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LIT-1435: Theory of Value: An Axiomatic Analysis of Economic Equilibrium

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Properties5
StatusActive — read in part 2026-10-23 (NOTE-1112): §4.6, "Continuity assumption on preferences" (pp. 55–59), and the notes to chapter 4 (pp. 72–73), from the PDF the Cowles Foundation posts as Monograph 17. Worth reading for Debreu's own statement of the utility representation theorem and its limits. A complete preorder on a connected subset of Rˡ (more generally, a connected separable space) with closed upper and lower contour sets has a continuous utility function. The lexicographic ordering of R² has no real-valued representation at all. A utility function is "arbitrary to a large extent". The rest of the book (commodities, producers, equilibrium, optimum, uncertainty) was not read.
Tagsmathematics, social-science
ExtendsLIT-1454 — Representation of a preference ordering by a numerical function
SupportsCLAIM-436 — An instantaneous ordering is a common currency on a finite set of options, where every complete preorder has a real-valued representation, but not in general: the lexicographic order on the unit square has no real-valued representation, while a continuous complete preorder on a connected separable space has a continuous one; so admitting an ordering is strictly weaker than admitting a currency, a graded lexicographic override lies in the gap while an exact threshold override does not, a threshold override with a discrimination threshold is a lexicographic semiorder whose choices can cycle and so is no ordering at all, and even where each instant has a currency, successive currencies need not share exchange rates
Cited as “paper” byNOTE-1112 — Theory of Value: An Axiomatic Analysis of Economic Equilibrium

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Gerard Debreu (1959), Theory of Value: An Axiomatic Analysis of Economic Equilibrium, Cowles Foundation Monograph 17, New York: Wiley; fourth printing, Yale University Press, 1971. URL: the Cowles Foundation’s PDF.

Key takeaways

Read for §4.6 and its notes only.

  • The question. Can each indifference class be given a real number “in such a way that, if a class is preferred to another, the number of the first is greater than the number of the second?” “The answer to the existence question is: not necessarily so (see an example in note 2)” (pp. 55–56).
  • Continuity assumption (a). For every x′, the sets {x | x ≾ x′} and {x | x ≿ x′} are closed (p. 56).
  • Theorem (1). “Let Xᵢ be a connected subset of Rˡ, completely preordered by ≾. Under assumption (a) there is on Xᵢ a continuous utility function” (p. 56). The proof builds an increasing map from a countable dense subset onto the rationals of an interval and extends it by suprema (pp. 56–59). Conversely, a continuous utility function implies (a).
  • The general form, and its credit. Only the countable dense subset is used, so the theorem holds for “a connected, separable topological space”. “In this form it is essentially a result of S. Eilenberg” (1941), and connectedness “has been removed for spaces satisfying the second axiom of countability in G. Debreu [3]” (the 1954 paper, p. 73).
  • The lexicographic counterexample. “An example of a complete preordering which cannot be represented by a real-valued function is the lexicographic ordering of R²” (p. 72). The proof sets up a one-to-one correspondence between the reals and a countable set of disjoint intervals (p. 73).
  • Utility is ordinal. Any increasing transform of a utility function is one too, so “a corresponding utility function is thus arbitrary to a large extent” (p. 56).

Standing in the record

Filed on 2026-10-23 as the lawful first-hand source for the theorem CLAIM-436 uses. The 1954 paper (LIT-1454) is Unavailable. §4.6 confirms the claim’s statement of the theorem and its lexicographic argument. It corrects the attribution of the connected-separable form, and it exposes a gap in how the claim and CLAIM-435 read a “step” in an indifference curve. See NOTE-1112.

Its general-equilibrium chapters are the work the record’s economics neighbours point to (LIT-067, Debreu 1952; LIT-218 on the Hicks–Allen to Debreu line). They were not read for this filing.

No instruction for machine-learning practice; nothing here belongs in the anthology.

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