Undecidability of Uzawa equivalence theorem and LLPO (Lesser limited principle of omniscience)

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The Uzawa equivalence theorem [H. Uzawa, Walras’s Existence Theorem and Brouwer’s Fixed Point Theorem, Economic Studies Quarterly 8 (1962) 59–62] showed (classically) that the existence of Walrasian equilibrium in an economy with continuous excess demand functions is equivalent to Brouwer’s fixed point theorem, that is, the existence of a fixed point for any continuous function from an n -dimensional simplex to itself. We examine the Uzawa equivalence theorem from the point of view of constructive mathematics, and show that this theorem, properly speaking, the assumption of the existence of a Walrasian equilibrium price vector in this theorem, implies LLPO (Lesser limited principle of omniscience), and so it is non-constructive.