# Increasing returns, implications?

If a firm has increasing returns to scale (i.e., doubling inputs more than doubles output) would that firm logically end up being the sole firm in its sector in the long run?

If not, what is the advantage of displaying IRTS?

• Ie, like Facebook or Google? Being a dominant firm is often more valuable than being the sole firm--you can buy out successful competitors without needing to pay the R&D costs of all competitors. – Mox Nov 4 '20 at 0:08

Provided that increasing returns to scale apply over the whole production function of the company it is likely that it would become natural monopoly. For example, Mankiw in Principles of Economics (pp 292) in some passages even defines monopolies in relation to their cost function:

When a firm’s average-total-cost curve continually declines, the firm has what is called a natural monopoly.

Generally the average-total-cost curve will continually decline when firm has increasing returns to scale.

However, there are models with increased returns to scale where monopolies are not inevitable. For example, Wang (2016) presents a model of banking competition where firms face increasing returns to scale, and even though the model predicts increase in market concentration it does not necessary predict that a single bank will become monopoly.

• I think the most important point here is that generally (but not always) IRTS leads to continually decreasing average-total-costs. This is because IRTS is a narrow concept related to only production function whereas decreasing costs considers many other aspects, as discussed here. – Dayne Nov 4 '20 at 12:33
• @Dayne yes that’s why I also said “Generally” at the start of the paragraph - but you are right to stress that – 1muflon1 Nov 4 '20 at 12:34

It can be shown that a firm with increasing returns to scale (IRTS) and no market power makes a negative profit (and may not be observed at all, in the long run). Conclusion: either subsidies, or market power is necessary for a firm with global IRTS to be sustainable.
With usual notations, the claim follows from the first order condition for an (inner) optimum with perfect competition: $$c'(y) = p$$. When multipling by $$y/c$$ we obtain the inverse of the rate of returns to scale: $$c'(y)y/c = py/c$$. With global IRTS, $$c'(y)y/c < 1$$, and this implies that profits are negative: $$py-c<0$$.

• +1 this is interesting result, is it a famous result form IO? – 1muflon1 Nov 3 '20 at 21:05
• It is difficult to me to find to whom attributing the paternity of this result. I spontaneously think to Pigou, Boiteux, or more recently to Baumol and Bradford (1970). – Bertrand Nov 3 '20 at 22:37
• thanks I will check those references it is very neat result would not expect that but it makes sense – 1muflon1 Nov 3 '20 at 22:40
• $c′(y)y/c<1 \implies c'(y)<c(y)/y.$ Varian (1992, p.88-89) discusses this issue and shows how it is related to the "production function based measure" of RTS. – Bertrand Nov 4 '20 at 14:53
• This says that MC is less than AC so AC must be declining. This is economies if scale and as we know that IRTS and economies of scale are different. As also pointed out by @1muflon1: generally IRTS leads to economies if scale (not always). – Dayne Nov 5 '20 at 0:49

A firm ends up being a sole player in the sector when market entry for others is restricted, which may not be guaranteed by IRTS or even continual economies of scale.

First, it is important to realize that, contrary to our first naive intuition, IRTS does not imply economies if scale. See this for counter example and related discussion.

Second, lets be generous and assume that there is continual economies of scale. Below is a counter example to show that even now, competition can exist:

Consider a cournot duopoly with two identical firms, and marginal cost:

$$\frac{\partial C}{\partial q_i}=50-q_i/2$$

Inverse demand function for the product market is:

$$p=100-q=100-q_1-q_2$$

Writing the profit function for firm 1:

$$\pi_1 = q_1 (100-q_1-q_2) - C(q_1)$$

FOC:

$$\frac{\partial \pi}{\partial q_1}=50-3q_1/2-q_2$$

So we have following best response function (for $$i\ne j$$):

$$q_i=\frac{100-2q_j}{3}$$

So, at equilibrium:

$$q_1=q_2=20$$

This is clearly a permissible solution and shows existence of cournot duopoly with continual economies of scale.

The idea is that if same tech is available to all, anyone can enter market with big enough investment and take a share if market demand. On the other hand, if there is first mover advantage (for example ,because of consumers' inertia to move to new seller or capital constraints, etc) economies of scale can help create a monopoly by restricting market entry.

So to summarize, IRTS, or even economies if scale for that matter, does not guarantee monopoly.

What is the benefit of displaying IRTS: it can certainly deter a market entrant by (possible a credible) threat that incumbent can increase investment significantly to make entry costly.

• Now that there are three pieces of the puzzle available, the natural questions are: which industry structure is the less inefficient (and under which conditions)? How to regulate the industry in order to sustain efficiency? The space available in this comment is unfortunatelly too small to survey this large literature. – Bertrand Nov 5 '20 at 7:42