Question

An entrepreneur has a venture that will make either $196 million or $0. The chance that this venture will make $196 million depends on the effort expended by the entrepreneur: If she tries hard, the chance of the $196 million outcome is 0.2. If she does not try hard, the chance of this outcome is 0.04.

The entrepreneur is risk averse with utility function: u(x)=√x-disutility of effort:

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

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)?

A risk-neutral venture capitalist is prepared to support this venture. Suppose also, for now, that effort is contractable so that there is no need to write an incentive contract. Instead, the venture capitalist simply specifies a base wage, B, and an effort level (high or low) for the entrepreneur and then takes the returns of the project for themselves. Assuming this venture capitalist is the entrepreneur’s only alternative to going it alone (doing whatever you determined was the answer to part a), what is the optimal contract of this sort for the venture capitalist to write? What effort level will be specified? What will be the venture capitalist’s net expected monetary value with this contract?

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. 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 $196 million the venture generates, if the venture succeeds (and $0 otherwise).   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 going it alone on this venture?  Specifically, what are the values of B and X that satisfy both the Participation and Incentive constraints? What are the expected profits to the VC from this contract?

Answer 1

ANSWER 1) The expected utility theory deals with the analysis of situations where individuals must make a decision without knowing which outcomes may result from that decision, this is, decision making under uncertainty. These individuals will choose the act that will result in the highest expected utility, being this the sum of the products of probability and utility over all possible outcomes. The decision made will also depend on the agent’s risk aversion and the utility of other agents.

FOR EXAMPLE:

A) $1 million

B) 50% chance of $3 million

We can construct a scale, called a utility scale in which we try to quantify the amount of satisfaction (UTILITY) we would derive from each option. Suppose you use the number 0 to correspond to winning nothing and 100 to correspond to winning $3 million. What number would correspond to winning $1 million?

Suppose you said  80.  This means that the difference between what you have now and a million extra dollars is four times as great as the difference between a million and three million extra dollars. We can express the original choice between A and B in terms of these units (UTILES) instead of dollars.

A) 80 utiles for sure

B) 50% chance of 100 utiles

You calculate expected utility using the same general formula that you use to calculate expected value. Instead of multiplying probabilities and dollar amounts, you multiply probabilities and utility amounts. That is, the expected utility (EU) of a gamble equals probability x amount of utiles.

So EU(A)=80. EU(B)=50. Expected utility theory says if you rate $1 million as 80 utiles and $3 million as 100 utiles, you ought to choose option A.

So here in this case if she is bearing all the risk there are two uncertainity:

A) the chance of the $196 million outcome is 0.2 with the disutility of effort is 1000

B) the chance of the $196 million outcome is 0.04 with the disutility of effort is 0

While all entrepreneurs face these types of risk and uncertainty, smart entrepreneurs are exceptionally effective at taking calculated risks and at transforming more risky contexts into less risky ones. So as she is risk averse she should go taking smart risk which results to her expected utility to $196 million outcome is 0.04 and net of the disutility of effort is 0.