The world’s two richest persons have been longtime business rivals, but now one of the two has the ear of the president elect of the United States. A few months back, a three-post exchange between Bezos and Musk on Mr. Musk’s social networking site X would have passed for petty sniping between business rivals. But times have now changed.

“Just learned tonight at Mar-a-Lago that Jeff Bezos was telling everyone that @realDonaldTrump would lose for sure, so they should sell all their Tesla and SpaceX stock,” Mr. Musk posted on X on Wednesday night, naming two of his companies. He attached an emoji for a snickering face, with a hand covering the mouth. “Nope. 100% not true,” responded Mr. Bezos on Thursday morning. “Well, then, I stand corrected,” Mr. Musk retorted back, with a laughing-crying emoji.

With President-elect Donald J. Trump’s history of hostility toward Mr. Bezos, the posts carried an implicit message about Mr. Musk’s growing power within the arriving administration. The exchange — fleeting, brassy and fairly typical of Elon Musk’s overwhelming presence on X — could foreshadow a rough next few years for Jeff Bezos and the companies he started, the rocket maker Blue Origin and multinational technology company Amazon. It was also a clear reminder that the power dynamics in the longtime rivalry between the world’s two wealthiest persons changed on Nov. 5. Plenty of tech magnates have drawn Mr. Trump’s ire over the last few years. Perhaps none more than Jeff Bezos, mainly because he owns The Washington Post, which has often written critically about Mr. Trump.

Few if any business barons have had as much influence over an arriving presidential administration as Mr. Musk appears to have. He hitched himself to Trump’s campaign, becoming a crucial financial backer, and has become a near consistent presence at Mr. Trump’s Mar-a-Lago estate. Trump named Musk a co-leader of what the president-elect has called Doge (the Department of Government Efficiency), giving him a broad mandate to find possible ways to slash federal expenditure and bureaucracy. Mr. Musk could have substantial sway over how the federal government interacts with Mr. Bezos’ companies.

Mr. Trump and Mr. Musk have even become travel companions. Over the weekend, they were seen standing next to each other at the Ultimate Fighting Championship event held in New York’s Madison Square Garden. And on Tuesday, Trump attended the test flight of Musk’s SpaceX’s Starship rocket in Texas.