A "public" university, I've found, is something slightly different depending on where in the world the University is located. This means I need to rephrase my question. So I'm going to change it to: How can you empirically measure the effectiveness of a public university in the United States?
This leads us to defining exactly what a public university is in the U.S. The landmark document in this case appears to be the Morrill act of 1862, which provided states with Federally owned land to sell in order to fund "... at least one college, where the leading object shall be, without excluding other scientific and classical studies, and including military tactics, to teach such branches of learning as are related to agriculture and the mechanic arts... in order to promote the liberal and practical education of the industrial classes in the several pursuits and professions in life."
Certainly not all "public" universities were created as a direct result of this act, and, it would be a mistake to apply the "leading object" above to every public university. Other public universities, for example, were started as teachers colleges or strictly agricultural schools, and were founded by individual state act, or as the result of private donations to the state. What we can say is that the Morrill act provided the precedent for universities to be governed by the state rather than a private body.
So, this makes my definition "An institution of higher learning, with teaching and research facilities, made up of colleges and professional schools, which grants both graduate and undergraduate degrees, and is governed by the state, and not a private body." But now, based on my definition, I need to determine what the measurable output(s) should be in order to answer my question.
More on this later, hopefully less than a month this time.