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I'm trying to create a formula that would show:
If every citizen in the world had an average carbon footprint of X.X kg per month, global temperatures could rise by X.X degC by 2050.
I know there will be quite a lot of variance in the answer, and it certainly won't be perfect, but assuming all other factors are equal, I think it hypothetically should be possible to create a formula...
If every citizen in the world had an average carbon footprint of 16.1 Mg (1 Mg = 1 metric ton) per year, the global average temperature would likely rise more than 6.0º C by 2050.
However, there are some important caveats to this answer which I will detail below.
The question you stated might be slightly different from the question you intended to ask.
The quantified emissions per capita of American citizens may under-represent the emissions associated with American lifestyle if America is buying things made in other countries. For example, if a car is manufactured in Taiwan but used in America, then the emissions associated with production will be counted in Taiwan, and not counted as part of the per capita emissions of the American citizen. Globalization and cross-border trade makes it quite difficult to assess the "true" environmental impact of products. For further reading, look up scope 3 emissions or product lifecycle analysis.
To simplify this question, I'll just go with Wikipedia's number of 16.1 metric tones per person, per year.
For simplicity, I'm only going to consider carbon dioxide (CO2) and not other greenhouse gases like methane (CH4). I would also note that in asking this question, it's important to be careful about the difference in mass of elemental carbon vs carbon dioxide. One ton of carbon equals 3.67 tons of carbon dioxide.
Global average temperature does not change instantly in response to changes of atmospheric composition. Higher concentrations of greenhouse gases in the atmosphere causes more radiative forcing (heat), which over time causes the slow accumulation of thermal energy. So the temperature at some point in the future could be very different depending on when emissions levels changed.
Some scenarios of inquiry might ask what the temperature would be like if that shift had happened in the past, or if it happened gradually over a period of decades. For simplicity, I'm going to assume that global emissions suddenly jump up to a much higher level starting next year and stay at that level until 2050.
This is the largest, most difficult part of the whole question. How much will global surface temperature change as a result of changing composition of the atmosphere? This is a fertile area for ongoing science and there is plenty of nuance. For example...
The relationship between carbon dioxide and radiative forcing is logarithmic, at concentrations up to around eight times the current value, and thus increased concentrations have a progressively smaller warming effect.
And that's just talking about concentration of trace gases in air, without even getting to other challenging questions like oceans and soils as carbon sinks. Those other factors mean there isn't an even linear relationship between emissions and atmospheric concentration.
But to keep this answer short here's a very rough benchmark.
Climate sensitivity is around 3ºC for a doubling of CO2.
Average per capita emissions of CO2 on a global basis are 4.981 metric tons; for the United States it would be 16.1 metric tons. Current global emissions of CO2 are around 37 Gt per year, which would change to about 120 Gt per year if everyone in the world had an average carbon footprint like an American.
For a first order estimate and sanity check, I turned some knobs in The Global Calculator and was able to produce a scenario that forecast global GHG emissions of more than 120 Gt per year. This produced a warning:
The top of the possible temperature range exceeds 6º C and you are at risk of triggering feedbacks and impacts not represented well in climate models. The IPCC does not assess a high/low temperature range at this extreme level of emissions.
Looks like we'll have to fall back on our very rough ballpark estimates.
Atmospheric CO2 levels are expressed in parts per million by volume (ppm). To convert from ppm to gigatonne of carbon, the conversion tables of the Carbon Dioxide Information Analysis Center advise that 1 part per million of atmospheric CO2 is equivalent to 2.13 Gigatonnes Carbon.
So if we emit 120 Gt CO2 each year and all of it ended up in the atmosphere, that would raise atmospheric concentration of CO2 by 56 ppm each year. Over 30 years, that's a delta of 1680 ppm. The current concentration of CO2 is about 400 ppm, and the 2050 concentration would be 2080. That's a bit more than two doublings, so we would probably be looking at 6-7º C of surface warming in 2050 relative to current day, or 7-8º C of surface warming in 2050 relative to the usual baselines.
The answer to your question is too frightening to consider. If everyone on Earth had the same carbon footprint as the average American, then it would be all over by now. We would be extinct and humanity would no longer be producing any greenhouse gasses at all, unless you count those emitted by our smoldering corpses.
The closest thing to the formula you are looking for is nowhere near as simple as you want it to be. It's a series of very complex models that try to take into account the feedback that each subsystem has on the enitire biosphere. It's a loop that spins out of control called "The Runaway Greenhouse Effect" and it's what puts us at risk of turning Earth into something like what Venus looks like now.
There's a great website that allows you to adjust every conceivable factor that has been considered in that formula. Go to tool.globalcalculator.org and create your own pathway to our destruction (or salvation) by adjusting levers for lifestyle, technology, food, land use, costs as a percentage of GDP, etc. and you can set each thing to whatever you want, from minimal abatement to extremely ambitious. When you are done you can compare your pathway to many others, from The Vegan Society to the World Nuclear Association.
If you get too carried away, the temperatures will run right off the chart to a level that can't be calculated by current models. You will also run into other constraints, such as there not being enough land in the whole world to do what you propose. It's great fun, and it's a lot more interesting than a simple formula.
