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Principal is interested in buying lake. The revenues of operating the lake as a camping site are determined by the weather (a random parameter $w$ distributed uniformly on $[0, 1]$) and by the development programme $d$. The revenue function is $wd$, the cost of development is $(d^2-1)/2$ and the agent outside option is $x>0$. Principal needs to specify a contract paying the agent an amount $π$ (if the site produces revenue $R$, the principal gets $R-π$ and the agent get $π$). Suppose that the agent have to accept or reject the contract before observing $w$, but choose the development programme after observing it.

First assume that the principal can observe the revenue $R$ but cannot separately observe $d$ or $w$. You may assume that the contract takes the form $π = aR$ where $a$ is a (not necessarily positive) parameter. Find the agent optimal development as a function of $a$, $x$ and $R$, and use this to compute the value of $a$ that maximises the investor’s expected utility. Also, find the optimised expected value of the lake as a function of $x$.

Need help solving this I don't want solution but just guidelines. First time on stack exchange so please ignore if I am breaking any rules while asking this question. I am struggling with how to form expected payoffs for agent and principal separately help will be appreciated thanks. As far as off topic is concern I think mechanism design is economic topic although little focused on but it is pure financial economics.

should agent expected payoff be $$\int_0^1 {[awd-(d^2-1)/2] dw} $$ and principal expected payoff be $$\int_0^1 {[wd-awd] dw} $$

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    $\begingroup$ Is this a homework question cleverly repackaged as a challenge? If so visit the Help Center, economics.stackexchange.com/help, Category "Asking", to see how questions should be asked here. $\endgroup$
    – Giskard
    Commented Apr 20, 2015 at 10:11
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    $\begingroup$ I'm voting to close this question as off-topic because it is a homework question without sufficient effort shown. $\endgroup$
    – FooBar
    Commented Apr 20, 2015 at 12:03
  • $\begingroup$ This is definitley a homework and not even a hard one. I also think this should be closed or OP should modify it asking for some specific help. $\endgroup$
    – BVJ
    Commented Apr 20, 2015 at 12:26
  • $\begingroup$ Off-topic is not really the deal here. The real concern is lack of effort shown on your side - I don't know why the site decided to leave off-topic as the reason for the hold. $\endgroup$
    – FooBar
    Commented Apr 23, 2015 at 15:35

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