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

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

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

<p>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: </p>

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

<p>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:</p>

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

<p>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 $\sigma&gt;0$. </p>

<p>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.</p>

<expected-utility><asymmetric-information><principal-agent>
LEN-Model equivalency
Fusscreme
# LEN-Model equivalency

(Asked by: username_0 on 2016-04-02)

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.

**Tags:** expected-utility, asymmetric-information, principal-agent


---

## 0 answers

