An entrepreneur has a venture that will make either $$100$M or $$0.$ The chance that this

venture will make $$100$M depends on the effort level expended by her: If she tries hard, the

chance of the $$100$M outcome is $0\mathrm{.}2.$ If she does not, the chance of this outcome is $0\mathrm{.}05.$

This entrepreneur is risk averse, with utility function: $$x $-$ disutility of effort, where the

disutility of effort is $0$ if the entrepreneur does not try hard and $1000$ if she does.

$($a$)$ Assuming this entrepreneur bears all the risk of this venture, will she try hard or not?

What will be her expected utility, net of the disutility of effort $($if any$)?$

$($b$)$ A risk$-$neutral venture capitalist is prepared to support this venture. Risk$-$neutrality

means that the venture capitalist cares only about expected profit $($average weighted by

probabilities of net gains$).$ Specifically, the venture capitalist will pay the entrepreneur

a base amount B up front, in return for which the venture capitalist will retain X out

of the $$100$M the venture generates, if the venture succeeds. Assuming this venture

capitalist is the entrepreneur$$s only alternative to doing it alone $($doing whatever you

determined the answer was in part a$),$ and assuming the venture capitalist can make

part of his contract with the entrepreneur a specification of her effort level, what is

the optimal contract of this sort for the venture capitalist to write? What will be the

venture capitalist$$s net expected monetary value with this contract?

Hint: Notice that the venture capitalist $($VC$)$ is risk neutral, while the entrepreneur

$($E$)$ is risk$-$averse. Thus the optimal contract will have the VC bear all the risk, with

X$=$$$100$M$.$ With the VC receiving all the profit in case of success, your task is simply

to find B that makes E just willing to go through the VC instead of bearing the risk of

this venture himself $($without VC$,$ as in Part $($a$)$ of the exercise$).$ You should do this

both for a contract that stipulates low effort and one that stipulates high effort, and

see which is most profitable to the VC$.$

$($c$)$ Unhappily, the venture capitalist cannot contractually specify the effort level of the

entrepreneur. If the venture capitalist wishes to motivate the entrepreneur to try hard,

he must do this with the terms B and X in the contract he provides. The purpose of

this question and the next is to figure out the best contract for the venture capitalist to

offer the entrepreneur, assuming that if the entrepreneur does not accept this contract,

she is stuck doing it alone on this venture.

Hint: The optimal contract $($B$,$X$)$ in this exercise must satisfy two key inequalities:

$($i$)$ E wants to perform high effort, and $($ii$)$ E prefers performing high effort under the

contract $($B$,$X$)$ instead of bearing the risk of this venture himself $($without VC$,$ as in

Part $($a$)$ of the exercise$).$ Write these two inequalities.

$($d$)$ Use $($c$)$ to fully solve for the optimal contract. To do this, remember that the two

inequalities you have found so far must be satisfied with an equality $($as we discussed

in the salesperson example$).$ So you get two equations that will determine the two

unknowns $($the B and the X$).$

$1$