ハーバート・E・スカーフ (Herbert E. Scarf), 1930-

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Prominent Yale economist who pioneered the use of numeric algorithms to facilitate the "computation" of equilibrium in general equilibrium systems. His approximation method to fixed points using simplicial subdivision was announced in 1967 and became the basis for famous 1973 monograph on Computation of Equlibrium, which launched the whole area of applied general equilibrium theory.

Scarf also provided the first proof of the non-emptiness of the Edgeworth's "core" in 1967. He had previously provided in 1962, followed by joint work with Debreu in 1963, the core convergence theorem for a replicated economy. His counterexamples of stability of equilibrium (1960) helped, in several ways, to bury that research program. He is also reknowned for his work on (S, s) inventory policy (1959).

In his later work, Scarf tackled the problem of production sets with indivisibilities (i.e. non-convexities), a problem that has bedeviled production and equilibrium theory for a while -- indivisibilities, after all, are the justification for technical increasing returns to scale. Resurrecting a result that he had found in 1963, Herbert Scarf (1986) noted that when such non-convex production sets are used, the non-emptiness of the core is not guaranteed. In order to handle non-convex production sets, Scarf has developed the method of "integral" activity analysis (e.g. Scarf, 1981, 1986, 1994).

Major Works of Herbert Scarf

• "On Differential Games with Survival Payoff", 1957, Annals of Mathematical Statistics.
• "Games with Partial Information", with L.S.Shapley, 1957, in Annals of Mathematical Statistics.
• Studies in the Mathematical Theory of Inventory and Production with K.J. Arrow and S. Karlin, 1958
• "The Optimality of (S, s) Policies in the Dynamic Inventory Problem." 1959, in Mathematical Methods in the
  Social Sciences
• "Some Examples of Global Instability of the Competitive Equilibrium", 1960, IER.
• "Optimal Policies for a Multi-echelon Inventory Problem," with A. J. Clark, 1960, Management Science
• "Approximate Solutions to a Simple Multi-Echelon Inventory Problem" with A.J. Clark, 1962, in Arrow et al, editors,
  Studies in Applied Probability and Management Science
• "An Analysis of Markets with a Large Number of Participants", 1962, Recent Advances in Game Theory.
• "A Survey of Analytic Techniques in Inventory Theory." 1963, in Scarf et al, editors, Multistage Inventory Models and
  Techniques
• "A Limit Theorem on the Core of an Economy", with G. Debreu, 1963, Econometrica.
• "The Approximation of Fixed Points of a Continuous Mapping," 1967, SIAM
• "On the Computation of Equilibrium Prices", 1967 in Ten Economic Studies in the Tradition of Irving Fisher.
• "The Core of an N-Person Game", 1967, Econometrica.
• "An Example of an Algorithm for Calculating General Equilibrium Prices," 1969, AER
• "On the Existence of a Cooperative Solution for a General Class of N-Person Games", 1971, JET.
• "The Limit of the Core of an Economy" with G. Debreu, 1972, in McGuire and Radner, editors, Decision and Organization
• The Computation of Economic Equilibrium (with collaboration of T. Hansen), 1973.
• "On Cores and Indivisibility", with L.S. Shapley, 1974, JMathE.
• "The Solution of Systems of Piecewise Linear Equations," with B. C. Eaves, 1976, Mathematics of
  Operations Research
• "An Observation on the Structure of Production Sets with Indivisibilities," 1977, PNAS
• "Production Sets with Indivisibilities, Part I: Generalities: & Part II: The Case of Two Activities", 1981, Econometrica
• The Computation of Equilibrium Prices" 1982, in Arrow and Intriligator, editors, Handbook of Mathematical Economics, Volume II
• "Fixed Point Theorems and Economic Analysis,", 1983, American Scientist
• "The Computation of Equlibrium Prices", 1984, in Scarf and Shoven, editors, Applied General Equilibrium Analysis.
• "Integral Polyhedra in Three Space," 1985, Mathematics of Operations Research
• "Neighborhood Systems for Production Sets with Indivisibilities," 1986, Econometrica
• Testing for Optimality in the Absence of Convexity", 1986, in Heller, Starr and Starrett, editors, Social Choice and Public Decision Making.
• "Notes on the Core of a Productive Economy.", 1986, in Hildenbrand and Mas-Colell, editors, Contributions to Mathematical Economics
• "A Theorem in Search of a Simple Proof." 1989, in Anderson et al, editors, Probability, Statistics, and Mathematics,
• "Mathematical Programming and Economic Theory," 1990, Operations Research
• "The Origins of Fixed Point Methods." 1991, in Lenstra et al., editors, History of Mathematical Programming
• "The Generalized Basis Reduction Algorithm," with L. Lovász, 1992, Mathematics of Operations Research
• "The Frobenius Problem and Maximal Lattice Free Bodies" with D.F. Shallcross, 1993, Mathematics of Operations Research
• "Shortest Integer Vectors," with D.F. Shallcross, 1993, Mathematics of Operations Research
• "The Allocation of Resources in the Presence of Indivisibilities," 1994, JEP
• "The Complex of Maximal Lattice Free Simplices," with I. Bárány and R. Howe, 1994, Mathematical Programming
• "Tjalling Charles Koopmans, 1910-1985 Biographical Memoir," 1995, at NAS (pdf file)
• "Matrices with Identical Sets of Neighbors", with I. Bárány, 1998, Mathematics of Operations Research
• "Optimal Inventory Policies When Sales Are Discretionary"
• "How to Compute Equilibrium Prices in 1891" with W.C. Brainard, 2000

Resources on Herbert Scarf

• HET Pages: Debreu-Scarf Core Convergence, Global Stability, Non-Convexities, Computation of Equilibrium
• Herbert Scarf Page at Yale University
• Herbert E. Scarf Page at the Cowles Foundation (another page)
• Scarf's Citation for Distinguished Fellow of AEA
• "Herbert Scarf's Contributions to Economics," by Kenneth J. Arrow and Timothy J. Kehoe, JEP (pdf file)
• Citation from 1983 John von Neumann Theory Prize from INFORMS
• Scarf's call for research assistance for "Production Sets with Indivisibilities" project

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