The lecture will be an introduction to the model of economic equilibrium. The basic concepts: preferences, initial endowments and market clearing prices will discussed - in general and by means of examples. I will indicate how fixed point theorems are used to demonstrate the existence of equilibrium prices and sketch an algorithm for Brouwers theorem. If time permits, there will be some remarks on equilibrium models with production.

The Walrasian model of economic equilibrium is a generalization to the entire economy of the basic notion that prices move to levels that equilibrate supply and demand. Although the model avoids some factors of economic significance, it is extremely useful in helping us evaluate the effects of changes in economic policy or the economic environment. A moderately realistic model designed to illustrate a significant economic issue typically involves a large system of highly nonlinear equations and inequalities. Existence of a solution is demonstrated by non-constructive fixed point theorems. The explicit numerical solution of such a model requires sophisticated computational techniques.

Part I of this paper introduces a general framework for the discussion of discrete production sets and the associated programming problems which arise when a particular endowment of factors is specified. In this part of the paper we shall apply these ideas to integer programming problems with two activities and bring to bear some of the basic considerations of the theory of computational complexity. The numbering of sections, figures, and equations will follow those used in Part I.

This paper and its sequel present a new approach to the study of production sets with indivisibilities and to the programming problems which arise when a factor endowment is specified. The absence of convexity precludes the use of prices to support efficient production plans and to guide the search for optimal solutions. Instead, we describe the unique minimal system of neighborhoods for which a local maximum is global, and discuss a related algorithm. The definition of this neighborhood system is based on techniques used in the computation of fixed points of a continuous mapping. In Part II of the paper this neighborhood system is investigated in the special case of two activities and it is shown that the algorithm may be accelerated so as to terminate in polynomial time.

I will discuss a specific class of simplicial complexes, K(Y), whose vertices are contained in a set Y ∈ R n . The vertex set is finite or denumerable and satisfies some genericity properties. The complex has been studied for many years under a variety of different names: ordinal bases, primitive sets, the complex of maximal lattice free bodies and most recently, Algebraic Geometers have used the name the Scarf Complex.

: We present two arguments, one based on index theory, demonstrating that the multicountry Ricardo model has a unique competitive equilibrium if the aggregate demand functions exhibit gross substitutability. The result is somewhat surprising because the assumption of gross substitutability is sufficient for uniqueness in a model of exchange but not, in general, when production is included in the model.

Irving Fisher's Ph.D. thesis, submitted to Yale University in 1891, contains a fully articulated general equilibrium model presented with the broad scope and formal mathematical clarity associated with Walras and his successors. In addition, Fisher presents a remarkable hydraulic apparatus for calculating equilibrium prices and the resulting distribution of society's endowments among the agents in the economy. In this paper we provide an analytical description of Fisher's apparatus, and report the results of simulating the mechanical/hydraulic "machine," illustrating the ability of the apparatus to "compute" equilibrium prices and also to find Multiple equilibria.

Inventory models customarily assume that demand is fully satisfied if sufficient stock is available. We analyze the form of the optimal inventory policy if the inventory manager can choose to meet a fraction of the demand. Under classical conditions we show that the optimal policy is again of the (S,s) form. The analysis makes use of a novel property of K-concave functions.

Irving Fisher's Ph.D. thesis, submitted to Yale University in 1891, contains a fully articulated general equilibrium model presented with the broad scope and formal mathematical clarity associated with Walras and his successors. In addition, Fisher presents a remarkable hydraulic apparatus for calculating equilibrium prices and the resulting distribution of society's endowments among the agents in the economy. In this paper we provide an analytical description of Fisher's apparatus, and report the results of simulating the mechanical/hydraulic machine, illustrating the ability of the apparatus to compute equilibrium prices and also to find multiple equilibria. Copyright 2005 American Journal of Economics and Sociology, Inc..

Let A be a real matrix of size (n + d + 1) × n. We assume that all n × n submatrices of A are nonsingular and define the condition number C = C(A) to be the ratio of the largest n × n subdeterminant of A to the smallest in absolute value. In addition we assume that there is a positive vector π such that πA = 0. This implies that for any b, the body Kb = {x∣Ax ≤ b} is bounded. Let f(A) be the number of subsets of the rows of A, of cardinality n + 1, for which a positive linear combination equals zero. The Banach-Mazur distance Ρ(Kb, Kc) for a pair of nonempty full dimensional bodies Kb and Kc is defined as follows: let λ1 be the smallest λ for which Kc ⊆ λ Kb + ξ1 for some ξ1 and λ2 the smallest λ for which Kb ⊆ λ Kc + ξ2 for some ξ2. Then Ρ(Kb, Kc) = log(λ1 ⋅ λ2). We show that for any ε > 0, there exists a subset of the bodies Kb, of cardinality not larger than f(A)⌈2 log2 (nC)/ε⌉d, such that every body is within distance ε from some member of the subset.

