Now, if the argument was "don't raise the minimum wage because we want people to keep their jobs temporarily while we spin up xyz programs to help for when then inevitably lose them to automation" then, OK, that seems reasonable. There's a plan to move forward from the inevitable.
http://i.investopedia.com/inv/tutorials/site/economics/econo...
is not a free market model! Not at all! It is a "perfectly competitive" model and has several very specific assumptions:
1) All firms sell an identical product
2) All firms are price takers - they cannot control the market price of their product
3) All firms have a relatively small market share
4) Buyers have complete information about the product being sold and the prices charged by each firm
5) The industry is characterized by freedom of entry and exit.
I'm not sure the labor market actually meets any single one of those criteria.Furthermore, if you are interested in how minimum wage actually effects employment I would check out the following report(here's a good quote).
The employment effect of the minimum wage is one of the most studied topics in all of economics. This report examines the most recent wave of this research – roughly since 2000 – to determine the best current estimates of the impact of increases in the minimum wage on the employment prospects of low-wage workers. The weight of that evidence points to little or no employment response to modest increases in the minimum wage.
The paper I linked mentions a dynamic monopsony model and also 11 different channels through which adjustment is possible. Basically the market is lots and lots of different forces (not just supply and demand) all originating from (not always rational) people, in this case we need a model which more accurately models this.
If you'd like to read more about how the proliferation of the basic supply and demand model has really hurt the layman's understanding of the economy there's a great book that came out recently called "Economism." Here's a short article which can give you a bird's eye view: https://www.theatlantic.com/business/archive/2017/01/economi...
There's three things that can give:
1) Unemployment (workers eat shit)
2) Inflation (customers eat shit)
3) Profits (owners eat shit)
I'll give you a clue which group takes the brunt of minimum wage increases. Take a wild guess who paid for this expensive times sq advert:
https://pbs.twimg.com/media/CNsVNeGWUAE3T7p.png
The low pay commission study actually found no effect on employment and no effect on prices except for a very slight non-commensurate rise in one industry (canteen services i think).