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"Resources are allocated by prices that form in the market after exchanges of private property. If there is no private property, there are no exchanges."
Shame that someone who claims to understand economics is so woefully wrong.
First off, yes, private property is a key element of a marketplace, but you have to first understand the purpose of private property. This is rather important, and often gets lost in the shuffle. The purpose is to allow for the efficient allocation of resources.
However, when a good is infinitely available, there is no question of the efficient allocation of that resource, because efficiency is easy: anyone who wants a *copy* can get it. This is not socialism, as you imply. It's pure free market capitalism. Your mistake is confusing the "copy" with the original good. It's a common mistake.
Thus, it makes little sense at all to apply property rights to infinitely available goods. In that case, all you are actually doing is making allocation INEFFICIENT by putting artificial and unnecessary limitations on things, and actually diminishing transactions, because the infinitely available works cannot be built upon to create new and valuable works.
So you have put the cart way before the horse here in focusing on the marketplace, rather than the efficient allocation of resources. We like markets for scarce goods because they make allocation efficient.
We can have a market for infinite goods, but since the supply is infinite, the price will get set at zero (this is just basic economics).
Now, the real reason given for IP rights is not about allocation of resources, but incentives to create (you seem to confuse these things in your comment). So, your question about "the calculation problem" is actually about incentives to create, which is separate from the issue you later brought up (the market for efficient allocation).
But, once again, here you seem to have a lack of understanding of the facts. There is tremendous evidence nearly across the board that there are significant and workable models to create plenty of new works (or new inventions) in the absence of IP protections. In fact, much of the evidence suggests that the end result would, in fact, be MORE works created, because so many creative and inventive works are actually built off of earlier works.
In other words, both creative and inventive works are an ongoing process of creation, and placing an unnecessary, inefficient and market limiting tollbooth at each stage of the process drastically slows down the incentives for creation.
So, let's try starting again from scratch. Look at the difference between the allocation problem from the creation problem -- and then look at alternative models and the research concerning how those alternative models do. Then, perhaps, you might want to tone down your false statements concerning "socialism" and maybe, just maybe, apologize to those you insulted with your own statements.