In addition to inventory policy, Scarf had at least four more major breakthroughs. Perhaps his most famous discovery is the Scarf algorithm. Arrow and Debreu had proved that the equations describing economic equilibrium always have a solution when goods are divisible, but they were baffled by the problem of how to find one, except in special cases. The Scarf algorithm always finds an equilibrium, no matter how complicated the economy. This gave applied economists the ability to work with much more realistic models of the economy and thus to predict the consequences of major policy reforms including NAFTA and the U.S. tax system.
The four breakthroughs:
- Computation of equilibria (Scarf algorithm)
- Production sets with indivisibilities
- Core allocation existence and computation
- Equivalence of core allocation and competitive equilibrium allocation (Debreu-Scarf Theorem, Edgeworth’s conjecture)
Here’s Scarf presenting “Fixed Point Theorems and Economic Equilibrium” at “The Economics of Kenneth J. Arrow,” The Institute for Advanced Studies, The Hebrew University, June 30, 2008: