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Starting position is a principal-agent-model with incomplete information (moral hazard) and the following properties:

  • Agent utility: $u(z)=-e^{(-r_az)}$
  • Principal utility: $B(z)=-e^{(-r_pz)}$
  • Effort levels $e\in \Bbb R $
  • Outcomes $x\in \Bbb R, x\sim N(\mu(e), \sigma), \mu'(e)>0, \mu''(e)\le0$
  • Contract: $w(x)=a+bx$,

where $r_A$ and $r_P$ is the Arrow–Pratt measure of absolute risk-aversion for the agent and the principal respectively.

I am looking for the optimal contract for the principal to offer to the agent when the agent's effort is not visible. The principal's utility can be written as follows:

$$U^P(e,a,b)=\int_{-\infty}^\infty-e^{(-r_P((1-b)x-a))}f(x\mid e) \, dx$$

I want to show that the following equivalence holds, meaning that the maximization of the principal's utility can be written as the RHS of the following equivalence:

$$\max_{\rm e,a,b}\int_{-\infty}^\infty-e^{(-r_P((1-b)x-a))}f(x\mid e) \, dx \Leftrightarrow \max_{\rm e,a,b}(1-b)\mu(e)-a-\frac{r_P}2(1-b)^2\sigma^2$$

where $f(x|e)=\frac{1}{\sigma\sqrt{2\pi}}e^{(-\frac{1}2(\frac{x-\mu(e)}\sigma)^2)}$ is the density function of a normal random variable $x\sim N(\mu(e),\sigma)$, with expected value $\mu(e)$ and variance $\sigma>0$.

I tried to use the explicit form of $f(x|e)$ in the LHS, manipulate it a bit and then itegrate but could not get the equivalence.

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The main point is that the principal's expected utility from a payoff $z$ conditional on a certain level of effort $e$ can be written as

$$ \text{E}[{z}|e] - \frac{r_p}{2}\text{Var}(z|e).$$

In other words, since wealth is normally distributed, exponential utility has a simple `mean-variance' representation. For a derivation, see here.

I take it that the principal's payoff $z$ equals $x - w(x) = (1 - b)x - a$. It is then straightforward to compute the (conditional) mean and variance of $z$:

$$ \text{E}[z|e] = (1 - b)\text{E}[x|e] - \text{E}[a] = (1 - b)\mu(e) - a,$$

$$\text{Var}[z|e] = (1 - b)^2 \text{Var}(x|e) - \text{Var}(a)= (1 - b)^2\sigma^2. $$

It follows that the principal's expected utility can be written as

$$ (1 - b)\mu(e) - a - \frac{r_p}{2}(1 - b)^2\sigma^2.$$

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