Everyone knows that health care costs are high. From 2009 to 2013, the United States spent about 18 percent of its gross domestic product on health care (Worldbank.org). It is also known that people tend to spend less on health care if they spend their own money, which motivated the creation of flexible spending accounts (FSAs). Roughly speaking, an FSA works as follows: 

(i) at the start of the year, an employee decides how much she will contribute to her FSA; 

(ii) the contribution is taken from pre-tax dollars, which means the employee does not have to pay payroll taxes on the contribution; 

(iii) the employee uses her contribution in the FSA to pay for qualified medical expenses during the year, such as eyeglasses and dental work; 

(iv) any money left in the FSA at the end of the year is lost. If you know that you will spend $500 on glasses during the year, it makes total sense to contribute $500 to your FSA. Because of the contribution, you pay $500 for the glasses and avoid paying payroll taxes on the same $500. 

The amount you save in taxes depends on your income. Say you are fortunate enough to be a high-income earner, which means that your marginal tax rate is around 40 percent—for every additional dollar you earn, you pay $0.40 in payroll taxes. In that case, if you pay for the glasses through an FSA, your total cost is just the $500 for the glasses. But if you don’t make contributions to your FSA, then you have to pay $500 for the glasses and $200 = 500 × .40 in taxes! Clearly, it can make good sense to use the FSA. But there is a catch. You have to decide on your contribution amount at the start of the year before you really know all of your expenses, and if you over contribute, you lose the excess amount! The government probably didn’t realize this when they created FSAs, but by doing so they have imposed a newsvendor decision problem on millions of Americans.
So what are the underage and overage costs in the FSA newsvendor problem if you have a 40 percent marginal tax rate? If you contribute $1 too few to your FSA, then you end up spending $1 outside your FSA on health care. Had you contributed the $1 to your FSA, you would have reduced your tax bill by $0.40 = .4 × $1. Thus, Cu = $0.40. If you contribute $1 too much to your FSA, then you lose the $1 at the end of the year. Had you known that you would not use the $1, you would not have contributed it to the FSA and you would still have the $1 at the end of the year. However, you also would have to pay $0.40 in additional taxes (because the $1 was not contributed to the FSA), so you are really only $0.60 = $1 – $0.40 better off. Thus, Co = $0.60. The critical ratio is then
CuCo+Cu=$0.4$0.6+$0.4=0.4

Hence, you should contribute an amount to your FSA such that there is a .40 probability your expenses will be less than your contribution and a .60 probability your expenses will be more than your contribution. In other words, it is slightly more expensive to over-contribute than to under-contribute, so the optimal contribution amount is somewhat conservative (but not so conservative that you don’t contribute).
You can also confirm that if your marginal tax rate is lower than 40 percent, then your optimal contribution to an FSA is more conservative. This is related to one of the critiques of these accounts: People with higher income (and in a higher tax bracket) enjoy a greater benefit than people with lower income.

Discussion Questions 
1). According to the above logic, assuming 20% of your marginal tax rate, what would be your critical ratio?
2). How would you anticipate and characterize a distribution of your medical expenses for next year? Minimum, average, maximum?
3). Given your answers to questions 1 and 2, what would be your declared amount contribution for medical expenses for next year, if you are in United States?