Are Identical Make-up Exams Fair?
Whether identical make-up exams are fair depends on the circumstances.  The key point is that you can't eliminate all potential for cheating: there are too many ways to cheat, and you can't even try to rule them all out without brutal and expensive oversight.  Ultimately, you have to make a trade-off, in which you take some anti-cheating steps but stop focusing on cheating once you reach the point of diminishing returns.  [For example, if you are really concerned about cheating, then you can't use take-home exams, and you'll have to police restroom visits very carefully for in-class exams.  These steps might be sensible under some circumstances, but counterproductive in others.]

So how does this apply to identical make-up exams?  If you're teaching a required, low-level class to a large group of unenthusiastic students who desperately want good grades, then cheating sounds like a real worry, and giving the same exam twice would be a foolish decision.  If you're teaching an advanced class to a small group of students you know and trust, then there may be no issue at all.  Reasonable people could differ as to where to draw the line between these scenarios, and it depends on their judgment of the likelihood and consequences of cheating.

There's more to this than laziness, although writing a second exam is certainly a pain.  Instead, there's a trade-off between several effects.  If you write a new exam, then you've eliminated the possibility of learning the questions from a classmate.  However, there's no way to calibrate the exams perfectly, so you've introduced the possibility that the make-up exam was inadvertently easier or harder, and the students may feel stress or bitterness over this.  Both the cheating and the differing difficulty are bad, and there's no way to completely eliminate them both, so you have to decide how much they worry you and balance between them.

There are also matters of educational philosophy.  I'm not fond of any approach that tells students I don't trust them or that challenges them to see if they can slip anything by me.  That doesn't mean I won't take countermeasures against cheating, but it means I consider the countermeasures to be problematic in and of themselves, so I try to use them as sparingly as I reasonably can.

In practice, how I design make-up exams differs from course to course.  Sometimes I write a new exam from scratch (while allowing some possibility of overlap with the previous exam, so students can't be sure something won't be on the make-up just because it was on the original exam).  In other cases, I just fiddle with some of the details or replace a few questions, while warning students not to discuss the exam until after the make-up exam.

Regarding outcomes, I find that students who take similar make-up exams rarely do surprisingly well, and on average they do a little worse than I would have expected.  Of course I don't know what's happening behind the scenes, and perhaps cheating saved them from scoring even lower.  However, I'm at least confident that students aren't using this as an opportunity to rack up lots of undeserved points.


  If it's not compromising the integrity of the exam, why not just have the same exam every year and collect the question sheet from students?


It's not a matter of absolutes, but rather of how much the integrity is compromised and what the consequences are.  Re-using the same exams every year offers much more scope and temptation for cheating than a single make-up exam does.  The longer you re-use an exam, the more likely it is that unscrupulous students will piece together and circulate a list of questions, even if you collect the question sheets.  If you do this for all the exams in the course, then the potential benefits of cheating are quite a bit higher than in just one make-up exam.  (If I saw the same student repeatedly requesting make-up exams, I would wonder why and might become suspicious.)  If the ease of cheating in every exam is known, then the course will attract dishonest students who might never have taken it otherwise.  There's a big difference between possibly being able to learn about one exam if you convince someone else to help you cheat, and being able to learn about every exam by downloading a file from the internet.  It's reasonable to draw a distinction between these scenarios.