYI here is 'supply and demand' [1] to help you with your understanding of what happens when prices change (demand does not change, possibly the amount of goods purchased does)
[1] https://en.wikipedia.org/wiki/Supply_and_demand