One other "classic rebuttal" is of the claim that increases in the minimum wage do not drive inflation. Now, we can either check the data, or we can think about what we would expect to see. Let's do the latter and see where it takes us.
If, as data assure us, less than 2% of the workforce is paid minimum wage, but increases in the minimum wage tend to put lower-skilled workers out of work, we would expect that those who employ them would strike a balance of higher efficiency (self-service gas stations, burger-flipping machines) and shifting a portion of the cost on to consumers. On the flip side, those lower skilled workers who have lost their jobs do not have as much to spend.
In short, we would expect something of a wash inasmuch as minimum wage could drive inflation (exactly what a quick review of historic data suggests), and that monetary policy, the overall state of the economy, tax policy, and spending policy would dwarf the effects of the minimum wage--it's at best #5 on the Pareto for variables influencing inflation.
Translated into non-statistician's terms, it means that designing an experiment to test the hypothesis is in itself a fool's errand almost guaranteed to confound the measurable mean shift of your test variable with the ordinary variance of your more important variables. Translated into parents' terms, it means that at least one more person with no clue about experimental design is probably at some college teaching economics to your kids, telling them from behind a tweed logic-proof jacket that it matters that increases in the minimum wage do not consistently result in inflation spikes.
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