Please show all work.

Please show all work.

Please show all work.

A simple banking model. A bank raises funds by accepting deposits D in order to make loans L. The central bank requires that the bank hold a minimum quantity of reserves R in proportion to its deposits: R = PD, (1) where p is the required reserve ratio. You should assume that the bank does not hold excess reserves and has no capital. The bank is competitive and takes the lending rate ry and the deposit rate ro as given. The supply of deposits is perfectly elastic at ro. The bank chooses a quantity of deposits to accept, loans to make, and reserves to hold in order to maximize its profit. a) (3 points) Solve the bank's profit maximization problem to find the equilibrium relation- ship between the deposit rate ro, the lending rate rL, and the required reserve ratio p. Show your work. b) (4 points) Suppose that a borrower uses borrowed funds from the bank to purchases an investment good / so that I = L. With certainty, the investment good generates income for the borrower according to the function: f(1) = 500 x 705. (2) Suppose that the deposit rate is ro = 0.036 and the required reserve ratio is p = 0.1. If the borrower seeks to maximize profit given by: profit (L, I) = f (1) - (1 + rL)L. (3) then what is the quantity of investment / that the borrower will choose to undertake? Show your work.Answer: a) The equilibrium relationship: TD TL 1 - p b) The borrower's first order condition for an optimal choice of / is: f'(I) =1+rL which implies: I* = 57,784.76