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Answer from David S. Rose, Managing Partner, Rose Tech Ventures; CEO, Gust.
When a stock begins trading for the first time, it has to start *somewhere*. That's the opening IPO price established by the underwriter(s) of the offering. The minute that first share is sold, however, it's a free market, and the price will rise or fall depending on what buyers and sellers in the market decide to do.
So technically, the first time a share of LinkedIn was sold during the IPO, the first purchaser of that share (who had previously agreed with the underwriter to make the purchase at the opening) paid $45. But the second purchaser, who bought it from that first purchaser on the open market the second after the bell rang, paid $83, because that's what the "market" price was at that exact second. The company received $45 less the 3-5 percent fee paid to the underwriter, the first purchaser (assuming he, she or it immediately turned around and sold), made a profit of $83-$45, or $38! Now, if the second purchaser also flipped the shares during that first day of trading, depending on exactly what time during the day he sold, he could have either made almost $40, or lost $3!
Given this structural situation (underwriter/company sets the IPO price, "inside" purchasers buy at that price and sell at the 'opening market', regular purchasers purchase at the "opening market" and sell at "market") there are a host of weird externalities that come into effect, with different players having different motivations to do different things.
The company tries to walk a very fine line. In THEORY, it should want the IPO price to be as close to the "opening market" price as possible, because that would mean that whatever value the market assigns to the stock will go directly into the company's bank account (since the only thing it gets directly are the proceeds from the first sale.) HOWEVER, the last thing it wants is for the market price to be *lower* than the IPO price, because that implies that the market thinks the company was overvalued, and once downward momentum starts, it may well be the only thing people remember about the IPO. So therefore, the company tries to err somewhat in the other direction, and price the IPO about 10-20% *below* where they think the market really is, because that way, if they guess correctly, the price of the stock will immediately rise, and the market will be signaling that it's a hot company, thereby making the overall company more valuable.
Meanwhile, the underwriter has another set of goals. They want to set a high IPO price, because that will make the most money for the company, and because they get a percentage of only that first sale, not the secondary ones. But at the same time, they want to be seen as being the "go-to guys" for hot companies...and hot companies are ones whose price quickly rises. They also want to take care of the buyers to whom they allocate those first sales, because if they set a low price and the market gives the shares a big "pop", that first purchaser will make a lot of money...and therefore will likely be willing to commit to investing in the next IPO led by this underwriter.