An entrepreneur has a venture that will make either $100 millionor $0. The chance that this venture will make $100 million dependson the effort expended by the entrepreneur: If she tries hard, thechance of the $100 million outcome is 0.1. If she does not tryhard, the chance of this outcome is 0.02. The entrepreneur is riskaverse with utility function:
u(x) = sqrt(x) - disutility of effort
in other words X1/2 - disutility of effort
where the disutility of effort is 0 if the entrepreneur does nottry hard and 500 if she does.
a. Assuming this entrepreneur bears all the risk of thisventure, will she try hard or not? What will be her expectedutility, net of the disutility of effort (if any)?
b. A risk-neutral venture capitalist is prepared to support thisventure. Specifically, the venture capitalist will pay theentrepreneur a base amount B up front, in return for which theventure capitalist will retain X out of the $100 million theventure generates, if the venture succeeds. Assuming this venturecapitalist is the entrepreneur’s only alternative to going it alone(doing whatever you determined was the answer to part a), andassuming a venture capitalist can make part of his contract withthe entrepreneur a specification of her effort level, what is theoptimal contract of this sort for the venture capitalist to write?What will be the venture capitalist’s net expected monetary valuewith this contract?
c. Unhappily, the venture capitalist cannot contractuallyspecify the effort level of the entrepreneur. If the venturecapitalist wishes to motivate the entrepreneur to try hard, he mustdo this with the terms B and X in the contract he provides. What isthe best contract for the venture capitalist to offer theentrepreneur, assuming that if the entrepreneur does not acceptthis contract, she is stuck going it alone on this venture?
