Howard is a great engineer, who works at a research centre, controlling a piece of very expensive

equipment. The center values the on$-$going project, the combined value of the equipment and

Howard$$s work, at $$1$ million. Unfortunately, Howard also loves to bring his dates to the lab to show

off the fancy equipment, which increases Howard$$s utility by $80.$ If Howard does not bring his date,

there is a $10\%$ chance that the equipment will malfunction and become garbage. In this case, the

research project $($and the equipment$)$ is valued at $$0$ to the center. If Howard brings any of his dates

to the lab, the chance of malfunction increases to $50\%.$ Howard has a utility function U$($M$)=$M$0\mathrm{.}5,$

and

he can always work at a different place for $$40000$ a year, though he will not be able to show$-$off

that place to any date. The center is risk$-$neutral, and it maximizes the difference between the value

of the project and the salary to Howard.

a$.$ Assume the research center cannot observe what Howard does in the lab, what is the

optimal contract the center should offer to Howard?

b$.$ Assume the center can install a security system and perfectly observe what Howard does in

the lab. What is the optimal contract the center should offer to Howard?

c$.$ If the security system costs $$6000$ annually to the research center, should the center choose

to install the security system?