Intellectual monopolies are not property
But surely it is clear, given the origin, justification, and function of property rights, that they are applicable only to scarce resources. Were we in a Garden of Eden where land and other goods were infinitely abundant, there would be no scarcity and, therefore, no need for property rules; property concepts would be meaningless. The idea of conflict, and the idea of rights, would not even arise. For example, your taking my lawnmower would not really deprive me of it if I could conjure up another in the blink of an eye. Lawnmower-taking in these circumstances would not be “theft.” Property rights are not applicable to things of infinite abundance, because there cannot be conflict over such things.
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Like the magically-reproducible lawnmower, ideas are not scarce. If I invent a technique for harvesting cotton, your harvesting cotton in this way would not take away the technique from me. I still have my technique (as well as my cotton). Your use does not exclude my use; we could both use my technique to harvest cotton. There is no economic scarcity, and no possibility of conflict over the use of a scarce resource. Thus, there is no need for exclusivity.
Ideas, unlike objects, are infinite -- my use of your idea (song, invention, formula) does not conflict with your use of your idea. That's really all you need to understand why you can't say you "own" an idea -- why "ownership" of ideas is an oxymoron that simply does not make sense.
This quotation, and select others, are there for you in the fantastic short book called Against Intellectual Property.
Your very first sentence is where the problem in your argument lies. Limiting the justification to scarce resources sounds plausible on the surface, but it neglects the reason why many people feel that property rights should extend, to some extent at least, to ideas. The reason has two parts: (1) Many ideas require much time, effort and expenditure to come up with the first time, but very little time, effort or expenditure to understand and apply once they have been observed/explained; (2) if one person comes up with an idea and applies it to a commercial venture in the hope of generating a profit, and another person sees the idea and also applies it, the first person's income stream will be reduced due to competition.
Because of the great disparity in investment, many feel that the original inventor should be entitled to a greater share of the resulting profit in this situation. (Unfortunately, how this "greater share" is decided must be somewhat arbitrary.) You might argue that the intuitive notion of "fairness" at work here is simply our emotions clouding our rationality, but it underlies a sound argument based on rational self-interest. If the result of one's investment in invention could be immediately duplicated at low cost, and if there are no legal obstacles to prevent this from happening, then it would simply never be profitable to research "easily duplicatable" solutions to problems -- anyone doing so would be rapidly outcompeted. Since many excellent solutions to problems fall into the "easily duplicatable" category, this would leave the world a comparatively barren place.