Germany, Finland, Norway, Denmark, Slovenia, Austria, Slovak republic, and so on.
Why do they allow non-EU/EEA students to have free tuitions?
What is their benefit as a country?
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There are at least three benefits:
That's not necessarily a strong reason to have absolutely no fees. For example, some Dutch universities position themselves as an alternative to the US and UK for students from China, India, and elsewhere by having lower fees but studying there is certainly not completely free.
Also, you asked about benefits and I focused on that but that's not the only consideration. For example, providing education to foreign citizens from poorer countries can also be regarded as a form of development aid.
Incidentally, strict visa requirements and teaching (mostly) in German means that the number of students in question is any case very limited so there is no risk to see German university swamped by legions of Asian students and costs running out of control.
As a former non-EU student at a Czech university who studied free of charge (and in fact received a stipend from the government), I can answer this for Czech Republic.
At current rates, the government spends around 50 thousand crowns per university student per year, which is around 2000 Euros. In 2016 there were 280 thousand students in total and out of those, 38 thousands were foreigners. Of those, around 13 thousand are from non-EU countries, so merely 4.5% of the student population can be counted as completely foreign. Taking those statistics in mind, here are the government's reasons for keeping the education free for all:
I believe the above points also apply to Germany, Finland, Norway, etc. You can also note that the UK charges extra fees to non-EU citizens precisely because English is the world's most popular language and therefore they have no issues with attracting willing students.
Counterquestion: Why exactly should people pay for education?
Before you think that it is a ludicrous counterquestion, think hard about it. Would you expect that parents should pay the same amount of tuition fees for their little children? True, a professor earns much more than a primary school teacher, but if you compare the relative amount primary school is still much cheaper.
The overall costs of education are in comparison to other ventures small. You need a place, a teacher which must be supported and teaching material like books. Almost all parents have the desire that their children should have a good life and that their children should be able to reach their goal on their own accord (and not because they do not have the money or connections).
So in the cultures you mentioned where education is almost free, the ulterior motive is simply that the population thinks education is a human right. It's really simple as that. So their country will offer an opportunity to study. It does not mean that you get state-of-the-art teaching, but you will be able to pursue your own education goals. In Germany e.g. there are Volkshochschulen which allow you to get the university-entrance diploma if you are still an unqualified adult in the evenings also almost free.
If, as people suspected, the countries offer it only because they need to attract students, then those instituitions would not exist. It it also contradicted by history because despite free education Germany grew to be the world leader in science, technology and history during the second part of the 19th and the first part of the 20th century. According to Carl Diehl, approximately 10 000 US students studied in Germany from 1815 to 1914, 5500 from Yale and Harvard alone, including Charles Eliot who reformed the Harvard education (I invite to read the Wikipedia article which quite frankly admits that US education during this point was...suboptimal).
If you see it from a pure economical standpoint, free education seems to be counterintuitive. If something exists which is valued, why give it away for free? I think you will run into a wall if you try to understand it from this perspective.
In order to answer your question about foreigners it helps to think about the practical reasons why governments are subsidizing higher education for their own people.
The main economic reason for doing so is to improve your workforce so that you have higher productivity, more added value, and a higher tax base.
An alternative to "making your own" highly skilled workforce is importing it. Many European countries want to attract skilled immigrants - immigration of cheap labor is a highly contested political topic, but there mostly is a consensus that with the current demographic situation attracting highly educated workers is desirable.
If you have people who come to your country for higher education (especially for masters and PhD programs), they have stayed here for multiple years, learned local language and habits, obtained a good education, and are quite likely to stay (or leave, but come back in the future). That is as valuable investment as educating your own people, and possibly more effective - since in order to prepare a local student you'd need to pay for 15-20 years of schooling, starting from kindergarten, but to prepare an "imported" student you just pay for the last couple years in university.
Germany is a welfare state: its constitution declares "human dignity is inviolable" and this has been interpreted as guaranteeing a state-provided minimum living standard for any human residing under German jurisdiction. So unemployable residents cause permanent costs that could possibly be averted after a temporary investment in education.
Now there are a number of education "tourists". It turns out that enough of them stay after completing education to offset the negative impact of those that leave before a significant net contribution to the economy. Now for one, those education tourists more often than not are dedicated, meaning that they finished their primary education outside of Germany and invested into learning the language. As a consequence, their total in-country stay causes a fraction of the costs of a native.
A number of those are engineering students. When they leave, they are schooled in German standards, language, and products. When they return to their home country and are able to work in positions with significant acquisition and trade responsibility, they will be most comfortable doing trade with Germany.
Germany is an export-oriented country. So they don't even lose a lot on such education tourists when looking at the grand bottom line.
Things are different for people moving to Germany without much of a motivation or talent for serious study, basically goofing around. They don't actually have a lot to win in Germany (as their education is not likely to help them a lot in life), and their number is limited. So while they don't constitute a net win, figuring out rules to make things harder for them while not hitting either constituents or others for which the education is not regaining the costs is likely to cause more damage than good.
There was a time when the U.S. was the land of opportunity and self-made men. This is history. Being born with a silver spoon in your mouth or not has an influence on where you make it in life much more than your intelligence and thirst for knowledge.
In Germany, a factory worker with mother at home can send four children to university if they are aptly talented while paying off a house. Yes, this will have an impact on the quality of life and take a long time.
And the respective tax rates make it much harder to move from well-off to wealthy to rich. But the nature of welfare state and public health support and others make moving from well-off to wealthy to rich a much less significant target than it is in the U.S.
I participated in a program when I was younger in Germany.
From their website:
DAAD supports the internationalization of German universities, promotes German studies and the German language abroad, assists developing countries in establishing effective universities and advises decision makers on matters of cultural, education and development policy.
DAAD actively fosters partnerships with its alumni, German universities and institutions of German language instruction. Here you will find links that describe the various connections the DAAD utilizes to promote study abroad and German studies
Both those are pretty reasonable explanations for why they want to do such things.
In addition, while participating I frequently heard about opportunities to work in Germany when I was finished with the program. There are a few reasons for this:
Ultimately, that entire program is effectively a giant recruiting program geared towards people that Germany wants to have there - PhDs, masters students, people who speak German, etc.
In the case of Hungary, due to historical reasons there is a large number of ethnic Hungarians in many countries neighbouring Hungary. In these countries these Hungarians are often second class citizens in the following sense. They pay the same amount of tax as the people from the majority, but they do not receive the same services, even if these countries signed international treaties to treat all their citizens equal and to give linguistic minorities the opportunity to study in their own languages. In particular, the higher education options for them these ethnic Hungarians are very limited in their native country, if there exist at all. Hungary tries to help them by providing free access to its higher education.