A risk$-$averse employer is looking to hire a risk$-$neutral worker. The employer$$s output Q depends on the worker$$s effort E in $[0,1],$ which has a disutility c$($E$)=100$ E$.$ Output can take two possible values: Q$1$ with probability $$ E and Q$2$ with probability $1$ E$,$ with Q$2$ Q$1.$ The employer offers a salary W$,$ and it has a utility over net profits given by ln$($Q $$ W$).$ Assume the worker$$s utility from the best alternative job is C $>0.($a$)$ Write down the expected utility of the employer and the worker assuming the employer pays W$1$ if output is high, and W$2$ if output is low. $($b$)$ Derive the optimal first$-$best contract C $=($W$1,$ W$2,$ E$)$ assuming that effort E is observable, and that Q$1=200$ and Q$2=100.($c$)$ Assume effort E is observed only by the worker, and determine the incentive compatibility and the participation constraints. Derive the second$-$best contract C $=($W$1,$ W$2,$ E$)$ with Q$1=200$ and Q$2=100.($d$)$ Compare the first$-$best C $$ and the second$-$best C $$ contracts emerging from $($b$)$ and $($c$),$ explaining briefly any differences and$/$or similarities between the two. Question $2(25$ marks$)$ A risk$-$averse employer is looking to hire a risk$-$neutral worker.

The employer's output $Q$ depends on the worker's effort Ein$[0,1],$ which has a dis$-$

utility $c(E)=100E.$ Output can take two possible values: $Q_1$ with probability $\sqrt[\phantom{2}]{E}$ and

$Q_2$ with probability $1-\sqrt[\phantom{2}]{E},$ with $Wln(Q-W)C>0W_1{W}_2{C}^{**}=(W_1^{**},W_2^{**},E^{**})E{Q}_1=200Q_2=100E{C}^{****}=(W_1^{****},W_2^{****},E^{****})Q_1=200Q_2=100C^{**}{C}^{****}{Q}_2.$ The employer offers a salary $W,$ and $it$ has

a utility over net profits given $byln(Q-W).$ Assume the worker's utility from the best

alternative job $isC>0.$

$(a)$ Write down the expected utility $of$ the employer and the worker assuming the $em-$

ployer pays $W_{1}if$ output $is$ high, and $W_{2}if$ output $is$ low.

$(b)$ Derive the optimal first$-$best contract $C^{**}=(W_1^{**},W_2^{**},E^{**})$ assuming that effort $Eis$

observable, and that $Q_1=200$ and $Q_2=100.$

$(c)$ Assume effort $Eis$ observed only $by$ the worker, and determine the incentive

compatibility and the participation constraints. Derive the second$-$best contract

$C^{****}=(W_1^{****},W_2^{****},E^{****})$ with $Q_1=200$ and $Q_2=100.$

$(d)$ Compare the first$-$best $C^{**}$ and the second$-$best $C^{****}$ contracts emerging from $(b)$

and $(c),$ explaining briefly any differences $an\frac{d}{o}r$ similarities between the two.