The plus side to this area is the rents have been climbing due to a shortage of units.

Not only a shortage of good deals, but there is a shortage of good help to work on them!   

There won't be any shortage of college students in that area!

That means the lender can come after the borrower for the shortage that results from the foreclosure sale.

The same inventory shortage is causing a price increase to owner occupy though, so I'd be surprised if this caused more people to decide to buy vs rent (for most areas). 

A similar alternative might be to ask the lender to accept a promissory note for the shortage.

As a Realtor I make money selling property so it's frustrating when there's a huge shortage of a certain asset class!

Rental rates in Miami go up really high very often because right now there is a shortage in hotel rooms for the amount of visitors they are receiving per year.

Whether the back taxes total $4, 5 or 6k is of no concern to you if your agreement is that you are buying the property free of all liens.If, on the other hand, you are purchasing the property free of all existing liens and paying $10k with a purchase money note for $10k, this would imply that the shortage would require the seller to bring money to escrow to pay the back taxes, and then walk away with a note from you for their trouble.

I recently heard on one of the local news stations that there's an estimated 10,000 home shortage.