d Write k as a function of , A, r.

2. Private debt and capital. (20pts) Consider a lender who lives for two periods, has an endowment y when young and nothing when old, and has access to two assets. First, the lender can issue loans (or borrow) l at interest rate r (taken as given). Second, the lender can invest in risk-less capital k, which has a return f(k) = Aka, for A 2 O, o: E (0,1). Capital must satisfy k 2 0 but loans .1 can be positive or negative. The preferences of the lender are as follows: (\"61162) = log Cl + log 02 (a) (41915) Formulate the budget constraints of the lender when young and when old. Use the fact that the lender can save in terms of both loans I and capital k. (b) (2pm) Formulate the objective of the lender as a functiOn of loans and capital (I, k). (c) (pts) Argue that the optimal choice of capital 19\" must be positive, i.e., there is an interior solution. (C!) (5pts) Obtain the rst order conditions with respect to k and l, Write k' as a fumc 1.2.... ...E_ JI _