Assumptions: everybody is risk-neutral, the interest rate is 0, and investors (lenders) are competitive.

One entrepreneur needs to borrow money for her project. The initial cost of the project is K. The entrepreneur has a quantity of cash of her own equal to E. Assume K > E, so if the entrepreneur wants to start the project, she will need to borrow K E from investors. The project, after one period, can be either a success and pay off Y, or a failure and pay off 0. The probability for the project to be a success depends on the entrepreneur's behavior: if she behaves the probability of success will be equal to pH, if she shirks the probability will be equal to pL < pH.

On the other hand, if the entrepreneur shirks, she will enjoy a private benefit (nontransferable to others) equal to B > 0 that can be summed to her monetary payoff

. Assume the following: pHY > pLY + B, pHY K > 0 and 0 > pLY + B K. Before the project is undertaken and before the entrepreneur makes the choice between behaving and shirking, the entrepreneur and the investors can enter a debt contract. The entrepreneur cannot commit to behaving and she enjoys limited liability. The sequence of actions is the following:

1. The loan agreement is signed (one-period zero-coupon bond)

2. The investment is made

3. The entrepreneur decides whether to shirk or behave

4. Payoff is realized

Overall payoff Y will be shared between the entrepreneur and lender. Y_l is the part of Y going to the lender (face value of the bond) and Y_B is the amount retained by the entrepreneur.

a) What is the face value (Yl) of the debt contract as a function of the probability of success, K, E and B?

b) What payoff YB should the entrepreneur receive, in case of success, in order to behave (always as a function of the probability of success, K, E and B)?

c) What is the largest Yl that can be paid out to lenders in case of success, that will keep the entrepreneur from shirking?

d) What is the minimum amount of cash E that the entrepreneur needs to have (and invest in the project) in order to credibly behave in case of debt financing? (Hint: ask yourself when the lenders would be willing to lend money to the entrepreneur, given that they have to provide her with enough incentives to behave.)