Kind of. The profit margins stay low by lowering wages and prices together, and the profits increase by having more volume. Price competition isn't about increasing volume, but REDUCING the volume of a competitor—in a microeconomic sense (note: I have not studied microeconomics and may be abusing the term here), it's about drawing more buyers and more profit; but in a macroeconomic sense, for the given quantity demanded, drawing more buyers typically means people recognize Brand X costs more than Brand Y and defect from Brand X to Brand Y if their brand loyalty has less marginal utility than the difference in price.
This is sort of an arms race: Brand X then reduces price—paying labor less is a good way to reduce cost, allowing such reduction. The backstop is the minimum wage.
Another thing that happens is people want $20/hr, but at $20/hr it's cheaper to switch to a lower-labor method involving higher average wages—say, $18/hr. So those laborers can be jobless or they can take $18/hr; if retraining to find a new career has a perceived cost of more than $2/hr, they'll just take lower wages. Eventually the job becomes a minimum wage job, and laborers can't lower their wage prices, so they are replaced by the new, more-productive method; raising the minimum wage hurries this along, and the relationship I'm looking at is the ratio between that cut-off wage and the minimum wage, and how that ratio changes with the relationship between minimum wage and per-capita income.
I don't care about GDP here. You can vastly increase GDP by expanding the labor force and not increasing productivity. GDP/C increases when your labor force participation rate increases OR productivity increases. I'm interested in productivity, i.e. GDP/hr for each hour of labor.
