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Posner & Weyl claim that property rights lead to monopoly rents:
Property Is Only Another Name for Monopoly
Radical Markets

Have their claims been elaborated, tested, or peer-reviewed?

In situations with assets with unique attributes that cannot be substituted (e.g. a plot on a high street) this seems to be intuitive. An owner cannot be compelled to subject her property to market competition but she can hold out for a higher price than she values the property at. This generates an economic rent resulting from her market power over the plot of land in question. The claim seems to be less obvious where assets can be easily replicated (e.g. a pint of milk). That would imply that property rights may only lead to monopoly rents where they grant property over assets that generate monopoly rents (e.g. land).

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In narrow sense of the word the article was definitely peer reviewed as it was published in Journal of Legal Analysis. But in this narrow sense it was most likely peer reviewed by jurists not economists.

In a broader sense of the word there are economists who had a look at the claims as well. For example, review of their book by Levine (2020) in Journal of Economic literature (one of the top ranking journals), examines their claims about monopolies and also other claims and policy proposals made in the book. The review is not very warm to the ideas of Posner and Weyl. Regarding the idea of property rights causing widespread monopoly power and misallocation Levine writes [emphasis mine]:

The chapter begins with a clear and careful description of the hold-up problem. Here, indeed, monopoly power is a problem. It then goes on to suggest that the hold-up problem is ubiquitous in ownership of property—that all property is monopoly and that the resulting misallocation might be on the order of 25 percent of income. Here as elsewhere the book is not strong on data. There is extensive research indicating that in some places at some times, misallocation is a problem possibly of this order of magnitude. Typically this research does not attribute misallocation to a problem with private property—generally the reverse.

... Nevertheless the economics profession does not generally view hold-up and private information problems as ubiquitous nor externalities as insoluble. In many markets, ample substitutes are available— for example in the housing market—so that monopoly power is not important. In some cases the market solves the hold-up problem relatively well—through tender offers and other ways of packaging properties.

Pushing on: the authors argue that the problem with the ubiquitous monopolies created by private property can be solved by a self-assessment wealth tax with mandatory sale at the self-assessed price. Self-assessment with a commitment to sell at that price is a useful and credible solution to problems involving assets for which there is substantial private information. This has been a particular problem for commercial real estate and for privately held businesses, where often artificially low transfer prices are used to avoid taxes. As the authors correctly say, top researchers such as Harberger (1965) and Cramton, Gibbons, and Klemperer (1987) have argued for self-assessment in these types of circumstances. It is not immediately clear why a proposal for solving a particular private information problem should be viewed as a broad solution to a problem of ubiquitous monopoly...

There is more critique on this in the paper. To give you a wrap up (as mentioned in the comments under your other question), the problem with Posner and Weyl is not that their analysis would be inaccurate in some specific cases. Rather they try to generalize well understood result that holds in some very specific circumstances onto property rights in general.

There is quite large consensus in literature that property rights do not generally lead to monopoly or even market power (see Belleflamme & Peitz Industrial organization: markets and strategies for overview of many models where even monopoly does not guarantee market power and models of competition where property rights are implicit but firms have no market power). In fact as Mankiw put it in his Principles of Economics, economists view property rights as "an important prerequisite for the price system to work".

Furthermore, Acemoglu and Robinson (top developmental economists) in their magnum opus Why Nations Fail, go even so far as to state:

To be inclusive, economic institutions must feature secure private property...

Acemoglu and Robinson go further on proving that sustainable growth and economic development without inclusive institutions (including property rights) is virtually impossible.

Additionally, this work is apparently even criticized in law profession itself (see Wyman, K. M.; 2019), but as I am not a law professional I will have no further comments on that (it just seemed relevant to point that out).

Hence given the above I think it is fair to say that the claims of Posner and Weyl are not received positively by economics profession, to put it very generously.

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  • $\begingroup$ Levine says that "ample substitutes are available [...] in the housing market—so that monopoly power is not important". Where does this leave land rent then? A rent-free plot on the edge of civilisation is not a substitute for a plot in the centre of the city for which the owner can ask far more than "the costs needed to bring that factor into production". $\endgroup$
    – sba222
    Nov 16 '20 at 6:31
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    $\begingroup$ @Steve222 as mentioned in this answer in some specific cases such analysis is correct. You can find some examples when it would work. Problem is that Posner and Weyl claim that property rights will virtually always lead into monopoly power and create misallocation whereas as mentioned in the excerpts from the paper evidence and literature suggests most of the time it is lack of property rights that leads to misallocations and holdout problems are only important in some specific cases $\endgroup$
    – 1muflon1
    Nov 16 '20 at 9:01
  • $\begingroup$ I can see that for most goods an owner’s self-assessed value would correspond to the cost of replicating the good in a competitive market (i.e. no monopoly). But a competitor cannot replicate a plot of land in the centre of a city. How does property in land not lead to monopoly? What else is land rent then? (Perhaps that should be a separate question?) $\endgroup$
    – sba222
    Nov 16 '20 at 9:41
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    $\begingroup$ @Steve222 I will give you an analogy. For example, economic profession accepts that government is necessary for provision of public goods (as narrowly defined in economics). But now imagine someone publishes book saying I think every good is a public good and since this valid economic research shows that public goods need to be provided by government we need to eliminate all private sector and have total gov. ownership. The fallacy of that statement is that it takes a result that is valid for a subset of goods for which claim is true and then changes definition of every good to public good $\endgroup$
    – 1muflon1
    Nov 16 '20 at 9:50
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    $\begingroup$ @user253751 I dont know how to respond to that surely you are just jesting now $\endgroup$
    – 1muflon1
    Nov 16 '20 at 19:03

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