The LinkedIn (LNKD) buying frenzy Thursday is a windfall for most of the business networking company’s longtime shareholders, led by founder Reid Hoffman and his private equity backers.

The 115% surge in LinkedIn shares at midday Thursday sent the value of Hoffman’s stake, for instance, soaring from a merely huge $852 million to a surreal $1.8 billion.

But one sizable LinkedIn investor, Goldman Sachs (GS), decided to sit out the secondary-market fun. Practically alone among big holders of the stock, Goldman decided to dump it all in Thursday’s debut, selling its entire 871,840-share stake at the IPO price of $45 a share. What gives?

Goldman declined to comment, and it’s not like the firm is taking a bath by selling out. Goldman stands to make something on the order of $34 million by selling the LinkedIn shares at the IPO price, three years after it bought the shares. In mid-2008 LinkedIn was paying $5.56 a share to buy back stock from its chief technology officer, so it stands to reason Goldman’s per-share profit on this deal is around $39.

Yet Thursday’s blastoff says Goldman could have made even more. The firm’s supposed to excel at this sort of value-maximizing activity, after all. And even if LinkedIn’s valuation is on the bubbly side, the massive for the stock this week seems to suggest that particular bubble is in no danger of imminent collapse.

Or is it? Goldman may be banking on a replay of this month’s other hot IPO, the May 4 debut of Chinese social networking company Renren (RENN). Its shares jumped more than 70% (see chart, right) when trading started, topping out at $24 early on their first day after pricing at $14.

But how quickly the Renren craze faded. The stock dropped 25% from its peak on opening day to close at $18, then dropped in each of the next six trading sessions before bottoming out at $12.60, 10% below the IPO price. The stock was up 4% in Thursday’s social networking party but is only just above $14.

LinkedIn mania may well last longer than Renren’s run did. But it’s easy enough to see the case for selling any stock that is trading, as LinkedIn is at a recent $108.55, at 666 times earnings.