Do price and value of degrees correlate much? Many American colleges and universities have very high tuition fees, in particular at prestigious universities.  For example, Harvard University in 2014-2015 is US$43,938 for tuition and US$58,607 for tuition, room, board and fees combined.  Not coincidentally, the total US study debt is currently US$1.34 * 1012 and growing.

Many European universities have very low or no tuition fees, but their degrees are by no means worthless.  To consider an arbitrary ranking, consider the list of Nobel laureates by university affiliation: tuition fees in the top ranking institution range from $0 to $50,000+.  Personally, I am a post-doc at a reasonably well-known north-American university.  I paid several 1000€ for my Bachelor's degree, 0$ for my Master's degree, yet I work along people who have likely paid far more.   Have those co-workers wasted money?  Or, to formulate the question more precisely:

Among high-ranking research universities, do those with high tuition fees perform significantly better than those with low or no tuition fees?  Do their alumni fare better?  “Better” may be measured by: current research output (high-impact papers published), number of winners of the Nobel prize, Fields medal, and other highly prestigious awards; and for those not staying in academia, lifetime earnings of alumni, or other measures.  This should be possible to investigate
High cost and high quality are not related in the United States.

First, remember that there is a large fixed base cost of room and board, which amounts to around $10K-$15K, depending on location.  Just a quick glance at a few top schools will show that the cost of tuition beyond that is correlated primarily with state-run vs. private rather than quality:


Harvard: $43,938
MIT: $43,720
Berkeley: $12,972
University of Michigan: $13,977 (in-state)


Also, remember that many of the high-priced colleges do need-based rather than merit-based financial aid, so these numbers should be seen as a "ceiling" rather than an expected price: even a super-genius will pay based on what their parents make, rather than based on how smart they are.  That same Harvard article linked in the original post notes:


  During the 2012-2013 academic year, students from families with incomes below 
   $65,000, and with assets typical for that income level, will generally pay nothing toward the cost of attending Harvard College.  Families with incomes between $65,000 and $150,000 will contribute from 0 to 10 percent of income, depending on individual circumstances.  Significant financial aid also is available for families above those income ranges.


Thus, even at a super-expensive place like Harvard, the cost of education is completely uncorrelated with its quality.