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Consider a bank that has total funds F to divide between its reserve account at the central bank and overnight lending. The bank receives an
Consider a bank that has total funds F to divide between its reserve account at the central bank and overnight lending. The bank receives an interest rate r on overnight lending. If the bank puts a sum R in its reserve account, it has (F-R) left to lend out overnight, giving earnings of (F - R)r. The central bank does not pay interest on reserves. After the end of the day. the central bank clears payments between banks, adding a net sum P to the bank's reserve account, where P can be a negative number. That leaves R + P in the bank's reserve account. From the bank's point of view, P is a random variable, uniformly distributed between a minimum value (the smallest possible net payment into the bank's reserve account) of -10, and a maximum value (the largest possible payment into the hank s reserve account) of + 10. The reserve requirement is 5. If the balance in the bank's reserve account falls below 5 after clearing, the bank must take an emergency loan from the central bank to cover the shortfall. The central bank charges an interest rate r_P for emergency loans to cover overdrafts. a) What is the smallest quantity of reserves that the bank will choose to hold if the market overnight rate r is equal to zero? b) What is the largest quantity of reserves that the bank will choose to hold if the market interest rate r is as high as the central bank's emergency lending rate r_P? c) Given a value of R somewhere between the values in a) and b), what is the probability that a bank will have a shortfall in its reserve account? Check: a higher value of R should make this probability smaller. d) Assuming a bank runs an overdraft in its reserve account, what is the expected value of the amount that the bank will have to borrow from the central bank
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