An ergodic theorem for multi-period mutual insurance

2026-08-14

AI Summary

This study addresses the determination of limit utility for self-enforcing contracts in multi-period mutual insurance with heterogeneous agents. Leveraging ergodic theory and asymptotic analysis, we establish an ergodic theorem for multi-period mutual insurance that reveals a collapse of degrees of freedom in large populations. We prove the uniqueness of the limit utility as population size approaches infinity and innovatively propose an explicit mechanism achieving this limit using only short-term contracts. By overcoming the design bottlenecks associated with long-term agreements, this work provides a practical theoretical foundation and mechanism design paradigm for optimal insurance systems, such as pension schemes, based on implementable short-term contracting.

Abstract

Suppose there are $N$ heterogeneous agents in a market with idiosyncratic risks but no uninsurable systematic risk factors. These agents may agree arbitrary financial contracts with one another, subject to the condition that contracts are self-enforcing under coalitions of agents in a common state. We show that, under mild conditions, this uniquely determines the limiting utility of every agent as $N$ tends to infinity. The result is an ergodic theorem: as the population grows, the number of degrees of freedom in the problem collapses, so that agents in the same state are treated identically in the limit. We exhibit an explicit, practically realisable mechanism achieving this limit using only short-dated contracts. The model can be applied either to an economy of heterogeneous agents pooling idiosyncratic risk through self-enforcing contracts or to the design of optimal insurance products such as pensions.

Problem

• Ergodic theorem
• Multi-period mutual insurance
• Heterogeneous agents
• Self-enforcing contracts
• Idiosyncratic risks

Innovation

• Ergodic theorem
• Self-enforcing contracts
• Heterogeneous agents
• Short-dated contracts
• Mutual insurance