Warranties, part 2: Warranty for You

In the previous post we discussed whether it’s ever a good idea to buy a product warranty such as AppleCare. To answer this, we used something called expected value. For instance, if an $829 iPad has a 5% chance of failing --- and assuming AppleCare will cover the problem if it does --- we say the warranty has an expected value of 0.05 x $829 = $41.45. If AppleCare costs more than this, we shouldn’t buy it. If it costs less, we should.
But it won’t cost less. Since our expected value is Apple’s expected loss, they’re going to set the price to guarantee they make a profit. Sure, some people will buy AppleCare, use it and conclude it was a good idea. But most people who buy it will never need it. In the end, buying a product warranty is a lot like going to Vegas...and Apple is the casino.
So does this mean we consumers should never buy product warranties? Yes.
And no. Because there is one warranty that may be worth buying even if it costs more than its expected value: health insurance. Like AppleCare, health insurance is a product warranty. The main difference is that the product is you.
Still, the underlying concept of expected value is exactly the same and can help us understand how the health insurance industry works...and how it doesn’t.
To help illustrate this, let’s imagine a country with just four people, and where the only medical procedure is a $40,000 emergency heart surgery. Since we’re suspending disbelief, let’s also assume that everyone knows exactly how likely they are to need the surgery.

Name & Age Alonso, 20 Beatriz, 47 Chris, 63 Delilah, 76
Health Status Healthy High B.P. Double bypass Mostly healthy
Odds of Surgery 1% 25% 50% 16%
E.V., Insurance $400 $10,000 $20,000 $6,400

As you can see, everyone has a different expected value for health insurance. The higher someone’s risk, the more insurance is worth to them. Since Alonso is young and relatively low-risk, he only “expects” to spend $400 on heart surgeries and is therefore only willing to pay $400 for health insurance. He still wants it --- who wouldn’t want AppleCare, after all? --- but only at a certain price. Chris, on the other hand, has a history of heart problems, is much higher-risk and is therefore willing to spend much more. (Oh, we’ll also assume that everyone could afford an insurance policy at any price. It turns out that Econ 101 is a slippery slope towards the absurd.)
Now we know how much everyone is willing to pay for health insurance. But this is only half the story. We also need to know how much the insurance company plans to charge.
For the sake of simplicity, let’s assume the insurance company --- like Apple --- charges everyone the same price for health insurance. In this case, how much will it cost?
In this magical country where everyone knows their likelihood of needing surgery, Alonso & Co. aren’t the only ones pulling out their calculators. The insurance company “expects” to spend $400 on Alonso, $10,000 on Beatriz, etc. If everyone buys insurance, the company expects to spend $36,800 on heart surgeries in total. To stay in business, the company must charge $9,200/person.
But at that price, will everyone buy it? This is where things start to get interesting...

For Beatriz and Chris, insurance is worth $10,000 and $20,000, respectively. To them, $9,200 is a bargain and they’ll buy insurance in Year 1. But Alonso and Delilah won’t; insurance costs more than their expected values, so they’ll decide to save their money and roll the dice.
From the consumers’ perspective, the whole system is working just fine. If they want insurance, they’ll buy it. If they don’t, they won’t.
From the insurance company’s perspective, though, things aren’t so great. The low-risk people have left the pool, leaving the company with higher-risk customers. This is what economists call adverse selection: the only people who buy insurance are the ones who really need it, i.e. the customers the insurance company wants the least.
Since Alonso and Delilah have opted out, the company now expects to spend $30,000 in heart surgeries. This is less than the $36,800 from before, but now it’s only divided between two people. What this means is that in Year 2 they’ll have to raise the price to $15,000/person...at which point Beatriz will bolt.

