After factoring in government subsidies (or lack thereof), is the total price for one year of undergraduate education in Europe increasing at a rate similar to America?
This answer is based on the situation in Sweden, I think similar principles would apply in the Nordic countries and possibly in north Europe. First cost of education is very low compared to US, as can be seen in the following references:
There are no tuition fees for European citizens (non Europeans pay a tuition fee which varies based on the University and the Education. For example a technical education for a non European may cost in the range of US$12000/year. The government also provides a grant and a loan on favourable conditions. Everyone who wants to study can get a seat.
It could be argued that even when the government subsidies something there is still a real price, and if there is a real market driver the price should go up.
The government provides resources to a University based on the number of students and the number of courses provided. The amount differs based on the course. For example for natural science and technology the amount is higher than for social sciences. Half the amount is given when the student signs up for a course and half when he or she passes the course. Since the government provides means based on the number of students, there is no reason for why supply and demand should not be met and there should not be any price driver as in US.
For certain very popular educations there are limited seats, and the college would then limit access to these seats based on scores. But money is still not the driver.
For commercial education such as a professional MBA at Stockholm School of Economics there are market price and there prices have increased dramatically over the years. The current cost is in the range of SEK 500K.
In the UK universities have received about £10,000 (15,000 USD) in tuition and fees per student per year. This has been pretty constant with small increases over the years. Students who are not part of the EU are expected to pay the fees in their entirety "up front". EU students who are not part of the UK pay £6000 a year up front. For UK students the rules vary amongst Scotland, Wales, and England. In Scotland university is free for Scots. I am not sure how much, if anything other Brits pay. In England, the fees just rose from £4000 to £9000. Students don't have to pay until after they graduate and begin making more than £25,000 a year.