GDP is composed of four components. One of those components is government spending, which can be negative (ie, taxes) or positive (subsidies or direct purchasing).
However, government spending doesn't come from nowhere (no, even if the money is printed or issued as debt). A change in government spending is always matched by an equivalent change in consumption, investments, and/or net exports, which are the source of that money.
The secondary effects of tax policies may be to ultimately change the shape of aggregate supply or aggregate demand functions (at least, that's the argument for doing so), but taxation does not directly have any effect on GDP.
So no, taxing air would not increase the GDP.
That's an oversimplification that honestly provides a more confusing definition than the real one, which is that GDP is the sum of consumption, investment, government spending (which can be negative) and net exports.
> If it was taxable (privatized), it would increase the GDP, whereas if not (as if the government performed and didnt tax it), it would not increase the GDP.
No. Whether a service is performed by the private or public sector has no bearing on how much it counts towards GDP.
GDP can be measured in several ways. In a perfectly observable system, all the measurements would be exactly equal. They are not different models and they do not represent different underlying concepts. It is wrong to say that taxes are included in one measurement of GDP and not the other. In some methods, they are measured directly, and in others, they are measured indirectly. In both cases, taxes have the same effect.
> If a government decides to not tax air, no economic activity. If they do, and a company sells you taxable air, there is economic activity. In the former case, it no money is required. In the latter, it generates economic activity, activity that cannot be swept under the rug and must be measured.
Your usage of "economic activity" isn't really well-defined. But the difference between the two situations you're describing is that, in one, air is free, and in the other, air is being sold. The fact that it's being taxed on top of the sale is irrelevant.
This is also a really pointless example because air is mostly a non-excludable resource, so talking about either "selling" it or "taxing" doesn't reveal anything meaningful.
> It does matter, gov vs private industry.
No, it does not. Taxation doesn't come from nowhere. From a macroeconomics perspective, the government isn't "special".
Let me try one last time: Remember when I said that GDP = consumption + investment + government expenditures + net exports? Some economists would instead say that GDP is really "just" consumption + investment + net exports. And guess what? Regardless of which school of thought you subscribe to, the end result is exactly the same. It doesn't matter whether you break out government expenditures separately from consumption, investment, and net exports or whether you include government consumption in "consumption", government investment in "investment" and so on. The end result is exactly the same.
It's true, if the gov't performs a service, it's included in GDP. And no, taxes don't increase GDP.
No, that's not how GDP works. The GDP measurement is independent of whether the action is performed by the private or public sector.
> Problem with using GDP, in my limited understanding, is that taxing things, even your breathing air, increases the GDP
No, that's not how GDP works either.