Although the study of market equilibria has occupied center stage within Mathematical Economics for over a century, polynomial time algorithms for such questions have so far evaded researchers. It has been customary to relegate such efficiency issues to the ``Invisible Hand'' of the market, a notion propounded by Adam Smith (Wealth of Nations, 1776).

In the context of Algorithmic Game Theory, a nascent area attempting to address new issues arising from the Internet, we provide the first polynomial time algorithm for the linear version of a problem defined by Irving Fisher in 1891.

Although the linear case provides an excellent starting point, it is not general enough to capture some of the more attractive aspects of pricing mechanisms. To study these, we define a new model which takes into consideration spending constraints among buyers. Our algorithms are inspired by the primal-dual schema from combinatorial optimization.

(First result joint with Devanur, Papadimitriou, and Saberi, and available at http://www.cc.gatech.edu/fac/Vijay.Vazirani)