A large part of the expense was due to the successful effort to apprehend him without killing him or bystanders.
Tweets aren't precisely geolocated, so that's not really relevant.
The US has a "value of a statistical life"[1] of around $7M, and they had already killed 4 people and injured over a hundred more, so $50M seems like it would be very easily justified.
My point was this was one person (okay, two brothers originally) what would have happened if it was a team of people who were organized and actually secretive and didn't use twitter or live out in the open?
No, the United States Government is saying that, by using the Value of a Statistical Life that they use. So I'm telling you that (by their own terms) they can justify spending a lot of their money on catching someone who has killed several people and intends to kill more.
I do not agree that it makes sense to value US lives at $7M each and developing world lives at apparently less than $2000 each (the approximate price of saving one according to GiveWell). But that's what the link I cited says is the case here.
I'm not sure this adequately captures it, though; the bombers are said to have talked in front of their carjacking victim about planning to escape to NYC. Shouldn't we include some expected value term of the cost of shutting down NYC, in case they moved there and had to be hunted there too? Seems like the entire country/world has an interest in seeing them caught here.