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 $p'(e) > 0 \geq p''(e)$ 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.
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$\begingroup$ So this looks like a standard MH problem, so first set up the first best case where effort is observable. Then find the 2nd best with the appropriate IR and IC constraints. Have you done these sorts of problems before? $\endgroup$– VCGCommented Sep 12, 2016 at 20:49
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$\begingroup$ Yes, I have set up and solved both problems but cannot find that result which relates them. $\endgroup$– Joaquín MayorgaCommented Sep 12, 2016 at 20:58
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$\begingroup$ Well one could just solve it given the info you provided. In the FB the principal insures the agent against risk with flat pay, but in the 2nd best, he can't do that perfectly so he must give him a higher payoff (you can show that using Jensen's inequality). All of this is in MWG if you have that. $\endgroup$– VCGCommented Sep 12, 2016 at 21:01
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$\begingroup$ Flat pay is ruled out by the IC. Plus, is this case, effort follows a linear function, while in Mas Colell effort is e={e_h, e_l}. $\endgroup$– Joaquín MayorgaCommented Sep 12, 2016 at 23:47
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$\begingroup$ Appendix A in 14 shows you the first order approach - how you deal with a continuous set of effort levels. I'll write something up if you get stuck with this. $\endgroup$– VCGCommented Sep 13, 2016 at 0:23
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1 Answer
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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:
$v'(q_h-w_h)_{first best}<v'(q_h-w_h)_{second best}$
Now v concave and (assuming inverse exists):
$q^f_h-w^f_h>q^s_h-w^s_h$
So the spread decreases,
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$\begingroup$ The problem with this result is that it requieres the second term to be equal in both cases. $\endgroup$ Commented Oct 15, 2016 at 6:56