Suppose you're not just popular because you're fun at parties or, like Mungo and me, look good in a string bikini. Suppose you're a well-connected celebrity academic, like Mungo, but not like me. Maybe you've got an extremely pleasant, tenured position at an institution you treasure in a part of the world that fits you just perfectly. Maybe it would take an Act of God to dislodge you from your big overstuffed comfy chair in your tastefully-appointed, yet delightfully cluttered office. Maybe you have no narrow, self-interested reason to obtain or maintain an account with a service like LinkedIn.
Should you do it anyway? Sure, you might not gain from it, but if you teach at an elite, or even just a middle-of-the-road graduate-degree-granting institution, your students, both current and former can use you as a particularly valuable node. If you even casually keep up with alumni, you have a much better idea of where to point students on the cusp of graduation. Moreover, you have a really good idea of who the best talent in the current crop of graduates is, much more so than can fit on a CV or can be easily communicated in a ten minute interview in a conference hotel room.
You're a club good. Your account is more than just merely euvoluntary. You generate a positive externality.
But the marginal cost to you of joining this probably exceeds the private marginal benefit. Ordinary relative price economics and Coase suggests that you should join and then be compensated by the beneficiaries. Of course, there's a bit of a dispersion problem that seems like it might wander a bit out of Ostrom territory. For most of the private common pool resource allocation solutions studied by Elinor and Vince Ostrom, there were good social ties that kept people from shirking. If you think it's worth it to subsidize Paul Krugman and Tyler Cowen's LinkedIn accounts, how would you design the subsidy? Who should bear the cost and how should it be administered?
If you give a good enough answer, I'll advertise it and who knows, you might get featured on MR.