http://holloway.co.nz/blog/2009/09/public-libraries-are-just-a-form-of-theft-brian-edwards/

Your exact question is here: http://freakonomics.blogs.nytimes.com/2007/07/10/if-public-libraries-didnt-exist-could-you-start-one-today/

In this parallel universe where public libraries hadn’t been invented, you’d start off with Amazon introducing the Kindle, other websites following suit with similar services. Public domain books would likely have been made freely available as PDFs over P2P networks for quite some time.
Eventually, you’d have widespread access to digital book purchase schemes (at fixed rates per book, probably over iTunes etc.). You’d also get services offering a monthly rate tariff for DRM’d books (along the lines of the netflix model methinx). Sometime after that model proved successful, you’d have someone suggesting government subsidy of such services in order to increase the availability of literary works (that aren’t public domain)- perhaps just intended for students taking English Lit. courses (government subsidised e-Book rental may turn out cheaper than schools needing to budget for the purchase and replacement of reading material).

What happens after that point? I’m not sure. Perhaps the scheme would only remain available to students. Perhaps people would push for it to be made available to the rest of the citizenry. But in essence, you’d have a public library- people (possibly just students/educators) would be able to check out books at no cost to themselves. Granted, it’d be entirely digital, but if someone were to suggest mass purchase of physical media in a time where digital distribution is possibly more environmentally friendly than physical distribution, who’d want to spend money on re-purposing existing buildings or building entirely new ones ?

@Ritulia: Apparently I’m capable of just a few short words every once in a while. Quite a shock, I know.

@Imnotgoing: Heh. Actually, it’s more copyrightist. Intellectual property rights are particularly interesting compared to regular property rights. More on that later.