A risk neutral principal is hiring a risk averse agent. When hired, the agent can either work hard $(a=1)$ or shirk $(a=0).$ The principal's gross profit $q$ depends on the agent's action a and

the state of the world, de noted by $in\{{}_1,{}_2,{}_3\}.$ The values of $q,$ as we $11$ as the probabilities of each state, are summarized in Table $1$ below. The principal's and agent's $($Bernoulli$)$ utility functions are given by $v(q,w)=q-w$ and $u(w,a)=\sqrt[\phantom{2}]{w}-2a,$ respectively, where $w0$ is the wage payment from the principal to the agent. By working elsewhere, the agent can get the reservation utility of $2.$ Throughout this problem, assume the realized value of $q$ is always contractible. Also assume that the agent carmor observe $$ when he chooses $a.$

$(1)$Suppose that the agent's action $a$ is contractible while the state $$ is not. What is the max imal expected utility the principal can achieve in this case? Answer the closest integer

to the maximal level multiplied by $100.(2$ points$)$

$(2)$Next suppose that neither a nor $$ is contractible, so that a contract specifies $(a,w_H,w_L),$ where $w_H$ and $w_L$ be the wage level in the case of $q=32$ and $q=0,$ respectively. Let

$(a^{**},w_H^{**},w_L^{**})$ be the optimal contract for the principal in this case. Answer the closest integer to $100(w_H^{*}-w_L^{*}).(2$ points$)$

$(3)$Although $a$ is still unverifiable, a scientist inve nted a radar that catches radio waves, which are generated only in state ${}_3.$ So it is now verifiable ex$-$post whe ther the state was ${}_3$ or

not. $($But continue assuming the agent has no access to this information upon choosing a$.)$ That is$,$ a contract can specify $(a,w_H,w_L,w_3),$ where $w_3$ is the wage payment when

${?}_{(}3)$ realizes while $w_H$ and $w_L$ are paid only when ${}_3.$ What is the maximal expected utility the principal can achieve with this new technology? Answer the closest integer to the maximal level multiplied by $100.(2$ points$)$