Consider an economy with two types of farmers, indexed by i {1,2}, that need an L amount to invest in a project that, with probability p_i, brings a return R_i and with probability (1-p_i) does not have no return (note that the probability of success of the project depends on the type of farmer). Suppose R2>R1>L e, p₁=1 and 0₂<1. Farmers do not have their own resources and, therefore, the only way to obtain the L amount is through a loan. In this economy there is only one lender who has exactly L to lend. In return for L the lender charges R from the farmer (ie R = (1 + r) * L, where r is the interest rate). When the return on the project is zero, both types of farmers do not pay the loan. The opportunity cost of the loan is zero for both types.

(a) Find the reserve interest rate of the two types of farmer.

(b) Suppose the lender knows the type of farmers (that is, he knows the probability of success for each farmer's project). What would be the value of R that the lender would charge in balance?

(c) Suppose now that the lender DOES NOT know the farmer's type. The chance of the farmer of type i {1,2} to take the loan is . Write the expression for the expected profit obtained by the lender if he lends to a type i farmer.