We have a principal-agent model with hidden actions in which the principal is risk averse and the agent is risk neutral; Assume also there are two levels of output, $x$ and $x'$ (with $x'>x$) and two actions $a,a'$. Define $p(a),p(a')$ the probabilities of $x'$ under actions $a,a'$ respectively. Also, the agent disutility from action $a'$ is $-1$. The wages associated to $x,x'$ are $w,w'$ respectively.
My problem is that I am not sure how to show that the optimal contract requires $x'-w' =x-w$, i.e. that the agent, being risk neutral, takes on all the variability associated with the project.
I formalize the problem (assume the principal wants to induce $a'$, otherwise my question is trivial)
$\max\limits_{\{w,w'\}} u(x'-w')p(a') + u(x-w)(1-p(a'))$
st
$w'p(a') + w(1-p(a')) - 1 \geq 0 $
$w'p(a') + w(1-p(a')) - 1 \geq w'p(a) + w(1-p(a))$
In particular, when I try to solve the problem by maximizing the principal expected payoff subject to the "standard" Individual rationality (with $\lambda$ multiplier) and incentive compatibility (with $\mu$ multiplier) constraints (I assume the principal is interested in the more costly action $a'$) I end up with two equations which are not consistent with the aforementioned result. In particular:
$ u'(x-w) = \lambda + \mu [1- \frac{(1-p(a))}{(1-p(a'))}]$
$ u'(x'-w') = \lambda + \mu [1- \frac{p(a)}{p(a')}]$
It is evident that $x-w = x'-w'$ holds iff $p(a) =p(a')$ which is not the case in this problem (here we have that $p(a') >p(a)$). Another possibility would be to assume that the Incentive compatibility constraint is slack (hence $\mu = 0$); however I cannot understand why that should hold, when the principal wants to induce the most costly action $a'$ (help here)
I have read online that another approach would be to assume that the principal "sells" the project to the agent and the agent, after having chosen which level of effort maximizes its expected utility, pays back a fixed amount to the principal (call it $\beta_{a}, \beta_{a'}$)
So we would have something like:
$w'p(a') + w(1-p(a')) - 1 -\beta_{a'} \geq 0 $ if the agent chose to undertake high effort and $w'p(a) + w(1-p(a)) -\beta_a \geq 0 $ otherwise.
But then how to go from there? How to insure that the agent is going to choose the action $a'$? How are the fixed amounts determined? Why are they optimal?