Stronger core results with multidimensional prices

Technical Deep Dive: Stronger Core Results with Multidimensional Prices

Overview

This work proposes a new solution concept called a lexicographic dividend equilibrium (LDE) for one-sided matching markets with endowments. The authors show that an LDE always exists and coincides with the rejective core as the economy grows large, suggesting LDEs are the only viable market outcomes.

Problem & Context

• One-sided matching problems without money involve assigning indivisible goods to agents with preferences over the goods.
• The seminal work of Hylland and Zeckhauser (1979) introduced the concept of competitive equilibrium from equal incomes (CEEI) for this setting, showing it always exists and has desirable properties.
• However, when goods are privately owned (not collectively owned), the CEEI solution may be undesirable as it gives all agents equal budgets regardless of their endowments.
• The authors study the problem of one-sided matching with endowments, where each agent contributes their own goods to the market.
• It is known that a competitive equilibrium may not always exist in this setting due to issues with agent satiation.

Methodology

• The authors propose a new solution concept called a lexicographic dividend equilibrium (LDE) that associates each good with a multidimensional price.
• An LDE always exists and resides within the rejective core, a stronger stability concept than the weak core.
• LDEs are defined as follows:
  • Allocation x, price matrix p, and dividend matrix α such that:
    • The first non-zero entry in each column of p is positive.
    • The dividend α for each agent and currency k is the maximum of 0 and the surplus the agent has in that currency.
    • Each agent receives her most preferred affordable bundle given p and α.
  • LDEs can also satisfy additional properties like simple prices and cheapest bundle properties.
• The authors show that the set of LDEs satisfying the strong cheapest bundle property coincides with the rejective core as the economy grows large.

Results

1. Existence of LDEs: The authors prove that an LDE satisfying the strong cheapest bundle property always exists, even without the assumption of strict positive endowments.
2. Convergence to Rejective Core: The set of LDEs satisfying the strong cheapest bundle property converges to the rejective core as the economy grows large.

Interpretation

• LDEs provide a solution concept that always exists and exhibits desirable properties like individual rationality and weak core stability, even in the presence of agent satiation.
• The convergence result suggests that LDEs are the only viable market outcomes in a strong sense, as they are the only allocations that a coalition cannot reject as the economy grows.
• The multidimensional pricing structure of LDEs allows them to circumvent the non-existence issues that plague competitive and dividend equilibria in economies with satiation.

Limitations & Uncertainties

• The authors leave open the question of whether two currencies are always sufficient to clear the market in an LDE.
• The linearity of preferences plays an important role in the authors' proofs. It is unclear to what extent the results extend to more general preference structures.

What Comes Next

The authors suggest that their techniques for establishing the existence of LDEs and their convergence to the rejective core may be applicable to more general economic settings beyond one-sided matching with endowments, provided certain conditions on preferences and consumption sets are satisfied.