The well known "Hungarian Algorithm" is capable of finding a matching in a weighted bipartite graph with minimal (respectively maximal) weight, and therefore can be used to find an optimal allocation of potential buyers to products, e.g. given valuations: $$ \begin{matrix} & x & y & z\\ A & 4 & 0 & 0\\ B & 3 & 0 & 0\\ C & 0 & 0 & 1\\ \end{matrix} $$ With A,B,C buyers, x,y,z products and buyer A has valuation 4 for product x, valuation 0 for product y and so on. Multiplying all elements by -1 and solving it, yields the assignement "A gets x, B gets y and C gets z": $$ \begin{matrix} & x & y & z\\ A & 0* & 1 & 2\\ B & 0 & 0* & 1\\ C & 3 & 0 & 0*\\ \end{matrix} $$ But how do I extract the market clearing prices when using the Hungarian algorithm?
(In my example, $x =\$ 3$, $y = z = 0$ dollars would be market clearing, as the utility of a buyer for product i is valuation(i) - price(i), and a buyer wants one of the items for which her utility is maximal).