A perennial debate in the UK concerns private, fee-paying versus public, government-funded schools. The former are often academically strong (but of course don't have to deal with the full range of students to which government schools must cater) but find themselves blamed for a number of social ills related to equality of opportunity, social mobility and so on.
Here's an attempt to at least describe the trade-offs involved in the schools issue. I'll start off by saying I don't know what the answer is, but am for now simply trying to pose the question.
It seems to me there are two issues here: (i) (academic) selectivity (selective or non-selective) and (ii) fees status (fee-paying or free). These dimensions are often discussed together, but I think it's useful to consider them one at a time. All four combinations are possible: German gymnasia and the old English grammar schools are examples of free, selective schools; equally, one can imagine fee-paying schools which accept any student who can bear the cost, regardless of academic ability (the other two combinations are the conventional ones: free, non-selective, and fee-paying, selective).
What I want to do first is consider the possible costs and benefits of selectivity and fees, with some first thoughts on how important those effects might be in practice. I'll then sum up with some (tentative) conclusions.
First, what are the costs of selectivity? One is an "external cost" to the remaining, non-selective schools. If we assume some fixed number of academically able students, simply by bringing together such students in one place, selective schools must reduce their proportion elsewhere. To the extent that the presence of such students benefits the class as a whole, there must be some cost associated with selectivity. On the opposite side of the ledger there is an external benefit: bringing together academically inclined kids may help them achieve their best, and protect them from some of the schoolyard problems such students sometimes face.
How important are these effects? My feeling is that the number of academically inclined kids is sufficiently small that the effect of spreading them out in a non-selective system must surely be small. How much difference can it possibly make to have one or two academic kids in a typical classroom? On the other hand, the positives of having at least some environments in which these kids can excel are considerable. We should all want smart kids to achieve their full potential: even someone who hates studying would surely want the person operating on their heart, or designing their car, or in charge of their country's energy policy to be as capable as possible. I would tentatively conclude then that selectivity is overall probably a benefit.
But what are the costs of selective schools being explicitly fee-paying? Suppose you hold academic ability constant and look at the effect of parental income. Under a fee-paying system, does having richer parents confer a better chance of attending a fee-paying selective school? If yes, then there is a serious issue to be addressed. We surely don't want to place yet another hurdle in the path of bright, poor kids, and equally we should want to find the most able students in the country, no matter what their parents financial status. How big is this effect? While most fee-paying schools have some scholarships for poorer students, I don't think anyone would argue that the chances of attending such schools is neutral to parental income in the sense I've described above.
So what we need is some way of making it easy for bright kids to attend selective schools regardless of parental income. One approach would be for the government to directly pay privately-run schools the fees for students who are good enough to get in but can't pay. The downside of this approach would be the usual difficulties of enforcement and means-testing: such things are clunky and lend themselves to being "gamed". Alternatively, selective schools might be government funded and free to attend. (A possible downside might be a loss of competition: fee-paying schools presumably have to work hard to keep parents happy. Some would say a government-run school might have a captive audience and not really care about how well it's run. But equally, there are plausible market-failure type arguments against fee-paying schools).
On to tentative conclusions. I think it's fees, and not selectivity per se, that is the key issue. Two possibilities are (i) government grants for poor kids to attend privately-run schools and (ii) selective schools which are free to attend. I would be in favour of having both approaches and seeing what works: it's hard to know in advance just how difficult grants would be to administer, or how effective government-run schools would be compared with their private counterparts.
Thus, I think an at-least reasonable solution is to have selective, government-run schools which are free to attend (but which students are free to choose between), and privately-run selective schools with full grants available for poorer students.