# What are examples of natural monopolies that don't have overwhelming economies of scale?

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?

• 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 $\hat{C}(y) = 1000 + y^{0.99}$? – Giskard Mar 14 '16 at 22:00
• @denesp I agree its vague, but it was a quote. I added a note after the quote explaining what I think it means. – B T Mar 14 '16 at 22:59