First, we, the people, are the government. Second, what makes you think the cost would be high? Where's your evidence or at least a hypothesis as to why?
I've given lots of reasons in other threads. I'll repeat them if you really want, but I'd really rather not. The short version is, it would be a regulatory nightmare. Two-person marriages are quite simple: one relationship, one contract, no fuss. Polygamy leads to marriage graphs of arbitrary size. Every decision that needs to be made regarding a marriage would be complicated exponentially (well, okay, geometrically, if you want to get technical).
Could the issues be resolved? Maybe. Would it be worth the effort and expense, not to mention the extra burden on the legal system? Given that I really don't think too many people are interested in polygamy, I seriously doubt it.
It saddens me to hear someone say, "I can't say I see a reason why it should be legalized." I prefer a society where we need damned good reasons to make things illegal.
If this were an issue of individual liberty, then I would agree 100 percent. Sleep with whoever you want, live with whoever you want, raise children with whoever you want. But this isn't an issue of liberty, even though I know it's tempting to couch it as one. This is about whether the government -- the people, as you point out -- should be forced to enter into a contract against its will. It's about whether you, as a participant in the government (and, presumably, a taxpayer), should be forced to enter into a contract against your will, and foot the bill for it to boot. I think that you shouldn't.