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An integrability condition with applications to utility theory and thermodynamics

1979; Elsevier BV; Volume: 6; Issue: 1 Linguagem: Inglês

10.1016/0304-4068(79)90019-3

1873-1538

Leonid Hurwicz, Marcel K. Richter,

Economic Theory and Institutions

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.