Marcel K. Richter

INVARIANCE AXIOMS AND ECONOMIC INDEXES

exhibited. We present axiom systems for indexes of several economic variables: inputs, outputs, prices, wages, inflation, and technological change. In addition to conventional smoothness and proportionality conditions, in each case an Invariance Axiom is proposed. For technological change this says, in a sense, that when there is no technological change there is no change in the index. One can prove that in each case there is a unique index satisfying the axioms, and this is a Divisia index.2 Since these continuous indexes are not generally independent of the path, the problem of how often to change index weights may be viewed in the light of a choice between invariance and independence. We show that the unique invariant measure of technological change is a natural generalization of Solows measure to the case of many commodity types.

Nontransitive-Nontotal Consumer Theory

Abstract We show the parallel nature of two approaches to nontransitive or nontotal consumers: through “weak” preferences and through “strict” preferences. This yields specific results (e.g., a new equilibrium existence theorem with weak preferences), general results (a metatheorem translating between the two approaches), and general concepts (a new notion of “rationality”). We give revealed preference axioms to characterize preferences, and prove equivalences among several axiom systems, showing that the apparent weakness of some axiom systems is illusory. We resolve a Weak Axiom conjecture, and we introduce a new axiom. We prove that our nonclassical consumers generalize classical equilibrium theory.

The core-walras equivalence

The equivalence of cores and competitive equilibrium sets in the very general framework of Boolean rings and algebras is proved. The results include most previous results as special cases. In particular, it is shown that Theorems 1 and 2 of M. K. Richter (J. Econom. Theory 3 (1971), 323–334) remain true without the usual assumptions of δ-algebras for coalitions and δ-additive measures for allocations. This permits economies with countably many agents, rather than requiring continuumly many.

An integrability condition with applications to utility theory and thermodynamics

We establish a criterion alternative to the classical Frobenius condition for the solvability of a system of nonlinear partial differential equations (or a corresponding total differential equation) that arises in consumer choice theory. The Frobenius condition, that a certain matrix be symmetric, translates in consumer theory into the rather unmotivated requirement of symmetry of the Slutsky or Antonelli matrix. In contrast, the alternative criterion presented here is easily motivated from revealed preference considerations, and thus provides an economic justification for the classical symmetry condition. Such partial differential equations systems are of economic significance not only in consumer theory, but also in general equilibrium analysis and welfare economics. In addition they arise in very different areas, such as classical physics and thermodynamics. Wherever they arise, their solvability is often a critical step in theory construction. As indicated by our applications in section 3 (to consumer theory) and section 4 (to thermodynamics), the solvability criterion presented in our theorem may provide more plausible axioms for such theories, and simplify the task of proving solvability from them.

Concave utility on finite sets

Abstract When does a preference relation on a finite set have a concave or a strictly concave utility function? We provide a complete answer. Our proof is an application of the Theorem of the Alternative, and constructs a concave utility if one exists.

Existence of nonatomic core-walras allocations☆

Abstract We prove the existence of Walras allocations, where allocations are understood as measures on a nonatomic Boolean algebra. The algebra need not be a σ-algebra, as usually assumed, and the measures need not be countably additive and need not have densities with respect to the endowment allocation. Except for a strengthening of the continuity hypothesis, this is the framework we used earlier to prove the equivalence of core and Walras allocations. Our proof rests on two propositions of special interest in their own right: one asserts extensibility of coalition preferences, and the other characterizes coalition preferences.

Rational choice and polynomial measurement models

Abstract Rational choice behavior is defined in a manner that subsumes rational choice theory as part of general measurement theory. The special class of polynomial measurement problems are defined, and their solutions are reduced to solving polynomial inequalities. An algebraic criterion is given for the solvability of arbitrary finite sets of polynomial inequalities, resolving a conjecture of Tversky.

Computable preference and utility

Abstract We introduce computability definitions for both preference orderings and utility functions, on both Euclidean spaces and computable real spaces. We prove that computable preferences have computable utility functions. Computable preferences are interesting for three reasons: (i) they model one aspect of bounded rationality; (ii) they tell us when computational economics is practical; and (iii) they provide a framework for developing complexity classifications of economic theories. Our proof uses a new utility-construction algorithm, extending the methods of Debreu and Bridges in several directions.

Implicit Functions and Diffeomorphisms without C1

We prove implicit and inverse function theorems for non-C 1 functions, and characterize non-C 1 diffeomorphisms.

Revelations of a Gambler

Abstract Knowing that a decision maker maximizes expected utility with respect to some (unknown) utility U and some (unknown) probability P , what can one tell about P by observing his decisions? We discuss this revealed preference question primarily in the simple case of a two-element ( H and T ) state space, and show that the possible revelations of P T / P H are precisely those of the form P T / P H e∪ K k =1 (γ k ,δ k ), for some algebraic numbers γ k ,δ k .