Because the exchange was brief and, basically, an aside to other issues being addressed at the time, I hesitate to make too much of it. But I must say I find Carol's position baffling. An institutional structure - whether common-property, individual private-property, or some other management system - obviously affects rates of extraction/resource-depletion. But it does not affect subtractability which depends exclusively on ecological circumstances.* Ecological circumstances can/should affect the applicable institutions, for reasons explained by Hardin (1968), Demsetz (1967) and many others. But institutions don't alter the fundamental ecological conditions. Put differently, a subtractable resource remains subtractable whether the current rate of demand, under prevailing institutional structures, is either zero or very high. (This is why Ostrom's social-ecological systems framework quite rightly pays as much attention to ecological conditions as to institutions.)
*Institutions can affect excludability, if only indirectly, by affecting incentives to innovate technologies that, themselves, can enhance excludability, at least with respect to some resources. Thus, the invention of barbed wire in the mid-19th century enhanced excludability of lands in the western US, where lack of timber had previously made enclosure too expensive (Anderson and Hill 1975).