<p>principal: $\max_{w_h,w_l,e}p(e)v(q_h-w_h)+(1-p(e))v(q_l-w_l) $ subject to</p>

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

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

<p>Replace the IC with the FOC approach:</p>

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

<p>Now consider the principal's foc wrt $w_h$:</p>

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

<p>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:</p>

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

<p>Now v concave and (assuming inverse exists):</p>

<p>$q^f_h-w^f_h&gt;q^s_h-w^s_h$</p>

<p>So the spread decreases,</p>

CC BY-SA 3.0
2016-10-15T06:56:29.733Z
The problem with this result is that it requieres the second term to be equal in both cases.
Joaquín Mayorga
CC BY-SA 3.0
2016-09-16T01:02:40.133Z
2016-09-16T01:02:40.133Z
VCG
<p>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.</p>

CC BY-SA 3.0
2016-09-12T20:49:42.103Z
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?
VCG
CC BY-SA 3.0
2016-09-12T20:58:32.027Z
Yes, I have set up and solved both problems but cannot find that result which relates them.
Joaquín Mayorga
CC BY-SA 3.0
2016-09-12T21:01:48.213Z
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.
VCG
CC BY-SA 3.0
2016-09-12T23:47:45.450Z
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}.
Joaquín Mayorga
CC BY-SA 3.0
2016-09-13T00:23:53.043Z
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.
VCG
CC BY-SA 3.0
2016-09-13T00:37:51.093Z
Problem is, MWG has a profit function linear in w_i, which greatly simplifies the exercise.
Joaquín Mayorga
CC BY-SA 3.0
2016-09-15T23:55:15.377Z
I'm stuck. Please write it up.
Joaquín Mayorga
CC BY-SA 3.0
2016-09-16T01:03:05.090Z
I added something as a first pass. I'll have someone check it for me.
VCG
<asymmetric-information>
Moral hazard with linear effort and two possible outcomes
Joaquín Mayorga
# Moral hazard with linear effort and two possible outcomes

(Asked by: username_0 on 2016-09-12)

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.

**Tags:** asymmetric-information

> **Comments:**
> **username_1**: 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?  
> **username_0**: Yes, I have set up and solved both problems but cannot find that result which relates them.  
> **username_1**: 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.  
> **username_0**: 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}.  
> **username_1**: 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.  
> **username_0**: Problem is, MWG has a profit function linear in w_i, which greatly simplifies the exercise.  
> **username_0**: I'm stuck. Please write it up.  
> **username_1**: I added something as a first pass. I'll have someone check it for me.  

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## 1 answers

### Answer by username_1 on 2016-09-16. Score: 1

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}

Now v concave and (assuming inverse exists):

$q^f\_h-w^f\_h>q^s\_h-w^s\_h$

So the spread decreases,

> **Comments:**
> **username_0**: The problem with this result is that it requieres the second term to be equal in both cases.  

---

