Why do universities have endowments?

Why do universities (in the US and Britain only) have endowments, and should they? And why does no one but Henry Hansmann [pdf, eBook version here] write about this question?

Because they can.  Tax law doesn’t stop them, and why should a University President spend down the fund?  An ongoing high balance in the fund means prestige, a good ranking, and an ability to make credible commitments to quality faculty and quality programs.
Current donors know that their support will feed into something long-run and grand.  Rationally or not, it is less persuasive for an alumni donor to hear a pitch like “We will spend down the corpus.  Penn State will rise seventeen spots in the ratings, for twenty years, and then fade into obscurity.”  Many givers care predominantly about the “here and now,” but they donate to political campaigns, or benevolent charities, not universities.
Ultimately we need a theory of segmented giving, and how board structures of universities support such giving.  University board members benefit most from a prestigious school with a high endowment and other prestigious board members.  In general those boards will support accumulating the endowment, at least if the school has any chance for prestige in the first place.  Spending money within the university instead distributes those benefits to current faculty and students, rather than to the decision-makers over the endowment.
Note that while the most visible colleges and universities usually have large endowments, the median and modal schools have endowments very close to zero.  They have no chance of accumulating their way to substantial prestige benefits.

Alternatively, you could drop the fancy institutional economics and apply crude price theory.  Universities can borrow or otherwise raise money tax-free, and at g > r you should expect ongoing and rising accumulation.
It is striking how much the list of top U.S. universities does not change over the last century, albeit with some new entries from the west coast.  Among other things, that suggests there has been no fancy, expensive and effective new product that a school might invest in and run down its endowment for.  This might change in the next twenty years.  One can imagine a middling school running down its endowment to spend its way to leadership in on-line education.

I have never seen a good paper on which non-profits accumulate endowments and which do not, and how that difference functions as both cause and effect.  I would think, for instance, that the Heritage Foundation has a substantial endowment, but many think tanks do not.

Here is a new paper on university endowments (pdf), by Gilbert and Hrdlicka, asking whether endowments are invested in too risky a fashion.  It also raises the question of how well endowment practices will survive in a time with low rates of return.  Here is a 2008 dialogue on endowment reform.  Here is a 2009 law review piece on university endowments, it is a little slow to load.  Here is a TIAA-CREF perspective (pdf) on the investment committees for university endowments; they tend to be run by donors.  Here is a look at mandatory payout proposals.  Here is a good 2010 paper (pdf) on what happens when endowment values decline, it is called “Why I Lost My Secretary.”