What makes buying insurance rational?

The expected monetary value of insurance is negative (or rather, negative in real dollars. It can be positive in nominal dollars but underperform inflation.)

But the utility is not linear in money. Losing e.g. $10,000 might be 20 times as bad as losing $1,000. If so, you should pay $1,000 100% of the time to avoid paying $10,000 8% of the time.

The insurance company averages out over many buyers, so their utility is roughly linear.

Insurance is just trading against different utility scales.

It can be rational because of diminishing marginal utility (happiness) of money. Imagine if I'm poor I get 100 units of pleasure from having an extra $10,000 whereas if I'm rich I get only 50 units of pleasure from having an extra $10,000. If I face some risk of becoming poor I would be willing to buy insurance that takes money from me in the state in which I'm rich and gives it to me in the state in which I'm poor even if on average I end up losing money, because on average I will end up gaining utility.

And apparently insurance companies can make money because the expected utility of buying insurance is lower than it's price.

No, the expected monetary value of insurance is lower than its price. (Assuming that the insurance company's assessment of your risk level is accurate.) You're equivocating between money and utility, which is the source of your confusion.

Suppose I offered a simple wager: we flip a coin, and if it comes up heads, I give you a million dollars. But if it comes up tails, you owe me a million dollars, and I get every cent you earn until that debt is paid. Is this bet fair?

Monetarily, yes. But even if I skew the odds in your favor a little, maybe 60/40, I'll bet you still don't want to take it. Why not? Isn't an expected return of $200,000 wildly in your favor?

Yeah, but that doesn't matter. An extra million dollars would make your life somewhat better; spending the next twenty years flat broke would make your life drastically worse. Expected utility is very negative.

The utility of money is sometimes claimed to be logarithmic. For small amounts of money you can use a linear approximation, but if the outcome can shift you to a totally different region of the curve, the concavity becomes very important.

Insurance for small consumer products are not rational for the buyer, for the very reasons which were presented in the question. If you can afford the loss of the item, it's better to not buy insurance and just buy the item again in the case it is lost or destroyed. Why insurance companies are still making money out of extended warranties for consumer products, is because they have good marketing and people are not perfectly rational. Gambling, lottery, etc. exist for the same reasons, despite having a negative expected value.

However, if you cannot afford the loss, it is advantageous to buy insurance. There are things which people own but cannot replace on short notice, and suffer greatly if they do lose it. For example, houses, or business-crucial items. You can afford to pay the insurance, but cannot afford losing the item in question. Taking a loan to replace it might be much more expensive than the insurance.

There are situations when losing something might cost you much more than its monetary value. Losing your house might make you homeless. Losing you car, if you require it for your job, might cost you your job. Having an expensive machine you make your living out of, losing it might put you out of business. Not having enough money to afford an expensive operation might cost you your life if you don't have the health insurance which would have paid for it.

For at least some kinds of insurance, there's an economy of scale involved that reduces overall cost and hassle for some kinds of event. At many doctors and hospitals, you really want the negotiated healthcare rates that insurance companies get rather than the list rates. It's very nice if you are involved in a car accident to have an insurance company deal with the other parties rather than you having to do so.

For some, this may even extend to small amounts - it may be convenient to have replacement protection for your phone rather than having to agonize over whether to repair or replace if you drop it.

Buying insurance is rational for low chance, high cost (i.e. bigger than what you have in your bank account at the moment) risks. It is not rational for low cost risks, like loosing your phone, unless you tend to loose your phone more often than insurance companies accounts for.

That might depend the the kind of insurance. For example, here in California, I am required by law to have personal liability and property damage coverage on my cars. If I take out a loan for a car, the lender will require I have collision and theft as well. So, if I decide I want to drive on public streets, buying insurance -- rational or not -- becomes a part of the cost of owning and operating a vehicle.

In addition to what everyone else said, I recommend Gwern's "Console Insurance". Also, Jacob from Early Retirement Extreme says the following about dental and vision insurance:

I don’t have dental or vision insurance. Paying insurance that covers “regular maintenance” like teeth cleaning or contact lenses which these kinds of insurance do makes no sense whatsoever. Suppose everybody pays $25/month for contacts. Now do you think that everybody paying those $25 through an insurance company will make it any cheaper? No, the insurance company will add a $5 administrative fee—they most definitely will not give away free money. As such this kind of insurance is nothing but a financing plan for people who can’t figure out how to save the money for a $200 dental visit. The point of insurance is to cover rare events with a six-figure cost, which dental or vision simply doesn’t have.

Insurance companies are in a much better negotiating position than private buyers, because they're dealing in bulk, so their expenses are based on paying much lower prices for services than their members would get if they bought individually.

Other commenters have already addressed the difference between expected utility and expected monetary return, but in fact having insurance can have a positive expected monetary return simply because you're forced to pay more when buying the services privately.

I have only glossed over the other answers, but there is a cool way to approach this question that nobody mentioned...

For the sake of this exercise, let's say that you are part of a household with 100 people. All cousins live together, I don't know. You're just one big family who share the budget.

If one family member gets sick, it's alright. The other 99 are still healthy.

Let's say there is a p=0.1 chance any one member of the family is sick for a year. That member cannot take money home during that full year. Let's see how many family members will be sick that year.

The answer is this binomial distribution: http://postimg.org/image/vkn76o9sd/

P(N <= 20) = 0.9991924. There is a 0.9991924 probability that 20 members of the family are sick in a given year. Or, in other words, P(N > 20) = 0.0008. After N>40, my 64-bit does not have enough precision to show the probability. It is effectively zero.

Now, I am going to flip the graphic so that is shows the probability of each person being healthy. I am also adding a x-axis at the top showing how much income each family member has: http://postimg.org/image/cprhxbum9/

Now, contrast that with living alone and having a probability=10% of getting sick where your income drops to 0, which would be this discrete probability distribution: http://postimg.org/image/5tdb44vhr/. (Sorry for the ugly graphics :P)

Nobody lives in huge families anymore. The huge family is the insurance company.

In the real world, this is why big rent-a-car companies do not buy insurance. They are like the big family. If some car goes to the shop, no worries, the other make up the income. But small rent-a-car companies to buy insurance.

What makes buying insurance rational?

Suitably high stakes and a combination of risk aversion and/or credit constraints. Next question?

Simply, it's not a zero value equation.

Just because insurance companies have a positive expected utility, doesn't mean you will have a negative expected utility.

apparently to figure out whether the risk is worth taking you multiply expected value of the outcome by it's probability.

Nope, that's how you calculate the expected value and not the risk. In fact, risk is entirely absent from this calculation.

But buying insurance is a slightly different question. Basically, insurance makes sense when a possible loss will have large secondary and tertiary effects the negative value of which is large.

For example, consider a retired couple without much savings living in a house with a paid-off mortgage. Let's say the house is worth $200K. What happens if the house burns down? Is their loss $200K? Nope, their loss is much bigger because they don't have a place to live, can't afford another house, and their life just got much worse -- more than the nominal loss of $200K would indicate.