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I have read the the IPO price of a stock is set by underwriters, who then sell to (typically) institutional and "special" investors. Once the stock is ready for the stock exchange, who decides what the stock will open at? And how is the price of that stock determined?

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When a stock is going to become public there's a level of analysis required to figure out the range of IPO price that makes sense. For a company that's somewhat mature, and has a sector to compare it to, you can come up with a range that would be pretty close. For the recent linkedin, it's tougher to price a somewhat unique company, running at a loss, in a market rich with cash looking for the next great deal. If one gives this any thought, an opening price that's so far above the IPO price represents a failure of the underwriters to price it correctly. It means the original owners just sold theirvshares for far less than the market thought they were worth on day one.

The day of IPO the stock opens similar to how any stock would open at 9:30, there are bids and asks and a price at which supply (the ask) and demand (bid) balance. For this IPO, it would appear that there were enough buyers to push the price to twice the anticipated open and it's maintained that level since.

It's possible to have a different system in which a Dutch auction is used to make the shares public, in theory this can work, it's just not used commonly.

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