4.2. THE RISK SHIFTING PROBLEM assume the agent's consumption is non-negative(limited liability). In the first best, we have seen that when the manager is risk neutral there is a number r>0 such that w(s)= maxR(s-T, 01, s, and the return to the principal is R(s)-(s)=min(R(s), r1,Vs With this payment structure, the entrepreneur chooses a to maximize his expected return ∑pa,s)m(s)=∑p(,)max{(s)-r,0 Suppose that the principal is restricted to offering an incentive scheme of this form. Then the(constrained) principal-agent problem max(a,r(a, s)V(minT, R(s))) r≥0 ∑、p(a,s)max{f(s)-r,0}≥∑,p(a,s)max{(s)-r,0},vs max R(s)-r,0} For any probability vector p=(p1, .,ps)let P()=∑ d A=∑P(R(a+1)-Ra) A distribution p is a mean-preserving spread of p if it satisfies one of the following equivalent conditions Proposition1 Suppose that∑、p,R(s)=∑。p,R(s). The following cona tion ≤∑。=04 (ii) for any non-decreasing function f: S-R with non-increasing difference∑spf(s)≤∑。Psf(s) (iii) p is obtained from p in a finite sequence of transformations at each step of which the probability of one outcome is reduced and the probability mass is redistributed to a higher and lower outcome in a4.2. THE RISK SHIFTING PROBLEM 7 assume the agent’s consumption is non-negative (limited liability). In the first best, we have seen that when the manager is risk neutral there is a number r > 0 such that w(s) = max{R(s) − r, 0}, ∀s, and the return to the principal is R(s) − w(s) = min{R(s), r}, ∀s. With this payment structure, the entrepreneur chooses a to maximize his expected return X s p(a, s)w(s) = X s p(a, s) max{R(s) − r, 0}. Suppose that the principal is restricted to offering an incentive scheme of this form. Then the (constrained) principal-agent problem is max(a,r) P s p(a, s)V (min{r, R(s)}) s.t. r ≥ 0 (IC) P s p(a, s) max{R(s) − r, 0} ≥ P s p(a, s) max{R(s) − r, 0}, ∀s (IR) P s p(a, s) max{R(s) − r, 0} ≥ u¯ For any probability vector p = (p1, ..., pS) let P(s) = Xs σ=0 pσ and As = Xs σ=0 Pσ(R(σ + 1) − R(σ)). A distribution p0 is a mean-preserving spread of p if it satisfies one of the following equivalent conditions: Proposition 1 Suppose that P s psR(s) = P s p0 sR(s). The following condi￾tions are equivalent: (i) Ps σ=0 Aσ ≤ Ps σ=0 A0 σ; (ii) for any non-decreasing function f : S → R with non-increasing differences P s p0 sf (s) ≤ P s psf (s) ; (iii) p0 is obtained from p in a finite sequence of transformations at each step of which the probability of one outcome is reduced and the probability mass is redistributed to a higher and lower outcome in a mean-preserving way