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College Admissions and Market-Clearing Prices

In the past, choosing a college was much simpler: you applied to a few schools and chose to attend the best school that accepted you.  However, with tough economic times and rising tuition prices, choosing the right college has become more complicated.  The demand for a college education still remains high, especially with today’s job market.  Even with a college degree, it can be hard to find a job after graduation, so choosing the right college is especially important.

Matching perspective college students to colleges can be thought of as a bipartite graph with buyers and sellers.  Colleges can be thought of the sellers, perspective students can be thought of as the buyers, and a college education is what is being sold.  Every perspective student places a different value on the college education being offered by each of the different colleges.  The higher the valuation that a student places on a college, the more that student wants to attend that college.  A student’s valuation for a college can be based on factors such as its location, its academic prestige, the size of the school, and so on.  A preferred-seller graph would place an edge between students and the college or colleges that they value highest and want to attend most.  If all colleges were priced the same, there would not be a perfect matching in the preferred-seller graph.  Students would all want to attend better schools since they would be the same price as lesser schools.

By increasing their prices, colleges effect the valuations that students place on the different college educations.  For example, a student A may have preferred a college X at first.  However, after the price of X increased, A then preferred college Y to college X.  Another student B, may have had a high enough valuation for college X, that even after the price was increased, he still valued X more than any other college.  As a final result, college prices lead to a preferred-seller graph that has a perfect matching.  These prices are the market-clearing prices.



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September 2012