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Individual and Collective Welfare in Risk Sharing with Many States

Federico Echenique, Farzad Pourbabaee

Abstract

We study efficient risk sharing among risk-averse agents in an economy with a large, finite number of states. Following a random shock to an initial agreement, agents may renegotiate. If they require a minimal utility improvement to accept a new deal, we show the probability of finding a mutually acceptable allocation vanishes exponentially as the state space grows. This holds regardless of agents' degree of risk aversion. In a two-agent multiple-priors model, we find that the potential for Pareto-improving trade requires that at least one agent's set of priors has a vanishingly small measure. Our results hinge on the ``shape does not matter'' message of high-dimensional isoperimetric inequalities.

Individual and Collective Welfare in Risk Sharing with Many States

Abstract

We study efficient risk sharing among risk-averse agents in an economy with a large, finite number of states. Following a random shock to an initial agreement, agents may renegotiate. If they require a minimal utility improvement to accept a new deal, we show the probability of finding a mutually acceptable allocation vanishes exponentially as the state space grows. This holds regardless of agents' degree of risk aversion. In a two-agent multiple-priors model, we find that the potential for Pareto-improving trade requires that at least one agent's set of priors has a vanishingly small measure. Our results hinge on the ``shape does not matter'' message of high-dimensional isoperimetric inequalities.
Paper Structure (20 sections, 11 theorems, 57 equations, 2 figures)

This paper contains 20 sections, 11 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Model
  3. Notations and Conventions
  4. Preferences and Uncertainty
  5. Exchange Economies
  6. Main Results
  7. Walrasian Equilibrium
  8. Optimality and Resource Utilization
  9. Prior Beliefs and Welfare-Improving Trade
  10. Discussion
  11. Behavioral Implications of Theorem \ref{['thm:small_belief_set']}
  12. On the Interpretation and Magnitude of .
  13. Methodology and Proofs
  14. Brunn-Minkowski Inequality
  15. Proof of the Results in Sections \ref{['sec:walras']} and \ref{['sec:pareto']}
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\mathcal{E}\colon I \to \mathcal{P} \times \mathbb{R}_+^d$ be an exchange economy, $\omega = \left(\omega_i\right)_{i \in I}$ be the endowment profile, and $f = \left(f_i\right)_{i\in I}$ be a Walrasian equilibrium allocation. Assume that there exists $\tau >0$ such that $\omega_i \geq \tau \bm

Figures (2)

  • Figure 1: Risk sharing with two states and two agents.
  • Figure 2: Individual welfare.

Theorems & Definitions (20)

  • Definition 1: $\varepsilon$-upper contour set
  • Definition 2: $\varepsilon$-Pareto optimality
  • Definition 3: Walrasian equilibrium
  • Theorem 1
  • Remark 1
  • Corollary 1
  • Theorem 2
  • Remark 2
  • Proposition 1
  • Proposition 2
  • ...and 10 more