Who Gets How Much in a Carbon Tax-and-Dividend?

I was curious about how a carbon tax-and-dividend would play out. The plan would collect a carbon tax at the wellhead, mine or (for imports) the port-of-entry, then distribute all tax collected back on an equal per capita basis.

Warning: Wonky Ahead

First, I'm sure this has been done by some economists already. At an AGU talk a few years ago James Hansen mentioned an economic study, and I asked him about it afterward and he told me the firm's name but now I can't find my notebook from then, but I think it started with an "E." 😒  Anyway I didn't Google much because I wanted to try to calculate it myself.

The most recent chart of income vs CO2 I could find was this 2017 study about the US, by Lutz Sagar of the Grantham Research Institute on Climate Change and the Environment, which splits 2009 households (HHs) into ten deciles:

Here the wealthiest decile emits about three times the CO2 as the poorest decile.

I'll assume the carbon tax-and-dividend has no administrative costs (it's easy to add them in if so). The US had 117.181 M households in 2009, so there will 1/10th this in each decile. I digitized the above graph with Web Plot Digitizer, which was fairly easy to use. Then it's just calculating, which I'll leave in a footnote below, but I should note here that something doesn't add up somewhere, because the chart implies US 2009 CO2 emissions were 3.81 Gt CO2, when they were actually 5.39 Gt CO2. That's a big difference, but I don't know what it's from -- maybe someone reading this can make suggestions. (One thing I'm assuming is that the study alloted the emissions from businesses, industries and land changes to households.) Meanwhile I'll plow onward.

Given that, here my results for a carbon tax of $40, $100 and $300 per metric ton of CO2, where a positive "net dividend" is how much a household receives above what they paid in carbon fees:

The inflection point is at about $46,000/HH/yr in 2009 dollars, which is about $55,000/HH/yr today.

Personally, $300/t CO2 sounds right to me, if not low. Perhaps $500/t would be even better. It would help the poor quite a bit, and the upper middle class and wealthy would barely notice it.

Footnote:
Here's my calculation for a carbon tax of $300/t CO2. (I didn't worry about significant figures here because this is just blog work):