Knowledge that Transforms

Axioms for an individual's preferences over time, taken from the present perspective, usually assume that the individual will live, or expects to live, throughout a given horizon span. This paper offers an axiomatization that explicitly recognizes the uncertainty of an individual's lifetime. It divides a horizon span into n periods and assumes that if death is not immediate then it will occur at the end of one of the periods. The theory is based on an unconditional preference relation over potential future consumption streams that accounts for uncertain lifetime, along with a conditional preference order that is based on the hypothesis that death will occur at the end of period i. There is a conditional order for each i from 1 to n. The utility representation involves an order-preserving utility function for each of the n conditional orders such that one potential consumption stream is unconditionally preferred to another if, and only if, the sum of the conditional utilities for the first stream exceeds the sum of the conditional utilities for the second. It is argued that the theory seems fairly reasonable only if probability of survival does not depend significantly on past consumption.

[A parametric procedure for linear programming is shown to solve the bilinear complementarity problem, of which a special case is the problem of computing a competitive equilibrium of a piecewise linear economic model.]

[It has recently been shown that the utility of playing a game with side payments depends on a parameter called strategic risk posture. The Shapley value is the risk neutral utility function for games with side payments. In this paper, utility functions are derived for bargaining games without side payments, and it is shown that these functions are also determined by the strategic risk posture. The Nash solution is the risk neutral utility function for bargaining games without side payments.]

H. Neudecker, Bounds for the Bias of the LS Estimator of σ 2 in the Case of a First-Order (Positive) Autoregressive Process when the Regression Contains a Constant Term, Econometrica, Vol. 46, No. 5 (Sep., 1978), pp. 1223-1226

THE MAJOR INTEREST in the recent literature on economies has lain in the results about but finite economies that have been derived from the results proved for economies. It is thus important to find simple yet general proofs for economies. In this article we wish to provide a simple yet general proof of the existence of a competitive equilibrium in an infinite, nonstandard economy with production. The simplicity of our proof comes from the fact that nonstandard analysis can deal with large and small quantities very much as ordinary analysis deals with finite quantities.2 As a result, it is possible to follow very closely the proof of existence for an economy with a finite number of traders, such as that in G. Debreu's classic Theory of Value [6]. In fact, if one is willing to believe that nonstandard analysis permits us to manipulate quantities as claimed above, then no further knowledge of nonstandard analysis is required in order to follow the proof. As examples of the simplicity of nonstandard analysis, it may be pointed out that no analogue of the Fatou-Schmeidler lemma [7, p. 69], a fairly difficult mathematical theorem, is required; nor is it necessary to prove separately that preserves upper-semicontinuity, a proof that Aumann [1] has recently simplified, because integration in the nonstandard model consists of an infinite summation, hence an appeal to 1.9.4 of Debreu [6] suffices to establish this point. As our main objective is to obtain results about but finite economies, it is a welcome bonus to find out that no further effort is needed to obtain these desired theorems. This arises because of the following property of nonstandard analysis. Consider a sequence of real numbers {an} which tends to zero. If we could extend this sequence to the integers, it would surely be a necessary property of the values of {an} at the integers that they are all infinitely close to zero. What makes nonstandard analysis powerful is that the above line of reasoning can be reversed, so to speak. Suppose we have a sequence which

We start with a study of the piece-wise linear function determining how the maximum of the objective varies with the right-hand members of the constraints. Generation of this function needs dual prices. The latter being used only in their validity sets, the two proposed procedures avoid the main difficulty to which the Kornar-Liptak procedure is subject. Basically, the subsystems propose at each step new production processes out of various projects which are often not well specified. The central agency sends prices or resource allocations. TwO LINEAR DECENTRALIZED procedures are proposed for finding the optimum of linear programming models. The first procedure operates through prices sent by the central agency to the subsystems; the second one uses resource allocations. The paper starts from the basic idea developed by Kornal and Liptak. That leads to a study of the piece-wise linear function which determines how the maximum of the objective function varies when the right-hand members of the constraints vary. For generating this function, the subsystems report a vector of dual prices to the central agency. These dual prices constitute efficiency indicators for the subsystems, namely marginal gains, only if the conditions of economic activity, set out by the central agency, do not vary too much. With dual prices, the subsystems will provide a validity set defining the set of allocations out of which the dual prices should be revised. This twofold information is equivalent to providing the vector of the coefficients of a production process which the considered subsystem intends to use henceforward (this vector will be the column vector entering the basic matrix). The central agency selects the best among the production processes proposed during the previous iterations. The subsystems are specialized in this way by the central agency. In the first algorithm, this specialization process is carried out by solving a program whereby a certain number of previous propositions are compared against each other. In the second algorithm, specialization is progressive whereas the central agency moves in the allocation space along the reduced gradient direction, that is according to the locally most efficient direction at each iteration. The computations carried out by the subsystems are simple and do not require any optimization. They are straightforward in the first algorithm which merely compares gains yielded by the various available production processes. These two procedures were first described in [5].

[Given a known demand schedule for a mineral at each instant of time and many deposits with different known extraction costs per ton (different qualities) and different known sizes, how should exploitation be organized? How does an exogenous change in the size of deposit i or in the extraction costs per unit in deposit i affect the program of exploitation? These questions are investigated for the case of extraction costs constant per ton for deposit i. The comparative static analysis parallels that for a problem with many income classes in location theory.]

[Techniques are described whereby the distribution of completed unemployment spell lengths may be inferred from the distribution of in-process unemployment spell lengths recorded each month in the Current Population Survey. An extension is proposed whereby the complete population joint distribution of labor market transition probabilities can be estimated using only Current Population Survey information.]