# First order conditions in a Moral Hazard problem

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.

• 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. Apr 29 at 17:09
• 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 Apr 29 at 17:18

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.
• @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. Apr 29 at 20:54
• Yes, I've clarified that$\ e \in \{e_H,e_L\}$ now. – Student Apr 29 at 21:17