3. A private lending market Consider the simple model of private loans from the lecture. Borrowers have nothing when young and 3,95; when old. Lenders have m when young and nothing when old. For the moment, assume there is no capital. Both agents have preferences of the form U{cl , Cg] = log c1 + log c2 (24) (a) Write the market clearing condition for loans that determines the interest rate? Use it to compute the loan rate 1'. Feel free to adapt the equation from lecture. (b) How does the loan rate r depend on the endowment ratio 3,13,31,51,? Provide economic intuition. (c) Let ,3 = 095,333 = 1, and y; = 1.5. Solve for the interest rate r and the loan quantityP L (d) Now suppose there is an active capital market with a gross return a: = 1.05. Thus, the lender will not lend below a rate of .r. Solve for the loan size and the amount held in capital. (Hint: solve for the total savings of the lender at rate :r. The difference between the total savings and the amount demanded by the borrower at :0 constitutes the amount held in capital.)