In this passage, Hoppe practically takes the Lange-Lerner position on knowledge and prices:
"the knowledge conveyed by prices actually can be centralized. But if price information is public information and thus can be centralized, then, according to Hayek's thesis that socialism's problem stems from the inefficiency of trying to centralize genuinely uncentralizable private knowledge, it would follow that the absence of prices, and hence of private property, has nothing to do with the plight of socialism."
This presents Hayek's arguments in essentially the same terms as Oliver Williamson's theory of the firm. To a certain extent I would agree with that characterization of Hayek, although I wouldn't consider it to be as unsatisfactory as Hoppe apparently does. The "knowledge problem" version of the "calculation problem" does turn on the practical inefficiency of socialism, rather than the logical impossibility of socialism (I can agree with Hoppe on that). But the Hayekian argument loses none of its force for being an accusation of practical inefficiency, rather than logical impossibility (here I disagree with Hoppe).
Hoppe prefers what he contends is the Misesian version of the argument (I have to take his word on that - I've never read Mises on the calculation problem). Unfortunately, this essay is almost entirely taken up with Hoppe's critique of Hayek, so little is offered in elaborating this property-based position. This is all we really have in the essay in terms of an exposition of his own position:
"Mises's well-known calculation argument states this: If there is no private property in land and other production factors, then there can also be no market prices for them. Hence, economic calculation, i.e., the comparison, in light of current prices, of anticipated revenue, and expected cost expressed in terms of a common medium of exchange-money-(thus permitting cardinal accounting operations), is literally impossible. Therefore, socialism's fatal error is the absence of private property in land and production factors, and, by implication, the absence of economic calculation."
My impression is that he's identifying property as the stake that actors have in their economic action - that knowledge of the situation alone is not sufficient, but that the incentive to act on that knowledge is required. I would agree with this too, and simply say that socialism fails on both counts: the problem with socialism is that it lacks both the knowledge to optimize decisions about allocation and production, as well as the incentive to act on whatever knowledge it does have. For all Hoppe's belly-aching about Hayek, I imagine Hayek would agree with him on this point too.
My feeling is that none of this changes the distinction I've been making (here, here, and here) between calculation problems and incentive problems. The market (or really, free society in general) usually has a considerable advantage over the state because it has both the decentralized knowledge that Hayek highlights and the incentive to act on it that Hoppe highlights. However, when this incentive to act is absent, all the knowledge of the market is useless because, as I said before, the market then "efficiently targets a biased solution". The solution might be to adjust the institutions so that the market can act - create property rights, for example. But when such solutions are problematic, using the state becomes a very plausible possibility. The classic externality of greenhouses gases is the most obvious: it seems more reasonable to impose a carbon tax than it does to assign property rights to the air we breath to solve the incentive problem. Of course questions still remain about what the tax should be (or if it should be a tax at all - maybe it should be a subsidy if climate change is a good thing and we want more balmy winters). These questions don't just go away. But the point is the decentralized information of the market is no better positioned to answer these questions than the state because the property rights simply aren't there - there is no incentive to act on the information that people have. Because there are no property rights, doing nothing is as much an imposition on human liberty as imposing a carbon tax is.
As a post-script, generally speaking I find Hoppe's views extremely problematic, if not dangerous (and as readers know, that's not really an adjective I apply to thinkers, even those I disagree with). I don't genuinely recommend him, but the distinction he makes here was worth highlighting in light of the issues I've been writing about recently.