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Constrained optimal transport with an application to large markets with indivisible goods

Koji Yokote

Abstract

We establish a variant of Monge--Kantorovich duality for a constrained optimal transport problem with a continuum of agents, a finite set of alternatives, and general linear constraints. As an application, we revisit the large-market model of indivisible goods in Azevedo et al. (2013), identify a flaw in the original equilibrium-existence proof stemming from an incorrect compactness claim, and recover equilibrium existence via our duality approach. We also characterize equilibrium prices as minimizers of a potential function, which yields a method for computing equilibrium prices.

Constrained optimal transport with an application to large markets with indivisible goods

Abstract

We establish a variant of Monge--Kantorovich duality for a constrained optimal transport problem with a continuum of agents, a finite set of alternatives, and general linear constraints. As an application, we revisit the large-market model of indivisible goods in Azevedo et al. (2013), identify a flaw in the original equilibrium-existence proof stemming from an incorrect compactness claim, and recover equilibrium existence via our duality approach. We also characterize equilibrium prices as minimizers of a potential function, which yields a method for computing equilibrium prices.

Paper Structure

This paper contains 14 sections, 6 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. A variant of Monge-Kantorovich duality
  3. Proof of Theorem \ref{['thm-1']}
  4. Proof of (D) $\geq$ (P)
  5. Proof of (P) $\geq$ (D) and existence of primal optimizer
  6. Application: large markets with indivisible goods
  7. AzevedoWeylWhite2013's (AzevedoWeylWhite2013) model
  8. Counterexample to AzevedoWeylWhite2013's (AzevedoWeylWhite2013) claim in the proof
  9. Recovering the existence proof
  10. Application of the new duality
  11. Competitive equilibrium and the potential function
  12. Translating optimal transport equilibrium into AzevedoWeylWhite2013's (AzevedoWeylWhite2013) equilibrium
  13. Computation of equilibrium prices
  14. Concluding remarks

Key Result

Theorem 1

Suppose one of the following two conditions hold: (i) theree exists $\pi$ that satisfies the constraints of (P), or (ii) the optimal value of (D) is finite. Then, the primal problem $(\mathrm{P})$ has a solution and $(\mathrm{P})=(\mathrm{D})$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (21)

  • Theorem 1
  • Claim 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2: Hahn-Banach (see, e.g., Theorem 5.67 of AliprantisBorder2006)
  • Claim 2
  • proof
  • Claim 3
  • proof
  • ...and 11 more