1. An entrepreneur has a venture that will make either $100 million or $0. The chance that this venture will make $100 million depends on the effort level expended by the entrepreneur: If she tries hard, the chance of the $100 million outcome is 0.1.

If she does not try hard, the chance of this outcome is 0.02. This entrepreneur is risk averse, with utility function square root x disutility of effort where the disutility of effort is 0 if the entrepreneur does not try hard and 500 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. Specifically, the venture capitalist will pay the entrepreneur a base amount B upfront, in return for which the venture capitalist will retain X out of the $100 million the venture generates, if the venture succeeds. Assuming this venture capitalist is the entrepreneur's only alternative to going 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? You may assume that the entrepreneur accepts the contract if she is indifferent between accepting the contract and going it alone. (Hint: What does efficient risk sharing requires?)

(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. What is the best contract for the venture capitalist to offer the entrepreneur, assuming that if the entrepreneur does not accept this contract, she is stuck with going it alone on this venture? Again assume that the entrepreneur accepts the contract if she is indifferent between accepting the contract and going it alond