Showing papers on "Tuple published in 1974"

A New Approach to the Uniqueness of Equilibrium

Abstract: A problem of major concern to economists is that of knowing whether certain phenomena are " likely " or not. For example, the econometrician would doubtless be happy if it were true that Cobb-Douglas production functions are very " likely " in the sense that any production function could be closely approximated by a function of the Cobb-Douglas type. Such results would provide excellent protection against those critics who claim that such functions are only " special cases ". The lack of these results is due largely to the idea of being " likely " having been used in too general and intuitive a way. In showing here that economies with a unique equilibrium are likely we will incidentally show that if being likely is interpreted carefully, within the context of a particular problem, then a surprising number of apparently improbable phenomena may in fact rigorously, be shown to be likely. We will now examine the fundamental problem of establishing whether uniqueness of equilibrium is " likely " (see Arrow and Hahn [1, Chap. 9]). The standard way to demonstrate the " likeliness " of some property is to show that it holds generically, i.e. on an open dense set with respect to some meaningful topology. We show that for a carefully chosen topology on the space of all economies, which we call the " y topology ", just such a statement is true for the uniqueness of equilibrium.3 A suggestion that such a statement might be true can be inferred from Dierker's result [4] that the number of equilibrium prices is generically odd. He also gives an indication of the significance of establishing the uniqueness of equilibrium. Following Debreu [3] we define an economy with a commodity space R' as a tuple (f1, *--, fm1~l ..., COm). Here m denotes the number of consumers involved, fi (resp. woi E R') denotes the C' demand function 4 (resp. the strictly positive resources) of the ith consumer. An equilibrium for this economy is a tuple (p, x1, ..., x.) of prices p E R', zPh = 1, and commodity bundles xi e R' , such that lixi = Sioi and xi = fi(p, p cioi) for each consumer i. In the following all topological notions are with respect to the y topology. The definition of this topology, however, is somewhat technical, so it is relegated to the appendix, as one of the proofs of the theorems. We now state our main result.

An approach to formal reasoning about programs

Abstract: This thesis presents a formal apparatus which is adequate both to express the termination and correctness properties of programs and also the necessary induction rules and axioms of their domains. He explore the applications of this formalism with particular emphasis on providing a basis for formalising the stepwise-development of programs. The formalism provides, in some sense, the minimal extension into a second order theory that is required. It deals with binary relations between tuples and the minimal fixpoints of monotone and continuous functionals on them. The correspondence between common constructs in programming languages and this formalism is shown in an informal manner. To show correctness of a program it is necessary to find an expression for its termination properties which will depend on the induction rules for the data structures of the program. We show how these rules may be formally expressed and manipulated to derive other induction rules, and give a technique for mechanically deriving from a schema an expression for its domain which may be expressed in terms of given induction rules by the manipulations referred to above. We give axiomatic definitions, including an induction rule, for some domains, which commonly occur in programs, these being finite sets, trees, structures, arrays with fixed bounds, LISP S-expressions, linear lists, and the integers. In developing a program one may start by defining the basic operations and domains in an axiomatic manner. Development proceeds by finding satisfactory representations for this domain in terms of more specific domains and their operations, until finally one has domains which are representable in a target language. We discuss what is meant by a representation in an attempt to formalise this technique of data refinement, and also mention the less general notion of simulation which requires that a representation is adequate tor a particular program to work. A program may have been developed in a recursive manner and if the target language does not contain recursion as a basic primitive it will be necessary to simulate it using stacks. We give axioms for such stacks, and give a mechanical procedure for obtaining from any recursive program, a flowchart program augmented by stacks, which simulates it.