Exercise 5

Due Wednesday 2/29

Suppose a researcher has invented a new process for growing algae that can be used to produce diesel fuel for vehicles (that is, biodiesel).  However, the process has only been tested in the lab and it's not clear whether it will work at the large scales necessary for commercial use.  To go beyond the lab scale, the reseacher is considering starting a small company.  

Here are some facts relevant to her decision.  If she starts the company, she'd have to give up her current job, which pays $90,000.  If the project succeeds, she'd end up with a valuable patent worth $2 million.  If it fails, she'd end up lower quality results worth $40,000.  She'd also have to decide how hard to work on the project.  Routine effort (R) would cost her $20,000 and would give the project a 10% chance of success.  However, she could choose to put in an especially high level of effort (H), which would cost her $40,000 and would raise the chance of success to 20%.

1. What is the expected value of the project?  If the researcher were risk neutral and only cared about expected value, would she undertake the project? What level of effort would she choose?  What would be her expected gain or loss?

Now suppose the researcher is actually risk averse and maximizes her expected utility.  The utility she gets from any given net payoff in dollars, c, is given by the square root of c: u=c^0.5.

2. Given her utility function, what will she choose to do?  Why?  What is her expected utility of starting the company and exerting routine effort? Of starting it and exerting high effort? Of staying in her current job?

Finally, suppose that the (risk averse) researcher is approached by a venture capitalist (VC) who offers her the following contract.  If she launches the startup, the VC will pay her $35,000 regardless of whether or not the project succeeds.  In exchange, if the project succeeds the VC will be entitled to 65% of the payoff (that is, $1.3 million).  However, if the project fails, the VC would get nothing -- the researcher would keep the $40,000 (although her net payoff would be reduced by the cost of effort). 

3. Please determine whether this contract satisfies the incentive-compatibility and participation constraints.  That is, determine whether the researcher would agree to the contract and then choose to exert high effort.  Please show your calcuations and report the following items: the expected value of the contract to the reseacher; the expected value to the VC; the certainty equivalent of the contract to the researcher; the total gain to the researcher and the VC together.