If an activity, like education, has positive externalities, it can be shown that relying on the pure free market will result in an allocation of it, and other goods, that is Pareto sub-optimal. That is to say, you will end up with an allocation where it's possible to make a trade (transaction), or series of trades, that make at least one person better off (higher utility), but no person worse off (as long as transactions costs and other frictions don't exceed the benefits)
Thus, a Pareto sub-optimal situation is clearly inefficient. And economists like to focus on Pareto optimality much too much to the exclusion of other measures of efficiency, or desirability, because it avoids the discussion of value judgments. People usually don't object to making someone better off if it makes no one else worse off, at least as opposed to the choice of doing nothing and leaving things as they are.
But for the vast majority of people there are a lot of other important economic criteria besides just Pareto optimality, and besides just inequality too, and some of these receive far too little attention in economics. One is just the total utility, or real dollar value of goods, of all of the people added together.
A situation where 1 billion people all have 0 utils (or units of goods), and one person has 100, is Pareto optimal if there's no way to make anyone better off without making at least one person worse off, like if you'd have to take goods from the person with 100. If you did move to a situation where now that one person had 99 utils, or units of goods, but all 1 billion other people went from 0 units to 98, then this is a Pareto sub-optimal move, but it explodes the total amount of goods in the world, or utils, from 100 to over 98 Billion!
It's not just that inequality decreased. Inequality can decrease with the total pie shrinking, or with the total pie only increasing a little. The point is there are economic criteria that are very important to the vast majority of people, myself included, besides just Pareto Optimality and inequality, that are given too little, or much too little, attention by the field, and one of them is total utils or total real dollar value of goods (I know the field doesn’t like to estimate utils cardinally, as opposed to just ordinally, because it's hard, but you can't avoid it -- your results will be consistent with some assumed cardinal numbers anyway, and you can often do far better if you actually put some thought into your estimate or assumption. It's like when a factor is hard to estimate, so many economists say, it's hard so we "won't estimate it"; then they call it zero. Hello?! Zero is an estimate. You can't avoid estimating, but if you had put some thought and effort into your estimate, in many cases you could have come up with one a lot better than 0, or a series of candidate estimates to examine in a sensitivity analysis.).
Ok, the point about education is that subsidizing it is not just to allay externalities. It is also that even beyond externalities doing so can tremendously increase the total pie, even if it's not Pareto optimal because some people might, even on net, lose a little from the taxes. And subsidizing education also decreases inequality greatly, as shown in the outstanding new book, "The Race between Education and Technology" by Harvard economists Claudia Goldin and Lawrence F. Katz.
An additional issue is inefficiency, shrinkage of the pie, etc. due to human imperfections. Many economic models assume that people have perfect information about what's best for them, and they have perfect self discipline too, so they will always make the decision that maximizes their utility. Such models can be useful aides if interpreted intelligently, rather than taken literally. Acting as though the real world really is exactly like those models is not intelligent. For many things these assumptions can be very far from the truth. Subsidizing and mandating education can get people – and especially children – to do things that greatly increase their utility, that they otherwise would not have done, because in the real world people do not always have the information, expertise, and/or self-discipline to do what is in their best interest. Often without government subsidization, nudging, encouragement, or mandating, taxation, discouragement, or outlawing, many people will do things that are far from utility maximizing for them.
Finally, there still are, however, enormous positive pecuniary externalities to education. The Vox EU article, and the papers it cites, miss much, and I have little doubt there are severe flaws with the models, econometric techniques, and assumptions behind them. But even in these studies, the article states, "The recent construction of state-level physical and human capital stock data has provided the opportunity to apply the macro-Mincerian model to US states. Chad Turner and his co-authors estimated a social return of 12% to 15%, while I estimated a slightly lower social return of 9% to 13%. The closeness of the estimates of the social return to the private return suggests that US schooling generates little to no external return.". Little return? What do you think that's 9-15% of? It's 9-15% of trillions of dollars every decade. That's not little, and it's still a gross underestimate of the true amount.
One thing which I hope to have time to write about in detail in the near future is the enormous positive time, or intergenerational, externalities. That is, when people become more educated today, advancing science, technology, and medicine more for future generations, that's a positive externality for those future generations. They benefit, but they are not involved in the transactions and payoffs which take place before their time, thus we can expect that their benefits are not fully considered.
I hope to find time to write more about all of this in the near future (and work on improving what I have written so far). It's very important.