Problem
Consider an economy populated by two types of risk-neutral borrowers. And suppose that all potential borrowers live throughout four periods: 0, 1, 2, and 3. At the beginning of each period, every potential borrower needs at least $45 in order to satisfy her basic necessities for the entire period. At date 0, each individual is endowed with $45, which is just enough to survive until date 1. At both dates 1 and 2, investment and job opportunities emerge. Each time, individuals can invest in a project which requires $100 and one period to yield a return. Any individual wishing to take advantage of the investment opportunities presented to them will thus have to obtain a loan. Suppose that the only lender is an NGO that just wants to break even. In particular, the NGO wants to cover its gross cost K = $120 for each $100 loan. If she qualifies for a loan, an individual of type 1 can invest and generate a gross return y1 = $230 with probability 90 percent, and nothing with 10 percent probability. If she does not borrow, she can work and earn $65. If she obtains a loan, a type 2 individual can invest and succeed with 50 percent probability, in which case her gross return is y2 = $360. The other half of the time, her investment fails and she earns nothing. Type 2's opportunity cost is $70. The population is made up of 60 percent type 1 individuals; the other 40 percent consists of type 2 individuals. Assume that the NGO cannot observe individuals' types.
Moreover, suppose that all individuals are very patient, that is, that their discount factor β = 1. All borrowers are protected by limited liability. At time 3 there is no investment. All individuals just consume the sum earned in periods 1 and 2. Show that the two types will invest in one project, in period 2 only.
The response should include a reference list. Double-space, using Times New Roman 12 pnt font, one-inch margins, and APA style of writing and citations.