Moral hazard with linear effort and two possible outcomes Suppose you have a risk averse principal and a risk averse agent with utility functions $v(q_i - w_i)$ and $u(w_i)$, respectively, where $i = \{H,L\}$, and $q$ is output and $w$ is the wage. The probability that output is high is given by $p(e)$ with $p'(e) &gt; 0 \geq p''(e)$ and effort is given by the linear function $e$. How do you prove that in the second-best case, the agent receives a larger (smaller) share of the profits when output is high (low) relative to the fist-best case? Bolton and Dewatripont (2005) mention this result but do not prove it.
principal: $\max_{w_h,w_l,e}p(e)v(q_h-w_h)+(1-p(e))v(q_l-w_l) $ subject to

(IR)       $~~~~~p(e)u(w_h-c(e))+(1-p(e))u(w_l-c(e))\geq \bar u$   

(IC)$~~~~~~e\in arg\max_{\tilde e}p(e)u(w_h-c(e))+(1-p(e))u(w_l-c(e))$

Replace the IC with the FOC approach:

(IC'))$~~~~~~p'(e)u(w_h-c(e))+p(e)u'(w_h-c(e))(-c'(e))-p'(e)u(w_l-c(e))+(1-p(e))u'(w_l-c(e))(-c'(e))=0$

Now consider the principal's foc wrt $w_h$:

$p(e)v'(q_h-w_h)(-1)+\lambda(p(e)u'(w_h-c(e))+\mu [p'(e)u'(w_h-c(e))+p(e)u''(w_h-c(e))(-c'(e))]=0$

So the difference between this and first best is that third term in the FOC. Using the standard assumptions, we know that that third term is positive and so:

$v'(q_h-w_h)_{first best}&lt;v'(q_h-w_h)_{second best}$

Now v concave and (assuming inverse exists):

$q^f_h-w^f_h&gt;q^s_h-w^s_h$

So the spread decreases,