A farmer has a production function f(L) where the input is capital (L). The cost of this loan is L(1+i). The farmer also has an outside option (loan from family member) which generates a profit of A . Suppose a farmer has N years where he can work productively, and then must retire. Each year, if he repays the loan, he may take out another loan, but, if he neglects to repay, the bank will never lend to him again. The farmer is also very poor and has limited liability.
A) If the farmer took out a loan L and only had for 1 year (N = 1) left to work, would the farmer pay back the loan? Why or why not?
B) Now, let's say the farmer took out a loan L and had multiple years left to work, which we will simply denote as N. Write down the condition under which the farmer would repay the loan. (Hint: Write down the benefit of repaying the loan and the benefit of defaulting)
C) If the bank somehow knew how many years that the farmer could work productively, and the production function the farmer faced, what interest rate would the bank set?
D) In this question, is the bank facing a problem of adverse selection or moral hazard?Explain why?
E) Now assume that the farmer still has N years of work ahead of him (same as in 3b), but now the farmer has tradable assets that can be use to partially collateral the loan L he takes out. Assume the value of this collateral for loan size L to be M. What interest rate would the bank set now? How does this interest rate compare to the one our found in 3c?
Overall, would interest rates be higher or lower when the farmer has collateral for the loan?