$(1).$ Consider the following principal$-$agent problem. The principal $is$ risk neutral and her utility

$isy-w$ where $y$ denotes the random output $of$ the project and $wis$ the wage paid $to$ the agent.

The agent $is$ also risk neutral and the agent's utility function $is$ given $asw-e$ where $e$ denotes

the agent's effort; which has two possible values $e_0=0$ and $e_1=3.$ Output may take two

distinct values $y=0$ and $y=8.Ife=e_1$ then

$Pr(y=$$$|e=e_1)=\frac{3}{4}$

Otherwise $ife=e_0$ then

$Pr(y=8|e=e_0)=\frac{1}{4}$

The agent's reservation utility $\frac{?}{\text{b}\text{a}\text{r}}(U)(outside$ option$)is$ normalized $to$ zero, $i.e.\frac{?}{\text{b}\text{a}\text{r}}(U)=0.$

$(a).$ Assume first that the level $of$ effort $eis$ verifiable. Solve for the first best incentive contract

that the principal offers the agent?

$(b).$ Now assume that the level $of$ effort $eis$ non$-$verifiable but the output level $yis.$ Assume

further that $w$ can take any value $in$ the interval $-100,100,i.e.$ win$[-100,100].$ Solve for the

second best incentive contract that the principal offers the agent?

$(c).$ Assume now that a Limited Liability constraint operates, $i.e.w$ must $be$ non$-$

negative $=>w0.$ Try $to$ solve for the second best incentive contract $in$ presence $of$ this

additional constraint?