3. 3. INCENTIVE EFFICIENCY ubject t ∑p(a,s((s)-v()≥ for some constant元 Proposition 1 Under the maintained assumptions, a contract(a, w())is Pareto-efficient if and only if it is a solution to the decision problem DP1 for sone u Suppose that(a, w()) is Pareto-efficient. Put i equal to the gent's pay off. By definition, the contract must maximize the principal's payoff subject to the constraint that the agent receive at least i. Conversely, suppose that the contract(a, w()is a solution to DPl for some value of i. If the contract is not Pareto-efficient, then there must be another contract that yields the same payoff to the principal and more to the agent. But then it must be possible to transfer wealth to the principal in some state, contradicting the optimality of (a,w() Suppose that the sharing rule satisfies w(s)>0 for all s. Then optimal risk sharing requires V(R(s)-(s)) 入 U((s) These are sometimes referred to as the borch conditions. If the action a belongs to the interior of A and if the functions p(a, s) and v(a) are differ entiable at a, then ∑m(a,s)(F(s)-(s)-X((s)+(a)=0 3.3 Incentive efficiency Now suppose that the agent's action is neither observable nor verifiable. In that case, the action specified by the contract must be consistent with the agent's incentives. A contract (a, w()) is incentive-compatible if it satisfies the constraint ∑p(a,s((s)-(a)≥∑p,s)(m()-v(b, s∈S s∈S3.3. INCENTIVE EFFICIENCY 3 subject to X s∈S p(a, s)U(w(s)) − ψ(a) ≥ u, ¯ for some constant u¯. Proposition 1 Under the maintained assumptions, a contract (a, w(·)) is Pareto-efficient if and only if it is a solution to the decision problem DP1 for some u¯. Suppose that (a, w(·)) is Pareto-efficient. Put u¯ equal to the agent’s pay￾off. By definition, the contract must maximize the principal’s payoff subject to the constraint that the agent receive at least u¯. Conversely, suppose that the contract (a, w(·)) is a solution to DP1 for some value of u¯. If the contract is not Pareto-efficient, then there must be another contract that yields the same payoff to the principal and more to the agent. But then it must be possible to transfer wealth to the principal in some state, contradicting the optimality of (a, w(·)). Suppose that the sharing rule satisfies w(s) > 0 for all s. Then optimal risk sharing requires: V 0 (R(s) − w(s)) U0 (w(s)) = λ, ∀s. These are sometimes referred to as the Borch conditions. If the action a belongs to the interior of A and if the functions p(a, s) and ψ(a) are differ￾entiable at a, then X s∈S pa(a, s) [V (R(s) − w(s)) − λU(w(s)] + λψ0 (a)=0. 3.3 Incentive efficiency Now suppose that the agent’s action is neither observable nor verifiable. In that case, the action specified by the contract must be consistent with the agent’s incentives. A contract (a, w(·)) is incentive-compatible if it satisfies the constraint X s∈S p(a, s)U(w(s)) − ψ(a) ≥ X s∈S p(b, s)U(w(s)) − ψ(b), ∀b