Here is a question from last year’s exam in the course “Basic Ideas of Mathematics”:
You buy a toaster for 200 NIS ($50) and you are offered one year of insurance for 24 NIS ($6).
a) Is it worth it if the probability that damage covered by the insurance will occur during the first year is 10%? (We assume that without insurance, such damage makes the toaster a “total loss”.)
b) Is it worth it if the probability that the toaster will be damaged is unknown?
As an additional test of your imagination, can you think of reasons why buying the toaster insurance would be rational?
Combinatorics and more 
Don’t we need to know the prevailing interest rate as well?
If the lemon rate is higher than 6/50, it’ll be rational every time 🙂 Given the rate of breakdowns and failure on modern laptops, I’d never buy one without an insurance policy good for its expected lifetime.
Well, if my budget is $56, and I absolutely need a toaster for the whole year, I’ll buy the insurance. That is to say, it’s not necessarily the expected expenditure that I’m minimizing.
Insurance will only be profitable if the premium is greater than the expected payout. Can there be such a premium at which the insurance is worth your while anyway?
As anon says, the question is “can you afford to replace the toaster if it breaks down?”. If the answer is yes, then insurance will never be a good idea. If the answer is no then there will be a price at which you will buy insurance.
Bigger question: is it rational to buy a $50 toaster? I mean, a while ago I saw a combination toaster and kettle (it was as large as both) for several hundred $, but, really. Let alone a $50 toaster with a 10% failure rate…
Another consideration is how easy it is to claim on the insurance: for example, several times when taking a broken device they decided they had to send it away for repairs and then give it back several weeks later. So, how does the expected value of the insurance outweigh the better part of an hour convincing the store clerk that I have an ex-toaster and several weeks without toast, as opposed to the time it would take me to go to the nearest department store and buy a new $20 toaster?
Ray: you need to consider opportunity costs as well.
tl;dr: Buy neither the toaster nor the insurance.
Give an n% chance of the toaster breaking. Suppose the insurance completely replaces our toaster with a new toaster valuing $50 if the first one breaks in a year. Then the amount of money we’d expect to have expended for the toaster after a year is
EE = (56(100 – n) + 6n)/100.
If that is a rational understanding of the situation, then insurance is worth buying if EE 12%. So it does seem irrational to buy the insurance at our 10% breakage rate.
But then you can increase the breakage rate in a rational way! I’d recommend buying the toaster and insurance. Then you can use it much more roughly than the average person would. I’m not sure if this would be dishonest – the insurance rate probably factors in for people like you ;). If your enthusiastic use doesn’t manage break it after a year, then by gum you’ve got a good toaster and everyone’s happy.
Hmm, my “less than” and “greater than” symbols must have been interpreted as enclosing a tag. I meant to say that if EE is less than $50, that is, if n is greater than 12%, then the insurance is worth buying.
But don’t underestimate your ability to (legally and rationally) ride the edges of the insurance company’s predicted bell curve.
“Man does not live by bread alone (even if it is toasted)”
I will insure the toaster because if I do so I will have nothing to worry about. I daresay I can use it carelessly 🙂
If this was an economics exam I would say that the question is unanswerable without knowing the utility function. Since it is a math exam I guess the interpretation intended is whether the expected value is positve.
If I buy the insurance I on average get 5 dollars payment from it which is less than the cost. So the expected value of paying for the insurance is negative. Thus you should not buy it.
Johan, One problem is that we hardly never really know the utility function. Anyway, would you regard buying an insurance for a toaster rational (or rationaizable) from the point of view of economics?
For the second part of the question: maybe we would interpret the offer by assuming the insurance company is assuming it is selling to rational customers, in which case the toaster has a 12% failure rate, per max above (?). But of course, that can’t be right: the insurance company must be making some profit on it—there’s no competition when it comes to these sorts of offers, right? And thus we’d conclude it’s a scam, confirming reality (at least for extended warranty type offers.)
For the purposes of a clean question this makes an assumption that is consistent with the times (but was not the case 40 years ago before toasters went out of vogue prior to their recent revival):
You can’t repair the toaster (or it would cost more to repair than replace).
What if the repair cost were $20 and repair increased the probability of a second failure from 12% to 20% for the rest of the year? Now what should you pay for it to be fair?
When I was growing up toasters tended to have pretty interchangeable parts and my father would regularly fix toasters for friends using the parts from an extra broken toaster or two. The problems would invariably be something simple to fix. The same thing held for many other appliances both large and small.
We now have more of a throw-away society. Some items are made more cheaply but other items are made to obsolesce because the new models have much more sophisticated digital controls and there is not a market for the replacement parts for the old controls.
Well, in principle it is rationaizable for some utility function but it would imply a ludicrous amount of riskaversion unless you were very poor, in which case you probably would not be buying a toaster anyway.
a. Giving i’m indifferent to risk I would not buy the toaster. the expected value is negative.
b. There is no rational way of deciding this case. someone mentioned taking in to account the fact that the insurance company seeks profit, but that wasn’t in the original data. what if you knew the insurance rate was decided randomly? Will you still refuse?
I would just guess. Not everything has to be rational 🙂
This reminds me of a question I recently came across in my friend’s blog. Following is the question:
There are two players, A and B, who play the game as follows. A fair coin is tossed 25 times in a row. If even one T comes up, the game stops, A gives B N$ (say 100$) and they part. However, if everything comes as heads, B gives A 2^32 $ (about 4 billion dollars). You are given a choice whether you want to be A, or you want to be B (you HAVE to choose or be beheaded by the king). Who would you rather be and why?
In the above problem, if you consider the expected value, you would get it positive for the player A to be positive. But I would prefer to be player B rather than A, especially since the game is played only ONCE.
It seems to be the similar case in the problem stated in the blog, considering one wouldnot be buying too many toasters, expectation doesnot appear to play a good measure of one’s ‘preference’.
I think one needs to consider some other quantifiers.
I thought like this. Suppose you buy 10 tosters and pay $500 and $60. I Since only one is expected to break, it does not seem a good idea to insure in purely economical considerations. Of course if one can not wake up without toast, you may want to insure it.
Now, if the probablity to break is more than 12%, it is financially feasible to insure neglecting the other factors like reduced value after using for some period etc.
It may be a good idea to insure in both the cases above as if it is broken, you have a hefty loss. To avoid that even you do not expect the item to be broken, one should probably insure.
Now, if you are insuring in India, it may not be of much use anyway as insurance companies in general find some way to avoid paying (or refuse to pay) you or to reduce the due amount :).
Interesting question which has a simple pat answer. The company selling the insurance expects to make money on the deal, and they have a lot more information on the reliability of the product than I do. This is a zero sum game so what they “win”, I lose. So no, on average the insurance isn’t a good deal for the consumer.
Furthermore, the consumer can not make a “rational” decision without a lot more information:
What is the store return policy? (replace or repair?)
What is the manufacture’s warranty? (replace or repair?)
What is the failure rate curve?
The problems with returns has also been noted. If he item is to be sent back for repair, am I willing to be without it for some indeterminate amount of time? Am I going to have to pay shipping costs back to manufacturer to get the item repaired?
Also what is my time worth to me? If I can make $50 an hour doing machine work in my machine shop, and I have more work than I can do, it doesn’t really make sense to spend 2 hours fighting over a $50 toaster.
I think the real basis for a decision is to look at the store warranty. Usually something like a 30 day warranty. Than be sure you use it frequently in the first 30 days. I’d really expect that if a toaster doesn’t fail in the first half dozen uses that it will probably be good for years.