It is late October and on my to-do list is wading through Open Enrollment options. That inevitably brings up the question of how much money to put into our Flexible Spending Account (FSA). Given that a lot of people face that question at this time of year, I thought I would recycle a post (from Dec 2011) on how to think of funding an FSA as an inventory problem:

FSAs allow US tax payers to set aside pre-tax dollars to pay for authorized expenses. One can have separate accounts for healthcare related expenses (think office-visit co-pays or dental work beyond what your insurance covers) and dependent care. Let’s focus on the medical one. Here is how a **Forbes** blog explained the pros and cons of the program (A Tax Break For Driving To Wal-Mart!, Dec 2).

The way you save with an FSA is this: If you divert $5,000 from taxable salary to pay for braces and your combined federal/state income tax rate is 40%, you save $2,000. You can use the money you stash in the account for medical, dental and vision expenses for yourself, your spouse or your kids. …

If you don’t spend your FSA money within the plan year (or a 2.5 month grace period in some cases), you lose it. The fear of forfeiture leads folks to underfund these accounts. Not paying attention to how expansive the list of eligible expenses is leads folks into forfeiting money. The average amount employees set aside into a healthcare flexible spending account is $1,500, and about half of participants lose an average of $75 at year-end, according to WageWorks. But even these employees who forfeit $75 are still better off with the FSA, says Dietel. Since the average election is just under $1,500, the employee has saved 25% to 40% of that, or $375 to $600, so they are still $300 to $525 ahead of where they would have been without a healthcare FSA.

So here’s the question: Do people, in fact, underfund these accounts?

Let me suggest that a simple inventory problem provides some insights into how much to set aside for an FSA. The newsvendor model assumes that one has a single opportunity to purchase inventory before uncertain demand is realized. The analysis is pretty simple. Suppose that *F*(x) is the known demand distribution. That is, *F*(*x*) gives the probability that demand is less than or equal to *x*. If we are selling newspapers (which is the motivating example that gives the model its name), *F*(*10*) = 60% means that 60% of the time 10 or fewer customers show up to buy papers. Note that *F*(*x*) will be increasing (i.e.,the chance that demand is less than or equal 11 has to be greater than the chance that it is less than or equal to 10).

Let *Q* be the quantity ordered. Let *MB* be the marginal benefit of stocking one more unit. The relevant thought experiment is to suppose that one stocks *Q* units but realized demand is *Q+1* and ask how much one would have earned if one had had that extra unit to satisfy that last unit of demand. In our newspaper example, suppose they retail for 50¢ but we have to pay the publisher 20¢. We then get *MB* = 50¢ – 20¢ = 30¢. Similarly let *MC* be the marginal cost of stocking one more unit. The thought experiment here is to suppose that realized is *Q-1*. If we can recycle papers at 5¢ a piece, we get *MC* = 20¢ – 5¢ = 15¢.

The model then says we should balance the expected marginal benefit with equal the expected marginal cost. If we stock *Q* newspapers, the probability we incur the marginal cost *MC* is *F*(*Q*) and the probability that we receive the marginal benefit *MB* is (1 – *F*(*Q*)). That means we want *Q* to solve:

*MC* × *F*(*Q*) = *MB* × (1 – *F*(*Q*))

The wonders of junior high algebra then give:

In words, we can think of the model as saying that the optimal inventory level is driven by picking a probability of covering all demand. In our newspaper example, we get so our newsvendor should not stockout (i.e., have newspapers left) on two out of three days.

OK, back to FSAs. Note that we’ve got a one-period inventory model. You decide in October how much to set aside for the coming year and dollars set aside for 2011 cannot be used in 2012. We just need to determine *MB* and *MC*. The tricky part here is that we need to be clear of when we are talking about pre-tax dollars and when we are talking about post-tax dollars. Let’s work with pre-tax dollars because that is the amount you specify when you enroll.

*MC* is then easy. If you put one too many pre-tax dollars into your FSA, you lose that dollar. Thus, MC = 1. For MB, if you are short a dollar in your FSA, you need to spend one *post*-tax dollar. That costs you pre-tax dollars, where τ is your marginal tax rate. The only way to avoid this would have been putting another pre-tax dollar in your FSA (note that this is the analog of paying the publisher for the newspaper). We then get

Now we just need to calculate that fraction. The optimal FSA amount *Q *should satisfy

So what do we see? First, not too surprisingly, the marginal tax rate matters a lot. The recommendation is that the tax payer should cover all of her qualifying expenses with probability τ. If her marginal tax rate is very low, say, 10%, she should use all of her FSA contributions nine time out of ten. If τ is very high, say, 90%, she will very often have excess cash in her FSA.

That clearly sounds right. The higher the marginal tax rate, the greater value in paying with pre-tax dollars. Said another way, FSAs have a regressive effect since they provide greater benefit to those in higher tax brackets.

This also lets us answer my original question: Do people, in fact, underfund these accounts? I would argue probably not. Marginal tax rates top out around 40% so even the highest paid Americans should max out their FSAs in most year. Indeed, if half of users end up with excess cash in their FSA (as the quote above suggests), they are putting TOO much money in their FSAs.

There is a caveat to all this analysis. I assuming here that customers are just endowed with some distribution of their spending. To some extent this is true. If you have a chronic condition that requires regular medication, you can forecast your co-pays and then add in kids getting ear infections or needing fillings. That is like forecasting demand for newspapers. However, FSAs also covers spending that is essentially discretionary. If you have a few hundred dollars left in your account in early December, you can always go buy new glasses even if your current ones are fine. Thus, as is pointed out in a recent **LA Times** article (FSAs encourage rather than reduce unnecessary healthcare spending, Nov 20), FSAs aren’t just regressive, they are inefficient. If one considers people buying unnecessary glasses to clear out their FSA and we still have half of users with leftover cash, then clearly average users are not underfunding their FSAs.

A final point, if you don’t like risk — that is, you would prefer greater certainty in your post-tax income — you may want to put some extra pre-tax dollars in your FSA. See here.

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