Infinite Number Of Commodities Research Articles

Optimal sharing with an infinite number of commodities in the presence of optimistic and pessimistic agents

We prove that under mild conditions individually rational Pareto optima will exist even in the presence of non-convex preferences. We consider decision-makers (DMs) dealing with a countable flow of pay-offs or choosing among financial assets whose outcomes depend on the realization of a countable set of states of the world. Our conditions for the existence of Pareto optima can be interpreted as a requirement of impatience in the first context and of some pessimism or not unrealistic optimism in the second context. A non-existence example is provided when, in the second context, some DM is too optimistic. We furthermore show that at an individually rational Pareto optimum at most one strictly optimistic DM will avoid ruin at each state or date. Considering a risky context, this entails that even if risk averters will share risk in a comonotonic way as usual, at most one classical strong risk lover will avoid ruin at each state or date. Finally, some examples illustrate circumstances when a risk averter could take advantage of sharing risk with a risk lover rather than with a risk averter.

Supporting weakly Pareto optimal allocations in infinite dimensional nonconvex economies

In this paper, we prove a new version of the Second Welfare Theorem for economies with a finite number of agents and an infinite number of commodities, when the preference correspondences are not convex-valued and/or when the total production set is not convex. For this kind of nonconvex economies, a recent result, obtained by one of the authors, introduces conditions which, when applied to the convex case, give for Banach commodity spaces the well-known result of decentralization by continuous prices of Pareto-optimal allocations under an interiority condition. In this paper, in order to prove a different version of the Second Welfare Theorem, we reinforce the conditions on the commodity space, assumed here to be a Banach lattice, and introduce a nonconvex version of the properness assumptions on preferences and the total production set. Applied to the convex case, our result becomes the usual Second Welfare Theorem when properness assumptions replace the interiority condition. The proof uses a Hahn-Banach Theorem generalization by Borwein and Jofre (in Joper Res Appl Math 48:169–180, 1997) which allows to separate nonconvex sets in general Banach spaces

First In, First Out Properties of a Commodity-Based Kinematic Wave Simulation Model

Computational efficiency and first in, first out (FIFO) are both important issues in applications such as dynamic traffic assignment. This paper first shows by example the FIFO properties of a computationally efficient commodity-based kinematic wave (CKW) model of network traffic flow. After the development of measurements of FIFO violation among commodities in location and time on the basis of the misplacement of vehicles in total and commodity traffic, it is argued theoretically that numerical CKW solutions converge to FIFO ones. Then, with numerical examples, it is shown that FIFO violation in CKW solutions is a function of the simulation time and the number of commodities but decreases with decreasing cell lengths, even for infinite number of commodities. Finally, the possible implications of this study are discussed.

The non-existence of smooth demand in general banach spaces

We study the questions of existence and smoothness of demand functions with an infinite number of commodities. The main result obtained, under some hypothesis, is: if a C 1 demand exists in a commodity space B, then B can be given an inner product structure. For example, if B is L p, 1≦ p≦∞, and if there exists a C 1 demand function defined on B then p must be 2. Another result is: if a demand function exists, defined for all prices p and income, then the commodity space must be reflexive. For example, if B is L p and a demand function exists on B, defined for all prices and incomes then 1< p<∞. We also study the cases L ∞ and L 1 with weaker assumptions. We finish the paper proving that the demand function is always defined for a dense set of prices and convenient incomes.

On a market equilibrium theorem with an infinite number of commodities

A direct proof is given of the market equilibrium theorem of Gale, Nikaido and Debreu for an infinite-dimensional commodity space. The theorem is closely related to a recent result of Aliprantis and Brown, but allows for excess demand correspondences rather than excess demand functions.