Knibb High football rules!
There has never been a study done to assess the true cost to government of administrating marriage. That doesn't mean the cost isn't there, it just means nobody has bothered to look at it.
Likewise, there has never been a study done to substantiate whether the sky is, in fact, blue.
However, if you use a bit of common sense, it is hardly a stretch to assume that the more work you ask the government to do on our behalf, the more resources they will need to expend to complete that work.