there are only two possibleoutcomes, R in the good state and 0 in the bad state, and if En provides effort the success probability is pH < 1. However, we now assume that En is risk averse. That is, En obtains utility u(Rb) if she is rewarded Rb, where u is an increasing and concave function. For simplicity, we maintain the assumption that En gets private utility B if she shirks and B is added to the utility she may get from her reward. Also, let E's reward be R_b^s and R_b^f in the good and bad state, respectively. Finally, we impose limited liability for En, but not for the lenders.

Suppose  that u(Rb) = log(Rb) (log is the natural logarithm).

question:Write up the program which the optimal contract must solve. Solve the program, i.e. derive (R_b^s ;R_b^f), and write up En 's (expected) utility, Ub.
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是我们学习TIROLE 得THEORY OF CORPORATE FIANCE时老师布置得一道题目。

我得思路是Ub = PH*u(R_b^s)+(1-PH)*u(R_b^f)-A.
要取得OPTIMAL，u(R-b^f)要是0。
（书上第三章讨论得时候，U=PH*R_b^s+(1-PH)*R_b^f-A. 在满足IC, IR得条件下，只有R_b^f=0
才可能得到optimal的U。）
这样就得到R_B^F=1,最后得到R_B^S= EXP(R+(A-I)/PH)。

不过，在后续带入具体数值的计算中，经常得到R_b^f 和R_b^s不能满足IC条件得情况。