In November 2016, Worker Jon must choose how much he wants to contribute to his Flexible Savings Account (FSA) for the year 2017. He can contribute up to $2,550, which will be deducted pre-tax from his 2017 income. He can then use the deposit (e.g., get reimbursed) for eligible health care expenses incurred during 2017. His tax rate is 30%, so for every dollar that he does actually spend from his FSA, he will get 30 cents tax benefit. However, since he does not know exactly how much he will spend in 2017, there is a risk involved. If he did not spend all the money he had deposited, the remaining balance will expire (i.e., it cannot be withdrawn or rolled over to next year).
Suppose Jon has estimated that his eligible expenses follows a Normal distribution with mean and standard deviation .
Guideline: Newsvendor calculations with continuous demand distribution.
a) What is the shortage cost? What is the excess cost? What is the news vendor critical ratio?
b) What is the optimal amount Q* to contribute his FSA?
c) If he deposits Q* found in part (b), what is the expected amount he needs to pay using his post-tax income (i.e., after he has spent all the amount in FSA)?
d) Following part (c), what is the expected amount he will actually pay using his FSA? What is the expected amount left unused in his FSA? What is his expected tax benefit?