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If a firm has increasing returns to scale, does that mean that "costs" will always decrease as production increases. If so, does that mean the firm will end up being a monopolist?

And what do we mean by "costs" in this context: are they average, fixed, total, marginal, or can all costs be interpreted for the IRTS firm as it expands?

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First, let's start with a definition of increasing returns to scale (IRS). A firm is said to have IRS if multiplying the quantity of each of its inputs by a common factor, $a$, causes its output to increase by a factor greater than $a$. For example, if doubling all of its inputs causes its output to more than double.

It can be shown that if the firm's production technology exhibits IRS then its average cost must be decreasing. This is why, in most introductory economics courses, you may encounter situations where the average cost decreases in the quantity produced described as exhibiting IRS. But, again, strictly speaking it is not decreasing average costs per se that constitute IRS, but rather the fact that output expands proportionally faster than inputs.

So, the answer to your last question is that by costs we mean average costs.

The other remark is that IRS is a "local" property, meaning that starting from the current level of output, an increase in $q$ would cause average costs to decrease. The average cost function can then be said to exhibit IRS at that point. But it is possible for an average cost curve to be decreasing in one region and increasing in another. E.g., the average cost curve in this figure exhibits IRS for low quantities, but decreasing returns to scale for high quantities:

enter image description here

This brings us to the last part of your question. To produce a "natural monopoly" it is not enough that costs exhibit IRS at one point. Rather, we need, roughly, the average cost curve to decrease across enough of its range that there is no way for a new firm to produce a subset of the monopolist's output at a lower average cost. A sufficient condition is that the average cost curve be decreasing across the entire relevant range.

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    $\begingroup$ The concept of a 'natural monopoly' is commonly defined by the subadditivity of the cost function. This definition originates from Baumol's seminal paper (1977)[jstor.org/stable/1828065]. For a firm with a single output, it is not strictly necessary for the average cost to be decreasing over the entire relevant range. For a single output firm, this point is however more technical than practical. Nevertheless, it is true and relevant for someone who wants to understand the economic theory of natural monopolies. $\endgroup$ Commented Jun 11 at 11:23
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    $\begingroup$ @JesperHybel Thanks, yes I revised the answer to be a little more careful that I am giving a sufficient rather than necessary condition. Indeed, we can have AC that is increasing over some part of the relevant rage, provided the resulting AC remains above that of the natural monopolist. Of course, from Baumol we also learn that a natural monopolist may not behave very much like a textbook monopolist if the market is contestable, but that's perhaps for another question… $\endgroup$
    – Ubiquitous
    Commented Jun 12 at 14:20
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    $\begingroup$ Yes, I think that with respect to contestable markets, one answer is that for the monopoly to be sustainable there must be some kind of barriers to entry. It is hard to be more specific because there can be many different mechanisms that ensure that the market is not perfectly contestable. I suppose it could be the presence of sunk costs or threats of lowering prices + some game-theoretic assumptions to ensure non-entry in perhaps a Baysian equilibrium depending on the assumptions made for expectations. But, I have not read this debate. $\endgroup$ Commented Jun 12 at 21:25

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