An entrepreneur needs funds for a project. He has funds of his own, but enough to cover the required investment of $100. If the entrepreneur doesn’t misbehave, the project will succeed with probability .9, yielding a gross return of 120 in one year. With probability .1, it fails and yields nothing. If the entrepreneur misbehaves, on the other hand, he obtains a private benefit of $14 in one year, while lowering the probability of success to .5. Lenders require a gross expected return of $1 in one year on $1 loaned today, and the lenders are competitive and so receive no more than this. So, for example, if they lend $1 and there is no misbehavior, they must be given 1/.9 = $1.11 dollars if the project succeeds.
Show that if the entrepreneur borrowed the entire $100, he or she would have the incentive to misbehave.
Since lenders realize this, they will not lend the entire $100. What is the most that they will lend and why? Put another way, what is the minimum amount the entrepreneur must put up in order to have the incentive to behave? Explain.