Your city produces base commerce. This base commerce is then divided into luxuries, science, and taxes. Then building bonuses take over.
Each citizen produces a certain amount of commerce. An unroaded square not by a river produces 0. Roaded squares not by a river or squares by a river but unroaded produce one (as do sea squares, I think). Roaded squares by rivers or coast squares produce two. Various resources can increase this.
Now, the answer to your question is that it greatly depends on the commerce of the city.
City 1: Size 6 with all 6 citizens working producing 10 base commerce. This gives 3 happy faces from lux tax. One born content and 2 MP and it's good to go in Monarchy. Change to Republic and the citizens produce 17 base commerce, none of which is lost to corruption (in this example). 30% luxes gives 5 happy faces from lux tax and one borne happy and it's fine.
City 2: Size 6, not as developed. All 6 citizens working but 3 are on undeveloped squares, so you have only 4 base commerce. That's one happy from lux tax, say 3 from MP, one born happy, and a temple. Change to Republic and base commerce is only 8 (only the three citizens getting commerce initially plus the central square gain the bonus). Lux tax gives 2 happy at 30%, which isn't enough in Republic, with the loss of MP.
City 3: Size 12. All citizens working developed squares gives 20 commerce in Monarchy. That's 6 happy from lux tax at 30%. One born happy means 5 happiness is coming from other sources (say 2 MP, a temple, and two luxes). Changing to Republic gives 33 base commerce, so 30% lux tax gives 10 happy faces. That's 14 with a temple, two luxes, and one born happy. You can almost get almost away with 20% luxuries (7 happy + 1 for temple + 1 born + 2 lux = 11 is CLOSE) in this city.
The key is the base income from worked squares, for happiness. Lux tax is a straight percentage of that and has no multipliers.
HTH,
Arathorn