Can't say if such studies exist, doubt that, so as doubt the usefulness of that for certain reasons from an economic point of view.
Fundamentally one of the reasons is that it like tax or inflation and there are simpler behavior patterns to be set in motion by that - the same as if there is inflation - people demand a pay rise and all that, raising prices of service to offset expenses and all that.
and not sure if there are possible some generalizations as sure there are commonalities but so as specific differences, country local environment-specific. it may affect differently not only different cities but different districts in one city.
Another reason is, look there always have been cheap and small cars as an example, but yeah their share stays small as well - those who look for alternatives they do that for a set of different reasons and gas price, not necessarily even top 10 reasons on their list.
So there would be hard to establish some magic numbers for a variety of reasons Dependency u looking for is hard to establish in a much simpler case of grocery prices.
So more important is where existing tax money is spent in? Are they invested in the development of energy-saving solutions etc? Do that money work in finding implementing solutions? is it enough money for the changes? etc.
Let's look at an ideal car - it is 100 percent efficient, very small but combustion engine so it burns gas anyway. And if one has to use the car for personal transportation needs, one has to pay the gas price no matter how much it is. And it not possible to improve it further no matter the efforts. if u like to investigate the mass behavior patterns it may be not that bad to start with the USA 1973 1979 oil crisis.
And then the question is not does the tax changes behavior of individuals, but attractive solutions available to them, do they have to do certain things or don't. people are lazy energy-saving creatures and if one offers better convenience people go for it, one of the reasons small cars share isn't huge.
Modern electric cars offer certain conveniences, in certain cases, but not always.
So it needs to look not for a golden number, but it has to be low enough to not trigger inflation behavior, and the created monetary stream has to be directed where it brings changes, in an efficient way.
work with energy-demanding industries to find solutions, energy-producing devices etc etc. So the effectiveness of carbon tax is limited from the start, it just another straw to a camelback, and it is important what else one does.
And it would be a mistake to think that all the solutions lay in where energy is consumed or produced. Changes in city planning, how the work is done, delivery of goods and services - that may have a greater effect on behavior changes. And remember nuclear energy is carbon-free.