# Optimal Risk Sharing in Society

## Abstract

**:**

## 1. Introduction

## 2. The Basic Risk-Exchange Model

#### 2.1. Some Basic References

#### 2.2. The No Arbitrage Requirement

**Theorem**

**1.**

## 3. Pareto Optimality

**Definition**

**1.**

**Definition**

**2.**

#### 3.1. The Characterization of a Pareto Optimum

**Theorem**

**2.**

**Theorem**

**3.**

**Example**

**1.**

#### 3.2. Risk Tolerance and Aggregation

**Theorem**

**4.**

#### 3.3. HARA-Utility Functions

**Example**

**2.**

**Example**

**3.**

#### 3.4. Affine Contracts

**Theorem**

**5.**

## 4. Equilibrium

**Definition**

**3.**

**Theorem**

**6.**

**Example**

**4.**

## 5. Syndicates I

**Example**

**5.**

#### 5.1. Homogeneous Probability Beliefs

**Theorem**

**7.**

#### 5.2. Example: Optimal Diversification

#### 5.3. Heterogeneous Probability Beliefs

**Theorem**

**8.**

**Theorem**

**9.**

**Example**

**6.**

## 6. Syndicates II

**Proposition**

**1.**

**Theorem**

**10.**

**Theorem**

**11.**

**Theorem**

**12.**

**Corollary**

**1.**

**Theorem**

**13.**

**Theorem**

**14.**

#### 6.1. An Example

## 7. Syndicates, Financial Markets, and General Equilibrium

**Proposition**

**2.**

## 8. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Theorem**

**1.**

## Appendix B

#### Appendix B.1. State Prices

#### Appendix B.1.1. A Thought Experiment

#### Appendix B.1.2. Equivalent Representations of the Value of a Project

## Appendix C

**Proof**

**of**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**5.**

**Proof**

**of**

**Theorem**

**6.**

**Proof**

**of**

**Theorem**

**7**

**Proof**

**of**

**property**

**(iii)**

**in**

**Section 5.3.**

**Proof**

**of**

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**2.**

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Aase, K.K.
Optimal Risk Sharing in Society. *Mathematics* **2022**, *10*, 161.
https://doi.org/10.3390/math10010161

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Aase KK.
Optimal Risk Sharing in Society. *Mathematics*. 2022; 10(1):161.
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Aase, Knut K.
2022. "Optimal Risk Sharing in Society" *Mathematics* 10, no. 1: 161.
https://doi.org/10.3390/math10010161