A principal is hiring an agent. Once hired, the agent chooses his effort level $ain{R}_{+}.$ The

principal's gross profit is given by $q=a+,$ where the noise term $$ is normally distributed

with mean $0$ and variance $1.$ The $($Bernoulli$)$ utility functions for the principal and agent are,

respectively,

$v(q,w)$:$=1-$exp$[-2(q-w)],$ and

$u(a,w)$:$=1-$exp$[-(w-\frac{a^2}{2})],$

where $w$ is the wage payment. Note that both the principal and the agent are risk$-$averse. As

usual, assume $w$ must be linear in the gross profit; $w(q)=s+bq$ for some $s$ and $b,$ which are

specified in a contract. By working elsewhere, the agent can get the reservation utility of $0.$

Suppose that the agent's effort level $a$ is contractible, and consider the optimal contract

for the principal $($subject to the participation constraint$).$ What is the optimal bonus rate

$b^{*}$ in this case? Answer the closest integer to $100b^{*}.[$Hint: As usual, first rewrite the

expected utilities into certainty equivalents as a function of $(a,s,b).](2$ points$)$

Next consider the optimal contract when $a$ is not contractible. What is the optimal bonus

rate $b^{**}$ in this case? Answer the closest integer to $100b^{**}.(2$ points$)$