I wonder how this relates to what economists call "cost disease?" As the cost goes down in some industries (through automation or other efficiency improvements), this increases wages in those industries, which spills over into other, traditionally lower-wage industries due to competition for workers, not through efficiency improvements.
But despite calling it a "disease," this doesn't necessarily seem like a bad thing. Often they're difficult jobs (high emotional labor), so they maybe they should pay more?
That's not exactly correct; the real issue with Baumol's Cost Disease isn't that it leads to wages in "traditionally lower-wage industries" — it's that wage competition leads to wage increases in sectors that don't undergo productivity gains.
To give an example: medical doctors aren't a low-skill/low-wage profession, but their productivity in crude terms of patients examined per hour appears stagnant — medicine is inherently labour-intensive (although the outcomes of care provided can improve over time as new and better treatments become available). If wages rise in other professionalized sectors that require huge amounts of training due to productivity gains (e.g. among engineers), wages will consequently rise for doctors because doctors and engineers are loosely coupled at source through the training/labour market.
TLDR: cost disease isn't about low-wage occupations, it's about pay in occupations that are refractory to performance improvement being coupled to pay in sectors where performance gains are possible via the labour market.
Charlie: the concept is highly analogous to Amdahl's Law, in some regards. The limit to performance increases in parallelisation is limited by the non-parallelisible portion of a process. The limits to productivity increases are limited by the irreducible labour component of a productive process.
The other element is that wages represent the provisioning costs of labour, or as Adam Smith puts it, "A man must always live by his work, and his wages must at least be sufficient to maintain him." (Further qualified upwards in subsequent text.) WoN, Book 1, Ch. 8.
Which is to say: wages are not defined by productivity, but by the general wage level. What is defined by productivity is instead the quantity demanded of a good.
If some quantity of a high-labour good is demanded, then that quantity will pay wages based on the minimum prevailing wage (and if that's a sustainable level: the minimum living wage), plus additional premia for various other factors elevating wages above that floor, as described by Smith in chapter 10 of book 1.
But despite calling it a "disease," this doesn't necessarily seem like a bad thing. Often they're difficult jobs (high emotional labor), so they maybe they should pay more?