CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM A contract is incentive-efficient if it is incentive-compatible and there does not exist another incentive-compatible contract that makes one party bet- ter off without making the other party worse off. We can characterize the incentive-efficient contracts using the following decision problem(DP2 max>p(a, s)V(R(s-w(s) (a,u() subject to ∑pa,s(m(s)-a)≥∑pb,s(m(s)-(), and ∑p(a,s)U(m(s)- Proposition 2 Under the maintained assumptions, a contract (a, w())is e-efficient only if it is a solution of DP2 for some constant i. A contract that solves DP2 is incentive-efficient if the participation constraint ng for every solution. The proof of the "only if"part is similar to the Pareto efficiency argument If (a, w() is a solution to DP2 and is not incentive-efficient, there exists an incentive-efficient contract that gives the principal the same payoff and the agent a higher payoff. But this contract must be a solution to DP2 that strictly satisfies the participation constraint The assumption of a uniformly binding participation constraint is restric tive: see Section 3. 7. 1 for a counter-example This DP can be solved in two stages. First, for any action a, compute the payoff V*(a) from choosing a and providing optimal incentives to the agent to choose a. Call this DP3 V"(a)=max)p(a, s)V(R(s)-w(s) s∈S b ∑p(a,s)(m(s)-v(a)≥∑p(b,)U((s)-v(b) ∑p(a,s)C(()-v(a)≥元 s∈S4 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM A contract is incentive-efficient if it is incentive-compatible and there does not exist another incentive-compatible contract that makes one party bet￾ter off without making the other party worse off. We can characterize the incentive-efficient contracts using the following decision problem (DP2): max (a,w(·)) X s∈S p(a, s)V (R(s) − w(s)) subject to X s∈S p(a, s)U(w(s)) − ψ(a) ≥ X s∈S p(b, s)U(w(s)) − ψ(b), ∀b, and X s∈S p(a, s)U(w(s)) − ψ(a) ≥ u. ¯ Proposition 2 Under the maintained assumptions, a contract (a, w(·)) is incentive-efficient only if it is a solution of DP2 for some constant u¯. A contract that solves DP2 is incentive-efficient if the participation constraint is binding for every solution. The proof of the “only if” part is similar to the Pareto efficiency argument. If (a, w(·)) is a solution to DP2 and is not incentive-efficient, there exists an incentive-efficient contract that gives the principal the same payoff and the agent a higher payoff. But this contract must be a solution to DP2 that strictly satisfies the participation constraint. The assumption of a uniformly binding participation constraint is restric￾tive: see Section 3.7.1 for a counter-example. This DP can be solved in two stages. First, for any action a, compute the payoff V ∗(a) from choosing a and providing optimal incentives to the agent to choose a. Call this DP3 V ∗ (a) = max w(·) X s∈S p(a, s)V (R(s) − w(s)) subject to X s∈S p(a, s)U(w(s)) − ψ(a) ≥ X s∈S p(b, s)U(w(s)) − ψ(b), ∀b, X s∈S p(a, s)U(w(s)) − ψ(a) ≥ u. ¯