[derf_]

Exactly. Even though Baumol himself used the phrase "Cost Disease", I think that framing distracts from the fact that it is a result of something desirable happening, namely increased efficiency in some sectors. You could also posit a case where some sectors become less efficient, due to badly conceived regulations, exhaustion of non-renewable resources, an unchecked monopoly, or some other factor, but you don't need a special mechanism to explain why prices rise in such a scenario.

> ...consider a case where finance becomes much more productive... leading to fewer people becoming doctors because finance is much more attractive.

This is the opposite of what one would expect from a sector whose efficiency increases, as modeled by Baumol. See the first bullet in the article: "The share of total employment in sectors with high productivity growth decreases, while that of low productivity sectors increases" (also see the detailed analysis in the Technical Description section). It might be theoretically possible that induced demand could still increase overall employment in a sector as its efficiency increases, but I think you have to make an argument why that would be true. During the industrial revolution, automation eliminated 98% of the labor required to produce a yard of cotton cloth, but between 1830 and 1900 the number of weavers in the US increased by a factor of 4, because demand increased due to lower prices [0]... although the US population also increased by a factor of 6, so as a percentage of the workforce weavers still declined, even as people consumed much more cloth per capita.

[0] James Bessen, Learning by Doing - The Real Connection between Innovation, Wages, and Wealth (2015), pp. 96–97.

[wat10000]

I picked finance for my example because demand is practically unlimited. People only need so many clothes, but when your business comes from making money directly, there’s a lot of room for growth.

Imagine some new math allows HFT to make more money. HFT firms wouldn’t start laying off quants. They’d probably hire more to try to capture more of that new money, and they’d have more money available for hiring.