Free AccessAboutSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack CitationsPermissionsReprints ShareShare onFacebookTwitterLinked InEmail Go to SectionFree Access HomeOperations ResearchVol. 50, No. 1 Inventory TheoryHerbert E. ScarfHerbert E. ScarfPublished Online:1 Feb 2002https://doi.org/10.1287/opre.50.1.186.17773 Previous Back to Top Next FiguresReferencesRelatedInformationCited byA robust model for the lot-sizing problem with uncertain demands1 April 2023 | Optimization Letters, Vol. 17, No. 6A survey of stochastic inventory models with fixed costs: Optimality of ( s , S ) and ( s , S )‐type policies—Discrete‐time case3 October 2022 | Production and Operations Management, Vol. 32, No. 1Ultra-short-term optimal dispatch for EH-IES considering uncertainty of delay in the distribution networkInternational Journal of Electrical Power & Energy Systems, Vol. 141A new framework for analyzing technological change23 October 2020 | Journal of Evolutionary Economics, Vol. 30, No. 4A Large Economic System with Minimally Rational Agents28 June 2019Significance of Nonlinearity and Many Goods Models28 June 2019Inventory Theory1 December 2016Risk-Averse Newsboy Problem with Incomplete Demand InformationBounds for path-dependent options19 September 2015 | Annals of Finance, Vol. 11, No. 3-4The distribution-free newsboy problem and the demand skew21 December 2014 | International Transactions in Operational Research, Vol. 22, No. 5On the distribution-free newsboy problem with some non-skewed demandsOperations Research Letters, Vol. 43, No. 2Continuous Real Time Dynamic Programming for Discrete and Continuous State MDPsTechnical Note—A Risk- and Ambiguity-Averse Extension of the Max-Min Newsvendor Order FormulaQiaoming Han, Donglei Du, Luis F. Zuluaga22 April 2014 | Operations Research, Vol. 62, No. 3Economic Theory and the World of Practice: A Celebration of the ( S , s ) ModelJournal of Economic Perspectives, Vol. 24, No. 1Inventory Control2 January 2010IFORS' Operational Research Hall of FameInternational Transactions in Operational Research, Vol. 12, No. 4Option bounds14 July 2016 | Journal of Applied Probability, Vol. 41, No. AOption bounds14 July 2016 | Journal of Applied Probability, Vol. 41, No. AAnalysis, Design, and Control of Queueing SystemsShaler Stidham, 1 February 2002 | Operations Research, Vol. 50, No. 1Sample Path Derivatives for (s, S) Inventory Systems with Price DeterminationCooperative Inventory Control Volume 50, Issue 1January-February 2002Pages iii-247 Article Information Supplemental Materials Metrics Information Published Online:February 01, 2002 © 2002 INFORMSCite asHerbert E. Scarf, (2002) Inventory Theory. Operations Research 50(1):186-191. https://doi.org/10.1287/opre.50.1.186.17773 KeywordsInventory/productionpersonal reflections. Professionalcomments onPDF download

In this paper I discuss various properties of the simplicial complex of maximal lattice free bodies associated with a matrix A . If the matrix satisfies some mild conditions, and is generic , the edges of the complex form the minimal test set for the family of integer programs obtained by selecting a particular row of A as the objective function, and using the remaining rows to impose constraints on the integer variables.

This thesis is concerned with practical computation algorithms for solving general convex mixed-integer programming problems. Integer programming is an important mathematical model for economic decision making problems in the presence of indivisibilities. Integer programming problems are particularly difficult to solve because the traditional tests for optimality fail when some or all of the decision variables are discrete. In this thesis, we present a new implementation of the Lovasz and Scarf generalized basis reduction algorithm (Lovasz and Scarf, 1992) for convex mixed-integer programming. Our method generalizes branch and bound algorithms by branching on general linear integral functions rather than the coordinate variables. The generalized basis reduction algorithm is used to determine good integral linear functions that make the number of branches at each node small. Our method has been very successful in solving a set of linear and nonlinear testing problems.