hidden action model的题目

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The owner is risk neutral while the manager has preferences defined over the mean and variance of his income w and his effort level e as follows: Expected utility=E(w)-φVar(w)-g(e),where g'(0)=0,for e>0,g'(e),g''(e),g'''(e)>0. Possible effort choices are e>0.Conditional on effort level e, the realization of profit is normally distrbuted with mean e and variance σ²;.
(1)Derive the optimal contract when e is observable
(2)Derive  the optimal linear compensation scheme when e is not observable, What effects do change in φ and σ²; have?

以上是题目，我的思路是目标函数是最大化owner的期望利润，当e可见的时候，参与约束是经理的EU>=0.
但是遇到的问题是owner的期望利润怎么表示呢？
e不可见的时候，约束该怎么表示呢？

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关键词：Hidden Action model mode CTI 题目 model Action Hidden

First best effort e* is given by 1=g'(e*): marginal benefit equal to marginal cost.When e is observable, contract is: ask the agent to exert effort e* and set w=g(e*).

A risk neutral principal hires a risk averse agent to produce output. The agent can exert costly effort to increase output. The output is

    y=μ+e+ε,

    where ε∼N(0,σ²). μ and σ² represent the exogenous expected payoff and the riskiness of the given production method separately; while e is effort exerted by the agent. The effort cost is c(e).
    Contract. The effort is not observable. The only observable and contractible variable is the output y. Assume that the principal uses a linear contract

    w=w₀+αy,

    where w₀ is the fixed salary and α is the power of incentive.
    Given the contract, the agent choose e to maximize his certainty equivalent

    CE=w₀+αμ+αe-((rα²σ²)/2)-c(e).

    F.O.C with respect to e gives the following incentive compatible condition

    IC₁:c'(e)=α.

    The participation constraint requires

    IR₁:CE>=0.

    The participation constraint should be binding. Otherwise, the principal can be better off by reducing the fixed salary w₀. By adjusting the fixed salary, the principal extracts all the surplus. His problem is then to choose the power of incentive α in order to maximize the total surplus:

     max_{α}  μ+e-c(e)-((rα²σ²)/2)
     s.t.IC₁.

    Solving this problem, one can get the optimal power of incentive:

    α₁^{∗}=(1/(1+rσ²c'')).


<proposition/>There are both a negative relationship between risk and performance pay and a negative relationship between agent's risk aversion attitude and performance pay.

    For the principal, there is a trade off between inducing effort and providing insurance. The benefit of a higher power of incentive is that it induces a higher effort. The cost is that the principal needs to pay the agent a higher risk premium. As the risk or the agent's risk averse attitude increases, a higher power of incentive becomes more costly. Accordingly, the principal will reduce the power of incentive.