Shares rose 11 percent Tuesday but fell almost 2 percent at midday Wednesday to $372.38. Later in the day it became clear that Musk was in fact serious about the bid to take Tesla private when Tesla published a blog post in Musk's name addressed to all employees and when Musk tweeted that he had already lined up the funding for such a move.
Tesla's board, a nine-member group that includes Musk's younger brother Kimbal, issued a belated statement Wednesday morning saying Musk had "opened a discussion" last week with the board about the benefits of taking Tesla private.
Bernstein analyst Toni Sacconaghi said Musk's statements puts Tesla under pressure to detail how it intends to proceed with the deal to go private.
The move to take the company public would remove Musk, who himself has a 20 percent stake, from the public arena - and contentious criticism. MarketWatch reported that information like the fact that the CEO is considering taking a public company private is generally supposed to be announced to investors and the public by way of official earnings reports and press releases.
Former SEC chair Harvey Pitt told CNBC that while the United States stock market regulator permits executives of publicly listed companies to use social media to make statements about their businesses, Musk's tweet was still "highly unprecedented".
"Basically, I'm trying to accomplish an outcome where Tesla can operate at its best, free from as much distraction and short-term thinking as possible, and where there is as little change for all of our investors, including all of our employees, as possible".
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More than a dozen states follow California's standards, amounting to about 40 percent of the country's new-vehicle market. Some Republican lawmakers supported the mileage freeze, but environmental groups and many states assailed it.
Tesla has moved to further formalise the prospect of the electric vehicle company undergoing a buyout with a statement from the company's directors.
Musk has promised a sustained net profit starting in the third quarter.
The key question following Musk's Tuesday tweet-storm was whether current investors, as represented by the board of directors, were interested in such a proposal.
China's Tencent Holdings Ltd (0700.HK), which took a 5 percent stake in Tesla a year ago, could also be a possible partner.
Any buy-out would require a shareholder vote.
On Wednesday, Tesla's board said it was evaluating the proposal. Tesla is a legendary cash-burner, with roughly $10 billion in debts and $2 billion in reserves, and Musk would probably need in the tens of billions of dollars to buy out shareholders at the right price.
"Just because" Musk wants it at $420 "doesn't mean that there aren't other people who might be willing to come in with another transaction that would be more beneficial to shareholders", Pitt said. He said funding was "secured", without elaborating.