Question

We consider a bank that offers savings accounts with interest rate r to its customers. We assume that the amount of money y(r) in million dollars that the customers deposit on their savings accounts is related to the interest rate by y(r) = 1000r. For example, if the banks offers an interest rate of 3%, the customers will deposit y(0.03) = 1000 · 0.03 = 30 million dollars in their savings accounts. The bank can reinvest the money that it gets from its customers at 4%. What is the interest rate r that the bank should pay to maximize its profit?
Hint: The bank’s profit is the interest the bank earns by reinvesting the money minus the interest the bank pays to its customers for the deposited money.

Answer 1

The amount f poit the bank will get is 4% have The monay depoaited, minus the money the to pay back to the customers , ohich is the ir de ponited interest ate ni times the money Protit Po.04 m-im Given m y 10007 Ym 1000 subshhite 100o Y foY m P-(0.04)(m) - ) since amount o money depoited to interest Tate () poportina P= (0.04) to00Y 10001) P401-1000Y 0 tor to tind the minimum ds

2 P 40Y -l000Y dP dr 40-2000Y =O 1 20001 4-0 =40 0.02 2000 YO.02 Y2 The interest ate 2%