<p>Notice that MWG choose a wage <em>function</em> <span class="math-container">$w(\pi)$</span> to maximize profit, not a wage <em>level</em> <span class="math-container">$w$</span>. As footnote 6 on p.481 of MWG says:</p>

<blockquote>
  <p>The first-order condition for <span class="math-container">$w(\pi)$</span> is derived by taking the derivative with respect to the manager's wage at each level of <span class="math-container">$\pi$</span> separately.</p>
</blockquote>

<p>In other words, you'd treat the wage function as a variable, e.g. <span class="math-container">$w(\pi)=z$</span>, and solve 
<span class="math-container">\begin{equation}
\max_{z} \int [\pi-z]f(\pi|e)\mathrm d\pi +\gamma \left[\int v(z)f(\pi|e)\mathrm d\pi-g(e)-\overline u\right],
\end{equation}</span>
which will yield the FOC 
<span class="math-container">\begin{equation}
-f(\pi|e)+\gamma v'(z)f(\pi|e)=0. 
\end{equation}</span>
Substituting <span class="math-container">$z$</span> with <span class="math-container">$w(\pi)$</span>, you'd have condition (14.B.3) in MWG. </p>

CC BY-SA 4.0
2019-04-29T19:35:17.450Z
Thank you. However, will the certainty equivalent level of wealth always exist?
Student
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2019-04-29T19:36:22.923Z
Moreover, the book I am referring to (Mas-Colell, Whinston and Green) uses first order conditions, which is why I proceeded that way.
Student
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2019-04-29T20:54:04.123Z
@Student: Certainty equivalent always exists (unless the set of wealth is non-convex). I didn't realize you were referring to the MWG; based on your description, I assume that $e\in\mathbb R$ whereas in MWG $e\in\{e_H,e_L\}$. I'll write up a revision later.
Herr K.
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2019-04-29T21:17:23.973Z
Yes, I've clarified that$\ e \in \{e_H,e_L\}$ now.
Student
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2019-04-29T22:29:38.477Z
@Student: I rewrote my answer based on your clarification. Let me know if it's still unclear.
Herr K.
CC BY-SA 4.0
2019-04-29T23:22:03.330Z
Ah yes, I think I see now. The function-level difference clarifies why exactly they take the FOC at each $\pi$. Thanks!
Student
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2019-04-29T19:29:51.603Z
2019-04-29T22:28:35.643Z
2019-04-29T22:28:35.643Z
Herr K.
<p>Consider a principal-agent problem where effort is observable. Let <span class="math-container">$\pi$</span> denote profit, let <span class="math-container">$w$</span> denote the wage to be paid to the agent, and let <span class="math-container">$f(\pi|e)$</span> denote the pdf of the profit, conditional on effort <span class="math-container">$e$</span>. Further, <span class="math-container">$e \in \{e_L,e_H\}$</span>. The problem for the principal is then as follows: For each level of effort, <span class="math-container">$e$</span>: <span class="math-container">$$ \min_w\int w(\pi) f(\pi|e)\, d\pi$$</span> subject to <span class="math-container">$$\int v(w(\pi)) f(\pi|e)\,d\pi - g(e) \ge \overline u$$</span>
(Here, <span class="math-container">$v$</span> is the utility function for the agent, <span class="math-container">$g$</span> is the cost function of effort and <span class="math-container">$\overline u$</span> is the outside option for the agent). Then, choose <span class="math-container">$e$</span> that gives maximum profit given the wage <span class="math-container">$w$</span>, which has now been solved for at each effort level.</p>

<p>I'm not sure how to take the first order conditions in the minimisation problem above. If <span class="math-container">$w$</span> were the argument for the integral, it'd be manageable.</p>

