<p>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:</p>
<blockquote>
<p>When a firm’s average-total-cost curve continually declines, the
firm has what is called a natural monopoly.</p>
</blockquote>
<p>Generally the average-total-cost curve will continually decline when firm has increasing returns to scale.</p>
<p>However, there are models with increased returns to scale where monopolies are not inevitable. For example, <a href="https://academic.oup.com/ej/article-abstract/125/585/989/5077363" rel="nofollow noreferrer">Wang (2016)</a> 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.</p>

CC BY-SA 4.0
2020-11-04T12:33:25.990Z
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](https://economics.stackexchange.com/q/40368/24967).
Dayne
CC BY-SA 4.0
2020-11-04T12:34:39.843Z
@Dayne yes that’s why I also said “Generally” at the start of the paragraph - but you are right to stress that
1muflon1
CC BY-SA 4.0
2020-11-03T19:54:12.817Z
2020-11-03T20:45:18.867Z
2020-11-03T20:45:18.867Z
1muflon1
<p>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.<br />
With usual notations, the claim follows from the first order condition for an (inner) optimum with perfect competition: <span class="math-container">$ c'(y) = p $</span>. When multipling by <span class="math-container">$y/c$</span> we obtain the inverse of the rate of returns to scale: <span class="math-container">$c'(y)y/c = py/c$</span>. With global IRTS, <span class="math-container">$c'(y)y/c &lt; 1$</span>, and this implies that profits are negative: <span class="math-container">$py-c&lt;0$</span>.</p>

CC BY-SA 4.0
2020-11-03T21:05:13.600Z
+1 this is interesting result, is it a famous result form IO?
1muflon1
CC BY-SA 4.0
2020-11-03T22:37:18.580Z
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
CC BY-SA 4.0
2020-11-03T22:40:27.907Z
thanks I will check those references it is very neat result would not expect that but it makes sense
1muflon1
CC BY-SA 4.0
2020-11-04T12:23:26.220Z
Sorry if this is a naive question: how does IRTS imply $c'(y)y/c <1$?
Dayne
CC BY-SA 4.0
2020-11-04T14:53:21.663Z
$c′(y)y/c<1 \implies c'(y)<c(y)/y.$ Varian (1992, p.88-89) discusses this issue and shows how it is related to the "production function based measure" of RTS.
Bertrand
CC BY-SA 4.0
2020-11-05T00:49:58.407Z
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
CC BY-SA 4.0
2020-11-03T21:01:40.823Z
2020-11-03T21:01:40.823Z
Bertrand
<p>Besides the great answers already given, adding my two cents:</p>
<p>A firm ends up being a sole player in the sector <strong>when market entry for others is restricted</strong>, which may not be guaranteed by IRTS or even continual economies of scale.</p>
<p>First, it is important to realize that, contrary to our first naive intuition, IRTS does not imply economies if scale. See <a href="https://economics.stackexchange.com/q/40368/24967">this</a> for counter example and related discussion.</p>
<p>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:</p>
<p>Consider a cournot duopoly with two identical firms, and marginal cost:</p>
<p><span class="math-container">$$\frac{\partial C}{\partial q_i}=50-q_i/2$$</span></p>
<p>Inverse demand function for the product market is:</p>
<p><span class="math-container">$$p=100-q=100-q_1-q_2$$</span></p>
<p>Writing the profit function for firm 1:</p>
<p><span class="math-container">$$\pi_1 = q_1 (100-q_1-q_2) - C(q_1)$$</span></p>
<p>FOC:</p>
<p><span class="math-container">$$\frac{\partial \pi}{\partial q_1}=50-3q_1/2-q_2$$</span></p>
<p>So we have following best response function (for <span class="math-container">$i\ne j$</span>):</p>
<p><span class="math-container">$$q_i=\frac{100-2q_j}{3}$$</span></p>
<p>So, at equilibrium:</p>
<p><span class="math-container">$$q_1=q_2=20$$</span></p>
<p>This is clearly a permissible solution and shows existence of cournot duopoly with continual economies of scale.</p>
<p>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.</p>
<p>So to summarize, IRTS, or even economies if scale for that matter, does not guarantee monopoly.</p>
<p>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.</p>

CC BY-SA 4.0
2020-11-05T07:42:38.617Z
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
CC BY-SA 4.0
2020-11-05T00:43:23.553Z
2020-11-05T01:06:28.720Z
2020-11-05T01:06:28.720Z
Dayne
<p>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?</p>
<p>If not, what is the advantage of displaying IRTS?</p>

CC BY-SA 4.0
2020-11-04T00:08:13.237Z
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
<production-function><markets><technology>
Increasing returns, implications?
user31043
# Increasing returns, implications?

(Asked by: username_0 on 2020-11-03)

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?

**Tags:** production-function, markets, technology
> **Comments:**
> **username_1**: 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.  

---

## 3 answers

### Answer by username_2 on 2020-11-03. Score: 4

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)](https://academic.oup.com/ej/article-abstract/125/585/989/5077363) 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.

> **Comments:**
> **username_3**: 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](https://economics.stackexchange.com/q/40368/24967).  
> **username_2**: @Dayne yes that’s why I also said “Generally” at the start of the paragraph - but you are right to stress that  

---

### Answer by username_4 on 2020-11-03. Score: 3

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: $py-c<0$.

> **Comments:**
> **username_2**: +1 this is interesting result, is it a famous result form IO?  
> **username_4**: 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).  
> **username_2**: thanks I will check those references it is very neat result would not expect that but it makes sense  
> **username_3**: Sorry if this is a naive question: how does IRTS imply $c'(y)y/c <1$?  
> **username_4**: $c′(y)y/c<1 \implies c'(y)<c(y)/y.$ Varian (1992, p.88-89) discusses this issue and shows how it is related to the "production function based measure" of RTS.  
> **username_3**: 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).  

---

### Answer by username_3 on 2020-11-05. Score: 3

Besides the great answers already given, adding my two cents:

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](https://economics.stackexchange.com/q/40368/24967) 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.

> **Comments:**
> **username_4**: 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.  

---

