Abstract
Herbert Scarf has made seminal contributions in inventory theory. In his paper “The Optimality of (S, s) Policies in the Dynamic Inventory Problem,” published in Mathematical Methods in the Social Sciences 1959, Proceedings of the First Stanford Symposium, Stanford University Press, 1959, he proved the optimality of an (S, s) policy for the stochastic dynamic inventory problem involving a fixed cost K of ordering. Such a policy was introduced in the classic 1951 Econometrica paper titled “Optimal Inventory Policy” by K. Arrow, T. Harris, and J. Marschak, and was suspected to be optimal. Scarf accomplished his proof by introducing the new concept of K‐convexity. This concept is in wide use for dealing with many variations of inventory problems involving fixed costs.
His paper “Optimal Policies for a Multi‐echelon Inventory Problem” (with A. J. Clark), Management Science, Vol. 6, No. 4, 1960 addressed an inventory problem in which N installations are linked in series. The paper demonstrated that the value function (arising in its dynamic programming formulation) which in general depends on the stock levels at each installation can, under certain assumptions, be decomposed into functions of single variable, each of which satisfies its own recursion and thus can be readily solved. The paper was designated as one of the ten most influential papers published in the first fifty years of the journal and was re‐published in a supplemental issue of Management Science, December 2004, with comments from Herbert Scarf and Andrew Clark. Scarf has written other influential papers on inventory models and his 1958 book Studies in the Mathematical Theory of Inventory and Production, Stanford University Press (co‐authored with K. J. Arrow and S. Karlin) is considered to be the bible in the production ‐ inventory literature.
One of Scarf's earliest papers in economics contains his well known example of a model of exchange in which the classical price adjustment mechanism is globally unstable. In 1962 and in a subsequent joint paper with Debreu, he demonstrated a major generalization of Edgeworth's conjecture that the core of a classical market economy tended to the set of competitive equilibria as the number of consumers became large. Among his notable works in economics and cooperative game theory is a seminal paper in which he showed that the core of an N person game without transferable utility was non‐empty if the game was balanced. Necessity and sufficiency of this condition had been previously shown by Bondareva and Shapley for games in which players were allowed to transfer utility between themselves freely, but the problem is much more difficult if the assumption of transferable utility is dropped.
Scarf pioneered the use of numerical algorithms to compute equilibrium prices in a general equilibrium model of the economy. These algorithms approximate fixed points of a continuous mapping using simplicial subdivisions of the unit simplex. The first method was announced in 1967 and became the basis for the famous 1973 monograph The Computation of Economic Equilibria, co‐authored with Terje Hansen, which launched the whole area of applied general equilibrium theory. Scarf received a 1983 Frederick W. Lanchester Award for this work.
In later work, Scarf tackled the problem of production sets with indivisibilities, a problem that has bedeviled production and equilibrium theory for many years. Indivisibilities, after all, are the justification for technical increasing returns to scale. In order to handle non‐convex production sets, Scarf developed a special simplicial complex for these sets, whose edges provide the minimal test set for the family of integer programming problems associated with this discrete production possibility set. Scarf's simplicial complex has found a surprising number of applications, including graph theory, reliability theory, the flow of traffic on the Internet and algebraic geometry.
He is the winner of the 1983 John von Neumann Theory Prize given by INFORMS. Among other things, the prize recognized his work on (S, s) policies in inventory theory. He is a Fellow of INFORMS, a member of the American Academy of Arts and Sciences, the National Academy of Sciences, and the American Philosophical Society. He is a Distinguished Fellow of the American Economic Association. 