Problem

Consider a project that needs a fixed investment, I, and which yields a gross return of y > I with probability p, and a gross return of zero with probability (1 - p). A risk-neutral borrower who has private wealth w is willing to borrow (I - w) in order to invest in the project. The lender knows y and p, but can only observe the final return with probability q. If the borrower refuses to repay and the lender knows that the return on her project is y, the lender can seize w. Suppose that the lender's cost of capital is zero, that it is competitive, and that it only wants to recuperate the expected value (I - w) attached to the loan. Compute the threshold w* below which the lender is unwilling to finance the project, and comment on how your result relates to microfinance institutions facing unbanked but wealthy potential clients.