I feel like I pre-addressed this with the second paragraph of the post you're responding to, so I'm going to repeat some of what I said there. It seems like we agree now that prices aren't going to rise proportionally with the minimum wage but would at most increase by the added cost of the increased minimum wage. And I had two things to say about that. First, this is not a huge price increase because the minimum wage is not a huge factor in costs. Second, not all (or likely even most) of this increase in cost due to the minimum wage will be passed on to consumers. You suggest here that basically all of it will be, but I note again that your claim is essentially that companies all operate as monopolies - your theory is that they effectively collude to avoid competing with each other on price and can increase profits at will by raising prices.
Also, your theory seems to imply that there's basically no way to reduce inequality. Like, suppose we start taxing very high incomes (including capital gains) at very high rates. Doesn't this look to the owners of capital like they're "losing a portion of their current profit to 'extra'" taxes? So since they're all seeing this happen don't they all decide to raise prices or cut jobs to make back what they're losing in additional taxes? This seems kind of ridiculous, right? I mean, there's a reason they lobby so hard for lower taxes - they think that lets them make more money.
I want to gently suggest that you're looking for a story to tell to justify opposition to a minimum wage increase rather than getting a position on increasing the minimum wage out of a consistent and informed theory of economics.
The reason I'm trying to argue using a story is that I can give first-hand opinion on it, rather than 100% speculation on what may happen at the top and bottom of the payscale (although we're all definitely offering some speculation). We agree that prices will, whatever the magnitude, rise across the board. I believe that rather than all acting as pseudo-monopolies, corporations are simply acting towards the best interest of their profit. If there is, in general, "more money" out there in the populace, why wouldn't a company raise prices? If their supply remains the same, and the ability for consumers to purchase that supply increases (due to more income), it would just make sense for them to raise prices, and people would be willing to pay it because "I'm making more now, so I can afford an extra $xx.xx for this."
Bringing this (which we agree on, just seemingly not to the same absolute amount of price inflation) back to apply it to my story, would hurt me financially. If my $24/hour right now is sufficient at the moment according to a market that sells a 2-L bottle of soda for $1, it would be less sufficient in a market that has adjusted to a higher minimum wage, selling a 2-L bottle of soda for $1.25.
I guess what I'm trying to argue is that raising minimum wage 100% positively for sure helps the workers earning minimum wage. It will make it much easier for them to live a less stressful life, with more breathing room in their finances. And I do think it will reduce inequality. However, it's going to harm the middle-class much more than the upper class. For instance, a change from $8 to $16 minimum wage (as an example), would be like going from 0.001% of the CEO's salary to 0.002% of it. For somebody making, say, $32/hour, it's like going from 25% to 50% of their wages. That is a huge relative hit to a mid-level employee's wages while not even draining a drop from the CEO's bucket.
Your example of higher taxations is exactly what I think needs to be implemented (along with whatever else actual economists think will help, I don't know crap about in depth economics). There's a big difference in the company itself losing money, and the consumer-base gaining money. A company can't just raise prices because it is making less money, because demand isn't going up. A company is totally capable of raising prices and succeeding with the new prices if demand does go up, which is exactly the case with a larger pool of expendable income.
I'm sure a lot of what I'm saying isn't being well-explained and I'm rambling. But I 100% stand by the fact that just raising minimum wage won't magically solve all of the country's current financial problems without creating others. And I know you specifically aren't advocating that, I just know that a very vocal group of people demand higher minimum wages and refuse to address the "but then what?" question that it requires.
And I'm probably sounding very selfish or close-minded with this, and I apologize for it! But it is a selfish issue. From my point of view I would say "yay I'm happy for those employees that are finally making a living wage, but now I'm living paycheck to paycheck rather than having some savings every month." Again, some hyperbole but the point still stands. In order to not harm the middle class while still helping the lower class, there would have to be wage bumps across the board (up to $100k/year, or $150k/year, or whatever people deem the "middle class"). And if that is the case, all we're doing is artificially inflating the dollar.
Edit: That was a really long reply, sorry