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Often we employ the use of the separating hyperplane theorem to prove existence of price vectors, when discussing infinite economies this proof is substituted for proving existence of linear functionals.

This all being said, I find it difficult to see a case in consumer theory, producer theory and general equilibrium where we cant have a price vector (unless the problem is a planners problem).

To clarify I'm asking in what mathematical environments do prices not exist and what would their economic interpretation be.

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    $\begingroup$ From a generic mathematics standpoint, lack of uniqueness also poses issues, and not just existence. If a mathematical problem has an infinite number of solutions, it is possible that no numerical algorithm would converge to any of them. Although multiple equilibria is a concern in economics, not sure whether that issue shows up (if the equilibria are separated, no effect on convergence). $\endgroup$
    – Brian Romanchuk
    Commented Oct 20, 2020 at 19:23
  • $\begingroup$ @BrianRomanchuk I agree, but the way I understand price is as a numerical condition of transaction. When we have an infinite number of solutions we have an infinite set of conditions. Qualitatively im asking what is the mathematical structure of problems where people are unable to transact. $\endgroup$
    – EconJohn ♦
    Commented Oct 20, 2020 at 21:37
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    $\begingroup$ "...the separating hyperplane theorem..."---non-existence of separating hyperplanes would (trivially) imply non-existence of equilibrium price. E.g. one can drop the convexity assumptions and get a counterexample (Edgeworth box with non-convex preferences). $\endgroup$
    – Michael
    Commented Oct 21, 2020 at 9:12
  • $\begingroup$ I have no idea what this question is supposed to be about. $\endgroup$
    – Michael Greinecker
    Commented Oct 21, 2020 at 20:54
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    $\begingroup$ @EconJohn: I think you should be more clear about what you're asking (or expecting from an answer). The existence of (equilibrium) price as a mathematical condition is substantively different from the existence of price as an economic institution whereby payments for goods and services are measured. Brian and Michael's comments address the former interpretation of your question and most of the current answers address the latter interpretation. The divergence in these responses also suggests that there is ambiguity in the question as is. $\endgroup$
    – Herr K.
    Commented Oct 22, 2020 at 1:35

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There are a bunch of examples for incomplete markets in the finance literature. The oldest (that I know of) is Hart (1975). In finance, the problem is that if you have two different assets they have different prices that make the market complete. But then equilibrium considerations make them have the same price, which forces the market to no longer be complete. (If the market was complete, then they would have the same price.)

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There are "missing markets" such as in the case of pollution. We can try to create a price by charging fines against polluters, but no natural price exists.

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