Analysis of free goods The normal supply and demand models take the supply and demand of a particular good and show that the equilibrium price is where the two curves intersect.  At this point, all the people below the equilibrium point value the good as worth less than the price; on the other hand, all the people above the equilibrium point value the good as worth more than the price; people at the equilibrium point value the good at the price value.
In my mind, following this logic, everyone should want to "purchase" a free good.  But that is not necessarily the case.  For example, in an undergraduate program, the university will give away t-shirts, pens, etc. to the students for free.  However, not all of the students will want these items.  However, since the price is free, it seems that everyone should want them.
How do we reconcile both of these observations?
You can imagine a demand curve continuing through the x-axis as monetary price becomes negative. A negative price can be interpreted as payment to pick up shirts, a kind of subsidy.* That might tempt those who are still unwilling at price of zero because there's still a opportunity cost of picking up the shirt, or because they only wear collared shirts, or because disposal is not free (the unwanted shirt would need to recycled).

In the simple model, the demand curve that represents willingness to pay is holding these other factors fixed. In a more complicated model, the price drop might induce congestion at the student union, so the demand curve would also shift down or up, depending on your desire to spend time in a crowd of college students.



*Obviously, you have to impose some sort of limit per student to keep quantity demanded from shooting off to infinity.