CC BY-SA 4.0
2019-04-29T17:09:20.547Z
You're not giving the full picture of the problem. What is the individual rationality constraint for the agent? Is the agent risk averse or risk neutral? Why does wage depend on profit when effort is observable? And, even if wage does depend on profit, shouldn't it include the pdf of profit, i.e. $\int w(\pi)f(\pi|e)\mathrm d\pi$?
Herr K.
CC BY-SA 4.0
2019-04-29T17:18:45.643Z
You are right, I've fixed the minimand. Risk preferences are arbitrary. By assumption, wage depends on profit. However if the agent is strictly risk averse, wage does not depend on profit as it turns out (but we only see that after the FOC have been taken I guess).
Student
<microeconomics><principal-agent>
First order conditions in a Moral Hazard problem
Student
# First order conditions in a Moral Hazard problem

(Asked by: username_0 on 2019-04-29)

Consider a principal-agent problem where effort is observable. Let $\pi$ denote profit, let $w$ denote the wage to be paid to the agent, and let $f(\pi|e)$ denote the pdf of the profit, conditional on effort $e$. Further, $e \in \{e\_L,e\_H\}$. The problem for the principal is then as follows: For each level of effort, $e$: $$ \min\_w\int w(\pi) f(\pi|e)\, d\pi$$ subject to $$\int v(w(\pi)) f(\pi|e)\,d\pi - g(e) \ge \overline u$$
(Here, $v$ is the utility function for the agent, $g$ is the cost function of effort and $\overline u$ is the outside option for the agent). Then, choose $e$ that gives maximum profit given the wage $w$, which has now been solved for at each effort level.

I'm not sure how to take the first order conditions in the minimisation problem above. If $w$ were the argument for the integral, it'd be manageable.

**Tags:** microeconomics, principal-agent
> **Comments:**
> **username_1**: You're not giving the full picture of the problem. What is the individual rationality constraint for the agent? Is the agent risk averse or risk neutral? Why does wage depend on profit when effort is observable? And, even if wage does depend on profit, shouldn't it include the pdf of profit, i.e. $\int w(\pi)f(\pi|e)\mathrm d\pi$?  
> **username_0**: You are right, I've fixed the minimand. Risk preferences are arbitrary. By assumption, wage depends on profit. However if the agent is strictly risk averse, wage does not depend on profit as it turns out (but we only see that after the FOC have been taken I guess).  

---

## 1 answers

### Answer by username_1 on 2019-04-29. Score: 3 | Accepted answer

Notice that MWG choose a wage *function* $w(\pi)$ to maximize profit, not a wage *level* $w$. As footnote 6 on p.481 of MWG says:

> The first-order condition for $w(\pi)$ is derived by taking the derivative with respect to the manager's wage at each level of $\pi$ separately.

In other words, you'd treat the wage function as a variable, e.g. $w(\pi)=z$, and solve
\begin{equation}
\max\_{z} \int [\pi-z]f(\pi|e)\mathrm d\pi +\gamma \left[\int v(z)f(\pi|e)\mathrm d\pi-g(e)-\overline u\right],
\end{equation}
which will yield the FOC
\begin{equation}
-f(\pi|e)+\gamma v'(z)f(\pi|e)=0.
\end{equation}
Substituting $z$ with $w(\pi)$, you'd have condition (14.B.3) in MWG.

> **Comments:**
> **username_0**: Thank you. However, will the certainty equivalent level of wealth always exist?  
> **username_0**: Moreover, the book I am referring to (Mas-Colell, Whinston and Green) uses first order conditions, which is why I proceeded that way.  
> **username_1**: @Student: Certainty equivalent always exists (unless the set of wealth is non-convex). I didn't realize you were referring to the MWG; based on your description, I assume that $e\in\mathbb R$ whereas in MWG $e\in\{e_H,e_L\}$. I'll write up a revision later.  
> **username_0**: Yes, I've clarified that$\ e \in \{e_H,e_L\}$ now.  
> **username_1**: @Student: I rewrote my answer based on your clarification. Let me know if it's still unclear.  
> **username_0**: Ah yes, I think I see now. The function-level difference clarifies why exactly they take the FOC at each $\pi$. Thanks!  

---

