3.7. EXAMPLES function U( is assumed to satisfy U(O)=0 and the reservation utility is 元=0. The inferior project can be implemented by setting w(s)=0 for s= 1, 2 Suppose the principal wants to implement a=2. The constraints can b written as (IC)(1-p(2,2)U((1)+p(2,2)U((2)-(2)≥ (1-p(1,2)U((1)+p(1,2)U(v(2) which simplifies to (p(2,2)-p(1,2)(U(2)-U(1))≥v(2) (IR)(1-p(2,2)U((1)+p(2,2)U((2)-v(2)≥0 In order to satisfy the (ir) constraint, consumption must be positive in at least one state. This implies that the expected utility from choosing low effort is strictly positive (1-p(1,2)U((1)+p(1,2)U((2)>0, so if the(IC) constraint is satisfied, the(IR) constraint must be strictly satisfied (1-p(2,2)U(v(1)+p(2,2U(v(2)-v(2)>0 Thus, if (w (1), w(2) is the solution to the optimal contract problem, the (IR) constraint does not bind. The principal's problem can then be written min(1-p(2,2)(1)+p(2,2)(2) st.(v(1),(2)≥0 (p(2,2)-p(1,2)(U((2)-U((1))≥v(2) Then it is clear that a necessary condition for an optimum is that w(1)=0 So the optimal contract for implementing a 2 is(0, w*(2)), where w*(2) solves the(IC) p(2,2)-p(1,2)U((2)=v(2) The payment w* (2) needed to give the necessary incentives to the manager3.7. EXAMPLES 9 function U(·) is assumed to satisfy U(0) = 0 and the reservation utility is u¯ = 0. The inferior project can be implemented by setting w(s)=0 for s = 1, 2. Suppose the principal wants to implement a = 2. The constraints can be written as (IC) (1 − p(2, 2))U(w(1)) + p(2, 2)U(w(2)) − ψ(2) ≥ (1 − p(1, 2))U(w(1)) + p(1, 2)U(w(2)) which simplifies to (p(2, 2) − p(1, 2))(U(2) − U(1)) ≥ ψ(2) and (IR) (1 − p(2, 2))U(w(1)) + p(2, 2)U(w(2)) − ψ(2) ≥ 0. In order to satisfy the (IR) constraint, consumption must be positive in at least one state. This implies that the expected utility from choosing low effort is strictly positive: (1 − p(1, 2))U(w(1)) + p(1, 2)U(w(2)) > 0, so if the (IC) constraint is satisfied, the (IR) constraint must be strictly satisfied: (1 − p(2, 2))U(w(1)) + p(2, 2)U(w(2)) − ψ(2) > 0. Thus, if (w(1), w(2)) is the solution to the optimal contract problem, the (IR) constraint does not bind. The principal’s problem can then be written as: minw(1 − p(2, 2))w(1) + p(2, 2)w(2) s.t. (w(1), w(2) ≥ 0 (p(2, 2) − p(1, 2))(U(w(2)) − U(w(1))) ≥ ψ(2). Then it is clear that a necessary condition for an optimum is that w(1) = 0. So the optimal contract for implementing a = 2 is (0, w∗(2)), where w∗(2) solves the (IC): (p(2, 2) − p(1, 2))U(w∗ (2)) = ψ(2). The payment w∗(2) needed to give the necessary incentives to the manager will be higher: