If your objective is to reduce carbon dioxide emissions, then the correct questions to ask are:
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What can each oil product producer do to be more efficient (in terms of carbon dioxide emissions per unit production) during the production process, and
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What can oil product customers do to be more efficient (in terms of carbon dioxide emissions per mile, or per ton-mile, or per kWh generated, whatever)
Almost all oil products, such as gasoline, are fungible. And the underlying market is inelastic over the short term.
Trying to influence point 2 by shaming just one of the oil producers in point 1 accomplishes nothing. If ExxonMobil stopped producing tomorrow, everyone in point 2 will simply buy from one of the other oil producers (at perhaps higher prices, but again, demand is inelastic over the short term). These non-ExxonMobil fuels will be consumed in the same piston engines, turbine engines, or boilers that would have burned the ExxonMobil stuff.
Oil product customers base their decisions on procuring and deploying new, more efficient machinery and processes based on a variety of inputs. But even once a customer makes a decision, it could take months or even years to realize the full carbon dioxide savings. At the same time, other oil product producers base their decisions on production levels based on a variety of inputs. It could take weeks or months to change production levels.
In terms of refinery capacity, look at https://corporate.exxonmobil.com/Locations/United-States and add up all the refinery capacity for the refineries shown. (Baton Rouge = 502,500 bbl, Baytown = 584,000 bbl, Beaumont = 366,000 bbl, Billings = 60,000 bbl, Joliet = 250,000, total = 1,762,500 bbl). Then go to https://archive.is/Mdh7c. Gross input to Atmospheric Oil Distillation Units = 13,846,000 bbl, Operating Capacity = 16,943,000 bbl. There is 3,097,000 bbl of extra operable capacity; the loss of ExxonMobil's 1,762,500 bbl will just mean someone else refines that 1,762,500 bbl. Long term there isn't expected to be any net changes in prices; our oil product customers will just pay a bit extra for a few months.
In terms of oil well capacity, there are plenty of oil wells that are currently idled because the market is oversaturated with crude. Owners of those wells can put them back into production once ExxonMobil's wells bow out of the market. Long term there will be a permanent price increase caused by ExxonMobil's departure from the market, but it will not be gigantic.
So lets say on Monday, January 11, 2021 at noon ET, ExxonMobil sends the order: Shut down every well, turn off every refinery. Prices will spike initially short term. But as the new wells come back up, and as the existing non-ExxonMobil refinery capacity handles the shift in demand, prices will soon go down to a slightly elevated level than they would be.
Note that I said "slightly elevated level than they would be." Fuel prices fluctuate naturally. It is entirely possible that February's fuel prices would be below where they were on January 11, 2021 at 11:59 AM. The point is ExxonMobil's news would only makes prices slightly higher than they otherwise would be (or in this case, the price would have been even lower had ExxonMobil stayed in the market).
Oil product customers might change plans as a result, but again it would take months or years to fully deploy a more efficient solution. Compared to a world where ExxonMobil stayed in the market, there would be no change in consumption in February, March, or April. If the overall price trend goes down in spite of ExxonMobil's announcement, it is entirely possible that oil product customers will make no changes.
We've established that in the short term, a decision by ExxonMobil to leave the market will not change demand (and, consequently, oil product customer carbon dioxide emissions). We've established that there is a chance that ExxonMobil's decision could create a long-term change, but only as a function of increased fuel prices. Fuel prices vary anyway, so establishing such a linkage compared to the counterfactual case is misleading.
Based on the above, ExxonMobil can't influence the demand of oil product customers. It can't influence the carbon dioxide emissions of its oil product customers. All it can do is influence which oil product customers will be ExxonMobil's oil product customers. That's great and wonderful, but doesn't actually do any good.
Everything written here is the truth. It remains the truth (albeit an inconvenient one) even if it gets downvoted due to its perceived blasphemy. Take care, y'all.