Abstract

We investigate competitive equilibria in a special type of incomplete markets, referred to as a comonotone market, where agents can only trade such that their risk allocation is comonotonic. The comonotone market is motivated by the no-sabotage condition. For instance, in a standard insurance market, the allocation of risk among the insured, the insurer and the reinsurers is assumed to be comonotonic a priori to the risk-exchange. Two popular classes of preferences in risk management and behavioral economics, dual utilities (DU) and rank-dependent expected utilities (RDU), are used to formulate agents’ objectives. We present various results on properties and characterization of competitive equilibria in this framework, and in particular their relation to complete markets. For DU-comonotone markets, we find the equilibrium in closed form and for RDU-comonotone markets, we find the equilibrium in closed form in special cases. The fundamental theorems of welfare economics are established in both the DU and RDU markets. We further propose an algorithm to numerically obtain competitive equilibria based on discretization, which works for both the DU-comonotone market and the RDU-comonotone market. Although the comonotone and complete markets are closely related, many of our findings are intriguing and in sharp contrast to results in the literature on complete markets in terms of existence, uniqueness, and closed-form solutions of the equilibria, and comonotonicity of the pricing kernel.

Highlights

  • 1.1 BackgroundThis paper studies risk sharing games in a special type of one-period exchange markets, called the comonotone markets, and compare them with those in the classic complete markets

  • We focus on rank-dependent expected utilities (RDU) preferences because, even in a complete market, finding competitive equilibria is very challenging; existence may not be guaranteed, and explicit forms are generally unavailable

  • We propose an algorithm to numerically obtain competitive equilibrium based on discretization, which works for both the dual utilities (DU)-comonotone market and the RDUcomonotone market

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Summary

Background

This paper studies risk sharing games in a special type of one-period exchange markets, called the comonotone markets, and compare them with those in the classic complete markets. We focus on RDU preferences because, even in a complete market, finding competitive equilibria is very challenging; existence may not be guaranteed, and explicit forms are generally unavailable. Our paper is essentially different from Xia and Zhou (2016) and Jin et al (2019), noting that our market is comonotone to begin with In the latter two papers, non-trivial technical assumptions are imposed to guarantee the existence of an RDU equilibrium, and this ensures counter-comonotonicity of the pricing kernel with the market risk. We show that the pricing kernel is not necessarily counter-comonotonic with the market risk, and we require only a very weak condition on the structure of the preferences to guarantee existence. In order for explicit calculation of competitive equilibria, some non-trivial technical conditions need to be imposed

Contribution of the paper
Related literature
Dual utilities and rank-dependent utilities
Comonotone market and competitive equilibria
Individual optimization
Competitive equilibria with dual utilities
Equilibrium allocations and prices
Pareto optimality and fundamental theorems of welfare economics
Competitive equilibria with rank-dependent utilities
General results
An analytical approach for the competitive equilibria
The RDU-exponential market
An algorithm to compute the competitive equilibria
Illustration of the algorithm
Concluding remarks
The RDU-power market
Convergence of the algorithm in a DU-comonotone market
Full Text
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