The Baumol effect says wages for doctors will also have to go up. Society can afford this because it now has commensurately more resources due to increased efficiency. It’s a tide that raises all boats, precisely because of this effect. This is why a taxi in London costs and pays better than the same service in Cairo.

> Society can afford this because it now has commensurately more resources due to increased efficiency.

Does it though? Suppose that Wall St has discovered a strategy, like high frequency trading, that produces nothing but allows the one doing it to extract a margin that would otherwise have gone to the second-fastest trader. Many people are employed in a competition to be the fastest because being the fastest is rewarded but it's a zero-sum game where nothing useful is produced and the players each have to continuously spend resources to keep running faster in order to stay in the same place.

What benefit is the person now paying more for healthcare getting in exchange for this?

> It’s a tide that raises all boats, precisely because of this effect.

What if it's not all boats? Suppose it causes doctors to get paid more because people who have the wherewithal to become doctors could also work in finance, but it doesn't cause retail clerks to get paid more because Wall St isn't hiring them away from their existing jobs, and in the meantime they now have to pay more for healthcare.

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.

It’s not like their wages will always go up exactly in proportion to your income. Goods and services that are afflicted will become less accessible if your own wages increase at a lower rate.

Will doctors’ pay go up enough to retain the same number of doctors?