To key or not to key? (Key economics)

About three months ago I misplaced the keys to my room. Of course, I had to get in to my room, so I went (for the fifth time), to Appel to check out a spare key. Now, if you are a Cornell student and you have a ‘busy’ mind, then you are probably familiar with the lock-out policy.  Once you have locked yourself out, two situations can occur: 1) you can pay an RA $5 (if you don’t have a first-time voucher) to unlock your door, and hope that your keys are inside your room. 2) You can check out a spare key for 5 days, which upon return, also costs $5. If you do not return it by the deadline, the rules say that Cornell will change the lock on your door and charge you $70.  This is the same cost incurred if you were to permanently lose your keys.

However, after paying $5 to have my door unlocked and checking out a spare key, my original keys return to me! During this whole fiasco, I found a very interesting tidbit regarding the inefficiency of Cornell bureaucratic systems: nobody checks the logs to see who has not returned their spare key after five days!

So, with my newly inquired Insider information, I have a big decision to make: do I return the spare key I checked out or not? While this decision may seem obvious, I believe it is a gamble the owner makes that could either cost them a lot of money, or save them even more!

If the owner returns the spare key, then the next time they get locked out they will have to go through the whole obnoxious process again and be out anywhere from $5-$10. If the owner does not return the spare key, however, than they could set up a back-up system in which they never ever get locked out of their room, because they have the spare to unlock the door.  Now, if the owner chooses to go with the latter option and not return the spare, they run the risk of losing the spare as well. I suggest that the owner does NOT return the spare, on the basis of a very simple mathematical / economical analysis.

Given:
I lost my original keys 5 times in the first semester; this problem was presented to me at the beginning of the second semester. It is therefore reasonable to conclude that there is a strong chance I will misplace my keys at least this many times again.
   Cost to unlock door: $5
+ Cost to return spare key: $5
= Total Cost to get locked out and get back in: $10

Cost to replace lock if spare is lost: $70

Total cost to unlock and check out/return a spare key for each lock-out (X) incident throughout semester 2: ($10 per lockout at a rate of 5 incidents per semester)=$10*5=$50 (C,return)

Cost to keep the spare and return it at the end of the year: Only $5! (C,safe)

Cost to keep the spare key and lose it, too!: $75 (C,lose)

So, when coming to a decision, it ultimately depends on what the owner thinks the probability of losing his keys a set amount of times is (P1), as well as the probability of losing the spare key (P2).  If we look at the probability of having to pay each total cost, we can determine which situation will probably cost less, and therefore which option we should choose:

Probable cost of getting locked out X times and each time returning the spare: (P1)*(X*10) =
C,return

Probable cost of keeping the spare and then losing it as well! = (P2)*(75) =
C,lose

as an added bonus…

Probable cost of keeping the spare key and not losing it, returning it at the end of the year = (1-P2)*5 = C,safe

So, in my situation (‘guesstimating’ the probabilities)
C,return=(.8)*(5*10)=40
C,lose=(.3)*(75)=22.5
C,safe=(.7)*5=3.5

Now purely from an economic standpoint, we see that (C,lose)<(C,return) and (C,safe)<<(C,return) so it is the most economically sound choice to not return your spare key! Unfortunately it is not that simple; the implementation of the model breaks down to a certain extent in the real life situation. The first issue is that there is no sure way of being able to estimate the probability of losing your keys. This could possibly become more accurate as time goes on however, and you have a greater sample of semesters to look at.  Also, at first glance it seems to make sense that the higher your probability to lose your original key is, (C,return) becomes larger so (C,lose) will be the better choice.  But, at the same time, if you are losing your original keys a lot, it is safe to say there is a greater chance that you will lose your spare as well, and therefore pay big time!

More problems arise under extreme conditions– lets say you lose your keys 20 times in a semester and there is an 80% chance you do so, your (C,return) value is 160. Even if you have 100% chance of losing your spare, the model would indicate it is wiser to not return it, because (C,lose) has a maximum of 75. Obviously this model isn’t flawless, but it think it justifies my decision in not returning my spare key!

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