Fuzzy economic spaces

Traditional spatial economic analysis is limited to the description of precise spaces. To say that an economic space is precise means: (1) that this space has, or else, has not given constituent characteristics and (2) that the agents located there prefer, or else, do not prefer one possible action to another. Proposition (1) implies that an economic space is perfectly delimited and that it can be clearly partitioned into homogeneous subspaces. Proposition (2) implies that economic agents undertake exact economic calculations and optimize, under rigid constraints of resource limitation, objective functions whose arguments are clearly defined. Thus, traditional spatial economic analysis is based on a binary logic: presence or absence of the space's characteristics, preference or non-preference of agents with respect to possible actions. This logic supposes the principle of the excluded middle. However, the real world is imprecise. The observed economic spaces (areas of influence, regions, attraction zones, market areas, etc.) have "more or less" the given characteristics; instead of having frontiers, they have ill-chiselled limits; they partially overlap one another and they do not allow themselves to be subdivided without ambiguity. Likewise, economic agents pursue vague objectives, sometimes incompatible or contradictory, and they appraise imperfectly the constraints which limit their resources. The analyst who admits that the lights and shades of expression "modify everything" and are, at the same time essential, intends to retain them in full. But he must go beyond the usual literary comments which are often juxtaposited to scientific analysis and whose purpose is to relativize the conclusions, in other words, to contest implicitly the results. He is bound to give a formalized expression of these nuances and gradations of the real world and he ought to reconcile the imprecision inherent in the latter with the precision of the mathematical model being used. 0.3. It is true that n-ary logics have been in use for some time now: POST (1921), LUKASIEWICZ (1937), MOISIL (1940). But, it is with the recent development of the theory of fuzzy subsets that the elaboration of a spatial economic study, perfectly rigorous and fully formalized, has become possible. This theory, presented for the first time by ZADEH (1965) is making great strides and penetrates every branch of mathematics. A few primers are now available. Since 1974, the Institut de Mathématiques Economiques of the University of Dijon, associated with the Centre National de la Recherche Scientifique (France), devotes an important part of its researches to the theory of fuzzy subsets and its applications, especially its applications to spatial economic analysis. In the Institute, research has followed four directions : 0.4.1. : Firstly, it was absolutely necessary to rigorously formulate the axiomatic framework of the theory of fuzzy subsets, to clearly distinguish between the latter and probability calculus, to assemble the principal mathematical results to present the concepts and theorems which are useful to economics and more importantly to spatial economic analysis and to resolve a number of algorithmic problems. 0.4.2. : Next, many types of fuzzy economic spaces were studied: attraction zones for sale-points, areas of fuzzy spatial interactions, fuzzy regions, french fuzzy regions defined by a fuzzy numeric taxonomy, fuzzy regional dynamic systems, fuzzy interregional relations, fuzzy hierarchy of a system of central places and fuzzy urban spaces. 0.4.3. : Then, analyses of fuzzy spatial behaviours of the consumer and of the producer led to a reformulation of the theories of partial equilibria which prepares the way for that of the theory of general spatial equilibrium and of the optimum [ under study ] . 0.4.4. : Finally, various contributions have been made to general economics: fuzzy multicriterion analysis, fuzzy decision theory and fuzzy econometrics. 0.5. The aim of the present study is not to summarize the totality of these works. The time has come for the presentation, with all the rigour called for in this new and, for some people, unwonted field, of the scientific foundations of the theory of fuzzy economic spaces in the course of elaboration. This reconsideration of the foundations of the theory should answer two series of questions: 0.5.1. : On what axiomatic framework is the description of economic universes based? Has it at its disposal specific and novel mathematical instruments, sufficiently pertinent and sophisticated ? 0.5.2. : Can the description of fuzzy spatial behaviours of economic agents rely on a coherent and an appropriate type of economic calculation? On what theory of value is a fuzzy economic calculation based? 0.6. The above set of questions command the plan which will be followed: 1 - Fuzzy economic universes; 2 - Fuzzy spatial behaviours. 0.7. Remark: In order to avoid any ambiguity in the notation of mathematical terms, ordinary concepts (non-fuzzy) are underlined, whereas fuzzy concepts are not. For instance, A C E is read: A is a fuzzy subset of the ordinary reference set E. Furthermore: g(x) designates an ordinary function, whereas f(x) defines a fuzzy function. Likewise: [a1,a2 ] designates a non-fuzzy interval, whereas [t1 ,t2 ] represents a fuzzy interval. For lack of space, the results of numerous theorems are cited without demonstrations, but the complete references indicate in what books and articles these demonstrations can be found.