Suppose a person has an amount A of spending money deposited to her savings account each month, where it earns interest at a monthly rate R. Assume she spends the entire amount throughout the month at a steady rate. When she makes cash withdrawals from the account, she incurs a transaction cost T (a combination of bank fees and the cost of her time). She would save money by making fewer withdrawals, but the more money she leaves in the account the more interest she earns. Suppose she makes cash withdrawals for the same amount during the month.
Then her average cash balance at any given time is A / (2n). (Why?) Find the value of that minimizes total costs (transaction costs and lost interest), and then show that the optimal average cash balance is √AT / (2R).