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On wikipedia, I read that the currently-used formal definition of a natural monopoly is where “[a]n industry in which multi-firm production is more costly than production by a monopoly”.

I tried looking up things related to monopolies and subadditivity, but couldn't find anything that made it easy to understand.

To further quote wikipedia, "Baumol also noted that for a firm producing a single product, scale economies were a sufficient but not a necessary condition to prove subadditivity."

Note that in the above quote, I'm almost certain the "scale economies" they're talking about are cases where the optimum scale is large enough to serve the entire market.

So my question is, what is another set of conditions that don't involve "scale economies" that is a sufficient condition to prove subadditivity, and thus show that a situation is a natural monopoly?

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    $\begingroup$ I think you really need to define what you mean by scale economies here. For example would the cost function: $C(y) = 1000 + y$ imply that there are decreasing returns to scale? What about $\hat{C}(y) = 1000 + y^{0.99}$? $\endgroup$ – Giskard Mar 14 '16 at 22:00
  • $\begingroup$ @denesp I agree its vague, but it was a quote. I added a note after the quote explaining what I think it means. $\endgroup$ – B T Mar 14 '16 at 22:59
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One answer I found are natural monopolies that form on the basis of significant network effects. While economies of scale decrease the cost per unit as you can produce more units, network effects increase the value per unit as you sell more units. These two things together seem like they cover everything, but I haven't read any sources that mention whether there are other situations than economies of scale and network effects. Btw, I'm including "economies of scope" in my mention of economies of scale, since they're so similar.

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You seem to be looking for a monopoly with barriers to entry other than capital. A patent grants the holder a monopoly that prevents other businesses from entering the market for a certain number of years, even if the product may be produced very cheaply. Crystal methamphetamine is illegal to produce but can be made very cheaply. The barriers to entry are not the scale or infrastructure, but rather the law enforcement agents and the likelihood that new entrants to the market will be murdered by current holders of the local monopoly.

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    $\begingroup$ Neither of those situations is a natural monopoly - both are coercive monopolies. One via government enforcement, and the other through both government enforcement and private coercion. $\endgroup$ – B T Mar 15 '16 at 18:37

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