To summarize, after two years only 25% of the population has health insurance, which leaves the rest vulnerable to a freak heart attack. Meanwhile, the price of a policy has jumped by over 60%. What a mess! Is there a way to fix this?
“Yes,” the insurance company CEO says. “The reason insurance is so expensive is because of high-risk people like Chris. If we didn’t have to sell insurance to him, we could make it cheaper for everyone else.” (Alternatively, the insurance company could sell insurance to Chris but simply not cover his heart condition. After all, Apple won’t sell AppleCare for an iPad that’s already broken, so why should a heart be any different? Since heart surgery is the only procedure in our hypothetical country, though, this pre-existing condition clause is somewhat moot.)

In this case, insurance does cost less in both Years 1 and 2. Like before, though, only 25% of the population is insured by Year 2; it’s just Beatriz this time. Looks like we’re still in trouble.
“But wait,” says Chris. “It's not my fault insurance is so expensive. It's Alonso's! If everyone were required to buy insurance, it would be a lot cheaper.”

As you can see, Chris is right: insurance is cheaper...for him. This individual mandate effectively saves him and Beatriz $10,800 and $800, respectively. On the other hand, it forces Alonso and Delilah to spend way more than they’d like. Each year Alonso has to pay 23 times what insurance is worth to him, even though he only has a 1% chance of needing it. The mandate extends a hand to high-risk people...but requires low-risk people to foot the bill!
Clearly there are still some kinks in our insurance market.
But wait. Throughout this process we’ve been treating health insurance as equivalent to AppleCare. In reality, of course, they’re very different. If your iPad breaks and you don’t have AppleCare, you might have to shell out for a new one. But if you break and you don’t have health insurance, you could go bankrupt or even die. Because the possible consequences of not having insurance are so great, people often buy it even when it costs more than its expected value.
Returning to our hypothetical country, let’s imagine that everyone is actually willing to pay 1.5 times their expected values for health insurance; instead of $400, Alonso is now willing to spend $600. Since everyone’s chances of needing surgery haven’t changed, of course, the insurance company still expects to spend the same as before.

Instead of 25%, now 50% of the population has insurance. It’s not perfect, but it’s definitely better than before. The higher-risk people are insured. The lower-risk people aren’t. And maybe it’d be okay to leave things like this. Okay, that is, until...
Alonso has a heart attack.
Unlike Apple and a broken iPad, the law requires hospitals to provide emergency care even when patients don’t have health insurance. (Even if the law didn’t require this, the Hippocratic Oath might.) To stay in business, the hospital will pass the $40,000 cost onto the insurance company, which will in turn pass the cost onto Beatriz and Chris. In Year 3 the price of an insurance policy will rise from $12,133 to $32,133. At which point nobody will buy it. At which point the insurance company will go out of business. At which point the hospital will, too.
Which is to say, if there’s a 1% chance that an uninsured Alonso will have a heart attack, then there’s a 1% chance that the entire healthcare market will suffer a nuclear meltdown. Hmph.
Of course, this is a very simplified model. In reality we don't live in a country with four people, and there's much more to insure against than just a heart attack. Furthermore, individuals may have very little idea of how likely they are to need medical care (although insurance companies have a very good idea of how much customers will cost and price their policies to guarantee that they not only break even but generate huge profits for their shareholders).
Unlike in our hypothetical country, the health insurance industry in the United States hasn't disintegrated beneath the weight of cause-and-effect, and lots of Americans have health insurance.
Then again, lots don’t. People with pre-existing conditions often can’t get insurance or can’t afford it. Young and otherwise healthy people do get into car accidents every once in a while. And because of this “free-riding,” people who do have insurance pay on average $1000 more for their policy than they would otherwise.
Which is to say, as simplified as our model is, it still highlights the general forces at play in any health insurance market. And as contentious as the discussion has been surrounding health care in America, the issue itself is relatively straightforward and comes down to basic math. If we can understand expected value --- if we can understand how insurance is similar to AppleCare and how it’s different --- we’ll be much more able to discuss how to address heart attacks without actually having one.

2 thoughts on “Warranties, part 2: Warranty for You”

  1. I have used this with my classes several times. It really, really puts the concept into perspective and has initiated some very usable conversations in my classroom. Thank you.

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