Infinite inequality systems and cardinal revelations

Abstract

Many economics problems are maximization or minimization problems, and can be formalized as problems of solving “linear difference systems” of the form \(r_i-r_j \geqq c_{ij}\) and rk-rl > ckl, for r-unknowns, with given c-constants. They typically involve strict as well as weak inequalities, with infinitely many inequalities and unknowns. Since strict inequalities are not preserved under passage to the limit, infinite systems with strict inequalities are notoriously hard to solve. We introduce a unifying tool for solving them. Our main result (Theorem 1 for the countable case, Theorem [2] for the not-necessarily-countable case) introduces a uniform solvability criterion (the \(\omega\)-Axiom), and our proof yields a method for solving those that are solvable. The axiom’s economic intuition extends the traditional ordinal notion of revealed preference to a cardinal notion. We give applications in producer theory, consumer theory, implementation theory, and constrained maximization theory.