(Edited by Zaifu Yang) Herbert Scarf is a highly esteemed distinguished American economist. He is internationally famous for his early epoch-making work on optimal inventory policies and his highly influential study with Andrew Clark on optimal policies for a multi-echelon inventory problem, which initiated the important and flourishing field of supply chain management. Equally, he has gained world recognition for his classic study on the stability of the Walrasian price adjustment processes and his fundamental analysis on the relationship between the core and the set of competitive equilibria (the so-called Edgeworth conjecture). Further achievements include his remarkable sufficient condition for the existence of a core in non-transferable utility games and general exchange economies, his seminal paper with Lloyd Shapley on housing markets, and his pioneering study on increasing returns and models of production in the presence of indivisibilities. All in all, however, the name of Scarf is always remembered as a synonym for the computation of economic equilibria and fixed points. In the early 1960s he invented a path-breaking technique for computing equilibrium prices.
This work has generated a major research field in economics termed Applied General Equilibrium Analysis and a corresponding area in operations research known as Simplicial Fixed Point Methods. This book comprises all his research articles and consists of four volumes: Volume 1 collects his papers in the area of Economics and Game Theory; Volume 2 collects his papers in the area of Operations Research and Management; Volume 3 collects his papers in the area of Production in Indivisibilities and the Theories of Large Firms; and Volume 4 collects Herbert Scarf’s papers in the area of Applied Equilibrium Analysis.
Given a1, a2,…,an in Zd, we examine the set, G, of all nonnegative integer combinations of these ai. In particular, we examine the generating function f(z) = Sum{b in G}zb. We prove that one can write this generating function as a rational function using the neighborhood complex (sometimes called the complex of maximal lattice-free bodies or the Scarf complex) on a particular lattice in Zn. In the generic case, this follows from algebraic results of D. Bayer and B. Sturmfels. Here we prove it geometrically in all cases, and we examine a generalization involving the neighborhood complex on an arbitrary lattice.
Given a1, a2,…,an in Zd, we examine the set, G, of all nonnegative integer combinations of these ai. In particular, we examine the generating function f(z) = Sum{b in G}zb. We prove that one can write this generating function as a rational function using the neighborhood complex (sometimes called the complex of maximal lattice-free bodies or the Scarf complex) on a particular lattice in Zn. In the generic case, this follows from algebraic results of D. Bayer and B. Sturmfels. Here we prove it geometrically in all cases, and we examine a generalization involving the neighborhood complex on an arbitrary lattice.
We present two arguments, one based on index theory, demonstrating that the multi-country 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.
We present two arguments, one based on index theory, demonstrating that the multi-country 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.
We provide two new, simple proofs of Afriat’s celebrated theorem stating that a finite set of price-quantity observations is consistent with utility maximization if, and only if, the observations satisfy a variation of the Strong Axiom of Revealed Preference known as the Generalized Axiom of Revealed Preference.
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.
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.
