Consider a principal$-$agent problem of moral hazard. A principal hires an agent to do a task, the output q takes two possible values

q$=\{10$ probability Pr$=$p $5$ probability Pr$=1-$p

The agent exert either low effort aL or high effort aH$,$ which will affect the probability of high output level, p$.$ And we have

p$=\{0\mathrm{.}7$ for effort aH $0\mathrm{.}4$ for effort aL

Therefore, it's more likely to achieve high output with high effort level. Effort is costly with the cost of

c$=\{1$ for aH $0$ for aL

The principal' utility function is Vq$,$t$=$q$-$t

where q is the output level and t the transfer to the agent. The agent is risk neutral and has the following utility function

Ut$,$c$=$t$-$c

The agent's reserveation expected utility is $0.$

Derive the first best contract under full information. Which action level is preferred by the principal, low effort aL or high effort aH$?(20$ marks$)$

Now assume asymmetric information for the following questions, where the principal can't observe effort levels. Therefore the wage paid to the agent can only be contingent on output, denoted as t in $\{$tL$,$ tH$\}.$

Under asymmetric information, what would be the optimal contact if the principal induces the agent to exert aH effort. $(30$ marks$)$

Under asymmetric information, what would be the optimal contact if the principal induces the agent to exert aL effort; which effort level would the principal implement under asymmetric information. $(20$ marks$)$

Now consider, agent is under the protection of limited liability, that is$,$ tL $>=0$ and tH $>=0.$ Suppose the principal wants the agent to exert aH effort, what would be the optimal contract? If the principal wants to induce aL$,$ what's the optimal contract? Which one is better, aH or aL$?(30$ marks$)$