# Proofs of propositions

## Marcel Fafchamps

## Extract

Proof of Proposition 4. 1: Each point on the boundary of the equilibrium set can be found by maximizing one agent’s expected utility subject to satisfying all participation constraints and maintaining other agents at a given expected utility level. By varying the expected utility of other agents and by repeating the process for all agents, we can span the whole boundary of the equilibrium set. Now, participation constraints with A1 are a restricted version of participation constraints with A2. Therefore, by Le Chatelier principle, the maximum expected utility with A2 lies weakly above what can be achieved with A2 for all expected utility levels of other agents. Every point on the boundary of ⍀(A1) thus lies weakly below every point on the boundary of ⍀(A2). This proves the proposition. Strict inclusion occurs whenever ␦ and A1 are low enough for some participation constraints to be binding. A strictly higher A2 then is sure to release the binding participation constraints somewhat and to strictly enlarge the set of equilibria. i Proof of Proposition 4.2: For any given IRSA s, the right hand side of the voluntary participation constraint VP decreases with ␦. Consider an arbitrary agent i and state of nature s. Let ␦ fall to the point where the VP constraint is binding for that i and s. In order to further decrease ␦ while i still ensuring payment of s, agent i has to be compensated in other states of nature sЈ. Compensation by other agents is possible as long as some of...

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